Momentum ux due to trapped lee waves forced by mountains

Q. J. R. Meteorol. Soc. (00), 18, pp. 167 173 Momentum ux due to trapped lee waves forced by mountains By A. S. BROAD Met Of ce, UK (Received 18 June 001; revised April 00) SUMMARY A simple, but general, horizontal momentum budget for inviscid ow is developed to understand how the vertical ux of horizontal momentum varies with height in a mountain-forced trapped lee-wave train. Taking a sinusoidal form for the wave eld, from the analytical solution for a two-layer Scorer-parameter atmosphere, the constant Bernoulli functional on a streamline is used to diagnose the momentum ux. It is shown that in an inviscid, steady wave train the magnitude of the momentum ux decreases with height as a sinusoidal function. The present theory clearly shows how this pro le of momentum ux with height is a direct consequence of the exact balance between the vertical derivative of momentum ux and the dynamic pressure difference across the mountain in steady state. The simple analytic pro le of ux with height shows a remarkable qualitative similarity with numerical-model results from idealized case-studies of ow over an isolated mountain ridge. KEYWORDS: Orographic gravity waves 1. INTRODUCTION When air ows over a mountain barrier, gravity waves may be generated in the lee of the mountain. Depending on the upstream atmospheric pro le of wind and stability, and the horizontal length-scale of the mountain, two generic wave trains may be forced: an upwardly propagating wave train, or a trapped lee-wave train which propagates horizontally. As a consequence of the generation of either type of wave train, or a combination of both, a cross-mountain pressure gradient exists, implying a pressure force on the mountain. To counteract this force on the mountain there exists an equal and opposite force acting to maintain the ow in steady state. According to classical gravity-wave theory, in regions of wave dissipation there is a loss of wave energy associated with a non-zero gradient of vertical ux of horizontal momentum with height which acts to decelerate the mean atmospheric ow. A primary source of dissipation is wave breaking of upwardly propagating wave trains which generally occurs at upper tropospheric or lower stratospheric levels. Such wave trains are characterized by a nonzero vertical ux of horizontal momentum which is constant with height until a reduction in magnitude over the vertical depth of the breaking region. An equally important source of dissipation comes from the removal downstream of trapped lee-wave trains. Various studies have shown that, averaged across the mountain, such a downstream gravity-wavetrain generates a non-zero vertical ux of horizontal momentum which decreases in magnitude with height. The classical Eliassen Palm theory (Eliassen and Palm 1960) of wave mean ow interactionwould suggest that this vertical gradient of ux acts to deceleratethe mean ow. However, it is assumed that the trapped wave train is steady and not yet dissipated, indicating that the mean ow has not been modi ed, and the Eliassen Palm theory is not applicable. In this case the assumption of horizontal periodicity, or perturbations vanishing at some distance downstream, is violated. The horizontal wave momentum budget cannot be closed without accounting for the upstream and downstream values of the dynamic pressure (p C ½u, where p is pressure, ½ is density and u is wind speed). In the present note, a simple general theory, which naturally extends the work of Bretherton (1969), is developed to include the dynamic pressure contribution to the momentum budget. For both trapped and upwardly propagating orographically generated gravity waves the dynamic pressure contribution is calculated from an analysis of the Bernoulli function on a streamline. It is shown how recent results (Durran 1991, 1995; Keller 1994; Lott 1998) indicating non-zero, and vertically divergent, wave momentum uxes in a trapped lee-wave train are due to the variation of the dynamic pressure perturbation across the wave train.. WAVE MOMENTUM FLUX DUE TO LEE WAVES (a) Momentum budget analysis Let us consider the speci c problem of atmospheric ow in a vertical plane without background rotation. In a two-dimensional Cartesian coordinate framework the full nonlinear equation of motion in Corresponding address: Met Of ce, London Road, Bracknell, Berkshire RG1 SZ, UK. e-mail: adrian.broadmetof ce.com c Crown copyright, 00. 167

