1. Introduction
One of the basic effects of subgrid-scale orography that must be parameterized in large-scale atmospheric models is orographic gravity wave drag. This force arises in stratified flow over mountains due to the upwind– leeward asymmetry of the pressure perturbation associated with stationary internal gravity waves generated by the mountains. With the same value and opposite direction to the drag exerted on the mountains, there is a reaction force acting on the atmosphere that tends to decelerate the flow. When integrated over the main mountain ranges, this force has an important impact on the global atmospheric circulation (McFarlane 1987). In order to develop physically sound drag parameterizations, it is necessary to know how the surface drag varies with the characteristics of the flow: the shape of the orography, its height and width, the stratification of the incoming flow, its velocity, and the associated velocity gradients.
As pointed out by Shutts (1995), surprisingly few studies exist that treat the effect of shear on mountain waves. Until now, most of these studies have either used numerical models or highly idealized analytical models, where the equations of motion are linearized with respect to the perturbations induced by the orography. A notable exception to this linearization is the model developed by Long (1953) where the equation for the streamline displacement is linear, while being valid for waves of arbitrary amplitude. Unfortunately, the validity of Long's solution, which has been used by Huppert and Miles (1969), Lilly and Klemp (1979), Smith (1985), and Durran (1992), is limited to unperturbed flows with constant velocity and 2D orography. Studies using numerical models often focus on the nonlinear regimes that cannot be simulated accurately using the linear approximation, such as resonance (Clark and Peltier 1984; Bacmeister and Pierrehumbert 1988; Scinocca and Peltier 1991; Miranda and Valente 1997) or lee vortices (Smolarkiewicz and Rotunno 1989). In these cases, the effect of shear cannot be easily separated from nonlinear effects.
Despite being limited in their application to flows over gentle orography, linear models are particularly useful for studying the dependence of the drag force on the parameters of the flow in an exhaustive way, because they have analytical solutions. A linear model has been used, for example, by Broad (1995), to derive a quite general extension to 3D of the Eliassen–Palm (EP) theorem.
However, linear models also have the important limitation that closed formulas for the drag can only be obtained for a small set of orography shapes and highly simplified wind profiles. Analytical expressions for the surface drag have been obtained, in the hydrostatic approximation, for flow over bell-shaped ridges or isolated mountains, in the cases of a constant wind (Smith 1979, 1980; Phillips 1984), a wind that varies linearly with height (Smith 1986; Keller 1994; Grubišić and Smolarkiewicz 1997), and a wind profile with directional shear where one of the velocity components varies linearly and the other is constant (Shutts 1995; Shutts and Gadian 1999). The analytical resolution of the equations of motion is practically limited to these simple types of flow, and consideration of more complicated velocity profiles requires the use of approximate methods. One such method is the Wentzel–Kramers–Brillouin (WKB) approximation.
The WKB approximation is applicable to waves propagating in slowly varying media, and has been used by Grisogono (1994) to calculate the surface drag for unidirectional flow with a hyperbolic-tangent profile over a Gaussian ridge. The expression for the surface drag derived by Grisogono does not depend on the first derivative of the velocity, and this is clearly incorrect, as the studies of Smith (1986) and Grubišić and Smolarkiewicz (1997) have shown, respectively, for flow over a ridge and flow over an isolated mountain. In the studies of Shutts (1995) and Shutts and Gadian (1999), the WKB approximation has also been used to determine the vertical momentum flux in an atmosphere with weak shear. But the expression obtained by them for the surface drag is equal to that valid for an atmosphere with constant wind. An aim of this study is to overcome these limitations and present a model based on the WKB approximation that is totally consistent and capable of reproducing the correct asymptotic behavior of the surface drag displayed by previous exact analyses.
The effect of shear on the drag is reflected essentially through a dependence on the Richardson number of the flow, Ri. This study was motivated in part by the numerical results of Grubišić and Smolarkiewicz (1997) and of Valente (2000), which show that the surface drag displays totally different behaviors for a velocity profile with a linear variation and for a wind that rotates with height, decreasing as Ri decreases in the first case and increasing as Ri decreases in the second. Another aim of the present study is to present a framework from which expressions for the drag can be obtained for more general wind profiles, therefore clarifying these differences in behavior.
