Atmospheric Boundary-Layer Flow Over Topography: Data Analysis and Representations

of Topography

YOSEPH GEBREKIDAN MENGESHA

A thesis submitted to the Faculty of Graduate Studies in partial fulfilment of the requirements for the degree of

Master of Science

Graduate Programme in Earth and Space Science York University Toronto, Canada

May 1999

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Atmospheric Boundary-Layer Flow Over Topography: Data Analysis and Representations of Topography

bY YOSEPH GEBREKIDAN MENGESHA

a thesis submitted to the Faculty of Graduate Studies of York University in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE

Permission has been granted to the LIBRARY OF YORK UNIVERSITY to lend or seIl copies of this thesis, to the NATIONAL LIBRARY OF CANADA to microfilm this thesis and to lend or seIl copies of the film, and to UNIVERSITY MICROFILMS to publish an abstract of this thesis. The author reserves other publication rights, and neither the thesis nor extensive extracts frorn it may be printed or otherwise reproduced without the author's written permission.

iv

Abstract

We analysed high resolution (-90 m) digital terrain data (1 deg by 1 deg) for Sand

Hills, Nebraska and Fergus, Ontario. The Sand Hills region was the location in which

NCAR's Queen Air collected low level flight data on August 20, 1980 with a sampling rate

of 20 Hz and an average aircrafi speed of 100 mls. Terrain slope, terrain height spectra,

fiactal dimension, and aspect ratio are used to characterize and assess the spatial variability

of the terrain. The aircraft observations of integral statistics of atmospheric variables such

as the second-order moments (heat fluxes, turbulent stresses), drag coefficient and

atmospheric velocity spectra were analysed and compared to turbulence values estimated

fiom published rneasurements over flat and homogenous terrain.

Our analysis shows that there is a scaIe break (at about 0.5 cycle/km) in the terrain

height spectra of the Sand Hills region. Norrnalized standard deviations ( with respect to

friction velocity) of horizontal velocity components were about 20 - 40 % higher than one

might expect over flat terrain. Flight level drag coefficients were compared to the root rnean

square slope of the underlying terrain heights detennined from aircraft observation dong the

flight path and results indicate a potential for a significant increase in the drag coefficient

with topographic dope. Wavenwnber-weighted power spectra of the three velocity

components were estimated and results show that for the most stable atmospheric conditions

the spectral shape is bi-modal, separated by a spectral gap with distinct peaks at about 0.19

and 2.5 cycle~km for u and v, and 0.3 and 8 cyclekm for W . Spectral peaks at wavenumbers

of the order of O. 1 and 0.5 cyclelkm are likely to be terrain induced.

Acknowledgements

I would like to express my profound gratitude to my supervisor, Dr. Peter A. Taylor

for helping me fiom day one since 1 joined the York University community. Without the

material and mord support he gave me, to teach myself some scientific programming, and

his continuous guidance throughout the year, this thesis would be meaningless.

1 am grateful to Dr. John Miller and Dr. Qiuming Cheng for their helpful suggestions

during the course of the research,

Many thanks to Dr. Dapeng Xu , Dr. Don Lenschow, and NCAR's data manager,

Ron Ruth for helping me in data retrieving. Last but not least my thanks goes to Dr. Jim

Salmon, Zephyr North, and Dr. Wensong Weng for providing some source codes and

continuous guidance and Mrs. Sally Marshall for much assistance during my stay at York.

Contents

Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction and Objective 1

Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Data Source and Acqusition 7

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Digital Topographie Data 7 . . . . . . . . . . . . . . . . . . . 2.1.1 1 -degree Digital Elevation Models (DEMs) 8

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 GTOP030 and other DEMs 1 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Aircraft Turbulence Data 14 . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Estimation of Aircraft offset location 18

Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Characterization of topography 29

3.1 Power Spectral Analysis of Terrain . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.1.1 Variograrns and the relation of spectral slope to

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . fiactal dimension D 38

3.2 Choice of topographic parameters and resolution dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2.1 Resolution dependence of synthetic topographies . . . . . . . . . . . . . 55

Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results from Aircraft Data Analysis 63

4.1 Review of dimensional analysis and similarity theory of the . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . boundary layer 63

. . . . . . . . . . . 4.2 Velocity variances and other turbulence statistics 66 . . . . . . . . . 4.3 Drag coefficient and estimation of roughness length 84

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Velocity Spectra 88

Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions 127

vii

Appendix A . A 1 Measurement of Air Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Appendix B ....................................................... 134

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

Chapter 1

Introduction and Objective

The Earth's atmosphere is semi-transparent to incoming solar radiation (insolation);

it obtains only 20% of its energy by absorption; about 30% of the solar radiation is reflected

or scattered back to the space. The rest of the energy passes through the atrnosphere and goes

directly to the surface. Later this energy is transferred back to the lowest few kilometers of

the atmosphere. This lowest portion of the atrnosphere, which exchanges heat, mas , and

mornentum with the surface on a tirne scaie of few hours is called the atmospheric boundary

layer. Typically over land, its depth is a few hundred to a few thousand meters in the

daytime, and 10 to 300m at night. Over the ocean, its depth is relatively more constant at

several hundred meters to a kilometer or more. It is in the boundary layer where almost al1

hurnan and biological activities with their consequences take place; for instance, it transports

carbon dioxide and oxygen to plants and animals for respiration and photosynthesis, it dilutes

waste products through dispersion and cleanses the atrnosphere via photochernical reaction.

It is a source of weather systems on al1 scales, fiom fine weather to severe convective storrns

and to large-scale planetary waves. The differential heating of the surface by solar radiation

creates pressure and temperature gradients which drive the atrnospheric flow. To

quantitatively describe and forecast the atrnospheric flow, a set of equations are derived fiom

2

three basic physical laws; conservation of mas , conservation of energy, and conservation

of momenturn. However; since these equations are non-linear, analytical solution is

practically impossible, and the problem is approached by numerical techniques to obtain

approximate soIutions. The flow incorporates various temporal and spatial scales; from the

large planetary scde (-1 000lun) to the local microscale (-1 00m). Each term in the goveming

equations has its own scale of influence; and scale analysis becomes important to simplifi

the equations. The boundary layer, for example, can be divided into two layers depending

on the order of magnitude of the forces afTecting the motion: (i) the surface layer (-50 - 100

m ) - in this layer, the Coriolis force (due to the rotation of the earth) and the pressure

gradient force integrated over the depth of the surface layer are negligible compared to shear

stresses arising from the surface and it is treated as a constant flux layer, and (ii) the Ekrnan

layer or planetary boundary layer (-1000 m) - in this layer, the Coriolis force, the pressure

gradient force and shear stresses gradient are in balance. Above the planetary boundary layer

is the fiee atmosphere, where turbulent stresses are neglected and geostrophic flow is

assumed.

Modeling of the atrnosphere, therefore requires knowledge of the structure of the

boundary layer whose flow is characterized by turbulence, which is chaotic and random in

nature and difficult to predict. Terrain irregularities M e r complicate the flow situation, so

analysis of spatial variability in topography can be of use in deterrnining the overall

contributions fiom various scales as they affect the fiow field. Three-dimensional models are

3

needed to mode1 the Muence of the horizontal variability of surface properties on boundary-

layer structure. At the present t h e most 3-dimensional regional scale models are used with

a horizontal resolution of the order of 10 - 30 km which means that horizontal variations of

the surface on a smaller scale must be treated as a sub-grid scale (sgs) effect which has to be

parameterized. The comrnonly used way of introducing the effect of resolved terrain is to

make a transformation of the height coordinate, e.g., zf= z-q(x,y), where z, is the elevation,

and transform the equations of motion; but the effect of the unresolved terrain should be

somehow parameterized, and represented in terms of mean values of variables at grid points.

Many of the previous studies on the structure of the boundary layer gave emphasis

to flow over flat and homogenous surfaces, and similarity theories were developed in the

1940's based on dimensionai arguments (see Chapter 4 for a review of dimensional analysis).

To validate the theories, extensive boundary-layer experiments, such as those in Kansas

(1 969), Minnesota (1 972), and Wangara (1 968) using micro-meteorological instruments

mounted on towers, were conducted. Turbulence statistics and velocity spectra for flat terrain

for different atmospheric stabilities, and at various levels in the boundary layer were studied.

Spectral analyses of the wind were found to be useful in parameterizhg eddy difisivities

for applications in air pollution studies. Later, analytical, numerical, and laboratory studies

of boundary-layer flows over isolated hills provided a stepping stone to the understanding

the effect of obstacles to boundary-layer flows (Jackson and Hunt 1975; Taylor 1981). These

studies show that for flows with neutral static stability (zero vertical potential temperature

4

gradient), hills exert a net drag (horizontal component of the force acting on the hill), arising

fiom pressure differences between upstrearn and downstream slopes, on the turbulent flow

over them, and which can have an effect on large-scale atmospheric flows. The scales of the

obstacles which contribute to such a net drag are usually less than the resolution of weather

prediction and climate models, and as a result, the drag must be pararneterized. These studies

also showed that the topographically induced speed-up of wind at the sumrnit of a hi11 is a

significant factor in site selection of wind energy machines and large structures since the

power output of a wind turbine varies as the cube and the pressure force on a building as the

square of the wind speed. Many of the previous studies of the effect of fi%tion and the

pressure drag of sub-grid scale (sgs) topography on air flow in meteorological modets have

focused on simple idealized sinusoidal topography (e.g., Taylor et al 1989). In applying this

work to a real complex terrain, we need some parameters to characterize topography so that

empirical relations can be investigated to represent the sgs topography. Some topographic

features, for example, are characterized by variations fiom ridge to valley in short distances

(e.g., the Grand Canyon), while others have gentle, slowly varying undulations. The

representative spatial scales of these features determine the resolution needed in a

meteorological model if realistic simulation results are to be obtained. That is to say, terrain

height variations with horizontal wavelengths greater than 2Ax , where Ax is grid spacing

of the model, c m be resolved directly by the model, whereas those with wavelength less than

2Ax (the sub-grid scale, sgs) are not resolved. It would be desirable to choose a value of Ax

5

small enough so that the effect of sgs variances are negligible, but because of computer

memory limitations, the sgs can only be parameterized. Availability of digitized topographie

data such as digital elevation rnodels @EM) and aircraft data has increased research interest

in these fields. Aircrd observation can cover a large distance in a short t h e (- 100km in

10-1 5 min being typical) and hence provide representative estimates of turbulent statistics.

The objective in this thesis is to analyse the spatial vmiability of high resolution

(-90m) Digital Elevation Mode1 (DEM) data of the Nebraska Sand Hills and of Fergus,

Ontario using various methods, define pararneters which characterize topography and assess

their resolution dependence. We also analyse data fiom NCAR's Queen Air, August 1980,

low level flights over parts of the Sand Hills and estimate the integral statistics of

atmospheric variables such as the second-order moments (heat flues, turbulent stresses),

drag coefficient and atrnospheric velocity spectra and compare these with representative

values over flat and homogenous temain.

To begin with, the acquisition of both digital terrain and aircraft turbulence data for

Sand Hills and Fergus is described in Chapter 2. This is followed by a discussion of the

andysis of digital terrain using spectral and fkactal methods. The pararneters that characterize

topography and their sensitivity and fwictional f o m to grid resolution, by re-sampling the

original DEM data and some synthetic data will also be discussed. Chapter 4 will focus on

the results and methods that are used to calculate the integral statistics and velocity spectra

of aircraft data. The effect of stability and underlying terrain on the turbulence parameters

6

and on the shape of the velocity spectra; comparison of turbulence values obtained fiom

field measurements over flat homogenous terrain with those of the present study over

complex terrain will also be discussed. Conclusions are given in Chapter 5.

Chapter 2

Data Sources and Acquisition

This Chapter describes the sources and types of our data for the study area, and data

that are being used by others; the rnethodology that we used; and also problems that we faced

to extract some of the usefùl parameters, and to project the data to a coordinate system other

than the raw data coordinate system.

2.1 Digital Topographie Data

Digital topographic data of the Sand Hills terrain in Nebraska (USA) were obtained

(fiee of charge) fiom the US Geological Survey (USGS), via the world wide web

(http://sunl .cr.usgs.gov/eros-home.h~), and can also be obtained via anonymous ftp fiom

edcftp.cr.usgs.gov (Nurneric IP address 152.61 -192.70) as public domain information. The

digital elevation data are stored at the EROS Data Centre in Sioux Falls, South Dakota on

a robotic mas-storage device. These data are encoded with the GNUWp compression

program, which makes transfer of files much faster than unçompressed files, though there

is an option for transfering uncompressed files. The files are about IMB compressed but are

around 10MB once uncompressed. Data for Canada are only available through Geomatics

Canada at a cost of $270 per file. One Canadian digital elevation data set (CDED), othenvise

known as Digital Terrain Elevation Data LeveI 1 (DTED-l), produced by the Defense

Mapping Agency @MA), was provided by Zephyr North for Fergus, Ontario, under a

collaborative research arrangement.

2.1.1 1-degree Digital Elevation Models @EMS)

The 1-degree DEMs cover a one degree by one degree block representing one-half

of a 1 degree by 2 degree 1 :250K scale quadrangle map. For the contiguous United States

the files are organized in directories (A-2) that represents the first character of the name of

the DEM with "-e" or "-w" appended for the east or west block. Each DEM consists of a

regular array of elevations referenced horizontally on the geographic coordinates (latitude,

longitude) of the World Geodetic System 1972 (WGS 72) or for a few DEMs, the WGS 84

Datum. Elevations are in metres relative to mean sea level, and are ordered from south to

north in profiIes that are ordered from west to east. Spacing of the elevations along and

between each profile is 3 arc-seconds, and the four data points at the corners of the

quadrangle are at the integer degrees of 1atitudeAongitude which results in 120 1 elevations

per profile. Three arc-seconds correspond to approximately 90 metres in the north-south

direction and variable spacing in the east-west due to convergence of meridians as latitude

increases (approximately 90 metres at the equator and approximately 60 metres at 50 degrees

latitude). Al1 1-degree DEMs are classified as Level 3 in terms of accuracy (root-mean-

square enor RMSE < 0.333 contour interval, Absolute Maximum Error < 0.666 contour

interval), and the majority of them are produced fiom cartographie and photographie sources.

Note that the 1-degree DEMs are derived fkom DMA DTED Level 1 data, but the DMA use

the term "level" to refer to spatial resolution, whereas USGS uses it for accuracy.

