Representing Gravity Current Entrainment
in Large-Scale Ocean ModelsRobert Hallberg (NOAA/GFDL & Princeton U.)
With significant contributions from Laura Jackson, Sonya Legg, and the Gravity Current Entrainment
Climate Process Team:
NOAA/GFDL: S. Griffies, R. Hallberg, S. Legg, L. Jackson*NCAR: G. Danabasoglu, P. Gent, W. Large, W. Wu*U. Miami: E. Chassingnet, T. Ozgokmen, H. Peters, Y. Chang*WHOI: J. Price, J. Yang, U. Riemenschneider*Lamont Doherty: A. GordonGeorge Mason: P. SchopfPrinceton U.: T. Ezer (Plus ~12 active collaborators)
*Postdocs funded by the CPT
http://www.cpt-gce.org
An Idealized Rotating OverflowDOME Test Case 1 (Legg, et al., Ocean Modelling,
2006)Near-bottom tracer concentration
with contours of buoyancy
x=500m, z=30m MITgcm Simulation
Tracer concentration just west of the inflow
An Idealized Rotating OverflowDOME Test Case 1 (Legg, et al., Ocean Modelling,
2006)
Tracer concentration just west of the inflow
x=500m, z=30m MITgcm Simulation
Near-bottom tracer concentrationwith contours of buoyancy
Shear instability & entrainment
Detrainment
Geostrophic eddies
xz
y
Downslope descent
Bottom friction
Physical processes in overflows
Important Processes in Overflows
Resolvable by large-scale models
1. Hydraulic control at sill2. Geostrophic adjustment of
plume along slope3. Downslope transport of dense
water (some model types?)4. Some geostrophic eddy
effects?5. Detrainment at neutral density
Require Parameterization1. Exchange through subgridscale
straits2. Shear instability and entrainment
(TURBULENCE!!!)3. Bottom boundary layer mixing and
drag processes (TURBULENCE!!!)4. Some eddy effects?5. Flow down narrow channels?
Hydraulic control at sill
Bottom-stress mixing
Overview
• A tour of overflows Oceanic Gravity Currents are important in the formation and
transformation of the majority of deep water masses.
• Important Processes in Typical Oceanic Dense Gravity Currents: Hydraulic or tidal control of source water flows, often in narrow
straits Downslope descent (gravitational, Ekman driven, and eddy induced) Shear-driven mixing at the plume top Bottom boundary layer mechanical stirring within the plume Thermobaric influences of the ocean’s nonlinear equation of state Detrainment at the neutral depth
• Challenges for representing overflows in large-scale models: Avoiding inherent problems with excessive numerical entrainment Source water supply (representing the unresolved) Studies of equilibrium stratified shear instability. A new shear-driven turbulence mixing parameterization A new bottom-turbulence mixing parameterization
Mediterranean Outflow Plume• Without its 3-fold entrainment, Mediterranean
Outflow water would fill the bottom of the Atlantic• Gibraltar itself exhibits rectified tidal exchange in
conjunction with hydraulic control• Because of thermobaricity, salty Mediterranean
water has a greater density at lower pressures, contributing to shallow detrainment.
Gibraltar Velocities over the Tidal Cycle(CANIGO cruises Send & Baschek, JGR 2001)
Faroe Bank Channel and Denmark Strait Outflows
Density along axis of Faroe Bank Channel
Denmark Strait: J. Girton; FBC: C. Mauritzen, J. Price
Denmark Strait Sea Surface Temperature
Abyssal Overflows – the Romanche Fracture Zone
Potential Temperature along Romanche Fracture Zone
Ferron et al., JPO 1998
Potential Temperature at 5000 m Depth
Shear instability & entrainment
Detrainment
Geostrophic eddies
xz
y
Downslope descent
Bottom friction
Physical processes in overflows
Steps in Adequately Representing Gravity Currents
1. Supply source water to the plume with the right rate and properties.
2. Model must be able to represent downslope flow without excessive numerical entrainment.
3. Parameterize entrainment & mixing to the right extent.4. Parameterize subgridscale circulations? (e.g. eddies, flow in
small channels).
Hydraulic control at sill
Bottom-stress mixing
Source water supply• Source Water properties depend on the right large-scale circulation and
properties.• Several important source waters enter through very narrow channels!
Gibraltar is ~12 km wide. Red Sea outflow channel is ~5 km wide. Faroe Bank channel is ~15 km wide at depths that matter.
Channels that are much smaller than the model grid require special treatment – e.g. partial barriers.
The topography around Gibraltar, with a 1° grid (black), and the coastline (blue) that GFDL’s 1° global isopycnal model uses.
Representing Straits with Partially Open Faces(Work with A. Adcroft, GFDL)
Partially open faces can dramatically improve simulations of overflows that pass through narrow straits.
