This chapter presents results from two groups of three-dimensional simulations. The experiments are distinguished by mountain shape. A circular mountain shape defines the first and a ridge of finite length with the long axis oriented perpendicular to the base state flow defines the second. The purpose of these experiments is to investigate the effects of surface heating on mountain wave flows in three dimensions.

These simulations can be directly compared with the numerical solutions for the similarly configured two-dimensional heated mountain wave flows. In addition, the circular and finite ridge shaped mountain flows are contrasted qualitatively with three-dimensional linear theory of Phillip’s (1984), Smith (1980, 1988, 1989), and numerical results of Reisner and Smolarkiewicz (1994). A cross to parallel flow mountain axis ratio of 5:1 defines the finite ridge shape. This ratio is similar to the observed north south to east west Front Range aspect ratio near Boulder, Colorado. A modified form of (4.2) defines the circular mountain profile:

For a circular mountain, the parameters and are equivalent. For the finite ridge simulations, (6.1) is replaced by the two-dimensional equivalent (4.2) between the north and south ends of the ridge. At the ridge ends, (6.1) is applied directly where is defined by the distance from each end of the ridge and is the distance from the ridge line in the direction of the base state flow. For these simulations, the base state flow is directed from west to east.

In each test group, designated by the NLP groups in Table 6.1, three experiments are conducted to test the affects of surface heating over a range of inverse Froude Numbers. The static stability is varied over a range that includes the standard atmosphere. These tests follow those defined in Chapter 5 and are distinguished by their non-linear measure = 3.0, 1.0, and 0.2. The domain size for all the circular mountain flow simulations was 197x117x75 and for the finite ridge = 3.0 and 0.2 tests was 197x157x75 grid points in the x, y, and z directions. The = 1.0 finite ridge simulation required a larger computational domain (237x193x103), owing to observed lateral and vertical boundary condition sensitivities. The initiation and duration of the heating period followed that used in the two-dimensional tests. Aside from the addition of the third dimension, all other experimental variables remained unchanged.

As before, time series of surface wave drag and maximum surface wind speed and vertical cross sections of selected model fields through the mountain centerline are used to measure the wave activity. The predictions are advanced to a pseudo-steady

state prior to the introduction of surface heating. In each test, the magnitude of the pre-heating period wave activity is smaller than the steady state infinite ridge analog and is a function of the blocking characteristics of the flow. According to Smith (1989) blocking along the y=z=0 centerline of a three-dimensional mountain with a ridge width to length ratio of 5:1 occurs at = 1. When blocking commences, flow is diverted around the mountain by a north-south pressure gradient oriented along the mountain y=0 centerline. For circular terrain shapes, a larger mountain is required to produce blocked flow. This is due to the fact that gravity waves generated at the north and south ends of a finite ridge are dispersive in the y-direction. In the three-dimensional problem, the group velocity in the east-west direction is now a function of the north-south wave number and is less than the environmental flow and consequently the disturbance is swept downstream by the base state current.

A summary of the surface wave drag computed along the mountain centerline as a function of and (mixed)/ for all of the three-dimensional heated mountain flow tests is presented in Figure 6.1. The three-dimensional = 3.0 tests recover only 10% of the normalized surface wave drag realized by the two-dimensional counterpart (see Figure 5.23). These values are far below those expected by the linear theory of Phillips (1984) and Smith (1989). Linear theory predicts the steady state three-dimensional surface wave drag for a circular mountain to be 70% of the two-dimensional limit. For a ridge with the long axis oriented perpendicular to the flow and

Figure 6.1. Normalized surface wave drag as a function of the ratio of the mixed layer depth (mixed) to the mountain height for the three-dimensional heated mountain tests. Values are normalized by linear hydrostatic two-dimensional theory.

a long to short axis ratio of > 4:1, the surface wave drag is expected to be >90% of the infinite ridge value. In this case, the mountain wave is weak, with the strongest perturbations occurring in the field as air moves around the mountain and converges downstream. In contrast to the two-dimensional simulations, the =3.0 circular and finite ridge experiments diverted a significant amount of flow around the mountain. Note the strong nearly horizontal flow around the peak and the lee-side convergence in the circular mountain simulation (Figure 6.2). The heating cycle had little impact on the surface wave drag and other flow characteristics above the boundary layer.

