Strong westerly winds in the lee of the Central Colorado Rockies termed "downslope winds" have been observed for many years. Windstorms are characterized by winds greater than 60mph with a duration on the order of several hours. This phenomenon is also observed near other mountain ranges such as the Andes in South America and the Alps in Europe. Until the 1970?s, severe wind events were not well understood. With the location of the National Center for Atmospheric Research Mesa Laboratory in Boulder Colorado in the early 1970?s, local atmospheric scientists became interested in this phenomenon. Several severe events were investigated and the strong winds attributed to gravity wave-related processes. In the years that followed, numerical and analytical studies addressed different aspects of this phenomenon. However, predicting the onset of strong downslope winds remains a forecasting challenge.

The purpose of this work, using analytical and numerical methods, is to investigate the effects of a developing mixed boundary layer (heating period of the diurnal cycle) on idealized mountain wave flows and to propose an explanation for the observed windstorm tendencies. The desire to investigate this topic developed after reading hundreds of articles on mountain waves and downslope windstorms. There are relatively few papers in the literature addressing the effects of a neutral boundary layer on mountain wave flow. In addition, there was no explanation of the observed diurnal windstorm bias (presented in the next section). This chapter includes a review of the Central Colorado windstorm observational record, mechanisms for downslope windstorms, studies on heated windstorms, and windstorm prediction. This chapter is closed with a statement of the project?s objectives and methods.

1.2 Windstorms Observations

Boulder Colorado is located on the lee slope of the Rocky Mountains and has experienced a significant number of high wind events. Three observational studies using data collected in the Boulder area suggest a diurnal variation in windstorm strength and occurrence. The study by Julian and Julian (1969) includes surface meteorological data, newspaper accounts of wind damage, and emergency calls to the local fire departments over the period 1906-1969. Their analysis reveals both diurnal and seasonal variability to windstorm occurrence. They found frequency minima in the months of June, July, and August and a frequency maximum during January (Figure 1.1). The annual peak in January is associated with a minimum in solar radiation and strong cross-mountain flow.

Brinkmann (1974) found 20 windstorm cases over the 1968-1971 period. Her criterion for windstorm occurrence is sustained winds of 22 m/s or wind gusts to 33 m/s (hurricane force). Her analysis produces a ratio of approximately 2.5 to 1 for the number of windstorms occurring at night versus during the day. In addition, the most exposed Boulder wind-recording site indicates a tendency for surface wind speed maxima at 3, 7, 15, and 20 LST. She found instances during the nighttime hours where

Figure 1.1. Boulder, CO monthly windstorm frequency distribution. The data source is Julian and Julian (1969).

the surface winds subside, only to increase dramatically just before sunrise. Brinkmann notes that the wind maxima propagate eastward down the lee slope. Although damaging winds have been documented during the mid-day hours, this observational study shows that the most severe winds occur during the night.

Whiteman and Whiteman (1974) analyzed data collected from 1869-1972 and obtained results similar to those of Brinkmann. Their study includes an hour by hour frequency distribution and is shown in Figure 1.2. The results show a four-fold difference between the frequency maximum at night and the daytime minimum.

Observations during the January 12, 1972 windstorm in Boulder (Figure 1.3) indicate winds speeds gusting to 100 mph, the instrument limit, with sustained winds of 60 mph for periods greater than an hour. From Figure 1.3, it is clear that winds associated with high wind events are inherently gusty. During the observation period, there are two distinct strong wind periods, from 1230-1330LST and 2030-2200LST (a three hour power failure occurred between 1330-1630LST and is not indicated on the strip chart).

Figure 1.2. Hourly frequency distribution for windstorms observed in Boulder, CO during the period 1869-1972. The data is taken from Whiteman and Whiteman (1974).

Insert Figure 1.3 here

anemometer trace from ? wind storm....

from Lilly and Zipser 1972.

