ATMOSPHERIC SCIENCE LETTERS Atmos. Sci. Let. 6: 152–159 (2005) Published online 27 September 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/asl.109 Comparison of four non-hydrostatic models for flow over the Isle of Arran Juma Al-Maskari* and Alan M. Gadian School of the Environment, University of Leeds, Leeds, UK *Correspondence to: Juma Al-Maskari, School of the Environment, Leeds University, Leeds, LS2 9JT, UK. E-mail: juma@env.leeds.ac.uk Received: 28 October 2004 Revised: 23 May 2005 Accepted: 10 June 2005 Abstract Four non-hydrostatic atmospheric models were compared for a case study of southerly airflow over the Isle of Arran in Scotland. The model results are contrasted with each other and aircraft wind observations. All resolve qualitatively similar the background gravity wave flow, despite different formulations. Copyright 2005 Royal Meteorological Society Keywords: numerical atmospheric non-hydrostatic model inter-comparison; flow over steep slopes 1. Introduction One of the challenges that faces small-scale nonhydrostatic models is the simulation of flow over steep terrain, i.e. where gradients exceed 45◦ , including the generation of gravity waves. Gravity waves generated by flow over hills propagate downstream, depending on the background winds and stratification. They can break or dissipate, allowing the deposition of momentum at higher altitudes. It is impossible to quantitatively evaluate different aspects of a model from an individual study such as this, but intercomparison studies do provide useful tools in the modelling community (Doyle et al., 2000) to analyse characteristics of the equation sets and numerical methods. Comparison with observed fields provides an estimate of their accuracy. Four meso-scale non-hydrostatic models are applied to the flow over the island. For brevity, the vertical velocity fields are compared with each other and aircraft observations. The models’ characteristics are summarised in the next section, and the set-up and the synoptic situations are described in Section 3. Analyses of the results are covered in Section 4, followed by the conclusions in Section 5. 2. Description of the models The four meso-scale models used in the comparison are the Clark, Smolarkiewicz, BLASIUS (Boundary Layer Above Stationary Inhomogeneous Uneven Surfaces), and the UK Met Office Unified Model (V5.3 ‘New Dynamics’). The characteristics of the model options employed in this study, selected purposely to maximize the range of numerical designs, are summarised in Table 1. The Clark and Hall model (Clark, 1977, 1979), which uses the anelastic equation set (Lipps and Copyright 2005 Royal Meteorological Society Hemler, 1982), has been applied to a wide range of meso-scale atmospheric processes including flow over complex terrain (Thielen et al., 2002; Clark et al., 1997). The model is non-hydrostatic time-dependent, with a regular Cartesian grid in the horizontal and a terrain following vertical coordinate (Gal Chen and Somerville, 1975). The same anelastic equation set is used in the Smolarkiewicz model. It has both semi-Lagrangian (Smolarkiewicz and Pudykiewicz, 1992) and Eulerian (used here) schemes, (Smolarkiewicz and Margolin, 1993). It has been used for a variety of problems including small and large-amplitude internal gravity wave problems and orographic precipitating clouds. This model is run (here) in an inviscid mode, and, as with the Clark model, assumes a free slip lower boundary condition and terrain following coordinates. The BLASIUS model (Mason, 1987; Wood and Mason, 1991) employs the same terrain following coordinate system and selectively solves Boussinesq or anelastic equations. The leapfrog numerical scheme used in the model is the same as the Clark model, but with the option of carrying turbulent kinetic energy as a prognostic variable. The model normally uses a boundary layer mixing length proportional to height near the ground and approaches a constant value (40 m in this study) with increasing height. Stratification effects are taken into account via a Richardson number dependency of the eddy viscosity. Above a critical value (0.3 in this study) the eddy viscosity is set to zero. The UK Met Office Unified Model differs, as it is a fully compressible model and has hybrid Gal Chen terrain following vertical coordinates but is used here in the same form as the other models. A SemiLagrangian advection scheme with semi-implicit time stepping, but no horizontal diffusion, is applied. The boundary layer scheme is turned off and there is no turbulence closure even in the boundary