AN ANALYTICAL MODEL OF MOUNTAIN WAVE FOR WIND WITH SHEAR

A 2-D analytical mesoscale hydrostatic model of a stably stratified orographic barrier has been considered. Expressions for surface pressure perturbation, mountain drag and energy flux across Pirpanjal mountains of Kashmir valley for variable wind have been derived. These are further evaluated for realistic vertical profile of wind and temperature. Also study has been extended to obtain all these parameters for different vertical wind profile. KeywordsEnergy flux, Mountain drag, Surface pressure perturbation and Pirpanjal Mountains.


INTRODUCTION
Lyra [2] first addressed the study of 2-D mountain wave problem.He considered a 2-D model and obtained solutions using Green's function.Later Queney [3 and 4] purposed a theory in a stratified and rotating atmosphere and applied this theory to the flow over a 2-D bell shaped mountain.Afterward many studies have been done related to mountain wave.
Recently Teixeira et al. [6] developed an analytical model to predict the surface drag exerted by internal gravity waves on an isolated axisymmetric mountain for a velocity profile that varies relatively slowly with height based on Wentzel-Kramers-Brillouin (WKB) approximation.They showed that drag is proportional to inverse Richardson number 1 − i R and it decreases as i R decreases for wind varies linearly with height.Afterward Teixeira and Miranda [7] modified the model of Teixeira et al. [6] to calculate the mountain drag exerted by a stratified flow over a 2-D mountain ridge.They showed that drag is strongly affected by the vertical variation of the background velocity than an axisymmetric mountain and calculated mountain drag and pressure perturbation at surface analytically.
In this paper, the aim is to evaluate the surface pressure perturbation, mountain drag and energy flux for variable wind across double ridge profile of Pirpanjal mountains of Kashmir valley using the model purposed by Teixeira and Miranda [7] for single ridge.Further study has been extended to obtain all these parameters for different vertical wind profile.

THEORETICAL MODEL
The analytical expression of Pirpanjal hills of Kashmir valley (Kumar et al. [1]), whose profile is shown in figure 1 is given by ( ) ( ) where, To evaluate surface pressure perturbation across Pirpanjal mountains of Kashmir valley, we take the following expression of surface pressure perturbation Now substituting equation (2) into equation (5) for real solution, we have Equation ( 6) is the analytical expression of surface pressure perturbation for 2-D profile of Pirpanjal mountains of Kashmir valley.Which contains two parts, first part is antisymmetric with respect to mountains and second part is symmetric with respect to mountains.Now, the expression of mountain drag is (Teixeira and Miranda [7]) Substitute equation (3) into equation ( 7), we get So, mountain drag for real solution becomes As , ( ) By its Fourier transform, we have Substituting ( ) Finally using equation (3) into equation (13) for real solution, we get      The above expressions for surface pressure perturbation contain two parts, first part is antisymmetric with respect to mountains, second part is symmetric with respect to mountains and its contribution is negligible.Also as surface wind decreases in result magnitude of antisymmetric part increases and magnitude of symmetric part decreases.Also if we assume that wind is constant with height, in that case symmetric part becomes zero.
It can be noticed from the expressions of mountain drag and energy flux that as 0 → f , the magnitudes of last two factors increases, which are due to valley between the ridges, thus valley role becomes important for mountain drag and energy flux in case of 0 → f and its profile is shown in figure 4. When the distance between the ridges increases, the valley's contribution started    9) and ( 14) respectively, we get This implies that normalized mountain drag is equal to normalized energy flux and independent on the orographic barrier.
If in case wind rotates with height at constant rate such that ( ) (Shutts and Gadian [5]) where, β is constant, so equations ( 6), ( 9) and ( 14) become the expression of energy flux at surface is

πρ
Using above expressions into equation (

(
its contour is shown in figure8.
Fig. 8, Contour of surface pressure perturbation for         − = c z z U U 1 0 Figure 9, Contour of surface pressure perturbation for    