TWO DIMENSIONAL HEATED MOUNTAIN WAVE SIMULATIONS
This chapter investigates idealized and observed two-dimensional surface heated mountain wave flows through the use of a mesoscale numerical model. Researchers have frequently applied two-dimensional numerical models to the observed downslope wind events in Central Colorado. The success of the two-dimensional studies is largely due to the fact that the Front Range of the Rocky Mountains are oriented in the north-south direction. Their north-south wavelength is substantial and acts as a two dimensional barrier to the predominantly westerly flow.
Six two-dimensional test groups are used to evaluate the heating and cooling aspects of the diurnal cycle. They include finite amplitude narrow and wide mountain profiles, mean state critical layer tests, non-dimensional parameter (Froude number) range experiments, and the observed January 9, 1989 Boulder Colorado windstorm. Each group approaches the windstorm problem from a different perspective, ranging from analytical comparisons to the January 9, 1989 highly variable recorded event. The tests are designed to isolate the impacts of the parameterized surface heating on the numerically generated mountain wave flows. In each test group, a comparison is made between the non-heated or control run and their heated counterparts. Direct evaluations are possible since the control runs are advanced the same length in time as the heated runs. In the heated cases, an approximate steady state is attained before the diurnal cycle is activated. One idealized cooling simulation is performed at the completion of the heating period.
As mentioned in Chapter 3, the redistribution of surface heating in this model is accomplished in one of two ways. In one method, a strong diffusion coefficient is applied to distribute the heating in the vertical. Due to its strong dependence on the turbulent kinetic energy budget and mixing length, this method is referred to as the parameterized approach. It is best suited for horizontal grid spacing equal to or greater than 1000m. The parameterized method works well for flow over long wavelength mountains. In the second technique, I attempt to explicitly resolve the convection in the developing boundary layer. The mixing length is an order of magnitude smaller than that used in the parameterized approach and allows for the development of a superadiabatic layer near the surface. A random number generator is applied to the heating term at the beginning of the diurnal cycle. These small spatial variations in the potential temperature field near the surface grow with time into efficient convection, responsible for nearly all of the heat redistribution in the boundary layer. The explicit technique requires resolution on the order of 100m in each dimension to resolve accurately the developing mixed layer eddies. This approach is better suited for short wavelength mountains that force non-hydrostatic gravity waves or lee waves. Lee waves require horizontal grid spacing of order 400m. Note that in all heated simulations, the upstream lower boundary is heated, allowing for a more realistic boundary layer evolution. Recent and relevant surface flux observations are presented in the next section.
5.1 Surface Heat Flux Measurements
The choice of a maximum heat flux for use in the numerical simulations is guided in part by a recent field study conducted in the Boreal forest of Saskatchewan and Manitoba, Canada. The Boreal Ecosystems Atmosphere Study (BOREAS) is designed to improve the understanding of energy exchanges between the boreal forest and the lower atmosphere. A description of the project can be found in Sellers et. al. (1995).
Data collection began in August 1993 and continued through 1996. The
majority of the measurements were taken in contiguous periods spanning
several days and included data from eddy correlation equipment on a surface
tower network. Flux measurements were enhanced with observations from four
instrumented aircraft. At the end of 1995, two years into the study, a
detailed wintertime boundary layer study was conducted. This period represents
the first in-depth study of the Boreal forest ecosystem during the winter
months. Tree-top sensible heat fluxes as high as 400 W/
were measured near the end of March 1996. Figure 5.1, courtesy of Black
(1996), displays a plot of sensible heat flux as a function of time at
a site in the Boreal forest and is typical for data collected during March
1996. These observations provide evidence that significant heating can
take place in the tree canopy above the snow-covered tundra.
(Add a chart w/data here)
place figure 5.1 here....obtain data from A. Black
plot of sensible heat flux
from the boreal forest project
Figure 5.1. Plot of sensible heat flux (top curve) and latent heat flux (bottom curve) for the period March 22, 1996 inclusive. The data were collected at the Old Aspen Site PANP in Saskatchewan, Canada.
5.2 Idealized Finite Amplitude Mountain Wave Flow
Three tests are conducted for finite amplitude wide and narrow mountain
forced flows. Referring to (4.2), the mountain quarter wavelength 
is
10km and 2km for the wide and narrow mountain tests, respectively. Two
narrow mountain simulations are offered to measure the sensitivity of the
gravity wave response aloft to the developing boundary layer circulation.
One prediction of a wide mountain flow is made and the resulting wave activity
contrasted with that predicted by linear theory. The results obtained in
this section will be used in all of the remaining numerical experiments.
5.2.1 Heated Narrow Mountain Tests
Two methods of heat distribution are investigated here using a narrow mountain profile. The non-hydrostatic effects can be measured by the ratio of the horizontal to vertical wavelengths, where values approaching unity indicate scale equivalence and signify substantial non-hydrostatic forcing. For this test the ratio is:
,
where 
=0.0108 
is
the static stability, 
=2
km the quarter wavelength parameter, and 
=20
m/s is the base state wind. The mountain profile is estimated by (4.2).
The non-linear effects are measured by the gravity wave strength (
)
normalized by the base state wind. For the narrow mountain case with 
=
300m, the non-linear effects are classified as moderate with:
.
Table 5.1 provides a summary of the model parameters used in the two-dimensional simulations presented in this Chapter. For this particular test, the model parameters are listed under the narrow mountain test group.
The approach to steady state of a mountain wave simulation can be evaluated
using a time series of the computed surface wave drag. This test indicates
a steady state at approximately 
=60
or 12,000 seconds. Therefore, in both the parameterized and explicit heated
runs the heating cycle begins at 12,000 seconds. Recalling (3.16), the
diurnal heating cycle is:
,
with a maximum value of 
and
a heating period of 12 hours. The heating cycle begins with zero amplitude
and grows to a maximum at 6 hours. The sine wave time representation produces
a shape similar to the observed diurnal cycle. As introduced in Chapter
1, the majority of windstorms occur during the winter months when the days
are shorter than 12 hours. The length of day chosen here is arbitrary and
is likely longer than would be experienced during the peak downslope windstorm
period.