Figure 1.3. Anemometer trace from the Southern Hills Junior high School of Boulder Colorado on January 12, 1972 from 1000-2300LST. A power failure interrupted the chart recording at 1325LST and times after this point should be increased by approximately 3 hours. Vertical scale is in mph and time runs from right to left. (From Lilly and Zipser, 1972)
 

1.2 Literature Review

This review focuses on the application of analytical methods to downslope windstorm formation and heated mountain flows. The meteorological literature contains hundreds of papers on gravity wave theory and its application to a variety of problems. In this section, gravity wave theory is used in a linear sense to help explain some of the observed mountain wave characteristics. Other theories such as the hydraulic flow analog are introduced to add another perspective for downslope windstorm development. Finally, heated mountain waves and forecasting efforts are discussed from the limited journal entries.
 

1.2.1 Mechanisms for Downslope Windstorm Generation

Over the past 25 years, a significant effort has been made to unravel the cause of occasional strong winds on the lee side of the Front Range of Colorado, using two and three-dimensional numerical models, analytical theory, and forecasting experience. Three analytical theories explaining the downslope windstorm amplification process are reviewed here.

Clark and Peltier (1977, 1984) and Peltier and Clark (1979) conducted a number of non-linear numerical simulations in which significant amplification in the surface wave drag was observed in combination with a wave-induced critical layer. A critical layer occurs when the phase speed of a wave equals the speed of the flow. For waves locked to terrain, this occurs when the wind speed is reduced to zero. In a critical layer, both the horizontal and vertical velocity components vanish. More information concerning critical layers is available in Bretherton (1966) and Gill (1982). Clark and Peltier suggest that strong downslope winds are coupled to the presence of a wave-induced critical layer, with high winds developing shortly after the critical layer appears. A critical layer generally forms after an intensifying wave overturns and breaks. Their numerical model results reveal surface wave drags of order 6 times greater than the linearized counterpart. Clark and Peltier propose, using resonant linear theory, that wave-induced critical layers develop when the distance between the critical layer and the mountain is:

                                                                                                 n = 0,1,2,3....

They contend that the wave-induced critical layer acts as a reflector of the vertically propagating wave energy.

Smith (1985) applies Long?s equation to a strongly forced mountain flow. The idealized configuration includes a dividing streamline with an initial upstream height. Above the dividing streamline height the flow is assumed to be undisturbed. Over the mountain, the region above the dividing streamline is assumed to be well-mixed. This method implicitly includes a critical layer and wave-overturning characteristics in the region above the dividing streamline. Smith?s theory predicts amplification when the critical layer height is between (¼ + n) and (¾ + n) vertical wavelengths. This amplification is tied to a specific mountain height. If the mountain height is greater than needed, then the upstream conditions may adjust and the theory no longer applies. Durran (1986) and Durran and Klemp (1987) tested this approach to the downslope windstorm problem through the use of a numerical model for a single layer atmosphere. Within the confines of a mean state critical layer required by Smith?s theory, the results were verified for a number of critical layer and mountain heights.

 
                                                                                            , (1.1)

                                                                                            , (1.2)

a relationship for the slope of the free surface can be developed:

                                                                                             . (1.3)

The variables in (1.1), (1.2), and (1.3) are  the horizontal velocity,  the acceleration due to gravity,  the thickness of the fluid, and  is the height of the topography. Flow over an obstacle can be divided up into two categories, subcritical and supercritical, according to the Froude number. In shallow water theory, the Froude number is defined by:

                                                                                             . (1.4)

The Froude number describes the ratio of the advection and pressure gradient terms. Referring to (1.1), the balance of forces for supercritical flow (>1) reveals that the advection term (first term) dominates the pressure gradient term (second term). The resulting acceleration acts to slow the parcel down as it approaches the mountain crest. The slope of the free surface is positive and the parcel increases its elevation on approach to the crest. In the lee of the obstacle, the free surface has a negative slope and the parcel accelerates down the lee slope. In subcritical flow, the pressure gradient due to the deflection of the free surface dominates the advection term (<1) and the fluid accelerates as it approaches the mountaintop. Following (1.3), the slope of the free surface is negative upstream of the mountain and positive downstream of the mountain. A transition to strong flow on the lee side of the obstacle is possible when subcritical flow becomes supercritical. This occurs when the decrease in the thickness of the fluid and increase in the velocity is sufficient to force the Froude number to greater than unity. A diagram of three types of shallow water flow is presented in Figure 1.4 courtesy of Durran (1990). This figure presents subcritical, supercritical, and hydraulic jump fluid flow patterns over an obstacle. A hydraulic jump is defined as a turbulent energy-dissipative region in which a supercritical flow pattern transforms to subcritical flow, and is commonly compared to severe downslope winds. In this case, potential energy is converted to kinetic energy the entire length of the mountain, creating strong lee side flow. Durran (1986) contends that the processes leading to strong winds in the lee of the mountain are explained most accurately by the hydraulic analog. This theory has its limitations, since the free surface assumption prevents vertical gravity wave propagation.

