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Gravity wave drag Parameterization of orographic related momentum fluxes in a numerical weather processing model Andrew Orr anmcr@bas.ac.uk Lecture 1: Atmospheric processes associated with orography Lecture 2: Parameterization of subgrid-scale orography

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Equation of horizontal hydrostatic motion Parameterized tendency related to the vertical divergence of sub-grid scale vertical momentum flux (N/m 2 ) (also called stress) (Note a prime denotes sub-grid scale deviation and an overline denotes gridbox average) Dyn: dynamics (resolved), e.g. advection, pressure gradient and Coriolis force (linearise around mean state) Param: physical parameterizations of sub-grid scale processes: e.g. radiation, convection, boundary layer, orography Drag effects acceleration through tendencies of the horizontal wind Tendencys due to sub-grid scale orography (SSO) Equation for resolved part of the horizontal momentum

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x h: topographic height above sea level (from global 1km data set) From Baines and Palmer (1990) σ: mean slope (along principal axis) Note source grid is filtered to remove small-scale orographic structures and scales resolved by model – otherwise parameterization may simulate unrelated effects Additional filtering so that only forced at scales the model can represent well At each gridpoint sub-grid orography represented by: μ: standard deviation of h (amplitude of sub-grid orography) 2μ approximates the physical envelope of the peaks γ: anisotropy (measure of how elongated sub-grid orography is) θ: angle between x-axis and principal axis (i.e. direction of maximum slope) ψ: angle between low-level wind and principal axis of the topography * * * * h: mean (resolved) topographic height at each gridpoint Specification of sub-grid orography

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Calculate topographic gradient correlation tensor Diagonalise Direction of maximum mean-square gradient at an angle θ to the x-axis Evaluation of required parameters

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Change coordinates (orientated along principal axis) If the low-level wind is directed at an angle φ to the x-axis, then the angle ψ is given by: (ψ=0 flow normal to obstacle; ψ=π/2 flow parallel to obstacle) Anisotropy defined as (1:circular; 0: ridge) Slope (i.e. mean- square gradient along the principal axis)

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From Scinocca and McFarlane 2000 Illustration of anisotropy and principle axis of sub-grid scale orography for a number of general- circulation model grid cells near the Tibetan plateau. Contours show the unresolved orography

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Resolution sensitivity of sub-grid fields Horizontal resolution: ERA40~120km; T511~40km; T799~25km mean orography / land sea mask standard deviation (μ) slope (σ)

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From Smith et al. (2006) mean orography (UK Met Office Unified Model)

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Lott and Miller (1997) parametrization scheme 2. Compute vertical distribution of wave stress accompanying the surface value (a) Gravity wave drag (b) Blocked flow drag 2. Compute drag at each model level for z < z blk See Lott and Miller (1997) z blk hz/z blk h eff h 1. Compute gravity wave surface stress exerted on sub-grid scale orography 1. Compute depth of blocked layer Contributions from (a) gravity wave drag and (b) low-level blocking drag

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Typically L 2 /4ab ellipsoidal hills inside a grid point. Summing all forces we find the stress per unit area (using a=μ/σ) G (~1): constant (tunes amplitude of waves) Increasing this increases gravity wave surface stress Gravity wave stress can be written as (Phillips 1984) Assume sub-grid scale orography has elliptical shape See Lott and Miller 1997 Determine where and how strongly waves are excited More, generally where is the amplitude of the displacement of the isentropic surface Effects of anisotropy on gravity wave surface stress Compute gravity wave surface stress

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Determine how strongly the waves are dissipated Strongest dissipation occurs in regions where the wave becomes unstable and breaks down into turbulence, referred to as wave breaking Convective instability: where the amplitude of the wave becomes so large that it causes relatively cold air to rise over less dense, warm air Kelvin-Helmholtz instability also important: associated with shear zones Lindzens saturation hypothesis: instability brings about turbulent dissipation of the wave such that the amplitude of the wave is reduced until it becomes stable again (Lindzen 1981), i.e. λ equals the local saturated stress λ sat i.e. dissipation is just sufficient to ensure that Which gives, i.e. strong dependence on U which shows that wave breaking is strong preferred in regions of weak flow. Similarly, the stronger/smaller N/ρ the more readily waves break (Palmer et al. 1986) Compute vertical distribution of stress