layer. The Comparison of four non-hydrostatic models for flow over the Isle of Arran 153 Table I. Summary of the main set-up of the four models in this case study Description Clark Smolarkiewicz Anelastic/Lipps and Hemler (1982) Gal Chen and Somerville (1975) terrain following coordinates Staggered Arakawa (1966) C-grid Anelastic/Lipps & Hemler Anelastic/Boussinesq Fully compressible(Cullen, 1993) Gal Chen terrain following coordinates Gal Chen terrain following coordinates Gal Chen terrain following up to an arbitrary height Non-staggered Arakawa A-grid Staggered Arakawa C-grid Advection scheme and time stepping Turbulence closure Explicit second-order accurate leapfrog scheme Smagorinsky (1963) scheme and free slip Semi-implicit & NFT Eulerian Free slip inviscid Model Explicit second-order accurate leapfrog scheme Linear mixing length and no slip lower b.c. (also a run with free slip) Arakawa C-grid staggering in the horizontal and Charney–Phillips in the vertical Semi-implicit semi-Lagrangian Lateral boundary relaxation 1 km 12 km 4.5 km Equations Coordinate system Grid structure UM is designed as an operational model as well as a research model. In this research mode, small time steps are used to satisfy the Courant Friedrich and Lewy (CFL) condition. The fully compressible set of equations is better for large-scale GCMs when modelling deep atmospheres, but not considered of a great advantage for shallow flows. The models use different grid structures. The Smolarkiewicz model has an A-grid which is generally fast, but not considered as accurate for modelling flows as the C-grid used in the other three models (Durran, 1998). The Charney–Phillips staggering grid in the vertical in the UM reduces spurious gravity wave generation. (Cullen, 1993). The Clark and BLASIUS models both use an explicit second-order accurate leapfrog scheme compared to the semi-implicit time integration used in the UM and Smolarkiewicz models. The leapfrog requires only one function evolution per unit time step, but the weakness is that undamped computational modes can slowly amplify and can, but not seen here, generate a time-splitting instability (Durran, 1998). The UM model uses a semi-Lagrangian scheme, which, when combined with a semi-implicit time stepping, brings greater stability (Cullen, 1993) and can be more efficient than Eulerian methods (Durran, 1998). Ritchie et al., 1995, show that the semi-implicit semiLagrangian method, with a 15-min time step, has equivalent accuracy and is more efficient than the semi-implicit Eulerian method, with a 3-min time step. However, small time steps are used in the UM simulation so that gravity wave modes can be accurately approximated and the CFL condition is satisfied. None of the four models uses time filters; they were all run with a 4-s time step. For all models, a Helmholtz or Poisson equation for the pressure must be solved at every time step. Both Smolarkiewicz and the UM use an iterative generalized conjugate residual (GCR) technique, accelerated by an Alternating Direction Implicity (ADI) preconditioner (Thomas et al., 2003), for improving efficiency. Copyright 2005 Royal Meteorological Society BLASIUS UM No turbulence closure 1 km No turbulence closure is used in the UM model, implicit diffusion being provided by the semi-implicit Semi-Lagrangian scheme. The BLASIUS uses a mixing length closure of 40 m, whereas the Clark model uses a first-order closure scheme of Smagorinsky (1963). The use of an inviscid model does not produce significantly different results from the viscous model for this type of gravity wave simulations, (Thielen et al., 2002). Smagorinsky (1963), proposed that the effective Laplacian viscosity due to unresolved scales should be proportional to the resolved horizontal deformation rate multiplied by the squared grid spacing. This scheme parameterises the effects of three dimensional isotropic turbulence and is defined by νT = (Cs g )2 (2 Sij Sij )1/2 Where, νT is the eddy viscosity, and Cs is the Smagorinsky coefficient, and g is the grid spacing and Ŝij is the resolved strain rate tensor. In the BLASIUS scheme, the first bracket is approximated by the mixing length l. 3. Setting up the case study The four models are initialised with a profile (see Table 2) based on a combination of the radiosonde ascent and aircraft data for 1200 UTC, 4 April 1996. The model domain is approximately 74 by 54 km with 266 by 196 grid points in the horizontal and 45 levels in the vertical, a grid spacing of 280 m, a smoothed topography field derived from a 50-m data set. Figure 1(a) shows a