(a)

(b)

(c)

Figure 1.4. Flow regimes for water flowing over an obstacle: (a) supercritical, (b) subcritical, and (c) hydraulic jump. (Taken from Durran, 1990)
 
 

1.2.2 Three Dimensional Theory

Studies of three-dimensional flows over mountains are relatively rare in the scientific literature. Only recently have attempts been made to explain the processes associated with flow over isolated mountains using analytical and numerical methods. The numerical approach is discussed later in the forecasting section.

Three-dimensional analytical mountain wave solutions are more difficult to obtain. There are far fewer papers related to three-dimensional analytical gravity wave solutions as compared to the two-dimensional equivalent. The three-dimensional studies include Wurtele (1957) and Crapper (1959, 1962) for non-hydrostatic modes and the recent work of Phillips (1984) and Smith (1980,1988, and 1989) for the hydrostatic modes. Smith?s (1980) analytical work discusses a number of issues not previously explored. He uses the Boussinesq linearized hydrostatic set of equations to obtain analytical solutions for flow over a circular mountain. The solution for the streamline deflection is of the form:

                                                                      , (1.5)

where,

                                                            , , , , , ,

                                                                                                  ,

and is the angle of the horizontal wave number vector. Figure 1.5 depicts the analytical vertical streamline deflections at different heights in the vicinity of the mountain. The low-level solution is quite similar in shape to the surface of the mountain. Flow is diverted around the mountain by a horizontal pressure gradient in the cross flow direction. Aloft, the solution magnitude over the mountain is reduced but the disturbance extends a significant distance downstream. The disturbance field widens in both horizontal directions with increasing z in response to the non-zero cross-stream group velocity. From (1.5) it is not clear that the magnitude should decrease with height. This reduction in magnitude is offset in a compressible atmosphere by the decrease in density with height. The wave for a three-dimensional problem would likely break but at a higher altitude than the two-dimensional case. Smith performs an asymptotic analysis far above the mountain to explain the solution results. The largest deflection corresponds to the region near the mountain peak. The decrease in the wave amplitude with height can be attributed to the dispersive properties of three-dimensional gravity waves. The disturbance energy propagates along straight lines with slopes:

                                                                                    , ,  .

The group velocities with respect to the mountain are:

                                        , ,  .

(b)

Figure 1.5. Plots of vertical streamline deflection for a three-dimensional linear hydrostatic Boussinesq mountain wave at (a) = /8 and (b) = /2. (Taken from Smith, 1980).

For a non-zero y-group velocity the wavelength in the y-direction must be less than infinity ( > 0). In the two-dimensional infinite ridge limit, the only non-zero group velocity with respect to the mountain is the vertical component. The slope of the group velocity defines the rate of widening with height. For small  and large  the slope is large and disturbance energy is transported vertically.

Phillips (1984) obtained an analytical expression for the surface wave drag for a three-dimensional elliptically shaped mountain. His results show that for a cross-stream to downstream mountain width ratio greater that 4:1, 90% of the two-dimensional surface wave drag is retained. Phillips also contends that since the difference of the maximum pressure perturbation between the infinite and finite ridge cases is about 10%, the three-dimensional problem can be reasonably approximated by the simpler two-dimensional solution. For a circular mountain profile, the surface drag is 30% lower than the two dimensional counterpart.
 

1.2.3 Heated Mountain Waves

Few researchers have addressed the effects of surface heating on mountain waves. Malkus and Stern (1953) performed a linear analysis for a stably stratified atmosphere with a heat source located over an island and the surrounding ocean defined as a heat sink. Their upper boundary condition only allowed lee wave motions in the solution and is not suitable for vertically propagating hydrostatic modes. In addition, their analysis neglected the direct application of diffusion of heat away from the lower boundary.

