Non-Hydrostatic Effects in the Interaction between Flow and Orography

arXiv:0802.3084v1 [physics.ao-ph] 21 Feb 2008 Under consideration for publication in J. Fluid Mech. 1 Non-Hydrostatic Effects in the Interaction between Flow and Orography By I V A N G L A D I C H1, F. S T E L2, D. G I A I O T T I2 AND G. F U R L A N3 1 Department of Mathematics, University of Trieste, Italy 2 ARPA-CRMA, via Cairoli 14, I-33057, Palmanova (UD), Italy 3the Abdus Salam International Centre for Theoretical Physics (ICTP) Strada Costiera, 11 I-34014 Trieste Italy (Received ?? and in revised form ??) The interaction between flows and orography is a fundamental aspect of theoretical fluid dynamics for its direct applications (e.g., in dynamical meteorology); a comprehensive description is nowadays still lacking in some aspects. In this work, in particular, the authors would like to face the problem of flow-blocking and of the streamlines pattern formation, examining the role of stratification (i.e., Brunt-Vaisala frequency) and Froude number on these problems. In particular this work wants to investigate the role of vertical advection on flow-blocking and on streamlines geometry. The importance of streamlines curvature and stratification for the formation of pressure perturbation, then their role in flow-blocking will be shown. Moreover it will be shown how flow-blocking cannot be easly predict using only a stratification parameter or the Froude number. 1. Introduction The interaction beetween flow and orography is an important topic of theoretical fluid dynamics because of its direct applications in everyday life. As an example, orographic rain is originated by a moist flow that, interacting with orography, gives rise to a vertical motion, then to condensation and precipitation formation. Even if these phenomena are very common, their explanation is nowadays not complete. Infact a full description requires the knowleadge of the solution of Navier-Stokes equation with complex boundary conditions (i.e. top of troposphere and the orography). The literature facing the interactions between flows and orography can be divided in: i) numerical works, ii) analytical works and iii) experimental works: Riley et al. (1976), Baines (1979), Hunt & Snyder (1980), Castro et al. (1983) and Snyder et al. (1985). Among the numerical studies there are several contributions produced using hydrostatic numerical models (Smolarkiewcz & Rotunno 1989) and more recently some contributions realized using non-hydrostatic numerical models (e.g., the Weather, Research and Fore- casting model, WRF) (Miglietta & Rotunno 2005). The problem with numerical models, both hydrostatic and non-hydrostatic, is that their output is extremely complex then, generally, very difficult to interpret weighting the physical role of every possible param- eter used in the tuning of the model (Giaiotti et al. 2007). On the contrary analytic works permit to keep a more complete control of the role of each parameter inserted into the analytical model even if some approximations need to be taken to reduce the mathematical difficulty of the starting equations. 2 I. Gladich, F. Stel, D. Giaiotti and G. Furlan Concerning the analytical works, several of them make use of the hydrostatic approxi- mation and of different kind of obstacles (Lilly & Klemp 1979), using both stratified and rotational fluids (Inttyre 1972), using thermal forcing (Reisner & Smolarkiewicz 1993) or imposing turbulent boundary layer at the surface (Carrunthers & Hunt 1990). Only a few analitical works avoid the use of the hydrostatic approximation, this because the vertical advective term makes the analytic approch more difficult. A comprehensive review of all these works can be found in Baines (1995). Among the analytical works, three of them, Smith (1989a), Wurtele et al. (1987) and Keller (1994) deserve a special mention. In particular Smith (1989a), developing the previous work of Smith (1988) and Smith (1989b), studies the interaction between hy- drostatic and stratified flow on an idealized 3-D topography. In his contribution the attention is focused on the case in which the flow stops its upward motion while moving on the topography (i.e., stagnation of the flow). Moreover, in this work, the use of Froude number as a discriminating factor between stagnation and non-stagnation, proposed by Sheppard (1956), is critically reviewed. In the work of Keller (1994) the study of inter- action between non-hydrostatic and stratified flows on an idealized 2-D topography is presented but, in this case, the attention is focused on the effects of non-hydrostaticity on the formation of downstream lee waves. Keller (1994) also analizes the behaviour of the flow with different vertical velocity profiles. The interesting aspect of Wurtele et al. (1987), instead, stays in the approach to the gravity wave propagation in stratosphere. Following the line defined by the three analytical works above introduced, the aim of this work is to study analytically the influence of non-hydrostatic effects on the geometry of streamlines, essen

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