9

The whole Sand Hills region comprises about six 1-degree DEMs (North Platte East

and West block, Alliance East and West block, Valentine East, and Scottsbluf East).Using

a Fortran code provided by Zephyr North and the Geographic Information Systems (GIS),

ARCm\JFO software, the DEM data format of the six blocks was converted to a composite

lattice/grid, of elevation points. However, the lattice consû-ucted fiom this USGS 1 deg by

ldeg DEM are not immediately suitable for extracting the topographie pararneters, such as

the analysis of slope, because the horizontal dimensions i.e., x and y coordinates, the ground

planimetric units, are measured in latitude and longitude (arc seconds), whereas the z values,

the elevation points, are measured in metres. The x, y, and z coordinates should have the

sarne units of measurements; and to make calculations of dope and other pararneters

possible, the DEM was projected to a non-angular unit of measurement, UTM (Universal

Transverse Mercator) of zone 14 with false easting and northing. For the UTM Cartesian

coordinate system, the globe is divided into 60 zones each spanning 3 degrees west and 3

degrees east. The lirnits of each zone are 84 degrees north and 80 degrees south . The origin

of each zone is the equator and its central meridian. Regions beyond this use Universal Polar

Stereographic projections. In the conversion of coordinates and resarnpling of the data on a

regular grid, we utilized bilinear interpolation. This method identifies first the four nearest

old grid point elevation values of the input data to the location of the desired grid point to

be interpolated. The new value at the point in question is a weighted average determined by

the values of the four nearest grid points and their relative position or weighted distance fkom

10

the location of the desired point. The following diagram shows how bilinear interpolation

calculates the Z (elevation) value of a point on a grid.

I Z

X

Fig 2.1 Labelling of points used in the bilinear interpolation method

The left intermediate point on the y-axis is linearly interpolated as:

The right intennediate point on the y-axis is linearly interpolated as:

Then, the required Z value of the point is calculated again by linear interpolation between

the right and left intennediate points as:

Note that in our subsequent analysis interpolation is bilinear uniess othenvise specified.

A shaded relief map of the site generated by Surfer (Golden Software) with horizontal and

vertical angles of illumination of 135" and 45" respectively is shown in Figure 2.2. The

Fergus region consists of three CDEDs with the same grid iresolution as the DEMs but

11

different file format and was projected to UTM zone 17. A contour map is shown in Figure

2.3. Three non-overlapping sub-regions of the Sandhills area, Al, A2, and A3 were taken for

analysis; details are explained in Chapter 3.

2.1.2 GTOP030 and other DEMs

The EROS Data Centre has recently compiled a global digital elevation mode1

GTOP030 covering the full extent of latitude fiom 90 degrees south to 90 degrees north, and

the full extent of longitude fiom 180 degrees west to 180 degrees east with a horizontal grid

spacing of 30 arc seconds (approximately 1 km). The horizontal coordinates are in decimal

degrees of latitude and longitude referenced to WGS84, and elevation points are in metres

above mean sea level ranging from -407 to 8,752 metres. GTOP030 is divided into tiles each

of which covering 50 degrees of latitude and 40 degrees of longitude. The data are being

used for large-scale regional and continental models such as climate and hydrological

models. Other DEMs are also available in the EROS Data Centre. These are the 7.5-minute

DEMs covering 7.5- x 7.5- minute blocks, with a grid resolution of 30 metres, and 30-minute

DEMs covering 30- x 30- minute blocks, with a resolution of 2-arc seconds (approximately

60 metres). Unlike other DEMs, the 7.5-minute DEMs are referenced to the Universal

Transverse Mercator (UTM) Cartesian CO-ordinate system of the North American Datum

(NAD 27) or NAD 83, and are classified as Level 1 (RMSE 15m, Absolute Maximum Error

50m)'. Al1 30 -minute DEMs derived fiom contour plots are classified as Level2 (RMSE <

' For the DEM file format and many more details, see the user's guide at the website of EROS Data Centre.

Figure 2.2 Shaded relief map of Sand Hills ( Nebraska )

Figure 2.3 Countour map of 25.6 X 25.6 km of FERGUS, Ontario. Countour interval 1 Om

14

0.5 contour interval, Absolute Maximum Error < 0.666 contour interval).

2.2 Aircraft Turbulence Data

Observations of wind speed, temperature, etc were obtained fiom the National

Centre for Atmospheric Research (NCAR) Queen Air flight (of 20 August 1980) surface

roughness experiment. The Queen Air data were gathered during flight conducted in a cross

pattern in the central part of the Sand Hills, with a sampling rate of 20 Hz and an aircraft

speed of 100 ms-'. There were two flights, one early in the morning, when the boundary

layer was in a stable condition, and the other one near to rnid-day in convective conditions.

The longest track going North-South was about 90km long (41°1 5'20" N, 101 "37'30" W to

42"35'56"N, 101°34'45"W), the south end being near the North Platte river. The E-W line

crosses the N-S line at roughly mid-point and has coordinates 4 1°5 1 ' 15 "N, 10 1°5' 1 0" W to

420010"N, 102O12'32"W. The trajectories, overlayed on a contour map with coordinates in

UTM, are given in Figures 2.4 and 2.5. The numeric labels on the trajectory are time in

minutes, and for the sake of clarity every 20 points were plotted and labeled, and the text

labels are flight legs (sarnple runs, for exarnple MN1 stands for morning northbound leg 1).

In the experiment, about sixty-five geographical and rneteorological parameters were

measured; however, we focused mainly on air velocity and temperature data. The velocity

of air with respect to the earth was obtained by adding the velocity of the aircraft with respect

to the earth and the velocity of the air relative to the aircraft. The air velocity relative to the

UTM coordinate (m) Easting

Figure 2.4 NCAR Flt#l Aug80 trajectory (5:40am - 7:40pm) over Sand Hills, Zone 14

Figure 2.5 NCAR Flt#2 Aug80 trajectory (10:27arn - 12:46pm) over Sand Hills, Zone 14

17

aircrafl was measured by a Pitot tube mounted at the tip of the front boom (- 5m) of the

aircraft. The velocity and attitude angles of the airplane relative to the earth were obtained

fkom an inertial navigation system (INS), which contains an orthogonally mounted triad of

accelerometers on a platforrn that is stabilized by means of gyroscopes, whose outputs are

integrated to obtain the aircraft velociiy, and integrated again to obtain position. Air

temperatures were measured by a 0.025mm platinum wire. The instrumentation and

techniques for measuring air velocity, temperature, humidity, etc are described by Lenschow

(1986). See also Appendix A for fùrther details.

The aircraft measurement data were stored in the Scientific Computing Division

(SCD) Mass Storage System (MSS) at NCAR. This is a central, large scale data archive that

stores observational data and data generated by climate models and other programming

activities on NCAR's computer servers. The objective of the MSS is to service client

supercomputers at data rates tolfiom the MSS in the 10's of Megabytes (Mbytes) per second

and has a storage capacity in the 100's of Terabytes (Tbytes). The observational data of the

two flights are about 400 Mbytes stored on two volumes. To get or read files fiom the SCD

machines to the Mesoscale Microscale Meteorology (MMM) workstation, where we have

an account, a series of batch jobs (scripts including remote copying, RCP, ninning executable

files, etc) were processed by the Masnethternet Gateway Server (MIGS). The first step that

we used to get the important parameters for our study was to download a one minute of data

18

fkom any of the two observational flight data sets and write a Fortran source code to extract

parameters of interest. One minute of data is approximately 1MByte. This was initially

transferred to York via FTP and stored as a single record on a PC. As a result our

FORTRAN source code could not read the file sequentially and had to be edited on a PC

editor. This is easy to do for few megabytes, but practically impossible for huge files. To

remedy this problem, we sent the source code to an NCAR super cornputer (CRAY Y-

MPICRAY J90) to read and write selected parameters fiom the files that were retrieved fiom

the MSS. This smaller files are then transferred via FTP to our PC. The following diagram

depicts the transfer process.

MIGS f j( RCP I I

Figure 2.6 Block diagram of data transfer

2.2.1 Estimation of Aircraft offset location

In addition to wind speed, direction, and temperature, the aircraft observations

included geographic location, latitude and longitude in decimal degrees, pressure heights in

geopotential metres above sea level, and radiometric height of the aircrafi in metres above

19

the local ground. Terrain heights along the flight path were obtained by subtracting the

radiometric hcight fiom the pressure height. At the beginning of the fiights, before the north-

south track, the aircraft was well above the local ground. Since the full scale of the radio

altimeter was 761 m, most of the records of the radiometric heights in this flight segment

were confined to 761 m. This produced erroneous results of elevation heights of the order

of 2 - 3 km, so we discarded the full scale records, and retained oniy those heights

determined fiom radiometric records that do not exceed full scale. In order to use these

heights for our future analysis, we had to make sure that they were consistent with the

heights of the topographie map, so we interpolated elevation points along the flight trajectory

fiom the DEM and compared to the aircraft measured heights. However although the ranges

and averages were comparable, there was no match between the two. Since the aircraft

observations was performed before the Global Positioning System (GPS) was available, the

locations were determined by an Inertial Navigation System (NS), that is, by twice

integrating the acceleration of the aircraft. The absolute accuracy after the Boulder to

Nebraska transit is approxirnately 1 km, but relative errors over a short section of the flight

path should be small (Lenschow 1986). In order to estimate the offset of the locations, we

set up first a window (4km X 4km) of offset points centred on the reported position at the

start of a section of the flight path with a grid resolution of Z5m (see Figure 2.7). Then, for

each trial offset start point, we interpolated elevation points along the subsequent flight track

20

and calculated the standard deviation (an estimate of error) between the DEM interpolated

temain heights and the flight data for terrain height to see if there exists a minimum value

fiom which we could infer the probable offset. The standard deviation error were calculated

as follows:

The overbars are the means of the elevations of the DEM interpolated and flight

(FLT) data, and N is the number of elevation points in the flight segment. For the North -

South track, we took four blocks each of 7 min. flight data (-42km). Since the aircraR speed

was about 100 m/s and the sampling rate 20 Hz, the spatial resolution of the elevations was

about 5 m, and interpolating each sample points dong the track takes excessive CPU time

and disk space. We then reduced the number of data points used, but chose at least one

sample point for every grid ce11 of the DEM. The first two blocks, which are at the beginning

and at the mid-point of the North flight track, north of North Platte river, show an offset

value of (2Sm West, 575m north) and (75m east, 600m north) respectively.

Grid of offset locations .-----.-----.----- &--,

L - - - - - L - - - l I

I I I I I I I I

I I A ircraft location I I

Topographic map

Figure 2.7 Schematic of window of offset locations

Block3, which is above the mid-point of the flight segment, has some problem with the

matching because the DEM shows some flat terrain features while the flight shows

undulations. Block 4, which is at the end of the N-S flight track is slightly better than block

3. The offset points are (200m east, 700m north) and (3001-11 east, 700m north) respectively.

The East-West flight segment, again for 7 min. blocks (block I and block 2), shows a very

good match with intemally consistent offset values of 45Om west, 575m north, and 325m

west, 675m north respectively. Sample results are shown for E-W block 1 and N-S block 1

in Figure 2.8 (a,b,c) and Figure 2.9 (a,b,c). The figures show tirne series of elevations

interpolated firom the DEM dong the flight path and the data fiom flight 2 (FLT#2), before

adjustment, a contour plot of the standard deviation error, and the adjusted elevation points.

The centre of the contour plot corresponds to the centre of the window of offset points at the

reported location of the aircraft. The above offset values are obtained by multiplying the grid

22

resolution 25 m by the difference of the central grid point, i.e., 8 1 , and the grid points at the

minimum value of the standard deviation. The error seems to slightly increase with time of

flight and t m s counter-dock wise in the E-W and N-S plane. The above analysis was for

the afternoon flight track (FLT#2). Similar results should be expected for the morning flight

track (FLT#l) as long as the error in the location is a systematic error.

- DEM - Flt#2

1080 ~ 1 1 , 1 , 1 1 1 , , 1 1 , 1 1 , 1 , 1 1 1 1 1 1 1 1 1 1 1

64 66 68 70 72 74

Time (min)

Figure 2.8(a) E-W cross section of Sand Hills, block 1 (unadjusted)

Figure 2.8(b) Contour plot of standard deviation (m) for block 1. Absolute minimum = 8.2 @ (63,104)

- DEM - Flt#2

1100 ~ , l l l l l l i i l i l l i i i l l l

64 66 68 70 72 74

T i m e (min)

Figure 2.8(c) E-W cross section of Sand Hills, block 1 (adjusted)

DEM Flt#2

Figure 2.9(a) N-S cross section of Sand Hills, block 1 (unadjusted)

Number of points in E-W direction

Figure 2.9(b) Contour plot of standard deviation (m) for block 1. Absolute minimum = 8.9 @ (80, 104)

Time (min)

Flt#2 DEM

Figure 2.9(c) N-S cross section of Sand Hills, biock I (adjusted)

Chapter 3

Characterization of topography

In this Chapter, we discuss some details of the methods that we used to analyse

digital terrain, and the parameters that characterize topography. Also as discussed in Chapter

2, there are many digital terrain data sets with various resolutions, accuracy and extent of

coverage used for modelling meteorological or hydrological purposes. We will investigate

the sensitivity and functional dependence of the parameters to grid resolution, by resampling

a 1-degree DEM and some idealized terrains.

3.1 Power Spectral Analysis of Terrain

At present, mesoscale or regional atrnospheric models generally work with a

horizontal resolution of the order of 5-50 km, which means that horizontal variations of the

surface on a smaller scale must be treated as a sub-grid scale forcing which has to be

pararneterized. Since any spatially varying surface can be approximated or represented by

Fourier series, to assess the spatial irregularities of an underlying terrain, a two-dimensional

terrain variance spectra can be used for numerical modelling grid design . Young and Pielke

(1983), Pielke and Kennedy (1980), and Young et al. (1984) use spectra of a number of

linear profiles through their terrain of interest and average them to represent the spectral

behaviour of the area. Ryner (1 WZ), and Steyn and Ayotte (1 985) perform two-dimensional

spectral analysis to show the effect of directionality of terrain in the spectra. The advantage

30

of two-dimensional terrain spectra over one-dimensional spectra is that they retain any

directionality in the terrain. However, for approximately isotropie terrain, one-dimensional

analysis along any cross-sections may sufice to represent the spectra and it is easier to

interpret the results.

For the three non-overlapping subregions of the Sand Hills and for Fergus, Ontario

shown in Figure 2.2 and 2.3, with N x N grids of equally spaced data points in a square with

linear size L and grid spacing of &=a= WN, we calculated one-dimensional complex

Fourier spectra along profiles in both the east-west and north-south directions. These were

then averaged using the power spectral density function, which is defined as:

P ( k ) =2LIF(k)I2 k = 1,2 ,....., (N2-1)

P(k)=LIF(k)IZ k = N / 2 , & k = O

Where F(k) is the discrete complex Fourier coefficient given by:

Here k is the wavenumber and fin) is elevation in physical space at location nllc. The power

spectral energy density, that is, the energy or the variance per unit wavenumber estimated

using Equation (3.1) is the one-sided presentation of the power spectrum. The factor 2 is

introduced because for real raw data, the amplitudes of Fourier coefficients greater than N/2

(Nyquist wavenumber) are identically equal to those at lower fiequencies. The power

31

spectral density gives an estimate of the mean square fluctuations at spatial wavenumbers

k, or length scales of 2 d k We used the standard FFT package fiom Numerical Recipes

(Press et al., 1986) to estimate the Fourier coefficients. The value at k =O represents the mean

of the terrain and was ignored as it does not give any information on the variance fiom the

mean. Also for each profile a linear trend was first calculated by the method of least squares

and then subtracted fiom the terrain height. To reduce leakage(variance introduced at the

sharp edges of discontinuity), a sine squared (bel1 tapered) srnoothing fùnction near the

beginning and ending 10% of the record was used, that is to say, the raw data was multiplied

sin2(§?tn/N) for 0': n 1 0.1N W(n) = elsewere

sin2(5nn/N) for 0.9N s n s N

Many window functions have been suggested in the literature. Equation (3.3) is window

fünction given in Stdl(1988). Estimates of the one-sided one-dimensional power spectra for

Sand Hills and Fergus on a log-log scale for various sample intervds are given in Figures

3.1 through 3.4. The sample intervals range fiom 50 m to 1600 m and were obtained by

interpolation of the original 3 arcsecond (90 m) data. As shown, the larger-scde elements

of the topography are by far the most important since the spectnun declines rapidly as scale

decreases (increasing wavenumber).

