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
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.
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)
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.
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)
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
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