According to Smith’s (1989) Figure 1, these tests lie in the region of flow splitting and wave breaking and no longer satisfy his assumptions and boundary conditions. It is apparent that flow separation has dominated the solution, with only a small portion of the flow traversing the mountain. These results are very different from the two-dimensional simulations in which wave breaking is observed above the mountain. A comparison of isentropes along the y=0 mountain centerline from the present model is made with those generated by Reisner and Smolarkiewicz (1994) (Figure 6.3). They used three-dimensional linear theory and a three-dimensional hydrostatic isentropic model to characterize and simulate heated mountain flow over isolated obstacles and the island of Hawaii. Their numerical predictions for = 3.0 with no heating are quite similar to those predicted by the current model for a similar Froude number in their numerical simulations.

Figure 6.2. Surface horizontal velocity vector plot at = 13.3 for the = 3.0 circular ridge test. The terrain contour interval (dotted lines) is 200m. The area displayed is in the vicinity of the mountain peak.

(a) (b)

Figure 6.3. Vertical X-Z cross-section of isentropes along the mountain centerline for (a) the present model at =20,000 seconds =3.0 and (b) from Reisner and Smolarkiewicz (1989) (reference solution) for the = 0.33 circular mountain flow test. The contour interval in (a) is 10K and unknown in (b). The cross section is taken from the mountain centerline y=0.

Both the = 1.0 circular and finite ridge tests achieve steady state surface drag states equal to or greater than the two-dimensional linear normalized values. The circular mountain profile test generated a similar gravity wave response to that expected by a linear infinite ridge. During the heating period, both three-dimensional tests underwent a significant reduction in wave drag and surface wind speed (Figures 6.4 and 6.5). The reduction of surface wave drag when the mixed layer depth is approximately 2km for the three-dimensional simulations (approximately 35%) is almost 1/2 that predicted by linear theory (67%) and larger than the value in the two-dimensional =1.0 experiment (25%). The corresponding maximum surface wind for the three-dimensional tests is 10% lower than the non-heated steady state values. Near the end of the heating cycle, the maximum surface wind actually increases due to unresolved convection near the surface.

Cross sections of total horizontal velocity and potential temperature prior to heating for the two and three-dimensional ridge cases display strong near-surface flow in the lee of the mountain (Figures 6.6 and 6.7). At = 60,000 seconds (= 40), the total horizontal velocity and potential temperature variables, portrayed in Figures 6.8 and 6.9, reveal significant differences between the infinite and finite ridge experiments. The infinite ridge case exhibits a stronger mountain wave flow, with alternating levels of strong and weak flow above the mountain. The infinite ridge potential temperature and total horizontal velocity fields are similar to those shown in the 17km critical layer tests presented in Chapter 5. Well-mixed critical layers define a significant portion of the flow above the two-dimensional mountain. The strong flow between the critical layers advects perturbations significant distances downstream. In the finite ridge case, wave breaking is only evident within a few kilometers above the mountain peak and the flow is considerably weaker. Plots of the vertical velocity at the k=2 and k=30 computational surfaces (Figure 6.10) show a spreading of the wave energy in the north-south and downstream directions with height, in accordance with three-dimensional linear theory. The spreading of the wave envelope with height plays an important role in the development of the flow above the mountain. As shown by Smith (1980), in a Boussinesq atmosphere, the wave amplitude of a three-dimensional gravity wave decreases with height above the mountain. On the other hand, the amplitude for the equivalent two-dimensional gravity wave is not a function of height. Therefore, the amplitude for a given mountain profile should be larger for the infinite ridge case. In the finite ridge tests, there is a slight increase in amplitude near the top of the model domain and is likely associated with a decrease in density with height.

