The Influence of Coriolis Forces on Flow Structures of Channelized Large-Scale Turbidity Currents and their

Depositional Patterns

by

Remo Cossu

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy

Department of Geology University of Toronto

© by Remo Cossu 2011

II

The Influence of Coriolis Forces on Flow Structures of Channelized Large-Scale Turbidity Currents and their Depositional Patterns

Remo Cossu

Doctor of Philosophy

Department of Geology, University of Toronto

2011

Abstract

Physical experiments are used to investigate the influence of the Coriolis forces on flow

structures in channelized turbidity currents, and their implication for the evolution of straight and

sinuous submarine channels.

Initial tests were used to determine whether or not saline density currents are a good

surrogate for particle-laden currents. Results imply that this assumption is valid when turbidity

currents are weakly-depositional and have similar velocity and turbulence structures to saline

density currents. Second, the controls of Coriolis forces on flow structures in straight channel

sections are compared with two mathematical models: Ekman boundary layer dynamics and the

theory of Komar [1969]. Ekman boundary layer dynamics prove to be a more suitable

description of flow structures in rotating turbidity currents and should be used to derive flow

parameters from submarine channels systems that are subjected to Coriolis forces. The

significance of Coriolis forces for submarine channel systems were determined by evaluating the

dimensionless Rossby number RoW. The Rossby number is defined as the ratio of the flow

velocity, U, of a turbidity current to the channel width, W, and the rotation rate of the Earth

represented by the Coriolis parameter, f. Coriolis forces are very significant for channel systems

III

with RoW ≤ O(1). Third, the effect of Coriolis forces on the internal flow structure in sinuous

submarine channels is considered. Since previous studies have only considered pressure gradient

and centrifugal forces, the Coriolis force provides a crucial contribution to the lateral momentum

balance in channel bends. In a curved channel, both the Rossby number RoW and the ratio of the

channel curvature radius R to the channel width W, determine whether Coriolis forces affect the

internal flow structure. The results demonstrate that Coriolis forces can cause a significant shift

of the density interface and the downstream velocity core of channelized turbidity currents. The

sediment transport regime in high-latitude channel systems, which have RoW << R/W, is therefore

strongly influenced by Coriolis forces. Finally, these findings are incorporated into a conceptual

model describing the evolution of submarine channels at different latitudes. For instance, the

Northern Hemisphere channels have a distinctly higher right levee system and migrate

predominantly to the left side and generally exhibit a low sinuosity. In contrast, low latitude

channel systems have RoW >> R/W so that centrifugal forces are more dominant. This results in

more sinuous submarine channel systems with varying levee asymmetries in subsequent channel

bends. In conclusion, Coriolis forces are negligible around the equator but should be considered

in high latitude systems, particularly when RoW ~ O(1) and RoW << R/W.

IV

Acknowledgements

There are many people I would like to thank for their help and support over the last four years. Without them I could not call my experience in Canada and the completion of my Phd an exciting one. • First, and foremost, my supervisor Mathew Wells for giving me the opportunity to come to Canada and to go on this adventurous journey. I am especially grateful for his truly unconditional support regarding all aspects of the scientific world. I enjoyed working with Mathew from the first to the very last minute of my PhD. With his guidance, teaching and understanding he was the most pleasant supervisor I could have asked for. Mathew, thank you for everything! • Members of my PhD Committee: Brian Greenwood, Nick Eyles, Joe Desloges and Julian Lowman for contributing timely and healthy criticism and inspiration. Special thanks go out to Brian for his help with the ADV, to Nick for extending my knowledge of glacial sedimentology/geology and Julian for his patient, nonchalant help with mathematics in the late stages of this thesis. • Bruce Sutherland for his helpful comments and suggestions for the thesis. Special thanks for his readiness and willingness to travel in from Edmonton to attend the thesis defense. • Jeff Peakall and Gareth Keevil for loaning me the the UDVP which I used for a major part of this thesis. Thank you Jeff, for all of your support, advice and input for my experiments and publications. • Anna Wåhlin for her mathematical modelling skills and contribution to this thesis. • Gary Parker and Carlos Pirmez for their intreguing discussions at the AGU Chapman conference in Oxnard. • I would also like to thank a few key influencers and mentors from back home who had guided me professionally in academia and industry: Birgit Brinkmann, Karl-Friedrich Daemrich and Joerg Osterwald. I would not have undertaken the step of going to Canada without your support, advice and help. • Ulrich Langer for proofreading gazillions of documents in preparation for my acceptance into the PhD program at UofT.

V

• The province of Ontario for its financial support of this thesis project. • All of the administrative staff in the Geology Department, as well as the Department of Physical & Environmental Sciences. • Bob Ush and Joan Mills for being so supportive during those long nights of writing and editing. • C.C. for countless free coffees on my days out at UTSC. • Joerg Bollmann for being a great and generous friend and counsellor in academia and the weird ways of life. • Christoph Schrank for being a great schoolmate during the first years in Toronto. • The wild bunch of grad students who started this journey with me in 2007. You made my first year probably the most memorable. Cheers mates, the next international dinner is on me! • My various officemates I had over the years, especially Bronwyn and Taronish for enduring the occasional odor of gym gear with only minor complaints. I also want to thank Taronish for becoming a companion, who I could always rely on for down time and being such an awesome workout partner. • Many longtime and most loyal friends back home. Although separated by an ocean from you, I did not get lost being so far from home because I knew you were still there for me. You know who you are! “Wir werden immer laut durchs Leben ziehen, jeden Tag in jedem Jahr…” • The honorary final position is reserved for those that I owe the most: Kristina, for standing beside me along this journey. You helped me grow through things and have kept me in balance (not only with the rotating table!). Thank you for who you are! I finally want to thank my family and relatives: especially my magnificent sister and my wonderful, loving parents. Leaving home is never easy…You all rule!

VI

Table of Contents

Abstract II-III

Acknowledgements IV-V

Statement of authorship XII

Chapter 1: Introduction

1.1 The significance of turbidity currents 1-8

1.2 The importance of the Coriolis force 9-13

1.3.1 Physical models 13-16

1.3.2 Velocity structure 16-17

1.3.3 Secondary flow cells due to centrifugal forces 17-19

1.3.4 Secondary flow cells due to Coriolis forces 20-22

1.4. Motivation and thesis overview 22-24

References 24-30

Chapter 2: A comparison of the shear stress distribution in the bottom boundary layer of experimental density and turbidity currents

Abstract 31-32

2.1 Introduction 32-35

2.2 Shear stresses in the BBL 36-38

2.3 Method 38-42

2.4.1 General flow properties 42-44

2.4.2 Velocity fluctuations 45-46

2.4.3 Reynolds stresses 47-51

2.4.4 Turbulent kinetic energy profiles 51-52

2.4.5 Drag coefficients 53-54

2.5. Discussion 55-60

2.6. Summary and Conclusions 60-61

Acknowledgements 62

References 62-65

VII

Chapter 3: Influence of the Coriolis force on the velocity structure of gravity currents in straight submarine channel systems

Abstract 66-67

3.1 Introduction 67-71

3.2.Theory 72-80

3.3 Method 80-83

3.4.1 General observations 83-85

3.4.2 Observations of downstream velocity U 85-89

3.4.3 Slope of the interface with changing f 89-91

3.4.4 Secondary flow cells and across-stream velocities 91-95

3.5 Discussion 95-99

3.6 Conclusions 100-101

Acknowledgements 101

References 101-105

Chapter 4: Coriolis forces influence the secondary circulation of gravity currents flowing in large-scale sinuous submarine channel systems

Abstract 106

4.1 Introduction 106-109

4.2 Theory 109-110

4.3 Experiments 110-112

4.4 Results and Discussion 112-117

4.5 Conclusions 118

Acknowledgements 118

References 119-120

VIII

Chapter 5: Flow structures and sedimentation processes in submarine channels under the influence of Coriolis forces: experimental observations in rotating gravity currents

Abstract 121

5.1 Introduction 122-124

5.2 Method 124-125

5.3 Results 125-128

5.4 Discussion 129-133

5.5 Summary and conclusions 133

Acknowledgements 134

References 134-135

Chapter 6: Final remarks

6.1 Summary and implication 136

6.2 1 Density currents as an analogue for turbidity currents 136-137

6.2.2 Coriolis forces in straight submarine channels 137-138

6.2.3 Coriolis forces in sinuous submarine channels and

their implication for the evolution of channel systems 138-140

6.2.4 Future work 140-141

References 141-142

IX

List of Tables

Chapter 2

Table 2.1: Summary of parameters for experimental gravity currents 44

Chapter 4

Table 4.1: Experimental conditions 112

X

List of Figures

Chapter 1

Fig. 1.1: Environment of large-scale turbidity currents 3

Fig. 1.2: Bathymetric map of the Zaire Canyon 4

Fig. 1.3: Cross section of the Upper Indus and the NAMOC 5

Fig. 1.4: Sinuosity of submarine channels versus the latitude 8

Fig. 1.5: Particle motion on a rotating platform 9

Fig. 1.6: Downchannel components of a density current in the Faroe Bank Channel 13

Fig. 1.7: Velocity and density profile of an experimental turbidity current 16

Fig. 1.8: Velocity component of density currents in the Black Sea and Baltic Sea 18

Chapter 2

Fig. 2.1: Experimental setup 38

Fig. 2.2: Size distribution for particles 39

Fig. 2.3: Velocity distribution of the downstream velocity component 42

Fig. 2.4: Time series of downstream velocity components u 45

Fig. 2.5: Reynolds stress and viscous stress in the BBL 47

Fig. 2.6: Relation between the velocity gradient du/dz and Reynolds stresses 49

Fig. 2.7: Mixing length of saline and sediment-laden gravity currents 50

Fig. 2.8: Vertical TKE profiles 52

Fig. 2.9: Calculated drag coefficients 53

Fig. 2.10: Velocity profiles between natural and experimental gravity currents 57

XI

Chapter 3

Fig. 3.1: Channelized density current flowing down a submarine channel 73

Fig. 3.2: Velocity profiles in the Ekman boundary layer for various f 76

Fig. 3.3: Shapes of the interface for various f 79

Fig. 3.4: Experimental setup 80

Fig. 3.5: ADV and UDVP velocity profiles 82

Fig. 3.6: Measured downstream velocity U for various f 86

Fig. 3.7: Rossby number RoW and the Froude number Fr for various f 88

Fig. 3.8: Photographs of the interface for various RoW 90

Fig. 3.9: Deflection ΔΔΔΔh and RoW for various f 91

Fig. 3.10: Across-stream velocities for various f 92

Fig. 3.11: Relation between s/(dh/dy) and RoW for various f 94

Chapter 4

Fig. 4.1: Flow structures in channelized gravity currents for various geometries 108

Fig. 4.2: Experimental setup 111

Fig. 4.3: Photographs of experimental gravity currents for various RoR 113

Fig. 4.4: Secondary velocity fields for various RoR 115

Chapter 5

Fig.5.1: Experimental set-up 125

Fig. 5.2: Relationship between Fr 2 / RoW

and the tilt of the interface for various f 126

Fig. 5.3: Interface, distribution of the velocity core, across stream velocities

and distribution of the bottom downstream component for various f 128

Fig. 5.4: Conceptual model for channel evolution at different latitudes 130

XII

Statement of authorship

Chapter 2 has been submitted with M.G. Wells to the Journal of European Mechanics-

B/Fluids (Cossu and Wells, 2011). R. Cossu conducted the experiments, R. Cossu and M. Wells

analyzed the data and wrote the paper. Both authors participated in the discussion and

interpretation of the results. The paper has been accepted for publication for the Journal of

European Mechanics- B/Fluids.

Chapter 3 is co-authored by M.G. Wells and A.K. Wahlin. A.K. Wahlin derived the

analytical solutions, R. Cossu ran the physical experiments, developed the data evaluation

routines, and wrote the paper with M.G. Wells. All authors participated in the discussion and

interpretation of the results. This paper is published in the Journal of Geophysical Research

[Cossu et al., 2010].

Chapter 4 is authored by R. Cossu and M.G. Wells. R. Cossu performed the analogue

experiments. Both authors participated in the discussion and interpretation of the results. This

paper has been published in Geophysical Research Letters [Cossu and Wells, 2010].

Chapter 5 is authored by R. Cossu and M.G. Wells. R. Cossu performed the analogue

experiments. Both authors participated in the discussion and interpretation of the results. This

paper will be submitted to a scientific journal shortly.

1

Chapter 1

Introduction

1.1 The significance of turbidity currents

The goal of this thesis is to investigate the influence of Coriolis forces on the internal

flow structure and depositional patterns of large-scale gravity currents in submarine channels.

Gravity currents are important environmental flows that are driven by a density difference with

the ambient fluid, usually caused by a difference in temperature, salinity or sediment mass

concentration. Gravity currents occur in the atmosphere and the ocean. For instance, sea-breeze

fronts and thunderstorm outflows represent gravity flows of cold, dense air. Atmospheric-

suspension gravity currents comprise snow avalanches or fiery avalanches and base surges from

gases and particles resulting from volcanic eruptions [Simpson, 1982; Middleton, 1993].

Subaqueous gravity currents consisting of suspended sediments are known as turbidity currents

and are important agents transporting sediments in lakes and oceans. On geological time-scales,

they form large-scale features such as submarine fan systems and thick layers of sedimentary

rock that are volumetrically the most important clastic accumulations within deep ocean basins

[Normark, 1993]. The study of turbidity current dynamics in these fan systems allows for

improving geologic models that help to understand deep-sea environments that are paramount for

deciphering the geologic record on the sea floor and exploring hydrocarbon energy sources.

On November 18th, 1929, a 7.2-magnitude earthquake caused a slope failure

approximately 200 km off the coast of Newfoundland. The earthquake triggered a debris flow

that ultimately turned into a turbidity current that continued down a main continental slope valley

and spread out on the abyssal plain. During its passage it broke several trans-Atlantic

communication cables [Heezen end Ewing, 1952]. Based upon timing of these breaks, a

2

maximum propagation speed from 19 m s-1 up to 25 m s-1 was estimated [Piper et al., 1988;

Wells, 2009]. The sediment that was deposited on the ocean basin floor during this incident

covered an area of 280,000 km2 [Piper et al., 1985] with an approximate volume of 150 to 175

km3 [Piper et al., 1999]. This event demonstrates the significance of turbidity currents to

delivering coastal sediments to the deep ocean basins, but also indicates the hazard and power

emanating from these flows.

The reasons why the study of turbidity currents has become more popular are manifold.

Today, turbidity currents are known to be important agents for sediment transport from shallow

coastal areas to deep sea environments (Figure 1.1a), where they form vast sediment

accumulations [Curray et al., 2003], and the deposits often form significant deep-water

hydrocarbon reservoirs [Weimar and Slatt, 2007]. The transport of sediments by turbidity

currents is also environmentally important as they fill in lakes and ocean basins [De Cesare et al.,

2001]. Additionally, turbidity currents are hazardous flows that can damage or destroy seafloor

equipment [Khripounoff et al., 2003], or submarine cables as in the Grand Banks earthquake

turbidity flow [Heezen and Ewing, 1952]. Finally, their deposits are important as they help us to

understand the sedimentation record on the seafloor, which can be linked to prevailing climate

conditions during times of deposition [Pirmez, 1994].

Turbidity currents can be triggered by numerous mechanisms, such as when a sediment-

laden river enters a lake, when sloping layers of sediment on the continental shelf become

unstable due to loading, due to underground gas release or seismic activity [Piper and Normark,

2009; Meiburg and Kneller, 2010] or resuspension by wave action, tides or storm induced

downwelling events on the continental shelf [Palanques et al., 2006a]. In addition, some events

are even man-made; for instance, when mine tailings are dumped into Lake Superior [Normark,

3

1989], into the Prince Rupert channel [Hay, 1987] or when material is dredged close to a canyon

head [Xu et al., 2004].

The main pathways for sediments to deep ocean basins are submarine canyons and

submarine channels (Figure 1.1a) that are formed by turbidity currents and that are significant

morphological features on the continental slope and the ocean floor [Amos et al., 2010]. The

location of the upper pathway, the submarine canyon, is generally fixed and incised into the

continental slope. These canyon systems can exhibit widths up to 20 km and generally occur on

steeper gradients [Normark et al., 1993; Clark and Pickering, 1996].

Fig. 1.1: a) Schematic overview of the environment of large-scale turbidity currents flowing from the con-tinental shelf to the abyssal plain [after Normark et al., 1993] b) Schematic of a submarine channel-levee system [Migeon et al., 2004].

4

In contrast, submarine channels form on the lower pathways and flow through

subaqueous channel systems that are confined by levees as sketched in Figure 1.1b. Further

downstream the channel geometry can disappear and turbidity currents are free to spread out

across the deep ocean basins so that they give rise to subaqueous fans and basin-plain deposits

[Middleton, 1993]. Deep submarine channels occur on slopes with gradients smaller than 0.5º

and can reach length scales of up to several thousand kilometres [Meiberg and Kneller, 2010]

such as in the Zaire Fan system in West Africa (Figure 1.2).

Fig. 1.2: Bathymetric map showing the general morphology of the Zaire Canyon and the modern Zaire meandering channel in the Gulf of Guinea [Migeon et al., 2004]

This thesis focuses on the lower pathways in mid-fan systems where submarine channels are free

to migrate by avulsion and lateral migration [Middleton, 1993] and their morphology reflects the

nature of sedimentary processes active on the fan [Clark et al., 1992]. Channel systems have

been documented from many sources, such as ancient sedimentary successions, the present-day

seafloor and from areas of subsurface hydrocarbon exploration and development [Clark and

Pickering, 1996]. The dimensions of channels can range from several kilometers in width, and

5

several hundred meters deep [Weimar and Slatt, 2007], like the Upper Indus Fan channel

[Kenyon et al., 1995a], to small distal-lobe channels that are only 75 m wide and have depths

smaller than 2 m such as channels on the Outer Mississippi Fan [Twitchell et al., 1991].

Many channels are confined by prominent levees, which form by deposition of suspended

sediment on the slower moving margins of a turbidity current [Parsons et al., 2007] as shown in

Figure 1.3a and 1.3b. These levees can grow rapidly; for instance, the average sedimentation

rates during the active growth phases of the levees of the Amazon channel during the Pleistocene

were 1 to 2.5 cm year-1 [Shipboard Scientific Party, 1995]. The rapid growth has been attributed

to continuous deposition of suspended load. Suspended sediment can be delivered by successive

turbidity currents that transit the channel and spill over the channel margins along their entire

length [Hiscott et al., 1997; Peakall et al., 2000; Straub et al., 2008]. As a result, channel levee

systems can reach widths up to 50 km and heights up to 300 m above the surrounding seafloor,

as is observed in the Amazon Channel [Damuth et al., 1988] or the Indus Fan channel shown in

Figure 1.3a.

Fig. 1.3: a) Cross section of a submarine channel in the Upper Indus Fan [Kenyon et al., 1995a] b) Air gun seismic profile across the North Atlantic Mid-Ocean Channel [Skene et al., 2002].

6

Levee deposits and their structure are described in many studies [e.g. Normark et al., 1980;

Damuth and Flood, 1985; Kolla and Coumes, 1987]. These overbank facies commonly consist of

fine-grained and thin bedded, current laminated sands and silts as well as graded mudstones.

Inside the channel the sediments are usually larger and deposits are more coarse-grained [Clark

and Pickering, 1996].

In plan view most channels are straight, sinuous or meandering or commonly show a

combination of the above with down-channel changes in their planform geometry. Sinuosity can

be defined as the ratio of the channel length to the valley length [e.g. Clark et al., 1992; Imran et

al., 1999]. Submarine channel sinuosity of approximately 1 to greater than 3 has been reported

[e.g. see Clark et al., 1992; Kolla et al., 2007; Wynn et al., 2007]. In the literature, sinuous

channels have been defined as having a sinuosity of greater than 1.2 [Wynn et al., 2007] and 1.15

[Clark et al., 1992; Clark and Pickering, 1996]. Previous classifications of submarine channels

and fans associate channel sinuosity with the slope gradient which in turn is linked to sediment

type and sediment cohesion [Clark et al., 1992; Reading and Richards, 1994; Piper and Normark,

2001], as is also classically observed for river channels [Schumm and Khan, 1972]. Clark and

Pickering [1996] differentiate between two end members in these classifications, one being a

high sinuosity, low slope gradient, fine-grained system, while the other is characterized by a low

sinuosity, high slope gradient, coarse-grained system.

Highly sinuous submarine channels are found in modern equatorial regions, for instance

the Bengal Fan, the Indus Fan, the Mississippi Fan, the Zaire Channel (Figure 1.2), or the

Amazon channel (Figure 1.4a) [Clark and Pickering, 1996; Abreu et al., 2003]. For instance, the

Amazon fan reveals a maximum (peak) sinuosity of 2.6 in its mid-fan region at 3°- 7° North

[Pirmez and Imran, 2003]. In contrast, high latitude systems reveal generally only small sinuosity

[Peakall et al., 2011] and can show distinct levee asymmetries [Menard, 1955]. Examples of low

7

sinuosity channels are the North Atlantic Mid-Ocean Channel (NAMOC) with a sinuosity of

1.01-1.05 at 53°-59° North (Figure 1.4b) [Klaucke et al., 1997] and the Bering Sea channels with

a mean sinuosity of 1.05 at 55° North [Clark and Pickering, 1996].

The aforementioned features of submarine channel systems raise two challenging

questions that this thesis aims to address. The first is the apparent levee asymmetry with one side

being consistently higher than the other as discovered in the NAMOC and illustrated in Figure

1.3b. Secondly, the recently found relationship between the sinuosity of modern submarine

channels and the latitude as shown in Figure 1.4c. It can be argued that this inverse relation can

be attributed to several reasons, such as the seafloor or slope gradient, the prevailing topography

of the ocean floor, the tectonic setting, the flow type, and climate conditions during the time of

deposition [Normark et al., 1993; Peakall et al., 2011]. Nonetheless, Figure 1.4c reveals a much

better correlation between peak sinuosity and latitudes with R2=0.74 compared to the peak

sinuosity versus slope gradient illustrated in Figure 1.4d (with R2=0.24). Peakall et al. [2011]

discuss this relation in greater detail and find the most causative controls on this global

distribution to be the Coriolis force, the latitudinal variation of the flow and sediment type. The

observed levee asymmetry is also often attributed to the systematic change of Coriolis force

between the equator and the poles and indicates that the Earth’s rotation might have a large

influence on the growth patterns of these channel systems.

8

Fig. 1.4: a) Seismic image of the Amazon Fan at approximately 5° North. b) Seismic image of the NAMOC at 60° North. The figures 1.4c and 1.4d show data of various submarine channels [modified from Peakall et al., 2011]. c) Peak sinuosity versus the Latitude. d) Peak sinuosity versus Slope gradient.

9

1.2 The importance of the Coriolis force

Figure 1.5a illustrates the trajectory of a particle with a speed u at two different times, t1

and t2, on a counter-clockwise rotating platform with Ω being the rate of rotation. To an observer

spinning with the platform, the particle seems to have described a trajectory curved to the right

from t1 to its location at t2. However, Figure 1.5a also shows that the trajectory is actually

perfectly straight when it is noted from an observer not rotating with the platform indicated by

the dotted line. The apparent deflection from the perspective by an observer in the rotating frame

of reference is known as the Coriolis effect.

Fig. 1.5: a) Particle motion on a rotating platform. b) Definition of a local Cartesian framework on a rotating spherical Earth. The coordinate x is directed eastward, y northward and z upward and the corresponding velocity components are u (eastward), v (northward) and w (upward) [taken from Cushman-Roisin, 1994]. c) Dependence of the Coriolis force on the latitude φφφφ. The components of the angular velocity Ω of the Earth in the local Cartesian sytem are Ωx = 0, Ωy = Ωcosφ and Ωz = Ωsinφ [Kundu, 1990].