The model equations need to be modified to be energetically consistent. E.g. Sadourny’s 1975 Energy conserving discretization of the shallow water equations:
Terms underlined in red are affected directly by using the partially open faces.
Terms underlined in blue are affected indirectly (i.e. no code changes).
uvf
hA
Aq jq
yiq
xji
h
jih
11,
,
hy
hx
hvy
vx
vuy
ux
u AAA
jvx
iuyjih
hvVhuUVUAt
h
01
j
vi
uhiu
x
ji
ux
vAuAA
MVqt
u 22
2
111
j
vi
uhjv
y
ij
vy
vAuAA
MUqt
v 22
2
111
Resolution requirements for avoiding numerical entrainment
in descending gravity currents.Z-coordinate:
Require thatAND
to avoid numerical entrainment.(Winton, et al., JPO 1998)
Suggested solutions for Z-coordinate models: "Plumbing" parameterization of downslope flow:
Beckman & Doscher (JPO 1997), Campin & Goose (Tellus 1999). Adding a separate, resolved, terrain-following boundary layer:
Gnanadesikan (~1998), Killworth & Edwards (JPO 1999), Song & Chao (JAOT 2000).
Add a nested high-resolution model in key locations? No existing scheme is entirely satisfactory!
Sigma-coordinate: Avoiding entrainment requires that
Isopycnal-coordinate: Numerical entrainment is not an issue - BUT• If resolution is inadequate, no entrainment can occur. Need
mHz BBL 502/ kmHx BBL 52/
BBLOcean HD
AmbientOverflow 21
Diapycnal Mixing Equations in Isopycnic Coordinates
• In isopycnic coordinates, diapycnal diffusion is nonlinear
• The discrete form leads to a coupled set of nonlinear differential equations
These can be solved implicitly and iteratively, with an arbitrary distribution of diffusivities to avoid the impossible time-step limit (Hallberg, MWR 2000)
• The work-diffusivity relationship is exact in density coordinates.
• Entrainment can also be parameterized directly, based upon resolved shear Richardson numbers and a reinterpretation of the Ellis & Turner (1959) bulk Richardson number parameterization (Hallberg, MWR 2000).
This parameterization gives entraining gravity currents that are qualitatively similar to observations, but has subsequently been improved upon.
z
z
t
1
11
1
11
21
21
11
k
kk
k
kk
kk
kk
k
kk
k
k
hhhht
h
8.00
8.051
8.01.0
Ri
RiRi
RiUwE
21
212
122 kkkk uuuuU
U
ghRi
gdgWork
z
hht
hwhere
212/ ht
DOME Model Intercomparisons and Resolution Dependence (Legg et al., Ocean Modelling 2006)
2.5 km x 60 m 10 km x 144 m 50 km x 144 m
10 km x 25 Layer 50 km x 25 Layer
MITgcm (Z-coordinate) with Convective Adjustment
HIM (isopycnal coordinate) with shear Ri# param.
Plume Entrainment as a Function of Resolutionfor 6 DOME Test Cases
Fin
al P
lum
e B
uoya
ncy
(m s
-2)
Ent
rain
men
t Rat
e N
ear
Sou
rce
(non
dim
.)Horizontal Grid Spacing (km) Horizontal Grid Spacing (km)
Solid lines: MITgcm (Z-coordinate)Dashed lines: HIM (Isopycnal coordinate)
For full details, see Legg et al., Ocean Modelling 2006.
Parameterizing Overflow Entrainment:Observations of Bulk Entrainment in Oceanic
and Laboratory Gravity Currents (J. Price)
A bulk entrainment law applies, provided the Reynolds number is not quite small.
Examples of Gravity Current Mixing Parameterizations:• Generic shear parameterizations – e.g. KPP (Large et al., 1994):
Typically calibrated for the Equatorial Undercurrent.
• Two-equation turbulence closures (e.g. Mellor-Yamada; k-;).• Plume-specific parameterizations – e.g. Ellison & Turner (1959) bulk Ri#
parameterization reinterpreted for shear Ri# (Hallberg, 2000):
This can be cast as a diffusivity, is over an unstable region:
8.00
8.051
8.01.0
Ri
RiRi
RiUwEntraining
22 / SNRi 32124 7.011050 Rism
Ri
Ri
Ri
g
Uo
51
8.01.0
3
May Need ResolutionDependence!
Simulated Mediterranean Outflow Plume(Papadakis et al., Ocean Modelling 2003)
Zonal Velocity
Salinity in 3 Isopycnal Layers
Salinity
LES and Parameterized Overflow Entrainment (Xu, Chang, Peters, Özgökmen, and Chassignet, Ocean Modelling in press)
Failure and Success of Existing Parameterizations
• A universal parameterization can have no dimensional “constants”. KPP’s interior shear mixing (Large et al., 1994) and Pacanowski and Philander
(1982) both use dimensional diffusivities.• The same parameterization should work for all significant shear-mixing.