5.2.2 Parameterized vs. Explicit: Results and Discussion
Each simulation is advanced to 
=55,000
seconds, or a non-dimensional time 
=
275. The comparison includes a time series of surface wave drag, vertical
profiles of vertical fluxes of horizontal momentum, and x-z cross-sections
of selected
| Parameter | Narrow | Wide | 7km Critical | 17km Critical | 2-Layer | NLP030,010,002 | Boulder |
| nx,ny,nz | 434,4,83 | 434,4,83 | 434,4,113 | 434,4,213 | 245,4,83 | 163/237,4,75/103 | 650,4,115 |
 (m) |
200 | 1000 | 1000 | 1000 | 1500 | 2000 | 1000 |
 (m) |
--- | --- | --- | --- | --- | --- | --- |
 (m) |
50 | 100 | 100 | 100 | 200 | 100,200,250 | 250 |
 (s) |
5.0 | 10.0 | 5.0 | 5.0 | 10 | 10,20,20 | 5.0 |
 (s) |
0.5 | 2.0 | 2.5 | 2.5 | 2.5 | 2.5,2.5,4.0 | 2.5 |
 (m/s) |
10 | 20 | 20 | 20 | 20 | 10 | Sounding+ |
 |
0.324 | 0.293 | 0.31,0.39,0.47 | 0.10,0.31,0.52 | 0.6,0.3 | 3.0,1.0,0.2 | --- |
 (1/s) |
0.0108 | 0.01956 | 0.01047 | 0.01047 | 0.02,0.01 | 0.03,0.01,0.002 | Sounding+ |
 ( K) |
--- | 250 | --- | --- | --- | --- | --- |
 (m) |
300 | 300 | 600,750,900 | 200,600,1000 | 600 | 1000 | Profile* |
 (m) |
2000 | 10000 | 10000 | 10000 | 10000 | 15000 | Profile* |
 ( ) |
100 | 100 | 200,-40 | 200 | 300 | 360,300,60 | 200 |
 (hrs) |
24 | 24 | 24 | 24 | 27.7 | 72,27.7,16.6 | 24 |
 Horz. |
0.00001 | 0.00001 | 0.0005 | 0.0005 | 0.0005 | 0.0005 | 0.0004 |
 |
0.0 | 0.0 | 0.0001 | 0.0001 | 0.0005 | 0.0005 | 0.00001 |
 |
0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 |
perturbation fields. Figure 5.2 limns a comparison of the normalized
vertical profiles of momentum flux at 
=
75, 150, and 225 for the parameterized run (solid line) and the explicit
run (dashed line). The curves are normalized by the two-dimensional linear
hydrostatic values. For the parameterized experiment, the profiles are
relatively smooth, with nearly vertical orientation and exhibit decreasing
wave activity with increasing time and mixed layer depth. Overall, the
mountain flux profiles for the explicit case compares favorably with the
parameterized pattern. At 
=
75 and 150, the profiles are nearly identical. At 
=
225, the explicit case generates flux near the top of the mixed layer (approximately
1.0 km) that is 35% greater than its parameterized counterpart. This is
likely due to the inclusion of the convective elements in the flux computation.
But as the distance above the inversion increases the difference between
the explicit and parameterized experiments is reduced to less than 10%
at the height of 2/3 vertical wavelength (approximately 4 km). Plots of
the perturbation horizontal velocity in Figure 5.3 and potential temperature
in Figure 5.4 contrast the results between the two classes of heat redistribution.
Figure 5.3 presents the horizontal velocity perturbation at 
=
225 near the mountain peak. The location of the maximum surface wind moves
upslope and approaches the mountaintop as the mixed layer deepens. In the
potential flow limit, the maximum wind is located at the mountain peak
upstream of the maximum associated with a mountain wave. Figure 5.4 contrasts
the simulated isentropes of each test for the entire model domain. As evident
in Figure 5.3, both simulations predict a nearly identical upstream boundary
layer height, to within 50 meters (see the 294
K
isentrope). The disturbances in the
Figure 5.2. Vertical profile of the vertical flux of horizontal momentum
for heated narrow mountain flow for the parameterized case (solid lines)
and explicit method (dashed lines) at non-dimensional times of 
=
75, 150, and 225.
(a) (b)
Figure 5.3. Perturbation horizontal velocity for a finite amplitude
heated short wavelength mountain flow for (a) the parameterized case and
(b) the explicit simulation at 
=
225 (40,000 seconds). Area depicted is in the vicinity of the mountain
peak extending to the model top. The contour interval is 0.5 m/s.
(a) (b)
Figure 5.4. Isentropes for a finite amplitude heated narrow mountain
flow for (a) parameterized and (b) explicit runs at 
=
225 (40,000 seconds). The area shown follows that of Figure 5.3. The contour
interval is 1.0K.
potential temperature field near the top of the mixed layer in the explicit
case exhibit a dominant wavelength on the order of 5 km or less. In these
experiments, the perturbations at the top of the mixed layer become lee
wave sources. The minimum vertical wavelength that could propagate vertically
(2
)
in this example is nearly 6 km. Thus, gravity waves with horizontal wavelengths
less than 6km are classified as evanescent waves and do not propagate vertically.
The majority of the disturbances measure less than 6 km in length and do
not contribute significantly to the momentum flux at higher elevations.
The ratio of the horizontal wavelength of the convection to the depth
of the mixed layer in Figure 5.3 (b) is approximately 4:5 to 1. The grid
spacing is 200 x 50 meters corresponding to approximately 20 grid points
per convective cell in each spatial direction. Note that the gravity wave
activity aloft is clearly visible in both cases although slightly modified
in the explicit experiment. Inspecting the predicted turbulent kinetic
energy field 
best illuminates
individual convective plumes in the explicit mixed layer. Figure 5.5 illustrates
the total e cross-section at 
=
225 for each test. The structures of the e field in these plots are markedly
different. The convective plumes in (b) extend to nearly 1.3 km above the
surface. This is due primarily to the method in which the mixing length
l is computed and the randomness of the initial temperature perturbation.
The vertically oriented e fields in (b) correspond to resolved updrafts
in the simulated two-dimensional dry convection. In these tests the primary
source of e is the buoyant production (see page 56):
.
(a) (b)
Figure 5.5. Model predicted turbulent kinetic energy for a finite amplitude
heated narrow mountain flow for the (a) parameterized and (b) explicit
tests at 
= 225 (40,000
seconds). Area depicted follows that in Figure 5.4. The contour interval
is 0.05
.
The mixing length in the parameterized method can be an order of magnitude larger than the explicit case. As a result, resolved convection can be produced in the parameterized case but the depth and magnitude will be hampered since the vertical mixing will reduce any developing vertical potential temperature gradients.
As portrayed in the explicit and parameterized simulations, the convective motions modify the surface winds significantly but the flow aloft remains largely unchanged. The kinetic energy from this moderately non-linear case is comparable to that found in boundary layer convective motions. Yet, both tests reveal a nearly 50% reduction in wave energy aloft at the conclusion of the diurnal heating cycle. The majority of wave degradation stems from the results of convection and not the convection itself. From a mountain wave perspective, the differences in the parameterized and explicit experiments are small. This result favors the use of the computationally efficient parameterized approach for the redistribution of heat in the convective boundary layer. Choosing the parameterized technique allows the simulation of large two and three-dimensional downslope windstorms with today?s computer resources. In support of the coarser grid spacing selection, Clark et. al. (1994) show that horizontal grid spacings on the order of 500-1000 meters are sufficient for resolving the hydrostatic modes and the majority of the shorter wavelength lee waves.
