Linear theory builds a foundation for the understanding of the response of a stratified flow over a mountain (Klemp and Lilly, 1975). Gravity wave strength is a function of the static stability, with higher stabilities achieving stronger wind speeds (up to the point of upstream blocking). The initial response in a gravity wave flow is attributed to a linear term, the vertical advection of the base state density. But as Durran (1992) shows, linear theory can either under or overpredict the wave response of a two-dimensional mountain wave, depending on the upstream wind and density profiles. In a neutrally buoyant environment, gravity waves do not exist. In this situation, potential flow (if other waveforms are absent) dominates the physical processes. The perturbations associated with a neutrally stable flow decrease with increasing distance from the obstacle. The anticipated result is that deeper neutral layers correspond to smaller deflections in an overlying stable layer.

The simplified two-dimensional linear analytical solution to a two-layer atmosphere presented in Chapter 2 demonstrates that a well-mixed boundary layer reduces the existing gravity wave activity in the stable layer aloft by an appreciable amount. The neutral layer is assumed to be horizontally uniform and the result of a well-mixed convective boundary layer. This application of the upstream boundary condition is notably different from Reisner and Smolarkiewicz (1994), where the heating rate is a function of the mountain height. The present form allows for heating far upstream of the mountain, simulating large-scale diurnal heating. For a hydrostatically forced mountain flow, linear theory predicts the decay of wave activity (Figure 2.2). For non-hydrostatic modes, the reduction of the wave activity is a function of horizontal wavelength with shorter mountains exhibiting higher reductions. The decreases in wave activity are modified significantly if a strong inversion is introduced at the top of the mixed layer. Depending on the strength of the inversion, the reduction in wave activity could be reversed, as was the case for a 10inversion. The results from the linear analysis, related to an inversion at the top of the surface layer, compare favorably with those of Klemp and Lilly (1975).

In the numerical experiments, the amplitude of the mountain wave flow aloft is largely insensitive to the method of heat redistribution. Tests indicate that either the explicit or parameterized turbulent mixing methods are adequate for distributing the heat in the mixed layer. Both the timing and magnitude of the integrated momentum flux in the layer aloft were found to be comparable between the two mixing length methods. The explicit method generated short wave length gravity waves at the top of the mixed layer. These gravity waves did not contribute significantly to the pre-existing vertically propagating modes. An important result is that the magnitude of the mountain wave response to parameterized surface heating is a function of the mixed layer depth, with deeper mixed layer producing larger reductions from the pre-heating steady state values. This is true for both the linear and strongly non-linear flow regimes. In experiments where high drag states developed, a mixed layer of appreciable depth inhibits the wave response, but an elevated drag state remained.

The difference between the two and three-dimensional tests were most notable for small Froude number flows, or block flow. In these situations, only weak mountain waves developed and surface heating had little impact on the solution. For non-blocked flows with cross to parallel flow axis ratios >4:1, the effects of surface heating on a three-dimensional mountain wave could be approximated by the two-dimensional case. For circular mountain shapes, the control and heated mountain wave statistics are significantly lower that the two-dimensional equivalent.

The primary objective of this work is to investigate the role of the diurnal heating cycle in downslope windstorm climatology. The tests and analyses were designed to focus on the heating portion of the diurnal cycle. Results from both the linear analysis and the numerous model simulations indicate that strong mountain waves and downslope windstorms are sensitive to parameterized surface heat fluxes. Both the analytical and numerical modeling efforts concur, surface heating decreases the strength of the windstorm. These results are in agreement with the observational studies of Whiteman and Whiteman (1974). The daily windstorm frequency distribution exhibits a distinct minimum just after the maximum solar radiation period, a time when the mixed layer is near its maximum depth. One numerical experiment included a parameterized nocturnal cooling period. The results from that test indicate a decrease in maximum surface winds during the heating period and an increase in wave activity during the cooling period, as measured by maximum surface wind speed and integrated surface wave drag. This test, although representing a small sample size, supports the observed day and nighttime windstorm frequency climatology.

The seasonal trends shown by Julian and Julian (1969) (Figure 1.1) are more difficult to fit to the results presented here, since model heating rates were chosen to match observations from January through March, and the simulations were held to less than 1.5 days in duration. The seasonal minimum observed during the summer months may be due to weak tropospheric stability and light cross-mountain flow.

In terms of specific windstorm events, only one experiment was performed. The January 9, 1989 Boulder, CO windstorm simulation, incorporating a smoothed two-dimensional mountain profile and observed base state atmospheric data, predicted noticeably weaker wind speeds (15%) and surface wave drag (15-20%) on the lee of the mountain at the conclusion of the heating cycle. The maximum surface wind speed time series for the heated test does not register a significant deviation from the control run until seven hours into the heating cycle. This may be due to a positive velocity perturbation associated with neutral layer development near the mountain peak combined with the base state wind sheer.