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Can show that waves impact on static stability and vertical shear gives rise to a minimum Richardson number (Palmer et al. 1986) Instability occurs when Ri min < Ri crit (=0.25) (i.e. less than the critical value) Kelvin-Helmholtz: numerator becomes small Convective instability: denominator becomes large Following saturation hypothesis, when the wave is saturated Ri min =0.25 and δh=δh sat Wave breaking occurs more readily in this formulation than in a purely convective overturning scheme :amplitude of wave :mean Richardson number Saturation hypothesis in terms of Ri

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Gravity wave breaking only active above z blk (i.e. λ=λ s for 0

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Set λ=λ s and Ri min =0.25 at model level representing top of blocked layer U k-2,T k-2 λ k-2 U k-1,T k-1 λ k-1 z=0; λ= λ s z k =z blk ; λ k = λ s Height Calculate Ri at next level Calculate Ri min If Ri min

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Compute depth of blocked layer Characterise incident flow passing over the mountain top by ρ H, U H, N H (averaged between μ and 2μ) H ncrit 1 tunes the depth of the blocked layer. If H ncrit is increased then Z blk decreases Blocking height z blk satisfies: (assume h=2μ) U p calculated by resolving the wind U in the direction of U H

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Compute blocked-flow drag For z

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From ECMWF T511 operational model Parameterized surface stresses winter summer

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Resolved/parameterized drag contributions T213 forecasts: ECMWF model with mean orography and the subgrid scale orographic drag scheme. Explicit model pressure drag and parameterized mountain drag during PYREX. From Lott and Miller 1997 measured drag resolved dragLarge parameterized drag (makes up difference between resolved and measured drag)

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Results from Canadian Climate model of northern hemisphere winter, showing westerly gravity wave drag (tendency in m/s/d)

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Impact of scheme Without GWD scheme Analysis (best guess) With GWD scheme Mean January sea level pressure (mb) for years 1984 to 1986 Icelandic/Aleutian lows are too deep Flow too zonal / westerly bias Azores anticyclone too far east Siberian high too weak and too far south alleviation of westerly bias better agreement From Palmer et al. 1986 Alleviation of systematic westerly bias

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Analysis (best guess) Zonal mean cross-sections of zonal wind (ms -1 ) and temperature (K, dashed lines) for January 1984 and (a) without GWD scheme and (b) analysis westerly bias cold bias Without GWD scheme less impact in southern-hemisphere From Palmer et al. 1986

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Sensitivity of resolved orographic drag to model resolution From Smith et al. 2006 resolved drag converging parameterization still required at high-resolution Relatively weak flow: flow blocking dominates Strong flow: short- scale trapped lee waves produce significant fraction of drag (Georgelin and Lott, 2001 Flow-blocking case Flow-over case Sub-grid scale contribution still significant

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Parameterization schemes in NWP models BlockingGravity wave ECMWF HIRLAM DWD Meteofrance ECHAM UK UM

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Zonal cross-sections of the differences in (a) zonal wind (ms -1 ) and (b) temperature (K) slowing of winds in stratosphere and upper troposphere poleward induced meridional flow descent over pole leads to warming / alleviation of cold bias parameterisation of gravity wave drag decelerated the predominately westerly flow With GWD scheme - control From Palmer et al. 1986

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Effect of rotation on flow over orography From Olafsson and Bougeault (1997) Rossby number Ro=U/fL f: Coriolis parameter L: mountain width Nh/U=2.7 and Ro= Nh/U=2.7 and Ro=2.5 Ro 100km (with U=10 ms -1, f=10 -4 s -1 ) symmetrical flow asymmetrical flow / diverted to left i.e. important for large mountain ranges (e.g. Alps), but not individual peaks / sub-grid scale parameterization ?? no rotation rotation

