Wind direction effects on orographic drag

Transcription

1 QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY Q. J. R. Meteorol. Soc. 134: (2008) Published online in Wiley InterScience ( Wind direction effects on orographic drag H. Wells, a * S. B. Vosper, a A. N. Ross, b A. R. Brown a and S. Webster a a Met Office, Exeter, UK b University of Leeds, UK ABSTRACT: A series of idealised numerical simulations is performed to investigate the effect of wind direction on the pressure forces exerted on a high elliptical mesoscale ridge in the presence of Coriolis effects. At the Rossby number considered here (Ro 13), rotational effects have a significant impact on the flow fields, however the primary effect of rotation on the drag is to provide the asymmetry required to initiate vortex shedding when the flow is perpendicular to the mountain ridge. It is found that linear theory, although not valid for such high mountains, provides a useful scaling for the variation of drag with wind direction. For a large range of wind directions, the flow is in a high- (super-linear) drag state and wave breaking, vortex shedding and upstream flow blocking are observed. However, when the flow is close to being parallel to the major axis of the mountain ridge, the drag becomes sub-linear, and none of the above processes are seen. We show that the change from a high-drag state to a low-drag state can be explained in terms of the aspect ratio of the mountain, that is the ratio of the across-flow mountain length to the along-flow length. Finally we demonstrate that the results found for the idealised elliptical mountains also apply to a real mountain of similar dimensions. Copyright 2008 Royal Meteorological Society and Crown Copyright 2008, KEY WORDS gravity waves; mountain wakes; flow blocking; parametrization Received 10 December 2007; Revised 6 February 2008; Accepted 14 March Introduction Understanding the air flow around high mountain ridges is of considerable importance for predicting the dispersion of pollutants, the occurrence of atmospheric turbulence and for weather forecasting in mountainous regions. Additionally, being able to predict the drag that is exerted on these ridges is of prime importance for the development and testing of orographic drag parametrizations for numerical weather prediction (NWP) and climate models. Many previous studies have investigated flow past a high finite-length mountain ridge oriented perpendicular to the incident flow (e.g. Peng et al., 1995; Ólafsson and Bougeault, 1996, 1997; Bauer et al., 2000; Wells et al., 2005). Here by high mountain we refer to places where the non-dimensional mountain height h m > 1, where h m = Nh m/u and N is the Brunt Väisäla frequency, h m is the mountain height and U is the upstream wind speed. In contrast, only a few studies have looked at the problem of different orientations of high ridges relative to the incident wind direction. Recently Zängl (2004) examined the flow upstream of an idealised ridge with similar dimensions to the Alps under a range of different upstream wind directions. However he did not describe the pressure forces exerted * Correspondence to: H. Wells, Met Office, FitzRoy Road, Exeter EX1 3PB, UK. helen.wells@metoffice.gov.uk The contribution of Wells, Vosper, Brown and Webster of Met Office, Exeter, was prepared as part of their official duties as employees of the UK Government. It is published with the permission of the Controller of Her Majesty s Stationery Office and the Queen s Printer for Scotland. on the flow in these experiments. Petersen et al. (2005) investigated the effect of wind direction on idealised flow past a large idealised elliptical mountain with similar dimensions to Greenland and a small Rossby number Ro = U/fa = 0.42, where f is the Coriolis parameter and a is the half-width of the idealised mountain. They found that the flow pattern varies greatly with wind direction, with Coriolis effects causing a significant proportion of this variation. One of the main results of Petersen et al. (2005) was that for a north south oriented ridge in the Northern Hemisphere the component of the pressure force exerted normal to the ridge is larger for southwesterly flow than for northwesterly flow, where the ridge normal force is defined by F = p h x dx dy, (1) where the integration is over the surface of the computational domain, p is the surface pressure perturbation caused by the mountain, h is the mountain height and x and y are the directions along the minor (short) and major (long) axes of the mountain ridge respectively. Petersen et al. (2005) attributed the difference in F to two processes. Firstly, the mountain-induced lift force, defined by L = p h dxdy, (2) y where x is the along-flow direction and y is the acrossflow direction, reinforces the background geostrophic Copyright 2008 Royal Meteorological Society and Crown Copyright 2008,

2 690 H. WELLS ET AL. pressure gradient for southwesterly flow. The reinforcement of the pressure gradient means that the low-level flow is more strongly accelerated away from the mountain, leading to the development of a deeper pressure low in the lee. In contrast, for northwesterly flow the lift force opposes the large-scale pressure gradient. Secondly, the wake is systematically displaced to the right of a line running through the mountain crest and parallel to the upstream wind direction. For southwesterly flow, the lee low lies close to the centre of the ridge and so the ridgenormal force is large, whereas for northwesterly flow the lee low lies close to the southern end of the ridge leading to a reduction in the ridge-normal force. Petersen et al. (2005) attributed the displacement of the lee low to the circulation within the low, which moves it to the right of the ridge centre. Both Zängl (2004) and Petersen et al. (2005) examined low-rossby-number flow past orography and therefore emphasise the importance of Coriolis effects in determining the flow characteristics and how they vary with incident wind direction. In this study we shall instead focus on a narrower isolated mountain (which implies a higher Rossby number than that used by Petersen et al. (2005) and Zängl (2004)) of a scale that would typically be only partially resolved in current global NWP models. This study is therefore directly relevant to the NWP parametrization problem. Current parametrization schemes attempt to represent directional effects by implementing the analytical linear derivations of the pressure forces exerted on a isolated mountain ridge presented in Phillips (1984) (Equations (5) and (6) below). This study attempts to investigate the validity of this approach. Note that in the majority of our simulations Ro = 13 and thus our experiments lie in a different region of parameter space to those of Petersen et al. (2005) and Zängl (2004). Theaimsofthisworkare: 1. To understand how wind direction affects the pressure forces exerted on orography that would be only partially resolved by current global NWP models and to investigate whether linear theory can provide a useful scaling for these forces. 