Figure 6.4. Plot of surface wave drag computed along the mountain centerline as a function of time for the = 1.0 two and three-dimensional circular and finite ridge mountain wave simulations. The solid lines represent the control runs and dashed lines the heated experiments. The heating cycle was initiated at 20,000 seconds with a maximum of 250. The vertical lines represent the approximate mixed layer depth.

Figure 6.5. Plot of maximum wind speed along the mountain centerline as a function of time for the = 1.0 two and three-dimensional circular and finite ridge mountain wave simulations. The solid lines represent the control runs and dashed lines the heated experiments. The heating cycle was initiated at 20,000 seconds with a maximum of 250. The vertical lines represent the approximate mixed layer depth.

(a) (b)

Figure 6.6. X-Z cross-section of total velocity for (a) infinite ridge and (b) finite ridge =1.0 mountain wave flows at = 20,000 seconds. The area depicted for the three dimensional test is the centerline of the ridge. The contour interval is 2.5 m/s.

(a) (b)

Figure 6.7. X-Z cross-section of potential temperature for (a) the infinite and (b) finite ridge = 1.0 mountain wave flows at =20,000 seconds. The area depicted for the three dimensional test is the y centerline of the ridge. The contour interval is 2.5K.

(a) (b)

Figure 6.8. X-Z cross-section of total u velocity for the (a) infinite and (b) finite = 1.0 heated mountain wave flows at = 60,000 seconds. The area depicted in the three-dimensional test is the centerline of the ridge. The contour interval is 2.5 m/s.

(a) (b)

Figure 6.9. X-Z cross-section of potential temperature for the (a) infinite and (b) finite = 1.0 heated mountain wave flows at = 60,000 seconds. The area depicted in the three-dimensional plot is the centerline of the ridge. The contour interval is 2.5K.

(a) (b)

Figure 6.10. X-Y plot of vertical velocity at = 20,000 seconds for (a) k=2 (surface) and (b) k=30 computational levels from the = 1.0 finite ridge simulation. The contour interval is 0.1 m/s and the terrain contour interval is 200m.

These tests, with a nonlinear measure = 0.2, are more closely related to linear theory than the two prior experiments and those presented in Chapter 5. The corresponding static stability is small and the vertical wavelength large, approximately 60km. The pre-heating steady state surface drag is nearly equivalent to the linearized analytical values for both the circular and finite ridge tests (see Figure 6.1). Both tests experience diminished surface wave drag and maximum surface wind speed (not shown). The loss in wave drag for the heated finite ridge and circular mountain profiles is 35% and 25%, respectively. The reduction in wave drag for the finite ridge is nearly identical to the two-dimensional counterpart and accounts for only 50% of that predicted by two-dimensional linear theory (Figure 6.11). The response in the surface wind field is similar between the two and three-dimensional tests (not shown). Due to the development of unresolved convection, an increase in maximum surface wind is noted near the end of the heating period. The similarity between the finite and infinite is due to the weak stratification. Smith (1988) shows that the perturbation pressure near the surface is a direct function of the stability. If the gravity wave perturbation decreases, the cross-stream velocity also decreases and more of the incident flow traverses the mountain.

Figure 6.11. Plot of surface wave drag along the mountain centerline as a function of time for the = 0.2 two and three-dimensional circular and finite ridge mountain wave simulations. The solid lines represent the control runs and dashed lines the heated experiments. The heating cycle was initiated at = 10,000 seconds.

The results suggest that the largest differences are associated with the development of flow around the mountain and not associated with the heating aspects of the simulation. For high Froude number tests (= 0.2), the two and three-dimensional finite ridge flow characteristics are very similar in both the pre- and post heating periods. These tests were found to provide the best fit to linear theory. This is expected since the non-linear measure () is small.