height contour map of the orography of the domain used with the axis being in model grid points. The solid line AB shows the approximate aircraft track from A (upwind) to B (downwind). The synoptic map, Figure 1(b), shows a blocking high pressure system over Denmark and the eastern North Sea and a stationary low pressure system situated in the mid-Atlantic west of the Bay of Biscay with a second low pressure to the south of Iceland. Atmos. Sci. Let. 6: 152–159 (2005) 154 J. Al-Maskari and A. M. Gadian Table II. The profile used to initialise the models Pressure (mb) T (C) Wind direction (degrees) Wind speed (m/s) Vapour (g/kg) 1026.3 1022.6 1015.4 1004.6 991.5 975.1 956.1 936 915 892.9 869.8 845.1 816.8 783.2 744.4 700.4 651.1 596.6 539.5 482.4 422.7 360.6 304 252.6 201.4 150.4 99.5 56.9 6.2 5.9 5.4 4.6 3.6 2.4 1.4 1.1 0.3 0.5 0.4 0 −0.8 −2 −3.6 −6.8 −9.9 −14.6 −20 −24.1 −30.2 −38.1 −48.3 −59.9 −68.6 −62.8 −61.4 −62.6 187.8 187.7 187.4 187.1 187.4 187.1 186.2 182.7 178.2 173.9 169.5 165.4 163.9 158.5 155.8 162.2 164.1 158.1 158.4 159.6 163.1 154 139.4 112.8 103.9 94.1 73.9 63.8 11.2 12.7 13.6 14.5 15.3 16.5 16.7 16.6 16.4 16.1 15.8 15.6 15.6 14.7 14.2 12.7 14 15.1 13.6 15 15.3 16.3 13.6 12.8 12 10.8 16 25.2 3.3 3.3 3.3 3.2 3.1 2.6 1.3 0.7 0.7 0.2 0.7 0.6 0.8 0.7 1.4 0.6 1.9 0.9 0.2 0.1 0.5 0.3 0.1 0 0 0 0 0 Consequently, the surface wind at Prestwick airport was around 10 m/s south-south-easterly midday on 4th April, veering to a fairly steady 15 m/s southerly wind between 1000 m and 3000 m. The boundary layer was capped with a 5 ◦ C inversion at around 1600 m. Convection was suppressed in regions of downward wave motion and enhanced in the upward wave, resulting in a banded cloud field extending to the boundary layer inversion. The UMIST Cessna aircraft flew in upwind/downwind direction as in Figure 1(a). Boundary layer cloud into ‘bands’ roughly perpendicular to the wind direction indicated that the direction of wave propagation was close to the direction of the wind. The path of the aircraft was chosen so as to sample as much of the downwind orographically disturbed air motion as possible. The aircraft flew at several levels in the vertical, but, for brevity, only one height is shown, while others are briefly commented on. The aircraft data was recorded with spatial resolution of ∼5 m. 4. Analysis of the results The four models are run for 50 min, outputing every minute. The model spin appears to take less than 30 min, but only values from 40 to 50 min are shown in the analysis. The Smolarkiewicz model was also run for up to 90 min (not shown here) and gave no significant change. Copyright 2005 Royal Meteorological Society Figure 2 are still images from animation of the last 11 min of the models run for the vertical velocity field at 1400 m for the four models. The Clark model (upper right, Figure 2b) has clear evidence of a low amplitude southern boundary induced mode, but all four models are similar. The unfiltered profile was used to enable an unbiased comparison. All produce similar patterns and with close agreement in terms of the magnitude of the vertical velocity and phase, as described below. Figure 3 shows a vertical velocity cross-section at 1400 m. However, the BLASIUS model results differ in terms of vertical velocity wavelength and amplitude in the downstream region. The reasons are not clear. When the lower boundary scheme in the BLASIUS model is changed to a free slip condition (not shown here), a similar wavelength pattern is produced, with approximately a 10% increase in amplitude of vertical velocity. Additional sensitivity tests performed with the Smolarkiewiz model (e.g. testing solution dependence on the domain size or vertical and temporal resolution) do not provide a conclusive insight. The four models have different outflow boundary relaxation lengths (see Table 1), the Smolarkiewicz has the largest (12 km). Increasing the domain size gave no effect on the interior flow. Therefore, it is unlikely that the smaller relaxation length (4.5 km) in the BLASIUS model is the cause of the larger downwind vertical velocity. With both the BLASIUS and the Clark models having the same advection scheme, it is plausible that the influence of the outflow boundary could be a factor, but there is a need for further tests to investigate this. Comparison between models and aircraft observations, Figure 3, shows differences in the fine details. The reasons for this are partly the inability for the models to resolve fine scale structures that are evident in the aircraft observations and, therefore, there is a need for