2. To study how the Coriolis force affects the pressure forces at these length scales, and in particular to ascertain whether the effects identified by Petersen et al. (2005) are significant at these higher Rossby numbers. 3. To see if the effects of wind direction on flow past simple, idealised orography are the same for flow past a more complex, real mountain. The remainder of the paper is set out as follows. Section 2 describes the numerical experiments performed. Results from the idealised orography simulations are presented in section 3, and in section 4 we present results from the real orography simulations. A summary and conclusions are presented in section The experimental set-up Numerical simulations are conducted of flow incident on an elongated high mountain ridge defined by ( x h(x,y ) 2 ( y ) 2 ] 3 2 ) = h m [1 + +, (3) a b where a, the mountain half-width, is 7.5kmandb, the mountain half-length, is 22.5 km. The undisturbed incident flow is horizontally uniform. The Brunt Väisäla frequency and horizontal velocity are constant with height, such that N = 0.01 s 1 and U = 10 m s 1 respectively. The mountain height, h m, is 2 km giving a nondimensional mountain height of h m = 2. The main set of simulations are performed on a f -plane with the Coriolis parameter, f = 10 4 s 1. The Rossby number for these simulations is Ro = 13, significantly higher than in the simulations of Petersen et al. (2005) (Ro = 0.42) and Za ngl (2004) (Ro = 2). To represent different incident wind directions, the orography is rotated about its major axis, with the incident wind always westerly. Note that the reference frame is fixed with respect to the incident wind so that the x direction is parallel to the incident flow and the y direction perpendicular to it. The advantage of rotating the orography, compared to that of varying the wind direction, is that it allows the same domain (of 720 km 720 km) to be used for all the experiments and a constant resolution (of 2 km) to be maintained in the along-wind direction. The main set of experiments are performed for eight orientations of orography where the major axis of the orography is rotated in an anticlockwise direction by 0, ±30, ±60, ±80 and 90. Throughout the paper these experiments will be referred to as ZERO, +30, 30, +60, 60, +80, 80 and +90 respectively. In ZERO, the flow is perpendicular to the major axis of the orography. Negative angles correspond to a clockwise rotation from ZERO and positive ones to an anti-clockwise rotation. Figure 1 illustrates the alignment of the orography with respect to the incident flow for some of the simulations performed. The simulations presented here may be compared to those of Petersen et al. (2005) by remembering that negative angles correspond to flows which are south of westerly and positive ones to flows which are north of westerly. The numerical model used is the three-dimensional, non-hydrostatic BLASIUS model described by Wood and Lx Ly ZERO Figure 1. Schematic illustrating the naming convention for the simulations. The arrow indicates the incident wind direction in each of the simulations and the ellipse shows the orientation of orography.

3 WIND DIRECTION EFFECTS ON OROGRAPHIC DRAG 691 Mason (1993). Moisture is not included in these simulations and the Boussinesq approximation is made and the density, ρ, is set to unity throughout. The pressure term is divided into large-scale and perturbation components, the former of which is fixed and in geostrophic and hydrostatic balance with the basic state flow. Note that in the calculations of the pressure forces we focus purely on the forces due to the mountain-induced pressure perturbation, p, and thus we neglect the contribution from the large-scale balanced flow. Analysis of our simulations reveals that the mountain-induced pressure forces dominate anyway. To prevent the growth of grid-scale instabilities, a small horizontal fourth-order smoothing term was applied to the velocity and perturbation temperature fields following Durran and Klemp (1983). A first-order Richardson-number-dependent mixing-length turbulence closure scheme is used. The lower-boundary condition is free-slip with a formulation following Durran and Klemp (1983). A free-slip boundary condition was chosen because it simplifies the problem, decreasing the number of controlling parameters. Furthermore the impact of the lower-boundary condition was found to be small since, in sensitivity tests with a no-slip lower-boundary condition (using a roughness length of z 0 = 0.1 m), the pressure forces exerted on the mountain changedbylessthan5%. The lateral boundary conditions are bi-periodic with Rayleigh damping columns placed at each of the lateral boundaries in order to suppress wrap-around effects. The damping is applied over 20 grid points and the damping coefficient increases smoothly from zero at the innermost edge of the damping column to a maximum at the boundaries. A Rayleigh damping layer is placed in the upper portion of the domain, from 12 km up to the model lid at 30 km, to prevent gravity wave reflections off the upper boundary. The vertical and lateral damping is applied in the same way so that damping is applied to the perturbation velocity and potential temperature fields. The model domain consists of 60 vertical levels with a grid spacing of 20 m near the surface which increases smoothly to around 200 m near the mountain top (at 2 km) and increases further to around 1000 m at 12 km. The horizontal resolution is 2 km and the domain size is 720 km 720 km. The simulations are run for t = s, equivalent to 67 mountain advection times for ZERO (t = Ut/a)or 22 mountain advection times for +90 (t = Ut/b). In simulations where vortex shedding is present, this long run time allows a number of shedding periods to be sampled, so that the pressure forces may be time averaged over a number of shedding periods. 