Strong downslope windstorms have been shown by Durran (1992) and others to be highly non-linear events. Analytical finite amplitude solutions that include a non-linear lower boundary condition are available with the aid of numerical procedures. Given this constraint, a linearized two-layer solution is presented that adds insight to the numerical experiments and observational record. In addition, a scale analysis is provided for mountain waves and boundary layer convection.

2.1 Linear Two-Layer Solution

In the context of this study, linear theory has been applied sparingly in the literature. Diffusion of heat away from the lower boundary complicates the linear analysis considerably by introducing a 4th order governing equation. Not only is it difficult to apply the diffusion term analytically, the results may have little effect on the solution. At the surface, diffusion is useful in transporting heat away from the boundary provided the mixing coefficient is sufficiently large, as is the case in the parameterized methods described later in Chapter 3. But once heat is transferred away from the lower boundary, other effects such as horizontal and vertical advection, with time scales much less than the diffusive time scale, dominate the flow.

A different approach is taken here in regards to the linear analysis. Steady state linear theory is applied from the standpoint that the convective motions, associated with the process of heat redistribution in the mixed layer, are neglected. The objective is to look at the effects of varying mixed layer depths on wave amplitude. From this perspective, the solutions in the overriding stable layer can be easily solved in terms of the mixed layer depth and the horizontal and vertical scales. This assumption is defended later in this chapter.

During the first part of the diurnal heating cycle, the steady state assumption is not defensible as the mixed layer height is strongly time-dependant. The steady state assumption is most likely to be valid in the late afternoon when the mixed layer height is changing slowly and the heat is distributed over a large vertical extent. A time dependent solution in terms of the mixed layer height is not investigated here but is possible through the application of similarity theory (Garratt, 1994).

As mentioned in the review of Chapter 1, most of the analytical work applies the assumption that the heating source decreases away from the mountain. This restriction is not used here. In the real atmosphere, surface heating is not confined to the mountain and extends far upstream. The result is a boundary layer height that is, on average, nearly uniform upstream of the mountain. Consequently, this analysis investigates the significance of different boundary layer depths on the overlying mountain wave flow.

A linearized two-dimensional Boussinesq equation set in terms of , and  is used in this analysis. Little generality is lost from the application of the Boussinesq set of equations, as the effects of decreasing density with height are well known (Gutman, 1991). For the analysis given here, the base state wind is constant () with height and equal in both layers. The stratification is defined as  and is set to zero in the lower layer () and to a positive constant () in the upper layer. The steady state version of the Boussinesq equation set is:

                                                                                                  , (2.1)

                                                                                                  , (2.2)

                                                                                                  , (2.3)

                                                                                                  . (2.4)

Figure 2.1 sets up the problem graphically. The bottom layer (layer 1) is neutrally stratified (= 0) and represents a well-mixed boundary layer. The upper layer (layer 2) is stratified with = constant and supports gravity waves. Equations (2.1)-(2.4) combine to give a single equation in perturbation vertical velocity :

                                                                                                  . (2.5)

Equation (2.5) defines in the stratified environment of layer 2. For the neutral stability of layer 1, (2.5) simplifies to:

                                                                                                  . (2.6)

and describes potential flow. Equations (2.5) and (2.6) are forced by introducing a base state flow () over a small amplitude mountain. The terrain is defined by

                                                                                                  , (2.7)

where  is the horizontal wave number and is the mountain height. This expression can be used to represent a single wave or be combined with other wave components in a Fourier series representation of a particular mountain profile. This example is confined to a single wave component without loss of generality.

                                                    Layer 2 

Figure 2.1. Graphical depiction of the two-layer linear problem. In the analysis, the mountain is chosen to be a cosine function.

Solutions to (2.5) and (2.6) are of the form:

                                                                      , (2.8)

                                                                      , (2.9)

where the wave numbers are:

                                                                      ,

                                                                      .