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Effect of rotation on drag Drag normalised by that predicted analytically for linear 2d non-rotating frictionless flow From Webster et al. (2003) (after Olafsson and Bougeault (1997)) High-drag hydraulic state discussed previously For more realistic conditions, i.e. rotation and friction 1)Constrains the drag to values predicted by linear theory 2)Deviates by no more than 30% from the normalizing value 3)Less dependence on Nh/U 3) Explains why observed pressure drags agree remarkably well with those predicted by linear 2d non-rotating frictionless flow

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Met Office Unified Model parametrization scheme Predicted drag is essentially an empirical fit to the idealized simulations of Olafsson and Bougeault (1997) Uses simple 2d linear approximations to predict the surface pressure drag Surface pressure drag partitioned into a blocked-flow component and a gravity wave component depending on the Nh/U of the flow Gravity wave amplitude is assumed to be proportional to the amount of flow-over, and the remainder of the surface pressure drag assumed to be due to flow-blocking Deposition of assumes saturation process as in Palmer et al. 1986 Vertical deposition of assumed to be uniform throughout depth h

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References Baines, P. G., and T. N. Palmer, 1990: Rationale for a new physically based parameterization of sub-grid scale orographic effects. Tech Memo. 169. European Centre for Medium-Range Weather Forecasts. Beljaars, A. C. M., A. R. Brown, N. Wood, 2004: A new parameterization of turbulent orographic form drag. Quart. J. R. Met. Soc., 130, 1327-1347. Clark, T. L., and M. J. Miller, 1991: Pressure drag and momentum fluxes due to the Alps. II: Representation in large scale models. Quart. J. R. Met. Soc., 117, 527-552. Eliassen, A. and E., Palm, 1961: On the transfer of energy in stationary mountain waves, Geofys. Publ., 22, 1-23. Georgelin, M. and F. Lott, 2001: On the transfer of momentum by trapped lee-waves. Case of the IOP3 of PYREX. J. Atmos. Sci., 58, 3563-3580. Gregory, D., G. J. Shutts, and J. R. Mitchell, 1998: A new gravity-wave-drag scheme incorporating anisotropic orography and low-level wave breaking: Impact upon the climate of the UK Meteorological Office Unified Model. Quart. J. Roy. Met. Soc., 125, 463-493. Lilly. D. K., 1978: A severe downslope windstorm and aircraft turbulence event induced by a mountain wave, J. Atmos. Sci., 35, 59-77. Lindzen, R. S., 1981: Turbulence and stress due to gravity wave and tidal breakdown. J. Geophys. Res., 86, 9707-9714. Lott, F. and M. J. Miller, 1997: A new subgrid-scale drag parameterization: Its formulation and testing, Quart. J. R. Met. Soc., 123, 101- 127. Palmer, T. N., G. J. Shutts, and R. Swinbank, 1986: Alleviation of a systematic westerly bias in general circulation and numerical weather prediction models through an orographic gravity wave drag parameterization, Quart. J. R. Met. Soc., 112, 1001-1039. Phillips, D. S., 1984: Analytical surface pressure and drag for linear hydrostatic flow over three-dimensional elliptical mountains. J. Atmos. Sci., 41, 1073-1084. Olafsson, H., and P. Bougeault, 1997: The effect of rotation and surface friction on orographic drag, J. Atmos. Sci., 54, 193-210. Queney, P., 1948: The problem of airflow over mountains. A summary of theoretical studies, Bull. Amer. Meteor. Soc., 29, 16-26. Scinocca, J. F., and N. A. McFarlane, 2000: The parametrization of drag induced by stratified flow over anisotropic orography. Quart. J. R. Met. Soc., 126, 2353-2393. Smith, S., J. Doyle., A. Brown, and S. Webster, 2006: Sensitivity of resolved mountain drag to model resolution for MAP case studies. Quart. J. R. Met. Soc.., 132, 1467-1487. Taylor, P. A., R. I. Sykes, and P. J. Mason, 1989: On the parameterization of drag over small scale topography in neutrally-stratified boundary-layer flow. Boundary layer Meteorol., 48, 408-422. Webster, S., A. R. Brown, D. R. Cameron, and C. P. Jones, 2003: Improvements to the representation of orography in the Met Office Unified Model, Quart. J. Roy. Met. Soc., 129, 1989-2010.

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