Depending on the percentage of the total variance we want to consider, we c m set

the minimum wavelength that must be resolved. For instance, for 80% of the terrain height

E ciY 1000.00 (b) E-W

100.00 Gnd512x5

10.00 -- Gdd 128x1 28

-+, Grid 64x64 1 .O0 + Grid 32x32

0.10 +, Grid 16x76

0.01

0.01 0.1 O 1 .O0 10.00 wavenumber k(cycle/km)

Figure 3.1 Mean power spectral density of Sand Hills (A l )

(b) E-W b

0.01 0.10 1 .O0 1 0.00 wavenum ber k(cycle/km)

Figure 3.2 Mean power spectral density of Fergus

1 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1 - - - - - (a) N-S - -

- - - - - - - - - - - - I - - - - - - - - d

- - d - - % Gnd512x512 - - - - -

- - - =. - -A- - G-"d-2g*-56- - - - 7

- - - - - - ,+ Gfid 128x128 - - - - - -

,-&- Grid 64x64 d

- - d - - - - - - --+- Grid 32x32 - -

- - - a

Grid 16x16 - - - -

- I - - I 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1

I I 1 1 1 1 1 1 1

0.01 0.10 1 .O0 10.00 wavenum ber k(cycle/km)

Figure 3.3 Mean power spectral density of Sand i-fills

0.01 0.10 1 .O0 10.00 wavenum ber k(cycle1km)

Figure 3.4 Mean power spectral density of Sand Hills (A3)

36

variance, the values of the minimum wavelength required are given in Table 3.1. Note that

the integral of the power spectral density gives us the total variance in the domain of interest

on scales greater than or equal to the resolution of the original data, so to estimate the

minimum wavelength, we integrate the spectral density up to the specified percentage (Pielke

and Kennedy 1980). Since the minimum wavelength that can be resolved in a numerical

model of grid size Ax is 2Ax (Young et. al, 1983), for a mesoscale model with a grid size Say

2.5km, according to Table 3.1, the Sand Hills terrain must be considered in the

parameterization of the subgrid-scale flux tenns or at least a grid resolution of -1 km rnust

be used to model without pararneterization, whereas Fergus may be resolved adequately with

a 2.5 km grid.

The above power spectral representations appear to obey a power law relation of the

wavenumber with the spectral density for the high wavenumbers of the form:

PF) = aka (3 -4)

The values of a , the spectral slope, over the whole range of wavenumbers deterrnined fcom

least-squares best fit are 2.85 (2.83), 2.82(2.68), 2.71(2.65), and 2.1(2.2) for North-South

(East-West) profiles of Sand Hills Al, A2, A3, and Fergus respectively. For a range of

scales, the values of a and a are given in Table 3.2. Note that the units for a are not given

since they are dependent on a . The value of a reported in the literature for the surface of the

Earth is 1 lsas3, with a = 2 being typical, Brown (1987). It is also =2 for the surface of the

Moon, Venus and Mars according to Bills and Kobrick (1 985). The intercept of the graph

at k = 1 cycle/krn (dotted line) is taken as the roughness amplitude, a measure of how

relatively rough a surface is, following Huang & Turcotte (1989). Note that we should not

conftise roughness amplitude with the aerodynamic roughness length z, which will be

discussed briefly in Chapter 4.

Table 3.1 Minimum resolved wavelength

-2.0 km

II Sand Hills (A2) 1 -2.0 km 1 -2.4 km II II Sand Hills (A3) 1 -2.2 km 1 -2.8 km II

Table 3.2 Parameters of the least-squares best fit for wavelength bands.

P(k) = aka (North-South) P(k) = ak" (East -West) kC0.5 cvcle/km k>0.5 cvclekm kc0.5 cvcle/km k>0.5cvcle/km a a a a a a a a

SandHills A l 313 0.24 23 2.97 129 0.89 14 2.9 SandHills A2 74 1.4 14 3.2 45 1.6 8.8 3.0 SandHills A3 90 1.7 18 3.1 85 1.5 2.7 2.95 Fergus 4.9 2.1 1.4 2.2 6.5 1.9 1.5 2.2

--- - - -

Some of the plots of the spectra are almost Iinear throughout the domain, e.g., Figure 3.2

(Fergus), whereas others seem to have a scaIe break, that is, two quite separate straight lines

may be drawn intersecting, for the Sand Hills topography, at about 0.5 cycle/km. These

breaks could represent characteristic horizontal scales where surface behaviour changes

substantially. The plots for Fergus seem to have no visible scale break and its roughness

38

amplitude is Iess compared to Sand Hills. We can also see from the figures that the spectral

slopes are quasi-invariant with grid resolution. But this invariance is a bit strange as we

should expect some change in the estimate of the power s p e c t m at each wavenumber as we

increase or decrease the sarnpling interval; however, Press et ai (1 986) point out that al1 the

information obtained by increasing the number of sarnple points goes into producing

estimates at a greater number of discrete wavenumbers.

Another way of presenting one dimensional power spectra is to use what is known

as the average vector spectnim. This is accomplished by summing the spectral densities fiom

2D spectra in semi-circular strips of constant radial wavenurnber, l=(P + rn3 1'2, and dividing

by the nurnber of ternis that are summed (Ryner 1972; Huang and Turcotte 1989). Here k and

rn are wavenurnbers in x and y direction respectively. The nurnber of terms that are surnrned

increases with increasing the radial wavenumber. This method is found to be usefid and ideal

for xepresenting the mean spectnim of temin and we have used it in section 3.2.

3.1.1 Variograms and the relation of spectral dope to fractal dimension D

Mandelbrot(l977,1983) introduced the terrn "actal specifically for temporal or

spatial phenomena that are continuous but not differentiable, and that exhibit partial

correlations over many scales, and suggested that fiactals are usefül mathematical models

of naturd phenomena fiom topography to turbulence. We are not going to discuss the subject

of fiactals extensively and thus only a brief introduction is given here. A fiactai is a rough

or fiagmented geometrical shape which, when subdivided into parts has the property that

each part will look like a reduced copy of the whole shape. It's a particular mathematical

mode1 of irregular geometry, in which the scaiing properties are described by the fiactal

dimension D. For regular shapes for instance, the perimeter scales with 2' , area with Z2 and

volume with l3 where 2 is a linear length. The exponents in the scaling are integer Euclidian

dimensions, but in the case of imgular shapes the exponents can be fractions. For instance

for a piecewise linear profile the fiactal dimension D is non-integer and may Vary between

1 (relatively smooth) and 2 ( rough and irregular that it fills up Euclidian space). For surfaces,

the corresponding range for D lies between 2 (absolutely smooth) and 3 (increasingly

ragged). There are varieties of methods used to estimate the fkactal dimension, arnong them

are the divider, variogram, spectral and box-counting rnethods. In the divider method, one

opens a pair of dividers to some fixed distance r and walks them dong the profile to estimate

its total length. The total nurnber of steps (total length of the line) is plotted as a fiinction of

r on a log-log plot. If linear, this curve has a slope of 1- D. Mandelbrot (1977), and

Goodchild (1980) have used this method to estimate the fiactal dimension of coastlines. Here

we will use the variogram (structure function) method, another important statistic to

characterize surfaces. The relation of spectral dope and fiactal dimension, and the box-

counting method will be discussed later.

For the profile of a discrete spatial or temporal series, the structure fùnction is dehed as:

Here, 2, is the topographie height, h is the horizontal grid index lag between points, N is the

number of point pairs and q is the order of the moment considered; in our case q=2

(variogram). For a fractal mode1 of terrain, generated by integration of random numbers with

Gaussian distribution, the scaling is given by:

" hqH ( 3 4

where H is called the Hurst exponent and is in the range O <H<1. Voss (1985) and Huang

and Turcotte (1 989) discuss the relation of H and D and show that D= 2-H for a profile.

Therefore the fiactal dimension D cm be estimated fiom the dope of a log-log plot of p, vs

h. For our terrain data sets, an average variogram was calculated for linear cross-sections

along east-west and north-south directions ignoring those distance lags with less than 25

point pairs since the statistic then becomes less reliable. Estimates of p, are given in Figures

3.5 and 3.6. As shown, a straight line could not be drawn to estimate D throughout the whole

range of scales; but for some sub-ranges, the data sets exhibit properties of the fiactal mode1

with the fractal dimensions given in Table 3.3. The dotted lines are the nugget effect, the

value of the variogram at the smallest lag distance (a measure of roughness or degree of

spatial correlation). The smaller the nugget effect the higher the spatial correlation (relatively

smooth); therefore, the Fergus topography shows good spatial correlation between

neighbouring points compared to Sand Hills A 1, A2, and A3. Note also the periodicity of the

undulating surface in the N-S cross section of Sand Hills Al, which is clearly seen with

maxima at lag distance of about lkm, 3km, 5krn etc. The structure fünction of Al(N-S)

h o u g h many scales seems to follow a function of the fonn (solid line in Figure 3.5):

Where C and k are adjustable constants with values of 600 and 4 respectively. Thus for the

whole range of scales, the statistical behaviour of the surface of A1 (N-S) is better predicted

by Equation 3.7 than by the fiactd model, Equation 3.6.

Table 3.3 Fractal dimensions for a range of scales

E- W N-S h < I k m 1 < h < 3 k m h > 3 k m h C lkm h > l k m

Sand Hills (Al) D =1.28 D= 1.83 periodic D=l ,28 periodic

h < lkm h > lkm h c l k m h > ]km Sand Hills (A2) D =1.32 D=1.88 D=1.3 1 D=1.85

h c t k m h > lkm h <Ikm h> 1 km Sand Hills (A3) D = I .3 1 D=1.92 D=1.33 D= 1.86

h< lkm 1 <hcl km h> 7krn h c Ikm I<h<7km h > 7km Fergus D =1 .25 D =1.5 periodic D=1.24 D=1.68 D=1.93

Table 3.3 indicates that for shorter scales, h < 1 km, the fiactal dimension of the Sand Hills

topography is about 1.3, and for most of the medium and longer scales the dimension is

about 1.87 with the exception of Al , where the fi-actal model breaks down and couId be

replaced by periodicity. Similarly for Fergus, the shorter scales have a dimension of about

1.25 and the medium scales about 1.6. These fractal analyses suggest that sdaces with

larger nugget effect and fiactal dimension D are relatively rough, and elevation points

cannot be accurately interpolated fiom the heights of neighbouring points. The converse is

Sand Hills ( A l )

Sand Hills (A2)

Sand Hills (A3)

Fergus

C[l -sin(kx)/kx]

Nugget effect

1 .O0 10.00 Lag distance h (km)

Figure 3.5 Variogram of Sand Hills and Fergus for the N-S cross-sections

Sand Hills (Al)

Sand Hills (A2)

Sand Hills (A3)

Fergus

Nugget effect

1 .O0 10.00 Lag distance h (km)

Figure 3.6 Variogram of Sand Hills and Fergus for the E-W cross-sections

44

tme for low nugget effect and fiactal dimension.

Both the variogram and the integrai of the spectral density function have units of

variance and cm be related. As discussed in the previous section, the spectral density has a

power law dependence on wavenumber, hence on the wavelength L and can be related to the

variance 02(L) = p2 (h) by :

P(L) = L 02 (L) - - La (3.3)

Therefore fiom (3.8) a = 1+2H, and since D=2-H for a one-dimensional profile we have

D = (5-a)/2 (3 -9)

For fiactal surfaces D = (7-a)/2, Voss (1985). According to the definition of fiactal

dimension for a profile and Eq.3.8, the spectral slope a must thus lie in the range 1 sa s 3.

For surfaces with spectral slope outside this range, the relationship of D and a doesn't hold

and other methods must be used to estimate D. We have seen that there is scale dependent

fiactal dimension D, and the statistical behaviour of our topographie data sets can be

characterized by the fiactal mode1 for a range of scales. Recent studies have shown that rnany

spatial and temporal variables could best be characterized by multifiactal (multiscaling)

rneasures with universal multifiactal parameters. Among the multifiactal techniques are the

mass exponent (Cheng and Agterberg 1996), trace moment (Schertzer and Lovejoy 1987)

and functional box-counting method (Lovejoy et al 1987). Here we use the functional box-

counting method to show the multifiactal behaviour of terrain. Note that at this stage we have

not attempted to calculate the universal parameters.

In the functiond box-counting method, first a new set of data points with elevation

values greater than or equal to a specified threshold elevation value are generated

(intersection of a threshold plane and elevation points). Then the resolution of the set is

increased systematically by draping the set with a mesh of square boxes of decreasing size

by a factor of 2 (see the schematic diagrarn in Figure 3.7). For each box size, the number of

boxes N(L) which cover at least a data point is counted. The fiactal dimension D is then

obtained as the negative slope of the log-log scale graph of N(L) vs. L(box size). This is

done for various threshold values and results for out- data sets are given in Figure 3.8 and 3.9.

As shown in the figures, the multifiactal behaviour of our terrain data sets is evident. The

fiactal dimension increases with decreasing threshold elevation value; for monofiactd sets

one should expect an approximately constant fractal dimension D with threshold values.

Thus the functional box counting approach suggests that our topography is multiscaling and

a single dimension D is not adequate to describe the scaling law of the spatial distribution

of elevation points.