168 NOTES AND CORRESPONDENCE the horizontal direction may be written as u t C uu x C w u z C 1 ½ o p x D 0 (1) in an unsteady, non-hydrostatic, non-viscous anelastic framework. Here u and w refer to the x- and z-direction velocities respectively, and ½ o D ½ o.z/ is the density, taken to be a function of vertical height only. Multiplying (1) by ½ o, and rearranging,leads to the ux form of the u-momentum equation, t.½ou/ C x ½ o u Substituting the anelastic continuity equation, C z.½ouw/ u p.½ow/ C D 0: () z x x.½ ou/ C z.½ ow/ D 0; (3) into () we nd, t.½ ou/ C x.p C ½ ou / C z.½ ouw/ D 0: (4) Equation (4) is the fully nonlinear x-momentum equation in an inviscid uid without background rotation. Partitioning the atmosphere into a basic state (denoted by the subscript o) in hydrostatic balance, and a wave perturbation component, (4) may be rearranged so that t.½ou0 / C x fp0 C ½ o.u o C u 0 / g C z.½ouow0 / C z.½ou0 w 0 / D 0; (5) where u D u o.z/ C u 0.x; z/; w D w 0.x; z/ and p D p o.z/ C p 0.x; z/. The anelastic continuity equation implies x.½ouou0 / C z.½ouow0 / D ½ ow 0 uo (6) z which may be substituted into (5) to leave the result t.½ou0 / C x fp0 C ½ o.u o C uou0 C u 0 /g C z.½ou0 w 0 / D ½ ow 0 u o z : (7) To leading order, locally the vertical shear in the mean wind may be eliminated so that t.½ ou 0 / C x fp0 C ½ o.u o C u ou 0 C u 0 /g C z.½ ou 0 w 0 / D 0: (8) Durran (1991, 1995) performed such a momentum budget analysis to investigate two-dimensional numerical-model integrations of mountain-forced trapped waves. As per Durran, further progress can now be made by using the horizontal averaging operator, hmi D Z XC X where X C > X. Applying this operator to (8) over a suf cient horizontal domain allows for the deceleration effect induced solely by the atmospheric ow perturbations to be accounted for. In the situation of gravity lee waves forced by ow over a mountain ridge, X is taken to be in the undisturbed ow upstream of the mountain, and X C a suitable distance downstream dependent on the type of gravity wave train generated. Applying the averaging operator directly to (8), M dx t h½ ou 0 i C P XC P X C z h½ ou 0 w 0 i D 0; (9) noting that P D p 0 C ½ o.u o C uou0 C u 0 /. Consequently,in steady ow where mountain lee-wave activity occurs, z h½ou0 w 0 i D P X P XC ; (10) with the wave eld being a combination of upwardly propagating and trapped horizontally propagating wave trains in full generality. It remains to diagnose P XC and P X in order to calculate the steady-state pro le of h½ o u 0 w 0 i, the vertical ux of horizontal momentum due to the mountain lee waves, with height.

NOTES AND CORRESPONDENCE 169 (b) Bernoulli functional For the purposes of calculating the dynamic pressure contribution to the momentum budget the fully nonlinear, non-hydrostatic Bernoulli functional on a streamline in steady state is analysed. Taking D z z o as the streamline displacement relative to its upstream undisturbed reference height z o, it can be shown that p 0 C u C w C N D constant D ½ou o ½ o ; (11) where N is the Brunt Väisälä frequency (see Smith (1989) and references therein for the derivation of the hydrostatic version of this relationship). Substituting for u D u o C u 0 in the Bernoulli functional we nd that p 0 C ½o.u0 C u ou 0 / C ½ow0 C ½oN D 0; ) p 0 D ½ow0 ½o.u0 C u ou 0 / when D 0. For evaluation of p 0 it is required that D 0 for substitution into (10), since (10) results from integration with respect to x along a level where z D constant. Equation (1) holds for any type of wave pattern, whether it be upwardly propagating, trapped and leaky in the vertical, or strongly trapped and horizontally propagating. To be able to evaluate fully P XC and P X in (10) the perturbation pressure needs to be known, which is found from the Bernoulli functional as de ned in (1). To evaluate P X we consider the conditions suf ciently far upstream of the mountain where D 0. At such a position no perturbation to the basic state ow exists and w 0 D 0; u 0 D 0, leaving the result that p 0 D 0. Therefore, from the de nition of P, P X D p 