10

Hence, equations referring to motion in a rotating frame (such as the Earth) need an additional

term to describe correctly the motion of a fluid or particle. For instance, the Navier-Stokes

equation for incompressible fluids for a rotating frame in vector notation can be written as:

1

2pt

τρ

∂ + ⋅∇ = − ∇ + ∇ ⋅ − ×∂u

u u Ω u [1.1]

[e.g. Tritton, 1989]. The terms on the left represent the temporal and local change of the velocity

field (u and Ω represent vector quantities of the three dimensional flow field and the angular

speed of the Earth respectively). The terms on the right represent forces that act on the fluid

particle. The first term is the pressure gradient force. The second term is the viscous force

defined by the stress tensor, 2ij ijτ µε= ɺ , where µ is the dynamic viscosity and

1

2ji

ijj i

uu

x xε

∂∂= + ∂ ∂ ɺ the strain rate tensor [e.g. Kundu, 1990]. Horizontal scales and velocities are

usually more important than vertical scales and motions in the ocean so that the term w

x

∂∂

is

known to be negligible. The third term represents the Coriolis force (2Ω×u). For a detailed

discussion of how the Coriolis terms arise in this momentum equation, the reader is referred to

common textbooks on geophysical fluid dynamics [e.g. Cushman-Roisin, 1994]. Horizontal and

unforced motion of a particle that is free of any external force (so that viscous and pressure

forces can be neglected) in a rotating frame as in Figure 1.5a can be expressed as:

2Du

vDt

= + Ω [1.2a], 2Dv

uDt

= − Ω , [1.2b]

where u and v describe particle motion in x and y directions, respectively, and Ω is the rate of

rotation perpendicular to the x-y plane. Large-scale geophysical problems should be solved using

a spherical polar coordinate system. However, the horizontal length scales in this thesis are small

11

enough so that the curvature of the Earth can be ignored and the motions can be expressed by

adopting a local Cartesian system on a tangent plane [e.g. Kundu, 1990]. Figure 1.5b illustrates a

rotating frame in a three dimensional Cartesian coordinate system with the axis of rotation being

in the north-south direction. Figure 1.5c shows that the Coriolis force changes with latitude φ and

the Coriolis parameter f is defined as f = 2Ω sinφ , where Ω = 7.29 ×10-5 rad s-1 is the Earth’s

rotation about its axis. By definition of the sense of rotation, f > 0 in the Northern Hemisphere

and f < 0 in the Southern Hemisphere, and f changes from f = ± 1.45 × 10 -4 rad s-1 at the poles to

f = 0 rad s-1 at the equator. The Coriolis effect vanishes at the equator as any horizontal motion in

the x-y plane simply translates with the Earth’s rotation at the equator, while the full effect of

rotation is experienced at the poles as here the axis of rotation is now perpendicular to the x-y

plane (Figure 1.5b). From Figure 1.5c it follows that the Coriolis force (and f) gets larger with

increasing distance from the equator and Eq. [1.2a] and Eq. [1.2b] can be rewritten as:

Dufv

Dt= [1.3a],

Dvfu

Dt= − . [1.3b]

For many types of fluid motion the Earth’s rotation can be neglected in the sense that

Coriolis forces are much smaller than the other forces such as friction, pressure gradient,

buoyancy or centrifugal forces. But for large-scale flows in the atmosphere, the ocean or the

Earth’s outer core, where the length scales are so large that transit time scales become

comparable to the timescale of the rotation period of the Earth, Coriolis forces are an important

term in the momentum balance. Assuming that Coriolis forces are large compared to the inertia

of the relevant motion implies that the two terms from Eq. [1.1] can be rewritten as:

|u·∇u| << |Ω×u| . [1.4]

Expressing these forces in terms of relevant scales yields:

U2/L << Ω U [1.5a] or U /LΩ << 1. [1.5b]

12

The dimensionless parameter in Eq. [1.5b] can be defined as the Rossby number Ro = U/Lf

(where U is the mean horizontal speed of the fluid and L a characteristic horizontal length scale

[Cushin-Roisman, 1994]). This is an important dimensionless parameter in geophysical flows

that compares the velocity and length scales of motion to the rotation rate of the Earth. When Ro

~ 1 the travel time is comparable to a rotation period and the trajectory will be influenced by the

Earth’s rotation. The smaller the Rossby number gets the more important the rotation and

Coriolis forces become.

Due to the spatial extent of submarine channel systems and the associated long travel

time, the flow properties and environments of large-scale turbidity currents are likely to be

deflected by Coriolis forces [Menard, 1955; Komar, 1969; Wells, 2009] in mid- and high-latitude

systems. For the Grand Banks turbidity current with L = 800 km, f = 9.5 × 10-5 rad s-1 at

approximately 45° North and U = 20 m s-1 a Rossby number of Ro = 0.25 is obtained, which

indicates that Coriolis forces deflected the flow [e.g. Nof, 1996; Wells, 2009]. Even in lakes this

effect can be observed where sediment deposition patterns suggest that the deflection of the

turbidity current by Coriolis forces leads to sediment layers being thicker on the right-hand side

of the lake or fjord in the Northern Hemisphere [Pharo and Carmack, 1979; Syvitski, 1989;

Crookshanks and Gilbert, 2008]. Many direct observations of large-scale oceanographic density

currents confirm that Coriolis forces deflect the bulk of the gravity currents to the right in the

Northern Hemisphere [Davies et al., 2006; Wells, 2009; Sherwin et al., 2010]. Figure 1.6 shows

such a density current in the Faroe Bank channel at 61.5º North [Fer et al., 2010] where Coriolis

forces are balanced by pressure gradient forces which causes a lateral tilt of the interface (∆h) of

about 250 m across the channel of width varying between 5 km to 15 km (Figure 1.6a-c). This

tilt means that overbanking sediment flows are more likely to occur on the right-hand-side of the

channel (looking downstream) for mid- and high latitude systems in the Northern Hemisphere,

13

leading to an asymmetry between levee bank heights [Menard, 1955; Komar, 1969].

Observations of channel systems at higher latitudes have found that the right-hand-side channel

levee is consistently higher in the Northern Hemisphere [e.g. Klaucke et al., 1997] while the left-

hand-side channel levee is higher in the Southern Hemisphere [Droz and Mougenot, 1987; Carter

and Carter, 1988] which correlates with the opposite direction of the Coriolis force in both

Hemispheres. These differences in levee height can reach more than 100 m, as observed in the

NAMOC (shown in Figure 1.3b). The NAMOC exhibits a higher right levee system along a 950

km long section with a mean right-to-left difference of 65 m [Klaucke et al., 1997]. To show that

this asymmetry can be mainly attributed to Coriolis forces is one focal point of this thesis.

Fig. 1.6: a) Measuring locations in the Faroe Bank channel. b) Downchannel component of velocity of a density current deflected by Coriolis forces at location A, B and C in the Faroe Bank Channel [taken from Fer et al., 2010]. The current is displayed by the red color. Flow direction is into the page.

1.3.1 Physical models

Natural turbidity currents usually occur infrequently in remote and hostile environments,

so that field studies have remained fairly limited in their measurements of flow dynamics

[Chikita, 1990; Zeng et al., 1991; Xu et al., 2004; Best et al., 2005; Crookshanks and Gilbert,

14

2008]. Hence, the majority of previous investigations to improve the understanding of turbidity

current dynamics have used small-scale analogue laboratory experiments [e.g. Keulegan, 1957;

Ellison and Turner, 1959; Garcia and Parker, 1993; Simpson, 1972; 1997; Kneller et al., 1999;

Sutherland et al., 2004; Gray et al., 2006; Mohrig and Buttles, 2007; Straub et al., 2008; Islam

and Imran, 2010; Cossu and Wells, 2010]. Such experimental models have offered much insight

into the dynamic characteristics of turbidity currents such as velocity, turbulence and density

structures, and also helped to verify and to improve numerical modelling techniques.

Analogue models are often limited in scale and hence use dimensionless similarity, where

the current is fully characterized by a number of dimensionless variables. As long as the

variables in laboratory currents are comparable with those of the natural current, the experiment

is adequately scaled with respect to the parameters included in that variable [Kneller and Buckee,

2000]. Two of the key parameters for gravity currents are the densiometric Froude number

Fr = U / g 'h (with g’ being the reduced gravity of the density interface and h the mean height

of the current) and the flow Reynolds number Re = U h/ ν, where ν is the kinematic viscosity of

water. These parameters represent the ratio of inertial to gravitational forces (Fr) and the ratio of

inertial to viscous forces (Re) that act on a gravity current. Dimensionless similarity implies that

the Froude number Fr, of the experiment takes the same value as the natural flow. However, it is

much more difficult to achieve similarity in Reynolds number, Re, as natural turbidity currents

often reveal Re at O(106) due to their spatial extent and larger propagation speeds. Viscous forces

can only be neglected if the laboratory flow is fully turbulent, which is generally achieved when

Re > 2000 in a Newtonian fluid [i.e. Leeder, 1982; Kneller and Buckee, 2000]. Large scale

turbidity currents often have speeds in the range of 1-10 m s-1, so if the laboratory flow has a

speed of 0.1 m s-1 the height can be reduced by a factor of 10-100 in order to achieve a similar

15

Froude number when the density difference between the experiment and the prototype is similar

[Middleton, 1993].

For physical models using a rotating framework the Rossby number, Ro, is also a

fundamental scaling parameter. Similar to Fr, the experimental and natural currents have to share

the same value of Ro, which can be achieved by properly scaling the flow velocity, length scale

and rotation rate in the experiment. In this thesis both the width, W, of a submarine channel and

the radius of curvature, R, of a channel bend can be used as a characteristic length scale, so that

two Rossby numbers,

RoW = U/Wf [1.6a] and RoR = U/Rf [1.6b]

are obtained. As U, W and R are usually much smaller in the experiment the rotation rate is

increased to achieve similarity in experimental and prototype Rossby numbers.

One specific problem that often arises is the scaling of sediment in experimental

sediment-laden flows as this also requires a similar dimensionless settling velocity between the

experimental and natural flow. This settling velocity is usually defined as the ratio between the

terminal settling velocity of the sediment and some characteristic velocity scale of the current

[Kneller and Buckee, 2000]. However, even if this parameter can be matched correctly,

electrostatic forces, different sediment particle sizes and concentrations might significantly

change the turbulence structure [Middleton, 1966a; Peakall et al., 1996]. Therefore, many studies

use brine or other dense solutions as a proxy for turbidity currents [e.g. Keevil et al., 2006; Islam

et al., 2008; Darelius, 2008]. This simplifies the current dynamics as deposition and erosion is

neglected in the flow. However, it is assumed that fine-grained turbidity currents are dynamically

similar to saline currents [Stacy and Bowen, 1988a] as low concentration and weakly

depositional turbidity currents exhibit concentration and velocity profiles that are very similar to

saline currents [Kneller and Buckee, 2000; Sequeiros, 2009]. This assumption is also made

16

throughout this thesis. However, so far it has only been shown for “bulk flow” and at the upper

interface in experimental gravity currents [Sequeiros, 2009; Islam and Imran, 2010]. It is

therefore necessary in this thesis to test if fine-grained turbidity currents reveal similar flow

properties near the bottom to saline currents so that this assumption is applicable for all parts of

gravity currents. This investigation is essential to verify the methodology and to transfer results

gained from experimental saline currents to fine-grained turbidity currents. Secondly, a similarity

between both flow types implies that observations from oceanographic studies could also be

applied to large-scale turbidity currents which are of larger interest to geologists.

1.3.2 Velocity structure

Figure 1.7a shows a photograph of an experimental turbidity current with a

corresponding sketch of a typical vertical profile of the downstream velocity (u) in Figure 1.7b.

Fig. 1.7: a) Experimental turbidity current [taken from Wells, 2011]. b) Schematic vertical velocity (u) and density (ρρρρ1111,,,,ρρρρ2222) profiles [modified after Meiberg and Kneller, 2010].

17

The vertical profile of the downstream velocity and turbulence structure of gravity currents has

been analyzed in numerous studies [i.e. Stacey and Bowen, 1988a; Kneller et al., 1999; Gray et

al.; 2006; Islam and Imran, 2010]. The largest velocity (umax) usually occurs close to the bottom

near the density maximum and separates the current into an inner and outer region. The height of

the velocity maximum is controlled by the ratio of the drag forces at the upper and lower

boundaries [Middleton, 1966c; Kneller et al., 1997, 1999]. Due to presence of those two

boundaries velocity profiles often exhibit a ‘bullet-nose’ profile [Middleton, 1993].

In addition, gravity currents exhibit density stratifications that can be classified into two

main profiles [Peakall et al., 2000]. A smooth density profile (ρ1), as depicted by the dashed grey

line in Figure 1.7b, which is usually seen in low-concentration, weakly depositional currents [e.g.

Garcia, 1994; Altinakar et al, 1996] and in saline currents [Ellison and Turner, 1959; Buckee et

al., 2001]. The second is a stepped or Rouse-type profile (ρ2), depicted by the dotted dark grey

line in Figure 1.7b, commonly observed in erosional currents [Garcia, 1993] which are not

discussed in this thesis. The concentration maxima are mostly close to the bottom and the

magnitudes can vary greatly depending on entrainment (autosuspension) or loss (deposition) of

sediment to the bottom. Hence, the description of the Reynolds stresses and drag coefficients at

the bottom are important in order to determine the transport capacity of turbidity currents and

their ability to travel long distances through channels systems.

1.3.3 Secondary flow cells due to centrifugal forces

The architecture, morphological evolution and associated depositional histories of channels is

highly influenced by flow dynamics within the channel, which determine where erosion and

deposition will occur. Hence, the main focus of more recent small-scale physical experiments

has been to investigate internal flow structures and particularly secondary circulations in channel

18

bends [i.e. Corney et al., 2006; Keevil et al., 2006, 2007; Peakall et al., 2007a; Straub et al.,

2008; Islam et al., 2008; Kane et al., 2008; Amos et al., 2010]. Figures 1.8a and 1.8b depict the

flow structure of a density current in a channel bend in the Black Sea [Parsons et al., 2010]. The

flow is forced toward the outer (right) channel bend and generates a cross stream velocity field

with a basal outward directed flow and a return flow above it as sketched in Figure 1.8b.

Fig. 1.8: a) Downstream (upper panel) and cross stream (lower panel) velocity component of a density current in the Black Sea [Parsons et al., 2010]. Flow direction is into the page. b) Location of the measurements and sketch of the corresponding secondary flow cell due to centrifugal forces acting in the channel bend. c) Downstream (upper panel) and cross stream (lower panel) velocity component of a density current in the Baltic Sea [Umlauf and Arneborg, 2009a]. Flow direction is out of the page. d) Location of the measurements and sketch of the corresponding secondary flow cell due to Coriolis forces acting in a straight submarine channel.

19

Previous studies have shown that a primary control on the sense of rotation of helicity is the

downstream velocity profile and especially the height of the downstream velocity maximum

above the bed [e.g. Corney et al., 2006, 2008] although other factors such as cross-sectional

geometry greatly influence flow processes [Islam et al., 2008; Straub et al., 2008]. Though there

has been a lively debate on the sense of rotation of the secondary flow structure in the bend apex,

both flow directions ultimately lead to comparable depositional features with a growth in

sinuosity of the channel [Amos et al., 2010].

The lateral momentum equation in a channel bend can be derived from Eq. [1.1] and is,

for instance, analyzed in greater detail in Nedziecko et al. [2009]. Under steady conditions and

the neglect of viscous forces and the non-linear advection terms it reduces to

2

0

1U pfU

R yρ∂= −∂

[1.7]

which represents the momentum balance between Coriolis forces (left-hand-side), centrifugal

forces and across-stream component of the pressure gradient forces on the right-hand-side

respectively. Sedimentation processes and channel evolution in bends have been attributed

mainly to the internal flow structure resulting from centrifugal and pressure gradient forces by

omitting Coriolis forces in Eq. [1.7]. For instance, centrifugal forces lead to a superelevation and

overspilling of the bank by the current and to a more prominent levee system at the outer bend

[e.g. Straub et al., 2008; Kane et al., 2010] and to sediment accumulations at the inner bend

inside the channel; the latter are called lateral accretion packages [Abreu et al., 2003].

20

1.3.4 Secondary flow cells due to Coriolis forces

As shown in Figure 1.6, Coriolis forces can influence gravity currents through the

deflection of the flow. In a straight channel (where no centrifugal forces act on the current) the

across stream momentum balance for quasi-steady motion can be expressed by:

2

2

1E

p vfU

y zν

ρ∂ ∂= − +∂ ∂

, [1.8]

where viscous forces are now considered through the eddy viscosity approach represented by the

second term on the right-hand side with νE being the eddy viscosity coefficient [Tritton, 1989].

Viscous forces are only important near the upper and lower boundary regions (Ekman boundary

layers, see Chapter 3), but they can give rise to secondary flows throughout the entire current

thickness. The Figures 1.8c and 1.8d illustrate observations of a density current in a straight

channel in the Baltic Sea [Umlauf and Arneborg, 2009a]. The flow exhibits a secondary flow

field similar in magnitude to the one in the Black Sea in Figure 1.8a and 1.8b, but now due to the

presence of Coriolis forces. In the Northern Hemisphere Ekman boundary layer flows will be

deflected to the left and flow in the interior of the current will be deflected to the right channel

wall so that a secondary transverse circulation is superimposed on the downstream flow, as

sketched in Figure 1.8d. Qualitatively similar secondary flows to Figure 1.8d driven by Coriolis

forces have been monitored in oceanic gravity currents, i.e. in the Vema channel [Johnson et al.,

1976], in the Faroe Bank Channel [Johnson and Sanford, 1992; Fer et al., 2010] or in the Ellet

Gully [Sherwin, 2010]. These oceanographic studies found that rotating gravity currents rapidly

come into a geostrophic balance, where the trajectory of the density current is determined by a

balance between buoyancy forces, friction and Coriolis forces as described in Eq. [1.8]. The

basic features of such rotating currents have also been seen in several laboratory experiments

21

[Benton and Boyer, 1966; Hart, 1971; Johnson and Ohlsen, 1994; Davies et al., 2006; Darelius,

2008; Wåhlin et al., 2008].

Turbidity currents in channels should also experience similar secondary flow patterns to

large-scale oceanic density currents in channels, when the flows are at large enough scales that

Coriolis forces become important. This is significant as secondary flow structures in turbidity

currents are known to play a major role for the growth and evolution of submarine channel

systems. Though the influence of Coriolis forces upon turbidity currents is acknowledged in

several studies [Middleton, 1993; Nof, 1996; Klaucke et al., 1997; Emms, 1999; Imran et al.,

1999; Pirmez and Imran, 2003] it has almost been neglected in the majority of geological studies

of submarine channel systems. When Coriolis forces are mentioned it is usually in the context of

the theory of Komar [1969] who introduced a momentum balance similar to Eq. [1.7] for gravity

currents flowing in a channel bend. This mathematical description of the tilt of the interface is

2 , dh hf h

Frdy U R

= +

[1.9]

where dh/dy represents the interface slope across the channel. This theory has been used to

derive flow parameters from morphological settings [Bowen et al., 1984; Klaucke et al., 1997]

and has been cited numerous times in the geological literature but no experiment has ever tested

its validity. In addition, the standard usage of Eq. [1.9] assumes that the Froude number remains

constant at a value of one and that rotation has no control on the velocity of the current.

However, this is contradictory to the observations of Cenedese et al. [2004] and Cenedese and

Adduce [2008] who report a large reduction in velocity when the Coriolis parameter is high.

Another significant issue in the theory of Komar [1969] is the neglect of boundary friction which

can be described more accurately by traditional bottom boundary layer dynamics in

oceanographic studies. Therefore this thesis aims to gain more insight into the applicability of

22

Eq. [1.9] in order to derive flow parameters more accurately from geological settings in

channels.

1.4 Motivation and thesis overview

Many observations of oceanographic density currents have shown that Coriolis forces are

important for the flow dynamics as they deflect the fluid to the right-hand-side in the Northern

Hemisphere. However, observations of levee asymmetries suggest that turbidity currents

transporting sediment through large channel systems are also influenced by the Earth’s rotation.

To date, geological studies refer mostly to the theory of Komar [1969], which led to reasonable

estimates of flow parameters of turbidity currents. However, as outlined in the previous section,

this simple momentum balance does not describe the internal flow structure adequately, as it

neglects frictional effects, nor has it ever been tested against a physical model. With conceptual

models that aim to explain the evolution and growth of submarine channels, the focus has

recently been on more accurate descriptions of the flow field in turbidity currents. But due to the

neglect of Coriolis forces, previous experimental studies are insufficient to explain levee

asymmetry at only one bank or varying sinuosity observed in channel systems at different

latitudes.

This thesis contributes to the understanding of the internal flow structure of large-scale

turbidity currents driven by the interplay of centrifugal, pressure gradient and Coriolis forces, by

using physical modelling. This helps to develop further conceptual models [Clark and Pickering,

1996; Peakall et al., 2000; Amos et al., 2010] that describe the evolution of submarine channel

systems that are influenced by the rotation of the Earth. This thesis consists of four independent

papers that aim to answer the aforementioned problems step by step.

23

The methodology is based on physical modelling techniques that underlie the assumption

that saline experimental currents are dynamically similar to weakly-depositional sediment-laden

currents. This has been proven predominantly for the center and upper part of gravity currents

[e.g. Sequerios et al., 2009; Islam and Imran, 2010] but not for the bottom boundary region.

Chapter 2 examines the turbulence structure between saline and sediment-laden experimental

flows at this lower boundary and systematically investigates if saline density currents are a good

surrogate for turbidity currents. The experimental results are described and compared with

similar experiments in other studies. Particular attention is given to the Reynolds stress

distribution and drag coefficients in the bottom boundary layer.

Chapter 3 analyzes the influence of Coriolis forces on the dynamics of gravity currents

flowing in straight submarine channels. Due to the straight channel geometry no centrifugal

forces arise so that the momentum balance is reduced to the simple form of pressure gradient and

Coriolis forces. The transverse velocity structure, downstream velocity and interface slope are

investigated and compared with the theory of Komar [1969]. A more complex mathematical

model using Ekman boundary layer dynamics [Wåhlin et al., 2008; Darelius, 2008] is introduced

and also compared with experimental measurements. The results are discussed in terms of

observations in modern submarine channels.

Chapter 4 transfers the problem of Chapter 3 to more complex channel geometries. The

internal flow structure in sinuous submarine channels under the influence of Coriolis and

centrifugal forces using laboratory experiments is examined. Both forces can drive the secondary

circulation of turbidity currents in sinuous channels and determine where erosion and deposition

of sediment occur. The relative importance of Coriolis and centrifugal forces is described in

terms of the Rossby number RoR based upon the radius of curvature, which can be used to

classify if a channel system is dominated by Coriolis or centrifugal forces.

24

Previous conceptual models [e.g. Clark and Pickering, 1996; Peakall et al., 2000; Amos

et al., 2010] are able to describe the evolution of submarine channels, the growth of sinuosity and

levee deposits, but are unable to explain the relation between sinuosity and latitude described in

Figure 1.4c [Peakall et al., 2011]. In Chapter 5 this relation is investigated by an analysis of how

strongly the internal flow structure in gravity currents is influenced by Coriolis forces. These

observations are combined with an existing conceptual model for sedimentation and erosion in

sinuous channels. Depositional patterns between high and low latitudes are contrasted by

including Coriolis forces and their implication for the evolution of submarine channel systems is

discussed.

The main conclusions of this thesis are summarized in Chapter 6 and suggestions are

made for future work.

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Chapter 2 A comparison of the shear stress distribution in the bottom boundary layer of experimental density and turbidity currents Remo Cossu and Mathew G. Wells Abstract

The internal stress distribution within weakly depositional turbidity currents has often been

assumed to be similar to saline gravity currents. This assumption is investigated by analyzing a

series of experiments to quantify and compare the shear stress distribution in the bottom

boundary layer (BBL) of saline and particle-laden gravity currents. Vertical profiles of Reynolds

stresses, viscous stresses and turbulent kinetic energy (TKE) were obtained from the mean

downstream velocity profiles and turbulent velocity fluctuations, and were broadly similar in

both flow types, suggesting that saline gravity currents are a good analogue to turbidity currents.

Maximum positive Reynolds stresses occur where the velocity gradient is largest in the BBL but

below this maximum the Reynolds stresses decrease significantly and are balanced by an

increase of viscous stresses. The bulk drag coefficient CD is defined for both flows using three

methods: i) a log-fit method based on the law of the wall, ii) the observed maximum total stress

and iii) direct measurements of turbulent velocities. The CD values of both flow types were

broadly similar but each method led to CD values of different orders of magnitude. The log-fit

method yielded the largest drag coefficients of O(10-2) whereas measurements of turbulent

velocities gave relatively small values of O(10-4). The best correlation with drag coefficients

observed in field measurements of O(10-3) was obtained by using the maximum total stresses

32

next to the wall. The variation of CD is discussed in relationship to parameterization methods in

experimental and numerical modelling.

