Gravity wave drag Parameterization of orographic related momentum

1. Gravity wave drag Parameterization of orographic related momentum fluxes in a numerical weather processing model Andrew Orr anmcr@bas.ac.uk 30 May 2012 Lecture 1: Atmospheric processes associated with orography Lecture 2: Parameterization of subgrid-scale orography

2. (linearise around mean state) Drag effects acceleration through tendencies of the horizontal wind Param: physical parameterizations of sub-grid scale processes: e.g. radiation, convection, boundary layer, orography Dyn: dynamics (resolved), e.g. advection, pressure gradient and Coriolis force (Note a prime denotes sub-grid scale deviation and an overline denotes gridbox average) Parameterized tendency related to the vertical divergence of sub-grid scale vertical momentum flux (N/m2) (also called stress)

3. Equation of horizontal hydrostatic motion Tendency’s due to sub-grid scale orography (SSO) Equation for resolved part of the horizontal momentum

4. General principles (After ‘Subgrid-scale orographic drag’, part IV: physical processes, IFS documentation) The influence of sub-grid scale orography on the momentum of the atmosphere is represented by a combination of lower-tropospheric drag created by the orography assumed to intersect model levels, and vertical profiles of drag due to absorption of vertically propagating gravity waves. The scheme is described in detail in Lott and Miller (1996). heff zblk h hz/zblk The assumption is that mesoscale-flow dynamics can be described by two conceptual models, whose relevance depends on the non-dimensional height of the mountain via, (1)

5. Suppose that the mountain has an elliptical shape and a height variation determined by a parameter b in the along-ridge direction and by a parameter a in the cross-ridge direction, such that then the geometry of the mountain can be written as (2) At small Nh/U all the flow goes over the mountain and gravity waves are forced. In the simple case when the incident flow is at right angles to the ridge the surface stress due to gravity wave has the magnitude Provided that the Boussinesq and hydrostatic approximations apply. (3)

6. At large Nh/U the vertical motion of the fluid is limited and part of the low-level flow goes around the mountain. The depth zblk of this blocked layer can be expressed as (4) where hncrit is a critical non-dimensional moutnain height of order unity. In each layer below zblk the flow streamlines divide around the obstacle, and it is supposed that flow separation occurs on the obstacle’s flanks. Then, the drag Dblk(z) (measured in N), exerted by the obstacle on the flow at these levels can be written as (5) Here l(z) represents the horizontal width of the obstacle as seen by the flow at an upstream height z and Cd is the drag coefficient which is a constant with value close to unity.

7. For an elliptical mountain, the width of the obstacle as seen by the flow at a given altitude z<zblk is given by (6) In (6) it is assumed that zblk is raised up to the mountain top, with each layer below zblk raised by a factor h/zblk. This leads effectively to a reduction of the obstacle width, as seen by the flow when compared to the case in which the flow does not experience vertical motions as it approaches the mountain. Moreover, the blocked layer results in a reduction of the mountain height which produces gravity waves, meaning that the mountain height using in (3) is replaced by a lower effective (or cut-off) mountain height heff.

8. Specification of sub-grid orography h: mean (resolved) topographic height at each gridpoint * * * * x Description of the scheme From Baines and Palmer (1990) h: topographic height above sea level (from global 1km data set)

9. Specification of sub-grid orography 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 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

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

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

12. 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 From Scinocca and McFarlane 2000

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

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

15. Characterization of depth of blocked layer Characterise incident flow passing over the mountain top by ρH, UH, NH (averaged between μ and 2μ) Blocking height zblk satisfies (assume h=2μ): (8) hncrit≈1 tunes the depth of the blocked layer. If hncrit is increased then zblk decreases Up calculated by resolving the wind U in the direction of UH

16. Blocked-flow drag For z<zblk flow streamlines divide around mountain Drag exerted by the obstacle on the flow at these levels can be written as (5) l(z): horizontal width of the obstacle as seen by the flow at an upstream height z. It can be shown that this can be written approximately as with

17. Blocked-flow drag It can further be shown that summing over number of consecutive ridges in a grid point gives the drag B,C: constants In practice this equation is applied to the horizontal momentum equations, quasi-implicitly level by level below zblk

18. Gravity wave drag Assuming the sub-grid scale orography has elliptical shape, thegravity wave stress can be written as (Phillips 1984) G (~1): constant (tunes amplitude of waves) Increasing this increases gravity wave surface stress Typically L2/4ab ellipsoidal hills inside a grid point. Summing all forces we find the stress per unit area (using a=μ/σ)

19. Compute vertical distribution of stress • 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 • Lindzen’s 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. λequals the local saturated stress λsat • 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)

20. :amplitude of wave :mean Richardson number Saturation hypothesis in terms of Ri • Can show that wave’s impact on static stability and vertical shear gives rise to a minimum Richardson number (Palmer et al. 1986) • Instability occurs when Rimin < Ricrit (=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 Rimin=0.25 and δh=δhsat • Wave breaking occurs more readily in this formulation than in a purely convective overturning scheme

21. Gravity wave breaking only active above zblk (i.e. λ=λs for 0<z< zblk) Above zblk stress constant until waves break, i.e. saturation hypothesis (Lindzen 1981) when stress exceeds saturation stress Excess stress dissipated → drag This occurs when the local Richardson number Rimin < Ricrit(=0.25). Ricrit can be used as a tuning parameter. Increasing it makes it easier for waves to break, so drag momentum deposition occurs at a lower altitude. zblk λ • Some rough figures based on a surface flux of 0.1 N m-2 (Palmer et al. 1986 This will saturate in the • Boundary layer if U< 5 m/s (preferred region of wave breaking) • Mid troposphere if U< 7 m/s (mid-troposphere winds generally like this) • Lower stratosphere if U < 15 m/s (preferred region of wave breaking)

22. Assume stress at any level: Calculate Ri at next level λk-2 Set λk-1=λk to estimate δh using Uk-2,Tk-2 λk-1 Repeat Uk-1,Tk-1 Calculate Rimin zk=zblk; λk= λs If Rimin<Ricrit set Rimin=Ricrit estimate h=hsat Estimate λ= λsat go to next level Height z=0; λ= λs set λk-1= λk go to next level Values of the wave stress are defined progressively from zblk upwards Set λ=λs and Rimin=0.25 at model level representing top of blocked layer If Rimin>=Ricrit

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

24. Parameterized surface stresses From ECMWF T511 operational model winter summer

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

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

27. 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 Without GWD scheme cold bias westerly bias less impact in southern-hemisphere Analysis (best guess) From Palmer et al. 1986

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

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

30. Parameterization schemes in NWP models

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

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

33. PYREX campaign: Pyrenees Good agreement ECMWF T213 forecasts showing measured pressure drag and resolved and parametrized drag (Lott and Miller, 1997)

34. 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

35. 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.