3.2 Choice of topographie parameter and resolution dependence

Other parameters characterizing topography are: the slope correlation, M, defined as

the mean of the product of the slopes in x and y; the K value, defined by the sum of the mean

square slopes in x and y divided by 2; and the L value, defined by the difference of the mean

square slopes in x and y divided by 2 (see Baines, 1995). These are given by :

Figure 3.7 Schema showing how h c t i o n a l box counting c m be used to estimate the fiactal dimensions at various thresholds T, &er Lavallee et. al (1993)

0.1 1 .O 10.0 100.0 Box size L (km)

Figure 3.8 Functional boxcounting. (a) Sand Hills (Al); (b) Fergus

o. 1 1 .O 10.0 100.0 Box size L (km)

Figure 3.9 Functional boxcounting. (a) Sand Hills (M); (b) Sand Hills

Where the overbar represents a spatiaI average over the domain under consideration. The

principd axis of a topography is the axis where the slope correlation (M) vanishes and it is

oriented at an angle 8 to the x-axis, where 8 is given by 8 = ?4 arctan(M;/L). This gives the

direction where topographic variation as measured by the mean squared gradient is largest,

and the direction for minimum variation is perpendicular to this. If the x and y coordinates

are rotated to the principal axis with new x' and y' coordinate then the new values of K, L,

and M relative to the axes and denoted by Kt , L', and Mt are given by;

These values are then used to calculate the aspect ratio y, which describes the anisotropy of

a terrain. Anisotropy is a rneasure of the variability upon rotation of axes. It is important to

note that the concept of aspect ratio is not the sarne as the common word aspect, which is the

direction of a slope used in terrain modelling. y is defined by:

Note that in the above definitions the slope is calculated using the centred finite difference

method. The Sand Hills terrain has elevation points ranging fi-om a minimum value of 701

m to a maximum of 1287 m. It shows average anisotropy with y = 0.766. The average

principal angle 8 is found to be about 103"; this means that the principal axis has to be

rotated by 1 O3O anticlockwise fiom the x axis to coincide with the line of maximum variation

in topography. This is evident fiom the shaded relief map, Figure 2.1, which shows that high

undulations are oriented in an approximately north-south direction whereas minimum

variations are oriented in an east-west direction. As can be seen fiom Table 3.4, which is a

list of the topographic parameters defined above for different grid resolutions for region Al,

most of the statistical topographic parameters are dependent on grid resolution, and for K,

L and M there is no indication of convergence as the resolution is increased. Some of the

topographic parameters, especially the maximum slope, increase with decreasing grid size

(increasing resolution). Similar results were obtained by Jenson (1993) in her analysis of

slope of the Chemung River (Finger Lakes, NY, USA) area. Whereas the average and

standard deviation of height converge rapidly as the grid resolution increases. 8 appears to

converge more slowly to a value of 102O. The idea behind this analysis is to see if the

parameters converge to a certain value so that a representative grid size couid be taken in the

simulation of the flow over complex terrain. It would also be good to choose parameters

which are quasi-invariant with grid resolution and physicaily realistic to represent the temain,

but the parameters that seem representative are found to be resolution dependent. It is

Table 3.4 Topographic parameters of the SAND HILLS region (A 1) for various grid resolutions

Topographic Grid 17x1 7 Grid 33x33 Grid 65x65 Grid 129x129 Grid 257x257 Grid 5 13x513 parameter Res. 1600m Res. 800m Res. 400m Res. 200m Res. 100.m Res. 50m

1133.9

23.3

- 2 16E-02

.975E-01

.242E-0 1

.243 E-O 1

.203E-O3

.116E-00

.3 10E-0 1

.3 10E-0 1

.773E-03

-. 185E-03

-. 152E-03

109.680

.773E-O3

.239E-03

0.000E+00

0.727E+00

.3 18E-0 1

The 0 's are standard deviations. S(x) and S(y) are maximum dopes in x and y

52

therefore important to quanti@ this resolution dependence for the parameterkation of

boundary- layer process for studies at various spatial scales with numerical models. The

parameters that we will focus on are the root-mean-square slope and the maximum slope

since the drag coefficient is primarily dependent on the slope (see next chapter). Figures 3.10

and 3.1 1 and Table 3.5 show the fùnctional forrn of the dependence of the parameters on grid

resolution for the three subregions of the Sand Hills and for Fergus. The grid sizes were

normalized by ax,=100 m, close to the available raw data resolution (90m), and the s1opes

were normalized by the corresponding value at %anci plotted on a log-log scale. The curves

are least squares power 1aw best fits for A d x , r 1. Both the RMS and the steepest slope

seerns to approximately obey power law relations as indicated in Table 3.5. If these relations

hold tnie for any topography, self-similarity of the topography would enable the

quantification of the relation between topographie parameters at various spatial resolutions,

Le., the variability at shorter spatial resolutions may be inferred from more available coarse

grid resolution.

Table 3.5 Coefficients of least-square best fit for the normalized m s and maximum dope. The values in brackets are for the north-south cross-sections.

Sand Hills A(l)

Sand Hills A(2)

Sand Hills A(3)

Fergus

1 .259 (1 -30 1) 0.6 19 (0.732)

1.179 (1.296) 0.575 (0.713)

1.245 (1 -032) 0.619 (0.641)

1.164 (1.032) 0.457 (0.405) - -

Average

l

1.212 + 0.04 0.568 * 0.06 (1.219) * 0.09 (0.623) O. 1

t-

1 .O3 (A X/ A >6) 41-41

--- 1.25 (A X/ A XJ -0-ô4 & Y - o.lo 1, O Sandhills Al

+ Çandhills A2 4 -j A SandhillsA3

Fergus 0.01 I I I I I I I I I 1 I t I I I I ~ ~ - T T T T ] - - -

N m l i z e d grid size

Figure 3.10 Nomialized RMÇ dope as a fundon of m l i z e d grid size. (a) for dope along the üTM north; (b) for dope al- the UilVl east A x = 100 m

54

1 0 . 0 0 I 1 l 1 l 1 ' 1 i i 1 I l I I 1 1 1 1 1 1 1 1 1

a 3 . - 8

--- 1.2 ( ~ d A%) -0.96 0 Sandhills Al + SandhillsA2 A Sandhills A3 + Fergus

'Q, . 2 @ 1 t

Sandhills~l 1 + Sandhills A2

Nomialized grid size Figure 3.1 1 Nomialized maximum dope as a function of nomlized grid çize. (a) for dope dong

the üTM north; (b) for dope along the üTM east A xo= 100 m

Note that power law relations were also obtained for K values with slightly higher P 'S.

3.2.1 Resolution dependence of synthetic topographies

One might reasonably ask if the above functional relations are found in any spatially

varying series. To examine this question, we did spatial analysis for idealized topographies.

One with a very smooth surface z = cos(loc)cos(my), having four waves in both direction; one

with completely random and rough surface, Figure 3.12; and the other one a filtered

topography generated fiom the random surface, Figure 3.13. For the random topography case

we set up a square grid of the same size as sample A l with a grid resolution of 100m. Each

grid ce11 was assigned a random nurnber fiom a unifom Gaussian probabiIity distribution.

This generates white noise; that is, there is no spectral roll-off with increasing wave nurnber.

The slope a of the power spectral density in this case is O. Its aspect ratio was cornputed as

y = -992, which is almost içotropic. For the case of the synthetic topography, which

superficially resembles real complex terrain, a two dimensional FFT of the random numbers

was performed first and then the complex Fourier coeffkient of the m d o m numbers were

filtered with a specified spectral roll-off (multiplied by td2 ); where Z is the radial

wavenumber i-e., each complex coefficient was assigned an integer radial wavenumber using

1 = (p +&)Il2 and a is the slope in the mean spectral density. Then the new complex

coefficients were inverse transforrned back to the physical space. When we used k and m

instead of Z in the algorithm of the filter fûnction, the synthetic topographies suffered fiom

directional bias, Le., they look like 2D terrain. The value of a that we used is 3.2, although,

56

this is above the value (a -2.6) that is recornmended by Huang and Turcotte (1989), and

Mandelbrot (1982) for synthetic topography. Values less than 3.2 looked too rough and

rugged, though this could be a subjective viewer bias.

Results of the resolution dependence of the idealized surfaces are given in Figures

3.14 and 3.15, and Table 3.6. As shown, for the sinusoidal (smooth) case, both the maximum

and RMS-dope converge rapidly with increasing resolution and the curves are flat. Note that

re-sarnpling beyond the 1600 m grid size for the sinusoidal surface creates a saw tooth type

surface and was ignored since our centred finite difference method used to calculate the slope

gives us zero RMS slope. For the completely random surface case, both the RMS and

steepest slope increase without any sign of convergence; and both obey the power law

relation. But it is important to note that the values p for the RMS dope differ by a factor of

2 compared to values for our real complex terrain. Another interesting feature of the analysis

that is worth mentioning is the relation of a (the spectral slope) and P. When cil approaches

O, p is r 1, and when p is -0.5, a is 2 - 3 in the nomalized RMS slope. The quasi-invariance

of the spectral slope a with grid resolution mentioned could be useful in deterrnining the

coefficients. Frorn these preliminaty studies we suspect that the constants A and P for any

real complex terrain should be similar to the values given in Table (3.5). Further analysis on

a much wider range of topographic data sets with various re-sampling techniques are

however recornmended. Once the relation of the topographic parameters to grid resolution

is quantified, then it could be useful for parameterizations based on parameters computed

57

fkom coarse grid resolution data, like the GTOP030 dataset.

Table 3.6 Coefficients of least-squares best fit of normalized rms and maximum dope for idealized terrains.

As discussed in the previous section, depending on numerical mode1 grid size and the

variance spectrum of terrain, one could determine a cut-off wavenumber above which the

terrain should be considered for parameterkation. Therefore it would be appropriate to

represent the RMS height and slope in tems of the power spectrum of the terrain. The

moments of the power spectrum are given by:

The values in brackets are for the north-south cross-sections.

Where m, is the jth moment, k is the wavenumber, P(k) is the power spectral density, k, is

the highest wavenumber measured (i.e., Nyquist fkequency), and k, is the cut-off

wavenurnber. Fourier analysis shows that the zeroth moment m, is the variance of terrain

heights, the second moment m, is the variance of slopes, and the fourth moment nq is the

variance of curvatures (Brown and Schoiz 1985). We have seen that the power spectral

density is of the f o m P(k) - ka . Substituting this into the above equation, we can estimate

the RMS height and slope over the wavenumber range of interest fiom:

Maxslope(Axlhx,)= A(Ax/Ax,)-f' A P

1.107 (1.223) 1.275 (1 -357)

1.136(1.189) 0.516(0.532)

Random (white noise) data

Filtered data

RMS(AxIAx,)=A(AxlAxJ-'' A P

1.045 ( 1 .O96) 1.12 (1.14)

i.124(1.089) 0.34(0.31)

RMS height = 6

RMS slope = Jm;

To integrate the above equation requires a to be constant throughout the spectral range or

through a limited wavenumber range. In our study a was not found to be constant in some

cases in the domain of integration, but taking it as a constant could give us some insight into

the scaling property of rms height and slope. For k, >> k , or taking the upper limit

approaching infinity, we can have a scaling property of the form:

RMS height - Ac ("'w , RMS dope - Ac (3.15)

where, AC is the cut-off wavelength of the terrain profile. Note that for the rrns dope to

converge a should be > 3. We discussed in the previous section about the relation of the

spectral slope a with the monofiactal dimension D, where, a=5-2D. Therefore, in ternis of

D the scaling laws yield:

RMS height - Ac (* - , RMS dope - h, (' - ') (3.16)

Since for one dimensional profiles the fiactal dimension is in the range 1<D<2, the rrns

height converges to zero as the cut-off wavelength decreases. Whereas the rrns slope does

not. This is a crude scaling law as the recent studies show that topography is multi-scaling

and multifiactal, but it gives some insight.

Figure 3.12 Gaussian (white noise) random surface

Figure 3.j 3 Synaietic topography generated with a = 3.2

4 + Random (rwgh) . coskxcosr?Ty(smooth)

1 + Filtered

I I i I I i I vi111[1111-

1 .O0 10.00 Nomialized grid size

Figure 3.14 Nomlized RMS dope alwig N-S (a) and E-W (b) cmsçsections for three various idealized surface irreguiarities. = 110 m

1 .O0 10.00 Norrnalized grid size

Figure 3.15 Nomralized maximum dope along NS (a) and E-W (b) cross-secüons for three vanous idealized sutfaœ il~egularities. 100 m

Chapter 4

Results from aircraft data analysis

In this chapter we discuss the methods used to calculate integral statistics and

spectra of the aircraft data and the effects of stability and the underlying terrain on turbulence

parameters. We d s o compare turbulence values obtained fiom field measurements over

homogeneous flat terrain with those of the present study over complex terrain. A brief review

of similarity theory and numerical studies of drag pararneterization will be discussed.

4.1 Review of dimensional analysis and similarity theory of the boundary layer

From the discussion presented by Sorbjan (1989) and Arya (1988), we note that

sirnilarity theory is based on the organization of variables into dirnensionless groups using

dimensional analysis. This is one of the simplest but most powerful methods of investigating

a variety of scientific phenornena and establishing useful relationships between the various

quantities or pararneters, based on their dimension. It was Buckingham (1914), who first

used this method based on the assumption that some physical quantities which are necessary

and sufficient to describe a certain physical phenornenon are dimensionally dependent on

others. One can then f o m dimensionless groups of these variables which are independent

of the system of units chosen. These dimensionless groups or pararneters are of special

significance in any dimensional analysis in which the main objective is to seek functional

relationships between the various dimensionless pararneters; they facilitate comparisons

64

between data obtained by different investigators at different locations and times; and also

reduce the number of parameters that are invoived in a functional relationship by applying

the Buckingham Pi theorem which states that if m physically relevant quantities such as

velocis, density, stress, etc., involving n fundamental independent physical dimensions,

such as length, time, mas, etc., forrn a dimensionally balanced equation, the relationship cm

always be expressed in terms of m - n independent dimensionless groups (n,, n,, ..., n,,,)

made of the onginal m quantities, such that n, = f (n2, 7c3 , ..., nnl-J. A group of variables that

is applied in order to nondimensionalize another particular variable is often called a scale

(Sorbjan 1989; Arya 1988). The pi theorem and dimensional andysis discussed above are

simple mathematical forms that do not deal with the physics of the problem. In order to use

them, one has to know or correctly guess, by using physicd intuition and available

experimental information, the relevant quantities involved in any desired mathematical or

empirical relationship. "Appropriate choice of groups is hoped to allo w empirical

relationships between these groups that are universal, meaning, that work everywhere al1

the time for the situation studied Therefore, to develop a sirnilarity theory, the following

points could be taken as a stepping Stone: (1) choose, intuitively, which variables are

relevant to the situation, (2) organize the variables into dimensionless groups, (3) perform

an experiment, or gather the relevant data fiom previous experiments, to determine the

values of the dimensionless groups, (4)fit an empirical curve or regress an equation to the

data in order to describe the relationship between groups. The ultimate result ofthis four

65

step process is an empirical equation or a set of curves which show the same shape, or the

curves look selfsimilar. Hence, the name sirnilarity theory. qthis ernpirical result is indeed

universal, then we can use if on days and at locations other than those of the experiment

itself," (Stull 1988). One of the first successful applications of sirnilarity and dimensional

analysis was presented by Kolmogorov (1 941) in the field of turbulence theory. Kolmogorov

suggested that for a steady-state turbulent flow, the rate of energy cascade fiom the large

eddies d o m the spectrum must balance the dissipation rate at the smallest eddy sizes. Hence,

there are only three variables relevant to this flow of energy: S, k, e. Where S is the spectral

energy density (kinetic energy per unit wave number), k is the wavenumber, and E is

dissipation rate. By applying the Buckingham Pi theorem, we can make only one

dimensionless group from these three variables: ?r,= S3k5/c2 . Since there are no other Pi

groups for n, to be a function of, it should be a constant. Therefore, soIving for the spectral

energy gives:

Sfl) = % e2'3k5'3 (4*1)

Where ak is known as the Kolmogorov constant. This is in agreement with many

observational data. Another milestone in boundary-layer meteorology is the Monin-Obukov

(M-O) surface-layer sirnilarity theory. They introduced two scaling parameters, essentially

independent of height in the surface Iayer, for velocity and length. These are the friction

velocity, u,, and the Obukov length L,, where their definition is given in the next section.