In GFDL’s HIM-based coupled model, Hallberg (2000) gives too much mixing in the Pacific Equatorial Undercurrent or too little in the plumes with the same settings.
• To be affordable in climate models, must accommodate time steps of hours. Longer than the evolution of turbulence. Longer than the timescale for turbulence to alter its environment.
• 2-equation (e.g. Mellor-Yamada, k-, or k-) closure models may be adequate. The TKE equations are well-understood, but the second equation (length-scale,
or dissipation rate, or vorticity) tend to be ad-hoc (but fitted to observations) Need to solve the vertical columns implicitly in time for:
1. TKE2. Dissipation/vorticity3. Stratification (T & S)4. (and 5.) Shear (u & v)
Simpler sets of equations may be preferable. Many use boundary-layer length scales (e.g. Mellor-Yamada) and are not
obvious appropriate for interior shear instability.
However, sensible results are often obtained by any scheme that mixes rapidly until the Richardson number exceeds some critical value.
3-DNS of Shear Instability(L. Jackson, R. Hallberg, & S. Legg in prep.)
Kelvin-Helmholtz instability
3D stratified turbulencez
x
Tem
pera
ture
(°C
)
z
x
Tem
pera
ture
(°C
)
Temperature during initial developmentof Kelvin-Helmholtz instabilities
Representative instantaneous along-channelCross-section in statistical steady state
Considerations for a Parameterization of Stratified
Shear Instability S = ||∂U/∂z|| [s-1] Velocity shearN2 = -g/ ∂/∂z [s-2] Buoyancy FrequencyH [m] Vertical extent of small Ri
Q [m2 s-2] Turbulent kinetic energy per unit massu* = (/)1/2 [m s-1] Friction velocity (for boundary turbulence)z* [m] Distance from boundary (for boundary turbulence)
• Mixing should vanish if the shear Richardson number (Ri = N2/S2) exceeds ~1/4 everywhere
• Vigorous mixing may extend past the region of small Ri.
• Homogeneous stratified turbulence is often characterized by the buoyancy length scale
• Kelvin-Helmholtz (K-H) saturation velocity scales are ~ H S.
K-H instabilities span the region of small Ri, i.e. length scales of ~ H.
Mixing-length arguments suggest peak K-H-type diffusivities scaling as ~ H2S.
• Near solid boundaries, length scales are proportional to the distance from the boundaries, and diffusivities are ~ 0.4u*z*.
22 / NQLBuoy
The diffusion of density can be linked to entrainment parameterizations by combining the density conservation equation:
with the continuity equation in density coordinates:
The latter equality is ill-behaved when ∂/∂z=0, but with constant stratification it reduces to
)(22
2
RiFzz
u
ET parameterisation (Hallberg, 2000)
zzDt
D
Translating “Entrainment Rate” parameterizations into diffusive parameterizations (L. Jackson)
RiFzzz
zz
t zz
u
u
2
11
Properties:
• Uses a length scale which is a combination of the width of the low Ri region (where F(Ri)>0) and the buoyancy length scale LBuoy = Q1/2/N.
• Decays exponentially away from low Ri region
• Vertically uniform, unbounded limit:
• Ellison and Turner limit (large Q) reduces to form similar to ET parameterisation
• Unstratified limit: similar to law-of-the-wall theories of parabolic diffusivity between two boundaries and log-like profiles of velocity near the boundaries.
S=||Uz||
)(222
2
RiSFLz Buoy
NQLBouy /2/1
= 0 at solid boundaries
Entrainment-law derived theory for Shear-driven mixing
2BuoySL
Assumptions:
• Q reaches steady state faster than background flow is evolving so no DQ/Dt term
• Assume Pr = 1 (for now)
• Q0 needed to avoid singularity in diffusivity equation (solution not sensitive to Q0 and 0)
• Parameterization of dissipation as c(Q-Q0)N
Q intended for use in diffusivity equation is due to turbulent kinetic energy only - difficult to compare to results from DNS because of internal waves.
Dt
DQNQQcNS
z
Q
z00
220
N
QL
NL bOZ
2/1
2/3
2/1
TKE Budget to Complement Proposed Diffusivity Equation
Equilibrium DNS of Shear-driven Stratified Turbulence
• Non-hydrostatic direct numerical simulations (MITgcm)• 2m x 2m x 2.5m with grid size ~ 2.5mm in centre• Molecular viscosity and diffusivity, Kolmogorov scale mostly
resolved.• Cyclic domain in x,y• Shear and jet profiles• Statistically steady state reached • Force average velocity profiles to evolve to given profile• Initially constant stratification and relaxed to initial density
profile• All profiles are spatially averaged in x and y and time
averaged
3-DNS of Shear Instability(L. Jackson, R. Hallberg, & S. Legg in prep.)