2.1 Introduction

Gravity currents are important environmental flows that are driven by a density

difference with the ambient water, usually caused in a difference in temperature, salinity or due

to suspended sediment. Gravity currents that are driven by suspended sediments are known as

turbidity currents and are important agents to transport sediments in lakes and oceans. Turbidity

currents can be triggered by numerous mechanisms. For example, from a sediment-laden river

entering a lake, or the sloping layers of sediment on the continental shelf becoming unstable

because of loading, underground gas release or seismic activity [Piper and Normark, 2009;

Meiburg and Kneller, 2010]. A key feature of turbidity currents is that they have the ability to

erode sediment if they are moving fast enough so that shear stresses exceed a critical threshold

appropriate to entrain bed material; a process known as self-acceleration and autosuspension

[Bagnold, 1962; Garcia and Parker, 1993; Sequeiros et al., 2009]. Due to this process turbidity

currents can travel large distances before they finally deposit sediment as the currents slow and

shear stresses decrease. On geological time-scales a series of such turbidity currents can form

large-scale features such as submarine fans and thick layers of sedimentary rock within deep

ocean basins [Normark et al., 1993; Weimer and Slatt, 2007].

As natural turbidity currents usually occur in remote and hostile environments, field

studies have been fairly limited in their measurements of flow dynamics [Chikita, 1990; Zeng et

al., 1991; Xu et al., 2004; Best et al., 2005; Crookshanks and Gilbert, 2008]. Hence, the majority

of previous investigations into the dynamics of turbidity currents have used analogue laboratory

experiments [e.g. Garcia and Parker, 1993; Simpson 1972, 1997; Kneller et al.; 1999; Gray et al.,

33

2006; Mohrig and Buttles, 2007; Straub et al., 2008; Islam and Imran, 2010; Cossu and Wells,

2010] to describe the relationship between the velocity and density difference, or to determine

the internal turbulence structure that leads to erosion and transport of sediment. The internal

turbulent dynamics and shear stress distributions of oceanic saline density currents have received

considerable attention [Johnson et al., 1994; Dallimore et al., 2001; Peters and Johns, 2006;

Umlauf and Arneborg, 2009a] in contrast to sediment-laden flows that are usually hard to

monitor in the field. An open question remains as to how relevant the internal dynamics of either

large-scale oceanographic overflows or laboratory-scale saline density currents are to

understanding the dynamics of sediment laden turbidity currents, as has often been implicitly

assumed in experimental studies that describe turbidity current dynamics using saline density

currents [e.g. Kneller et al., 1999; Garcia and Parsons, 1998; Keevil et al., 2006]. The important

issue is to determine the differences between saline and turbidity currents that arise due to the

potential settling of particles in turbidity currents. For instance, in experiments with similar

density currents and turbidity currents, Gray et al. [2006] found larger Reynolds stresses and

turbulent kinetic energy (TKE) in the turbidity currents. In these experiments they used glass

ballotini with a mean diameter of 70 µm. In contrast, Islam and Imran [2010] noticed a good

agreement of Reynolds stresses and TKE between similar turbidity and density currents, when

using silica powder with a mean diameter of 25 µm for the particulate gravity currents. This

discrepancy in findings between the two studies might be attributed to the use of different

particle sizes and suggests a necessity to further distinguish turbidity currents, e.g. between

strongly depositional and non-depositional or weakly depositional currents based on the grain

size they transport. A question that motivates the present study is to determine how similar the

near bed turbulent shear stresses in turbidity currents are to the more widely studied saline

currents.

34

Most previous experimental studies of gravity current dynamics [e.g. Kneller et al., 1999;

Gray et al., 2006; Islam and Imran, 2010; Buckee et al., 2001] have mainly focused on the

turbulence structure at the upper boundary, and only a few measurements have been reported

very close to the bottom, so that flow properties in the bottom boundary layer (BBL) are still

poorly understood. The area between the maximum downstream velocity (umax) and the bottom

of gravity currents is called the inner region, and here the velocity generally reveals a non-linear

profile caused by friction at the lower boundary [Kneller et al., 1999; Altinakar et al., 1996]. For

instance, some studies on gravity currents [Garcia and Parker, 1993; Kneller et al., 1999;

Dallimore et al., 2001; Peters and Johns, 2006; Odier et al., 2009; Sherwin, 2010] have shown

that the BBL is only a small fraction of the total flow depth d of gravity currents and hence is

generally difficult to resolve in laboratory experiments. We note that other experimental studies

[Sequeiros et al., 20010a,b] report a larger inner region based on a subcritical Froude number

(Fr) or a rougher surface of the boundary. The BBL occupies the lowest part of inner region

where a significant increase in velocity can be observed [Tritton, 1989]. Despite having a small

proportion of the total depth, the turbulence structure and resulting Reynolds shear stresses in

this layer are very important because they define whether turbidity currents will have a

depositional or an erosive character [Garcia and Parker, 1993] which in turn strongly influences

the dynamics and the persistence of turbidity currents. Based on our observations of the velocity

profiles described in section 4.1 (of Chapter 2) we assume that the thickness of BBL is less than

10% of the total thickness h of our experimental flows so that z/h < 0.1 with z being the vertical

distance from the boundary.

The magnitude of the bottom stress is often parameterized in terms of a dimensionless

drag coefficient (CD) which relates the friction velocity u* to the mean downstream velocity U of

a flow. For instance, empirical values of CD have been used to estimate the bottom stress and to

35

determine the erosional capacity of density currents in depth-averaged mathematical models.

However, laboratory estimates of CD of gravity currents [e.g. Garcia and Parker, 1993; Straub et

al., 2008] are often of O(10-2), which is much higher than suggested by direct field observations

[Peters and Johns, 2006; Umlauf and Arneborg, 2009a] where CD = 10-3. Part of this difference

could be due to the variations in the nature of the boundary or due to the differences in how CD is

estimated in the experimental flow.

In this study we investigate flow properties for both turbidity currents and saline gravity

currents by making high resolution velocity and turbulence measurements within the BBL. The

shear stress distribution in the BBL of turbidity and saline density currents is investigated using

several measurement points within the bottom centimetre of the flow with a high resolution

Acoustic Doppler Velocimeter (ADV). In addition, we determine the Prandtl mixing length in

the BBL which relates the Reynolds stresses to the mean velocity gradient and compare it with

results observed outside the BBL by Odier et al. [2009]. Finally we determine CD values using

three different methods: (i) a log-fit method based upon the law of the wall [Kneller et al., 1999;

Straub et al., 2008], (ii) the maximum total bottom stress and (iii) direct observations of turbulent

velocities near the base as used by Dallimore et al. [2001]. We compare the CD values and

discuss each method in terms of its applicability. Some basic physical boundary layer processes

associated with turbulence structure are introduced in section 2. The experimental setup is

described in section 3. Results for the velocity, Reynolds stress and TKE distribution as well as

CD values are presented in section 4. We finish with a discussion of our observations and

compare them with field observations of oceanographic scale gravity currents in section 5 and

our conclusions in section 6.

36

2.2 Shear stresses in the BBL

Reynolds stresses play a dominant role in the mean momentum transfer by turbulent

motion and reflect stresses that are imposed by the turbulent velocity fluctuations on the mean

flow [Gray et al., 2006]. The magnitude of the shear stress in the BBL determines the sediment

transport capacity of turbidity currents and whether deposition or erosion processes dominate the

regime at the bottom boundary. In the literature the shear stress term in the momentum equation

of gravity currents is often related to the downstream velocity gradient using

( )R v E

u

zτ τ τ υ µ ∂= + = +

∂ [2.1]

[Stacey and Bowen, 1988] where τ is the total shear stress, τR the Reynolds and τv the viscous

stress respectively, µ is the molecular viscosity, νE is the eddy viscosity coefficient of

momentum and u the downstream velocity. The BBL can be further subdivided into a very thin

viscous sublayer at the base, where viscous forces are dominant, and a layer above it, where

viscous forces can be neglected and instead the velocity gradient depends on the shear stress and

the eddy viscosity [Tritton, 1989].

The shape of the velocity profile at the bottom of a gravity current in the viscous boundary layer

(log-boundary layer) is often described by a logarithmic profile of the form:

( ) *

o

u zu z ln

k z

=

[2.2]

where κ = 0.41 is the von Kármán constant, and u(z) is the velocity taken at the height z above

the bed and z0 represents the zero-velocity roughness height [Tennekes and Lumley, 1972;

Kneller at al., 1999]. We note that u* is a constant that represents a bulk value (vertically

averaged) of the friction velocity in the inner region. The oceanographic observations of Johnson

et al. [1994] showed that in a 200 m deep density current only the bottom 20 m had a well-

37

resolved logarithmic velocity profile where there was a constant shear-stress layer which is

assumed in deriving the logarithmic profile (and hence u*) in Eq. [2.2]. If the same proportions

hold in laboratory density currents (which are usually less than 0.3 m in thickness) that exhibit a

similar velocity structure as observed by Johnson et al. [1994], the log-boundary layer described

by Eq. [2.2] would only occupy a small region above the bottom unlike in channel flows where

the full depth of the flow is a log-boundary layer. Hence it is more difficult to resolve accurately

such a logarithmic boundary layer in a laboratory scale gravity current.

The friction velocity u* of gravity currents is related to the bottom stress τB near the wall

and the density ρ by using

τρ

*= Bu , [2.3]

[e.g. Tennekes and Lumley, 1972]. The relation between u* and the vertically averaged mean

flow velocity U of gravity currents can then be used to define CD as:

CD = u *2

U2, [2.4]

[e.g. Garcia and Parker, 1993]. In this study we determine the friction velocity u* both from log-

fit curves (using Eq. [2.2]) of the measured velocity and from direct measurement of the

maximum bottom stress τB (following Eq. [2.3]).

In addition, we use a third method reported in Dallimore et al. [2001] to calculate CD values

where Reynolds stress terms u'w' (along the flow) and v 'w' (perpendicular to the flow) were

measured to determine a turbulent velocity with

[ ] 41

22 )''()''( wvwuut += . [2.5]

In order to use Eq. [2.5] to calculate CD and compare it with the other methods we treat this

38

turbulent velocity as a bulk parameter by averaging it over the thickness of the BBL and use the

notation ût to distinguish it from u* hereafter.

2.3 Method

To investigate the BBL dynamics such as the velocity and turbulence structure in both

saline and particulate-laden gravity currents we conducted a series of lock-release experiments in

a horizontal tank. This popular experimental set-up has been used in many studies [Middleton,

1966; Kneller et al., 1999; Shin et al., 2004; Monaghan et al., 2009] to study the dynamics of

gravity currents. The tank consists of two compartments separated by a vertical lock as shown in

Figure 2.1. A fluid with a higher density is generated in the smaller separated compartment, and

when the lock is removed this denser fluid exits and flows as a gravity current beneath the lighter

fluid. The experimental apparatus used for the present work consisted of a horizontal tank with a

lock exchange section and a smooth floor (Figure 2.1).

Fig. 2.1: Schematic of the dam-break experimental setup.

The tank had a width of 0.30 m and total length of 6 m, of which the lock compartment

occupied about 1.40 m. The tank was filled with tap water at 13-15 °C to a depth of 0.28 m so

that the volume of the lock compartment comprised approximately 120 L. The aspect ratio

defined by tank width to water depth was therefore comparable to the previous laboratory

39

experiments studying turbidity currents [e.g. Kneller et al., 1999; Buckee et al., 2001; Gray et al.,

2006; Islam and Imran, 2010; Monaghan et al., 2009; Gladstone and Pritchard, 2010].

In order to compare meaningfully the shear stress dynamics in the BBL between turbidity

and saline density currents, the initial density contrast was kept constant in all experiments. The

denser fluid was produced by adding either sediment or salt to the lock compartment and stirring

it with two agitators until a well mixed suspension or brine was achieved. The agitators were

switched off 15 s before the lock was removed and the denser fluid was then released into the

main body of the tank. In all the experiments either 1.5 kg of sediment or salt were added to the

lock compartment so that initial density contrast was almost identical for each flow with ∆ρ =

11.9 kg/m3. The sediment used for the sediment-laden flows was silicon carbide (SiC), which has

a density of 3220 kg m-3 and a mean diameter of 27.5 µm. The particle size distribution is

illustrated in Figure 2.2.

Fig. 2.2: Distribution for the SiC particles. The mean grain size diameter is approximately 27.5 µm.

40

The particle settling velocity wp of these fine sediments can be estimated using the

empirical equation

ρ ρ

ν× × ×= ×

2 0.281 ( - )18P

s or g Ew [2.6]

[Dietrich, 1982] with r the mean radius of the particle, g the gravitational acceleration, ρo the

density of the ambient fluid, ρs the density of the sediment,ν the kinematic viscosity of water and

E the Janke shape factor of the particle [Dietrich, 1982]. For non-spherical particles this factor is

always smaller than unity but for an upper boundary we used the maximum shape factor E = 1.

This yields a maximum settling velocity of approximately wp = 9.1 × 10-4 m s-1. Hence, the

settling velocity is of the same order as in Monaghan et al. [2009] who also used SiC with a

mean diameter of 26 µm (wp = 8.18 × 10-4 m s-1) but with a smaller settling velocity than

wp = 4 × 10-3 m s-1 in Gray et al. [2006]. Flow velocities were measured at a point with a

NORTEK Vectrino ADV at a sampling rate of 100 Hz. The ADV is an acoustic velocity sensor

(consisting of one transmitter and three receivers) which uses the Doppler shift of the reflected

pulses to determine the fluid velocity. The receivers are slanted at 30o from the axis of the

transducer and focus on a common sample volume of 160 mm3 (averaging vertically over 5-6

mm), that is 50 mm away from the probes. This ensures nearly non-intrusive flow measurements

within the flow field of the current. The ADV makes simultaneous measurements of the three

velocity components at high sampling rates [Voulgaris and Throwbridge, 1998]. All ADV

measurements were taken 1.8 m away from the lock gate. The accuracy of the ADV instrument

is 0.5% of the measured velocity and hence the error is negligible for the velocity range

described in the next section.

A typical experiment lasted for 150 s after the lock gate was lifted and the higher density

fluid started to exit the compartment. The front of the dense current formed a head with the

41

typical overhanging nose [Simpson, 1969] that reached the measuring location after

approximately 30-35 s. Behind the head the density current was thinner, which is associated with

the characteristics of the body of the current. After the passage of the head the flow conditions

remained fairly constant. Several test runs demonstrated that both the distance and the time

between the release of the current and the start of the recording were sufficiently large enough so

that the gravity currents were not biased by any initial conditions, for instance, the removal of the

lock gate or the stirring of the agitators. For instance, we found very good agreement of velocity

profiles and turbulent parameters between the measuring location at 1.8 m and 2.8 m away from

the lock which showed typical distributions known from other studies [Gray et al., 2006; Islam

and Imran, 2010]. On the other hand, at a 0.8 m distance in particular, the Reynolds stress

distribution had not fully developed as it was still biased by initial conditions.

In most of the analyses the focus was on the near-bed downstream velocity profile,

consisting of 10 measurement points at 3, 4, 5, 6, 8, 10, 20, 30, 40 and 50 mm above the bottom

of the tank. For some experiments additional measurements at 60, 80, 100 and 120 mm above the

bottom were added. The depth of the gravity currents within the sampling interval of 30 s was h

= 0.12 m which was determined using the integral definition of depth from Ellison and Turner

[1959] where 0 0

( ) ( ) h u z z dz u z dz∞ ∞

= ∫ ∫ . Hence, the measurements represent the bottom 1/3-1/2

of the gravity current. To obtain one velocity profile required that 10 equivalent runs be made,

and that the ADV was raised between each run. The u-velocity reflects the streamwise, the v-

velocity the across stream and the w-velocity the vertical velocity component. Turbulent

properties were then extracted by defining the turbulent fluctuations from the mean flow, as

described in more detail in section 4.2.

42

After the experiments we observed only very thin deposits from the sediment-laden flows

on the tank floor near the measuring location with most of the sediment being deposited in the

sump. In the discussion section we show that within the main body of the current, where our

measurements were focused, velocities and velocity fluctuations remained large enough to keep

particles in suspension. We therefore assume that the main deposits on the tank floor must have

settled out in the tail section of the current where velocities decreased significantly. As this

sedimentation occurs outside of the measuring interval this had no affect on our results.

2.4.1 General flow properties

Figure 2.3a depicts experimentally determined velocity profiles of the saline and

particulate gravity currents. The level of umax occurs between z/h = 0.25-0.35 for both saline and

the sediment-laden flows. The velocity maxima can be considered as the top of the inner region

[Kneller et al., 1999] whereby the BBL only occupies a fraction of this depth [e.g. Dallimore et

al., 2001] at z/h < 0.1 (Figure 2.3a).

Fig. 2.3: Vertical distribution of the downstream velocity component. The horizontal bars indicate the standard deviation of the ADV data. a) Velocity distribution over the entire thickness of the current. b) Various curves added to the velocity data using the law of the wall approach in the BBL. R2 represents a least-squares fit of the data. Note that the graph in b) is plotted with a semilog axis.

43

Both the saline and particulate laden gravity currents had similar downstream velocity profiles.

Our analysis focuses hereafter on the inner region between the bottom z/h = 0 - 0.4 as shown in

Figure 2.3a. Figure 2.3b shows the velocity measurements at a higher resolution within the BBL

between z/h = 0 - 0.1. The dotted lines describe a possible fit of the logarithmic velocity profile

of Eq. [2.2] to the experimental data for the saline (R2 = 0.84) and turbidity current (R2 = 0.82)

obtained by a least squares fit of the data.

Lock-release gravity currents usually have Fr of order one [Shin et al., 2009], which is

defined as / 'Fr U g h= where g’ is the reduced gravity ′g = g (ρu-ρa) / ρa with ρu the density

of the underflow, ρa the density of the ambient water. The mean downstream velocity in our

experiments was U = 0.05 ± 0.001 m s –1, giving Fr ~ 0.43 for both the saline and particulate

gravity flows. We note that this estimation is based on the initial density contrast. Due to mixing

processes and entrainment of water the density difference g’ decreases as the flow evolves so

that the actual Fr will be slightly higher. However, such a small increase will not affect the flow

properties significantly so that this value of Fr should be treated as a lower limit. Such a value of

Fr is broadly similar to many previous experimental observations of gravity currents where 0.2 <

Fr < 1.3 [Garcia and Parker, 1993; Simpson, 1997; Kneller et al., 1999; Gray et al., 2006; Islam

and Imran, 2010] and in good agreement to the theoretical model introduced by Benjamin

[1968]. For the velocity profiles shown in Figure 2.3a, the bulk Reynolds number is

approximately Re = 6000 where Re= U h/ν, with ν is the kinematic viscosity. This value of Re

implies that the flow was fully turbulent and comparable to other experimental turbulent gravity

currents, e.g. Re = 2.7 × 103 Kneller et al. [1999], Re = 2 - 6 × 103 Gray et al. [2006], Re = 7.2 ×

103 Monaghan et al. [2009] or Re = 1.4 × 104 Islam and Imran [2010]. A detailed description of

all our relevant experimental parameters is given in Table 1.

44

Tab. 2.1: Summary of parameters for experimental gravity currents

45

2.4.2 Velocity fluctuations

We characterize the distribution of shear stress and turbulent kinetic energy (TKE) by

analyzing the velocity fluctuations in the body of the gravity current, i.e. after the passage of the

flow front and before the tail arrives at the measuring location. For our experiment the body of

the gravity current lasted over 30 s. The data shown in Figure 2.4a and 2.4c are time series of

downstream velocity (u) measured approximately 4 cm (z/h = 0.33) above the tank floor for a

saline and sediment-laden gravity currents. Generally, the downstream velocity is one order of

magnitude larger than the across-stream and bed-normal velocities. Saline gravity currents

showed similar velocity time series data compared to the sediment-laden gravity currents, with

the velocity fluctuations at the same order of magnitude as in similar experimental flows

described by Kneller et al. [1999], Gray et al. [2006] or Islam and Imran [2010].

Fig. 2.4: Time series of downstream velocity components (u) for saline and sediment-laden gravity current taken approximately z/h = 0.33 a,c) raw data reflecting the instantaneous velocities and b,d) averaged velocity ū over a 100 ms time window and turbulent velocity fluctuations u’ after subtracting u- ū.

Velocity fluctuations were obtained by subtracting a smoothed time series from the

measured data set, e.g. as u’ = u – ū, where u indicates the raw data obtained by the ADV

measurement at a sampling rate of 100 Hz and ū the smoothed local velocity component. The

smoothing of the data was done by filtering the raw data with a 100-point moving average to

46

determine ū, as in Kneller et al. [1999]. Hence, the temporal length of the filter used for the

calculation was 1 s and yielded comparable turbulent fluctuations obtained in previous studies

[Kneller et al., 1999; Islam and Imran, 2010]. This decomposition of time series data into a mean

ū and a fluctuating component u’ for the downstream velocity u(z) is plotted in Figure 2.4b and

2.4d. Usually the turbulent fluctuations are at least one order of magnitude smaller than the mean

(smoothed) values.

The sampling volume of the ADV was chosen to be large enough that potential bias due

to noise was kept at a minimum which resulted in a signal data quality for each flow with a

correlation of > 95 %. In addition, the time series data were analyzed and filtered with a spike

detection routine so that any corrupted data larger than 2.5 times of the standard deviation was

removed from the time series.

47

2.4.3 Reynolds stresses

The bed normal Reynolds stresses are obtained from the turbulent fluctuations of

downstream u’ and bed-normal velocity w’ components according to

τ R=-ρ u'w' [2.7]

[Tennekes and Lumley, 1972]. The mean Reynolds stress distribution using Eq. [2.7] for z/h <

0.4 is shown in Figure 2.5a for turbulent saline gravity currents, and in Figure 2.5b for sediment-

laden gravity currents.

Fig. 2.5: Profiles of Reynolds stress and viscous stress in the BBL for saline (a) and sediment-laden (b) gravity currents. All profiles are normalized by ρρρρU2. The horizontal bars indicate the standard deviation. The horizontal dashed line indicates the viscous boundary layer.

In both cases the Reynolds stresses are normalized by ρU2. Both the experimental saline and

particulate gravity currents have the highest positive Reynolds stresses near the bottom (z/h ~

0.1). The saline gravity current has a maximum normalized Reynolds stress of 1.5 ± 0.5 ×10-3,

which is significantly larger than the maximum of sediment-laden gravity current with 0.8 ± 0.45

×10-3. Below the maxima at z/h < 0.1 the Reynolds stresses decrease significantly towards the

bottom. Around and above umax at z/h = 0.25-0.35 (Figure 2.3a) the Reynolds stresses are close to

zero, and stresses become negative in the interfacial region where the velocity gradient is

48

negative. Those values are consistent with Reynolds stresses reported by Gray et al. [2006] and

Islam and Imran [2010]. A new experimental observation is the distinct decrease in Reynolds

stresses for z/h < 0.1, which has not been described in any previous gravity current experiments.

The viscous stress τ

v= µ ∂u

∂z is normalized by ρU2 and is also shown in Figure 2.5a and 2.5b

using the measured velocity gradient and µ = 1.17 ×10-3 N s m-2. Maximum velocity gradients

occur near the base, but away from the viscous boundary layer (VBL) there is a dramatic

decrease in vτ between z/h = 0.025-0.1 with increasing height. We can determine the thickness

of the VBL (δVBL) with the empirical formula δVBL = 8 µ/(ρ u*) [Tritton, 1989] where u* is

calculated using Eq. [2.3] with a depth averaged τB = 2 × 10-3 for saline and τB = 2.5 × 10-3 for

sediment-laden flows (from Figure 2.5a and 2.5b). This yields δVBL = 0.0042 m and δVBL =

0.0037 m (see Figure 2.5a and 2.5b) indicating that VBL occupies only a small region of the

entire BBL. According to Eq. [2.1] the total stress is obtained by addition of the two stress

components as R vτ τ τ= + . The largest positive total stresses are found close to the bottom,

where with increasing height Reynolds stresses show a significant increase which balances

decreasing viscous stresses so that the total stress is roughly constant. Above the Reynolds stress

maxima (at z/h > 0.1) the total stress gets significantly smaller.

The strong dependence of the Reynolds stress upon the velocity gradient is depicted in

Figure 2.6 where Reynolds stresses and velocity gradient are related to each other. Only around

umax, where du/dz is almost zero, are small stresses prevalent. The observed Reynolds stress

changes sign from the positive (below umax) to negative (above umax) corresponding to changes in

the velocity gradient du/dz. The largest positive Reynolds stresses coincide with the large

positive velocity gradient which is close to the bottom (Figure 2.5). Generally, we find a

49

favourable linear correlation between the Reynolds stress and the velocity gradient between -2 s-1

< du/dz < 3 s-1 for the particle-laden flow while there is a larger variance for the saline flows. For

du/dz > 3 s-1 a few stress values deviate from the linear correlation (depicted by gray squares and

dots in Figure 2.6) which we attribute to the close proximity to the bottom where du/dz has

maximum values and where viscous stresses dominate over Reynolds stresses (Figure 2.5).