5.2.3 Heated Wide Mountain Flow Test
The impacts of a parameterized diurnal cycle on a longer wavelength mountain profile of moderate height are presented here. The mountain quarter wavelength is five times that given in Section 5.2.1 and therefore forces mainly hydrostatic gravity wave modes. The stability and base state wind are nearly twice those of the previous case. The resulting hydrostatic measure is:
,
where 
= 0.0195 (isothermal), 
=
10 km, and 
= 20m/s. For 
=
300 m, the non-linear measure or inverse Froude number is:
,
approximately the same as given in the narrow mountain simulations.
A summary of the model parameters used in this test is provided in Table
5.1 under the wide mountain test group. A steady state solution for the
wide mountain case is obtained at 
=
60 or 
= 30,000 seconds,
after which the diurnal heating is activated via the parameterized method,
and the simulation advanced to 80,000 seconds (
=
160). The horizontal velocity (Figure 5.6) and potential temperature (Figure
5.7) are well mixed in the neutral layer and the maximum and minimum perturbations
aloft are approximately 5-10% lower than the steady state values.
Both the narrow and wide heated mountain wave simulations exhibit reduced
surface wave drag as the heating cycle increases the depth of the mixed
layer (Figure 5.8). The largest deviations in the momentum flux from the
wide mountain control run steady state values are observed at the end of
the experiment, in conjunction with the maximum depth of the mixed layer
(Figure 5.9). The decrease in momentum flux for the parameterized run is
approximately 18%, nearly one third of the reduction experienced by the
heated narrow mountain test (49%). Most of the discrepancy can be explained
by the fact that for the narrow mountain run, the static stability is approximately
one-half that used in the wide mountain simulation. This point is illustrated
in Figure 5.10. The resulting mixed layer depth for the wide mountain case
is about half that of the narrow ridge simulation. As illustrated in Chapter
2, the reduction of wave activity in the stable layer aloft is a function
of the mixed layer depth (H). Referring to Figure 2.2, linear theory predicts
for a horizontal wavelength of 80 km (the main contributor of the 
=
10 km mountain shape) and mixed layer depth of 0.6 km, a reduction of integrated
wave activity (
) on the order
of 16%. The corresponding linear theory estimate for reduction in wave
activity for the narrow ridge is 43%. Linear theory captures nearly all
the simulated reductions in wave activity associated with heated mountain
wave flows of this variety. These results lend support for linear theory
in estimating the wave behavior for moderately non-linear mountain flows
under the influence of surface heating.
(a)
(b)
Figure 5.6. Perturbation horizontal velocity for a finite amplitude
heated wide mountain flow at (a) 
=30,000
and (b) 
=70,000 seconds.
Area depicted is in the vicinity of the mountain peak upward to the model
top. The contour interval is 1.0 m/s.
(b)
=30,000
and (b) 
=70,000 seconds.
Area depicted follows that of Figure 5.6. The contour interval is 2.5 K.
Figure 5.8. Plot of surface wave drag as a function of time for the
parameterized heated wide (solid) and narrow (dashed) ridge flow tests.
Heating curves are provided at the bottom of the plot for the wide (solid)
and narrow (dashed) mountain tests. The maximum heating rate for both tests
is 100
. The vertical lines represent
the approximate mixed layer depth.
Figure 5.9. Vertical profile of the vertical flux of horizontal momentum
at 
= 60, 80, 100, 120,
and 140 for the heated wide ridge flow test. Profiles are normalized by
the linear hydrostatic Boussinesq value. One vertical wavelength is approximately
6.4 km.
(a)
(b)
Figure 5.10. Vertical profiles of potential temperature upstream of the mountain at the beginning, middle and end of the heating cycle for (a) wide and (b) narrow ridge parameterized tests. Line labels are in kilo-seconds.
5.3 Mean State Critical Layer Experiments
Downslope windstorms frequently display elevated regions of flow reversals and enhanced turbulence with weak or neutral stability. These regions are commonly referred to as critical layers. Current numerical models are quite capable of producing such features as evident in the January 11, 1972 Boulder windstorm simulation illustrated in Chapter 4. A critical layer exists when the phase velocity of the wave equals that of the transport medium. For the case of airflow over mountains, as alluded to earlier, this occurs when the cross-mountain wind speed is reduced to zero. These experiments are designed to classify the sensitivity of mountain wave flow to surface heating in the presence of varying mean-state critical layer heights. One experiment is extended to include the response due to a parameterized nocturnal cooling period. Simulations with a mean state critical layer are similar to those found in downslope windstorms: they both involve a critical layer, above which reduced wave activity is observed.
Studies of mean state critical layers in mountain wave simulations are presented by Durran (1986) and Durran and Klemp (1987). Durran investigates the amplification mechanisms of strong downslope windstorms. He compared numerical predictions of flow over a mountain with varying critical layer heights to the linear amplification model of Peltier and Clark (1979, 1983), to Smith?s (1985) hydrostatic non-linear analytical theory, and to the hydraulic analog. He found, for a Boussinesq atmosphere, the numerically predicted low and high drag states followed Smith?s non-linear theory and to a lesser, yet significant, degree the hydraulic analog. He concluded that the height of the wave-breaking region is sensitive to subtleties in the flow, including upstream inversion heights and that the prediction of the onset and placement of the wave-overturning layer can only be addressed through the use of numerical models. Two of Durran?s critical layer test groups are investigated here. The two groups are defined by the height of the mean state critical layer, 7 km and 17 km. Following Durran?s work has at least two benefits. The first is the evaluation of the model in another non-linear environment by comparing the numerical solutions with Smith?s predictions and Durran?s numerical results. The second allows for a direct assessment of the mature windstorm?s sensitivity to changes in the low-level stability given a simplified mountain shape and base state wind profile.
Motivated by the results of Chapter 4, the resolution in the present study is enhanced over what Durran used. Grid spacings were dx =1000 m and dz =100 m as compared to Durran?s dx =1500 m and dz =333 m. The Boussinesq option was invoked in the model for all of the critical layer tests. The details of the Boussinesq modification are summarized in the finite amplitude test description in Chapter 4.
5.3.1 7 km Mean State Critical Layer Results
The experimental set-up follows Durran (1986). The base state wind is
reduced in a shear layer from 20m/s at 5 km to zero at 7 km. From 7 km
to the top of the model domain (11 km) the base state wind is set to zero.
All simulations are performed without surface friction parameterization.