0 C ½ o.u o C uou0 C u 0 / D ½ ou o: (13) Being an upstream value of the perturbation dynamic pressure this result obviously holds for either type of downstream wave train. Now consider the conditions downstream of the mountain, either within the trapped wave train, or far downstream of an upwardly propagating wave train where the wave perturbations are once again zero. For an upwardly propagating wave train suf ciently far downstream of the mountain source (large X C ) the wave perturbation to the streamline has vanished and at any height D 0, w 0 D 0 and u 0 D 0. Substituting these values into the Bernoulli relationship given by (1), p 0 D 0, and hence P XC D ½ ou o. Alternatively, for a trapped, horizontally propagating wave train the vertical-velocityperturbations take the form w 0 / sin kx, for which u 0 / cos kx from the continuity equation and / cos kx. It is assumed for the purposes of the present calculation that the wave train does not diminish in amplitude with downstream distance from the mountain source in the inviscid environment under consideration. Then, when D 0 it follows that u 0 D 0 and w 0 6D 0. In fact w 0 is a maximum (D bw.z o /) when D 0 from the functional form w 0.x; z/ D bw.z/ sin.kx/ for a trapped lee wave, and from (1), P XC D ½ ou o ½ow0 = far downstream. In summary, given the form and propagation properties of the two generic wave types forced by an isolated mountain, or mountain ridge, the dynamic pressure perturbation due to the downstream wave train has the following form. For the case of an upwardly propagating wave train, the direction of wave propagation will result in an absence of wave perturbations suf ciently far downstream of the mountain so that P XC D P X. In contrast, for an inviscid trapped, horizontally propagating wave train the wave train will still be present far downstream, and P XC D P X ½ o bw.z/=. (c) Wave stress dependence with height Having calculated the pressure perturbationdownstream of the mountain from the Bernoulli functional, and hence the perturbation dynamic pressure either side of the mountain, the steady state horizontal momentum budget can be examined. From (10) the rate of change with height of the vertical ux of horizontal momentum is equivalent to the difference between the perturbation dynamic pressure upstream and far downstream of the mountain. For the case of an upwardly propagating wave train P XC D P X as discussed in the previous section, and (10) reduces to the result, (1) z h½ ou 0 w 0 i D 0: (14) This result is in accordance with the Eliassen Palm theorem (Eliassen and Palm 1960) as is to be expected. For an upwardly propagating wave train, all wave-inducedperturbations have decayed to zero at a suf cient distance far downstream of the mountain. This lateral boundary condition is appropriate for the use of the Eliassen Palm theorem, and allows the diagnosis of zero pressure perturbation far downstream of the mountain, as used in formulating (14).

170 NOTES AND CORRESPONDENCE The alternative generic situation of a trapped, horizontally propagating wave train produces the result, z h½ ou 0 w 0 i D ½ obw.z/ (15) from the form of the dynamic pressure perturbation in the non-dissipated downstream wave. Here bw.z/ is the perturbation vertical velocity induced by the wave train at a position far downstream of the mountain (x D X C) and evaluated at the same height (z D z o) as the unperturbed mean ow upstream of the mountain (x D X ). Quite clearly, the existence of the assumed inviscid, non-dissipated trapped wave train has an important impact on the wave momentum ux with height. In the steady state limit considered in this contribution the vertical derivative of wave momentum ux is exactly balanced by a non-zero dynamic pressure perturbation in the downstream wave train. A number of papers, both recently and more historically, have shown a non-constant pro le of vertical ux of horizontal momentum with height in the presence of trapped mountain lee waves. Bretherton (1969) and Keller (1994) have both produced theoretical and analytical workings to this effect, although Bretherton used an incorrect rigid lid upper boundary condition. Additionally Durran (1991) and Lott (1998) produced this generic wave momentum ux behaviour with a nonlinear and