Fig. 2.6: Relation between the velocity gradient du/dz and the normalized Reynolds stress ττττR/ρρρρU2 across the flow thickness. The error bars indicate the standard deviation.

The relationship between stress and velocity gradient was also investigated by Odier et al.

[2009]. They conducted supercritical (Fr >1) experimental saline density currents along an

inclined plane to investigate mixing processes outside the BBL. Their flow thickness was h = 6

cm and U = 6 cm/s, so that Re ~ 3000 and their flow was similarly turbulent (but supercritical) to

our experimental gravity currents. They found the Prandtl mixing length model to be a good

approximation to describe mixing processes in stratified gravity currents by relating Reynolds

50

stresses to the mean velocity gradient with:

z

u

z

uLwu m ∂

∂×∂∂= 2'' [2.8a],

z

u

z

uwu

Lm

∂∂×

∂∂

= ''2 [2.8b].

The Prandtl mixing length model implies that the eddy viscosity coefficient νE in Eq. [2.1] is a

linear function of the velocity gradient, so that νE = Lm2×∂u/∂z [2.8a] where Lm is the mixing

length. Lm can then be calculated using Eq. [2.8b] and resulting length scales are plotted

in Figure 2.7.

Fig. 2.7: Mixing length Lm of saline and sediment-laden gravity currents using Eq. [2.8b]. The error bars indicate the standard deviation.

In the BBL (z/h < 0.1) and in the upper part of the flow (z/h > 0.5) the mixing length is

significantly smaller than in the interface region previously studied by Odier et al. [2009] who

observed a fairly constant Lm = 0.45 ± 0.1 cm throughout the entire flow outside the boundary

regions. In the BBL the inferred mixing lengths were Lm = 0.01-0.15 cm, i.e. of a comparable

order of magnitude to the viscous boundary layer. Only between 0.1 < z/h < 0.5 is Lm = 0.05-0.35

cm, i.e. similar to measurements by Odier et al. [2009]. As Lm is not constant in the BBL, but

51

rather increases with height the mixing length model would not lead to a constant eddy

diffusivity in the BBL layer, unlike in the interfacial region studied by Odier et al. [2009]. The

higher values of Lm in the interfacial region in the saline flow of Odier et al. [2009] might be due

to the larger slope of 10o along which the current flowed leading to a higher Fr than in our

experiments. Due to the larger Fr in their supercritical flow, their density current would have had

more interfacial entrainment [Wells et al., 2010] due to the higher turbulence levels at the upper

interface associated with Kelvin-Helmholtz billows.

2.4.4 Turbulent kinetic energy profiles

The turbulent kinetic energy (TKE) in the gravity current represents the intensity of the

turbulence associated with velocity fluctuations from the turbulent eddies [e.g. Tritton, 1989].

The total TKE per unit volume is calculated from time averages of the velocity fluctuations

according to

TKE =

1

2ρ(u'2 + v'2+w'2) [2.9]

[Gray et al., 2006]. Experimental observations of TKE normalized by ρU2 for saline and

sediment-laden flows are plotted in Figure 2.8. Both flow types show a strong normalized TKE

increase in the BBL whereby the saline gravity current has a maximum normalized TKE value of

0.018 ± 0.001 at z/h = 0.08, which is approximately 30% larger than the normalized TKE value

of 0.013 ± 0.0005 for the particle gravity current at z/h = 0.08. Above and below their maxima

the normalized TKE decreases constantly and reaches minimum values of 0.004 at z/h = 0.35

which correlates with the flow velocity maximum at z/h ~ 0.3-0.35. Above this minimum the

normalized TKE tends to increase again. The TKE distribution of both flow types is

quantitatively comparable with the distribution described by Gray et al. [2006] and Islam and

52

Imran [2010]. Both previous studies report normalized TKE maxima of 0.02 at z/h = 0.1 as in

our study and a distinct normalized TKE minimum of 0.004 ± 0.0005 around umax. In addition,

our data show a continuous decrease of the TKE in the BBL for z/h < 0.1.

Fig. 2.8: Vertical profiles of TKE for saline and sediment-laden gravity currents. All values are normalized by ρU2. The horizontal bars indicate the standard deviation.

53

2.4.5 Drag coefficients

Figure 2.9 compares the calculated values of CD obtained from the three different

methods introduced in section 2 for saline and sediment-laden gravity currents. Generally both

flow types show similar CD values but each method yields a different order of magnitude. The

log-fit method gives the largest CD at O(10-2) with CD = 0.02 ± 0.002 for saline and 0.048 ±

0.005 for turbidity currents. Drag coefficients derived from the maximum total stress in the BBL

in Figure 2.5 is 3.65 ± 0.4 × 10-3 for saline and 2.95 ± 0.3 × 10-3 for turbidity currents and hence

one order of magnitude smaller than values obtained with the log-fit method. The direct

measurements of the averaged turbulent velocity ût using Eq. [2.5] lead to the smaller value of

CD = 4 ± 3 × 10-4 and CD = 0.95 ± 3 ×10-4 for turbidity and saline gravity currents respectively.

Fig. 2.9: Calculated drag coefficients using Eq. [2.2] (light gray), Eq. [2.3] (dark gray) and Eq. [2.5] (black) for saline and sediment-laden gravity currents. To treat the drag coefficient determined by using Eq. [2.5] as a bulk drag coefficient the data of the single measurements of ût were averaged between z/h = 0.025 - 0.085. The horizontal bars indicate the standard deviation.

54

The estimates of the CD obtained from the maximum total stress in the BBL and by using

a log-fit are consistent with the range of data found in the literature where CD values range from

10-2 [Garcia and Parker, 1993; Kneller et al., 1999; Straub et al., 2008] to CD = 10-3 [Bowen et

al., 1984; Pope et al., 2006]. However, we note that only the estimate of CD using the maximum

bottom stress approach gives values at the same order of magnitude as the observed CD values in

oceanographic gravity currents that range in thickness from 5 m to 150 m [e.g. Dallimore et al.,

2001; Peters and Johns, 2006; Umlauf and Arneborg, 2009a]. The range of reported values could

be attributed to a variety of reasons, most importantly differences in the calculation method used

to estimate u* and CD respectively.

55

2.5 Discussion

Our new observations of shear stresses within the BBL now complement the previous

Reynolds stress profiles measured outside the BBL [e.g. Buckee et al., 2001; Gray et al., 2006;

Islam and Imran, 2010]. Our observations have focused on the Reynolds stresses below umax and

especially below z/h < 0.1. Largest positive Reynolds shear stresses occur at z/h ~ 0.1 and

decrease rapidly in the region dominated by viscous stresses (Figure 2.5). A similar decrease of

the near bed Reynolds stresses close to the bottom boundary was also observed with high

frequency ADCP measurements of the turbulence structure in the bottom layer of the Red Sea

outflow plume [Peters and Johns, 2006]. As shown in Figure 2.5, we find that the Reynolds shear

stress (at a distance of z/h = 0.1) is the main contributor to the total stress budget outside the

viscous boundary layer [Tritton, 1989].

There has been a considerable discussion on the potential difference in turbulence

dynamics between saline and turbidity currents, as well as the influence of particle size [Stacey

and Bowen, 1988; Kneller and Buckee, 2000; Gray et al., 2006]. Using the Rouse number Z of

,max

1'

p

rms

wZ

w= < [2.10]

[Bagnold, 1962; Leeder et al., 2005], where w’rms, max is the root mean square of vertical turbulent

velocity and wp = 9.1 × 10-4 m s-1 the settling velocity defined by Eq. [2.6], we found that the

experimental turbidity currents (with an average value of w’rms = 1.4×10-3 m s-1) had Z=0.638 < 1

so were sufficiently turbulent to keep the particles in suspension. This calculation of the

suspension criterion is also consistent with observations of negligible deposition near the

measuring location (see section 3), so that our turbidity flows can be classified as non-

depositional or weakly depositional. This explains our observation that we find a similar

Reynolds stress (Figure 2.5) and TKE distribution (Figure 2.8) for both flow types. Beyond the

56

broad similarity, there are minor differences in that saline flows exhibits larger Reynolds stresses

and TKE values outside the BBL, as was also found by Islam and Imran [2010]. This suggests

that the presence of particles has some influence, as sediment-laden flows are often described as

having significant density stratification near the lower boundary compared to saline currents

[Garcia and Parker, 1993; Peakall et al., 2000]. For instance, Sequeiros et al. [2010a,b] observed

regions near the bed in subcritical flows where excess densities varied relatively little, while

supercritical flows exhibited density profiles that declined exponentially. Particles can

accumulate near the base of the current and exhibit a Rouse-type density profile where coarse

material is concentrated towards the lower part of the flow whereas fine-grained material is more

evenly distributed throughout the depth of the flow [Kneller and Buckee, 2000]. This could

account for an attenuation of turbulence in the BBL represented by smaller TKE and shear

stresses at z/h < 0.1. Despite these minor differences we conclude that the saline flows are a good

analogue to non-sedimenting turbidity currents. Our experimental observations of velocity and

turbulence structure give more weight to this implicit assumption in the numerous analogue

experiments where saline density currents are used to model the dynamics of turbidity currents

[e.g. Keevil et al., 2006; Islam and Imran, 2008; Sequeiros et al., 2009; Cossu et al., 2010].

57

Figure 2.10 depicts normalized velocity profiles of several field measurements of large

scale gravity currents and our experimental gravity currents. The comparison reveals that the

basic features of the velocity profile of these field gravity currents are in accordance to our

laboratory currents.

Fig. 2.10: Comparison of downstream velocity profiles between natural and experimental saline and sediment - laden gravity currents. Data are normalized by umax and the height of umax (hu)

Based on our results it seems reasonable to compare the observations from laboratory

gravity currents in section 4.5 with the direct measurements of field scale gravity currents. The

CD values found in natural density currents are of O(10-3), e.g. CD = 2 × 10-3 to 8 × 10-3 in the

150 m deep Red Sea outflow [Peters and Johns, 2006], CD = 5 × 10-3 in a 5 m deep underflow in

an estuary in Japan [Dallimore et al., 2001] and CD = 3 × 10-3 from ADV measurements in a

20 m deep density current in the Baltic sea [Umlauf and Arneborg, 2009a]. In all these cases

direct observations using turbulence microstructure profiles of the turbulent friction velocity lead

to estimates of order CD = 10-3. The CD values in Figure 2.9 stem from three different approaches

and differ by almost two orders of magnitudes between O(10-4) and O(10-2). The largest values at

58

O(10-2) in Figure 2.9 and in the literature were obtained based on a log-fit method. To have

confidence in this technique requires a good fit of the velocity profile to the logarithmic curve,

which needs a very high and accurate spatial resolution of the velocity profile to obtain a reliable

fit [Biron et al., 1998; Pope et al., 2006; Baas et al., 2009]. However, even with several

measuring points in the BBL estimates of u* based on the velocity profile in Figure 2.3b vary

between 2.5 - 5 × 10-2 and lead to values that are significantly larger than field observations.

Similarly, CD values derived from the turbulent velocity ût using Eq. [2.5] yield a significantly

smaller value with CD = 4 - 9.5 × 10-4 that are also not consistent with field data. On the other

hand, the Reynolds stress distribution (Figure 2.5) is in good agreement with other recent

experimental studies [Gray et al., 2006; Islam and Imran, 2010] so that we can say with

confidence that the estimates of u* using Eq. [2.3] with the maximum total bottom stress should

also be correct. Drag coefficients obtained with this method are of O(10-3) which agrees well

with observations from natural gravity currents. It can be concluded that while it is possible to

estimate u* reliably with a log-fit at field scale [Johnson et al., 1994], it is difficult to obtain a

reliable estimate at a laboratory scale. Hence, using only 2 or 3 vertical measurements of the

velocity within the BBL [Kneller et al., 1999; Straub et al., 2008] could lead to an overestimate

of CD while we found the maximum bottom stress to be the most robust method to calculate the

bottom drag coefficient.

Apart from the calculation method, the range of reported values in the literature could be

attributed to a variety of other reasons, e.g. different flow properties in experiments (different

Reynolds or bulk Richardson numbers), different bed surfaces (bed form structures like ripples

or dunes, width of the channel etc.) and different measurement techniques. Hand [1974, 1975]

and Komar [1975] discussed the importance of these factors in terms of their different estimates

for CD which deviated by 800 %, e.g. CD = 0.04 Komar [1975] and CD = 0.005 [Hand, 1975].

59

Providing conditions of a hydraulically smooth bed CD values in the range of O(10-3) seem to be

a reasonable estimate [Hand, 1975] and collapse well with our data and various field

observations. The variation of CD and values at O(10-2) for gravity currents can further be

attributed to the parameterization of the overall drag, consisting of the bottom drag described

above and the interfacial entrainment of overlying water.

The momentum budget for a steady gravity current that is subject to bottom stress and

entrainment of ambient fluid can be written as a balance between buoyancy forces and the sum

of the bottom drag and interfacial entrainment as:

′g S= (CD

+ E)U 2

d [2.11]

where S is the bottom slope and E is the entrainment coefficient [Ellison and Turner, 1959;

Stacey and Bowen, 1988; Price and Baringer, 2009]. E is a strong function of Fr, varying

between 0.1 for very high Fr values, down to values as low as 10-4 for Fr of 0.5. For the typical

Fr of gravity currents that are in the range of 0.4 to 1.2 [e.g. Simpson, 1997; Kneller et al., 1999;

Gray et al., 2006] the entrainment ratio E varies between 5 × 10-4 to 5 × 10-2 [Wells and

Wettlaufer, 2005; Wells et al., 2010]. These values of E are almost the same range as the values

of CD presented here and in the literature, which leads to the conclusion that in at least some

cases the influence of interfacial entrainment has been implicitly included into the estimation of

CD, which is then used to estimate the bottom drag and shear stresses at the bed. For instance,

Garcia and Parker [1993] determined CD in the range of CD = 2 - 9 × 10-2 by back-calculation

from experimental velocity profiles and density concentrations with a layer-integrated

momentum equation. The use of such an approach implicitly considers the drag over the whole

thickness of the current including the interfacial drag at the upper boundary of the density flow.

60

Including these high values of E may then lead to an overestimate of CD, and hence overestimate

the shear stress at the bed.

The rate of entrainment of bottom sediment by gravity currents is often calculated based

upon the empirical formulae of Garcia and Parker [1993], as in the numerical models of

Blanchette et al. [2005], Huang et al. [2005] and Hu and Cao [2009]. This empirical relation for

bed entrainment requires a definition of u* which can be estimated if U and CD are known. Even

numerical models [e.g. Huang et al., 2007] that derive their shear stress from the simulated flow

field (which does not require the specification of CD) use a calculation of entrainment of

sediment to suspension, following a relation of Smith and McLean [1977], which in turn has

been tested against data by Garcia and Parker [1991]. In their calculation of the erosion rate

Garcia and Parker [1993] used a CD of O(10-2). Their empirical formulae has a very strong

dependence of erosion rate on the CD value, so if bed erosion is modelled using bottom stresses

estimated with a CD that is an order of magnitude too high, then this method can overestimate the

entrainment rate and erosion at the bed by a factor of O(101-102). The solution would be to

assume simply that the shear stress at the bottom of their gravity current was lower and then

recalculate the empirical constants in their equation that predicted the observed erosion in their

original experiments.

2.6 Summary and Conclusions

Our experiments show that saline and weakly-depositional sediment-laden gravity

currents have a generally similar turbulence structure in the BBL, with both flow types revealing

very similar Reynolds stresses and TKE distribution. Outside of the BBL we find maxima in

both Reynolds stresses and TKE at z/h ~ 0.1 and minima around umax. These results support

findings in previous works [e.g. Gray et al., 2006; Islam and Imran, 2010] which focused on data

61

outside of the BBL. A new result of our measurements is that Reynolds stresses decrease rapidly

below their maximum and that Reynolds stresses balance viscous forces at the bottom and lead

to large positive stress maxima near the base of gravity currents (Figure 2.5).

The velocity, Reynolds stress and TKE distribution of saline and sediment-laden currents

suggests that both flow types are dynamically similar when turbulent velocities near the bed are

so large that the turbidity current is only weakly depositional. Figure 2.10 demonstrates that we

find a very good agreement between velocity profiles of field gravity currents and our laboratory

gravity currents. The useful implication of this is that much of the recent work by

oceanographers on the interior dynamics of density overflows [Johnson et al., 1994; Dallimore et

al., 2001; Peters and Johns, 2006; Umlauf and Arneborg, 2009a] could also be applied to

understanding non-depositional large-scale turbidity currents.

Three different methods to determine CD values for gravity currents were tested against

our experimental data. While the log-fit method and measurements of turbulent velocities at the

bottom yield either very large or small values of CD respectively, we find the most reasonable

results of O(10-3) by using the maximum total bottom stress for both flow types. This data range

is similar to that of oceanographic studies, but lower than previous indirect estimates from

laboratory studies. When a log-fit approach is used to estimate the bulk friction velocity near the

bed, we note that there must be a very high resolution of the velocity profile to avoid a potential

overestimate of CD if the whole velocity profile below the velocity maxima is assumed to be well

represented by a logarithmic velocity profile.

62

Acknowledgements

Funding for this work was provided by NSERC, the Canadian Foundation for Innovation and the Ontario

Research fund. We thank Brian Greenwood and Joe Desloges for help with the laboratory work.

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Chapter 3 Influence of the Coriolis force on the velocity structure of gravity currents in straight submarine channel systems Remo Cossu, Mathew G. Wells and A. K. Wåhlin Abstract

Large-scale turbidity currents in submarine channels often show a significant asymmetry in the

heights of their levee banks. In the Northern Hemisphere there are many observations of the

right-hand channel levee being noticeably higher than the left-hand levee, a phenomenon that is

usually attributed to the effect of Coriolis forces upon turbidity currents. This chapter presents

results from an analogue model that documents the influence of Coriolis forces on the dynamics

of gravity currents flowing in straight submarine channels. The observations of the transverse

velocity structure, downstream velocity and interface slope show good agreement with a theory

that incorporates Ekman boundary layer dynamics. Coriolis forces will be important for most

large-scale turbidity currents and need to be explicitly modeled when the Rossby number of

these flows (defined as RoW = U/Wf where U is the mean downstream velocity, W the channel

width and f the Coriolis parameter defined as f=2Ω sin(φ), with Ω being the Earth’s rotation rate

and φ the latitude), is less than order 1. When |RoW| << 1 the flow is substantially slower than a

non-rotating flow with the same density contrast. The secondary flow field consists of

frictionally induced Ekman transports across the channel in the benthic and interfacial boundary

layers, and a return flow in the interior. The cross-channel velocities are at the order of 10% of

67

the along-channel velocities. The sediment transport associated with such transverse flow

patterns should influence the evolution of submarine channel levee systems.

3.1 Introduction

Submarine channels are the most significant morphologic features of the submarine

landscape on the continental slope and are known to be the main conduits for turbidity currents

to transport sediments to the deep ocean basins [Meiburg and Kneller, 2010]. There are still very

few direct observations of turbidity currents moving through submarine channels [Hay, 1987;

Khripounoff et al., 2003; Xu et al., 2004] since their occurrence in great water depths and high

current velocities make measurements difficult to obtain. Due to the spatial extent of submarine

channel systems and the associated long travel time, the flow properties and environments of

those currents are likely to be deflected by Coriolis forces that arise due to the Earth’s rotation

[Menard, 1955; Komar, 1969; Wells, 2009] for mid- and high latitude systems. While the

influence of Coriolis forces upon turbidity currents is acknowledged in several reviews

[Middleton, 1993; Huppert, 1998; Imran et al., 1999; Kneller and Buckee, 2000] and theoretical

studies [Komar, 1969; Bowen et al., 1984; Nof, 1996; Emms, 1999; Ungarish and Huppert,

1999; Kampf and Fohrmann, 2000], there have been very few previous experimental studies

specifically focusing on turbidity currents on a rotating platform [Wells, 2009].

One of the most noted effects of the Coriolis force upon well-developed levee systems is

that the deflection of the turbidity current by Coriolis forces leads to an asymmetry between

levee bank heights [Menard, 1955]. The right-hand-side channel levee (looking downstream) is

consistently higher in the Northern Hemisphere [Chough and Hesse, 1976; Klaucke et al., 1997,

1998; Wood and Mize-Spansky, 2009] and the left-hand-side channel levee is higher in the

Southern Hemisphere [Droz and Mougenot, 1987; Carter and Carter, 1988; Bruhn and Walker,

68

1997]. These differences in levee height can be large, for instance, in the NAMOC, there is a

difference in levee height that can reach more than 100 m and has an average difference of 65 m

along a 950 km long section [Klaucke et al., 1997]. Such observations of levee asymmetry are

usually described in the context of the theory of Komar [1969], although this theory has never

been tested in laboratory scale experiments on a rotating platform. In this paper the first rigorous

test of this theory is presented as well as a complementary theory that quantifies the downstream

and cross-stream velocity components.

The basic description of rotationally influenced turbidity currents that is most widely

used was presented by Komar [1969]. This theory is based upon a simple momentum balance

across the channel, whereby the pressure gradient forces that result from the surface slope of the

turbidity current are balanced by the Coriolis and centrifugal forces. Turbidity currents are often

constrained to flow in channelized systems, and it is usually assumed that the maximum levee

heights are a good indication of the maximum thickness of the turbidity currents that formed the

levee systems. If the difference in levee heights is a good measure of the maximum slope of the

interface of the turbidity current, then the theory of Komar [1969] can be used to infer the

magnitude of the mean current speeds of the typical turbidity currents that would have originally

formed the channel levee system. There are several important simplifications made in the theory

of Komar [1969] to describe the influence of Coriolis forces upon turbidity currents. Firstly, it is

assumed that the Froude number Fr = U / g 'h (where U is the mean downstream velocity, g’

the reduced gravity and h the thickness of the current), remains constant at a value of one. The

use of a constant Froude number implicitly assumes that the rotation rate has no control on the

velocity of the current, counter to the observations of Cenedese et al. [2004] and Cenedese and

Adduce [2008] where laboratory experiments clearly show a large reduction in velocity when the

69

Coriolis parameter is high. Another significant assumption in the theory of Komar [1969] is the

neglect of boundary friction, which means that the Ekman boundary layers are ignored even

though the Coriolis forces are assumed to be high. In a series of theoretical and experimental

papers by Wåhlin [2002, 2004], Davies et al. [2006] and Darelius [2008] these boundary layers

were shown to play a critical role in determining the sense of the secondary circulation in

rotationally controlled gravity currents. Even in cases where Coriolis forces are not dominant,

frictional effects have been shown to be important in determining the slope of a density interface

flowing in a channel in estuaries [Ott et al., 2002; Chant and Wilson, 1997; Fugate et al., 2007;

Nidzieko et al., 2009].

In contrast to the rarely observed turbidity currents, there are many direct observations of

large-scale oceanographic density currents [Ivanov et al., 2004] that can inform our

understanding of the circulation patterns in turbidity currents. In the oceanographic literature

there a number of previous experimental studies on how Coriolis forces deflect cold or salty

currents of dense water as they flow down the continental margins of oceans [Griffiths, 1986;

Price and Baringer 1994; Etling et al., 2000; Hallworth et al., 2001; Cenedese et al., 2004;

Davies et al., 2006; Cenedese and Adduce, 2008]. These experimental studies found that rotating

gravity currents rapidly come into a geostrophic balance, where the trajectory of the density

current is determined by a balance between buoyancy forces, friction and Coriolis forces. Both

density currents and turbidity currents are examples of gravity currents, in that the flows are

primarily driven by density differences [Huppert, 1998]. Low concentration and weakly

depositional turbidity currents exhibit concentration and velocity profiles that are very similar to

saline currents [Kneller and Buckee, 2010]. For this reason many laboratory measurements using

saline currents [e.g. Keevil et al., 2006; Islam et al., 2008; Darelius, 2008] have made important

contributions to defining both the structure of the flow field and the turbulence intensities

70

associated with these gravity currents and therefore helped to develop a general understanding of

the fluid dynamics also for turbidity currents. In this context we use experimental density

currents for this study to understand the first order effects of how Coriolis forces influence the

circulation of channelized gravity currents, which will be also relevant to geologists wishing to

comprehend the interaction of turbidity currents and channels they build. Nonetheless we note

that further experimental work considering how sediment dynamics influence transport in Ekman

boundary layers, will be needed to gain more insight into evolution of submarine channel

systems subjected to Coriolis forces.