Table 5.1 presents a listing of the pertinent model parameters under the
7 km Critical test category. This experimental group is composed of three
separate simulations in which only the mountain height varies. The mountain
heights used in these tests are 600m, 750m, and 900m, which corresponds
to a non-linear factor or inverse Froude number 
=
0.314, 0.392, and 0.471, respectively. These tests are similar to those
given in the second line of Durran?s Table 1. The mountain profiles are
defined by the "Witch of Agnesi" profile in (4.2). The mountain quarter
wavelength is 
= 10 km.
The heating cycle for each test began at 
=
35,000 seconds (
= 70) and the
prediction advanced to 
=
80,000 seconds or a non-dimensional time of 
=
160. The 
= 750 m case is
extended to 
= 240 using
an estimated minimum surface cooling rate of -40 
.
The time dependant cooling function is given in Section 3.3. The present
model?s control runs reproduced the results of Durran?s Table 1 reasonably
well, although the computed surface wave drags are lower. Figure 5.11 compares
the normalized surface wave drag detemined from the present model with
those from Durran (1986). Figure 5.12 displays the computed surface drag
time series for all the 7 km critical layer tests. Both the 
=750
m and 
=900 m tests achieve
high drag states approximately 3.5 times that predicted by linear hydrostatic
theory, while the 
=600
test produces significantly sub-linear theory surface wave drag. These
results agree with those illuminated by Smith?s theory. The surface wave
drag, measured at the end of the heating period, for the heated 
=900
m and 
=750 m simulations
were reduced by 28% and 22%, respectively. In both cases, the mixed layer
achieved a depth of approximately 2.0 km. During the 
=750
m cooling period (between 80,000 and 120,000 seconds) the surface drag
increases but remains considerably lower than the control run. An experiment
initialized with a neutral surface layer depth equal to the maximum mixed
layer depth obtained for the 
=
750 m heated case was performed with the results presented in Figure 5.12
for
Figure 5.11. Normalized surface wave drag as a function of mountain
height and critical layer height. Results from Durran?s (1986) results
are indicated by the far right bar in all but the far right group. The
far left bar in each group represents the non-heated present model results
and the center bar depicts the heated results.
Figure 5.12. Time series plot of computed surface wave drag for all
tests with a mean state critical layer at 7 km. The solid lines represent
the control runs and the dashed lines the heated simulations. The heating
curve is provided at the bottom of the plot with a maximum heating rate
of 200
and a minimum rate of
-40. The vertical lines represent
the approximate mixed layer depth.
comparison. No heat is applied and the simulation is advanced to 
=
80,000 seconds. The graph shows that for nearly 14 hours the wave drag
remains nearly an order of magnitude lower than the control and heated
runs. At 
= 80,000 seconds
the heated and neutral layer simulations exhibit nearly identical surface
wave drags and peak surface winds. Two different paths are used to achieve
the same result. Since the growth rate was small, a low drag state could
have been mistakenly estimated prior to 
=
40,000 seconds. In this test, no gravity waves are present in the neutral
surface layer. The distance from the top of the neutral layer to the base
of the critical layer is 5 km, of which 2 km involves a linear decrease
in the base state wind. The vertical wave number increases and the vertical
wavelength decreases in the shear layer. Reports by Blumen (1965) and Klemp
and Lilly (1975) present a case for linear resonance. The 
=
600 m heated simulation exhibits a low steady state drag and undergoes
a near 50% reduction at the completion of the diurnal heating cycle.
Figure 5.13 displays the normalized surface wave drag curve for the
analytical two layer 80 km wavelength solution from Chapter 2 (line) and
the final surface wave drag for the three heated 7 km critical layer tests
(circular points). Note that all three critical layer runs points rest
above the curve, indicating that for a 80 km wavelength mountain profile,
linear theory overestimates the wave suppression due to a well mixed surface
layer. The best estimate by linear theory is made for the 
=
600 m low drag state condition in which the error is approximately 10%.
For the high drag states of the 
=
900 m and 
= 750 m tests,
linear theory under estimates the wave amplitude by factor of two. The 
=
900 and 
= 750 m cases exhibit
a 10% reduction in the maximum predicted surface wind at the conclusion
of the heating cycle (Figure 5.14).
Figure 5.13. Plot of the linear analytical steady state surface wave drag curve as a function of mixed layer depth and horizontal wave length (80 km) and 7 km heated critical layer tests. The values are normalized by the H=0 steady state values. Plotted points represent the normalized surface drag at the conclusion of the heating cycle for the simulations indicated in the box above.
Figure 5.14. Time series plot of the maximum surface wind speed with
a mean state critical layer at 7 km for all control and heated tests. The
solid line represents the control runs and the dashed line the heated tests.
The heating curve is provided at the bottom of the plot with a maximum
heating rate of 200 and a minimum
rate of -40. The vertical lines
represent the approximate mixed layer depth.
Most of the reduction in surface 
take
place late in the heating cycle, corresponding to the maximum mixed layer
depth. Note the hydraulic jump-like structure downstream of the mountain
peak at x= 320 km and wave induced critical layer at approximately z= 4.8
km (Figures 5.15 and 5.16). As anticipated, very little wave activity is
present above the critical layer (z=7 km).
The 
= 750 m test was
extended 40,000 seconds past the end of the heating cycle. The purpose
of this experiment is to gain insight on the effects of nocturnal cooling
on a heated mountain wave. Although the main goal of this study is to investigate
the diurnal trends from the heating perspective, the cooling period is
also a likely contributor to the observed trends, since gravity wave magnitude
is a direct function of stability (
).
A stable near-surface layer develops as a result of the parameterized cooling
function (Figure 5.17). The stability in the surface layer is similar to
the original base state profile and is approximately one-third the depth
of the mixed layer height (not shown). In response to the increase in static
stability at the surface, the wave drag and maximum wind speed increase
during the simulated nocturnal period, recovering nearly one-third of the
reduction attributed to the heating period. But, the remnants of the heating
cycle are clearly visible, with the presence of an elevated mixed layer
of appreciable depth (Figure 5.18). It is clear from this experiment that
the mixed layer continues to restrain the mountain wave response, albeit
from an elevated location. The overall character of the simulation remains
unchanged from that at 
=
80,000 seconds. Additional cooling period simulations were not conducted
in order to focus on heating portion of the daily trend.
Figure 5.15. Numerical model perturbation horizontal velocity for a
mean state critical layer at 7 km for the 
=
750 m heated case at 
= 80,000
seconds. Area depicted is the entire model domain. The contour interval
is 5.0 m/s.
Figure 5.16. Total potential temperature for a mean state critical layer
at 7 km for the 
= 750 m
heated case at 
= 80,000
seconds. Area depicted is the entire model domain. The contour interval
is 5K.