linear numerical model respectively. Durran and Klemp (198) and Keller (1994), amongst others, provided analytical solutions for speci c upstream conditions giving rise to trapped lee waves. In the two-layer Scorer-parameter situation examined by Durran and Klemp the vertical velocity for the trapped lee-wave train in the lower layer takes the form, bw.z/ D A sin 1.H C z/ sin.kx/; (16) where H is the depth of the lower atmospheric layer and 1 is a vertical wave number associated with the resonant horizontal wave number k of the trapped wave (tan 1H D i 1= ; 1.k/ D.l 1 k / 1= ;.k/ D.l k / 1= where l is the Scorer parameter: see Durran and Klemp (198) for further details). This functional form is very similar to that proposed by Bretherton (1969) except that at the top of the lower layer (z D 0 in Durran and Klemp s analysis), bw is non-zero (Note: bw. H / is the vertical velocity at the boundary and is correctly zero). Therefore, the trapped lee wave is actually leaky into the upper layer. The vertical ux of horizontal momentum can be calculated by using the functional form given by (16) substituted into (15) and integrating in the vertical. It should be remembered that the streamline displacement.d z z o / / R bw.z/ dx / cos.kx/. Therefore, when D 0 for the purposes of substitution into (15) cos(kx) D 0 and sin(kx) D 1, for which bw.z/ A sin 1.H C z/. Performing the integration results in the formula, h i D h½ ou 0 w 0 i D A ½ o Z D K C A ½ o 4 sin 1.H C z/ dz z 1 sin 1.H C z/ ; (17) 1 where K is a constant of integration and A the amplitude coef cient. For an idealized, purely trapped leewave train there is no leakage of wave energy, and hence momentum ux, above H and K 0. For more realistic situations, such as the troposphere stratosphere situation investigated by Durran and Klemp (198) and Keller (1994), K 6D 0 resulting from the constant vertical ux of horizontal momentum associated with an upwardly propagating wave mode which exists above the capping level of the trapped lee-wave train. In the two-layer atmosphere investigated by Durran and Klemp (198) and the troposphere stratosphere situation analysed by Keller (1994) the Scorer parameter decreases with height, leading to a trapped lee-wave train and an upwardly propagating wave train. Figure 1 shows a schematic representation of the gravity-wave eld generated by such a Scorer parameter decrease (as depicted in Figs. 4 and 5 of Durran and Klemp (198)). Figure portrays the resulting ux pro le with height given by (17). The vertical depth, H, of the trapped lee-wave train is taken to be 5 km and K is non-zero above that height, representing the momentum ux in the upwardly propagating wave train. Focusing on the variation of ux with height in the region of the trapped lee wave (below 5 km), the pro le in Fig. bears remarkable qualitative similarity to the numerical results, both linear and nonlinear, shown by Lott (1998) and Durran (1991, 1995) respectively. The shape of the sin 1.H C z/ relationship found for the momentum ux change with height derived in (17) is strikingly similar to the quoted steady state numerical model results. Keller (1994) calculated the line-averaged gravity wave momentum ux at various heights and various downstream positions. Her results show a similar pro le of momentum ux with height to that represented by (17), and shown in Fig., just downstream of the mountain. However, the further downstream the horizontal momentum ux integration was taken, the more constant with height the pro le of momentum

NOTES AND CORRESPONDENCE 171 Figure 1. Schematic representation of a gravity-wave eld forced by an isolated mountain ridge. The wave eld is composed of a horizontally propagating trapped lee-wave train and an upwardly propagating wave train. The dotted line at X D X C indicates the position of the pro les shown in Fig.. Figure. Pro le of the vertical velocity, w, and vertical ux of horizontal momentum,, at the downstream position indicated by X D X C from the composite gravity-wave eld schematically represented in Fig. 1. The magnitude of the momentum ux is non-dimensional and is chosen to highlight the variation with height of the steady state wave eld through the depth of the trapped lee-wave eld and in to the upwardly propagating wave train above.