The wave number in layer 2 is a function of the stability, base state wind, and horizontal wave number. Equation (2.8) is valid for  real. If  becomes imaginary then the solutions no longer admits gravity waves modes and follows the form of (2.9). Assuming hydrostatically forced gravity wave flow in the upper layer ( >0), the wavelike solution of (2.8) holds. The wave number in the bottom layer is equivalent to the horizontal wave number. The complex coefficients ,,, and  are determined from application of the boundary and matching conditions. The upper boundary condition requires energy to propagate out of the domain. Following Eliassen and Palm (1960) this is true when , where the overbar represents the horizontal average. The matching conditions require the displacement and pressure at the layer interface height (z=0) to be equivalent. A number of horizontal wavelengths and mixed layer depths are investigated here. In addition, a discontinuity in density is introduced into the solution. This is intended to represent an inversion placed at the interface between the two layers. Inversions are commonly observed at the top of the mixed layer. The application of the interface pressure condition follows that of Klemp and Lilly (1975) and can be obtained by integrating the hydrostatic relation. These conditions are presented in terms of the vertical velocity:

                                                                           , at z = 0

                                                                           . at z = 0

The second matching condition requires continuity of the vertical derivative of the vertical velocity and is equivalent to matching the horizontal pressure gradient term in the two layers. The term  represents the change in potential temperature across the inversion. Physically, the inversion represents external gravity waves along the layer interface. For the non-inversion case this term vanishes. The bottom boundary condition is linearized using:

                                                                           =  .

Enforcing upward energy propagation away from the mountain in the upper layer requires in (2.8), since the phase () emits only downward propagating energy when sgn() = sgn(). The three remaining coefficients are determined from application of the matching and lower boundary conditions:

                                                                              ,

                                                                              ,

                                                                              ,

where  is the depth of the mixed layer. Solving for the complex coefficients and incorporating them into (2.8) and (2.9) and taking the real parts gives:

                                ,    (2.10)

                                                  , (2.11)

where the constants are given by:

                                                                              ,

                                                                              ,

                                                                              ,

                                                                              ,

                                                                              ,

with,

                                                                              .

Results from the two-layer solution are presented in terms of normalized surface wave drag. For a linear hydrostatic Boussinesq system, the steady state surface wave drag is equivalent to the vertical flux of horizontal momentum (Eliassen and Palm, 1960):

                                                                              , (2.12)

                                                                               ,

where  denotes the complex conjugate and  is the average over one wavelength. The vertical flux of horizontal momentum can be shown to be:

                                                                               ,

or,

                                    .

The sensitivity of the surface wave drag to the mixed layer depth, horizontal wavelength, and for the hydrostatic case the inversion strength, is illustrated by the colored curves in Figure 2.2. For this analysis, the base state wind is = 20m/s and the static stability in the upper layer is = 0.01. The flux is normalized by the = 0 case. As indicated in the figure, the steady state momentum flux in the upper layer is significantly affected by changes in the thickness of the mixed layer and to a lesser extent by the horizontal wave number. The results reveal a nearly 80% reduction of the wave activity in layer 2 for a mixed layer depth of 3km. The wave number dependence is small, with only a 6% decrease in the remaining wave activity for the non-hydrostatic case versus the hydrostatically forced flows for a 3km deep neutral layer. Interestingly, the plot also shows that an inversion acts to offset the reduction of mountain wave activity due to the development of a neutral layer. For a 1.5K inversion, a mixed layer of 0.5km depth is required to remove the inversion layer mountain wave enhancement. A 10K inversion, although not likely to be observed during windstorms, greatly enhances the mountain wave activity. The enhancement is likely due to surface wave effects, but a detailed study has not been performed to confirm this. This analysis can be extended to systems with more than two layers.

Figure 2.2. Plot of the analytical steady state vertical flux of horizontal momentum curves as a function of mixed layer depth, horizontal wave number (16, 80, and 160 km) and inversion strength. The values are normalized by the =0 steady state values.
 
 
 

2.2 Energy and Scaling Considerations

2.2.1 Downslope Windstorm Horizontal Velocity Limit

A simplified system can be used to estimate the maximum winds generated by flow over an obstacle. One such method involves the Bernoulli equation (Fiedler, 1992). Consider the case in which the terrain elevations upstream and downstream of the mountain are equal and the entire upstream flow is reduced to a thin layer in the lee of the mountain. For irrotational flow upstream of the mountain, the Bernoulli relation can be applied along the surface streamline:

                                                                         . (2.13)

The subscripts 1 and 2 represent the upstream and downstream values, respectively. Since the height of the streamline is approximately the same upstream and downstream, (2.13) reduces to:

                                                                         , (2.14)

where  and the density is assumed to be constant for a Boussinesq atmosphere. Solving for the downstream wind (), we obtain:

.