3. The pressure forces exerted on the mountain Pressure forces for linear hydrostatic flow past a mountain of the shape described by Equation (3) have been derived analytically by Phillips (1984). The drag (or along-wind) component of the pressure force is defined as D = with the linear prediction given by p h dxdy, (4) x D lin (θ) = G { B(cos θ) 2 + C(sin θ) 2} ρunh 2 mb, (5) whilst the linear prediction for the lift (across-wind) component of the pressure force is L lin (θ) = G {(B C)sin θ cos θ} ρunh 2 mb, (6) where θ is the angle between the flow direction and the major axis of the orography (θ = 0 for flow perpendicular to the major axis of the orography and θ = 90 for flow parallel to the major axis). For the values of a and b used in the simulations described above, B = 0.94, C = 0.18 and G = 1. Figure 2 shows the modelled pressure forces plotted against the angle of the orography. Looking first at the ridge-normal component of the pressure force, F, shown in Figure 2(b), it is apparent that the force is symmetric about an angle of 0. Petersen et al. (2005) found that this component of the force was larger for southwesterly flow (corresponding to negative angles of orography) than for northwesterly flow (positive angles). Indeed for h m = 2, Petersen et al. (2005) found that F was around 20% larger for orientations of 30 than +30.As mentioned in section 1, Petersen et al. (2005) attributed this force difference to two processes: the mountaininduced lift force and the systematic displacement of the wake downstream of the mountain. For low-rossby-number flows, like those modelled by Petersen et al. (2005), these physical arguments are convincing. Moreover, it has been possible to replicate their results with the BLASIUS model. However, at the higher Rossby number considered here, there is no asymmetry in F although, as we will discuss later, rotation does affect the flow. Clearly the F asymmetry must depend on Rossby number, therefore Figure 3 shows the ratio of F at +30 to F at 30, plotted against Rossby number. These results have been obtained via a series of simulations in which the Coriolis parameter, f, was varied and all other parameters were held constant and as defined previously. The Rossby number was thus varied between 0.42 (the Rossby number used by Petersen et al., 2005) and 13.3 (the Rossby number used here). For values of Ro where the asymmetry in F is significant, the points on Figure 3 drop below unity. It is clear that, for h m = 2, the asymmetry in F is only noticeable when Ro 2. However, it is worth mentioning that Petersen et al. (2005) found that the asymmetry in F is also dependent on h m, increasing for larger h m, so it is possible that the

4 692 H. WELLS ET AL. Figure 2. Time-averaged pressure forces exerted on the mountain normalised by D lin (0) = N m 2. The pressure forces are time averaged from to s and the bars indicate the maximum/minimum force in that time period. For ease of viewing, the bars for ZERO and NC ZERO PERTURBED (f = 0 perturbed, to be discussed in section 3.3) have been displaced to the left and right respectively (as indicated by the arrows). The plots show (a) normalised drag, D/D lin (0), (b) normalised ridge-normal force, F /D lin (0), (c) normalised lift, L/D lin (0) and (d) normalised ridge-parallel force F /D lin (0). asymmetry will persist at higher Ro in the case of higher h m Ṡince the processes described by Petersen et al. (2005) do not make a significant contribution to the ridgenormal force for the higher-rossby-number simulations described here, D and F are symmetrical about an orientation of 0 (Figures 2(a) and (b)). The shape of the drag and lift curves (Figures 2(a) and (c) respectively) are similar to the shape described by the linear predictions for drag and lift (indicated by the dotted lines in Figure 2), suggesting that the linear predictions for each orientation of orography may be useful quantities to scale the drag and lift forces. Figure 4(a) shows the drag normalised by the linear drag prediction for each orientation of the orography. If the linear prediction was perfect, then the normalised Figure 3. The ratio of the ridge-normal force, F, for the +30 orientation to that for the 30 orientation plotted against Rossby number. The crosses indicate simulations using the set-up described in section 2, with the Rossby number varied using the Coriolis parameter. The star indicates the value found by Petersen et al. (2005) (for h m = 2). Note that the ridge-normal force in the standard simulations is time averaged from to s.

5 WIND DIRECTION EFFECTS ON OROGRAPHIC DRAG 693 Figure 4. (a) Time-averaged drag exerted on the mountain normalised by the linear drag prediction for each orientation of orography, D lin (θ), and (b) as (a) but for lift. The forces are time averaged from to s and the bars indicate the maximum/minimum force in that time period. drag would equal unity for all of the experiments. Figure 4(a) shows that this is not the case. However, the normalised drag is approximately constant at about 1.3 for a large range of orography orientations. Therefore, although the linear theory does not provide an accurate quantitative prediction of the drag, it does describe much of the drag variation with orography orientation. However, as the ridge becomes more aligned with the flow, the drag begins to decrease so that D/D lin 1.2 for ±80, before dropping to 0.9 for +90. Figure 4(b) shows the lift normalised by the linear lift prediction for each orientation of the orography for which the linear prediction is non-zero. The linear prediction also describes the shape of the lift variation for a wide range of orography orientations (for which the normalised lift is around 1.5), however again the ±80 simulations do not fit this pattern since the normalised lift in these cases increases to around 2 or 3 (although note that L lin becomes very small at these angles). Figure 4(a) indicates that the drag is substantially higher that that predicted by linear theory (by 30%) for a large range of orography orientations, and lower (by 10%) when the orography is aligned with the flow. Therefore it seems natural to split the discussion of the simulations into two flow regimes, a high-drag regime and a low-drag regime The high-drag regime The simulations where the orography orientation is between 0 and ±60 are in the high-drag regime, where the modelled drag exceeds the linear prediction by around Figure 5. The flow at 500 m above sea level for ZERO at s, with shading indicating wind speeds (contour interval 5 m s 1 )andthe vectors indicating direction. 30%. The high-drag regime is characterised by periodic vortex shedding and continuous gravity wave breaking. Since all of the simulations in this regime are broadly similar, the regime will be illustrated by a single simulation, ZERO (θ = 0 ). Figure 5 shows the horizontal winds at 500 m above sea level. Flow stagnation on the upslope is evident, with the flow splitting around 10 km south of the centre-line of the mountain. The winds are stronger to the north of the ridge than to the south of the ridge as a result of Coriolis effects. Figures 6 and 7 show the low-level flow for +60 and 60 respectively. In the absence of the Coriolis force, these simulations have flow fields which are mirror images of each other.