Many submarine fan channels are confined by prominent levees, which form by

deposition of suspended sediment on the slower moving margins of a turbidity current. These

levees can grow rapidly, for instance, the average sedimentation rates during the active growth

phases of the levees of the Amazon channel during the Pleistocene were 1 to 2.5 cm year-1

[Shipboard Scientific Party, 1995]. The rapid growth has been attributed to deposition of

suspended load as successive turbidity currents transit the channel and spill over the channel

margins along their entire length [Hiscott et al., 1997; Peakall et al., 2000; Straub et al., 2008]. In

addition, recent non-rotating experiments on channelized turbidity currents have shown that the

morphological evolution and associated depositional histories of submarine channel systems are

highly influenced by the secondary flow structures within the channel, which determine where

erosion and deposition will occur [Corney et al., 2006; Keevil et al., 2006; Straub et al., 2008;

Islam et al., 2008; Islam and Imran, 2008]. The main focus in these non-rotating experiments has

been to investigate the secondary circulation due to an imbalance of centrifugal and pressure

gradient forces in channel bends, which plays a major role in the formation of super-elevation of

levee systems at the outer bend [e.g. Straub et al., 2008]. Coriolis forces can also give rise to

secondary flows within turbidity currents through the generation of Ekman boundary layers at

71

the upper interface and the base of the flow. In the Northern Hemisphere these boundary layer

flows will be directed to the left when looking downstream. To conserve volume there is return

flow directed to the right in the interior of the flow. The basic features of such Ekman boundary

layers in channelized current have been seen in several laboratory experiments [Benton and

Boyer, 1966; Hart, 1971; Johnson and Ohlsen, 1994; Davies et al., 2006; Darelius, 2008; Wåhlin

et al., 2008]. Qualitatively similar secondary flows driven by Ekman boundary layer dynamics

have been seen in oceanic gravity currents, such as reported in the Vema channel [Johnson et al.,

1976], in the Faroe Bank Channel [Johnson and Sanford, 1992; Fer et al., 2010], in the Ellet

Gully [Sherwin, 2010] and in the Baltic Sea [Umlauf and Arneborg, 2009 a,b]. Turbidity currents

should also experience similar secondary flow patterns, when the flows are at large enough

scales that the Coriolis force becomes important.

Though the theory of Komar [1969] has been cited almost one hundred times in the

geological literature [i.e., Bowen et al., 1984; Klaucke et al., 1997; Imran et al., 1999], no

experiment has ever tested its validity. The goal of this study is to look at the flow structure in

experimental rotating gravity currents and to relate it to a general geological context. Of primary

importance for geologic applications is to infer the speed of the turbidity current that would have

formed the asymmetric levees in a submarine channel, as this information can be used to

determine the likely evolution of sediment deposition in turbidite beds that may be rich in

hydrocarbons [Weimer et al., 2000]. Using an analogue experiment mounted on a rotating

platform we are able to determine the dependence of the secondary flow structure, downstream

velocity and interface tilt upon rotation rate. These observations are compared with the theory

initially developed by Wåhlin [2002, 2004].

72

3.2 Theory

The most widely used description of how Coriolis forces influence the dynamics of

turbidity currents was proposed by Komar [1969], in which a force balance between the Coriolis

force, the centrifugal force induced by flow curvature, and the pressure gradient force is

assumed. Assuming that the upper interface has a constant slope and that friction is negligible,

the momentum balance across the channel can then be written as:

−g '

dh

dy= fU +

U 2

R, [3.1]

where U is the mean downstream velocity, R is the radius of curvature of the channel and f the

Coriolis parameter (defined as f =2Ω sin(φ), with Ω being the Earth’s rotation rate and φ the

latitude). The reduced gravity is g ' = g ρ

2− ρ

1( ) ρ1, where the gravity current has the density ρ2

and ρ1 is the ambient density of the seawater. Equation [3.1] can be rewritten in terms of a

Froude number ( Fr = U / g 'h ), as:

−

dh

dy= Fr 2 fh

U+

h

R

. [3.2]

This momentum budget does not include any turbulence drag from the boundary and so

implicitly assumes that stratification suppresses turbulent motions [e.g. Chant and Wilson, 1997].

Based upon the force balance shown in Eq. [3.2] and assuming that Fr = 1, Bowen et al. [1984]

expressed the difference in levee height due to the Coriolis-effect in a straight channel as:

∆h = Wh f / U , [3.3]

where ∆h= W dh/dy, with W being the channel width and h being the depth of the main body of

the flow. This equation can also be written as / 1/Wh h Ro∆ = where RoW = U/Wf. Equation

[3.3] is valid only for the straight sections of channels, where centrifugal effects are absent.

73

Equation [3.3] has been used to describe observed channel height asymmetries, such as in the

Amazon channel [Imran et al., 1999], the Navy fan in California [Bowen et al., 1984] or the

NAMOC described by Klaucke et al. [1997, 1998].

In a rotating system, the inclusion of friction into the momentum equation means that the

Ekman boundary layers must be properly described. The resulting flow structure of a gravity

current in a rotating system has been studied previously in the oceanographic literature [Wåhlin,

2002] and is illustrated in Figure 3.1.

Fig. 3.1: Schematic sketch of a density current flowing down a submarine channel with the gradient s, the channel height D and the channel width W, looking upstream. The density of the ambient fluid and the gravity are ρρρρ1 and ρρρρ2 respectively, with ρρρρ2 > ρρρρ1. The main downstream flow is uG while there is also a significant transverse motion consisting of the interior flow vG and bottom and interfacial currents ve. The thickness δ δ δ δ of the Ekman boundary is small in comparison to the entire thickness of the flow h(y).

In the Northern Hemisphere the currents are deflected to the right side of the channel

(looking downstream) until the flow reaches a geostrophic balance. Friction at the upper and

lower boundaries leads to the formation of Ekman boundary layers, which drive a transverse,

secondary circulation within the flow [i.e. Darelius, 2008]. Figure 3.1 shows a sketch of the flow

74

structure. The secondary circulation consists of an across-channel flow in the Ekman boundary

layers next to the bottom and next to the interface, and an oppositely directed across-channel

flow in the interior away from the Ekman boundary layers. The thickness of the Ekman layers is

δ , and the velocity there is ve. In the interior the across-channel velocity is vG. Similar secondary

flows have been described in previous rotating laboratory experiments by Hart [1971], Johnson

and Ohlsen [1994], Davies et al. [2006], Wåhlin et al. [2008] and Darelius [2008] and were also

reported in oceanic gravity currents, [e.g. Johnson and Sanford, 1992; Sherwin, 2010; Fer et al.,

2010]. In Umlauf and Arneborg [2009a,b] a different secondary flow structure was observed,

with a thin jet in the interfacial layer and no clear Ekman layer at the bottom. This may be due to

the fact that the Ekman boundary layer was comparable in thickness to the flow itself.

The Ekman boundary layer dynamics of a gravity current have been previously examined

for V-shaped [see e.g. Davies et al., 2006; Darelius and Wåhlin, 2007; Darelius, 2008], cosine-

shaped [Wåhlin, 2002; Darelius and Wåhlin, 2007] and parabolic shaped [Darelius and Wåhlin,

2007] canyons and ridges. In this paper a square channel is investigated for the first time, which

permits analytical expressions for the across and along channel components of the velocity and

the transverse slope of the interface, as a function of the rotation rate and mean slope of the

channel.

Consider a rectangular channel of width W, which slopes downward at angle s to the

horizontal (Figure 3.1). Assuming a force balance between Coriolis, bottom friction, and

pressure gradient, and a 1.5 layer system, the momentum equations can be written as:

2

2' E

ufv g s

zν ∂− = − +

∂ [3.4]

2

2' E

h vfu g

y zν∂ ∂= − +

∂ ∂, [3.5]

75

where f is the Coriolis parameter, v the velocity in the across-channel (i.e. y) direction, g' the

reduced gravity, νE the molecular viscosity, u the velocity along the channel and h(y) the

thickness of the dense layer. Note that Eq. [3.5] in similarity with Eq. [3.1] expresses the

interface slope as a function of the along-channel velocity. The difference is that Eq. [3.5]

includes the viscous term (last term on right-hand-side) but does not include the centrifugal term

which is absent in straight sections. Away from the Ekman boundary layers the viscous terms

can be neglected and Eq. [3.4] and Eq. [3.5] reduce to the geostrophic velocities, i.e.:

v → v

G= g 's

f , [3.6]

u → u

G= − g '

f

∂h

∂y. [3.7]

The Eq. [3.4] and Eq. [3.5] can be solved using Ekman theory [see e.g. Cushman-Roisin, 1994,

p. 66; Darelius, 2008], for which the velocity in the directions along- (u) and across- (v) the

channel are given by:

ue(z) = u

G(1− e

−z

2δ cos(z 2δ )) − vGe

−z

2δ sin(z 2δ ) [3.8a]

ve(z) = u

Ge

− z

2δ sin(z 2δ ) + vG(1− e

− z

2δ cos(z 2δ )) [3.8b]

where δ = υ 2 f is the Ekman layer thickness, uG and vG are the geostrophic velocities given

by Eq. [3.6] and Eq. [3.7] in the interior and it has been assumed that h>> δ (we note that often a

different definition of δ = 2υ f can be found in the literature, e.g. Cushman-Roisin [1994]).

Expression [3.8a,b] has been plotted in Figure 3.2 for uG = 0.07 m s-1, vG = -0.01 m s-1 and three

different values of f. As can be seen, far above the boundary layer the velocity approaches the

geostrophic velocities, and in the boundary layer the velocity rotates and decreases exponentially

76

to zero. For vG<<u G the maximum value of v(z) is v

MAX=

uG

2e

−π4 ≈ 0.3u

G, approached at

z ≈ πδ

2. The mean velocity in the Ekman boundary layer is 0.1 M EAN Gv u≈ . The boundary

layer thickness δ is indicated by thin horizontal lines in Figure 3.2.

Fig. 3.2: Dependence of the flow velocity u and ve in the Ekman boundary layer on the Coriolis parameter f. Note that also the thickness δδδδ of the Ekman boundary layer varies with f and is indicated by the horizontal lines. The net transport across the channel is found by vertical integration of Eq. [3.8b] from the

bottom (z = 0) to the top of the dense layer (z=h), to give:

`

v(z)dz=0

h

∫ hvG

+ δ (uG

− vG

) , [3.9]

where we have used a standard result from Ekman theory, namely that the transport in the

Ekman boundary layer is given by δ(uG – vG) when h>> δ (which will be used for the rest of the

theory). Classical Ekman theory pertains to the flow over a solid boundary, but there are many

examples of observations of interfacial Ekman layers, [e.g. Darelius, 2008; Sherwin, 2010]. The

77

basic dynamics are the same, although the thickness of the layer may be greater than in the

bottom boundary layer. The interfacial stress gives rise to an Ekman layer next to the interface,

with the Ekman transport directed to the left of the main flow direction in the Northern

Hemisphere (looking downstream). By vertical integration across the interfacial Ekman layer an

expression similar to Eq. [3.9] is obtained. The net across-channel transport in the presence of

both an interfacial and a benthic Ekman layer is hence given by:

v(z)dz=

0

h

∫ hvG

+ 2δ (uG

− vG

) [3.10]

where the factor 2 comes from the effect of including both boundary layers. We note that the δ

here is still the viscous Ekman boundary layer, and will discuss the possible influence of a

turbulent boundary layer later. When the Ekman boundary layers meet the vertical side walls the

horizontal flow will be transported vertically within Stewartson boundary layers [Duck and

Foster, 2001] and then returned to the interior flow. Provided there is no net transport across the

channel the horizontal flow in the geostrophic interior balances the flow in the Ekman boundary

layer, so that:

hvG

+ 2δ (uG

− vG

) = 0. [3.11]

Using Eq. [3.6] and Eq. [3.7], Eq.[3.11] can be rewritten as:

∂h

∂y− s

h

2δ= −s, [3.12]

which gives a solution for the position of the interface across the width of the channel as:

h( y) = Cesy

2δ + 2δ . [3.13]

78

In Eq. [3.13] C is a constant of integration with units of length that must be determined by a

boundary condition. If we use the along-channel transport Q as the boundary condition

(assuming that the dense layer is thick compared to the Ekman layer and vG<<u G) we get:

Q = hu

G0

W

∫ dy = hg 'f

∂h

∂y0

W

∫ dy , [3.14]

or using Eq. [3.13]:

Q = 1

2

g '

f

∂∂y

(h2)0

W

∫ dy

= 1

2

g '

fh2(W) − h2(0)

= 1

2

g '

fC2(e

sW

δ − 1) + 4δC(esW

2δ −1)

.

[3.15]

From Eq. [3.15] we can express C in terms of Q. For most cases, in particular for high rotation

rates, δ << h and the relationship between C and the volume flux Q is given by:

C = 2 fQ

g '1

(esW

δ − 1)

. [3.16]

79

Figure 3.3 shows the interface for various rotation rates, looking downstream. In order to

compare this expression for h with Eq. [3.3] we can determine ∆h = h(W) -h(0) using Eq. [3.13].

Fig. 3.3: Calculated position of the interface in the channel after Eq. [3.13] for varying Coriolis parameter f.

The mean along-channel velocity U is given by the volume flux divided by the cross-sectional

area A of the flow:

U = Q / A, [3.17]

where the area is calculated using Eq. [3.13]:

0

2

0

2

( )

( 2 )

2 ( 1) 2'( 1)

2

W

W sy

sW

sW

A h y dy

Ce dy

Qe W

s ge

f

δ

δ

δ

δ

δ δ

= ⋅

= + ⋅

= − +−

∫

∫ [3.18]

80

and Eq. [3.16] has been used to define C. The mean downstream geostrophic velocity uG will be

referred to as U in the rest of the manuscript so that meaningful comparison can be made with

the non-rotating environment.

3.3 Method

The physical experiments were carried out on a computer-controlled rotating platform

with a diameter of 1 m. The rotation rate was varied from f = -1 rad s-1 to f = 1 rad s-1 (including

f = 0) to represent a range of different latitude systems. All experiments were conducted in a

1.85 m × 1.0 m × 0.35 m rectangular tank that was placed on the rotating platform (Figure 3.4).

Fig. 3.4: a) Layout of the experimental setup. The velocity of the density currents was measured approximately 50 cm from the inflow at the upstream end of the channel model. b) Position of the UDVP and ADV used for the measurements.

81

Inside this tank a channel model was placed and the tank was filled with tap water up to a level

of approximately 0.30 m, so that the entire channel system was submerged by approximately 0.1

m at the inflow point. The channel model had a constant, rectangular cross-sectional shape with a

width of 0.1 m and a height of 0.08 m. The length of the straight channel was 0.6 m and was

elevated 0.12 m above the tank floor. When the density current leaves the channel the dense

water flows into the sump region below the channel, which restricts the influence of gravity

current reflections from the sides of the tank. In order to have a constant velocity of the inflow

the saline water passed through a 0.10 m thick diffuser made of drinking straws and foam to

damp any irregularities in the inflow velocity. The slope of the channel axis s was 1:50.

In the experiments the turbidity current was modelled by a dense saline mixture as in

previous studies [e.g. Keevil et al., 2006; Imran et al., 2007; Islam et al., 2008] that have used

saline gravity currents to gain insight into the flow field and secondary circulations in turbidity

currents. The justification for this is that the dynamics of large-scale non-depositional turbidity

currents are similar to density currents. The density contrast was generated with salt, and 30 L of

the saline mixture was pumped in at the upstream end of the straight channel section using a

pump with a constant discharge of Q = 16 L/min, so that each flow lasted for about 120 s after

the pump was turned on and the current entered the channel model. The excess density was 1%,

giving a density for the saline current of 1010 kg m-3. The tank was spun up for at least 30 min in

order to achieve solid body rotation of the water, after which the experiment was initiated. In

order to visualize the density current and the slope of the density interface blue food dye was

added to the mixture in some experiments.

The across-stream and along-stream velocities were measured at a distance of 0.5 m from

the flow diffuser at the start of the channel. Two different acoustic velocity instruments were

used: a Metflow Ultrasonic Doppler Velocity Profiler (UDVP) and a Nortek Acoustic Doppler

82

Velocimeter (ADV). Arrays of UDVPs have been frequently used in geometrically similar non-

rotating experiments [e.g. Best et al., 2001; Corney et al., 2006; Peakall et al., 2007a]. An array

of UDVP probes is ideal for making transverse measurements of the flow. Each UDVP probe

can record single-component velocity data along a profile of 128 points along the axis of the

ultrasound beam at a frequency of 4 Hz. Vertical velocity profiles were obtained from an array of

6 transducers to monitor the velocity at heights of 0.5, 1.5, 2.5, 3.5, 4.5 and 5.5 cm above the

bottom (Figure 3.4b). Representative values were obtained by averaging the velocities over 30-

35 s, starting immediately after the head of the current had passed the instrument (Figure 3.5).

Fig. 3.5: The time averaged horizontal velocity is plotted at different heights above the bed for three different rotation rates. These velocities were measured using the UDVP. The error bars depict the standard deviation over the 30 second measurement period. Most of the volume flux of the density current is between 0.5 cm and 4 cm above the bottom. Hence, the point measurements with the ADV were taken at 2.5 cm above the bottom, to represent the significant velocities between 0.5 and 4 cm. This region of the flow best represents the geostrophically adjusted velocity ug used in the theory section, e.g. Eq. [3.7] to Eq. [3.14]. The open markers showing the ADV data are taken at the same rotation rates as the closed markers for the UDVP data, e.g. diamonds reflect a Coriolis parameter f=0 rad s-1.

83

The second instrument was a Nortek ADV. This can sample simultaneously 3

components of velocity at frequencies up to 200 Hz, but only measures at a single point, so is a

good complement to the UDVP. The ADV consists of one transmitter and three receivers which

are slanted at 30o from the axis of the transducer and focus on a common sample volume of 80

mm3 that is 50 mm away from the probes [Voulgaris and Throwbridge, 1998]. This ensures

nearly non-intrusive flow measurements within the flow field of the current. The velocity

components measured by the ADV were recorded in the centre of the channel, 2.5 cm above the

bottom at a frequency of 50 Hz (Figure 3.4b). The data were taken over the same sampling

interval over which the UDVP data was averaged.

In order to measure the interface position, a digital camera was used. In those

experiments blue food dye was added to the salt water. The camera was mounted on the table

looking upstream so that the thickness of the current and slope of the interface could be

measured (Figure 3.4b).

3.4.1 General observations

The aspects of the flow that have been analyzed are the flow velocity U of the interior

bulk flow (measured with the ADV), the secondary flow cells (measured with the UDVP) and

the deflection of the density interface (photographs).

In the absence of rotation (f=0 rad s-1) the front of the dense current formed a head with

the typical overhanging nose that covered the whole cross section of the channel. Behind the

head the density current was significantly thinner, but the flow still occupied the whole channel

width. After the passage of the head the flow conditions remained fairly constant. A small

amount of mixing of ambient water into the current was noticed through the more transparent

color at the upper interface. Vertical velocity profiles from several experiments with different

84

values of the Coriolis parameter are shown in Figure 3.5 together with the point ADV

measurements. The velocity profiles show the typical “bullet nose” profile [Middleton, 1993;

Kneller and Buckee, 2000] with the maximum velocity close to 0.5-1 cm above the bottom.

Within the bottom boundary layer the velocity increases non-linearly between the base and the

velocity maximum Umax as also observed in Kneller et al. [1999]. Above the velocity maximum

the velocity decreases continuously to U < 0.02 m s-1 at 5.5 cm above bottom. Based on these

profiles the point measurements using the ADV were positioned 2.5 cm above the bottom,

approximately representative of the mean downstream velocity. As can be seen in Figure 3.5

there is a favorable agreement between the UDVP measurements and the ADV measurements,

which later were used to determine U.

In the experiments with rotation the formation of the head and its transition to the body

with a distinct thinning of the flow was less obvious, as the gravity current was pushed towards

the wall after it had entered the channel. With positive f the currents were deflected to the right-

hand-side of the channel (looking downstream) and for negative Coriolis parameter (-f) to the

left-hand-side. This deflection became greater as the rotation rate increased. The propagation

speed of the gravity currents was also observed to decrease significantly as the rotation rate

increased. In the non-rotating experimental gravity currents shown in Figure 3.5, the mean

velocity averaged over the entire thickness was approximately Umean= 0.04 m s –1. This flow had

a thickness of approximately h = 0.06 m so that the Froude number was / 'meanFr U g h= =

0.52. Such a Froude number is broadly similar to many previous experimental observations of

gravity currents where 0.2 < Fr < 1.3 [Garcia and Parker, 1993; Kneller et al., 1999; Baas et al.,

2005; Gray et al., 2006]. Flows with such small Froude numbers are also expected to have very

low interfacial entrainment rates [Wells and Wettlaufer, 2007; Wells and Nadarajah, 2009; Wells

et al., 2010]. The flow had a Reynolds number of Re = 2400 where Re = U h/ν, where ν is the

85

molecular viscosity. The value of Re indicates that the flow was turbulent and can be compared

to other experimental gravity currents such as Kneller et al. [1999], Amy et al. [2005] or Davies

et al. [2006] with similar Reynolds numbers where the flow was turbulent.

The distance over which the flow is expected to adjust geostrophically is in the order of

the internal Rossby radius of deformation defined as Rdef = (g’h)1/2/f [e.g. Darelius, 2008; Wells,

2009]. For the rotating gravity currents used in our experiments the value of Rdef is about 0.4 m

for low rotation rates of f = 0.1 rad s-1. The Coriolis force gets larger for higher rotation rates

(e.g. f > 0.1 rad s-1) so that these flows come more quickly into a geostrophic balance which

means that the radius of deformation decreases. As all of the velocity measurements were

obtained 0.5 m from the inflow point we expect that all flows will have reached geostrophic

conditions and reproduce sufficiently characteristics associated with rotating gravity currents.

3.4.2 Observations of downstream velocity U

The downstream velocities measured with the ADV for different rotation rates are plotted

in Figure 3.6 and compared to the mean velocity based on the theory incorporating Ekman

dynamics (using Eq. [3.17] and Eq. [3.18]) which predicts a significant decrease of the

downstream velocity between small and large Coriolis parameters f. The error bars indicate the

standard deviation in the averaging period. The experiments reveal that for f < 0.2 rad s-1 the

velocity is not significantly influenced by rotation as the velocity remains relatively constant at U

~0.07 m s-1, which is close to the velocity computed based on the density difference and a

constant Froude number. As f increases the measured downstream velocity decreases. At large f

the downstream velocities are up to 40% smaller than for the non-rotating case (f=0 rad s-1). The

experimental data show good agreement with the theory using Ekman boundary layer dynamics

(Eq. [3.17] and Eq. [3.18]).

86

In Figure 3.6 there is a slight offset in the downstream velocity between small positive

and negative Coriolis parameter (e.g. between f = 0.1 – 0.25 rad s-1). We can attribute this

velocity difference mainly to two reasons: the alignment of the measuring device might not have

been exactly at the centerline so that the maximum speed of current was not captured perfectly

during those experiments. Secondly, due to the experimental set-up at the inflow point those

currents were not entirely uniform and more pronounced at either the right-hand or left-hand side

of the channel so that the maximum flow velocity was off-set to the position of the measuring

device. However, we can consider those differences as negligible as the overall data set reveals a

distinct and consistent trend, with symmetry at large f (small RoW).

Fig. 3.6: Dependence of the mean downstream velocity U with varying Coriolis parameter f. The graph shows the mean velocity U based on the initial Froude number, and U calculated using Eq. [3.17] and Eq.[3.18] with a turbulent and a laminar boundary layer and direct ADV measurements. The ADV data were taken at a height of 2.5 cm above the base and in the middle of the channel approximately 0.5 m away from the injection point.