Figure 5.17. Total potential temperature for a mean state critical layer
at 7 km for the 
= 750 m
cooling portion at 
= 120,000
seconds (
= 240). Area depicted
is the entire model domain. The contour interval is 5K.
5.3.2 17 km Mean State Critical Layer Results
This section presents experiments with a mean state critical layer at
17 km and mountain heights of 
=
200, 600, and 1000 m. These tests differ from the previous work since multiple
waves in the vertical are possible. The non-linear effects are estimated
by 
= 0.104, 0.314, and
0.523 for the 
= 200, 600,
and 1000m tests, respectively. As before, each case is brought to a steady
state and the diurnal heating cycle enabled. The steady state was estimated
at approximately 
= 35000
seconds or 
= 70 for each
test (Figure 5.18), even though there was a slight increase with time of
the surface wave drag to the end of the simulation. For the 
=
1000, and 600 m tests a significant reduction of 43% and 37% from the non-heated
run is noted at the end of the heating cycle. For each heated run the mixed
layer developed to a height of 2 km by the end of the heating period. For
the 
= 200 m case, a high
drag state was not attained and the drag reduction due to heating is approximately
26%. Reductions for each case are plotted on Figure 5.13. The graph indicates
linear theory overpredicted the reduction by about a factor of two. The
weakly forced case (
= 200 m)
exhibited the largest deviation from linear theory. This is contrary to
earlier results in which moderately non-linear hydrostatic and non-hydrostatic
heated simulations followed linear theory reasonably well. The peculiar
behavior for the 
= 200
m case may be due to enhanced resonance, as the effective gravity wave
guide depth changes with the height of the mixed layer. The vertical wavelength
in each test is 12 km. The shear layer below the critical layer is a likely
candidate for reflecting part of the wave
Figure 5.18. Time series plot of the surface wave drag for the control
(solid lines) and heated (dashed lines) runs with a mean state critical
layer at 17 km for 
= 1000,
600, 200m, and 
= 1000m
constant base state wind with height (1000m-nc). The heating curve is provided
at the bottom of the plot with a maximum heating rate of 200.
Vertical lines indicate the approximate depth of the mixed layer.
energy back toward the surface. The reduction in maximum wind speed
for the h=1000 m and 600m tests are on the order 10 % (Figure 5.19). There
is actually an increase in the maximum surface wind speed in the 
=
200 m case. This is due to the generation of poorly resolved convective
cells in the boundary layer. In each of the heated tests, small-scale features
are present when the heating is strongest and are an artifact of poorly
resolved convection. The turbulent parameterization scheme is unable to
properly mix the near surface super adiabatic layer and the horizontal
grid spacing is too coarse to properly resolve the convective motions.
Location of the maximum 
perturbation
field (not shown) for the 
=
1000 m and 600 m simulations move down the lee slope of the mountain and
onto the downwind plain during the heating period. The magnitude of the
surface wind maximum is about 10-15% lower in the heated tests as compared
to the control case. In the 
=
200 m test maximum surface winds were observed to increased and can be
attributed to the convective boundary layer motions. The perturbation horizontal
velocity and total potential temperature fields at 
=
75,000 seconds for the 
=
1000 case are illustrated in Figures 5.20 and 5.21. As alluded to earlier,
both fields show signs of poorly resolved convection far downstream of
the mountain crest.
Figure 5.19. Time series of the maximum surface wind speed for all tests
with a mean state critical layer at 17 km. The solid and dashed lines represent
control and heated solutions, respectively. The heating curve is provided
at the bottom of the plot with a maximum heating rate of 200.
Vertical lines indicate the approximate depth of the mixed layer.
Figure 5.20. Perturbation horizontal velocity for the heated mean state
critical layer test at 17 km for the 
=
1000 m case at 
= 75,000
seconds for the entire model domain. The contour interval is 2.5 m/s.
Figure 5.21. Isentropes for the heated mean state critical layer at
17 km 
= 1000 m case at 
=
75,000 seconds for the entire model domain. The contour interval is 5K.
5.3.3 Discussion
In every experiment, the introduction
of the diurnal heating cycle reduced the gravity wave activity in the stable
layer aloft. The surface wave drag was reduced approximately 20-25% with
the exception of the 7 km 600 m simulation, which realized a reduction
on the order of 50%. Thus, it appears that the response is fairly predictable
regardless of the placement of the critical layer. For the critical layer
tests, linear theory continually overestimated the actual wave reduction
by 10% to 100%. From a qualitative standpoint, the non-heated critical
layer simulations compared favorably to Smith?s theory and to other published
numerical results. Smith?s work predicts amplification for critical layer
heights between (1/4+n)
and
(3/4+n)
for n
=0.
A notable exception is the 7 km 
=
750 m test case. The numerical model predicted a high drag state, whereas
Smith?s theory does not. Other simulations with lower mountain peaks failed
to generate a high drag state (see the 7 km, 
=
600 test). Yet as heat is introduced to the 
=
750m case, a nearly 2 km deep mixed layer develops. The mountain wave response
remains in the high drag regime with a normalized flux > 2.0 (see Figure
5.13). The 7 km 
= 750m
2 km deep neutral layer test requires nearly 20 hours to achieve a steady
high drag state. The neutral layer test verifies the heated 
=
750 m simulation but also exposes a slow yet significant growth mode. This
particular result suggests that, given sufficient time, the 2 km neutral
layer test can achieve a high drag state similar to that displayed by the
heated run. A mean state critical layer (with respect to terrain features)
is not common in the atmosphere, while a 2 km deep mixed layer is observed
frequently.
The weakly forced 
=
200 m, 17 km critical layer test defies linear theory. This may be due
to partial reflections below the critical layer, which the linear solution
does not include. The reduction of the surface winds follows linear theory,
as most tests produced a 10%-15% decrease in the predicted maxima at the
end of the diurnal heating cycle. As shown in Chapter 2, the magnitudes
of the perturbation velocity fields are functions of the mixed layer depth.
Since the drag is a quadratic quantity in terms of the perturbation velocity
fields, the reduction of wave drag should be more dramatic than that in
each individual wind field. During the cooling period for the 
=
750 m simulation, the wave drag rebounded, recovering only a fraction of
the control runs value. This is not surprising since only a portion of
the mixed layer nearest to the surface layer reestablishes stable stratification.
In the mid-latitudes, localized mean state critical layers are rarely
observed. With this in mind, a numerical exercise was conducted for the 
=
1000m 17 km critical layer test. The simulation is performed with a constant
non-zero base state flow extending to the top of the domain. The results
indicate a wave drag approximately 50% of the critical layer counterpart
and are included in Figure 5.18 for comparison. Clearly, the presence of
a critical layer enhances the response for certain atmospheric profiles.