6 694 H. WELLS ET AL. However, rotation clearly affects the flow, which is more strongly blocked in +60 than in 60. Therefore Coriolis effects on the flow fields are not negligible, despite the fact that they are not sufficient to generate asymmetry in the ridge-normal force. Downstream of the mountain in Figures 5, 6 and 7 there are recirculating wake vortices. Figure 8 shows snapshots of the vertical component of vorticity taken at s intervals during ZERO. At s or t = 27 (Figure 8(a)), the vertical vorticity consists of two wavy banners, one of negative vorticity to the north and one of positive vorticity to the south. After s or t = 40 (Figure 8(b)), the wake has become more unsteady and vortices are shed alternately from the northern and southern ends of the mountain and then advected downstream. Figure 9 shows the time evolution of the drag and lift (normalised by the linear drag prediction). After around s, quasi-periodic variations appear in the lift force. These oscillations are intimately related Figure 6. As Figure 5, but for +60. Figure 7. As Figure 5, but for 60. to the shedding vortices, with lift maxima occurring when a vortex is on the verge of being shed from the northern end of the mountain and minima just before vortices are shed from the southern end. Thus the period of the oscillation in the lift force corresponds to the period of the vortex shedding. As seen by Schär and Durran (1997) and Vosper (2000), the onset of vortex shedding is marked by a significant increase in the drag. Here the drag increases by around 15% between s and s as the shedding becomes established, before oscillating about the new (higher) value for the remainder of the simulation. The drag exhibits an oscillation with a period that is one half of the shedding period. Figure 10 shows the flow along the centre-plane (y = 0) for ZERO. On the upslope, flow reversal (associated with low-level flow blocking) is evident. Above the mountain, there is strong gravity wave activity, with vertical isentropes and flow deceleration indicative of gravity wave breaking at low levels. Above the downslope of the Figure 8. Snapshots of the vertical component of vorticity for ZERO at the surface taken at (a) s and (b) s. The contour interval is s 1, with solid (dotted) contours indicating positive (negative) values. The mountain is marked in bold, with contours at 100 and 1000 m.

7 WIND DIRECTION EFFECTS ON OROGRAPHIC DRAG 695 Figure 10. The flow along the centre-plane for ZERO at s, with shading indicating wind speeds (contour interval 5 m s 1 ) and line contours indicating potential temperature (contour interval 1 K). Figure 9. Time series of (a) drag and (b) lift for ZERO normalised by the linear drag prediction. mountain, strong downslope winds are observed, with a near-stagnant wake region further downstream. Analysis of time series for the other high-drag simulations (not shown) reveals that shedding is established much earlier when the flow is not normal to the ridge axis. Schär and Smith (1993) showed that the transition to vortex shedding is initiated by the growth of unstable modes which are asymmetric about the incident flow direction. When the orography is symmetrical about the incident flow direction (ZERO), the Coriolis force is the main cause of flow asymmetry and, due to the relatively high value of Ro, shedding takes around s or t = 40 to develop. In contrast, when the orography is asymmetrical about the x-axis, the flow is strongly asymmetrical from the start of the simulation and thus shedding develops much more quickly and is well established s (t = 27) into the integration. In summary, analysis of the flow fields and pressure forces in the high-drag regime has shown that the presence of vortex shedding makes a significant contribution to the drag, increasing it by around 15%. Additionally, low-level gravity wave breaking creates a stagnant region aloft which probably leads to nonlinear enhancement of the drag. These effects combine to give a drag force that is larger than the linear drag force The low-drag regime In contrast to the high-drag regime, vortex shedding, wave breaking and upstream stagnation are not observed in the low-drag regime (the +90 orientation). Instead the flow is only weakly perturbed by the mountain with the low-level flow diverted laterally around the flanks of the obstacle, leading to much weaker gravity wave activity. Figure 11 shows the mountain-induced surface pressure perturbations for the (a) ZERO and Figure 11. Mountain-induced surface pressure perturbations for (a) ZERO at a mountain advection time Ut/a = 27 (t = s) and (b) +90 at a mountain advection time t = Ut/b = 24 (t = s). The contour interval is 25 Pa, with solid (dotted) contours indicating positive (negative) pressure perturbations. The mountain is marked in bold, with contours at 100 and 1000 m.