87

The analytical expressions (Eq. [3.10] and Eq. [3.18]) are based on purely laminar flow

conditions [Wåhlin, 2002; Darelius, 2008]. However, a large Reynolds number of Re = 2400

suggests that the currents are turbulent rather than laminar. In both the turbulent and laminar

flows, the thickness of the boundary layers represents the length scale over which friction is

important. These equations can be estimated by rearrangement of the Ekman number, which is

defined as Ek = ν ΩH 2 [Cushman-Roisin, 1994], where H is a characteristic vertical length

scale. If the Ekman number is of order 1, then frictional influences are important so that

H ~ ν Ω . The expression for the turbulent boundary layer thickness comes from a

consideration of the turbulent viscosity. For instance, in Figure 3.2 the theory ( δ = υ 2 f )

predicts thicknesses of the laminar boundary layer of order 0.1 cm. As will be demonstrated in

section 4.3 and 4.4 the observations of the experimental currents suggest that there are larger

boundary layers, in particular turbulent mixing with the ambient water at the upper boundary

layer led to a thick interfacial boundary layer (Figure 3.10b). When comparing our theory using

Ekman dynamics (Eq. [3.17] and Eq. [3.18]) with the observations of velocities and interface

slope, we make the simple assumption that the turbulent Ekman boundary layers are thicker than

the laminar Ekman boundary layers. This is consistent with the scaling of a turbulent boundary

layer, which can be defined in terms of the turbulent velocity scale as *

* 0.4u

fδ = [Weatherly

and Martin, 1978], where u* represents the friction velocity. The friction velocity can be defined

in terms of the mean velocity as u* = C

DU . Thus with f = 0.25 rad s-1, where the velocity U is

0.06 ± 0.005 m s-1 (Figure 3.6), a drag coefficient of CD = 0.003 [Umlauf and Arneborg, 2009a]

implies u* = 0.0032 m s-1 so that the turbulent Ekman boundary layer is δ* ~ 0.4 cm, which is

approximately 2.5-3 times larger than the thickest theoretical laminar boundary layer thickness

88

of δ = 0.14 cm (black dashed line Figure 3.2). Empirically we find the best fit to our

experimental data where we use * 1.7 1.7 / 2fδ δ ν= = . This is a reasonable value as it is larger

than the laminar model and smaller than turbulent scaling which can overpredict the thickness in

strongly stratified fluids [Weatherly and Martin, 1978]. Using a thicker turbulent Ekman

boundary layer thickness yielded velocities that were significantly smaller than with a laminar

boundary layer. and are in much better agreement with the experimental data in Figure 3.6. We

use this estimated turbulent boundary layer thickness in Eq. [3.18] and in the rest of the

manuscript so that more meaningful comparison can be made with the experimental density

currents.

The experimental velocity data can be expressed in a more general way by the use of a

dimensionless Rossby number RoW and the Froude number Fr, which are plotted in Figure 3.7.

Fig. 3.7: Relation between Rossby number RoW and the Froude number Fr for the channel model with varying Coriolis parameter f. RoW and Fr (using h=0.06 m from Figure 3.5) are based on the measured mean velocity U (in the interior of the flow), the calculated mean velocity U using Eq. [3.17] and Eq. [3.18] (solid line) and U based upon the assumption of a constant velocity with Fr = 1 (dashed line).

89

We note that we now use the averaged geostrophic velocity of the interior of the current (from

Figure 3.6) to define Fr. For |RoW| < 2 there is generally a good agreement between the theory

using Ekman boundary layer dynamics (Eq. [3.17] and Eq. [3.18]) and the measured velocities.

The poorer agreement for |RoW| > 2 suggests that Eq. [3.17] and Eq. [3.18] are less applicable for

large |RoW|. This is expected as we assumed that f >> 0 rad s-1, in order for the flow to be

geostrophically adjusted, and for the Ekman boundary layers to be fully developed. Due to the

small velocity differences between small positive and negative Coriolis parameter f (Figure 3.6)

we observe a slight asymmetry in the Froude numbers (the normalized velocity) when the flow is

not strongly rotationally controlled, i.e. for |RoW| > 2 . Based on the experimental results in

Figure 3.6 and 3.7 we conclude that only for |RoW| < 2 does the Coriolis-effect become

significant for our experimental gravity currents. Such a threshold is consistent with

oceanographic literature where RoW of order 1 is usually the criterion when currents start to feel

the effect of rotation [e.g. Nof, 1996; Wells, 2009].

3.4.3 Slope of the interface with changing f

The strong influence of the Coriolis force on the lateral density interface can be seen in

Figure 3.8. The photographs show the shape of the across-stream interface for RoW ~ ∞ (f = 0)

and RoW = 0.83 (f =±0.6 rad s-1) looking upstream. For the case without rotation (f = 0 rad s-1,

Figure 3.8b) the density interface is horizontal. Those conditions, where f is almost zero, would

be relevant to large-scale turbidity-currents that occur in areas close to the equator, or for which

the scales are such that |RoW| > O(1). With smaller Ro, illustrated in Figure 3.8a (for f = +0.6 rad

s-1) and Figure 3.8c for (f = -0.6 rad s-1), a significant deflection of the slope of the across-stream

interface can be observed. For a positive Coriolis parameter (Northern Hemisphere), a deflection

of the current to the right-hand-side (looking downstream) can be seen. For a negative f

90

(Southern Hemisphere), the flow is deflected to the left-hand-side of the channel. The height

difference of the interface between the left and the right channel wall is negligible for an infinite

RoW, but for RoW = ±0.83 the difference is about 4 cm, which accounts for more than half of the

depth of the non-rotating gravity current. This tilt of the interface due to the rotation was also

observed in the low Reynolds number experiments by Darelius [2008] and with a numerical

model by Imran et al. [1999].

Fig. 3.8: Deflection of the interface of the density currents for a) RoW = 0.83 (with f=+0.6 rad s-1) b) RoW = ∞ (with f=0 rad s-1) and c) RoW = -0.83 (with f= -0.6 rad s-1). Note that the perspective is upstream and hence a deflection to the left-hand-side means a deflection to the right-hand-side from the downstream perspective.

Figure 3.9 shows the relation between RoW and the observed deflection ∆h of the

interface, normalized by the current thickness h. The deflection of the interface is defined as the

height difference between the left- and right-hand-side of the interface at the channel walls. The

experimental results demonstrate that with decreasing |RoW| the deflection increases. In addition,

Figure 3.9 compares the experimental results with the theoretical approaches to define the

deflection according to the theory of Komar [1969] expressed in Eq. [3.3] (where

/ 1/ Wh h Ro∆ = ) and the theory involving Ekman boundary layers of Eq. [3.13], respectively.

For small |RoW| the measured height difference is approximately 50% smaller than predicted by

Eq. [3.3] and there is generally a better agreement with the theory that used turbulent Ekman

boundary layer dynamics. The lack of viscous effects and Ekman boundary layers in Eq. [3.3]

91

tends to overestimate the slope of the interface when rotation is important as the velocity is also

overestimated (Figure 3.6 and Figure 3.7).

Fig. 3.9: Comparison of the deflection ∆∆∆∆h and RoW for various values of f after Eq. [3.3], Ekman boundary layer dynamics following Eq. [3.13] and the measured height differences. ∆∆∆∆h is normalized by the current thickness h (Figure 3.5). Error bars of the experimental data depict a variance of 10%.

3.4.4 Secondary flow cells and across-stream velocities

The secondary circulation patterns in the channel are illustrated in Figure 3.10 for various

Rossby numbers. Without rotation there are two adjacent flow cells in the channel that spin in

opposite directions, with a flow convergence at the surface and divergent flow at the bottom of

the density current (Figure 3.10a). This is essentially the same as the classic “helicoidal flow”

that was first observed in rivers at the end of the 19th century, and reported in open channel

flows [e.g. Rhoads and Welford, 1991; Colombini, 1993]. In a rotating density current the

secondary circulation changes dramatically, as demonstrated in Figure 3.10b and Figure 3.10c

for RoW = ± 2.4 (f= ± 0.25 rad s-1). The positive rotation leads to a deflection of the current to the

right-hand side (Figure 3.10b) and a corresponding interior flow vG with maximum velocities up

92

to 0.005 m s-1 towards the right-hand wall (looking downstream). This interior flow is bordered

by two return flows ve at the upper interface and the bottom of the density current, which are the

Ekman boundary layers. The maximum velocities of these return flows are approximately ve =

0.015 ± 0.005 m s-1 and hence larger than the interior flow towards the wall. Using Eq. [3.8a]

and Eq. [3.8b] the magnitude of the velocities in the Ekman boundary layer can also be

estimated. As the mean downstream velocity for f = 0.25 rad s-1 is U = 0.06 m s-1 we predict that

the mean velocity in the Ekman boundary layer is ve = 0.017 m s-1. This theoretical velocity

conforms to the observed velocity in Figure 3.10b where the transverse velocities have a

magnitude ve = 0.015 ± 0.005 m s-1 and are approximately 25 % of the downstream velocity U.

Fig. 3.10: Across-stream velocities for the experimental flows in the submarine channel (looking upstream) measured with the UDVP. a) RoW = 2.4 (with f = 0.6 rad s-1) b) RoW = ∞ ( with f = 0 rad s-1) and c) RoW = -2.4 (with f = -0.6 rad s-1). The sense of the rotation is sketched on the right hand panels. Note the upwelling mechanism denoted by the upward directed arrow in the right hand panels in b and c.

93

As the bulk flow vG in the interior towards the right wall of the channel (looking

downstream) occupies about 60 % of the depth of the current (near the right wall of the channel),

the return flows ve that occur in the thinner Ekman boundary layers have a higher velocity than in

the interior bulk flow (Figure 3.2 and Figure 3.10) in order for the integral of the cross-stream

velocity to be zero. When our experimental measurements of the transverse velocity were

integrated over the whole depth of the density current, we found an almost balanced relation

between the interior flow and the opposite velocities in the boundary layers. For larger f this

balance between vG and ve is even more prominent as the Ekman boundary layers tend to get

thinner (see section 2) while the interior flow expands further over the whole thickness. This is

accompanied by smaller bulk velocities vG in the interior and faster return flows ve in the

boundary layers. Figure 3.10c illustrates basically the same flow field as in Figure 3.10b, but for

negative f and hence mirrors the flow field of Figure 3.10b. Similar flow fields to those shown in

Figure 3.10 have been described by Johnson and Ohlsen [1994], Davies et al. [2006] and

Darelius [2008] from experiments in rotating fluids. Analogueous secondary flow patterns have

also been reported in several natural gravity currents, e.g. in the Faroe Bank Channel [Johnson

and Sanford, 1992; Fer et al., 2010], in the Ellet Gully [Sherwin, 2010] and in the Baltic Sea

[Umlauf and Arneborg, 2009a,b].

The magnitude of the secondary flows can be estimated from the geostrophic velocities

by rearranging Eq. [3.6] and Eq. [3.7]. Division of Eq. [3.6] by Eq. [3.7] gives an estimate of the

ratio of the interior transverse velocity vG to the mean downstream velocity U as vG/U =

s/(dh/dy). In our experiments the downstream slope of the channel s was constant, whereas the

cross-channel slope of the interface dh/dy increases for decreasing |RoW| (Figure 3.9), so that

vG/U should also decrease with |RoW|. Figure 3.7 showed that the theory incorporating Ekman

boundary dynamics appeared to be particularly suitable for |RoW| < 2.

94

Figure 3.11 depicts the measured values of the observed relation s/(dh/dy) for |RoW| < 4. With an

increasing deflection, the geostrophic interior of the flow becomes proportionally thicker, while

the Ekman boundary layers decrease in thickness, which results in a reduction in the velocity vG

and consequently a reduction of the ratio vG/U. For large rotation rates or |RoW| < 0.5 the ratio

vG/U is approximately 0.3. Figure 3.11 shows a partial regression line from which the

relationship U=0.5 sW2f/∆h can be derived. This simple empirical relationship could be used as a

first order approximation to determine the mean downstream velocity U of a turbidity current if

the levee height difference (or vice versa) and the slope of the submarine channel system is

known.

Fig. 3.11: Relation between s/(dh/dy) and RoW for varying Coriolis parameter f. The graph compares experimental data and the Ekman boundary layer dynamics calculated after Eq. [3.13].

There has been a considerable effort recently to look at the secondary circulations in

turbidity currents flowing in submarine channels, in particular in the bends of sinuous channels

[Keevil et al., 2006; Islam et al., 2008; Islam and Imran, 2008; Straub et al., 2008]. In channel

95

bends, the curvature-induced centrifugal acceleration of the flow balances an inwardly directed

radial pressure gradient leading to secondary flows that are of the order of 10% of the mean flow.

It is worth noting that the secondary flows driven by Coriolis-effects have a similar magnitude to

these previously studied flows but quite a different internal structure.

3.5 Discussion

An understanding of flow dynamics and sediment transport processes in submarine

channels is essential to interpret and analyze their morphology and architecture and to develop

process models, e.g. Peakall et al. [2000] or Pirmez and Imran [2003]. The growth of channel

levee systems is thought to arise from extensive overbank flow and over-spill, based upon direct

observations of overbank flow [Normark and Dickson, 1976b; Normark, 1989], and from

observations of the grain size distribution taken from silt and fine-sand beds on levee crests

[Hesse et al., 1987; Hiscott et al., 1997]. The height of turbidity currents can extend vertically

beyond the channel depth up to a factor of 4 [Normark, 1989] and fine grained sediment in this

upper part residing above the channel can spread laterally and be deposited on the overbank

surface [Straub et al., 2008]. Hence, in straight channel sections finer sediments are usually

found along channel levees rather than inside the channel and this continuous overspill accounts

for levee growth along submarine channels, e.g. in the NAMOC [Klaucke et al., 1998] or the

Amazon channel [Pirmez and Imran, 2003]. In channel bends however, the levee deposits can

also consist of coarse-grained sediments that are upwelled by centrifugal forces. These flow

dynamics in channel bends have been studied extensively in non-rotating sinuous submarine

channels [e.g. Corney et al., 2006; Keevil et al., 2006; Peakall et al., 2007; Straub et al., 2008].

Channel bend levees at the outer bank result predominantly from overspill deposition caused by

centrifugal forces, and the secondary flow cells. The deflection of the interface and the

96

movement of sediment by the secondary circulation, leads to an outer bank upwelling towards

the overbank region and promotes subsequent growth of high outer bank channel levees [Corney

et al., 2006; Straub et al., 2008].

Our results show that strong secondary circulations are present in straight channel

sections for |RoW| < 2 (f > 0.25 rad s-1) due to the interaction of Coriolis force and pressure

gradient force. A continuous deposition of sediments that are carried by these secondary

circulations in turbidity currents could lead to the asymmetry between right-hand and left-hand

levee banks, which has been described for several submarine channel systems in the Northern

and Southern Hemisphere [Menard, 1955; Carter and Carter, 1988; Klaucke et al., 1997]. These

secondary flows could contribute to overbanking in straight channel section where the

centrifugal force is absent. The secondary circulation consists of an interior flow towards the

right-hand-side of the channel (looking downstream) in the Northern Hemisphere and a return

flow along the density interface (Figure 3.10b). Hence this secondary flow cell could lead to an

upwelling of sediment at the right-hand-side of the channel and promote the growth of the right-

hand levee by continuous deposition (denoted by the vertical arrow in Figure 3.10b). This

upwelling is a similar mechanism that has been reported in channel bends without rotation,

where there was pronounced levee formation on the outer bends of submarine channels [Corney

et al., 2006; Keevil et al., 2006]. In a rotating system, we hypothesize that similar upwelling of

sediment, driven by the observed secondary flows, causes a constantly larger sediment flux

towards the right-hand-side of straight channel sections, leading to a consistent increase in levee

heights on the right-hand-side of submarine channels. This upwelling might also lead to

deposition of coarser sediment which is usually transported at the base of the current onto the

right-hand levee. Such deposition patterns and grain size distributions have been observed by

Straub et al. [2008] driven by a similar mechanism with superelevation and upwelling at an outer

97

channel bend. In the Southern Hemisphere the negative Coriolis parameter forces this upwelling

of sediments onto the left-hand-side of the channel (looking downstream), leading to greater

levee heights on the left-hand-side of the channel (Figure 3.10c).

In addition, the experiments showed good agreement with our theory (Eq. [3.13], Eq.

[3.17] and Eq. [3.18]) that incorporated Ekman boundary layer dynamics. In particular there was

good agreement with the predicted reduction in downstream velocity, U, and for the magnitude

of the cross-channel interface difference, ∆∆∆∆h. Our theory agreed with our observations better than

the theory of Komar [1969] in straight submarine channels. This is an important result, as the

new theory can now be used as an analytical method to derive flow properties from field data of

submarine channel systems.

The experiments indicate that the transition between the regimes when the two different

models should be used can be expressed in terms of the Rossby number so that the transition

occurs approximately for |RoW| = O(1). For example, in the NAMOC the right levee bank

exceeds the left levee bank up to 100 m, so that the ratio between ∆ ∆ ∆ ∆h to the entire depth D of the

channel is often large with ∆∆∆∆h /D > 0.5. The use of Eq. [3.3] for the NAMOC channel predicts

downhill velocities between 0.1-1.06 m s-1 [see table 1 in Klaucke et al., 1997]. The

corresponding Rossby numbers for these velocities at a latitude 53° N are small ranging from 0.2

- 0.5. These small values of RoW indicate the Ekman boundary layer dynamics cannot be

neglected in this system. At the equivalent Rossby number for our experimental analogue in

Figure 3.6, we found that the effective downhill velocity is over 30% smaller than obtained

following Eq. [3.3]. Hence, we conclude that the actual velocities in the NAMOC in straight

channel sections are generally smaller than proposed in Klaucke et al. [1997] by a factor of 0.7.

For smaller submarine channel systems such as the Navy fan in California, where the

estimated downstream velocities in the narrow upper fan are of order 0.75 m s-1 [Bowen et al.,

98

1984] there would be large Rossby numbers of about 10-18, for a latitude 33.5° N and a width W

= 0.5-1 km. Consequently rotation appears to be less important in this channel and no significant

asymmetry of the channel banks has been reported in the upper fan valley. However, when the

system widens towards the mid and lower fan system to 3 km to 8 km the velocities drop

significantly to 0.12-0.3 m s-1 and |RoW| < 1 is obtained. In this region Bowen et al. [1984]

observed an asymmetry on the mid-fan with the right-hand-side levee of the fan-system (looking

downstream) being up to 30 m higher than the left-hand-side.

Another example of a gravity current flowing in a submarine channel is the

Mediterranean inflow into the Black Sea [Flood et al., 2009]. Here a density current bearing a

small amount of sediment, continuously flows through a submarine channel system. When the

channel is on the inner shelf it has a width between 0.5–1 km and a depth of 10- 35 m [Flood et

al., 2009]. For the velocity range of about 0.2-0.4 m s-1 [Özsoy et al., 2001], the predicted

Rossby numbers are in the range RoW =2-5 suggesting that Coriolis forces will likely influence

the secondary circulation of any gravity currents in this channel system. Specifically we would

expect that the density interface tilts to the right-hand-side, and any sediment will be deposited

dominantly on the right-hand levee. However, due to the lack of significant sediment load in the

density current this system shows no prominent levee systems. Based on our results we predict

that the density interface would exhibit a tilt resulting in the interface being up to 5 m higher at

the right-hand-side (looking downstream). With a downstream velocity of 0.2–0.4 m s-1 we

expect that the mean velocity of the Ekman boundary layers will be of order 0.05–0.1 m s-1 and

will be directed to the left-hand-side of the channel looking downstream. At high discharge rates

this might also lead to intensive overspill of dense saline water, which could build features like

antidunes outside of the channel that have been observed and linked to overbank flow [Flood et

al., 2009].

99

Near the equator the Coriolis parameter is small and any asymmetry in levee heights is

expected to be less prominent compared to high latitudes. Nonetheless, the deep sea submarine

channel system offshore of Trinidad and Tobago at latitude 10.5° displays a consistently higher

right levee system, with cross channel differences of up to 20 m. This asymmetry in the

approximately 600 m wide and 200 m deep channel is attributed to the regional southward dip of

the northeastern South American basin [Wood and Mize-Spansky, 2009]. Nonetheless, we note

that for small downstream velocities, that are likely to occur in the upper portion of the velocity

profile of a gravity current (as in Figure 3.5) Rossby numbers of order 1 can be obtained. Fine

sediments transported as suspended load in the upper profile of a turbidity current [Peakall et al.,

2000] would predominantly be deflected to the right-hand-side of the channel system. In this

light, the observed asymmetry of up to 10% of the whole channel depth could also be attributed

to Coriolis forces. It follows that thicker, slower and finer-grained turbidity currents could give

rise to levees with more prominent asymmetry than thinner and faster, coarser-grained turbidity

currents, even at low latitudes.

100

3.6 Conclusions

This work demonstrated that the Coriolis force plays an important role in determining the

velocity structure in gravity currents running through straight submarine channels. The

frequently observed asymmetry in depositional elements in large-scale submarine channels,

where the right-hand (left-hand) levee banks tend to be higher in the Northern Hemisphere

(Southern Hemisphere), can be linked to the Earth’s rotation and arising rotational effects on

large-scale turbidity currents. In geological applications the theory of Komar [1969] as expressed

by Eq. [3.3] has been used to derive flow parameters from submarine channel systems. In

contrast, oceanographic studies mostly include Ekman boundary layer dynamics to describe the

characteristics of gravity currents in the ocean.

The major findings were that the theoretical model using Ekman boundary layer

dynamics is more accurate in describing the downstream velocity U and the interface tilt ∆∆∆∆h for

channels that have scales with Rossby numbers smaller than 2. In addition, it can be used to

describe the secondary across-stream velocities vG and ve. The downstream velocity decreases

significantly as the rotation rate increases, so that the theory after Eq. [3.3] overestimates the

velocity by more than 30% for large rotation rates. Similarly, the theory Komar [1969]

overestimates the tilt of the interface by as much as 50%.

Significant secondary circulations develop that are driven by Ekman boundary layer

dynamics. Those flow cells promote an upwelling at the right-hand-side of the channel (looking

downstream) in the Northern Hemisphere and an upwelling at the left-hand-side of the channel in

the Southern Hemisphere which governs most likely sedimentation transport processes and the

evolution of submarine channels.

The Rossby number RoW = U/Wf , where U is the mean downstream velocity, W the

channel width and f the Coriolis parameter, can be used to determine whether the rotational

101

effects are significant. The results show that particularly for |RoW| < 2 the flow properties are

better reflected by the theory incorporating the Ekman boundary layers dynamics.

Acknowledgements

We gratefully acknowledge the support of Jeff Peakall who loaned a Metflow UDVP system of the NERC

supported Sorby Environmental Fluid Dynamics Laboratory at the University of Leeds for the use in these

experiments. The rotating table was built on a design generously provided by Karl Helfrich. Helpful conversations

with Ilker Fer and Lars Umlauf are acknowledged. Mathew Wells received financial support from NSERC, the

Canadian Foundation for Innovation and the Ontario Innovation Trust. Remo Cossu was partially supported in this

work by a travel grant from the Centre for Global Change Science at the University of Toronto.

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Chapter 4 Coriolis forces influence the secondary circulation of gravity currents flowing in large-scale sinuous submarine channel systems Remo Cossu and Mathew G. Wells Abstract

A combination of centrifugal and Coriolis forces drive the secondary circulation of turbidity

currents in sinuous channels, and hence determine where erosion and deposition of sediment

occur. Using laboratory experiments we show that when centrifugal forces dominate, the density

interface shows a superelevation at the outside of a channel bend. However when Coriolis forces

dominate, the interface is always deflected to the right (in the Northern Hemisphere) for both left

and right turning bends. The relative importance of either centrifugal or Coriolis forces can be

described in terms of a Rossby number defined as RoR = U/Rf, where U is the mean downstream

velocity, f the Coriolis parameter and R the radius of curvature of the channel bend. Channels

with larger bends at high latitudes have |RoR| < 1 and are dominated by Coriolis forces, whereas

smaller, tighter bends at low latitudes have |RoR| >> 1 and are dominated by centrifugal forces.

4.1 Introduction

Recent non-rotating experiments on channelized turbidity currents have shown that the

morphological evolution and associated depositional histories of submarine channel systems are

highly influenced by the secondary flow structures within the channel, which determine where

erosion and deposition will occur [Corney et al., 2006; Keevil et al., 2006; Straub et al., 2008;

Islam et al., 2008]. The main focus in these non-rotating experiments has been to investigate the

107

secondary circulation due to an imbalance of centrifugal and pressure gradient forces in channel

bends, which plays a major role in the formation of superelevation of levee systems at the outer

bend [e.g. Straub et al., 2008; Kane et al., 2010]. The circulation in such a curved channel is

shown in Figure 4.1a. At the level of the downstream velocity maximum the centrifugal forces

are at maximum. The velocity maximum usually occurs relatively close to the base of the gravity

current due to drag induced by mixing processes at the upper interface [Turner, 1973; Meiburg

and Kneller 2010]. For such velocity profiles [e.g. Corney et al., 2006; Keevil et al., 2006;

Corney et al., 2008] the secondary flow near the base is directed towards the outer bend and

return flow is near the surface, in contrast to river flows. However, in some experiments using

square channels there is an additional secondary flow directed towards the outside bend below

the velocity maximum [Imran et al., 2007; Islam et al., 2008a,b].

Coriolis forces deflect the bulk of the gravity currents to the right (in the Northern

Hemisphere) [Hacker and Linden, 2002; Davies et al., 2006; Wells 2009], which causes a lateral

tilt of the interface in a confined, straight channel. This tilt (and any secondary circulation)

means that overbanking sediment flows are more likely to occur on the right-hand-side of the

channel (looking downstream) for mid- and high latitude systems in the Northern Hemisphere,

leading to an asymmetry between levee bank heights [Menard, 1955; Komar, 1969].