In this study, an improved linear solution for the surface gravity wave drag on an isolated mountain is developed, assuming that the velocity profile, and hence the vertical wavenumber of the waves varies slowly in the vertical. Adopting a WKB approximation, the vertical wavenumber of the internal gravity waves is expanded as a power series of a small parameter ε inversely proportional to the square root of the Richardson number. Unlike in the WKB treatments of Grisogono (1994) and Shutts (1995), this expansion is extended here to second order in ε, with the consequence that the first derivative of the velocity profile is now correctly predicted to have an impact on the drag, and the correction due to the curvature of the velocity profile is smaller than predicted by the approach of Grisogono, in better agreement with data. Although the WKB approximation is formally valid for high Richardson numbers, it will be seen that the model reproduces reasonably well the results of nonhydrostatic mesoscale numerical models not only for high but also for Richardson numbers of order one, for the three simple flows used as test cases.
This paper is organized as follows. Section 2 presents the theoretical model used in this study. The model is derived for nonhydrostatic conditions but is then simplified for hydrostatic flow. Section 3 presents the results, where the model is tested for three idealized flows. Finally, section 4 contains the main conclusions of this study.
2. A second-order WKB linear model
a. Hydrostatic flow
3. Results
When using polar coordinates to integrate (43), it becomes clear that p̂0, p̂1, and p̂2 do not depend explicitly on the value of the wavenumber k12, but only on the azimuthal angle [see (39)–(41)], so the integration over k12 is equal for D0, D1, and D2 (i.e., amounts to multiplication by the same constant, determined by the shape of the orography). This means that, although the absolute values of D0, D1, and D2 depend on the shape of the orography, their relative magnitude does not, as long as the orography is axisymmetric. This property adds much relevance to the present calculations, which therefore are applicable to any axisymmetric mountain. In nonhydrostatic conditions, this property does not hold, because nonhydrostatic effects depend on the value of the wavenumber, as is evident by looking at the second terms between square brackets in (36)–(38), and this dependence is different for p̂0, p̂1, and p̂2. This property does not hold either for nonaxisymmetric orography, because in that case η̂ depends on the azimuthal angle and this modifies the integration over this variable in (43).

The mathematical form of (50)–(51) remains too complicated and general to be readily understood. For that reason, three simplified flows will be considered next to test the accuracy of these equations against numerical simulation results. These results are taken from Grubišić and Smolarkiewicz (1997) and also calculated using the nonhydrostatic mesoscale model NH3D, developed by Miranda (see Miranda and James 1992).
Since the comparisons will focus on the dependence of the drag on the variation of the mean velocity profile, and this is quantified by the Richardson number, flows with a Richardson number that is constant in height will be addressed for simplicity.
a. Unidirectional wind shear: Velocity with a linear variation
Figure 1 shows the drag force D = Dx normalized by the drag in the absence of shear D0 = D0x as a function the inverse Richardson number Ri−1. Since in (53) the departure of the drag from the unsheared case is inversely proportional to Ri, the corresponding curve is a straight line. Numerical simulation data obtained in approximately linear and hydrostatic conditions by Grubišić and Smolarkiewicz (1997), and results from the exact formula derived by the same authors, which represents finite shear rates exactly, are also displayed as the diamonds and the dahsed curve, respectively. Numerical simulation data from the NH3D numerical model are shown as the circles. It can be seen that the present analytical model reproduces quite well the behavior of the numerical data, even better than the exact theory for relatively low Ri in the case of Grubišić and Smolarkiewicz's data, which is surprising. This is fortuitous and presumably due to factors that are not taken into account by either analytical models. One possibility is that â is too small in the numerical simulations of Grubišić and Smolarkiewicz (1997). This is apparently confirmed by the fact that their exact formula is in better agreement than (53) with data from the NH3D model, where conditions are more nearly hydrostatic. However, results are qualitatively similar and, due to its simplicity, (53) seems more appropriate for guiding estimates of the effect of shear on the drag.