8 696 H. WELLS ET AL. (b) +90 orientations at similar non-dimensional times. Comparison of Figure 11(b) with Figure 11(a) reveals that the surface pressure perturbations are much weaker for the +90 orientation than for ZERO and do not extend downstream of the mountain. In the intermediate drag simulations (±80 orientations) wave breaking is observed, but not vortex shedding or upstream stagnation. These simulations have D/D lin 1.2, lower than the high-drag simulations, but higher than seen in +90. The large drop in normalised drag when the mountain is oriented parallel to the flow (+90 ) is associated with a significant change in the flow fields. As already mentioned for the high-drag cases, gravity wave breaking, vortex shedding and upstream flow stagnation were observed, whereas for +90 none of these flow features are seen. This difference is in agreement with the numerical modelling study of Bauer et al. (2000), which found that the occurrence of these flow features is dependent on the aspect ratio, β = L y /L x,wherel x and L y are the half-lengths of the mountain in the along-flow and across-flow directions respectively. (These parameters are illustrated in Figure 1.) They found that gravity wave breaking, upstream stagnation and wake vortex formation are all suppressed in flow past low-aspect-ratio mountains (β = 0.25). In our simulations both a and b are kept constant, however the along-flow and across-flow mountain lengths are varied as a result of rotating the orography so that L x = asin α cos θ + b cos α sin θ, (7) and L y = b cos γ cos θ a sin γ sin θ, (8) respectively, where α = arctan(cos θ/sin θ) and γ = arctan( sin θ/cos θ). Note that the apparent complexity of these equations arises from the mountain having a finite width that has to be taken into account in the calculation of aspect ratio. Our high-drag simulations cover a range of aspect ratios 0.5 β 3, for which the work of Bauer et al. (2000) suggests that we might observe upstream stagnation and lee vortex formation, although their work suggests that we might not expect to observe wave breaking for β<1. Our low-drag-state simulation (+90 ) has β = 0.3, for which the work of Bauer et al. (2000) suggests that all of these features will be suppressed and that increased horizontal deflection around the mountain will result in drag values that are overestimated by linear theory. To test the idea that the effects of wind direction on the normalised drag are due to the aspect ratio of the mountain, several additional simulations were run for mountain ridges aligned either perpendicular or parallel to the incident flow. In these simulations we varied the aspect ratio by varying b/a and all other parameters were kept the same as in the main set of simulations. Simulations were run with b fixed at 22.5 km and a set to 13.6 km (β = 1.67, equivalent to ±30 ), 22.5 km (β = 1.00), 37.4 km (β = 0.60, equivalent to ±60 ), Figure 12. The drag normalised by the linear prediction plotted against aspect ratio. The crosses indicate simulations using the set-up described in section 2, where the aspect ratio has been varied by rotating the mountain. The stars indicates the simulations where the aspect ratio has been changed by varying the dimensions of the mountain. In all cases the drag is time averaged from to s km (β = 0.36, equivalent to ±80 ) and 67.5 km (β = 0.33, equivalent to +90 ). Additionally, simulations were run where b was fixed at 7.5 km and a set to 12.5 km (β = 0.60, equivalent to ±60 ) and 7.5 km (β = 1.00). The linear drag was then calculated for each of these mountain shapes and used to normalise the modelled drag. Figure 12 shows the normalised drag plotted against aspect ratio for both the simulations where the aspect ratio has been varied by rotating the mountain (indicated by crosses) and those where the aspect ratio was changed by altering the dimensions of the mountain (indicated by stars). The same pattern is clearly visible in both sets of simulations, with the normalised drag greater than unity for simulations where β 0.6, and the normalised drag less than unity when β 0.3. There is some disagreement between the two sets of simulations near the transition from a high-drag state to a low-drag state, with a low-drag state occurring when β = 0.36 as a result of changing the mountain width and length, and an intermediate drag state occurring when β = 0.36 as a result of changing the mountain orientation. However, although the agreement is not perfect, the variation in normalised drag with wind direction is consistent with that due to a change in the aspect ratio of the orography. Indeed the results in Figure 12 suggest that, in this parameter space, one could use the aspect ratio of the mountain to have a first guess at whether the flow will be in a high- or a low-drag state. It is interesting to compare the area-averaged momentum flux, τ = u w dxdy, observed just above the mountain top in the two different regimes. Figure 13 shows the momentum flux calculated 100 m above the mountain summit (at 2100 m) for each of the simulations, normalised by the modelled surface pressure drag and then time averaged over four snapshots taken at s intervals between and s. Gravity waves make a significant contribution to the total drag in both the low- and high-drag regimes. In the low (+90 ) and intermediate drag (±80 ) cases, the gravity

9 WIND DIRECTION EFFECTS ON OROGRAPHIC DRAG 697 Figure 13. The area-averaged mountain-top momentum flux (u w ) normalised by the total modelled drag plotted against the angle of the orography. The momentum flux is normalised by the modelled drag and then time-averaged. The bars indicate the maximum and minimum values calculated for the snapshots at , , and s. wave momentum flux is responsible for around 90% of the total drag, whereas in the high-drag regime its contribution is much smaller at around 50% of the total drag, with processes occurring below the mountain crest generating the remaining drag. The above analysis suggests that the aspect ratio of the orography largely determines whether the flow will fall into the high- or low-drag state, with high-drag states occurring when β 0.6 and a low-drag state when β = 0.3. However Bauer et al. (2000) observed a similar dependence of the drag on aspect ratio without including the Coriolis force in their experiments. Combined with the symmetrical drag curve seen