Observations at higher latitudes have found that the right-hand-side channel levee is consistently

higher in the Northern Hemisphere [Klaucke et al., 1997] while the left-hand-side channel levee

is higher in the Southern Hemisphere [Carter and Carter, 1988; Bruhn and Walker, 1997]. In

addition, Coriolis forces generate Ekman boundary layers in gravity currents [Wahlin, 2002] as

illustrated in Figures 4.1b and 4.1c, for the case of the Northern and Southern Hemisphere

respectively. In previous theoretical and experimental studies by Wahlin [2004] these boundary

layers were shown to play a critical role in determining the sense of the rotationally controlled

108

secondary circulation in gravity currents flowing down straight channels. These secondary

circulations dominated by Ekman boundary layer dynamics are also observed in oceanic gravity

currents [Johnson and Sanford, 1992].

Fig. 4.1:a) In a curved channel the velocity maximum of the gravity current is close to the base of the flow, the secondary circulation consists of a basal flow towards the outer bend and a return flow near the surface and sometimes below the velocity maxima [Keevil et al., 2006; Islam et al. 2008]. b) In a straight channel Coriolis forces deflect the upper density interface and drive secondary circulations due to the presence of Ekman boundary layers. In the Northern Hemisphere (f > 0) the interface is deflected to the left-hand-side of the channel, when looking downstream, whereas c) in the Southern Hemisphere (f < 0) the interface is deflected to the right-hand-side and the secondary circulation is in the opposite sense.

There are few direct observations of the velocity structure within turbidity currents

because their occurrence in great water depths and high current velocities make measurements

difficult to obtain [Xu et al., 2004, Meiburg and Kneller, 2010]. The dynamics of large-scale

non-depositional turbidity currents are often assumed to be similar to gravity currents, so that

previous experiments in sinuous channels [e.g. Keevil et al. 2006; Imran et al., 2007; Islam et al.,

2008] have used studies of saline gravity currents to gain insight into the secondary circulation in

turbidity currents. An open question is whether the secondary circulation in large-scale turbidity

currents flowing down a sinuous channel will be dominated by centrifugal forces or by Coriolis

forces. We will address this question through the use of analogue laboratory experiments

mounted on a rotating platform that can produce Coriolis forces. In particular we will investigate

how the interface slope and secondary circulation cells in a saline gravity current change in terms

109

of a dimensionless Rossby number RoR, and so determine when the flows are dominated by

Coriolis or centrifugal forces.

4.2 Theory

The geological observations of levee height asymmetry are usually described in terms of

the cross-channel tilt (dh/dy) of the upper interface of the turbidity current, using the theory of

Komar [1969]. Assuming that tangential friction is small and turbulence is absent, the

momentum balance across the channel can then be written as:

g '

dh

dy= fU +

U 2

R, [4.1]

where U is the mean downstream velocity, R is the radius of curvature of the channel and f the

Coriolis parameter, defined as f = 2Ω sin φ with Ω the Earth’s rotation rate and φ the latitude. R

is defined as positive when the bend is to the left (looking downstream), so that the force is in the

same direction as the Coriolis force in the Northern Hemisphere, while turns to the right will

have negative R. The Coriolis parameter f is positive in the Northern Hemisphere and negative in

the Southern Hemisphere, so that the sign of dh/dy depends upon both the latitude and the

curvature of the channel. The reduced gravity is g ' = g ρ

2− ρ

1( ) ρ1, where the gravity current

has the density ρ2 and ρ1 is the ambient density of the seawater. Equation [4.1] can be re-

arranged to give an equation for the interface slope whereby:

dh

dy= Fr 2 fh

U+

h

R

, where Fr 2 =

U 2

g 'h. [4.2]

Hence if the flow velocity remains constant, the interface deflection due to the Coriolis forces

increases with latitude. We note that Eq. [4.2] has never been previously tested in a rotating

experiment with a channel bend. The interface of the gravity current will be flat (dh/dy = 0) when

110

Coriolis forces and centrifugal forces balance, which occurs when fh/U = -h/R for a bend to the

right in the Northern Hemisphere. This condition can be re-written in terms of a Rossby number

defined as:

1R

URo

Rf≡ = − . [4.3]

We hypothesize that in sinuous flows with |RoR| >> 1, the interface always slopes in towards the

inner bend, while flows with |RoR| << 1 the interface will always slope to the right-hand-side

(left-hand-side) in the Northern Hemisphere (Southern Hemisphere). We note that a complete

description of the flow dynamics around a bend would require a full momentum budget

including cross-stream frictional influences and non-local adjustment terms as discussed in

Nidzieko et al. [2009]. However, we focus on Eq. [4.1] to Eq. [4.3] and point to future work that

has to incorporate these features in a more thorough mathematical model.

4.3 Experiments

All experiments were conducted in a channel placed within a 1.85 m × 1.0 m × 0.35 m

rectangular tank. This tank was rotated at a constant rate, with Coriolis parameters from

f = 0 to ± 0.5 rad s-1. Before the experiment began, the tank had to be spun up for at least 30 min

in order to achieve solid body rotation of the water. The channel had a constant, rectangular

cross-section with a width of 10 cm and a height of 8 cm (similar in dimensions to Keevil et al.

[2006]) and was submerged by 0.1 m of water at the inflow point. As shown in Figure 4.2a, the

channel consisted of a straight section of length 0.64 m, joined to a single channel bend of length

of 0.9 m and with mean radius of R = +0.36 m (representing a sinuosity of 1.09).

111

Fig. 4.2: Our experiment consists of a curved channel that can be rotated in either the Northern or Southern Hemisphere sense.

A constant velocity of the inflow was achieved by using a 0.10 m thick diffuser. We used

a saline gravity current as an analogue to a turbidity current. To visualize the slope of the gravity

current interface at the bend apex, fluorescein dye was added to the saline mixture and the flow

was illuminated by a thin sheet of light. The down-stream (u), across-stream (v), and vertical (w)

velocity data were recorded in the apex of the left-turning channel bend (Figure 4.2b) using a

Metflow Ultrasonic Doppler Velocity Profiler (UDVP), as used in Keevil et al. [2006]. Each

UDVP probe records simultaneous single component velocity data along a profile of 128 points

along the beam axis at a frequency of 4 Hz. Measurement time per profile was 11 ms, with a 15

ms delay between the recording of each profile. Vertical velocity profiles were obtained from an

array of 6 transducers at heights of 0.5, 1.5, 2.5, 3.5, 4.5 and 5.5 cm above the bottom. The

velocities were averaged over an interval of approximately 30 s after the head of the current had

passed the instrument. Experimental conditions are summarized in Table 1.

112

Tab. 4.1: Experimental conditions. The flow thickness h was estimated visually at the point upstream of the channel bend, and is a good estimate of the centre line channel depth as shown in Figure 4.2.

4.4 Results and Discussion

The strong dependence of the cross-stream slope of the density interface in the channel

bend to changes in the Coriolis force is shown in a series of photographs in Figure 4.3 for

various values of the Coriolis parameter f. For f = 0 rad s-1, the Rossby number is infinite and the

density interface slopes up towards the outer bend due to the centrifugal acceleration (Figure

4.3a). Such a superelevation is characteristic for non-rotating density currents in channel bends

as described by Corney et al. [2006], Keevil et al. [2006] and Straub et al. [2008]. For a positive

Coriolis parameter f = +0.25 rad s-1, the Rossby number is +0.55 and the tilt of the interface

towards the outer bend increases as now the Coriolis and centrifugal forces act in the same

direction (Figure 4.3b). The experiment shown in Figure 4.3c has a negative Coriolis parameter f

= -0.25 rad s-1 so that the Rossby number is -0.42, so that the Coriolis force acts in opposition to

the centrifugal force and the superelevation at the outer bend of the interface is largely reduced.

As the Rossby number is close to RoR = -1 there is an almost horizontal interface in the bend

apex. For a larger negative Coriolis parameter f = -0.5 rad s-1 with RoR = -0.2, the current shown

113

in Figure 4.3d now ramps up towards the inner bend and is completely reversed compared Figure

4.3a. The observations in Figure 4.3 are typical of the 100 experiments conducted using a range

of g’ and f. In particular the horizontal interface observed in Figure 4.3c is always seen when

0.35 < |RoR| < 0.45 in other experiments using different g’ and f.

Fig. 4.3: The photographs of the tilting interface of the gravity current are taken looking upstream at the apex of the bend. The inner and outer bends are marked as IB and OB respectively and the channel is 10 cm wide. In a) f = 0 rad s-1 and the Rossby number is infinite, in b) f = +0.25 rad s-1 giving RoR = +0.55, in c) f = -0.25 rad s-1 giving RoR = -0.42 and in d) f = -0.5 rad s-1 giving RoR = -0.2.

114

The resulting secondary circulation patterns are presented in Figure 4.4 for different

values of the Coriolis parameter f. In the non-rotating experiment with f =0 rad s-1 the cross-

stream velocity structure is broadly similar to experiments of Corney et al. [2006] or Keevil et al.

[2006]. Near the base of the gravity current (between 1 and 2.5 cm) the flow is directed from the

inside to the outside of the bend (in the region near the inside bend) while above this flow the

fluid moves from the outside towards the inside bend which causes an upwelling close to the

outside. The downstream flow velocity maximum occurs at depths between hu = 1-2 cm. The

total depth of the gravity current is h = 6 cm, so that the ratio of hu /h = 0.16-0.33. In non-rotating

gravity currents, the secondary circulation pattern is determined by the position of the velocity

maxima [Corney et al., 2008], so that when the velocity maximum is near the base (i.e. hu/h <

0.4), there is a flow towards the outer bend at intermediate depths, as seen in Figure 4.4a. Very

close to the bottom at 0.5 cm there is another flow directed toward the inner bend. This thin

bottom boundary flow occurred below the downstream flow maximum and is essentially the

same feature observed by Imran et al. [2007] using a similar rectangular channel cross section.

For a positive Coriolis parameter f = +0.25 rad s-1 (Figure 4.4b) the corresponding flow field has

a similar structure to the non-rotating experiment. However, as the centrifugal and Coriolis

forces act together the tangential and upwelling velocities are intensified as is the superelevation

(Figure 4.3b) at the outer bend.

115

Fig. 4.4: (left) The centre line downstream velocity profiles (u). (right) The cross-stream velocity field V is shown by the contoured colours, while the vectors show both V (cross stream) and W (vertical) velocities measured in two sets of identical experiments at the bend apex. The dashed boundary represents the spatial extent of the UDVP measurements, and the dark straight line represents the interface profiles from Figure 4.3. In a) the Coriolis parameter is f = 0 rad s-1, b) f = +0.25 rad s-1, c) f = -0.25 rad s-1 and d) f = -0.5 rad s-1.

The flow pattern changes dramatically if the Coriolis parameter is negative and hence the

Coriolis force counteracts the centrifugal force, as seen in Figures 4.4c and 4d for f = -0.25 and

f =-0.5 rad s-1 (RoR = 0.42 and RoR = -0.2 respectively). In both cases, the flow towards the

outside bend at mid-depth has decreased significantly compared to Figure 4.4a, but there is still a

flow between heights of 3 to 5 cm directed toward the inner bend. The vertical flows are now

almost absent in the current, suggesting that there is non-local adjustment of the velocity

116

occurring in the downstream direction. In both Figures 4.4c and 4.4d the basal flow towards the

outer bend is almost absent, suggesting that the opposition of Coriolis and centrifugal forces

suppresses this feature in rotating currents in left-turning channel bends in the Northern

Hemisphere. This is in contrast to the case when the Coriolis forces act in concert with

centrifugal force, and the basal flow is increased (Figure 4.4b). Thus, when |RoR| >> 1, in a

sinuous channel there would be a strong asymmetry in secondary circulation patterns between

successive left and right turning channel bends.

In addition, Figure 4.4 shows the mean downstream velocity profiles measured at the

centerline in the bend apex. Maximum velocities range from 0.075 m s-1 for f = 0 rad s-1 to 0. 05

m s-1 for f = - 0.5 rad s-1 which indicates that an increase of the rotation rate gradually decreases

the velocity. This is consistent with our observations in the straight channel in Chapter 3.

However, the influence of rotation on the downstream velocity core is discussed in more detail in

the following chapter. In previous studies [Keevil et al., 2006; Imran et al., 2007; Straub et al.,

2008] secondary flows were reported to be at the order of 10% of the mean flow. For our rotating

experiments we find similar values with the secondary flows being 5-15 mm s-1, which is about

10-20% of the mean downstream velocity. As sketched in Figure 4.1c the currents in Figure 4.4

will have Ekman boundary layer flows directed towards the inner bend for positive f. For a value

of f = +0.25 rad s-1 the Ekman number (Ek = υ / fH 2 ) is one for a thickness of H = 0.2 cm,

suggesting the Ekman boundary layer is much less than 1 cm in our experiments and hence may

not be detectable with the UDVP in Figure 4.4. Despite being unable to resolve explicitly these

thin boundary layers at large f, these Ekman flows are central to driving the significant cross-

stream flows as shown in the previous chapter for straight channels.

The importance of the Coriolis force on gravity current dynamics becomes evident when

we compare the Amazon submarine channel, located at latitudes between 3°-7° North, with the

117

NAMOC located at latitudes of 53°-59° North. The low latitude Amazon submarine channel

system exhibits a strong sinuosity over several hundred kilometers and consistently has the

highest levee banks on the outside of channel bends. Pirmez and Imram [2003] reported a mean

radius of curvature of between 1-2 km and estimated mean streamwise velocities of turbidity

currents between 1-3 m s-1. The Coriolis parameter for this range of latitudes is between f =

0.076 - 0.177 × 10-4 rad s-1 so that the magnitude of the Rossby number for the submarine

channel system is approximately between 30-130, i.e. |RoR| >> 1. These large Rossby numbers

indicate that Coriolis forces are negligible and the flow dynamics in the channel bends are

dominated by centrifugal forces. Hence, the superelevation is on the outside bend and the

resulting secondary flow field promotes upwelling at the outside (like in Figure 4.4a), leading to

the levee asymmetry between inner and outer bends.

By way of contrast the NAMOC channel has low sinuosity with observations [Klaucke et

al., 1997] suggesting that the mean radius of curvature of the channel is mostly 10-20 km and the

predicted mean velocity is in the range 0.2-1 m s-1. At latitudes between 53°-59°N the Coriolis

parameter is an order of magnitude larger at f = 1.16 - 1.2 × 10-4 rad s-1 so that the resulting

Rossby number is between 0.05-0.5, with an average value of 0.2, i.e. RRo << 1. The NAMOC

channel system shows a continuous higher right levee system irrespective of left or right turning

bends [Klaucke et al., 1997] consistent with Coriolis forces rather than centrifugal forces

dominating the secondary circulation and movement of sediment within the channel. Figure 4.3

and Figure 4.4 clearly demonstrate that the Coriolis forces are important for high latitude

submarine channel systems and give rise to flow patterns that explain the observations of levee

height asymmetry in these channel systems.

118

4.5 Conclusions

The influence of Coriolis forces upon the flow dynamics that we have reported in this

paper will have implications for the nature of depositional units such as the “outer-bank bars” or

“point bars”, as the orientation of secondary flows affects the position and geometry of these

deposits [Peakall et al., 2007; Amos et al. 2010]. The outer-bank bars are depositional units that

are likely to be sand prone and hence of high porosity which has significant implications for

prediction of hydrocarbon reservoirs [Nakajima et al., 2010]. Based upon the data in Figures 4.3

and 4.4 we predict that Coriolis forces will affect the secondary circulation in successive bends

of a sinuous channel differently; directing flow first towards the outer bank and then the inner

bank then back to the outer, as the channel turns to the left then right then left in the Northern

Hemisphere. Thus the relative position and geometry of inner (point-bar) and outer (OBB)

accumulations would vary between successive bends. We expect that this asymmetry in

deposition patterns between left and right bends will increase as a function of latitude, changing

from a symmetric distribution at low latitudes to a highly asymmetric distribution at high-

latitudes.

Acknowledgements

RC was partially supported by the CGCS at the University of Toronto. MGW acknowledges support from NSERC,

CFI and the Ontario MRI. The Metflow UDVP system was borrowed from Jeff Peakall of the NERC supported

Sorby Environmental Fluid Dynamics Laboratory at the University of Leeds.

119

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Chapter 5 Flow structures and sedimentation processes in submarine channels under the influence of Coriolis forces: experimental observations in rotating gravity currents Remo Cossu and Mathew G. Wells

Abstract

Submarine channels are the main conduits by which turbidity currents transport sediments from

the continental shelf to deep ocean basins. Many of these channels extend over thousands of

kilometers so that Coriolis forces can become important at higher latitudes and influence the

processes by which submarine channels evolve. Submarine channels at low latitudes often show

sinuous planform geometries similar to meandering rivers, whilst at high latitudes they are less

sinuous. The Coriolis-effect can strongly influence where erosion and deposition occur within

submarine channels by deflecting gravity currents and changing their flow properties, leading to

significant asymmetries between left and right channel levee heights at higher latitudes. We

present results from experimental gravity currents to show that the density interface and position

of the downstream velocity core shift for high values of the Coriolis parameter representative of

high latitudes. We combine our new observations of the velocity structure with an existing

conceptual model for sedimentation and erosion in sinuous channels to elucidate potentially

different sedimentation patterns at different latitudes. Based upon the internal velocity structure,

we predict that high latitude channel systems will exhibit depositional patterns that oppose a

strong lateral growth in channel bends, in contrast to low latitude systems.

122

5.1 Introduction

There has been considerable effort to explain the architecture of sinuous submarine

channels and to reconstruct their evolution as their depositional units have significant

implications as hydrocarbon reservoirs [Abreu et al., 2003]. Highly sinuous submarine channels

are found in modern equatorial regions, e.g. off the coast of West Africa and Brazil [Clark and

Pickering, 1996; Abreu et al., 2003]. For instance, the Amazon fan reveals a maximum sinuosity

of 2.6 in its mid-fan region at 3°- 7° North [Pirmez and Imran, 2003]. In contrast, high latitude

systems reveal generally only small sinuosity [Peakall et al., 2011] and can show distinct levee

asymmetries [Menard, 1955]. Examples of low sinuosity channels are the NAMOC with a

sinuosity of 1.01-1.05 at 59° North [Klaucke et al., 1997] and the Bering Sea channels that have

a mean sinuosity of 1.05 at 55° North [Clark and Pickering, 1996].

Due to the large dimensions of submarine channel systems and the associated long travel

time, turbidity currents are deflected and slowed by Coriolis forces at higher latitudes [Wells,

2009]. The Coriolis parameter f = 2Ω sinφ (with Ω being the Earth’s rotation rate and

φ being the latitude) represents the magnitude of the Coriolis force and is defined positive in the

Northern Hemisphere (+f) and negative in the Southern Hemisphere (-f). The Rossby number

RoW = U/Wf (where U is the mean downstream velocity, W the channel width) can then be used

to determine whether a submarine channel system is influenced by Coriolis forces [Cossu et al.,

2010]. Komar [1969] introduced a momentum balance for gravity currents flowing in a channel

bend as:

2 dh hf h

Frdy U R

= +

[5.1a], 2 1

W

h WFr

h Ro R

∆ = +

[5.1b]

123

where dh/dy is the interface slope across the channel, Fr is the Froude number (defined

as Fr = U / g 'h with g’ being the reduced gravity of the density interface), h is the mean height

of the current and R is the radius defined as positive if it turns left in the Northern Hemisphere.

Equation [5.1a] can be expressed in terms of RoW with ∆h representing the height difference of

the left and right channel levee (Eq. [5.1b]). This height difference can be observed in many

cross channel asymmetries and is attributed to the deflection of turbidity currents to one side so

that sediment is also predominantly deposited at one side of the channel levee system due to

overbanking [Menard, 1955; Klaucke et al., 1997]. For a given channel geometry where the ratio

W/R is fixed, the cross channel slope dh/dy depends on the magnitude of RoW. In high latitude

systems with /WRo R W< turbidity currents are more strongly deflected, resulting in larger

levee asymmetries [Komar, 1969]. It follows that less sinuous channels at high latitudes are

dominated by Coriolis forces, whereas sinuous submarine channels at low latitudes often have

/WRo R W>> and hence are dominated by centrifugal forces [Cossu and Wells, 2010]. The

motivation of this study is to investigate how the deflection of the velocity core in gravity

currents affects the growth of channels and their sinuosity due to depositional processes inside

the channel.

Recent laboratory studies [Amos et al, 2010 and references therein] have analyzed how

the internal flow structure of gravity currents determines the locations of erosion and deposition

in sinuous submarine channels. Amos et al. [2010] introduced a conceptual model that explains

the morphological evolution and associated depositional histories of submarine channel systems

at low latitudes (RoW ~∞). In this study we extend the process model of Amos et al. [2010] to

systems where /WRo R W< using new observations of the dependence of the position of the

downstream velocity core on RoW. We discuss how Coriolis forces could influence the evolution

124

and architecture of submarine channels. Our updated process model predicts that sediment

transport processes will change with latitude so that channel sinuosity, as well as levee

deposition, could differ significantly between the equator and the poles.

5.2 Method

The experiments were conducted in a sinuous channel model placed within a 1.85 m ×

1.0 m × 0.35 m rectangular tank. This tank was rotated at a constant rate, with Coriolis

parameters ranging from f = 0 to ± 0.5 rad s-1. The tank was spun up for at least 30 min in order

to achieve solid body rotation of the water. The channel had a constant, rectangular cross-section

with a width of W = 0.1 m and a height of H = 0.08 m (similar in dimensions to Amos et al.,

2010) and was submerged under 0.1 m of water at the inflow point. The channel consisted of a

0.64 m long straight section, joined to a 0.9 m long single left turning channel bend with a mean

radius of R = 0.36 m and sinuosity of 1.09 (Figure 5.1). The slope was either 1:50 or 1:25. A

constant velocity of the inflow was achieved by using a 0.10 m thick diffuser. We used saline

gravity currents which are a good surrogate for fine mud turbidity currents [e.g. Keevil et al.,

2006; Amos et al., 2010; Sequeiros et al., 2010]. To visualize the slope of the gravity current

interface at the bend apex, fluorescein dye was added. Photographs were taken looking upstream

into the channel bend. Downstream and cross stream velocity data in the channel bend were

recorded using a Metflow Ultrasonic Doppler Velocity Profiler (UDVP, as in Keevil et al., 2006)

and additional downstream data were taken along the channel with a Nortek Acoustic Doppler

Velocimeter (ADV, as in Cossu et al., 2010). The UDVP probe records simultaneous single

component velocity data along a profile of 128 points along the beam axis at a frequency of 4

Hz. Vertical velocity profiles were obtained from an array of 6 transducers at heights of 0.5, 1.5,

2.5, 3.5, 4.5 and 5.5 cm above the bottom. The ADV consists of one transmitter and three

125

receivers and measures the 3 components of velocity at a frequency of 50 Hz in a sample volume

of 80 mm3 that is 50 mm away from the probes [Cossu et al., 2010]. The velocity components

measured by the ADV were recorded 0.5, 1, 2, 3, 4 and 5 cm above the bottom (Figure 5.1). The

ADV and UDVP velocities data were averaged over an interval of 30 s after the head of the

current had passed the instrument.

Fig. 5.1: Schematic diagram of the laboratory channel showing location of velocity measurements.

5.3 Results

In the non-rotating flow (f = 0 rad s-1, RoW = ∞) the vertically averaged velocity at the

centre line was U = 0.052 m s–1. This flow had a mean thickness of h = 0.06 ± 0.005 m and a

reduced gravity of g’= 0.098 m s-2, giving a subcritical Froude number Fr = 0.68 ± 0.15, similar

to comparable experiments [Straub et al., 2008; Amos et. al., 2010]. The Reynolds number of the

flow was Re = 3100 (where Re= U h/ν, with ν being the molecular viscosity) again of a

comparable magnitude as in similar non-rotating experiments [Kane et al., 2008; Amos et al.,

2010].

126

Our observations of the interface slopes from 20 experiments in the sinuous channel and

11 experiments in a straight channel (data taken from Cossu et al., 2010) are summarized in

Figure 5.2. There is good qualitative agreement with Eq. [5.1b] in that for the straight channel

(where W/R = 0) we find ∆h / h ∝ Fr 2 / RoW

.

Fig. 5.2: Relationship between Fr 2 / RoW

and the lateral tilt of the interface ∆h / h for different rotation

rates, channel slopes and for sinuous and straight channel sections. In the straight channel the data collapses

to the dashed line where ∆h / h = 1.5 × Fr 2 / RoW

. The experiments in the sinuous channel collapse to the

solid line that has the same slope.