Kelvin-Helmholtz instability
3D stratified turbulencez
x
Tem
pera
ture
(°C
)
z
x
Tem
pera
ture
(°C
)
Temperature during initial developmentof Kelvin-Helmholtz instabilities
Representative instantaneous along-channelCross-section in statistical steady state
DNS data
New parameterisation (Jackson et al.)
ET parameterisation (Hallberg 2000)
F(Ri) = 0.15*(1-Ri/0.25)/(1-0.9*Ri/0.25), c=1.9
F(Ri) = 0.15*(1-Ri/0.8)/(1+1.0*Ri/0.8), c=1.7
F(Ri) = 0.12*(1-Ri/0.25)/(1-0.9*Ri/0.25), c=1.24
DNS Shear-Instability Results and the Proposed Parameterization
Buoyancy
flux (
m2/s
3)
DNS of Shear Instability and Existing 2-equation closures
Jackson et al., proposed parameterization:
Black: DNS ResultsGreen: GOTM kBlue: GOTM kRed: Mellor-Yamata 2.5
Buoyancy
flux (
m2/s
3)
DNS Jet results
Buoyancy
flux (
m2/s
3)
DNS data
New parameterisation (Jackson et al.)
ET parameterisation (Hallberg 2000)
F(Ri) = 0.15*(1-Ri/0.25)/(1-0.9*Ri/0.25), c=1.9
F(Ri) = 0.15*(1-Ri/0.8)/(1+1.0*Ri/0.8), c=1.7
F(Ri) = 0.12*(1-Ri/0.25)/(1-0.9*Ri/0.25), c=1.24
Existing 2-equation Closures Compared to DNS Jet
Black: DNS ResultsGreen: GOTM Blue: GOTM Red: Mellor-Yamata 2.5
Jackson et al., proposed parameterizations:B
uoyancy
flux (
m2/s
3)
Dia
gnos
ed d
iapy
cnal
dif
fusi
vity
(m
2 s-1)
Gradient Richardson number
Diapycnal Diffusivities Diagnosed from 3-D DNS
500 m x 30 m MITgcm
Ellison & Turner Mixing Only10 km x 25 layer HIM
At the start of the CPT, with thick, nonrotating plumes entering ambient stratification, GFDL’s Isopycnal coordinate model (HIM) would give plumes that split in two.
Such split plumes do not occur in nonhydrostatic “truth” simulations.
Illustrating the powerof the CPT paradigm
Observed profiles from Red Sea plume from RedSOX (H. Peters)
Well-mixedBottom BoundaryLayer
Actively mixingInterfacial Layer
Shear Ri# Param.Appropriate Here.
Bottom Boundary Layer Mixing
• Diapycnal mixing of density requires work.
• The rate at which bottom drag extracts energy from the resolved flow is straightforward to calculate.
• Assumptions: 20%? of the extracted energy is available to drive mixing. Available work decays away from the bottom with e-folding scale
of Mixing completely homogenizes the near bottom water until the
energy source is exhausted.
Legg, Hallberg, & Girton, Ocean Modelling, 2006
gdgWork 3BBLDo ucg
f
u*4.0
f
uch BBLDBBL24.0
exp2.0
500 m x 30 m MITgcm
Ellison & Turner + Drag MixingEllison & Turner Mixing Only10 km x 25 layer HIM
With thick plumes, both Interfacial andand Drag-induced Mixing are needed.(Legg et al., Ocean Modelling, 2006)
Double Mediterranean plumes without bottom-drag mixing
Year 5 salinity along 38.5°N in GFDL’s 1° Global Isopycnal Model
Adding the Legg et al. bottom-drag mixing parameterization leads to dramatic improvements in an IPCC-class ocean model.
Summary• Overflows are critical in the formation of most deep-ocean water masses.
• Turbulent mixing with the right rate is critical for models to obtain the right properties.(Otherwise in a stratified ambient environment, the plunging plume entrains the wrong water.)
• Large-scale models require parameterizations of such mixing that capture both the equilibrium turbulence and (sometimes) its equilibrium modification of the resolved flow.
• Mature Kelvin-Helmholtz-like mixing is significant in the interfacial layers atop gravity currents. Existing parameterizations do not appear to work very well in detail based on
comparisons with DNS (although they may work well enough for some overflows).
Laura Jackson (Princeton/GFDL CPT postdoc) has a new 2-equation (diffusivity – TKE) shear-driven turbulent mixing parameterization that looks very promising.
• Bottom-stress driven turbulence is significant for homogenizing the bottom boundary layer, and must be parameterized.
http://www.cpt-gce.org