According to the Monin-Obukov hypothesis, various statistics of the atmospheric parameters,

66

when nomaiized by proper powers of u, and L, are universal functions of z/L, where z is the

height above local ground. The M-O surface layer similarity theory has been tested

extensively in the famous Kansas field experiment (e.g., Businger et al. 197 l), and many

universal curves were obtained. For instance the dimensionless vertical velocity standard

deviation was found to be proportional to (-idL)'I3 in the convective layer. Note that the M-O

sirnilarity theory is based on the assurnption of steady state, horizontally homogeneous mean

flow over flat terrain with constant turbulent fluxes.

4.2 Velocity variances and other turbulence statistics

For the calculations of turbulence statistics, fiom sections of the NCAR Queen Air

data discussed in Chapter 2, the first step was to average the time series of the velocities and

--- temperature to produce the mean wind vector, with components u,, v,, w , and mean

potential temperature 8 . The rectangufar Cartesian coordinate system was then rotated in

such a way that the x - coordinate was aligned dong the direction of the mean wind

(strearnline coordinate). This is done by first horizontally rotating the x - axis to the

projection of the mean wind in the xy plane to get x, and y,. The new velocity components

are then given by,

where

67

Then the new x coordinate is vertically rotated around the new y-axis to the mean wind

direction in order to give zero 'vertical' mean .The fmal velocity components are then given

by9

where

The values of the horizontal (P ) and vertical rotation angles (4 ) for each data block are

listed in Table 4.1. Note that the 4 angles are generally small (< 0.5").

The variables in the timehpatial series are treated as the superposition of the mean

quantity and fluctuations, u, v, w, 0 . Heat fluxes and shear stresses at the flight level were

computed using the eddy correlation method, i.e., calculating the covariances of velocity and

temperature fluctuations. The averaging time used fox the samples ranges fiom 5 - 12 min.

(corresponding to -30 - 72 km with the aircraft average speed of 100 ms-'). This is long

enough to minimize any trend in the tirne series of the wind speed and temperature

associated with large-scale fluctuations. Because of shorter flight paths in the E-W directions

and to avoid curved flight paths we did not take a longer averaging time. The standard

deviations of horizontal and vertical velocity components, and potential temperature,

together with other turbulence parameters are given in Table 4.1. Most of the turbulence

statistics were normalized by appropriate scales to give dimensionless quantities in order to

68

see if similarity laws apply to them. The scaling parameters are defined in the following

pages. Vertical soundings of the morning and afternoon boundary layers were obtained while

the aircraft was in ascending and descending paths. For the remaining analysis, in stable

conditions (L, > O) we selected low level flights below 200 m above the terrain, which is

approximately the height of the stable boundary layer (roughly the height where the

nocturnal jet is located). Above the 200m level we found positive heat and momentum

fluxes, which are not usually associated with the stable boundary-layer. For the convective

case, few samples had very low level flights; most were above 200m as c m be seen in Table

4.1

The profiles for the moming soundings of potential temperature and wind speed and

direction are shown in Figure (4.1) for heights (z) up to 761 m above ground level, the full

scale of the radio altimeter. The wind speed profile exhibits a low level maximum near 200m

above ground, which is approxirnately the height of the boundary layer. The wind direction

increases with height (veering) and may be associated with wann air advection. The potential

temperature increases with height throughout the boundary layer, and has negative curvature.

According to André et al. (1 982), negative curvatures are most likely associated with cases

where the bulk of the inversion layer is dominated by clear-air radiative cooling. Profiles of

the afternoon soundings are s h o w in Figure 4.2. Potential temperature decreases with height

in the layer very close to the surface (Fig. 4.2a), then there is a deep mixed layer (ML) where

the potential temperature is nearly constant with height. The top of the ML is evident by the

TABLE 4.1 List of observations of turbuience parameters, Sand Hilis (Ahraft Data)

S.

Fiight Time Z u. L, ou Q, 0, 0 e P 4) k g period (m) (msw1) (m) (m-') (ms'l) rK) @cg) @cg)

MN] " 133 0.34 63 1.53 0.71 0.38 0.64 -17.8 0.15

MN2 133 0.33 76 1.44 0.75 0.42 0.55 -20.1 -0.26

MN3 " 123 0.25 127 0.93 0.68 0.34 0.24 -18.9 -0.19

MN4 6" 150 0.06 -7.6 1.52 0.31 0.22 0.27 -21.8 0.20

MS1 " 129 0.17 -28 1.90 0.72 0.39 0.46 -18.9 0.08

MS2* 6:46:00 104 0.39 248 1.81 0.99 0.64 0.54 -15.4 -0.47 658:ûû

ml* 6:58:00 91 0.45 382 2.04 0.97 0.67 0.47 -19.6 -0.31 7:lO:W

ME1 159 0.29 345 1.19 0.92 0.56 0.56 -17.5 0.32

M W 2 321 0.16 -27 1.58 1 .O5 0.33 0.62 -31.4 -0.44

AN] 11:14:41 121 1122:oo

0.70 -169 1.34 1.60 1.12 0.32 -20.6 0.53

A N 2 317 0.73 -274 1.63 1.42 1.22 0.20 -181 -0.14

AS1 I[:ziW 417 0.76 -279 1.49 1.24 1.42 0.32 -17.3 0.23

AWl* "'46:00 377 0.74 -406 1.73 1.54 1.36 0.26 -25.5 -0.08 11:58:00

AW2* 337 0.98 -389 1.93 1.99 1.56 0.32 -31.9 0.52 1205: 14

AEI f$yiM 191 0.96 -378 1.71 1.48 1-37 0.24 -25.7 0.09

AE2 12:'0:00 296 0.49 -89 2.24 1.52 1.34 0.39 -20.8 -0.56 1222:oo

AW3 f,iW 290 0.72 -175 1.90 1.85 1.59 0.44 -20.9 -0.22

AS2 357 0.94 -978 1.57 1.58 1.38 0.38 -25.6 0.26

Jott: Timc is tocal Standard Time (LST) . ,

MN1 = Moming. No& block 1 AN 1 = Afkrnoon. Nonh. block 1 . etc. L, = Obukov h g t h . u. is iocd fiction vclocity (sec ncxr section for thcir dcfinirions). and a's arc siandard deviations. Z is hcighi abovc local tcrrain. The *'s arc the curved flight paihs

260 280 300 320 wind direction

296 300 304 308

0 (OK) Figure 4.1 Profiles of the morning sounding (6:OO - 6: lO)

240.00 280.00 320.0 wind direction

Figure 4.2 Profiles of FLT#2 (laie moming and afternoon) sounding

strong temperature inversion, and its height increases with time as seen fiom Figures (4.2 a

and b ) because of suface heating and strong mixing. Note that the vertical axis in Figure

4.2a,b is height above the Iocai ground level and extends above the 76Im, the limiting scale

of the radio altimeter, with the intent to estimate the boundary-layer height. This is done

using measurements of the pressure heights and the latitude and longitude location of the

aircraft. The position of the aircraft was converted into UTM and corrected according to the

offset values obtained in Chapter 2. Terrain heights derived fiom the DEM were interpolated

and subtracted fiom the pressure heights to get heights above local terrain. Wind speed and

direction profiles show a lot of scatter compared to the stable profiles; however on average

they seem to be constant with height in the rnixed layer.

For our turbulence analysis, we used 18 data sets of various averaging times (5 - 12

min.). Of these, 12 cases were in unstable and 6 cases were in stable conditions. The

dimensionless groups of turbulence statistics that are formed were norrnalized by the

folIowing scaling pararnetexs:

Here u, is the velocity scale known as the local fiction velocity, L, is a length scaie known

as the Obukov length; it could be interpreted as the length above which buoyancy dominates

over shear production of turbulent kinetic energy; 0. is a characteristic temperature scale, 8

is the mean potential temperature, g is the gravitational constant, and K is the von Karman

constant, usually taken as 0.4. It is important to note that unlike the usual surface similarity

scales, which make use of surface values of heat flux and shear stress, the above scaling

parameters are ai1 calculated, of necessity, at the flight height. This type of scaling is often

called local sirnilarity scaling. Nieuwstadt (1984) applied local scaling for the stable

nocturnal boundary layer and found that the nondimensionai quantities that are forrned

approach a constant value as stability increases. It is proposed that for strongly stable

conditions, vertical fluctuations are suppressed; turbulence quantities are decoupled fiom

surface forcing and become independent of height.

Using the above scaiing parameters, for each run, the standard deviations of the

velocity components were normalized by local fiiction velocity and are plotted in Figures

4.3, 4.4 on a log-log scale as a function of normalized height dL,, which is the stability

parameter. As can be seen in Figure 4.3c, for the unstable case, the standard deviation of the

vertical velocity obeys similarity laws for 0.3 s -z/L,< 5 (10 unstable data points) and the

least squares best fit is,

O, l u. = 1 .~~(-z/L,)O.~~~ (4.4)

For the lower part of the convective layer over homogenous terrain Wyngard et al. (1 971)

found, for large values of (-z/LJ the following fiinctional relation:

- - - - - - - 7

5

4 - 3 -

2 - dl 3

9= 8 - 7 - 6 - 5

O 1 10 -z/L,

Figure 4.3 Normalized Standard deviation of velocity (unstable) as a function of fi,

I I I 1 I I I I I 1 I 1 1 1 1 1

c w = 1.75 (-z/b - --- 1.95 (-zlL)'m /

LL

& &.

(CI I I 1 1 1 1 1 1 I 1 I I I I l l

2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9

Figure 4.4 Normalized Standard dlviation of velocity (stable) as a function of z/L,

a, / u. = 1.95(-z/L)'" (4.5)

where u. is the fiction velocity and L is the Obukov length for the surface layer. Eq. (4.5)

slightly overestimates our data but some points agree well as seen in Figure 4% The

normalized standard deviation of the horizontal velocity components a, /u. and a, lu. shown

in Figure 4.3a,b exhibit some scatter, but in general they tend to increase with increasing

instability. In general the u and v components seem not to clearly obey surface layer

sirnilarity laws (Lumley and Panofsky 1964); however, Panofsky et al (1 977) suggested that

where zi is the height of the capping inversion. Since we didn't have zi values for al1 the

flight legs, and it is expected to vary, we were not able to effectively test the scaling

mentioned above.

For the stable boundary layer, the ratios of the standard deviations of the velocity

components are shown in Figure 4.4. For the relatively few data points that we have, most

of the values are confined within a limited range, and on average:

uJu.= 4.3, a,/u,= 2.5, and aJ u , = 1.5. (4.7)

Sorbjan (1987), for the stable case, estimated the foilowing values from BA0 (Boulder

Atmospheric Observatory) tower data, 25 km east of the foothills of Rocky Mountains for

measurement heights ranging from 10 m - 300 m:

oJus=3.1 , a,,/u.= 1.6 (4.8)

Observations at Cabauw, a flat and homogenous terrain in The Netherlands fiom

measurements between 20 - 200 m, fkom Nieuwstadt's (1984) paper for the stable case

indicate:

a J U+ z 2.6, a,J u+ = 1.5 (4.9)

Comparing our stable resuIts against the above values it appears that, a, Ju. seems to be

unaffected by changes in terrain conditions. A list of estimates of standard deviation ratios

fi-om observation at various locations characterized by uniform terrain in neutral conditions

is given in Table 4.2, taken fiom Panofsky and Dutton (1984), for cornparison. The same

insensitivity of a Ju, to terrain effects is observed. One possible reason for this is that

vertical velocity fluctuations are dominated by small eddies, with diameters of the order of

the height above ground. These small eddies rapidly adjust to terrain changes. If they are in

local equilibriurn with the terrain, then the ratio of variance to shear stress should remain

almost constant. However the horizontal fluctuations have significant contributions fiom

large quasi- horizontal eddies and they adjust to the terrain very slowly (Panofsky and Dutton

1984). From our analysis, we c m see that the ratio of the standard deviation of the horizontal

velocity component to local friction velocity in the stable case are increased by up to - 20 -

40% compared to uniform flat terrain (although we also have to consider stable versus

neutral stratification effects for the v component).

Table 4.2 Ratios of standard deviations of velocity components to friction velocity fiom Panofsky and Dutton (1 984)

Uppsak swcdtn Erie, CO Rock Springs. PA

Nonmotratain Mountain

Rat Terruin 2.42 1.73 - 14s 1.90 1.25 2.20 1.90 1.40 230 1-90 1-35 2-50 2.20 1.20 2.50 1.85 1.22 239 1.79 1.26 2.30 210 1.10 2.43 1.93 1-20

239 * 0.03 1.92 2 0.05 1-25 2 0.03

-

'12-17 min avcragc no vtnd mnovaL ' b u g b e s s kngths 0.7-1.7 rn

79

The shear stresses and heat f lues together with other turbulence parameters are given

in Table 4.3. The net shear stresszfor al1 the flight legs is negative, as expected, indicating

that there is downward flux of momenturn (mornentum loss to the ground) and this loss is

enhanced during the afternoon unstable cases. Unlike G, the sign of changes irregularly

and in rnost cases its value is about 10% o f E with its contribution to u, increasing in some

of the aftemoon flight legs. The net vertical heat f lw a for the morning flight leg is

negative (downward) except on two occasions, which could indicate the intermittency of

- - turbulence. Cross - correlation coeficients of uw , w 0, and ;8 were also calculated and

compared to values over homogeneous flat terrain.

- - ruw = uw /aUuw, rd = ~ û / a , u ~ , (4.1 O)

- and rd = uû huae

The a's are the standard deviations. The correlation coefficients plotted as a function of

stability parameter are shown in Figure 4.5. They do not seem to obey clear similarity

relations, but on average, ru,= -0.23, r,, = 0.27 for unstable cases compared to -0.35,0.5

respectively given in Kaimal and Finnigan (1994); r, = 0.8 for stable cases, and it is a bit

higher than the flat surface value which is 0.6 (Kaimal and Finnigan 1994). The decrease in

ru, value is most likely associated with the increase in the standard deviation of the u velocity

component over the uneven surface.

