Horizontal reduction of pressure to mean sea level

Transcription

1 Horizontal reduction of pressure to mean sea level Henrik Feddersen, DMI October 4 Abstract. A horizontal reduction of pressure to mean sea level used in DMI-HIRLAM is documented. This method has been introduced because the conventional vertical reduction to mean sea level yields mean sea level pressure fields that reflect the orography rather than synoptic weather patterns over complex terrain, such as the Alps or eastern Greenland. With the new method the mean sea level pressure field is much smoother over complex terrain. 1 Introduction Mean sea level pressure (MSLP) is one of the most widely used diagnostics in weather forecasting. However, over elevated terrain the reduction to mean sea level is somewhat arbitrary, and the result over high mountains is not always satisfactory in the sense that the constructed MSLP field reflects the orography more than synoptic weather patterns. It has been demonstrated in Mesinger and Treadon (1995) that this problem can be avoided by using a horizontal reduction of pressure to mean sea level. Instead of using an arbitrary lapse rate to define a virtual temperature under ground, they interpolate temperature horizontally under elevated terrain using temperatures at the sides of the elevated terrain as boundary conditions. The interpolated (virtual) temperature is subsequently used in the vertical reduction of surface pressure to mean sea level. A different approach which also involves a horizontal reduction to mean sea level was developed by Pielke and Cram (197) who related mean sea level pressure to surface geostrophic wind. In DMI-HIRLAM the conventional method for MSLP calculation has been replaced by a pragmatic method which contains elements of both of the above horizontal reduction methods. Reduction of pressure to mean sea level The conventional reduction of pressure to mean sea level is described by the formula (Källen, 1) { [ Φs p msl = p s exp 1 αφ s + 1 ( ) ]} αφs, (1) R d T R d T 3 R d T 73

2 where Rd is the gas constant for dry air, ps is the surface pressure, Φs is the surface geopotential, T is the surface temperature, ps T = Tnlev + αtnlev pnlev 1, () and α is a temperature reduction factor, α = ΓRd /g, (3) where Γ is a standard temperature lapse rate, Γ = 6.5 K/km. For a cold surface temperature (T < 55 K) or a warm surface temperature (T > 9.5 K) the surface temperature is modified as described in Ka llen (1). Figure 1 shows an example for the.15 version of DMI-HIRLAM where MSLP is calculated using the standard reduction (1) to mean sea level. Note the influence of orography over mountaineous regions such as, e.g., eastern Greenland Mon 3 Aug 4 Z +h Mon 3 Aug 4 Z Fig. 1. Mean sea level pressure calculated using the standard reduction (1) to mean sea level..1 Horizontal interpolation The first and second derivatives of pressure at the lower model levels provide good approximations to the corresponding derivatives of pressure at mean sea level over sea and flat terrain. Over complex terrain mean sea level pressure is not well defined, so we may choose to specify the mean sea level pressure π by requiring π = z pilev. (4) 74

3 Here pilev denotes the pressure at model level ilev, and z indicates that the derivatives should be evaluated for locally constant geopotential height corresponding to model level ilev. In practice, model level nlev-3 is used which is approximately 7m above the ground for the 4 vertical levels that are used in DMI-HIRLAM. Equation (4) is an elliptic equation which is solved numerically by approximating the derivatives on the right hand side with finite differences and solving the equation iteratively, using pmsl (from Eq. (1)) as initial guess. In DMI-HIRLAM we apply iterations of the Gauss-Seidel method (Press et al., 199), πi,j = 1 (n) (n) πi 1,j + πi+1,j + πi,j 1 + πi,j+1 (pi 1,j + pi+1,j + pi,j 1 + pi,j+1 4pi,j )ilev (5) 4 where πi,j is updated during the loop over the grid points (i, j). The ( )ilev notation is used to indicate that the pressure terms pi 1,j, pi+1,j etc. have been interpolated to the height corresponding to the height of centre grid point (i, j) for level ilev. The GaussSeidel method converges relatively slowly, but the resulting MSLP fields indicate that the method is adequate here. Figure shows the MSLP field derived from the same model prediction as was used for Fig. 1, but using the procedure described above instead of the standard reduction to mean sea level. We note that the orographic effects over mountaineous regions, such as the Alps and eastern Greenland, have been eliminated Mon 3 Aug 4 Z +h Mon 3 Aug 4 Z Fig.. As Fig. 1, but using the horizontal reduction (4) to mean sea level. In Fig. 3 the difference over Greenland is highlighted by overlaying horizontally reduced MSLP on standard MSLP. 75

4 Mon 3 Aug 4 Z +h Mon 3 Aug 4 Z Fig. 3. Mean sea level pressure calculated using standard reduction (blue) and using horizontal reduction (red) over Greenland. 3 Geopotential on pressure levels A similar approach is applied to the calculation of geopotential on constant pressure surfaces. If the pressure surface lies above the ground, geopotential on the model levels just above and below the constant pressure level is interpolated to obtain the geopotential on the pressure level (Källen, 1). But if the constant pressure level lies below the ground, an elliptic equation for geopotential similar to (4), Φ p = pφ ilev (6) is solved numerically. Figure 4 shows hpa geopotential height using the horizontal interpolation overlaid on hpa geopotential height using standard extrapolation when the surface pressure is less than hpa. Like for mean sea level pressure we find that the alternative calculation of geopotential yields a meteorologically more satisfactory result. Acknowledgements The method of horizontal interpolation of mean sea level pressure and geopotential on pressure levels was introduced into DMI-HIRLAM by Jess U. Jørgensen. The present note describes a slightly simplified approach which was introduced in order to allow for parallelization of the Fortran code. 76

5 Mon 3 Aug 4 Z +h Mon 3 Aug 4 Z Fig. 4. hpa geopotential height calculated using standard extrapolation (blue) and horizontal interpolation (red) when the hpa surface is below the ground. References Ka lle n, E.. Hirlam documentation manual. System.5. Tech. rep.. SMHI. Norrko ping, Sweden. 1. Mesinger, F. and Treadon, R. E Horizontal reduction of pressure to sea level: Comparison against the NMC s Shuell method. Mon. Wea. Rev.. 3, Pielke, R. A. and Cram, J. M An alernate procedure for analyzing surface geostrophic winds and pressure over elevated terrain. Wea. Forecasting., Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vetterling, W. T.. Numerical Recipes. chap. 17, Partial Differential Equations. Cambridge University Press