One method used to measure the effects of surface heating is to present
the results as a function of non-dimensional parameters (Figure 5.22).
For this study, the relevant parameters are the static stability (
),
base state wind (
), mountain
height (
), heat input (
),
and the depth of the mixed layer (
).
This particular configuration was chosen because the ease of comparing
it with linear theory from Chapter 2 and to radiosonde observations. Figure
5.22 displays a nearly linear decrease in wave drag as compared to the
exponential decrease forecast by the linear solution of Chapter 2. Another
option is to plot the normalized surface wave drag as a function of heat
input (
) to the system normalized
by the perturbation kinetic energy. This approach and others were attempted
but deemed unsuitable for a variety of reasons.
Figure 5.22. Normalized surface wave drag as a function of the ratio
of the mixed layer depth to the mountain height for the critical layer
experiments. The wave drag was normalized by the linear hydrostatic value.
5.4 Non-Linearity Parameter Study
In this section, a wider range of flow conditions is used to further
our understanding of the effects of heating on the gravity wave environment.
The work discussed earlier in this chapter involved inverse Froude number
flow between 0.1 and 0.5. This section introduces results from three classes
of inverse Froude number with a non-linear measure of 
=
0.2, 1.0, and 3.0. The simulations presented here were conducted in the
absence of a mean state critical layer. For 
=
1.0, the flow is nearly blocked from the kinetic energy argument of Sheppard
(1956). Smith (1988) contends that the flow unblocked remains up to 
=
1.3. As before, each simulation is allowed to achieve a pseudo steady state
and then surface heating is introduced. The magnitude of the heating is
different for each case, according to the base state static stability.
In these tests the only variable that is changed is the static stability.
Each simulation is designed to produce a minimum 2 km deep-mixed layer.
For the 
= 3.0 test, this
minimum thickness was not reached in a timely fashion (less than 1.8 days)
owing to strong base state static stability. Table 5.1 contains a summary
of the model parameters used here under the NLP column. In each simulation
the mountain height is 1000m and the base state wind is 10m/s. The mountain
quarter width 
= 15 km is
chosen to force mainly hydrostatic gravity wave modes. A horizontal Rayleigh
type sponge was placed near the downstream lateral boundary to minimize
the boundary effects due to sustained strong perturbation fields located
near the boundary.
5.4.1 Results
Each simulation was advanced to the point where the depth of the boundary
layer is equal to or greater that 2 km. For the 
=
0.2 test, a 3 km deep mixed layer was established only after a short integration
period of 10 hours or 
=
27 using a relatively small maximum heating value of 60 
.
The 
= 1.0 and 3.0 tests
required 
= 96 and 80 and
reached a depth of 2.7 km and 1.4 km, respectively. The required heating
rate maxima and period lengths for the 
=
1.0 and 3.0 tests were 300 
and
40000 seconds, and 360 
and
130000 seconds, respectively. Following the format given in Figure 5.22,
the normalized surface wave drags are given as a function of normalized
mixed layer depths in Figure 5.23 for each of the heated experiments. The
control run wave drags were used to compute the 
(mixed)/
= 0 values. The 
= 0.2 test
did not reach a high drag state or its linear equivalent. The 
=
1.0 and 3.0 cases reached an elevated drag states and were found to be
sensitive to the development of the mixed layer depth. Both tests displayed
significant reductions (40% of the steady state control value) in computed
surface wave drag by the end of the heating cycle.
Figure 5.23. Normalized surface wave drag as a function of the ratio
of the mixed layer depth to the mountain height for the two-dimensional
non-linear parameter experiments.
5.4.2 Discussion
Results from this section indicate that for a meteorologically significant
range of flows over a mountain, the final drag state is sensitive to the
development of a well-mixed boundary layer. In each case wave activity
was reduced by nearly 40% from the control case values. The slope and shape
of the surface wave drag curves are more closely related to the linear
theory solution than the critical layer tests. The slope of the 
=
1.0 and 3.0 curves is non-linear as compared to the linear reduction trends
displayed by the critical layer tests. The 
=
1.0 and 3.0 tests undergo similar reductions in wave drag for differing
mixed layer depths and associated energy input.
5.5 Two Layer Experiments
This section investigates the sensitivity of a strongly forced two-layer flow to surface heating. The purpose is to determine how the non-linear effects of scorer parameter layering are influenced by parameterized surface heating. Previous works assists the choice of the layering configuration. Durran (1986) performed a number of simplified multiple layer tests and found one particular case in which the expected linear response is small and the actual non-linear solution was large. This case involves a stable lower layer ½ vertical wavelength thick with an overlaying less stable upper layer with an assumed infinite depth. The expected linear response of this configuration is approximately ½ of the single lower layer analytic value. In his test, the non-linear effects created a surface wave drag nearly 6 times the expected linear value. Note that there are infinitely many multi-layer configurations to choose from and that only one of the most intriguing is investigated here.
5.5.1 Results and Discussion
The input data for these tests follow Durran?s (1986) Table 1 case 2
entry and are summarized in Table 5.1 under the two-layer test group. The
bottom layer stability is 
=
0.02 and the upper layer stability is 
=
0.01. The control and heated cases were extended to 60,000 seconds or a
non-dimensional time of 
=
120. The heating cycle was enabled at 
=
20. As evident by a plot of the surface wave drag in Figure 5.24, an elevated
drag state develops after approximately 10,000 seconds. Significant oscillations
are present through the first half of the control solution, but during
the last third of the simulation the wave drag is very nearly steady. The
normalized surface wave drag for this test compares favorably with the
results presented in Table 1 of Durran (1986).
The application of the heating cycle reduces the wave drag by almost 50% and the maximum surface wind speed by 15% (Figure 5.25). In terms of the wave response at the surface, these results compare favorably with previous non-critical layer findings. Figure 5.26 provides a comparison of the potential temperature field and indicates a 1.2km deep mixed layer (b). This equates to an expected reduction of 40% by the linear theory presented in Chapter 2. The decrease in the surface drag is approximately 20-25%. Note that the flow downstream of the mountain is weaker in the heated case. This test shows that strongly non-linear flows are just as susceptible to surface heating as their moderately non-linear and linear counterparts.
Figure 5.24. Time series of computed surface wave drag for the two-layer
control (solid) and heated (dashed) simulations. The wave drag is normalized
by the linear hydrostatic lower layer value. The heating cycle began at
10,000 seconds. The heating curve is provided at the bottom of the plot
with a maximum heating rate of 300.
Vertical lines indicate the approximate depth of the mixed layer.