I have found only one study that includes surface heating or cooling in a non-linear analytical approach. Raymond (1972) uses a modified approach to Long?s equation, which forces non-adiabatic near-surface heating and cooling. His analysis includes a single layer atmosphere with a constant upstream wind and stability profile as required by Long?s method. The solution procedure involves solving the lower non-linear bottom boundary condition via an iterative numerical method. The source terms are introduced by an arbitrary function in x and z located in close proximity to the surface. This particular source function is defined by a Fourier integral in x and a decaying exponential function in z. Raymond?s flux profiles were positioned symmetrically over the mountain with the maximum located at the mountain crest. Results suggest that heated mountains weaken the gravity wave response while cooled mountains enhance the wave activity. These results are limited in scope due to the unrealistic application and spatial arrangement of the source terms in the equation set and from the limited base state conditions. But, more importantly, his analysis did not consider the effects of the airflow on the heat source.

Reisner and Smolarkiewicz (1994) extend Smith?s (1980) three-dimensional analysis by including a surface heating term. In their analysis, the magnitude of the heating function is set to follow the mountain height, with the maximum corresponding to the mountain peak. A result is that the heating portion of the solution contributes only positive perturbations to the solution (Figure 1.6). The heat generated low pressure near the mountain peak creates a horizontal pressure gradient force that accelerates upstream parcels towards the mountain peak. On the lee side, the heat induced pressure gradient force decelerates the previously accelerated flow. The trajectory related minimum pressure perturbation is located just downstream of the peak and corresponds to the peak in the horizontal perturbation velocity. The placement of this minimum is due to the combination of the low pressure associated on the lee slope from the wave response and the advection of the thermally induced pressure minimum from the mountain peak. They found, by comparing linear theory to the numerical predictions of heated flow over an isolated mountain, that linear theory is in error by as much as a factor of two. The linear three-dimensional analytical solutions are useful as an interpretative guide but are not quantitatively applicable to non-linear problems.

In related works by Durran and Klemp (1983) and Smith and Lin (1982), the sensitivity of mountain wave flow to elevated heat sources was investigated. Their results show that mountain waves are sensitive to latent heat releases, with upstream cloud formation reducing mountain wave activity.

An important issue regarding the work presented in the literature needs to be addressed. In each of the above studies involving surface heating, the heat source is located directly over the mountain (except for Durran and Klemp, 1983) and was not a part of the upstream condition. Therefore, the amount of time the parcel spends over the mountain is small compared to the total trajectory time. In the real atmosphere surface heating occurs far upstream of the mountain as well as near the mountain. On length scales of the mountain width, a nearly horizontally uniform mixed layer develops without the assumed mountaintop bias.

1.2.4 Windstorm Prediction

In terms of forecasting the onset, strength, and dissipation of downslope windstorms, there are few non-numerical avenues available to the meteorologist. The analytical methods discussed above are limited in the value added to a specific forecast. Durran (1990) provides a summary of potentially useful suggestions to forecasters for the prediction of downslope windstorms. One of the most relevant issues is an evaluation of the upstream sounding data. The presence of an upstream near-mountaintop inversion and moderate cross-mountain winds (20-40m/s) in the mid-troposphere were found observationally by Brinkmann (1974) and theoretically by Klemp and Lilly (1975) and Durran (1986) to be important to windstorm development. These conditions were shown to favor windstorm development and are commonly observed upstream of the Boulder area during severe windstorm events. Following Clark and Peltier (1977) and Smith (1985), the existence of a critical layer enhances the development of low level high winds. This condition is not very common but is thought to play an important role in the windstorms of the Wasatch Front in Northern Utah and in the Bore of the Yugoslav coast. Along the Wasatch Front, strong easterly winds at the surface are likely when a synoptic scale closed low pressure is situated to the south of Salt Lake City. With this configuration, a critical layer is generally present in the stratosphere.