in Figure 2, this suggests that, in this region of parameter space, Coriolis effects are relatively unimportant in determining the variation of drag with wind direction. Therefore, in the following section we shall investigate the pressure forces in the absence of rotation The pressure forces in the absence of Coriolis effects The analysis performed so far suggests that, although the flows observed are affected by Coriolis effects, they are relatively unimportant in determining the drag. Therefore a set of experiments was performed with the Coriolis parameter set to zero. These experiments will be denoted by a prefix of NC. Figure 2 shows the time-averaged pressure forces in these simulations plotted alongside the pressure forces in the original simulations. Figure 2 reveals that there is only a significant difference between the NC experiments and the original experiments when the orography is aligned perpendicular to the incident flow (ZERO) and even then only for the drag, D, and ridge-normal, F, components of the pressure force. The drag in the NC ZERO experiment is around 20% smaller than that in ZERO. Ólafsson and Bougeault (1997) showed that Coriolis effects increase the drag for ridge-normal flow when h m > 1andRo = 2.5. In their parameter space, Coriolis effects increased the accumulation of low-level dense air above the upstream slope of the mountain. This caused large differences in the surface pressure distribution upstream of the mountain between the rotating and non-rotating cases, while the pressure perturbations in the lee were very similar. Figure 14 shows the variation of p ( h/ x) dy normalised by D lin (0) for ZERO and NC ZERO (and another simulation, NC ZERO perturbed, which will be discussed later). In contrast to the findings of Ólafsson and Bougeault (1997), there is only a small increase in upstream drag in ZERO compared to NC ZERO. However, there is a significant difference in the drag downstream of the mountain crest in these two simulations. In section 3.1 it was noted that the initiation of vortex shedding made a significant contribution to the drag, of around 15%. It is hypothesised that the difference in drag between ZERO and NC ZERO is solely due to the absence of shedding in the latter experiment, where shedding is suppressed since the flow is symmetrical about the centre-plane. An additional simulation, NC ZERO perturbed, was performed to test the hypothesis that the Coriolis force only has a significant effect on the drag because it breaks the symmetry of the flow (about the centre-plane) allowing vortex shedding to develop. In this simulation the Coriolis parameter was set to zero. The symmetry about the centre-plane was broken by perturbing the initial potential temperature field using exactly the same approach as Schär and Durran (1997) and Vosper (2000). The perturbation used was θ = A max(0, 1 r), (9) where here θ is the potential temperature, A = 3Kisa constant and r = { (x x0 ) 2 + (y y 0) 2 + (z z } 1 0) 2 2, (10) r x r y with (x 0,y 0,z 0 ) = (0, 2b, 0) and r x = a, r y = b and r z = 2h m. The pressure forces for this simulation are also shown in Figure 2. It is evident that the drag in this simulation is significantly higher than that in the NC ZERO simulation and is similar to the drag exerted in ZERO. Figure 14 also reveals that the drag and pressure profiles after s are much more similar to those in ZERO than those in NC ZERO. Flow fields (not shown) indicate that, as expected, vortex shedding is taking place in this simulation. It has been shown that, to first order, the effect of the Coriolis force on the drag in this study is only to introduce asymmetry into the flow when everything else is symmetrical about an axis aligned along the incident flow. This result suggests that NC ZERO is a special case that has little relevance to the real world since real mountains and flows will never be exactly symmetric. 4. Results for real orography The simulations which will be described in this section are of flow past South Georgia. Other than the model r z

10 698 H. WELLS ET AL. Figure 14. Profiles taken at s of normalised drag integrated over the y direction for simulations ZERO, NC ZERO and NC ZERO perturbed. The drag is normalised by the linear prediction for ZERO and the shape of the mountain is indicated by a faint solid line. orography, the set-up of the real orography simulations is identical to those with idealised orography (described in section 2). The orography data came from the Global Land One kilometre Base Elevation (GLOBE) dataset (GLOBE Task Team, 1999), and was smoothed in an effort to minimise grid-scale noise. The resulting model orography is plotted in Figure 15(a). In order to compare different orientations of the orography, the major axis of the orography is found using Equation (A.2) of Lott and Miller (1997). To represent different incident wind directions, the island is rotated about its major axis, with the incident wind always coming from the left-hand side of the domain. The main set of experiments consist of 12 simulations with the orientation of the orography varied through 360 at intervals of 30. For real orography, it is more difficult to define an aspect ratio, however from Figure 15(a) we estimate that β 160 km/40 km=4 for the 0 and 180 orientations, decreasing to β 0.25 for the ±90 orientations. Note that the idealised orography described in the previous sections was specifically designed to have a similar maximum height, aspect ratio and volume to the model real orography, so that the two sets of experiments are reasonably comparable. The purpose of this work is not to provide a detailed analysis of flows round South Georgia. It is instead to use South Georgia as an example of complex, high orography that would be unresolved by a typical global NWP model. Figure 15(b) shows the horizontal winds at 500 m above sea level for the 0 orientation of the real orography. As in the ZERO idealised mountain simulation, there is flow stagnation on the upslope and the flow splits to the south of the centre-line. However, here the barrier jet is stronger than that seen in ZERO. Immediately downstream of the mountain there are strong downslope winds, associated with gravity wave breaking aloft. Further downstream, there is a large recirculating vortex. Flow fields from later times reveal that the wake region is unsteady, with quasi-periodic