Figure 5.25. Time series of the maximum surface wind speed for the
two-layer control (solid) and heated (dashed) simulations. The heating
cycle was initiated at 10,000 seconds. The heating curve is provided at
the bottom of the plot with a maximum heating rate of 300.
Vertical lines indicate the approximate depth of the mixed layer.
(a)
(b)
Figure 5.26. Plot of the potential temperature field for the two-layer
test at 40,000 seconds for the (a) control and (b) heated runs. The contour
interval is 4
. Only a portion
of the domain is shown.
5.6 January 9, 1989 Boulder, Colorado Windstorm
All previous tests involved simplified base state conditions and idealized terrain profiles. In this section, a numerical experiment is posed using the January 9, 1989 Boulder windstorm event. These simulations include observations taken upstream of the Front Range and a realistic two-dimensional mountain profile. As mentioned in Chapter 4, two-dimensional simulations of observed events have been investigated for years with the purpose of expanding our understanding of observed windstorm characteristics. The ultimate goal is to predict the timing and magnitude of windstorm features with reasonable accuracy. The intent of the simulations presented here is to classify the effects of surface heating on a more realistic atmospheric flow pattern. Comparisons are made to observations, but with the expectations that the details of the observed event are not well represented by the model. The accurate prediction of windstorm onset, magnitude, and dissipation requires a far more sophisticated numerical model and is beyond the scope of this study. Data collected during the Boulder windstorm initializes the control run. This particular event is chosen for two reasons: first, limited observational data are available for verification purposes and secondly, a comparison can be made with the published numerical results of Clark et. al. (1994).
5.6.1 Model Initialization
The model is initialized with the 2305 UTC atmospheric sounding collected from Craig, Colorado. An additional control run was performed on data collected prior to the windstorm (0505 UTC). This test (not shown) did not produce significant surface winds or surface wave drag. Forecast Systems Lab and National Severe Storms Lab personnel collected the data and the final sounding data was provided by Clark et. al. (1994). For each experiment, the base state variables at the lateral boundaries are held fixed with respect to the horizontal advection terms for the duration of the prediction. A more complete investigation of this windstorm event, in which this scenario and others, including those with time dependent lateral boundary conditions, is found in Clark et. al. (1994). Their study contrasts the two and three-dimensional numerical model predictions to the observed surface winds, wind profiler information, and lidar data collected in the Boulder area. Their primary goal was to assess the ability of the numerical model to predict the onset and general windstorm features.
The ARPS terrain pre-processor provided a smoothed terrain profile from
the raw global 5-minute resolution data set supplied by NCAR Data Services.
The 5-minute data are smoothed and matched to the model grid using a multi-pass
Barnes (1964) analysis technique. In this particular application, the Barnes
scheme is applied twice and the resulting data field is available for direct
insertion in the model. The Barnes response function for the final smoothed
data field is determined from a preset first pass response and selected
wavelength. Figure 5.27 displays the response function for these experiments
as a function of model horizontal wavelength (in terms of 
x).
The analysis package operates from the model grid reference and not from
the terrain data spacing. The response for an 8
wave
and 28wave is approximately
2% and 90%, respectively, of their initial values. A detailed description
of the multi-pass Barnes analysis technique and response function is available
in Chapter 8 of the ARPS Users Guide Version 4.0. The terrain profile is
taken along the 40
N latitude
line and passes
Figure 5.27. Barnes two-pass response curve as a function of model
grid spacing for the terrain used in the two-dimensional January 9, 1989
Boulder windstorm experiment. The model horizontal grid spacing is 500m.
through the City of Boulder, Colorado. The model domain is 650 x 28
km and extends from Eastern Utah eastward to the western portion of the
Kansas-Nebraska state line. The predictions do not include the Coriolis
force but incorporate parameterized surface friction. As determined from
the control run, heating begins at 
=
25,000 seconds and the solution advanced to 
=
70,000 seconds. The other pertinent model parameters are summarized in
Table 5.1 under the Boulder test group.
One item that needs explanation is the choice of the upper boundary
condition. Clark et. al. (1994) used a sponge layer in combination with
a rigid lid. Bacmeister and Schoeberl (1989) presented a detailed numerical
study on the impacts of breaking waves in the stratosphere on the near
mountain level flow. Their results indicate that breaking waves aloft can
alter the existing steady state mountain wave flow dramatically by reflecting
upward propagating gravity wave energy downward. Their simulations clearly
show propagation of the momentum flux reduction downward with time. For 
<
0.15, the decrease of the vertical momentum flux computed over the mountain
is found to be periodic in time. For 
=
0.8, reductions in the computed vertical momentum flux originated in the
breaking wave aloft and traveled downward, influencing the flow above the
mountain, but the near surface flow is only minimally affected. Bacmeister
and Schoeberl note a minimal impact of the downward moving disturbance
in the momentum flux field when a breaking layer is observed near the mountain
peak. For cases in which a breaking wave exists near the mountain, the
importance of the upper boundary condition choice is reduced. Given their
result, a linear hydrostatic radiation condition is applied in these simulations.
This method allows wave breaking to occur up to the top of the model domain.
Even though errors due to the non-linear terms at the top boundary are
produced, the majority of the wave energy is absorbed and/or reflected
back towards the surface by breaking waves when they are present below
the model top.
5.6.2 Windstorm Observations
Aside from the collection of data upstream of the Front Range (Craig, Colorado), observations were taken in the Boulder area using surface wind instruments and lidar. The observed winds at the top of the NOAA Building in Boulder are plotted in Figure 5.28. Note the abrupt increase in wind speed near 1100 UTC January 9, representing the onset of the windstorm. The winds remain above 30m/s for several hours but show a steady decrease during the afternoon. The wind speed drops below 20m/s by 0000 UTC January 10, marking the dissipation stage. Figure 5.29 displays a vertical cross section time series of Doppler Lidar observations taken in Boulder and reveal an elevated jet with maxima on the order of 30m/s. An observational study by Neiman et. al. (1988) report the existence of an elevated jet region during windstorm events. The base state wind and temperature profiles measured from the Craig, CO sounding location and used to initialize the model experiments are plotted in Figure 5.30. The data show a strong jet near a height of 10.5 km, corresponding to the level of the tropopause.
Figure 5.28 5-minute peak wind gust
from boulder NOAA building.
Figure 5.28. Peak 5-minute wind gusts as a function of time as measured from the roof of the NOAA Building (20m AGL). Taken from Clark et. al. (1994).
(a)
(b)
(c)
(d)
Figure 5.29. Doppler Lidar vertical cross sections from the Boulder
area at (a) 0015 UTC, (b) 0052 UTC, (c) 0143 UTC, (d) 0230 UTC. Shaded
regions represent velocities >24m/s. Plots taken from Clark et. al. (1994).
The contour interval is 4.0 m/s.
