A Study of mountain waves using a 2D analytical model

Mountain waves are formed on the leeward side when a uniform and steady air flowimpinges a mountain. Unstable air and severe winds on the leeward side are hazardous tothe flight of aircrafts, change the distribution of aerosols, and cause damage to agricultural products. The purpose of this study is to examine the air flow over the entiremountain region using a two-dimensional analytical meso-scale model. The model wasrun for various conditions by changing one of the main factors or the number of mountainridges. These factors included Lyra parameter (c=2 U/N, where the initial state zonal flow U and buoyancy frequency N are both assumed to be constant), Froude number (Fr=c/2hm, where hm is the amplitude of the mountain ridge), height and width of mountains. For simplicity, we made many assumptions. The initial air flow on thewindward side was assumed to be stable, steady and frictionless with no rotation. Also,the horizontal temperature gradient and thereby the vertical wind shear for maintaining athermal wind balance are constant. These conditions cause the non-linear equations tobecome linear. By running the model, the Helmholtz equation is solved using the first andsecond Bessel equations and the horizontal wind which impacts the mountain nearly at aright angle is transformed to the Lyra parameter.Our results showed that all the factors mentioned above could substantially affect thecharacteristics of the air flow over the mountain ridge as well as at the leeward side. Inthis regard, streamlines as well as the horizontal and vertical components of the windwere investigated. For fixed height and width of a single mountain, the maximumamplitude of the streamlines and thereby the maximum upward motions were weakenedby increasing the Lyra parameter, whereas there was no specific change in the horizontalcomponent of the wind. Alternatively, for a fixed mountain height and the Lyraparameter, the maximum amplitude of the streamlines and the maximum horizontal windwere increased in the wide mountain ridge compared to the narrow one. Since streamlines tilted toward the west with height in the leeward side, the maximum upward motions were decreased somewhat. In the next experiment, for a fixed mountain width and the Lyra parameter, all the characteristics of the air flow in the leeward side (the maximum amplitude of streamlines and the maximum values of horizontal and vertical components of wind) were intensified largely when the mountain height was increased.In general, the effect of the mountain height on the characteristics of the air flowimpinging a mountain is the dominant one among the other factors. It can be mainly dueto the increase of streamline gradient in the windward side of the mountain. Extremelylarge surface and upper-level winds (horizontal and vertical components) were seen in allthe experiments. This is because the friction term has been neglected in the equations. In the real atmosphere, friction would moderate these winds quickly.Now consider a mountain associated with two ridges having equal heights. It isobserved that the existence of the second ridge leads to more intensification of thehorizontal and vertical components of the wind in the leeward side of the second ridge.Also, the lee cyclones produced by the wide ridges, especially over the second ridge,were more intense than those by the narrow ridges. The other noticeable point was thatthe more westward vertical tilt of streamlines in the wide ridges were associated withsmaller vertical motions, compared to the narrow ridges. This is because the verticalgradient of streamlines for the narrow ridges is much more than in the wide ridges. Butgenerally, the upward and downward motions became maxima in the downstream andupstream sides of the troughs, respectively.