higher model resolution grid spacing. Repeating the comparison for level 1960 m produces the same conclusion. Table 3 shows a correlation between the four models and aircraft observation. Further, close agreement is reliant on an ‘accurate’ initial profile. One sounding is unlikely to produce a good profile, and this is reflected in the model results. Meso-scale models, e.g. Thielen et al., 2002, are known to be sensitive to initial profiles. Both the UM and the Smolarkiewicz are significantly better correlated with each other and the observations, as shown in Table 3, than the other two models. The only significant similarity between these two models is the pressure solver. A power spectrum analysis (Bendat and Piersol, 1986) for vertical velocity at 1400 m of frequency versus spectral energy is given in Figure 4. A resolution of 280 m used in the models is unable to resolve the observed 400-m feature evident from the aircraft. However, the power spectrum analysis shows that all models are able to reproduce observed low-resolution feature of wavelength ∼5 km. Atmos. Sci. Let. 6: 152–159 (2005) Comparison of four non-hydrostatic models for flow over the Isle of Arran 155 (a) B 250 700 600 200 150 400 100 Height (m) Y-index 500 300 A 200 50 100 20 40 60 80 100 120 140 160 180 X-index (b) Figure 1. (a) Height contour map of the orography of the domain used in the models and the line AB represents the approximate aircraft track starting from point A. (b) Mean sea level synoptic chart analysis at 1200 UTC for 4 April 1996 (Crown copyright 1996 Published by the Met Office) Table III. Correlation coefficient between models and aircraft observation at t + 40 min at 1400 m Clark BLASIUS Smolark. UM Aircraft 0.77 BLASIUS 0.87 0.64 Smolark. 0.74 0.62 0.90 UM 0.20 0.22 0.37 0.29 Copyright 2005 Royal Meteorological Society Figure 5 is a wavelet power spectrum for vertical velocity at 1400 m for (A) aircraft observation, (B) Smolarkiewicz model and (C) the UM model. The Clark and BLASIUS models give similar wavelets to the UM. Wavelet analysis (Torrence and Compo, 1998)) can be used to obtain the dominant wavelength of a spatial, but equally spaced, data set. The black triangle represents the cone of influence, which distinguishes between the area where the spectrum is affected by the padding. The right side of Figure 5 gives the global wavelet spectrum. The solid line Atmos. Sci. Let. 6: 152–159 (2005) 156 J. Al-Maskari and A. M. Gadian Smolarkiewicz Model. Orography contours on top of w at 1400m. 4/4/ 1996, t+40min Clark Model. Orography contours on top of w at 1400m. 4/4/ 1996, t+40min 250 250 200 100 50 0 50 100 100 50 0 150 0 50 X-Index Blasius Model. Orography contours on top of w at 1400m. 4/4/ 1996, t+40min 200 200 5.0 4.0 3.0 2.0 1.0 0.0 −1.0 −2.0 −3.0 −4.0 −5.0 150 100 50 0 100 150 Y-Index 250 ms−1 Y-Index 250 50 150 X-Index UM Model. Orography contours on top of w at 1400m. 4/4/ 1996, t+40min 0 100 ms−1 5 4 3 2 1 0 −1 −2 −3 −4 −5 150 5.0 4.0 3.0 2.0 1.0 0.0 −1.0 −2.0 −3.0 −4.0 −5.0 150 100 50 0 0 50 100 ms−1 0 Y-Index 150 ms−1 Y-Index 200 5.0 4.0 3.0 2.0 1.0 0.0 −1.0 −2.0 −3.0 −4.0 −5.0 150 X-Index X-Index Figure 2. Orography contours on top of vertical velocity (m/s2 ) at 1400 m for t + 40 min for the four models. (Upper left — a) (Smolarkiewicz), (upper right — b) (Clark), (lower left — c) (UM), (lower right — d) (BLASIUS). The animation files can be found on the supplementary materials page shows the normalized wavelet spectrum, and the dashed line is the mean red-noise spectrum. As with the power spectrum, the model wavelet energy diagram reproduces the dominant wavelength pattern for the observations. However, the models overestimate the wavelet power. At approximately 2 km, there is a second peak in the global wavelet energy spectrum, which is poorly resolved by the models. The patterns are similar in all diagrams, confirming that the models are replicating the observed energy spectra and the ability of the models to simulate the important wave features. Semi-Lagrangian schemes are considered by some to be at a disadvantage to Eulerian schemes because the accuracy considerations often require that both methods use similar time steps (Durran, 1998). In the case of flow over steep orography, forcing that is stationary in the Eulerian system is Doppler shifted to a higher frequency in the Lagrangian coordinate system (Pinty et al., 1995; Hereil and Laprise, 1996). Any spurious orographic resonance can be treated with a space and time average of the forcing term, together Copyright 2005 Royal Meteorological Society with a modification of the surface pressure equation as given in Ritchie and Tanguay, 1996. However, Salerno and Balsamo (2003) argue that in some cases this treatment can be unsuccessful, especially at very high resolution (<10 km), unless a small time step is used. They show that a smoothing filter partially overcomes this problem, but reduces the representation of the orographic waves. The UM results show no evidence of spurious resonances for the semi-Lagrangian semiimplicit scheme and these problems are probably avoided by using a small time step of 4 s. Table 2 suggests the UM is better than the Clark and BLASIUS models, and that none of these concerns are apparent. 