For the low Froude number flows tests ( = 1.0 and 3.0), the differences in the solutions were dominated by whether the low-level flow circumvented or traversed the mountain profile. The factors that determine the near surface flow include the base state stability (), the base state wind () and the mountain profile (, , ). The = 3.0 experiments produced the largest deviations from the infinite ridge case. The three-dimensional runs produced significantly reduced perturbations aloft and surface wave drag when compared to the two-dimensional counterparts. A significant portion of the upstream flow is directed around the mountain (see Figure 6.2). Reisner and Smolarkiewicz’s (1994) simulations of flow over an obstacle with = 0.33 compare favorably to the present model’s = 3.0 results. Despite the differences in the heating function strength and spatial orientation, the model predicted lee wake regions are qualitatively similar. Only a weak gravity wave is present above the mountain, as indicated in the potential temperature field displayed in Figure 6.3.

For the marginally non-blocking flow case ( = 1.0), finite and infinite ridge experiments were qualitatively similar in terms of the pre-heating gravity wave response with some notable quantitative differences. The finite ridge pre-heating wave drag is nearly 50% of the infinite ridge counterpart. At the conclusion of the heating period the potential temperature and total horizontal velocity display large differences in the flow field aloft, but the gap in surface wave drag is considerably smaller. The decrease in wave drag due to heating is qualitatively similar in terms of percentage (25% vs. 35%) from the pre-heating values in the two and three-dimensional tests, respectively. Phillips (1984) suggests that an elliptically shaped ridge with a long to short axis ratio > 4 will realize >90% of the infinite ridge surface wave drag. In the strongly non-linear tests presented here, a mountain width to length ratio of 5:1 produces a pre-heating steady state wave drag approximately 75% of the infinite ridge solution and two times the two-dimensional linear normalized estimate. Linear theory does a fair job in the marginally blocking cases.

The circular mountain shape tests failed to produce high drag states over the range of Froude numbers examined here. For the configurations that force strong mountain wave responses, the parameterized surface heating significantly impairs the mountain wave flow. The two-dimensional approximation is qualitatively similar to the finite ridge experiments but notable differences remain which are not well represented by linear theory. If blocking is an issue, the resulting three-dimensional flow pattern can be markedly different from the two-dimensional case. In this case, the introduction of surface heating has little effect on the existing weak mountain wave flow aloft.

Several attempts were made to simulate the January 9, 1989 windstorm in three dimensions using smoothed topography. The results within the first 10 hours are quite good, with a strong response generated in the lee of the mountain along the Front Range, but the simulation degrades during the heating cycle, preventing any useful interpretation. The problem is believed to be associated with the model mass balance. Approximately 10% of the model mass is lost during the second half of the experiment. Both the control and heated tests experienced similar losses in mass and a domain wide deceleration of the flow. The problem can be traced to significant perturbations in the normal velocity component at the upstream boundary. The decelerated flow on the windward side of the Front Range decreases the amount of mass entering the model domain. At the same time, air moves around the mountain range towards Wyoming to the north and exits the model domain. Equations (3.17) and (3.18) in combination with the other predicted variables do not conserve mass at the lateral boundaries. The vast horizontal extent of the mountains forced a compromise between computer resources and safe modeling practices. The resulting three-dimensional computational domain is not able to prevent strong perturbations at the lateral boundaries. Methods exist that can reduce boundary influences. These include nested grid techniques and specifying the lateral conditions from another larger scale model. Both are viable options but are not a part of present model’s framework. The issue of strong forcing at a nested grid boundary or a pre-specified condition is only now being investigated and demands further study. The difficulty with accurately representing the flow over the Front Range stems from the vast extent of the Rocky Mountains in three of the four lateral directions. For a high-resolution simulation (order of 400-meter horizontal grid spacing) a minimum of two nested grids is required inside a larger grid spanning hundreds of kilometers on a side. This configuration would allow strong flow in the lee of the mountain to be resolved and the important upstream conditions to remain relatively undisturbed. Other models, such as the one used by Clark et. al. (1994), incorporate multiple nested grids and update the lateral boundaries from a larger domain model. The physical domain is frequently set to 1200 by 1200 km in order to reduce the lateral boundary influences (personal communication with Bill Hall at NCAR).