 

                                                                          .

If z =0, and the static stability is assumed constant and defined as , the resulting relation for the downstream wind is:

                                                                          . (2.15)

Equation (2.15) relates the downstream wind speed to the static stability and the change in height of the fluid from its upstream value. For a fluid depth on the order of =10km and static stability = 0.01, the downstream wind speed is on the order of 100m/s. All known observations of windstorms in the lee of the Colorado Rockies lie within the above limit, with wind gusts from 30-60 m/s commonly observed, but no observations of wind speeds approaching 100m/s.
 
 

2.2.2 Mountain Wave Scale Analysis

The scaling of such events can be estimated using the vertical and horizontal time and length scales. The length of the forcing mechanism, the mountain wave number , gives the horizontal scale in this case. For a linearized Boussinesq atmosphere with a constant base state wind, the vertical scale (wave number) is a simple function of the base state wind, static stability, and the horizontal wave number. The Scorer (1955) parameter is:

                                                                            .
 
 

One time scale can be determined from the velocity and the length scales () and represents the time a parcel takes to move through the standing wave. For hydrostatically forced mountain flow with a standard atmospheric temperature and wind profile, the vertical length scale is on the order of 10km. The corresponding time scale is approximately 1-2 hours. In numerical predictions, a steady state value is often obtained after the non-dimensional time of , where a is the horizontal length scale. This measure refers to the time it takes for a parcel to pass through the wave. A second time scale can be defined which involves the Brunt-Vaisala frequency . Taking the reciprocal gives dimensions of time on the order of . Typical values of  in the troposphere are on the order of 0.01  and equate to an approximate oscillation period of 10 minutes. The velocity scale can be defined by the vertical displacement times the stability. This equates to the maximum vertical distance an upstream parcel could be displaced before all the kinetic energy of the parcel has been converted to potential energy. For a 1km tall mountain and stability of 0.01 1/s, the velocity scale is on the order of 10 m/s. From this perspective, a 1km tall mountain can force significant perturbations on the base state to the point in which non-linear effects become important. From the above scaling arguments, it is apparent that the key variables include the static stability, base state wind, mountain height, and horizontal wavelength. These quantities form a basis that will be used in the numerical experiments of Chapters 5 and 6.
 

2.2.3 Convective Boundary Layer Scaling

Motions in the convective boundary layer can often be estimated using similarity theory. This theory is based on the characteristic length and time scales associated with the development of the heated boundary layer. Following a standard text on atmospheric convection, such as Emanuel (1994), it is shown that the velocity scale in a convective boundary layer can be estimated by:

                                                                                          . (2.16)

For a mixed layer depth of 1km and a buoyancy flux  (equivalent to a heating rate of 100), the convective scale velocity is on the order of 1m/s. A number of modeling and observation studies support this scaling result. The horizontal scale of the most unstable convective motion () in a three-dimensional Rayleigh convection problem with free slip boundaries can be formulated in terms of the fluid depth .

Thus, for a three-dimensional heated surface problem, the horizontal scale is approximately six times the vertical scale. The time scale for convection can be estimated from the velocity and length scales.

2.2.4 Discussion

For linear waves, the mountain height is small and the resulting contribution in the boundary layer from the convective motions is on the same order of magnitude as the gravity wave perturbations. For large amplitude mountain waves and downslope windstorms, the convective motions are an order of magnitude smaller than those generated by the gravity wave. The convective time, length, and velocity scales are significantly smaller than those associated with a strong gravity wave. From this perspective, the convective motions should play a minor role in the gravity wave solution. From a gravity wave perspective, as illustrated in Figure 2.2, the impact of the boundary layer convective motions is most important in terms of the development of the mixed layer height with time.

To test these scaling arguments, a simplified two-dimensional mountain wave simulation is presented in Chapter 5 for an explicitly resolved convective boundary layer solution and a less resolved diffusive approach.