80

Another important parameter worth mentioning is the turbulent kinetic energy per unit mass

--- TKE defined as E =(u +v +w 2)/2 . As seen in Table 4.3, the value of TKE increases in the

unstable boundary layer. It would give us insight to the generation and dissipation of TKE

in the boundary layer if we express its simplified (ID) budget equation for uniform terrain

(see Stull 1988 for mathematical derivation), which reads as:

In Eq. (4.11) the left-hand side is the storage or tendency of TKE, the first two terms on the

right-hand side represent shear production (mechanical turbulence), the third terrn represents

buoyancy production or los , the fourth turbulent transport, and the fifth viscous dissipation

of turbulent energy. In the stable boundary-layer, we see h m Figure 4.6a that there is an

almost linear relationship between turbulent kinetic energy, TKE and & with the ratio

TKE/-== 14. This relatively high value is again probably the result of high standard

deviations of the horizontal velocity components, compared to an expected value of 4 in the

surface layer over flat terrain, but height within the boundary layer will also be a factor. In

Figure 4.6b TKE does not show a clear dependence with a.

s, Sand Hills (Ail r

TABLE 4.3 List of observations of turbulence parametc iraft Data)

7

MWI* 1 6:58:00 1 -0.018 1 0.816 1 O. 124 7:lO:OO

Flight 1%

11:14:41 ANI 1 1 0 . 1 5 8 I -0.036 1 0.079 1 1 :22:00

ut3

(ms" "K)

V B

(ms" "K)

Tirne period

W B

(ms-' K)

Figure 4.5 Correlation coefficients vs stabihy parameters. Lables are flight legs

0.00 0.02 0.04 0.06

-we (rns-' O K)

Figure 4. 6 Relation of TKE with (a) shear stress and (b) heat flux (stable)

84

4.3 Drag coefficient and estimation of roughness length

As mentioned briefly in Chapter 1, sub-grid scale terrain variation can exert

significant additional drag on the atmospheric boundary-layer flow and as a result can have

a significant influence on large-scale atmospheric flows. Therefore, its effect should

somehow be represented in the regional or large-scale meteorological models. Taylor et. al

(1989) discussed the representation of the effect of topography (simple 2D sinusoidal

terrain) using data fiom non-linear models of neutrally stratified flows and suggested that

this can be achieved by adjustrnent to the nondimensional parameter called drag coefficient.

Xu and Taylor (1995) modelled the dependence of the drag coeff~cient with the normaiized

topographic length scale of idealized terrain for various turbulence closures, and showed that

it increases monotonically for E-kz closure while for higher-order closures, it increases first

and then decreases gradually. The form drag (the horizontal force acting on the surface of the

topography) exhibited an increase with the maximum slope of the temain. Zhou et. al (1 997)

using their simplified linear PBL mode1 studied topographic drag in stably stratified

boundary-flows for cases with Froude number, FL =UWN > 1, and F, < 1, where k is the

wavenurnber and N is buoyancy fiequency, and fowd that topographic drag is limited to the

lowest layers in the case of FI- > 1, hence increased surface drag would suffice to parametrize

the effect of topography for large scale models. For the FL < 1 case, there exists a net drag

on the boundary layer as well as vertically propagating gravity waves.

In our study, we focus on ernpirical observational results of the drag coefficient CD,

which is defined here as:

Where in our case v = O, since we have rotated the x-coordinate to the mean wind direction.

The above definition incorporates the effects of both (skin) friction drag and the form drag

introduced by flow perturbations around uneven topography. In the literature, CD(lO), the

drag coefficient base on wind speed at 10 m or the geostrophic drag coefficient based on

surface shear stress, and geostrophic wind are usually used instead. In our case we only have

- the aircrafl data to work with so CD = u.* / u 2 , where both the parameters are measured

at the flight level (- 100 m). The dependence of C,IFLv with stability is shown in Figure

(4.7a). C,(10) for neutral case ranges fi-om 1 to 2 xIO-~ (Garratt 1977). For the range of

stability given, CD (';L~ decreases sharply between unstable and stable cases, though there is

a lot of scatter; the scatter is most Iikely from the various roughness elements that contribute

different shear stresses along the flight path. The labels are the flight legs. We have also

compared CDfFLg with the slope of the underlying terrain, the root mean square (RMS) slope

fiom the aircrafl determined terrain heights along the flight path, and this is shown in Figure

4.7b. It suggests a potential for an increase of the drag coefficient with the topographie slope.

Note that the stability is also included, and it can be seen that most of the high CfLV values

have fiernoon labels as well as relatively high rms slopes. This in fact precludes us fiom

0.06 0.08 0.1 O O. 12 RMS slope

Figure 4.7 Drag coefficient vs Stability parameter and RMS slope

-

-

4E-3 I

-2 O 2 4 z/L

I I

-

3E-3 -

-

2E-3 - -

1 E-3 - -

OE+O

-

-

-

4E-3

*a (a)

Pz2 AN%W2 AS1

A W j a AWI

e MW1

MS2

9 MN1 ME1 MN3

I O

I

I I

l

I

-

2E-3 -

-

OE+O

- 1 1 3

(b) .AN1 O A E ~

m A ~ ~ ~ 2 ~ A W Z .AS1

A m

AWl

0 MW1 9 A E ~

MS2

a M N 1 ~ M N J ~ M N ~ Q M E I

~ M S I ~~ m MN^

I

87

giving a definitive statement on the dependence of drag coefficient on the RMS slope.

However, separating stable and unstable regions still suggest a potential for increased CDm7)

with rms slope. Although the flight trajectories in both the moming and afternoon were

almost the same, the runs (flight legs) were not. The averaging time for most of the

afternoon runs were higher and they have relatively higher RMS slope values.

We also estimated the aerodynamic roughness length Z,, defined as the height at

which U(z) vanishes. The concept of roughness length cornes from the similarity hypothesis

in the neutrally stratified, homogenous surface layer (see APPENDIX B for mathematical

formulation). Based on surface-layer similarity relations we estimated the roughness length

2, using the expression,

z, = z e W du8'-$ (fi)) m (4.1 3)

Values of 2, for various flight legs are given in Table 4.3. Note that Eq. (4.1 3) is sensitive

to errors, for instance, assuming that u. is the only parameter subject to error, for values of

U and u, , say 10mIs and 0 . 5 d s respectively, a 1% relative error in u. gives us about 20%

error in 2,. Many of the values given in Table 4.3 are unrealistically low, probably because

the above expressions are based on velocity and length scales calculated from surface values

of heat and momentum fluxes. Fielder and Panofsky (1972) suggested a method of

estimating the effective roughness length (roughness length which hornogenous terrain

would have to give the correct surface stress over a given area) using low level aircraft data

which is appropriate for near-neutral conditions. For flight leg MW1, which is somewhat

close to near-neutral, the value of 2, estimated from the formula {In(dzJ = 0.48

(aw/;+0.004) 1

given in Fielder and Panofsky (1972) is higher (the value in bracket in Table 4.3).

4.4 Velocity Spectra

In this sub-section, the behaviour of turbulence spectra in both stable and unstable

atmospheric conditions, the effect of the underlying terrain on the spectral shape, and the

Monin-Obukov similarity of the velocity spectra will be examined. As mentioned in the

review part of this chapter, it was Kolmogorov in the 1940's who gave a comprehensive

mode1 of turbulence, based on the energy cascade process in the atmosphere and

dimensional analysis.

For the analysis of velocity spectra, first the tirne series of the three velocity

components are examined and any linear trends, whkh are presurnably associated with

large-scale fluctuations, are removed. From the 18 flight legs, 9 were found to have trends

of varying degree, especially in the u and v components. As in Chapter 3 for the terrain

spectra, each time series of the velocity cornponents was multiplied by a window function

to reduce side lobe variances. Since the averaging time was not constant and the number of

data points were not always powers of 2, zero padding was also necessary in order to

implement the FFT routine. This does not pose a serious problem as long as we remove the

mean and the trend, but the estimated spectral density is reduced by a factor of (MN); where

Mis the number of original data points and N is the total number of data points including the

zeroes ( K m a l and Finnigan 1994). Since a one-minute record has 1200 data points, M for

the flight legs ranges fiom 6024 - 14400.

Figures 4.8 to 4.16 show the spectra of the three velocity components plotted as kS(2)

vs k, on a log-log scale, where Sfi) is the spectral energy density and k is the wavenumber

dong the flight path; Hfi) has a units of variance, but the area under the curve doesn't give

us the total variance of the data because of the logarithmic scale used for &?fi)). Because of

discrete sampling, the raw power spectra Vary significantly and for graphical purposes we

used some smoothing techniques. For the first 16 data points, low wavenumber region, we

applied the Hanning filter, a triangular moving average, of the form, Si = S,,/4 + S,/2 + Si+,

/4, since the variability of the spectra is relatively small; and the rest, the highwavenumber

region with relatively high variability by averaging every consecutive 32 data points.

Figure (4.8a) shows the spectral plot for the first flight leg in the morning MN1 at

133 m above the local terrain. Except for a narrow band in the high wavenumber region, the

u spectrum is larger than the v and w spectra in most of the spectral domain. The spectra

exhibit a bi-modal shape separated by a spectrai gap with distinct peaks at about 0.1 9 and 2.5

cycle/km for u and v, and 0.3 and 8 cycle/km for w; also the line with dope -213 is plotted

to show how the spectra rolls-off with wavenumber in the inertial sub-range. This -213 power

law behaviour is quite consistent with theoretical and observed values for unifonn flat

terrain, ( see Kaimal et al. 1972, and Kaimal and Finnigan 1994) except in the dissipation

range, in the very high wavenumber region, where the spectra rolls-off rapidly like a

moustache. A similar spectral shape in the very high wavenumber region was observed by

90

Bradley (1980) for Black Mountain, Canberra. Panofsky and Dutton (1984) argue that

Bradley's spectnun is caused by insufficient instrument response. According to

Kolmogorov, since the smaller eddies are the result of the successive breakdown of larger

eddies, they will be far removed fiom the initial cascade process and will ultimately be

unaffected by the orientation of the mean wind shear and buoyancy which produce the

turbulence. Hence the smaller scales will be isotropic, Le., S, = Sv = (4/3)S, 'assurning the

direction of sarnpling is along the u component, (see Panofsky and Dutton 1984 for

mathematical derivation of the 4/3 factor required for isotropy).

The ratio S,+/S,, is plotted on the right-hand side of Figures (4.8 to 4.16) to see if the

local isotropy conditions are satisfied in the inertial range; the dashed line is the 4/3 and the

solid line is the 1 value. Though there is scatter in the data, a 'quasi- isotropy' condition

seems to hold above about h l 0 cycle/km. As the boundary layer evolves ( see consecutive

figures), the liniiting value of k above which 'quasi-isotropy' occurs decreases and reaches

about 0.2 cyc le /h at 12:30 LST. Note that since we had a north-south sampling direction,

we should expect S , = Su = (4/3)S, in those cases, and it is so especially for ANI, AS 1, and

AS2. A feature worth mentioning is the spectral gap which is clearly evident in the first three

flight legs, and it is most likely associated with the stability of the atmosphere and the

undulation of the underlying terrain. The spectral gap however gradually disappears with

time of day and is replaced in this wavenumber range by a peak and at times by the inertial

sub-range. We will discuss the influence of the terrain on the velocity spectra later. Note that

0.01 0.10 1.00 10.00 100.00 0.01 0.10 1.00 10.00 100.00

k (cyclelkm) k (cyclelkm)

0.01 0.10 1.00 10.00 100.00 0.01 O. 1 O 1.00 10.00 100.00

k (cyclefkm) k (cyclelkm)

Figure 4.8 wavenumber-weighted u,v, and w spectra for MN1 and MN2

0.01 0.10 1.00 10.00 700.00 0.01 0.10 1.00 10.00 100.00 k (cyclelkm) k (cycle/km)

0.01 0.1 O 1.00 10.00 100.00 o. 01 o. 1 O 1.00 10.00 100.00 k (cyclelkm) k (cycle/krn)

Figure 4.9 wavenumber-weighted u,v, and w spectra for MN3 and MN4

0.01 0.10 1.00 10.00 100.00 0.01 0.10 1.00 10.00 100.00 k (cyclelkm) k (cyclelkm)

0.01 0.10 1.00 10.00 100.00 0.01 0.10 1.00 10.00 100.00 k (cyclelkrn) k (cyclelkm)

Figure 4. 10 wavenumber-weighted u,v, and w spectra for MS1 and MS2'

0.01 0.10 1.00 10.00 100.00 0.01 0.10 1.00 10.00 100.00 k (cyclelkm) k (cyclelkm)

0.01 0.1 0 1.00 10.00 ~00.00 0.01 O. 1 O 1.00 10.00 ~00.00

k (cyclelkm) k (cyclelkm) Figure 4. 1 1 wavenumber-weighted u,v, and w spectra for MW1 and ME1

0.01 0.10 1.00 10.00 100.00 0.01 0.10 1.00 10.00 100.00 k (cyclelkm) k (cyclelkrn)

0.01 0.10 1.00 10.00 100.00 0.01 0.10 1.00 10.00 100.00 k (cyclelkm) k (cydelkm)

Figure 4.12 wavenumber-weighted u,v, and w spectra for MW2 and ANI

0.01 O. 1 O 1.00 10.00 100.00 0.01 O. 10 1.00 10.00 100.00 k (cyclelkm) k (cyclelkm)

0.01 0.10 1.00 10.00 100.00 0.01 0.10 1.00 10.00 100.00

k (cyclelkrn) k (cyclelkm) Figure 4.1 3 wavenumber-weighted u,v, and w spectra for AN2 and AS1

0.01 0.10 1.00 10.00 100.00 O. 01 0.10 1.00 10.00 100.00 k (cyclekm) k (cyclelkm)

0.01 0.10 1.00 10.00 100.00 0.01 0.10 1.00 10.00 100.00 k (cyclekm) k (cycle/km)

Figure 4.14 wavenumber-weighted u,v, and w spectra for AWI* and Aw2*

0.01 0.10 1.00 10.00 100.00 0.01 O. 1 O 1.00 10.00 100.00 k (cyclelkm) k (cycte/km)

0.01 0.10 1.00 10.00 100.00 0.01 O. 1 O 1.00 10.00 100.00 k (cyclelkm) k (cycleJkrn)

Figure 4. 15 wavenumber-weighted u,v, and w spectra for AEI and AE2

0.01 0.10 1.00 10.00 100.00 0.01 0.10 1.00 10.00 100.00 k (cydelkm) k (cyclelkm)

0.01 0.10 1.00 10.00 100.00 0.01 0.10 1.00 10.00 100.00 k (cyclelkm) k (cyclelkm)

Figure 4.16 wavenumber-weighted u,v, and w spectra for AW3 and AS2

1 O0

the graphs for MW1, MS2, AWI, and AW2, which are curved flight paths, could be

misleading since the wavenumber is not calculated based on a linear distance. Another

feature to note is the relatively narrow spectra of the w component of the cross-wind (north-

south) flight legs compared to slightly broader spectra of the along wind (east-west) legs in

the afternoon, convective flight legs, AN 1 & AE2 , AW3 & AS2. A somewhat similar change

in shape was also observed in Kaimal et al (1982) for the convective boundary layer

experiment conducted over a gently rolling terrain near the Boulder Atmospheric

Observatory site at flight heights of 150 m. They suggest that the along-wind sampled

spectrum is viewed as the distorted counterpart of the crosswind sarnpled spectrum because

of the redistribution of energy resulting fiom the stretching of the longitudinal eddies by

mean shear. Since we had different flight heights and terrain features for the spectra in

question, we cannot give the same reasoning, but it does suggest that large eddies or coherent

structures (maybe longitudinal vortices) are elongated along the wind direction.