vortex shedding occurring. The same features were seen in the equivalent idealised orography simulation (ZERO). The linear drag was calculated numerically for each orientation by computing a two-dimensional Fourier transform of the orography and integrating the contribution over all wave modes. The drag expression used for each mode is the analytical expression for two-dimensional, non-rotating, hydrostatic, frictionless flow over periodic sinusoidal orography (Equation (2.41) of Smith, 1979). Figure 16 shows the modelled pressure forces plotted against the angle that the orography has been rotated from its orientation in Figure 15(a). As before, positive angles correspond to an anti-clockwise rotation of the orography from the orientation shown in Figure 15 and negative angles to a clockwise rotation. The most striking result is that the variation of the drag with angle of orography is very similar to that observed in the idealised orography experiments (Figure 2(a)), with a drag maximum when the orography is aligned perpendicular to the incident flow (0 or 180 ) andadrag minimum when the orography is aligned parallel to the incident flow ( 90 or 90 ). Comparison of the drag in the simulations including the Coriolis force (crosses in Figure 16) with the no-coriolis simulations (diamonds in Figure 16) reveals that Coriolis effects do not have a significant impact on the drag for any orientation of orography, despite having a significant effect on the low-level flow fields. However, in contrast to the idealised orography case, the drag is the same with and without rotation when the ridge is aligned perpendicular to the flow (0 or 180 ). This is consistent with the arguments made in section 3.3, since when real orography is used the flow is never symmetric. The ridge-normal component of the pressure force (not shown) and the drag (Figure 16(a)) are asymmetrical about 0. For example, the drag is almost 60% greater for the +60 orientation (60 anti-clockwise) than the 60 orientation (60 clockwise). Comparison of other pairs of simulations where the orography has been rotated by the same angle from the orientation shown in Figure 15(a) but in opposite directions, reveals that the drag differs significantly in these pairs. However, the difference in drag is in the same sense as the difference predicted by linear theory (squares in Figure 16(a)). More specifically, linear theory predicts that the +60 simulation will have more drag than the 60 one, in agreement with the model results. The similarity of the variation of the linear drag with orography orientation to the variation of the modelled drag (seen in Figure 16(a)) suggests that linear theory captures a significant proportion of the drag variation. Figure 16(b) shows the drag normalised by the linear drag prediction for each orientation of the orography. The majority of points lie close to a normalised drag of 1.5. Since the normalised drags do not cluster around

11 WIND DIRECTION EFFECTS ON OROGRAPHIC DRAG 699 the idealised orography experiments, described in section 3.2, is striking. With real orography, a low-drag regime is observed for flow parallel to the major axis of the orography (the +90 and 90 orientations) and in this regime D/D lin = 0.9. For the idealised orography, a high-drag state, where D/D lin = 1.3, was observed for most of the other orography orientations. Again, the similarity to the real orography results is clear, with all but one of the remaining experiments falling into a high-drag state with D/D lin = 1.5. Thus the normalised drag in the high-drag regime for the real orography experiments is about 15% higher than for the equivalent idealised experiments. In section 3.2, it was shown that, for the idealised orography simulations, gravity wave momentum flux contributed around 50% of the total drag in the highdrag simulations, rising to 90% of the total drag in the low-drag regime. Figure 17 shows the momentum flux calculated 100 m above the maximum orography height (1900 m) normalised by the total modelled drag. From Figure 17 it is clear that the high- and low-drag regimes behave in the same way for the real orography. For the high-drag regime, between 45% and 65% of the total drag is attributable to the mountain-top momentum flux, which is generally a slightly higher proportion than in the idealised simulations. For the low-drag regime, this rises to between 70% and 75% of the total drag. However this is much lower than the 90% seen in the low-drag regime in the idealised simulations. It is hypothesised that the smaller contribution of the mountain-top momentum flux to the total drag in the low-drag regime for the real orography (relative to the idealised orography) is due to enhanced wave breaking below the mountain top, which reduces the mountain-top momentum flux. It is interesting to note that, with idealised orography, wave breaking was not observed for a 90 orientation. Here the presence of channelling effects and steep slopes forces gravity waves to break, even for an aspect ratio as low as β Concluding remarks Figure 15. (a) The smoothed, rotated orography used in the numerical model for the 0 simulation, plotted with a contour interval of 500 m. (b) The flow at 500 m above sea level at s for the 0 orientation, with shading indicating wind speeds (contour interval 5 m s 1 )andthe arrows indicating wind strength and direction. unity, linear theory clearly fails to provide a quantitatively accurate value for the drag. However, away from θ =±90, the normalised drag is roughly constant and linear theory predicts much of the variation of drag with wind direction, as it did for the idealised mountain case. For two simulations ( 90 and 90 ), the normalised drag, shown in Figure 16(b), is significantly lower than 1.5. Again, the similarity to the low-drag regime in Analysis of the simulations of flow past an idealised mountain revealed that linear theory, although clearly not valid for such high mountains, describes the variation in drag with wind direction when the orography is not parallel to the incident flow. However, the drag in these cases is approximately 30% larger than the linear prediction due to nonlinear processes including gravity-wave breaking and vortex shedding. In contrast, when the orography is aligned with the incident flow, the drag is reduced below the linear prediction and none of the nonlinear processes described above are observed. This regime change was attributed to the small aspect ratio of the mountain in this orientation and some additional experiments were performed to confirm this interpretation. It was found that although Coriolis effects caused significant changes in the observed flow fields, they did not