The simulation was limited in many respects since is was two-dimensional, used a single sounding to initialize the domain wide model variables, and incorporated fixed inflow boundary conditions. Yet, it was able to reproduce a number of the observed windstorm characteristics, such as elevated jet region on the lee slope and strong downslope winds similar in magnitude to those observed in Boulder. The time series of the observed maximum surface wind speed peaks near noon January 9, 1989 and steadily decreases through the afternoon hours. This observation supports the results presented here, but may be fortuitous, as the windstorm may have been adjusting to other upstream influences.

As presented in Chapter 1, forecasting the onset, duration, and dissipation of downslope windstorms remains a challenge. With time scales on the order of a day, disturbance energy can be transported to great heights above and downstream of the terrain feature, requiring a large model domain. For numerical predictions conducted over a relatively short time period (a few hours), the onset and amplitude of the downslope windstorm remains a strong function of grid resolution and boundary conditions. Results presented in Chapter 4 stress the need for significantly enhanced vertical resolution (minimum 250m) in wave breaking regions. Tests of the lateral boundary conditions (not shown), the bulk of which were reported by Durran et. al. (1993), were found to have a profound effect on the windstorm development phase. In some instances, a specific lateral condition on the normal velocity component (Orlanski, 1976) prevented the development of a high drag state entirely.

For strong mountain wave responses, those that are most important to forecasters along susceptible mountainous regions, linear theory overpredicts the decrease of non-linear wave activity due to a developing mixed layer. This overprediction of the wave reduction varied in magnitude from 10% to 100% in the numerical simulations. Including a small inversion in the linear analysis, similar to that developed by the parameterized turbulent mixing, improves its use as a forecasting tool. It is quite capable of predicting the trends in measurable mountain wave quantities such as wave drag, velocity perturbations, and momentum flux transfers.

Results from the analytical and numerical studies presented here provide a guide to improving windstorm forecasts. Whether it is an empirical approach or a three-dimensional time dependant numerical model, the diurnal cycle contributes to the strength of the windstorm. The numerical tests show that parameterized turbulent mixing is sufficient for capturing the time dependant mixed layer height. This is especially useful to mesoscale models, as it relaxes the horizontal resolution requirement. In terms of global climate modeling, the analytical result may be quite useful. Climate models, due to lack of computer resources, are unable to resolve gravity waves and thus, parameterize the transfer of mountain generated momentum flux. The parameterized momentum transfer formulation could be modified, following the analysis in Chapter 1, to include the contribution from a neutral surface layer.

This study did not thoroughly investigate the cooling period of the diurnal cycle. Only one test, as an extension of a heated experiment, included parameterized nocturnal cooling. The results indicate that increases in the low-level static stability, associated with nocturnal cooling, forced amplification of the mountain wave, in support of windstorm climatology. Additional tests are needed verify this result.

The sensitivity to the vertical boundary condition was not thoroughly tested in the strong windstorm cases, although the use of the upper radiation condition and larger vertical extent of the modeling domain is supported by Bacmeister and Schoeberl (1989). They investigated the importance of wave breaking structures in the stratosphere on the flow near the mountaintop and found a strong sensitivity of the near surface flow to breaking waves in the stratosphere. Further work is needed to validate the upper radiation boundary condition in long-term mountain wave simulations.

Satisfactory results for the three-dimensional simulations of the observed January 9, 1989 windstorm were difficult to obtain under the current model configuration. Problems with the lateral boundary conditions prevented any useful comparisons with the two-dimensional tests and the observations. A more substantial three-dimensional modeling study, using grid-nesting procedures, is posed for the future that addresses the upstream and boundary conditions in a more reliable manner.

A summary of the significant contributions is provided below.

Analytical and numerical solutions indicate the reduction in mountain wave

activity is a function of mixed layer depth, with deeper layers producing

larger responses.

The numerical simulations and analytical results support the hypothesis that

the observed diurnal windstorm bias is at least partially attributed to the

response from surface heating.

Linear theory is useful in determining the reduction of wave activity due to a

developing mixed layer to within a factor of two.

Linear analysis shows that the presence of an inversion inhibits the mixed

layer effect on mountain waves. The contribution from the inversion

enhances the usefulness of linear theory when compared to the non-linear

numerical model results. Enhanced vertical resolution is needed in the

inversion.

Parameterized turbulent mixing is sufficient for predicting the height of the

mixed layer.

Mountain wave activity decreases after the development of a surface bound

mixed-layer, yet in the highly non-linear events, a high drag state remains.

Results from experiments using real data follow the idealized counterparts.

Onset and strength of downslope windstorms are sensitive to vertical

resolution and lateral boundary conditions. Vertical resolution on the order

of 250m is required to adequately resolve developing critical layers.

For non-blocking situations, results from the two-dimensional experiments

can be applied to three-dimensional mountains of sufficient cross-flow width.

When strong upstream blocking is present ( =3.0), the three-

dimensional solutions differ significantly from the two-dimensional case and

surface heating has little impact on the solution.