We also tested the Monin-Obukov similarity theory for the u, v and w spectra for

selected flight legs. The parameters involved in the absence of any topographie effects would

be the spectral energy density Sfi), the wavenumber k, the height z, the local friction

velocity u., and the Obukov length L, Hence we can have three non-dimensional parameters

related by an equation of the form:

kSfi)/ um2 = F&, ziL2 =F(n, c) . (4.14)

Here P is expected to be a universal function and to be determinecl empiricdly. For fixed

101

values of the stability parameter 5: , an empirical expression for flat uniform terrain is:

F = A n/(B +Cn)-SI3 , (4.15)

where A, B, and C are adjustable constants, (see Panofsky and Dutton (1984), and Kaimal

and Finnigan (1 994)). The u, v, and w nomalized, wavenumber-weighted spectra are given

in Figures (4.17 - 4.19 ). The solid curve is the function F for neutral cases over flat temain

given in Kaimal and Finnigan (1994). Note that in addition to u . ~ , au3 is used in the

denorninator; where ee is a dimensionless dissipation defmed as 4 ,= m / u 2. 4 ,is a function

of the stability parameter C= z/L whose functional forms, determined kom the Kansas

experiment by Businger et al. (1 97 l), are given by:

Since in the inertial subrange, after normalizing by the Kolmogorov hypothesis, (multiplying

Equation 4.1 by Wu? ), there is a factor of 4eu3 on the right hand side. In principle, after the

ordinate is divided by 4eU3 al1 the spectra will collapse to a single line with slope -213. Our

data could not support this fiilly; possibly due to inappropriate scaling parameters. The w

spectra, Figure 4.19, are much better than u and v to veriQ the surface Iayet similarity for

spectra. AIthough there is no evident systematic variation of the spectra with instability,

careful inspection reveals that the spectral peak in the w spectra shifts toward lower n values

as the instability increases. Since we did not have a neutral case, the transition frorn stable

to unstable case cm be taken to match the normalized spectra with the solid curve. As seen

Figure 4.17 Normalized wavenumber-weighted u spectra for various stability cases

Figure 4.1 8 Normalized wavenum ber-weighted v spectra for various stabiiity cases

Figure 4.19 Normalized wavenum ber-weighted w spectra for various stability cases

105

the w spectra almost match the solid curve except for very hi& wavenumbers, which could

be due to instrument response. There is a lot of scatter in the u and v spectra but they

generally suggest that the very low wavenumber spectra are enhanced for u and v.

As we mentioned earlier, to the left of the spectral gap, Figures (4.8 to 4.10), the

energy is most likely associated with the influence of the underlying terrain. Since the w

component has less energy than u or v at the first peak value, the graphs suggest that

fluctuations associated with flow distortion in the horizontal component of the wind by the

terrain are the most significant. To see the effect of the terrain, we plotted, for selected flight

paths, the wavenumber-weighted power spectra of both the terrain elevation and slope and

compared these to the velocity spectra, on semi-log plots, Figures (4.20 to 4.23). In this case

the area under the curve is proportional to the energy contributions fiom a band-width. The

ordinates are normalized by the respective variances. As shown, the terrain height spectra

generaily have peaks at -O. 1 and -0.3 cycles/km in most of the figures and the power rapidly

falls thereafter, whereas the primary peaks of the slope spectra shift towards higher

wavenumbers. For MN1, the across wind terrain profile, Figure (4.20) both u and v spectra

seem to have peaks close to those of the terrain elevation, near -0.1 cycle/km, and w seems

to have peaks at both -O. 1 and -0.3 cycle/km although the 0.3 cycle/km is stronger, and

corresponds to a low wavenumber maximum in the slope spectrum. The same feature is

clearly seen for MN4 in Figure 4.21. Since MN4 and MN1 have relatively low flight levels,

it is expected the eddies for w will be small (higher wavenumber) and that they will respond

106

to the slope fluctuations. In Figure 4.22, the dong wind profile for flight AEl shows that,

the main peak in w is found near O.Scycle/km and corresponds to that of the terrain spec tm,

while the u and v main peaks are slightly offset and are less pronounced. For ANI, Figure

(4.23), the location of the main peak in the terrain spectrum is somewhat similar with that

of AEl, except there is slightly enhanced peak in the slope spectnun at -0.5 cyclekm and

it conesponds mainly with the v spectral peak; and also the weak low wavenurnber peak at

-0.07 cyclelkm in AE 1 shifts to -O. 1 cycle/km in AN 1. This differences could be attributed

to the anisotropy of the terrain and the difference in flight heights.

To firther assess the effect of the terrain on the velocity spectra, we estimated the

cross-spectral parameters defined for two variables A and B as follows, Stull(1988):

Cospectnim, Co(k) = FApBr + FAiFBi Quadrature spec tm , Q(k) = FAiFBr - FarFBi

Amplitude spectrum, Arn(k) = Q2 + Co2 Coherence spectnun, Coh(k) = A m / S , S ,

Phase spectnim, @(k) = arctan(Q/Co)

Where F and S are the Fourier coefficients and amplitudes of the variables and the subscripts

r and i are the real and imaginary parts respectively. The Cospectnim is a spectral covariance

because the surn over al1 cospectral amplitudes is equal to the covariance between the

variables A and B. The Quadrature spectnim is that part of the cross-spectrum that is out of

phase (i.e. one variable leads or lags the other by d 2 rad). A peak in the amplitude spectrum

,4 terrain height

terrain dope

0.80

0.40

0.00

0.01 0.10 1 .O0 10.00 100.00 wavenumber k (cyclelkm)

% u - velocity

+ v - velocity

w-velocity

I " " " " 0.01 0.10 1 .O0 10.00 100.00

wavenumber k (cyclekm) Figure 4.20 Nonalized wavenumber-weighted spectra of terrain and velocity for flight leg MN1 (6:04:59 --- 6:lO:OO)

terrain height

-@- terrain slope

0.80 (V

4? n

V 2

0.40

0.00

0.01 0.1 O 1 .O0 10.00 100.00 wavenumber k (cyclekm)

LI - velocity

.4 v - velocity

w - velocity

0.01 0.1 O 1 .O0 10.00 100.00 wavenumber k (cyclekm)

Figure 4.21 Normalized wavenumber-weighted spectra of terrain and velocity for flight leg MN4 (6:22:00 --- 6:29:36)

4~ terrain height

terrain siope

CV

9 h

Y 0.40

SX

0.00

0.01 O. 10 1 .O0 10.00 100.00 wavenumber k (cyclekm)

+. u - velocity

0.40 +, v - velocity CV

4 w - velocity h

Y 9

0.20

0.00 I I I I

0.01 0.1 O 1 .O0 10.00 100.00 wavenumber k (cyclelkm)

Figure 4.22 Normalized wavenumber-weighted spectra of terrain and velocity for flight leg AE I (12:05:15 --- 121 0:OO)

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 l

.+ terrain height

IIK +~ terrain dope

0.01 O. 10 1 .O0 10.00 100.00 wavenumber k (cycle/km)

u - velocity

,+, v - velocity C

0.01 o. 10 1 .O0 10.00 100.00 wavenumber k (cycle/km)

Figure 4.23 Normalized wavenumber-weighted spectra of terrain and velocity for flight leg AN1 (1 l : i4:4 l -- i 1:22:00)

111

indicates a strong correlation of the variables regardless of their phase differences. Results

of a cross-spectral analysis for velocity components and terrain parameters for two flight

legs, MN1 and AN1 are given in Figures 4.24 through 4.30. For MN1, Figures 4.24 to 4.26,

the amplitude spectra suggest that the velocity components are strongly correlated with the

terrain height around 0.1 cyclelkm and -0.4 c y c l e h . The 0.1 cycle/km being relatively

strong for the u and v component and the 0.4 cyclelkm for w component. For velocity and

terrain slope, the main peak in the amplitude spectnun is around 0.4 cycle/km for al1 the

three components. The coherence also shows similar results. The net cospectrum (the sum

over al1 the wavenumber range) for the u component with the terrain height is negative

whereas for v and w it is positive. This is confirmed with the covariance results given in

Table 4.4. Note that apart from a few negative values, the covariance of w with the terrain

height is positive in most of the fiight legs. For u and v the sign of the covariance changes

randomly, which could suggest that influence fkom other foot pnnts plays a role. The phase

and quadrature spectrum do not help much in the interpretation, but the near zero and +180

phase values correspond to a positive CO-spectrum and strong correlation. Since for smaller

CO-spectral amplitudes the phase fluctuates rapidly, and for clarity, the high wavenumber

regions were not plotted. For ANI, Figure 4.27 to 4.29, the location of the peaks in the

amplitude are somewhat similar to that of MN1 but the value of the amplitude in u is reduced

by a factor of 2. It is dificult to interpret each cross-spectral point, but the general

impression is that the wavenumbers -0.1 cyclelkm and - 0.5 c y c l e h in the velocity are

112

most likely induced by the terrain. In the aftemoon eddies become larger, and the spectral

peak shifk toward small wavenurnber, so there could be a chance for the peak in the velocity

to comespond with the peak in the terrain, although this does not necessarily mean that the

influence of terrain is demonstrated in both the aftemoon and morning data.

TABLE 4.4 II List of observatia 1 & velocitv covariance. Sand Hills (Aircraft Data) s of term 11 Fr 1 T h e

period

Z, Terrain height Z,@ Tenrainslope

Terrain height 4E+1 I

0.01 0.10 1 .O0 wavenum ber k (cyclelkrn)

0.01 0.10 1 .O0 wavenum ber (cyclelkm)

Figure 4.24 Cross-Spectra of u-velocity component and Terrain height & dope for flight k g MN1 (6:04:59 - 6: lO: lO)

I Terrain height 1

0.01 0.10 1.00 10.00 100.00 wavenumber k (cyclelkm)

I Terrain dope O I

0.01 0.10 1 .O0 10.00 100.00 wavenumber k (cyclelkm)

-

1 1 1 1 1 1 1 1

2.00 - OE+O -

1.00 -

0.00 - -5E3 -

-1.00 - -2.00 1 -1 E-2

1 E-2 6.00 1 1 1 1 1 1 1 1

ic

, l l l l l 1 l l ~ 1 1 1 1 1 1 1 1 ~ 1 1 1 1 1 1 1 1 ] 1 1 1 1 1 1 1 1

1

1 1 1 1 1 1 1 1

0.01 0.10 1.00 10.00 100.00 0.01 0.10 1 .O0 10.00 100.00 wavenumber K (cyclelkm) wavenumber k (cyclelkm)

Figure 4.24 ( Cont ....)

1 1 1 1 1 1 1 1 I l 1 1 1 1 1 1 1 l l l l u - I II~IIII

- 5.00 -

1 1 1 1 1 1 1 1

4.00 - - 5E-3 - 3.00 -

I Terrain heigit

0.01 0.10 1.00 wavenumber k (cyclelkm)

0.01 o. 10 1 .O0 wavenum ber (cycle/km)

Figure 4.25 Cross-Spectra of v-velocity component and Terrain height & dope for flight leg MN1 (6:04:59 - 6:10:10)

0.01 0.10 1 .O0 10.00 100.00 wavenumber k (cyclelkrn)

0.01 0.10 1.00 10.00 100.00 wavenumber K (cyclelkrn)

Terrain dope

0.01 0.1 0 1.00 10.00 100.00 wavenumber k (cydelkm)

0.01 0.10 1.00 10.00 100.00 wavenurnber k (cyclelkrn)

Figure 4.25 (Cont ...)

i Terrain hei@

- Y

0.01 0.10 1.00 wavenum ber k (cycle/km)

0.01 0.10 1 .O0 wavenumber (cyclelkm)

Figure 4.26 Cross-Spectra of w-velocity component and Terrain height & slope fir flight leg MN1 (6:04:59 - 6:10:10)

Phase (k) Coherence (k) Amplitude (k)

0.01 0.10 1.00 10.00 100.00 0.09 0.10 1.00 10.00 100.00 wavenumber k (cyclelkrn) wavenumber k (cycle/km)

4E-3

0.01 0.10 1.00 10.00 100.00 0.01 0.10 1-00 10.00 100.00 wavenumber K (cyclelkm) wavenumber k (cyclelkm)

Figure 4.27 ( Cont ... )

1 1 1 1 1 1 1 1 1 lllllil

Terrain dope

1 IlIIIII l lllllll

Terrain hei&t 1

0.01 0.10 1 .O0 wavenum ber k (cycle/km)

Terrain dope T

0.01 o. 1 O 1 .O0 wavenumber (cycle/km)

Figure 4.28 Cross-Spectra of v-velocity component and Terrain height & slope for flight leg ANI (1 1 :14:41 - 11 :22:00)

0.01 0.10 1.00 10.00 100.00 0.01 0.10 1.00 10.00 100.00 wavenum ber k (cyclelkm) wavenumber k (cyclelkm)

0.01 0.10 1 .O0 10.00 100.00 wavenumber K (cyclekm)

-

0.01 0.10 1.00 10.00 100.00 wavenurnber k (cyclelkm)

I 1 1 1 1 1 1 1 4.00

Figure 4.28 (Cont ...)

1 E-2 I 1 1 1 1 1 1 1

C

-

0.00 -

-1.33 -

7 OE+O -

- -5E-3 -

I 1 1 1 1 1 1 1

Terraindope 2.67 -

1.33 -

' 1 1 1 ' 1 1 '

Terrainhei*t 7 - O SE-3 -

4.00

Terrain height

wavenumber k (cyclelkm)

0.01 0.10 1.00 10.00 10.0.00 wavenumber K (cycle/km)

wavenumber k (cyctelkm)

0.01 0.10 1 .O0 10.00 100.00 wavenumber k (cyclelkm)

Figure 4.28 (Cont ...)

0.01 0.1 O 1 .O0 wavenumber k (cycielkm)

Figure 4.29 Cross-Spectra of w-velocity component and for flight k g ANI (1 1 :14:41 - 11 :22:00)

0.01 O. 10 1 .O0 wavenum ber (cyclelkm)

Terrain height & slope

Terrain hei&t

wavenumber k (cyclelkrn)

0.01 0.10 1 .O0 10.00 100.00 wavenumber K (cyclelkm)

0.01 0.10 1.00 10.00 100.00 wavenumber k (cyclelkm)

0.01 0.10 1.00 10.00 100.00 wavenumber k (cyclelkm)

Figure 4.29 (Cont ...)

Chapter 5

Conclusions

As contributions to the study of boundary-layer flow over topography and its

pararneterization for larger scale models, we first analysed high resolution (-90 m) digital

terxain data (1 deg by 1 deg) for Sand Hills, Nebraska and Fergus, Ontario. The Sand Hills

region was the location in which NCARYs Queen Air collected low level flight data on

August 20, 1980 with a sarnpling rate of 20 Hz and average aircrafl speed of 100 d s . We

have analysed those data to M e r characterize and assess the spatial variability of terrain

and especially to estimate the integral statistics of atmospheric variables such as the second

order moments (heat f lues, turbulent stresses), drag coefficient and atmospheric velocity

spectra. We then compared our results for the Sand Hills flight with the turbulence values

estimated fiom published measurements over flat and homogenous terrain, in order to assess

the impact of topography on boundary-layer flow.