Another method used to forecast a high wind event is to characterize the synoptic scale weather patterns that favor windstorm development. Five of the most typical synoptic situations were compiled by Scheetz et. al. (1976). The common theme in each of these categories is the presence of moderate to strong mid-tropospheric westerly flow over the Front Range. Figure 1.7 displays the 500mb chart with the surface low and frontal positions for the configuration most likely to produce the most intense windstorm in terms of wind speed and duration. The stability profile that favors strong windstorms involves a stable layer extending above the mountaintop and a deeper less stable layer in the mid- to upper troposphere. Sounding data collected at Grand Junction, CO on the morning of January 9, 1989 are plotted in Figure 1.8. This figure represents what is thought to be a classic Boulder windstorm sounding. A configuration similar to this was found to be very effective in generating high winds near the surface in the numerical simulations of Durran (1986).

Over the past several years the meteorological community in Boulder, CO have developed an expert forecast system. It is largely an empirical approach put forth by Brown (1986) and Brown et. al. (1992) and is based on a combination of numerical model output and windstorm climatology. The forecast pyramid is built upon upstream atmospheric variables including the geostrophic wind at 1000, 700, and 500mb, the temperature difference between 500 and 300mb, the sign of the vorticity advection at 500mb, and the potential for a surface based stable layer in the lee of the mountains. During the 1990-92 windstorm seasons, it was evaluated and found to predict no greater than a 35% probability of high winds for any 6-hour period in the Boulder area. This system is much better at predicting when high winds would not occur. A similar system was applied to the Fort Collins area with better

Insert Figure 1.7 here

Preferred 500mb map

from Brown, 1986.

Figure 1.7. Type 3 windstorm composite chart from Scheetz et. al., (1976). Solid lines represent 500 mb height contours and the dashed lines the surface fronts. The dash-dotted line represents the lee side trough.

Insert Figure 1.8 here

Grand junction sounding

1/9/89 12z.

Figure 1.8. Grand Junction, CO sounding data collected 1200UTC January 9, 1989.

results. This expert forecasting system is designed to predict the maximum wind at a specific location, but fails to provide information about the time of onset and dissipation as well as the duration of the event. I contend (with Durran and Klemp, 1987) that these issues are better addressed through the use of numerical prediction methods.

Clark et. al (1994) performed a number of two and three-dimensional simulations of the January 9, 1989 Boulder wind event and compared the results to observations. They noted significant differences between the two and three-dimensional simulations, with the majority of the differences attributed to fine scale structures. In particular for the lee side gust structures, a near equal partitioning of the energy spectrum near the 3-km horizontal length scale is evident in the three-dimensional case. The two-dimensional study displays more energy at larger horizontal scales, with a number of peaks in the wave number spectrum not present in the three-dimensional simulation. Energy spectra for the gust structure in the north-south direction are centered near the 10-km wavelength, following the general observed variability of the terrain and large east-west oriented canyons. Their simulations predicted a windstorm but the location and timing of the event was inaccurate. The forecast location of the jump structures is west of the observed features. They also found that the gust structures were sensitive to model resolution and the surface drag formulation. In support of previous two-dimensional downslope windstorm modeling studies, propagating gusts were predicted by their model in both the two and three-dimensional simulations. The modeled gusts were found to be similar in structure to those observed with Doppler Lidar.

The general void of detailed three-dimensional numerical simulations of downslope windstorms in the literatures is obviously due to the large domain and resolution requirements. As discussed later in this paper, lateral boundary conditions severely restrict the usefulness of small domain runs. Large computational domains and long time integration are required to fully understand the onset, duration and dissipation of strong windstorms.
 
 

1.3 Objectives

As revealed in the literature review, little effort has been focused on the impacts of surface heating on mountain induced gravity waves. Observations (Figure 1.2) indicate that windstorms occur during all hours of the day but are much more frequent at night. Previous work fails to explain the observational record. Raymond?s study provides insight to the observed diurnal cycle but is limited in its application due to the constraints associated with Long?s finite amplitude theory and the placement of the source terms. Brinkmann, from a relatively small sample size of 20, reported a daytime maximum which is not explained by Raymond?s? preliminary results. The purpose of this study is to investigate the Central Colorado observed downslope windstorm diurnal bias via analytical and numerical means. Specifically, I will address the following questions:

1) Is the observed diurnal downslope windstorm frequency distribution attributable to the diurnal heating cycle?
 