(a)
(b)
Figure 5.30. Vertical profile of base state (a) E-W wind component and
(b) potential temperature measured from the Craig, Co 2305 UTC rawindsonde.
5.6.3 Results
The mountain induced surface wave drag and maximum surface wind speed
as a function of time for the 2305 UTC control and heated runs are contrasted
in Figures 5.31 and 5.32, respectively. The wave drag computed from the
2305 UTC heated run (dashed line) is nearly 20% lower than the control
run (solid line). The maximum surface wind speed is reduced by 20% from
the control run and the depth of the mixed layer reached 1.5 km. The simulated
maximum surface wind speed and observed wind measurements are not directly
comparable, since the model did not include the change in the upstream
conditions with time. But, using data collected upstream at a 2305UTC during
the downslope wind event, the simulated maximum surface wind speed is similar
in magnitude to those measured near the end of the observed storm. Figure
5.33 displays cross sections of potential temperature for the control and
heated tests for the region near the Front Range and the Boulder community.
The heated run exhibits a relatively well-mixed boundary layer approximately
1.5 km deep near the mountain peak. The near surface total horizontal velocity
is disclosed in Figure 5.34 for the 2305 UTC (a) control and (b) heated
runs at t=70,000 seconds. Note in each case that the strongest winds are
not at the surface but elevated a few hundred meters above the surface
and are consistent with the result of Miller and Durran (1991). In the
heated case, the vertical gradient of 
is
weakened, likely owing to the unrealistically strong parameterized vertical
turbulent mixing.
Figure 5.31. Graph of the surface wave drag for the January 9, 1989
Boulder windstorm simulation for the 2305 UTC control (solid line) and
heated (dashed line) runs. The heating curve is provided at the bottom
of the plot with a maximum heating rate of 200.
Vertical lines indicate the approximate depth of the mixed layer.
Figure 5.32. Plot of the maximum surface wind speeds for the Boulder
January 9, 1989 windstorm simulation for the 2305 UTC control (solid line)
and heated (dashed line) runs. The heating curve is provided at the bottom
of the plot with a maximum heating rate of 200.
Vertical lines indicate the approximate depth of the mixed layer.
(a)
(b)
Figure 5.33. Plot of potential temperature at 
=
60,000 seconds for the (a) control and (b) heated 2305 UTC Boulder windstorm
simulations. The contour interval is 2.0 
.
(a)
(b)
Figure 5.34. Horizontal velocity plots for the January 9, 1989 Boulder
windstorm simulation at 
=
70,000 seconds for the 2305 UTC (a) control and (b) heated tests. Area
shown is the lower 6 km in the vicinity of the mountain peak and city of
Boulder. The contour interval is 5.0 m/s.
Figure 5.35 displays the perturbation 
velocity
at selected times for the 2305 UTC control experiment and shows that the
highest winds remain near the foot of the mountain. In addition, a strong
vertical gradient in wind speed is present. This is qualitatively similar
to that shown by the lidar observations (Figure 5.29). Figure 5.36 presents
the perturbation 
field
for the heated case and reveals a general decrease in the elevated jet
magnitude as the mixed layer develops. Overall, surface heating acts to
decrease the lee side horizontal velocity roughly 10-15%. A comparison
of isentropes field for the control run (Figure 5.37a) with the January
11, 1972 simulation given in Chapter 4 reveals two distinctly different
windstorm types. The deviation of the potential temperature surfaces and
related wave induced critical layer directly over the mountain peak are
significantly reduced in the January 9, 1989 case from those of the observations
and idealized predictions of the January 11, 1972 Boulder windstorm event.
This case exhibits characteristics more like the hydraulic flow analog
than the critical layer amplification theory of Peltier and Clark.
(a)
(b)
(c)
(d)
Figure 5.35. Total horizontal velocity on the lee slope for the January
9, 1989 Boulder windstorm control simulation 2305 UTC at 
=
(a) 40,000 seconds, (b) 50,000 seconds, (c) 60,000 seconds, and (d) 70,000
seconds. The contour interval is 5.0 m/s.
(a)
(b)
(c)
(d)
Figure 5.36. Total horizontal velocity on the lee slope for the January
9, 1989 Boulder windstorm heated simulation 2305 UTC at 
=
(a) 40,000 seconds, (b) 50,000 seconds, (c) 60,000 seconds, and (d) 70,000
seconds. The contour interval is 5.0 m/s.
(a)
(b)
Figure 5.37. Model predicted isentropes at 
=
70,000 seconds for the January 9, 1989 Boulder windstorm 2305 UTC (a) control
and (b) heated simulations. The area shown is 647 x 14 km with a contour
interval of 10
K.
5.7 Discussion
In all but one test, parameterized surface heating reduces the established mountain wave activity, as measured by surface wave drag, horizontal velocity at the surface and aloft, and by the vertical flux of horizontal momentum. Reductions on the order of 30% for the wave drag and 10-15% for the maximum surface winds were common in the heated simulations. The response to surface heating is found to be a function of the mixed layer depth, with deeper surface layers forcing larger reductions in the wave activity aloft. The decrease in the steady state flow for the critical layer experiments is approximately linear in terms of the mixed layer depth. This is contrary to the results from the non-critical layer simulations and linear theory where the impact is observed to be a non-linear function of the neutral layer depth.
In tests displaying wave-breaking characteristics, such as those from
the critical layer and 
=
1.0 and 3.0 experiments, the surface drag is weakened but a high drag state
remains. The transition from the steady high drag state to a low drag state
did not occur when influenced by a moderate amount of surface heating.
Most of the tests performed here applied surface heating at a rate similar
in magnitude to that observed in Central Canada in March. In only a few
of the experiments, especially the low mountain height cases did the mixed
layer motion approach that expected from potential theory. In all of the
high drag simulations, the flow in the boundary layer is dominated by the
upper layer wave response. This is expected since the depth of the mixed
layer was rarely greater that 1/4 of a vertical wavelength. For the two-layer
tests, the surface wave drag was reduced 45% by the end of the heating
period. This configuration is especially vulnerable to surface heating,
since the lower layer is only 3km deep. Strong solar heating could all
but wipe out the low-level stable layer. The surface wind speed did not
undergo such a large decrease. It is likely due to two factors: the drag
is a quadratic quantity in perturbation variables and will respond to changes
in the flow more rapidly. Secondly, potential flow theory requires an increase
in the flow on approach to an obstacle.
In strongly forced flows, the linear theory presented in Chapter 2 over predicts decreases in wave activity as a result of a neutral boundary layer. The largest differences with linear theory were noted in the critical layer tests. For two-dimensional flow, linear theory, as presented in Chapter 2, is quite useful in both qualitative and quantitative terms for the moderately forced heated mountain flows.