5.1 Terrain Analysis

For the terrain data, spectral and fkactal methods were used to quanti@ the spatial

variation of elevation. Power spectral density functions (variance per wavenumber ) were

estimated for three non overlapping regions of the Sand Hills (Al, A2, and A3,) and for

Fergus. It is shown that the large-scale contributions to elevation are the most important

since the power spectnun declines rapidly as scale decreases (increasing wave nurnber). The

spectra of Sand Hills shows a scale break, that is, two distinct spectral slopes could be drawn

128

intersecting at about 0.4 - 0.5 cyclekm (- 2 km). This scale break could represent a

characteristic horizontal length scale, at which surface behavior changes substantially;

whereas the Fergus topography exhibited almost no scale break. The high wavenumber

regions have average spectral slopes of 2.8(2.7) and 2.2(2.1) for North-South (East-West)

profiles of Sand Hills and Fergus respectively. Most landscapes have spectral slope of order

2, hence the scale break and high spectral slope gives the Sand Hills terrain a unique

characteristic. Since the minimum wavelength that can be resolved in a numerical model of

grid size Ax is 2Ax, depending on the percentage of the total variance we want to consider,

we can set a minimum wavelength above which the terrain is resolved by the model. For

example in order to resolve 80% of the total variance in terrain height, assuming the

dependence of the air flow on the variance in terrain height is significant, the minimum

wavelength required for the Sand Hills region is - 2 km, where as for Fergus it is - 5 km.

Mode1 grid sizes could then be selected based on these values. Note that if variance in terrain

dope are considered to affect the air flow significantly, then the minimum wavelength

required should be less since dope has high spatial variability.

The variogram malysis revealed that for a range of scales, the spatial characteristics

of the terrain c m be described by a fiactai mode], where the scaling properties are expressed

by fiactal dimension D. For scales below 1 km, the Sand Hills and Fergus have average

fiactal dimensions of 1.3 and 1.25 respectively, and for large scales, above 1 km, the

dimension is higher indicating that heights of large scales cannot be accurately interpolated

fiom the heights of neighboring points.

Topographic parameters, such as the maximum and RMS slope, principal angle 8,

and aspect ratio y (degree of anisotropy), etc were also calculated. The average anisotropy

y and principal angle 0 were found to be 0.723 and 102" for the Sand Hills terrain. Many of

the statistical pararneters are dependent on grid resolution, and slopes, especially the

maximum slope and RMS slope clearly increase with increasing resolution. These resolution

dependencies pose a problem since some sub-grid-scale topography parameterizations use

RMS slope as a significant parameter. Therefore, a representative grid resolution could not

be specified. To quanti@ the resolution dependence, normalized RMS and maximum slopes

as a fhction of nomalized grid resolution were plotted and results show approximate power

law relations. The average value of the exponent j3 in the power law relation of the

nomalized RMS slope for real terrain is 0.6 and lies between extremely random surface (P

= 1) and extremely smooth surface (Po O). QuantiQing the resolution dependence of the

dope pararneters is important result of our study since the power law relations suggest that

spatial variability at finer resolutions could be inferred fiom available coarser grid resolution

data, which in turn could be useful for parameterization purposes of sub-grid-scale

topography .

5.2 Aircraft Observation

NCAR's Queen Air observations are comprised of many geophysical pararneters. For

our analysis however, we focused on the velocity components, temperature, flight height and

130

location, and topographic height. The aircraft used the Inertial Navigation System (INS) and

we found that it has an error of the order of 1 km. Because of the uncertainty in the location

of the aircraft, terrain heights determined fiom the aircraft measurements were not consistent

with heights of topographic map; hence, detailed analysis and interpretation of the data has

never done before. The data provided a unique opportunity to understand the influence of

terrain on air flow.

There were 18 flight legs. Out of tbese, 6 cases were in stable atmospheric condition.

By using eddy correlation methods, we estimated integral statistics ( shear stresses, heat

fluxes, etc) on a coordinate system which is rotated to be along the mean wind direction.

Throughout the flight legs, the shear stress E i s found to be negative, an indication of

momentum loss to the ground, and is enhanced in unstable atmospheric conditions. The

upward heat flux and turbulent kinetic energy (TKE) are also found to increase in the

afternoon. TKE in the morning (stable) is merely a result of the mechanical (shear)

production but with the ratio of TKE/-uw equal to 14 compared to 4 for near-surface neutral

atmospheric conditions over flat and homogenous terrain. Normalized (with respect to u.)

standard deviations of the horizontal velocity components were about 20 - 40 % higher than

one might expect over flat terrain. The normalized standard deviation of vertical velocity,

however, was found to be almost insensitive to the terrain. Al1 three components of the

normalized standard deviations were found to increase with increasing instability, and the

w component seem to obey local similarity laws with less scatter compared to u and v. Flight

131

level drag coefficients CDnT were calculated and compared with the root mean square slope

of the underlying terrain heights detemiined fiom aircraft observation dong the flight path.

The results indicate a potential for a significant increase in flight level drag coefficient with

topographic slope, which in turn suggest that enhanced surface drag could represent the

effect of sub-grid-scale topography. CDFLT was afso f o u d to increase with increasing

instability. We have not tried to link the aerodynamic roughness length 2, to the RMS slope

or C c T because of the unrealistic values of 2, obtained for most of the flight legs. The

formulation that is used to estimate 2, is sensitive to errors in the wind speed and fiction

velocity, and it is also based on surface layer similarity relations, which could not be

applicable to flight level (local) data.

Wavenurnber-weighted power spectral density functions of the three velocity

components were estimated for al1 the runs. The results show that for most of the stable runs

the spectral shape is bi-modal, separated by a spectral gap with a distinct peaks at about 0.19

and 2.5 cycle/km for u and v, and 0.3 and 8 cyclelkm for W. As the instability increases the

spectral gap is replaced by a peak or the inertial subrange. The inertial subranges also

approximately obey the Kolmogorov hypothesis; local isotropy with -2/3 spectral roll off.

The range of local isotropy increases with height and instabiiity. The bi-modal shapes of the

spectrsi are most likely a result of flow distortion by the underlying terrain. To help assess

the effect of the terrain on the spectral shape, we compared spectra of the terrain height and

slope with the velocity spectra and estimated the cross-spectral parameters. Results show that

132

wavenumbers of order O. 1 cycle/km and 0.5 cycle/km are likely to be terrain induced since

high cross-spectral amplitude and coherence values are found at those wavenumbers.

APPENDIX A

A.l Measurement of Air Velocity

The velocity of the air with respect to the earth, V = iu + j v + kw, is obtained by

adding the velocity of the aircraft with respect to the earth, V, , and the velocity of the air

with respect to the aircraft, Vu . That is:

v = v, + va ( A 4

The magnitude of Va is measured by pitot-static tube mounted on the forward boom of the

aircrafi. The components of Y, are obtained fiom integrated accelerometer outputs on an

inertial navigation system (INS), but to convert to an earth-based coordinate system, the

angular velocity of the airplane and of the earth must be added. Thus,

Where a is the measured aircrafi acceleration, me and are the angular velocities of the

earth and platform, respectively, and g is the gravitational acceleration. For air velocity

sensors located far fiom the base of the boom where the INS is located, the term Q, x R,

where n, is the angular acceleration of the aircraft, and R is the distance between the M S and

air velocity sensing platform should be included in equation A.2 and integrated to give:

Since the measured components of Vu and V, are based on the aircraft coordinate system, it

is necessary to rotate the coordinates into a local earth coordinate system (meteorological

fiame of reference). To do this, the aircraft's true heading Jr, angle of attack a, sideslip, pitch

and roll angles should be measured simultaneously, Fig A.1. For small roll and pitch angles,

after using the appropriate transformation equations, the approximate calculations of the

three velocity components reduce to:

u = - u,sin( Jr + f3) + up

v = - u,cos(ql+ p) + vp (A.4)

w = - u,sin(B + cl) + wp

Where u, is the magnitude of the air velocity measured by the pitot-static tube, and the

subsrcriptp is for the speed of the airplane.

.-----

(a) Airplana Axas

(bl. lnsrtiol Piatform Axas

$I DIRECTION O F AIRSTREAM 5: ir

/

/

/'

East --- \

TOP VlEW SIDE VlEW FRONT VlEW

Fig A. 1 Top: coordinate systems used in deriving equations for calculating the air velocity components. Bottom: airplane aîtitude angles and axes used in equations for calculating the air velocity components. (Lenschow, 1986)

APPENDIX B

If we propose that in the surface layer, the wind shear ùUl& (where the x - axis is

aligned to the mean w ind), is only dependent on the height z above the surface, the

surface drag, and the fluid density, i.e.,

aulaz =XZ, T, ,p) (B-1)

Both r, and p give us the characteristic velocity scale u. =(zJp)"*, and the only

characteristic length scale is z. Applying Buckingham Pi theorem, we have one

dimensionless quantity (dimensionless wind shear) which is constant.

( ~ u I ~ z ) ( z I u * ) = const. =i IK (B.2)

Where K is the von K m a n constant. The above relation is verified in many laboratory

boundary layers and near-neutral atrnospheric observations. However, it tells us that at

z=0, the wind shear is infinite; this is contrary to reality because wind shear rernains

finite. Therefore a suitable reference plane near the surface, zo, a dimensional constant of

integration is introduced in the above relation. Integration of Eq. (B.2) fiom zo to height z

yields, the well-known logarithmic profile law,

LI =(u* /~)ln(z/z~) (B.3)

Extending the above similarity hypothesis to include the stability of the atmosphere, the

Monin-Obukov length L will be an additional pararneter in Eq. (B. 1). Therefore we have

two dimensionless parameters, the dimensionless wind shear and the stability pararneter

c=z/L. From the discussion in the review, therefore, one dimensionless parameter will be

137

a universal fùnction of the other, i.e.,

(~UI~Z)(KZIU.) = 4, (O (B-4)

4, should be determined experimentally. The generally accepted empirical form for 9,

are that of 1968 Kansas Experiment, Businger et al. (1 97 1).

(1 -15c)-'" for C < O (unstable) 4 m =

1+4.7Ç for 2 0 (stable)

Integration of Eq.(B.4) with respect to height yields,

Eq. (B.6) is a modified logarithmic law, where the diabatic terni q, is the integral of

(1-@,/c) over limits z,,& to dL. i& is usually srnall and can be replaced by zero.

Therefore, using Eq. (B.5) we obtain

Where x = (1-15C)lt4

REFERENCES

André, J. C., and L. Mahrt, 1982: The nocturnal surface inversion and influence of clear-air radiative cooling. J Atmos. Sci., 39,864-878.

Arya S. P., 1988: Introduction to micrometeorology, Academic Press, 303pp.

Baines, P.G., 1995 : Topographie Eflects in Stratifed Flows, Cambridge Monographs on Mechanics, Cambridge University Press, 482pp.

Bills, B. G., and Kobrick, M., 1985: Venus topography: A harmonic analysis. J Geophys. Res., 90, 827-836.

Bradely, E. F., 1980: An experimental study of the profiles of wind speed, shearing stress and turbulence on the crest of a large hiil. Q. J R. Meteorol. Soc., 106, 10 1 - 123.

Brown, S. R., 1987: A note on the description of surface roughness using fiactal dimension. Geophy. Res. Lett., 14, 1095- 1098.

Brown, S. R., and Scholtz, C. H., 1985: Broad bandwidth study of the topography of natural rock suriaces. J. Geophys. Res., 90, 12575-12582.

Businger, J. A., Wyngard, J. C., Izumi, Y., and Bradely, E. F., 1971: Flux-Profile relationship in the atmospheric surface layer. J Atmos. Sci., 28, 1 8 1 - 189.

Cheng, Q., and Agterberg, F. P., 1996: Multifiactal modelling and spatial statistics. Math. Geol. 28, 1-1 6.

Fiedler, F., and Panofsky, H. A., 1972: The geostrophic drag coefficient and the 'effective' roughness length. Q. J R. Meteorol. Soc., 98,213-220.

Garratt, J. R., 1977: Review of drag coefficients over oceans and continents. Mon. Wea, Rev.,lOS, 915-926.

Goodchild, M. F., 1980: Fractal and the accuracy of geographical measures. Math. Geol., 12, 85-98.

Huang J., and Turcotte, L. D., 1989: Fractal mapping of digitized images: Application to the Topography of Arizona and Comparisons With Synthetic Images. J. Geophys. Res., 94,749 1 - 7495.

Jackson, P. S., and Hunt, J. C. R., 1975: Turbulent wind flow over a low hill. Q. J R. Meteorol. Soc., 101,929-955.

Jenson, S. K., 1993 : Application of hydrologic information automatically extracted fiom digital elevation models, Terrain analysis und distributed modelliizg in hydrology, John Wiley & Sons, 35-47.

Kaimal, J. C., Wyngaard, J. C., Izumi, Y., and Coté, R., 1972: Spectral characteristics of surface layer turbulence. Q. J. R. Meteorol. Soc., 98,563-689.

Kaimal, J. C., and Finnigan, J. J., 1994: Atmospheric Boundary Layer flows, Oxford University Press, 289pp.

Kolmogorov, A. N., 1941 : The local structure of turbulence in compressible turbulence for very large Reynolds nurnbers. C. R. Akad. Nauk, SSSR, 30,30 1-365.

Lavallée, D., Lovejoy, S., Schertzer, D., and Ladoy, P., 1993: Nonlinear variability and landscape topography: Analysis and simulation. Fractals in Geography. Eds. L. De Cola, N. Lam, pp 158-1 92, Prentice Hall.

Lenschow, D. H., 1986: Probing the Atmospheric Boundary Layer, American Meteorological Society, 269pp.

Lovejoy, S., Schertzer, D., and Tsonis, A. A., 1987: Functional box counting and multiple elfiptical dimensions in rain. Science, 235, 1036- 1038.

Lumely, J. '., and Panofsky, H. A., 1964: The structure of atrnospheric turbulence. Interscience, New York.

Mandelbrot, B. B., 1977: Fractals: form, chance and dimension. W. H. Freeman, San Fransisco.

Mandelbrot, B. B., 1983 : Thepactal geometry of nature. W. H. Freeman, San Fransisco.

Nieuwstadt, F. T. M., 1984: The turbulent structure of the stable, nocturnal boundary layer. J Atmos. Sci., 41,2202-22 16.

Panofsky, H. A., Tennekes, H., Lenschow, D. H., and Wyngaard, J. C., 1977: The characteristics of turbulent velocity components in the surface layer under convective conditions. Boundary-Layer Meteorol., 11,354-36 1.

Wyngard, J. C., Coté, O. R., and Izumi, Y., 1971: Local fiee convection, similarity, and the budgets of shear stress and heat flux. J Atmos. Sci. 28, 1 17 1-1 182

Xu, D., and Taylor, P. A., 1995: Boundary layer parameterkation of drag over small scale topography. Q. J. R. Meteorol. Soc., 121,433-443.

Young, G. S., and Pielke, R. A., 1983: Application of terrain height variance spectra to mesoscale modelling. J. Atmos. Sci. 40,2555-2600.

Young, G. S., Pielke, R. A., and Kessler, R. C., 1984: A cornparison of terrain height variance spectra of the fiont range with that of a hypothetical mountain. J. Atmos. Sci., 41, 1249-1250.

Zhou, J., Taylor, P. A., and Qi, Y., 1997: On stably stratified planetary boundary-layer flows over topography. Submitted to Boundary-Layer Meteorol.