2) To what extent can linear theory be used to predict the non-breaking mountain wave and downslope windstorm response to a well-mixed surface layer?

3) Can the heated gravity wave response in Central Colorado be approximated by two-dimensional simulations or are three dimensions required?

4) Are large eddy motions in the convective boundary layer needed to accurately predict the diurnal response of strong mountain waves and downslope windstorms?

5) Is the surface heat flux budget important in improving the predictability of strong mountain waves?

1.4 Methods

This study applies both analytical and numerical methods to investigate the questions posed in the previous section. The analytical approach involves a simplified linear two-layer solution to assess gravity wave responses to variable horizontal forcing wavelengths and mixed layer depths. The simplified two-layer linear approach is chosen for two reasons. Linear theory captures the basic gravity wave structure and the two-layer configuration allows for the introduction of a neutral surface layer. Other methods used in this study include a scale analysis of the convective boundary layer motions and mountain forced gravity waves. A theoretical limit to downslope windstorm strength is also reviewed.

The majority of the reported results are obtained from application of numerical methods. A numerical model is used to simulate two and three- dimensional mountain wave and downslope windstorm responses to parameterized diurnal heating cycles. The desire to incorporate three-dimensional aspects is brought about by the results of Reisner and Smolarkiewicz (1994) and Clark et. al. (1994). Initially, the model is applied in idealized two and three-dimensional configurations. A more realistic two-dimensional downslope windstorm experiment is included for comparison purposes. The strength of the mountain wave response is measured in terms of first and second order gravity wave properties. Following Eliassen and Palm (1960), computed surface wave drag and vertical profiles of the horizontally integrated vertical flux of horizontal momentum are compared. In most cases, the maxima in horizontal surface wind speeds are used to assess the gravity wave response. All numerical simulations presented in this study adhere to the following protocol:

a) Obtain a steady state non-heated mountain wave solution.

b) Calculate surface heat fluxes and assess the response in terms of wave properties.
 

An alternative modeling approach that could be performed begins with a characteristic atmospheric profile and applies the cooling portion of the diurnal cycle. A potential problem with this procedure is that the depth of the stable surface layer is small compared to the daytime boundary layer. The anticipated effects would be small since only a shallow stable layer is created overnight. The advantage to the approach is the generation of near surface stable air upstream of the mountain. It is not clear if the stable air settles in the valleys or is able to pass over the mountain and enhance the wave activity. This problem could be addressed in future work.

In order to keep the analysis simple, the earth?s rotation is not included in the experiments. The time scale for the hydrostatic waves (1/) is significantly smaller than for the rotational modes (1/), justifying the non-rotating assumption. For details on the effects of the Coriolis term on the solutions see Lilly (1983) and Clark et. al. (1994). The numerical experiments are categorized in terms of the dimensional arrangement and initial conditions.

1) Two Dimensional Idealized Mountain Profile

            - Non-linear narrow and wide mountain shapes in a single layer atmosphere using two different heat distribution methods (parameterized
                turbulent diffusion vs. explicit convection).

            - Mean state critical layer simulations for a simple one-layer  atmosphere and wide mountain shape.

            - Non-linear parameter range study.

            - Two-layer tuned atmosphere simulations.

  2) Two Dimensional Central Colorado January 9, 1989 Windstorm

            - Numerical experiment with a smoothed terrain cross section through Boulder, Colorado (40N latitude) and Craig, Colorado 12Z sounding.

  3) Three Dimensional Idealized Mountain Profile

            - Non-linear parameter range experiments for circular and  finite ridge mountain shapes (compared to 2-D tests).
 

This study is organized as follows: Chapter 2 presents a two-layer linear analytical solution and reviews the energetics of mountain waves and the convective boundary layer. Chapter 3 describes the numerical model formulation and Chapter 4 displays model verification test results. Chapter 5 and Chapter 6 focus on the two and three-dimensional numerical simulations and Chapter 7 provides a summary of the results. The appendices give additional information on the model?s vertically implicit time marching method, upper  radiation boundary condition, computational efficiency, streamline and trajectory computations, and atmospheric sounding profiles.