Estimating the Effects of Vertical Wind Shear on Orographic Gravity Wave Drag

Orographic gravity waves occur when air flows over mountains in stably stratified conditions. The flow of air creates a pressure imbalance across the mountain, so a force is exerted on the mountain in the same direction as the flow. An equal and opposite force is exerted back on the atmosphere, and this is gravity wave drag (GWD).

GWD must be parametrized in Global Circulation Models (GCMs), as it is important for large-scale flow. The first parametrization was formulated by Palmer et al. (1986) to reduce a systematic westerly bias. The current parametrization was formulated by Lott and Miller (1997) and partitions the calculation into 2 parts (see figure 1):

  1. The mountain waves. This is calculated by averaging the wind, Brunt-Väisälä frequency and fluid density in a layer between 1 and 2 standard deviations of the subgrid-scale orography above the mean orography.
  2. The blocked flow. This is based on an interpretation of the non-dimensional mountain height.
Fig 1
Figure from Lott and Miller (1997).

The parametrization does not include the effects of wind shear. Wind shear is a change in the wind with height and it alters the vertical wave length of gravity waves and so alters the drag. It has been shown (Teixeira et al., 2004; Teixeira and Miranda, 2006) that a uniform shear profile (i.e. a change in the magnitude of the wind with height) decreases the drag whereas a profile in which the wind turns with height increases the drag. This effect was seen by Miranda et al. (2009) to have the greatest impact over Antarctica, where drag enhancement was seen to occur all year with a peak of ~50% during JJA. Figure 2 shows this.

Fig 2
Figure 2: Annual mean linear GWD stress (1992-2001). Vectors show the surface stress with shear. Shading indicates the anomaly of the modulus of the surface stress due to shear. Computed from ERA-40 data. Taken from Miranda et al. (2009).

The aim of this work is to test the impact of the inclusion of shear effects on the parametrization. The first stage of this is to test the sensitivity of the shear correction to the height in the atmosphere at which the necessary derivatives are approximated. We carry out calculations using 2 different reference heights:

  1. The top of the boundary layer (BLH). This allows us to avoid the effects of boundary layer turbulence, which are not important in this case as they are unrelated to the dynamics of mountain waves.
  2. The middle of the layer between 1 and 2 standard deviations of the sub-grid scale orography (SDH). This is the nominal height used in previous studies and in the parametrization.

All figures shown below focus on Antarctica and are averaged over all JJAs for the decade 2006-2015. We are interested in Antarctica and the JJA season for the reasons highlighted above. All calculations are carried out using ERA-Interim reanalysis data.

We first consider the enhancement assuming axisymmetric orography. The advantage of this is that it considerably simplifies the correction due to terms related to the anisotropy becoming constant (see Teixeira et al, 2004). Figure 3 shows this correction calculated using both reference heights. We can see that the enhancement is greater when the SDH is used.

Fig 3
Figure 3: Drag enhancement over Antarctica with shear corrections computed at the BLH (left) and SDH (right), during JJAs for the decade 2006-2015, using axisymmetric orography.

We now consider the enhancement using mountains with an elliptical horizontal cross-section. This is how the real orography is represented in the parametrization. Again, we see that the enhancement is greater when the SDH is used (figure 4).

Fig 4
Figure 4: Drag enhancement (left) and enhancement of drag stress (in Pa) (right) over Antarctica with shear corrections calculated at the BLH (top) and SDH (bottom), during JJAs for the decade 2006-2015, using orography with an elliptical horizontal cross-section.

It is interesting to note that at both heights the enhancement is greater when axisymmetric orography is used. This occurs because, in the case of elliptical mountains, the shear vector is predominantly aligned along the orography, resulting is weaker enhancement (see figure 5).

Fig 5
Figure 5: Histograms of the orientation of the shear vector relative to the short axis of the orography over Antarctica for JJAs during the decade 2006-2015, using the BLH (left) and SDH (right).

We also investigate the fraction of times at which the terms related to wind profile curvature (i.e. those containing second derivatives) dominate the drag correction. This tells us the fraction of time for which curvature matters for the drag. We see that second derivatives dominate over much of Antarctica for a high proportion of the time (see figure 6).

Fig 6
Figure 6: Fraction of the time at which terms with second derivatives dominate the drag correction relative to terms with first derivatives over orography with an elliptical horizontal cross-section, for JJAs during the decade 2006-2015, calculated using the BLH (left) and SDH (right).

In summary, the main findings are as follows:

  • The drag is quantitatively robust to changes in calculation height, with the geographical distribution, seasonality and sign essentially the same.
  • The drag is considerably enhanced when the SDH is used rather than the BLH.
  • Investigation of the relative magnitudes of terms containing first and second derivatives in the drag correction indicates that second derivatives (i.e. curvature terms) dominate in a large proportion of Antarctica for a large fraction of time. This leads to an average enhancement of the drag which is larger over shorter time intervals.
  • Use of an axisymmetric orography profile causes considerable overestimation of the shear effects. This is due to the shear vector being predominantly aligned along the mountains in the case of the orography with an elliptical horizontal cross-section.

These results highlight the need to ‘tune’ the calculation by identifying the optimum height in the atmosphere at which to approximate the derivatives. This work is ongoing. We expect the optimum height to be that at which the shear has the greatest impact on the surface drag.


Lott F. and Miller M., 1997, A new subgrid-scale orographic drag parametrization: Its formulation and testing, Quart. J. Roy. Meteor. Soc., 123: 101–127.

Miranda P., Martins J. and Teixeira M., 2009, Assessing wind profile effects on the global atmospheric torque, Quart. J. Roy. Meteor. Soc., 135: 807–814.

Teixeira M. and Miranda P., 2006, A linear model of gravity wave drag for hydrostatic sheared flow over elliptical mountains, Quart. J. Roy. Meteor. Soc., 132: 2439–2458.

Teixeira M., Miranda P. and Valente M., 2004, An analytical model of mountain wave drag for wind profiles with shear and curvature, J. Atmos. Sci., 61: 1040–1054.

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