In the sinuous channel (with W/R ~ 0.28), the interface is flat at a negative value of Fr2/RoW = -

0.4 ± 0.1 indicating that in left-turning bends Coriolis forces and centrifugal forces counteract

and for ∆h / h = 0 balance each other in the Southern Hemisphere (f < 0). Without rotation

(Fr2/RoW = 0) we observe a height difference ∆h / h = 0.5 ± 0.1 which reflects the dominance of

the centrifugal force in the momentum balance at the equator and low latitudes. The data for

observed ∆h h for all 31 experiments collapses well to the dashed and solid lines which have a

slope of 1.5, which is in favorable agreement to the predicted slope of 1 from Eq. [5.1b].

127

The main experimental results for the internal flow structure are shown in Figure 5.3a-d,

where we compare the slope of the interface (Figure 5.3a), the position of umax (Figure 5.3b) and

across-stream velocities (Figure 5.3c) in the bend apex, for various Coriolis parameters f. The

spatial variations of the measuring locations of umax are shown in Figure 5.3d. The changes in the

position of the density interface clearly correspond to changes in the position of the location of

umax, so when the interface is deflected to the right, so is the location of umax (Figure 5.3b). For f

= 0 rad s-1 (RoW ~∞) the velocity core is predominantly close to the centre line whilst for f = -0.5

rad s-1 (RoW = -0.65) a significant deflection of the velocity core towards the inner bend can be

seen. For f = +0.25 rad s-1 (RoW = +1.5) the velocity core is shifted towards the outer bank, and at

the base of the current at the outer bend the downstream velocity has increased to 0.085 m s-1,

while there is almost no downstream flow (< 0.01 m s-1) near the inner bend. In Figure 5.3c we

observe a basal outward helical circulation for f = 0 rad s-1 (RoW ~∞) but the velocity field

changes significantly for f = ±0.5 rad s-1 (RoW = ±0.65) as velocities at the base decrease by a

factor of 4 and the entire helical flow cell is lifted up. For f = -0.5 rad s-1 the flow cell is reversed

compared to f = 0 rad s-1 with an inward directed flow and a return flow above it [see Cossu and

Wells, 2010]. The locus of umax is marked by a dashed line in Figure 5.3d. For f = 0 rad s-1 it

alternates between left and right turning bends, similar to patterns described in Keevil et al.

[2006] or Straub et al. [2008]. However, with f = -0.5 rad s-1 we observe that the location of the

downstream velocity maximum is also pushed to the left-hand-side in the channel upstream and

downstream of the bend apex (Figure 5.3d). Similarly for f = +0.5 rad s-1 both the velocity core

and the location of the velocity maximum are shifted to the right-hand-side along the channel

pathway.

128

Fig. 5.3: a) Lateral tilt of the density interface in the bend apex for various f. b) Corresponding distribution of the velocity core in the bend apex for various f. c) Across stream velocities in the bend apex for various f. d) Distribution of the bottom downstream component in a sinuous submarine channel for f = 0 rad s-1 and f = ± 0.5 rad s-1. The locus of the highest velocity is indicated by the dashed line.

129

5.4 Discussion

In submarine channels at high latitudes |RoW|< |R/W|, where Coriolis forces control the

internal flow structure of turbidity currents, it seems likely that the spatial distribution of

sedimentation and erosion could be different than in low latitude systems. As outlined by Amos

et al. [2010] sedimentation regimes can be classified as bed-load (Figure 5.4a-c) or suspension

fall-out dominated (Figure 5.4d-f). In non-rotating bed-load dominated turbidity currents with

/WRo R W>> (Figure 5.4a) sediment is eroded from the left-hand-side upstream of a bend and

accumulates on the inner bank downstream of the bend as inner lateral accretion packages [LAP,

e.g. Peakall et al., 2007a; Amos et al., 2010]. A similar growth of LAPs in channel bends is also

seen in river bends and attributed to perturbations of the primary flow field by the centrifugal

force [e.g. Johannesson and Parker, 1989]. Subsequent growth of LAPs continuously increases

the sinuosity of these channels [Abreu et al., 2003; Babonneau et al., 2010]. However, in high

latitude systems in the Southern Hemisphere ( /WRo R W< , Figure 5.4b) centrifugal forces are

outbalanced by Coriolis forces and the location of the velocity maximum is always on the left-

hand-side promoting erosion rather than deposition on the inside bank in the bend apex. This

effect would be even more pronounced in right turning bends where Coriolis force and

centrifugal force magnify the erosion on the outside bend. This is also evident as across stream

velocities near the base (Figure 5.3c) have reduced significantly, so that we expect a less

pronounced sediment transport across the channel. It can be concluded that this shift leads to an

overall lateral migration of the channel system to the left (dashed line). In the Northern

Hemisphere ( /WRo R W< , Figure 5.4c) we observe highest velocities always on the right-

hand-side suggesting a mirrored erosion depositional pattern as in Figure 5.4b.

130

Fig. 5.4: Predicted sedimentation pattern in bed-load and suspension fall-out dominated turbidity currents in a left- turning submarine channel for a,d] RoW = ∞, b,e] |RoW| < |R/W| in the Southern Hemisphere and c,f) for |RoW|< |R/W| in the Northern Hemisphere. The dashed lines delineate a possible channel plan form evolution with time.

131

Hence, in high latitude systems we predict that the formation of LAPs on both sides of

the channel will be suppressed and that deposition generally will occur in regions of minimum

downstream velocities, which are continuously on the same side upstream and downstream of a

bend apex as sketched in Figure 5.4b and 5.4c. This will minimize the potential for increases in

channel sinuosity and promote a lateral migration of the entire channel system, e.g. to the right-

hand side in the Northern Hemisphere.

In contrast, in suspension fall-out dominated regimes (Figure 5.4d-f) there is a vertical

accretion of the submarine channel [Nakajima et al., 2009; Amos et al., 2010]. Without rotation

(RoW = ∞) the deposition occurs predominantly along the outside bend where the downstream

velocity and suspended sediment concentration is largest [Amos et al., 2010]. Continuous

deposition in outer bends leads to vertical aggradation of the channel and little change in the

channel planform geometry [Straub et al., 2008; Kane et al., 2008]. In the Southern Hemisphere

(for /WRo R W< ) we found that the location of umax and the largest flow thickness and hence

the maximum sediment concentration will always be on the left-hand-side (Figure 5.4e), so that

in fine grained turbidity currents suspension fall-out would occur also on the left side along the

channel pathway. Particularly in right-turning bends in the Southern Hemisphere, when Coriolis

and centrifugal forces work in the same direction and amplify the inertial run-up, more sediment

will be deposited on the outside bend. Over time a continuous deposition on the left-hand-side

could decrease the sinuosity as sketched in Figure 5.4e. In the Northern Hemisphere where (for

/WRo R W< , Figure 5.4f) the sedimentation pattern will mirror that in the Southern

Hemisphere as the flow velocity maximum and maximum sediment concentration are shifted to

the right-hand-side of the channel (Figure 5.3d).

Between 53°- 59° North the NAMOC exhibits RoW < 0.5 [Cossu et al., 2010], and

132

/WRo R W>> [Klaucke et al., 1997] so that /WRo R W< . Using Figure 5.2 with Fr = 1 it

follows that the tilt of the interface is strongly dominated by Coriolis forces [Klaucke et al.,

1997; Cossu et al., 2010] so that the velocity core will also be strongly shifted to the right-hand-

side. The stratigraphy of the NAMOC shows fine sediments on its levees and coarser material

inside the channel [Klaucke et al., 1998; Skene et al., 2002] so we expect the depositional pattern

to be similar to the models in Figure 5.4c and 5.4f. The strong asymmetry between the right and

left levees, the low sinuosity and the minor lateral migration to the left in the NAMOC [Klaucke

et al., 1998] can be explained if suspension-fall out deposition predominantly takes place on

right-hand-side of the levee system and a deflection of the velocity core inside the channel to the

right inhibits a strong growth of sinuosity in channel bends. In contrast, for the Amazon fan we

obtain 30WRo > [Cossu and Wells, 2010] and / 1R W < [Pirmez and Imran, 2003], so that

/WRo R W>> and thus centrifugal forces will be dominant. The high sinuosity has been

attributed to the shift of umax due to centrifugal forces [e.g. Imran et al., 1999] leading to

formations of LAPs at inner banks behind bend apexes [Abreu et al., 2003].

Despite the significant role of Coriolis forces upon the flow properties in turbidity

currents, there are several other factors that may influence the evolution of sinuosity at high and

low latitude channel systems. Present day flows on continental margins are unlikely to be in

equilibrium with their channel morphology, as those channels were primarily formed during the

last glacial low-stand and subsequent sea-level rise [Imran et al., 1999; Parsons et al., 2010].

Particularly at times of rapid ice retreat glacially fed channels at high latitudes [Peakall et al.,

2011] may exhibit significant variations in flow and sediment type, and hence behave differently

to systems with direct inputs of large equatorial rivers such as the Amazon or Zaire fan.

However, as the morphology in submarine channel systems is mostly controlled by the

133

hydrodynamics of channelized flow [Clark and Pickering, 1996] our observations of the change

in hydrodynamics with latitude due to varying f and hence RoW might be the major control of

sinuosity in large-scale depositional channels in continental shelf and fan systems.

5.5 Summary and conclusions

We used a physical model to demonstrate the influence of Coriolis forces on flow

properties in gravity currents in order to explain the observed latitudinal dependence on sinuosity

in submarine channels [Peakall et al., 2011]. Based upon the locations of the velocity maxima

and the dependence of erosion and deposition on the velocity structure we suggest that Coriolis

forces will introduce a shift at high latitudes. When the width W, the radius R and the velocity U

of channel systems are known, the ratio RoW to R/W can be used to determine if depositional

patterns are influenced by Coriolis forces. In bed-load dominated flows at high latitudes we

predict that LAPs are built only on one side, thus inhibiting the growth of channel bends. In

suspension fall-out regimes sediment will mainly be deposited on the side to which the high

velocity core is shifted resulting in significant levee asymmetries due to overbanking. In both

sedimentation regimes, the shift of the velocity core to either the left- or right-hand-side of the

channel should lead to a gradual decrease in sinuosity in mid- and high latitude systems and a

lateral migration of the entire channel pathway. For channel systems at high-latitudes in the

Northern Hemisphere we predict /WRo R W< , so that channels exhibit a low sinuosity, have a

distinct higher right levee system and migrate predominantly to the left side. In channel systems

at low latitudes we usually find /WRo R W> implying that Coriolis forces are negligible. LAPs

are then formed on the inside of bends enabling an increase in sinuosity. Therefore equatorial

channel systems can become very sinuous, show a distinct lateral migration to both sides and

alternating levee asymmetry in subsequent channel bends.

134

Acknowledgements

RC was partially supported by the CGCS at the University of Toronto. MGW acknowledges support from NSERC,

CFI and the Ontario MRI. The Metflow UDVP system was borrowed from Jeff Peakall of the NERC supported

Sorby Environmental Fluid Dynamics Laboratory at the University of Leeds. We thank Nick Eyles, Brian

Greenwood and Joe Desloges for advice and discussions.

References

Abreu, V., Sullivan, M., Pirmez, C., and Mohrig, D., 2003, Lateral accretion packages (LAPs): an important

reservoir element in deep water sinuous channels. Mar. Pet. Geol., 20, 631-648.

Amos, K.J., Peakall, J., Bradbury, P.W., Roberts, M., Keevil, G., and Gupta, S., 2010, The influence of bend

amplitude and planform morphology on flow and sedimentation in submarine channels. Mar. Pet. Geol.,

27, 1431-1447.

Babonneau, N., Savoye, B., Cremer, M., and Bez, M., 2010, Sedimentary architecture of a submarine channel:

detailed study of the present Congo Turbidite Channel (Zaiango project). J. Sed. Res., 80, 852-866. doi:

10.2110/jsr.2010.078

Clark, J.D., and Pickering, K.T., 1996, Submarine channels: processes and architecture. Vallis Press, London, 231

pp.

Cossu, R., Wells, M.G. and Wahlin, A.K., 2010, Influence of the Coriolis force on the velocity structure of gravity

currents in straight submarine channel systems. J. Geophys. Res., 115, C11016 doi:10.1029/2010JC006208

Cossu, R., and Wells, M.G., 2010, Coriolis forces influence the secondary circulation of gravity currents flowing in

large-scale sinuous submarine channel systems. Geophys. Res. Letts., 37, L17603 doi:10.1029

/2010GL044296

Imran, J., Parker, G., and Pirmez, C., 1999, A nonlinear model of flow in meandering submarine and subaerial

channels. J. Fluid Mech., 400, 295- 331.

Islam, M.A., Imran, J., Pirmez, C., and Cantelli, A., 2008, Flow splitting modifies the helical motion in submarine

channels. Geophys. Res. Letts., 35, L22603, doi:10.1029/2008GL034995.

Johannesson, H. and Parker, G., 1989, Linear theory of river meanders. In River Meandering (ed. S. Ikeda and G.

Parker), AGU, 181-213.

Kane, I.A., McCaffrey, W.D., and Peakall, J., 2008, Controls on sinuosity evolution within submarine channels.

Geology, 36, 287-290.

Keevil, G.M., Peakall, J., Best, J.L., and. Amos, K.J., 2006, Flow structure in sinuous submarine channels: Velocity

and turbulence structure of an experimental submarine channel. Mar. Geol., 241, 241-257.

Klaucke, I., Hesse, R., and Ryan, W.B.F., 1997, Flow parameters of turbidity currents in a low-sinuosity giant deep-

sea channel. Sedimentology, 44, 1093-1102.

Klaucke, I., Hesse, R., and Ryan, W.B.F., 1998, Seismic stratigraphy of the Northwest Atlantic Mid-Ocean Channel:

growth pattern of a mid-ocean channel-levee complex. Mar. Pet. Geol., 15, 575-585.

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Komar, P.D., 1969, The channelized flow of turbidity currents with application to Monterey deep-sea fan channel. J.

Geophys. Res., 74, 4544-4548.

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Nakajima, T., Peakall, J., McCaffrey, W.D., Paton, D.A., and Thompson, P.J.P., 2009, Outer-bank bars: a new intra-

channel architectural element within sinuous submarine slope channels. J. Sediment. Res., 79, 872–886.

Parsons, D.R., Peakall, J., Aksu, A.E., Flood, R.D., Hiscott, R.N., Besiktepe, S., and Mouland, D., 2010, Gravity-

driven flow in a submarine channel bend: Direct field evidence of helical flow reversal. Geology, 38, 1063–

1066, doi: 10.1130/G31121.1

Peakall, J., Amos, K.J., Keevil, G.M., Bradbury, P.W., and Gupta, S., 2007, Flow processes and sedimentation in

submarine channel bends. Mar. Pet. Geol., 24, 470-486.

Peakall, J., Kane, I.A., Masson, D.G., Keevil, G., McCaffrey, W.D. and Corney, R., 2011, Global (Latitudinal)

variation in submarine channel sinuosity. (accepted for publication in Geology)

Pirmez, C., and Imran, J., 2003, Reconstruction of turbidity currents in Amazon Channel. Mar. Petrol. Geol., 20,

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bed resistance associated with density and turbidity currents. Sedimentology, 57, 1463–1490,

doi:10.1111/j.1365- 3091.2010.01152.x

Skene, K.I., Piper, D.J.W., and Hill, P.S., 2002, Quantitative analysis of variations in depositional sequence

thickness from submarine channel levees. Sedimentology, 49, 1411 – 1430.

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136

6 Final remarks

6.1 Summary and implication

In this thesis, the role of Coriolis forces on channelized large-scale turbidity currents

flowing through straight and sinuous submarine channels is investigated with physical

modelling. This is approached systematically: i) verification of the methodolog; ii) laboratory

experiments in a straight channel; and iii) laboratory experiments in a sinuous channel. The

results are summarized in a conceptual model contrasting the features and different growth

patterns of submarine channels at low and high latitudes due to the magnitude of Coriolis forces.

6.2.1 Density currents as an analogue for turbidity currents

Studies of the dynamics and characteristics of large-scale turbidity currents rely on

laboratory experiments, numerical techniques and field studies of turbidites. In particular, the

applicability and scalability of analogue experiments hinges on the constraints of the

experimental set-up, such as the spatial scale or the similarity of saline to sediment-laden flows.

Small-scale laboratory experiments of rotating density currents were conducted to simulate and

investigate the flow characteristics, which are most likely to be found in large-scale turbidity

currents. As discussed in Chapter 2, saline and weakly-depositional sediment-laden gravity

currents have generally a similar turbulence structure near the bottom with both flow types

revealing very similar vertical velocity, Reynolds stress and TKE distributions. Similar results

have been reported for the bulk and upper part of density and turbidity currents [Stacey and

Bowen, 1988; Islam and Imran, 2010 and partly by Gray et al., 2006]. These observations

suggest that saline and sediment-laden flows are dynamically similar when the settling velocity

is outbalanced by high turbulent velocities near the bed. In addition, a very good agreement

137

between velocity profiles of field gravity currents and laboratory gravity currents was observed.

This implies that findings of several recent studies [Johnson et al., 1994; Dallimore et al., 2001;

Peters and Johns, 2006; Umlauf and Arneborg, 2009a] on the interior dynamics of density

overflows could also be applied to understanding non-depositional large-scale turbidity currents.

However, not all turbidity currents are weakly-depositional or entirely fine-grained and might

show completely different behavior [Kneller and Buckee, 2000]. Hence, the outlined results

remain limited for the specific class of fine-grained and weakly-depositional turbidity currents.

6.2.2 Coriolis forces in straight submarine channels

Geological studies related to submarine channel systems usually refer to the theoretical

model of Komar [1969]. This model has been used predominantly to derive mean flow

parameters of turbidity currents flowing through submarine channels from channel geometries

and channel grain sizes [e.g. Bowen et al., 1984; Klaucke et al., 1997; Imran et al., 1999] using a

simple momentum balance between Coriolis, centrifugal and pressure gradient forces. However,

flow characteristics of channelized turbidity currents are far more complex as recent studies of

secondary flow fields in channel bends have demonstrated [i.e. Keevil, et al., 2006; Peakall et al.,

2007a; Islam and Imran 2008; Straub et al., 2008]. There is, however, a substantial analysis of

the importance of Coriolis forces on turbidity currents still missing as these earlier studies refer

to non-rotating environments, omitting potential influences from the Earth’s rotation. Therefore,

the experimental data discussed in this thesis contribute to a better understanding of turbidity

currents and turbidite systems. In Chapter 3 it was demonstrated that the downstream velocity

decreases significantly and the interface tilt increases with increasing Coriolis force. In addition,

secondary circulations develop throughout the whole thickness of the current that are driven by

Ekman boundary layer dynamics. These secondary flow cells promote an upwelling at the right-

138

hand-side of the channel in the Northern Hemisphere (Figure 3.10) which governs most likely

sedimentation transport processes and the evolution of levee systems. The Rossby number RoW =

U/Wf can be used to determine whether the rotational effects become important. As discussed in

Chapter 3 for systems with RoW ~ O(1) a theoretical model using Ekman boundary layer

dynamics [Wåhlin, 2004; Darelius, 2008] is more powerful in describing the flow field of

turbidity currents than the theory of Komar [1969]. Consequently, for geological applications

Ekman boundary layer dynamics should be used to describe the characteristics of gravity

currents rather than the theory of Komar [1969] for channel systems at high latitudes.

6.2.3 Coriolis forces in sinuous submarine channels and their implication for the evolution

of channel systems

Ultimately, the analysis of the internal flow structure in rotating density currents gives

rise to a conceptual model that helps to describe sedimentation patterns and evolution in straight

and sinuous submarine channels. The new results extend and improve an existing model [Amos

et al., 2010] by incorporating Coriolis forces so that it becomes more suitable to explain features

such as different asymmetries and sinuosity in channels at various latitudes.

Previous studies [Keevil et al., 2006; Peakall et al., 2007a; Islam and Imran 2008; Straub

et al.; 2008; Kane et al., 2008; Amos et al., 2010] observed secondary flow fields that are solely

driven by centrifugal and pressure gradient forces. The experiments in Chapter 4 showed that

Coriolis forces also have a significant effect on flow dynamics in sinuous channel systems which

will influence their evolution and growth at high latitudes. Depending on the magnitude of

curvature (magnitude of centrifugal forces), the latitude (magnitude and sense of rotation) and if

the channel bend turns to the left or the right, Coriolis forces can either balance, counteract or

magnify centrifugal forces. For small Rossby numbers RoR, where the radius of curvature is used

as a length scale, Coriolis forces significantly change secondary circulations in channel bends. In

139

Chapter 5 it was found that Coriolis forces can introduce a shift in the velocity core and the

superelevation of the interface and therefore change locations of erosion and deposition. At high

latitudes the results implied that LAPs [Abreu at al., 2003] are built only on one side, thus

inhibiting the growth of channel bends. Moreover, finer sediments from suspension fall-out

regimes will be deposited mainly where the high velocity core has shifted to result in significant

levee asymmetries due to overbanking. Hence, large Coriolis forces can lead to a decrease in

sinuosity in mid and high latitude systems and can cause a lateral migration of the entire channel

pathway which could explain the observed latitudinal dependence on sinuosity in submarine

channels [Peakall et al., 2011]. The ratio RoW to R/W can be used to determine if depositional

patterns are dominated by Coriolis forces. If |RoW|< |R/W|, high-latitudes systems will exhibit a

low sinuosity. In the Northern Hemisphere these channel systems have a distinctly higher right-

hand levee system and migrate predominantly to the left-hand side. In contrast, submarine

channels in low latitudes usually have |RoW| > |R/W| implying that Coriolis forces are negligible

so that centrifugal forces dominate the flow characteristics. Hence, equatorial channel systems

can become very sinuous, show a distinct lateral migration to both sides and alternating levee

asymmetry in subsequent channel bends [Abreu et al., 2003; Pirmez and Imran, 2003; Peakall et

al., 2011].

For geologic studies the conceptual model could be employed to further investigate and

characterize levee build-ups of channel systems. For instance, Klaucke et al. [1998] analyzed

core samples of the western and eastern levees of the NAMOC and found substantially thicker

Bouma divisions TC-TE on the western (right-hand-side) levee which was attributed to Coriolis

forces [Klaucke et al., 1997]. If similarly extensive and continuous asymmetries are found in

ancient channel levee deposits at low latitudes it could be concluded that the channel system had

been deposited in a high latitude setting and had been moved to its current location by plate

140

tectonics. Hence, the Coriolis force and resultant depositional pattern (e.g., Figure 5.4) could be

used as a key indicator to interpret and better understand facies models of modern or ancient

submarine channel deposits.

6.2.4 Future work

This thesis points towards interesting future research in submarine channels. Due to the

general difficulties in obtaining field data of large-scale channel systems, testing of the findings

against environmental data is desirable, but seems unlikely in the nearer future. Therefore, the

next step would be to repeat similar experiments at a larger scale which enables larger

experimental Reynolds numbers so that a larger dynamic similarity between experiment and

natural current can be achieved. One conceivable set-up would be the use of a deformable

bottom boundary similar to Peakall et al. [2007] or Amos et al. [2010]. Larger laboratory scales

would also allow the use of sediment-bearing currents as in Straub et al. [2008] or Islam and

Imran [2010] so that sediment properties as well as other types of turbidity currents (such as

erosional or strongly depositional flows) could be investigated. Furthermore, processes like flow

spilling and stripping and the related sediment delivery to areas adjacent to the channel could be

analyzed. As demonstrated in Chapter 3 to Chapter 5, Coriolis forces promote sediment transport

only to one side of the channel, but direct measurements of overspilling or density profiles could

lead to a more quantitative analysis of levee evolution. Another question that requires more

research relates to sediment processes inside the channel. Though there was a good agreement

between saline and sediment-laden flows (see Chapter 2), the analysis of Reynolds stresses in the

flow field of rotating experimental currents was not covered in this thesis. Hence, further work

that gives more insight into the turbulence structure of rotating turbidity currents to complement

findings of sediment-transport processes inside the channel as reported in Chapter 4 and V, is

141

necessary. Lastly, studies investigating channelized flow characteristics [i.e. Keevil et al., 2006;

Paekall et al., 2007; Imran et al., 2007; Islam and Imran, 2008; Straub et al., 2008; Kane et al.,

2008] used models that differ in size, aspect ratio and cross-sectional geometry. This might be an

important factor influencing flow processes [Islam et al., 2008; Straub et al., 2008] and it still

remains to be clarified to what extent the results in this thesis are affected by the rectangular

channel geometry and its aspect ratio. It is recommended therefore that future studies consider

using different channel geometries (aspect ratios) to shed light on the relevance of morphological

channel geometry for flow structures.

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