12 700 H. WELLS ET AL. affect the pressure forces very much. In particular, there was no significant difference in the ridge-normal pressure force between southwesterly and northwesterly flow, as identified by Petersen et al. (2005). It was shown that this effect only becomes measurable when Ro 2(and h m = 2), much lower than the Rossby number (Ro = 13) of the flows considered here. In our simulations, the main effect of the Coriolis force on the pressure forces is to introduce asymmetry into the otherwise symmetrical case where the orography is aligned perpendicular to the incident flow, allowing vortex shedding to develop. The presence of shedding increases the drag force exerted on the mountain by around 15%. Thus, for cases where the idealised orography is aligned at an angle to the incident flow, Coriolis effects have a negligible impact on the pressure forces. The latter part of this paper described results for flow past South Georgia, used here as an example of real high orography. Comparison of the pressure forces exerted on the real orography with those exerted on the idealised orography revealed surprising similarities. In both cases Coriolis effects did not make a significant contribution to the pressure forces. Also for both real and idealised orography the simulations separated out into high-drag and low-drag regimes depending on the aspect ratio of the orography perpendicular to the flow direction. Low-drag states were observed for low aspect ratios, corresponding to situations where the major axis of the orography was oriented parallel to the incident flow. The normalised drag associated with the high-drag state was seen to be about 15% higher for the real orography than for the idealised orography, possibly as a result of increased wave breaking. However, the pressure forces exerted on the real and idealised orography compared here are very similar. This similarity is reassuring for the modelling community since it supports the view that simulations Figure 16. The drag force exerted on the mountain normalised by (a) D lin (0) and (b) the linear drag prediction for each orientation of orography, D lin. The drag is time averaged over the period s and the error bars indicate the maximum/minimum force in that time period. In (a), the crosses indicate the drag for simulations including Coriolis effects (linked by a solid line), the diamonds show the drag in the No Coriolis simulations and the squares indicate the linear prediction for the drag for each orientation of orography (linked by a dotted line). Figure 17. The area-averaged momentum flux (u w ) at 100 m above the maximum orography height (1900 m) normalised by the total modelled drag and plotted against the angle of the orography. The momentum flux is normalised by the modelled drag and then time averaged. The bars indicate the maximum and minimum values calculated between s and s.

13 WIND DIRECTION EFFECTS ON OROGRAPHIC DRAG 701 of flow past simple symmetrical idealised orography are valuable in understanding flow past complex real orography. The work presented here shows that, although linear theory does not capture the absolute value of the drag, it does capture the majority of variation of drag with wind direction. This is encouraging for the parametrization of subgrid-scale orographic drag since it is shown that parametrizations based on linear theory (e.g. Lott and Miller, 1997; Scinocca and McFarlane, 2000; Webster et al., 2003) may be effective outside the range of validity of linear theory. This is of course dependent on the correctness of their assumption that complex subgridscale orography can be represented as a single isolated ridge. References Bauer MH, Mayr GJ, Vergeiner I, Pichler H Strongly nonlinear flow around a three-dimensional mountain as a function of the horizontal aspect ratio. J. Atmos. Sci. 57: Durran DR, Klemp JB A compressible model for the simulation of moist mountain waves. Mon. Weather Rev. 111: GLOBE Task Team The Global Land One-kilometer Base Elevation (GLOBE) Digital Elevation Model, version 1.0. National Oceanic and Atmospheric Administration, National Geophysical Data Center: 325 Broadway, Boulder, Colorado , USA. and CD-ROMs. Lott F, Miller MJ A new subgrid-scale orographic drag parametrization: Its formulation and testing. Q. J. R. Meteorol. Soc. 123: Ólafsson H, Bougeault P Nonlinear flow past an elliptic mountain ridge. J. Atmos. Sci. 53: Ólafsson H, Bougeault P The effect of rotation and surface friction on orographic drag. J. Atmos. Sci. 54: Peng MS, Shang-Wu L, Chang S, Williams RT Flow over mountains: Coriolis force, transient troughs and threedimensionality. Q. J. R. Meteorol. Soc. 121: Petersen GN, Kristjánsson JE, Ólafsson H The effect of upstream wind direction on atmospheric flow in the vicinity of a large mountain. Q. J. R. Meteorol. Soc. 131: Phillips DS Analytical surface pressure and drag for linear hydrostatic flow over three-dimensional elliptical mountains. J. Atmos. Sci. 41: Schär C, Durran DR Vortex formation and vortex shedding in continuously stratified flows past isolated topography. J. Atmos. Sci. 54: Schär C, Smith RB Shallow-water flow past isolated topography. Part I: Vorticity production and wake formation. J. Atmos. Sci. 50: Scinocca JF, McFarlane NA The parametrization of drag induced by stratified flow over anisotropic orography. Q. J. R. Meteorol. Soc. 126: Smith RB The influence of mountains on the atmosphere. Adv. Geophys. 33: Vosper SB Three-dimensional simulations of strongly stratified flow past conical orography. J. Atmos. Sci. 57: Webster S, Brown AR, Cameron DR, Jones CP Improvements to the representation of orography in the Met Office Unified Model. Q. J. R. Meteorol. Soc. 129: Wells H, Webster S, Brown AR The effect of rotation on the pressure drag force produced by flow around long mountain ridges. Q. J. R. Meteorol. Soc. 131: Wood N, Mason PJ The pressure drag force induced by neutral turbulent flow over hills. Q. J. R. Meteorol. Soc. 119: Zängl G Upstream effects of an Alpine-scale mountain ridge under various flow angles. Meteorol. Zeitschrift 13: