A bulk theory of airflow over a mountain ridge allowing for a stable overlying atmosphere

Abstract

A steady-state, one-dimensional, mesoscale, non-linear problem of the airflow over a mountain ridge is considered in the framework of the bulk theory in an attempt to develop a meteorological generalisation of the hydraulic problem formulated and solved previously by Houghton and Kasahara. In essence, only three modifications of their formulation are made in the present paper: (a) inversion strength is introduced and considered as an additional dependent variable; (b) the atmosphere above the flow is assumed to be stably stratified with a constant vertical potential temperature gradient; and (c) the condition requiring the flow to return to its initial state after descending and moving away from the ridge is removed. The problem is reduced to a transcendental algebraic equation, the solution of which describes all the possible types of flow, including a discontinuous one. A classification and meteorological criterion graphically presented in the form of a map is proposed for the realisation of each type of flow. If the ridge is higher than some threshold, a jump arises above the windward slope and propagates against the flow. As a result, the initial parameters change to new values for which a secondary steady-state solution exists. An iterative method has been developed to calculate these new initial parameters. The general features of the different types of flow, including those with a jump and secondary flows, are discussed. It is shown that if the ridge height exceeds some second threshold there are no solutions at all, which means that the flow is totally blocked. Concrete calculated examples are presented of all possible types of steady-state flows including secondary ones. It is shown that after introduction of these modifications, the hydraulic model of Houghton and Kasahara acquires more real meteorological meaning.