Figure 2 displays horizontal cross sections of the pressure perturbation at the surface calculated with the present model. The zeroth-;, first-;, and second-order contributions to the pressure field are separated to emphasize their different roles. This has been done using the expressions (39)–(41), which were inversely transformed back to physical space using a fast Fourier transform (FFT) algorithm (Press et al. 1992). Figure 2a shows the zeroth-order pressure perturbation, which is antisymmetric relative to the mountain, having a positive maximum on the upwind side and a negative minimum on the downwind side (cf. Fig. 3 of Smith 1980). This causes a positive drag on the mountain, as is well known. The first-order pressure perturbation, on the other hand (Fig. 2b), is symmetric, having a negative minimum, elongated in the spanwise direction, exactly above the mountain top and therefore producing no drag. It is for this reason there is no term proportional toRi−1/2 in the expression for the drag (53). The second- order pressure perturbation (Fig. 2c) is antisymmetric like p0, but has the opposite sign, with a negative minimum on the upwind side of the mountain and a positive maximum on the downwind side. The effect of the p2 field (which is also somewhat elongated in the spanwise direction), is therefore to oppose p0, making the drag decrease. This is of course consistent with the negative term proportional to Ri−1 in (53).
Figure 3 compares the sum of these three pressure contributions, for a flow given by (52) where α < 0 (negative shear) and Ri = 0.5, with a similar flow computed by Grubišić and Smolarkiewicz (1997) using their nonlinear numerical model. The total pressure perturbation shown in Fig. 3a is wedge shaped, with the pressure maximum shifted toward the mountain top. Note how these qualitative features are in close agreement with those quoted by Grubišić and Smolarkiewicz (1997) and visible in Fig. 3b. Figure 3a is also remarkably similar to Fig. 6b of Grubišić and Smolarkiewicz (1997), where the same field is computed using their exact analytical model. This confirms what was already suggested by Fig. 1: that the present model gives accurate and useful results for Ri = O(1).
b. Directional wind shear: Velocity with a linear variation
Figure 4 shows the drag expressions in (55), respectively, as the solid and dotted lines, against data from the NH3D numerical model, respectively, represented by the circles and the squares. It can be seen that, although (55) tends to overestimate the values of the data for Dx/D0x at low Ri, the agreement is quite good, especially for high Ri, consistent with the WKB approximation. The relative dependence of the two components of the drag on the Richardson number is particularly well captured, indicating that the angle between the drag force and the surface wind should be fairly well predicted.
Note that, in this situation, as well as in the situation treated in the previous section, the value taken by the drag does not depend, at second order in ε, on the sign of the shear (the Richardson number only depends on
Figure 5 shows the pressure perturbation for the present flow situation (54). While the p0 field, presented in Fig. 5a, is exactly identical to that in Fig. 2a rotated by 45° (due to the rotation of the surface wind by that angle), the p1 and p2 fields, presented in Figs. 5b and 5c, are qualitatively different from those of Figs. 2b and 2c. Over the mountain p1 still has a negative minimum, therefore producing no drag, but is rotated relative to Fig. 2b by an angle that is smaller than 45°. Field p2 is rotated by an even smaller angle. There are also some differences in the magnitude of the first- and second- order terms of the pressure perturbation, with p1 and p2 being somewhat smaller in Figs. 5b and 5c than in Figs. 2b and 2c. The important point in interpreting (55) is that the relatively small rotation of p2 relative to Fig. 2c counteracts the p0 pressure field more effectively along x than along y. For that reason, the coefficient multiplying Ri−1 is larger in Dx than in Dy. This is an interesting effect, which deserves more detailed investigation.
Figure 6 compares the total pressure perturbation, which is the sum of these three pressure contributions, with results from the NH3D model, for Ri = 0.5. Again, the total pressure field shown in Fig. 6a is wedge shaped, with the maximum slightly shifted toward the mountain top, and it is rotated from the x direction due to the oblique orientation of the surface wind, but this rotation is by more than 45°. The pressure pattern is more asymmetric than in Fig. 3a, due to the different orientations of p0, p1, and p2. This configuration compares very well with the pressure field computed using the NH3D numerical model, shown in Fig. 6b.
c. Directional wind shear: Velocity that turns with height
Figure 8 shows cross sections of the zeroth-, first-, and second-order contributions to the pressure perturbation; p0 is identical in shape and intensity to that presented in section 3a (because the flow is aligned in the x direction at the surface), but p1 and p2 are different; p1 is weak and has positive maxima and negative minima placed, respectively, in the first and third quadrants, and in the second and fourth quadrants of the plot. The symmetry of these pressure patterns implies that there is again no drag at first order in ε, as (57) confirms. The p2 field, on the other hand, is perfectly antisymmetric in the x direction, justifying why there is no drag component along y. The pressure field is elongated in the y direction, and tends to reinforce p0, having a positive maximum on the upwind slope of the mountain and a negative minimum on the downwind slope. Besides having the opposite sign, the maximum and minimum in Fig. 8c are approximately 5/3 larger in magnitude than those in Fig. 2c, reflecting the difference in the coefficients in the corresponding drag formulas (respectively, 3/32 and 5/32).
Figure 9a presents the total pressure perturbation, for (56) with β > 0 and Ri = 0.5. The pressure field is asymmetric, due to the shape of the p1 field, and its asymmetry bears some resemblance to that found in Fig. 6. This is presumably due to the fact that, in both cases, the wind rotates anticlockwise as z increases. However, the asymmetry does not cause here any misalignment of the drag with the surface wind, because both the maximum and the minimum of the pressure perturbation are displaced to the right of the surface wind by a similar amount. This configuration is caused by the p1 field, which weakens the p0 field in the first and second quadrants of the plot but reinforces it in the third and fourth. Figure 9b shows that, although most of these qualitative features are confirmed by numerical simulations of the NH3D model, the pressure pattern is slightly rotated, suggesting a small drag component along y.
d. Discussion
The agreement of the present analytical model with the numerical simulations in the three examples treated above suggests that the pressure field at the surface is primarily influenced by the flow field near the surface and that, for example, critical levels play a relatively unimportant role in determining the surface drag, at least in the parameter regimes considered here. The pressure perturbation that causes the drag is given in general by a 3D Poisson equation that is obtained by taking the divergence of the momentum equation, and using mass conservation. This Poisson equation has a source term depending directly on the unperturbed velocity profile. Due to the properties of the Laplacian operator, the pressure perturbation strictly depends on the velocity field in the whole domain. The fact that, in the WKB approximation, the surface drag only depends on the characteristics of the incoming flow at the surface means that the pressure is determined locally when the wind changes sufficiently slowly with height. This raises two important questions. 1) How slow must this wind variation be? 2) What should, in a real situation, be taken as the “surface” velocity appropriate for using in an inviscid model such as the one developed here?
The answer to question 1 appears to be, not very slow, as is corroborated by the accuracy of the model predictions for Richardson numbers as small as 0.5. One crude explanation for the good performance of the model at Ri = O(1) is that the coefficients multiplying Ri−1 in the drag formulas are relatively small (e.g., 3/32, 1/32, 5/32), so the series expansion for the drag is asymptotic even when Ri is relatively small. More generally, previous studies have shown that the WKB approximation can in fact be applied to situations where it would not, in principle, be strictly valid (Grisogono 1995).
Regarding question 2, boundary layer theory suggests that the inviscid pressure is transmitted almost unaltered through the region where the surface velocity varies due to the no-slip boundary condition. With this reasoning, the velocity (and velocity derivatives) to use in the present model should perhaps be those in the middle or upper part of the atmospheric boundary layer. In their study of the effects of form drag versus gravity wave drag, Belcher and Wood (1996) suggest, more specifically, that the velocity to use in inviscid models should be that at the top of their “middle layer,” which separates the region where shear (but not turbulence) is important from the region where the unperturbed flow is not only inviscid but also approximately irrotational. While Belcher and Wood's (1996) scaling analysis is useful for having an idea of the order of magnitude of this height, their treatment is limited, since they only take into account shear in a logarithmic surface layer, disregarding aspects such as the Ekman boundary layer.
The present study complements the study of Belcher and Wood (1996) by showing that shear outside the boundary layer has an important impact on the surface gravity wave drag. Of course, before the present model can be used in parameterizations of gravity wave drag in large-scale models, the height at which to evaluate the unperturbed velocity and its derivatives has to be defined more objectively. That is a suitable topic for future investigations.
4. Conclusions
An analytical linear model based on the WKB approximation has been developed to investigate the gravity wave drag exerted on mountains by stratified flows where the velocity changes relatively slowly with height. By extending the WKB approximation up to higher order than previously considered, general analytical formulas for the drag have been derived, in the hydrostatic approximation, that show the impact on the drag of the first and second derivatives of the wind velocity at the surface.
To test the drag expressions derived in the present study, three simplified situations were considered, all of them characterized by a Richardson number constant in height: 1) unidirectional shear flow consisting of wind with a linear variation, 2) directional shear flow consisting of wind where one of the velocity components varies linearly and the other is constant, and 3) directional shear flow consisting of wind that turns with height at a constant rate, maintaining its magnitude. In these three cases, the variation of the surface drag with the Richardson number was found to be proportional to Ri−1. For flow 1, the drag is aligned with the surface wind and given by D0[1 − 3/(32Ri)], where D0 is the corresponding value in the absence of shear. In case 2, where the surface wind is at an angle of 45° to the x and y axes, the drag is given by Dx = D0[1 − 3/(32Ri)] and Dy = D0[1 − 1/(32Ri)]. It is, therefore, not aligned with the surface wind, especially for relatively small Ri. In these two cases the drag decreases as Ri decreases. For flow 3, the drag increases as Ri decreases, being given by D0[1 + 5/(32Ri)], and is aligned with the surface wind. All these predictions are shown to be in fairly good agreement with data from numerical simulations of the nonlinear, nonhydrostatic equations of motion, especially for large values of Ri, but also even for values of Ri of O(1).
Cross sections of the pressure perturbation at the surface calculated using the linear model reveal the three leading-order contributions to the pressure and their effects on the surface drag. The zeroth-order contribution is equal to that valid for a wind profile with constant velocity; the first-order contribution is symmetric with respect to the mountain and therefore produces no drag. It is only the second-order contribution (proportional to Ri−1) that modifies the drag, either weakening or reinforcing the zeroth-order contribution. This explains why previous models based on the WKB approximation, which consider terms only up to first-order, fail to detect any differences between the drag for flow with shear and flow with a constant velocity.
In the hydrostatic approximation, the absolute value of the drag in the absence of shear depends on the shape of the orography. But the coefficients of the relative corrections due to the variation of the unperturbed flow do not, as long as the orography is axisymmetric. This happens because the expressions for the pressure at zeroth-, first-, or second-order, are independent of the wavenumber magnitude. Hence, the formulas presented in this study for flows 1, 2, and 3 are valid for any axisymmetric mountain. This property does not hold for nonhydrostatic flows, because the terms associated with nonhydrostatic effects depend on the horizontal wavenumber of the orography.
A natural next step in the line of research initiated by this study is to consider nonhydrostatic effects, which are certainly important for sufficiently narrow mountains, and to extend the analysis to more complicated orography geometries [e.g., elliptically shaped mountains, following Phillips (1984)]. Such developments would facilitate the formulation of parameterization schemes, since in these schemes the anisotropy of the orography must generally be taken into account.
Acknowledgments
This work was supported by Fundação para a Ciência e Tecnologia (FCT) under project BULET/33980/99, cofinanced by the European Union under program FEDER. MACT acknowledges the financial support of FCT under Grant SFRH/BPD/3533/ 2000. Suggestions and comments made by B. Grisogono and two anonymous referees are also gratefully acknowledged.
REFERENCES
Bacmeister, J. T., and R. T. Pierrehumbert, 1988: On the high-drag states of nonlinear stratified flow over an obstacle. J. Atmos. Sci, 45 , 63–80.
Belcher, S. E., and N. Wood, 1996: Form and wave drag due to stably stratified turbulent flow over low ridges. Quart. J. Roy. Meteor. Soc, 122 , 863–902.
Bender, C. M., and S. A. Orszag, 1999: Advanced Mathematical Methods for Scientists and Engineers. Springer, 593 pp.
Broad, A. S., 1995: Linear theory of momentum fluxes in 3-D flows with turning of the mean wind with height. Quart. J. Roy. Meteor. Soc, 121 , 1891–1902.
Clark, T. L., and W. R. Peltier, 1984: Critical level reflection and the resonant growth of nonlinear mountain waves. J. Atmos. Sci, 41 , 3122–3134.
Durran, D. R., 1992: Two-layer solutions to Long's equation for vertically propagating mountain waves: How good is linear theory? Quart. J. Roy. Meteor. Soc, 118 , 415–433.
Grisogono, B., 1994: Dissipation of wave drag in the atmospheric boundary layer. J. Atmos. Sci, 51 , 1237–1243.
Grisogono, B., 1995: A generalized Ekman layer profile with gradually varying eddy diffusivities. Quart. J. Roy. Meteor. Soc, 121 , 445–453.
Grubišić, V., and P. K. Smolarkiewicz, 1997: The effect of critical levels on 3D orographic flows: Linear regime. J. Atmos. Sci, 54 , 1943–1960.
Huppert, H. E., and J. W. Miles, 1969: Lee waves in a stratified flow. Part 3. Semielliptical obstacle. J. Fluid Mech, 35 , 481–496.
Keller, T. L., 1994: Implications of the hydrostatic assumption on atmospheric gravity waves. J. Atmos. Sci, 51 , 1915–1929.
Lilly, D. K., and J. B. Klemp, 1979: The effects of terrain shape on nonlinear hydrostatic mountain waves. J. Fluid Mech, 95 , 241–261.
Long, R. R., 1953: Some aspects of the flow of stratified fluids, a theoretical investigation. Tellus, 5 , 42–58.
McFarlane, N. A., 1987: The effect of orographically excited gravity wave drag on the general circulation of the lower stratosphere and troposphere. J. Atmos. Sci, 44 , 1775–1800.
Miranda, P. M. A., 1991: Gravity waves and wave drag in flow past three-dimensional isolated mountains. Ph.D. thesis, University of Reading, 190 pp.
Miranda, P. M. A., and I. N. James, 1992: Non-linear three dimensional effects on the wave drag: Splitting flow and breaking waves. Quart. J. Roy. Meteor. Soc, 118 , 1057–1081.
Miranda, P. M. A., and M. A. Valente, 1997: Critical level resonance in three- dimensional flow past isolated mountains. J. Atmos. Sci, 54 , 1574–1588.
Phillips, D. S., 1984: Analytical surface pressure and drag for linear hydrostatic flow over three-dimensional elliptical mountains. J. Atmos. Sci, 41 , 1073–1084.
Press, W. H., B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, 1992: Numerical Recipes in Fortran. Cambridge University Press, 992 pp.
Scinocca, J. F., and W. R. Peltier, 1991: On the Richardson number dependence of nonlinear critical-layer flow over localized topography. J. Atmos. Sci, 48 , 1560–1572.
Shutts, G. J., 1995: Gravity-wave drag parameterization over complex terrain: The effect of critical-level absorption in directional wind- shear. Quart. J. Roy. Meteor. Soc, 121 , 1005–1021.
Shutts, G. J., 1998: Stationary gravity-wave structure in flows with directional wind shear. Quart. J. Roy. Meteor. Soc, 124 , 1421–1442.
Shutts, G. J., and A. Gadian, 1999: Numerical simulation of orographic gravity waves in flows which back with height. Quart. J. Roy. Meteor. Soc, 125 , 2743–2765.
Smith, R. B., 1979: The influence of the earth's rotation on mountain wave drag. J. Atmos. Sci, 36 , 177–180.
Smith, R. B., 1980: Linear theory of stratified hydrostatic flow past an isolated obstacle. Tellus, 32 , 348–364.
Smith, R. B., 1985: On severe downslope windstorms. J. Atmos. Sci, 42 , 2597–2603.
Smith, R. B., 1986: Further development of a theory of lee cyclogenesis. J. Atmos. Sci, 43 , 1582–1602.
Smolarkiewicz, P. K., and R. Rotunno, 1989: Low-Froude number flow past three-dimensional obstacles. Part I: Baroclinically generated lee vortices. J. Atmos. Sci, 46 , 1154–1164.
Valente, M. A., 2000: Effects of directional wind shear on orographic gravity wave drag. Ph.D. thesis, University of Reading, 140 pp.
Normalized drag as a function of the inverse Richardson number for the flow given by (52). (solid line) Present model (53); (dotted line) analytical model of Grubišić and Smolarkiewicz (1997); (diamonds) numerical simulations by Grubišić and Smolarkiewicz (1997; â = 5, ĥ ≤ 0.1); and (circles) numerical simulations using the NH3D model (â = 23, ĥ = 0.01).
Citation: Journal of the Atmospheric Sciences 61, 9; 10.1175/1520-0469(2004)061<1040:AAMOMW>2.0.CO;2
Cross sections of the contributions to the normalized pressure perturbation at the surface given by the present model for the flow (52). (thick solid line) Terrain elevation equal to 0.5h0; (solid contours) positive values; (dashed contours) negative values. (a) Zeroth-order pressure, contour spacing 0.05; (b) first-order pressure, contour spacing 0.025; and (c) second-order pressure, contour spacing 0.005.
Citation: Journal of the Atmospheric Sciences 61, 9; 10.1175/1520-0469(2004)061<1040:AAMOMW>2.0.CO;2
Normalized total pressure perturbation at the surface for the flow given by (52) when Ri = 0.5. Shading similar to that used in Figs. 6 and 10 of Grubišić and Smolarkiewicz (1997). (a) Present model, (thick solid line) terrain elevation equal to 0.5h0; and (b) numerical simulations of Grubišić and Smolarkiewicz (1997; reproduction of their Fig. 10b)
Citation: Journal of the Atmospheric Sciences 61, 9; 10.1175/1520-0469(2004)061<1040:AAMOMW>2.0.CO;2
Normalized drag as a function of the inverse Richardson number for the flow given by (54). (solid and dotted lines) The Dx and Dy from the present model (55); (circles and squares) Dx and Dy from numerical simulations using the NH3D model (â = 23, ĥ = 0.01)
Citation: Journal of the Atmospheric Sciences 61, 9; 10.1175/1520-0469(2004)061<1040:AAMOMW>2.0.CO;2
Cross sections of the contributions to the normalized pressure perturbation at the surface given by the present model for the flow (54). (thick solid line) Terrain elevation equal to 0.5h0; (solid contours) positive values; (dashed contours) negative values. (a) Zeroth-order pressure, contour spacing 0.05; (b) first-order pressure, contour spacing 0.025; and (c) second-order pressure, contour spacing 0.005
Citation: Journal of the Atmospheric Sciences 61, 9; 10.1175/1520-0469(2004)061<1040:AAMOMW>2.0.CO;2
Normalized total pressure perturbation at the surface for the flow given by (54) when Ri = 0.5. (thick solid line) Terrain elevation equal to 0.5h0; (solid contours) positive values; (dashed contours) negative values, contour spacing 0.05. (a) Present model; and (b) NH3D numerical model
Citation: Journal of the Atmospheric Sciences 61, 9; 10.1175/1520-0469(2004)061<1040:AAMOMW>2.0.CO;2
Normalized drag as a function of the inverse Richardson number for the flow given by (56). (solid line) Present model (57); (dashed line) WKB model analogous to that of Grisogono (1994); (dotted line) corresponding first-order asymptotic expansion (58); (symbols) numerical simulations by Valente (2000; â = 7.6,ĥ = 0.08)
Citation: Journal of the Atmospheric Sciences 61, 9; 10.1175/1520-0469(2004)061<1040:AAMOMW>2.0.CO;2
Cross sections of the contributions to the normalized pressure perturbation at the surface given by the present model for the flow (56). (thick solid line) Terrain elevation equal to 0.5h0; (solid contours) positive values; (dashed contours) negative values. (a) Zeroth-order pressure, contour spacing 0.05; (b) first-order pressure, contour spacing 0.025; and (c) second-order pressure, contour spacing 0.005
Citation: Journal of the Atmospheric Sciences 61, 9; 10.1175/1520-0469(2004)061<1040:AAMOMW>2.0.CO;2
Normalized total pressure perturbation at the surface for the flow given by (56) when Ri = 0.5. (thick solid line) Terrain elevation equal to 0.5h0; (solid contours) positive values; (dashed contours) negative values, contour spacing 0.05. (a) Present model; and (b) NH3D numerical model
Citation: Journal of the Atmospheric Sciences 61, 9; 10.1175/1520-0469(2004)061<1040:AAMOMW>2.0.CO;2