17 NOTES AND CORRESPONDENCE ux became (Keller (1994) Fig. 1). This does not agree with the ux pro les calculated from a simple trapped lee-wave train by Lott (1998). His wave-train analysis shows a decreasing wave momentum ux pro le with height which is invariant with the distance of the downstream station from the mountain. The discrepancy between the ndings of Keller (1994), and those of Lott (1998), Durran (1991, 1995) and Durran and Klemp (198) comes from the degree of trapping of the lower wave train. If the rate of leakage of wave energy through the trapping layer is small then the trapped lee-wave train will continue to exist for a substantial distance downstream of the mountain. Correspondingly, the wave momentum ux pro le with height will be similar to that depicted in Fig.. When the wave energy leakage through the trapped layer increases, the amplitude of the trapped wave train decreases faster with downstream distance. This is represented by A in (16) being a function of x which reduces with increasing x. Correspondingly, the low-level contribution to the wave momentum ux in (17), from the trapped lee-wave train, decreases with increasing x. The gravity-wave eld calculated by Keller (1994) has a greater rate of energy leakage through the trapping layer than the wave elds considered by Lott (1998) and Durran (1991). 3. DISCUSSION Current understanding of trapped, resonant lee waves is not as advanced as for leaky, upwardly propagating wave modes. Bretherton (1969) was one of the rst to attempt to understand the momentum ux variation associated with a trapped resonant lee-wave train. In order to calculate explicitly the change of ux with height he proposed an analytic model in which a rigid upper lid acts as a wave re ector, and hence trapping mechanism. Bretherton proceeded to calculate a decrease in the magnitude of vertical ux of horizontal momentum through the depth of the lee wave but with an unphysical upper boundary condition. The present theory removes Bretherton s rigid-lid assumption from this calculation. The pro le of wave momentum ux with height is shown to decrease (in magnitude) through the depth of the trapped layer. The physical reason for this can be seen from the Bernoulli equation whereby the dynamic pressure across the mountain is non-zero because of the downstream trapped wave train. This non-zero dynamic pressure difference is exactly balanced by the rate of change of the wave momentum ux with height, leading to the decrease in ux magnitude with height. Recent papers investigating trapped lee waves (Durran 1995; Lott 1998) have appealed to more sophisticated pseudo-momentum diagnostics to account for the pro le of wave momentum ux with height. The present contribution shows this approach is not necessary. The acid test of the current work is to relate the idealized theory to real atmospheric situations. Recently, Georgelin and Lott (001) performed some linear, steady state, non-dissipative numerical-model integrations for IOP3 of PYREX (15 October 1990). They have shown that the observed wave eld was comprised of a short wavelength trapped, resonant mode, and a longer wavelength upwardly propagating wave. Their calculations of wave momentum ux show a decrease in magnitude with height through the depth of the trapped resonant wave, and a constant ux value with height above due to the longer-wavelength wave. The pro le of ux with height is almost identicalwhen the momentum ux calculation was continued out to a distance of 70 km or 140 km downstream of the Pyrenean mountains (Fig. 10 in their paper). The reason for this consistency of pro le is simply due to the resonantly trapped nature of the shortwavelength wave train from their model integrations. The trapped wave train had no energy leakage in the vertical, and continued downstream with almost constant magnitude. This modelling evidence from a real gravity-wave situation supports the theory presented in this paper for a resonant trapped lee-wave train. ACKNOWLEDGEMENTS Many thanks to Glenn Shutts for useful comments on an earlier draft of this paper, and to the reviewer and editor for making useful suggestions for improvement. REFERENCES Bretherton, F. P. 1969 Momentum transport by gravity waves. Q. J. R. Meteorol. Soc., 95, 13 43 Durran, D. R. 1991 Orographic wave drag on the lower troposphere: The importance of trapped waves. Pp. 377 380 in Proceedings of the eighth conference on atmospheric and oceanic waves and stability, 14 18 October 1991, Denver Colorado, USA 1995 Do breaking mountain waves decelerate the local mean ow. J. Atmos. Sci., 5, 4010 403 The third Intensive Observing Period of the Pyrenees Experiment.

NOTES AND CORRESPONDENCE 173 Durran, D. R. and Klemp, J. B. 198 The effects of moisture on trapped mountain lee waves. J. Atmos. Sci., 39, 490 506 Eliassen, A. and Palm, E. 1960 On the transfer of energy in stationary mountain waves. Geophys. Pub.,, (3), 1 3 Georgelin, M. and Lott, F. 001 On the transfer of momentum by trapped lee waves: Case of the IOP3 of PYREX. J. Atmos. Sci., 58, 3563 3580 Keller, T. L. 1994 Implications of the hydrostatic assumption on atmospheric gravity waves. J. Atmos. Sci., 51, 1915 199 Lott, F. 1998 Linear mountain drag and averaged pseudo-momentum ux pro- les in the presence of trapped lee waves. Tellus, 50A, 1 5 Smith, R. B. 1989 Hydrostatic air ow over mountains. Adv. Geophys., 31, 1 41