Miguel A. C. Teixeira 1 and Branko Grisogono 2

1. Miguel A. C. Teixeira1 and Branko Grisogono2 1GGUL, IDL, University of Lisbon, Lisbon, Portugal 2Department of Geophysics, University of Zagreb, Croatia The barotropic Rossby wave drag is given, for a circular mountain with the shape assumed here as (cf. Thompson and Flierl 1993): Introduction For parametrization purposes, mountain wave drag has been studied mostly at the mesoscale. At the planetary scale, the atmosphere has often been treated as unstratified and barotropic, in Rossby wave drag calculations. This study addresses the drag produced by Rossby-gravity internal waves in a continuously stratified atmosphere (see Teixeira and Grisogono 2008). Drag behaviour Fig. 2 shows the drag normalized by its non-rotating value, for a circular mountain, as a function of the two parameters controlling its behaviour, Ro-1=f0a/U and N=a2/U. The drag takes values near 1 when Ro-1 and N are small, is small when Ro-1 is large and N is small, and is particularly large when Ro-1 is large and N=O(1). Fig. 4 shows the ratio of the QG internal gravity wave drag, calculated here for a circular mountain, and the barotropic drag as a function of the two parameters controlling this quantity: N and NH/(f0a) (H is the height of the tropopause in the barotropic model). ROSSBY-GRAVITY WAVE DRAG PRODUCED BY LARGE-SCALEMOUNTAIN RANGES IN A CONTINUOUSLY STRATIFIED ATMOSPHERE The wave equation and solution The inviscid, adiabatic, steady, linearized equations of motion with the Boussinesq, hydrostatic, and beta-plane approximations, may be expressed in Fourier space and combined, yielding: It can be seen that the internal wave drag may be either much smaller or much larger than the barotropic drag, since their functional dependences (on N, for example) are different. However, the ratio can be of order one. This is better illustrated in Fig. 5, where the drag is presented for various latitudes as in Fig. 3, for the fairly typical parameter values N=0.01s-1, U=10m/s and H=10km. Fig. 5a shows the QG internal wave drag. Its behaviour is very similar to that in Fig. 3b, except for narrow mountains, at which the drag approaches zero. Fig. 5b shows the corresponding barotropic wave drag, and Fig. 5c the ratio between the two. It is confirmed that, for typical atmospheric values, the internal wave drag may be as large as the barotropic drag. Fig. 2. Normalized drag as function of Ro-1 and N (solid lines). Long dashed lines: =b/f0, short dashed lines: curves for different latitudes (labelled) with U=10m/s. In Fig. 3, similar results are presented as a function of the mountain’s zonal half-width a, for different latitudes and a wind velocity of U=10m/s. Fig. 3(a,b,c), present results for mountains as in cases 1, 2 and 3 of Fig. 1, respectively, where the aspect ratio of the elliptical cases (1 and 3) is 2. The drag enhancement at large values of a is considerable, being larger for flow across the elliptical mountain, and for higher latitudes. The (zonal) wind and static stability are assumed to be constant with height. The boundary condition at z=0 is: and the wave energy is also required to be radiated upward aloft as z→+. A solution of the following form is assumed: where Internal wave drag The drag is calculated for mountains of elliptical horizontal cross-section, as shown in Fig. 1. The drag is given by: Fig. 5. (a) QG internal wave drag and (b) barotropic wave drag, normalized by non-rotating drag. (c) Ratio of these two drags. Conclusions It was shown that the wave drag produced by internal Rossby-gravity waves generated by orography is, for typical atmospheric conditions, of the same order of magnitude as the barotropic wave drag that is more often calculated, but its dependence on flow parameters, particularly N, is different. where the pressure perturbation is obtained using Fig. 3. Normalized drag as a function of a for various latitudes and U=10m/s. (a) a/b=0.5, (b) a/b=1, (c) a/b=2. Fig. 1. Schematic diagram of the considered situation. The flow is assumed to be constant and zonal. The mountains are elliptical, aligned zonally or meridionally. Quasi-geostrophic flow Internal wave drag enhancement at high Ro-1 can be understood in the quasi- Fig. 4. Ratio of QG internal wave drag to the barotropic wave drag, for a circular mountain. References Teixeira, M. A. C. and Grisogono, B. (2008) Quarterly Journal of the Royal Meteorological Society, 134, 11-19. Thompson, L. and Flierl, G. R. (1993), Journal of Fluid Mechanics, 250, 553-586. and the Fourier transform of the terrain elevation is given by: geostrophic (QG) approximation. This drag can be compared with the barotropic Rossby wave drag, derived for unstratified airflow. for a bell-shaped mountain with half-widths a and b along x and y.