5. Conclusion An inter-comparison of vertical velocity of four mesoscale non-hydrostatic models and aircraft observations is completed. Model results are compatible with each other. The overall gravity wave structure is well simulated in Atmos. Sci. Let. 6: 152–159 (2005) Comparison of four non-hydrostatic models for flow over the Isle of Arran 157 5.0 4.0 3.0 2.0 W (m/s) 1.0 UM_1400 Clark_1400 0.0 Smol_1400 Aircraft −1.0 BLA_1400 −2.0 −3.0 0.8 km 0.6 Orography underneath flight path from left to right 0.4 0.2 0.0 Figure 3. Vertical velocity cross section at 1400 m for the 4 models and the aircraft observation (yellow). As in line AB Figure 1(a) Power Spectrum for w at 1400 m 102 Aircraft UM Smola. Clark Blasius 100 Spectral energy 10−2 10−4 10−6 10−8 10−10 10−12 10−4 10−3 10−2 10−1 100 101 500 m 50 m 5m Frequency 50 km 5 km Figure 4. Vertical velocity power spectrum at 1400 m for the four models along the line AB (see Figure 1a), and aircraft observation (blue) Copyright 2005 Royal Meteorological Society Atmos. Sci. Let. 6: 152–159 (2005) 158 J. Al-Maskari and A. M. Gadian b) Global wavelet spectrum a) Vertical velocity wavelet power spectrum 30 16 20 8 10 4 2 1 0 10 20 30 40 50 60 0 Wavelet power spectrum (m2 S2) Horizontal wavelength (km) (a) 0 20 Distance (km) a) Vertical velocity wavelet power spectrum b) Global wavelet spectrum 30 16 20 8 10 4 2 1 0 10 20 30 40 50 60 0 Wavelet power spectrum (m2 S2) Horizontal wavelength (km) (b) 0 Distance (km) a) Vertical velocity wavelet power spectrum b) Global wavelet spectrum 16 20 8 10 0 10 20 30 Distance (km) 40 Power 30 4 2 1 20 40 50 60 0 Wavelet power spectrum (m2 S2) Horizontal wavelength (km) (c) 40 Power 0 20 Power 40 Figure 5. Vertical velocity wavelet power spectrum for 1400 m. (A) Aircraft observation. (B) Smolarkiewicz model. (C) UM model. The cone of influence is marked by the black triangle. The left side of the image shows the normalized global wavelet spectrum (solid line), and the mean red-noise spectrum (dashed line) all models. The inter-comparison with aircraft data is consistent, and the Smolarkiewicz and the UM model results correlate with a factor of 0.9, although using different equation sets. The initial profile taken from radiosonde and aircraft soundings introduces uncertainties. The profile is not necessarily truly representative of the actual state of the atmosphere at the time of the flights, 2 h later. This asks to what degree we can hope to accurately model trapped lee waves with a fixed single upwind profile. Finer scale modelling is required to resolve the smaller scale features, but with ever increasing computer resources, higher resolution should improve the model results. Acknowledgements The authors would like to thank the UK Met Office for allowing them to use the new version of the New Dynamics and for the Copyright 2005 Royal Meteorological Society support given to set up the model. Special thanks should be given to Richard Forbes and Simon Vosper. Also, we would like to thank Chang-Gui Wang from UWERN support team for the help with Jview, and to Terry Davies for his useful comments. The authors are very grateful for the help and assistance provided by Piotr Smolarkiewicz and for the aircraft observations made by Ian Stromberg and Robert Wood. References Arakawa A. 1966. Computational design for long-term numerical integration of the equations of fluid motion: two-dimensional incompressible flow. Part I. Journal of Computational Physics 1: 119–143. Bendat JS, Piersol AG. 1986. Random Data: Analysis and Measurement Procedures, 2nd edn. John Wiley & Sons: New York; 566. Clark TL. 1977. A small-scale dynamic model using a terrainfollowing transformation. Journal of Comparative Physiology B 24: 186–215. Atmos. Sci. Let. 6: 152–159 (2005) Comparison of four non-hydrostatic models for flow over the Isle of Arran Clark TL. 1979. 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