Parametrization of orographic processes in numerical weather processing Andrew Orr Lecture 1: Effects of orography Lecture 2: Sub-grid scale orographic parameterization

History of orography parameterization 1.Pioneering of studies on linear 2d gravity waves (e.g. Queney, 1948) 2.Gravity wave drag recognised as important sink of atmospheric momentum (e.g. Eliassen and Palm, 1961) 3.Observational and modelling studies of non-linear waves (e.g. Lilly, 1978) 4.Modelling of 3d nonlinear waves 5.Development of envelope orography (not satisfactory technique for representation of large-scale flow blocking) 6.Alleviation of systematic westerly bias in numerical weather prediction models through gravity wave drag (GWD) parameterization (Palmer et al. 1986) 7.High-resolution numerical modelling 8.Alleviation of inadequate representation of low-level drag through ‘blocked flow’ drag parameterization (Lott and Miller 1997). This is the ECMWF orography parameterization scheme.

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

Alleviation of systematic westerly bias Analysis 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 (from Palmer et al 1986) flow is too strong temperature too low Without GWD scheme less impact in southern-hemisphere

Alleviation of systematic westerly bias 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 Parameterisation of gravity wave drag decelerated the predominately westerly flow

High-resolution numerical modelling From Clark and Miller 1991 Sensitivity of pressure drag and momentum fluxes due to the Alps to horizontal resolution No GWD scheme large underestimation of drag

Specification of sub-grid orography xx h: topographic height above sea level (from global 1km data set) * * * * h: mean topographic height at each gridpoint - From Baines and Palmer (1990) At each gridpoint sub-grid orography represented by: μ: standard deviation of h (amplitude of sub-grid orography) 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) 2μ approximates the physical envelope of the peaks Note source grid is filtered to remove small-scale orographic structures and scales resolved by model – otherwise parameterization may simulate unrelated effects

Specification of sub-grid orography Calculate topographic gradient correlation tensor Direction of maximum mean-square gradient at an angle θ to the x-axis Diagonalise

Specification of sub-grid orography Change coordinates (orientated along principal axis) 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: (ψ=0 flow normal to obstacle; ψ=π/2 flow parallel to obstacle)

Resolution sensitivity of sub-grid fields ERA40~120km T511~40km T799~25km

Sub-grid scale orographic parameterisation 1.Compute surface pressure drag exerted on subgrid-scale orography 2.Compute vertical distribution of wave stress accompanying the surface value Gravity wave drag Blocked flow drag 1.Compute depth of blocked layer 2.Compute drag at each model level for z < z blk Scheme used for: ECMWF (Lott and Miller 1997), UK Met UM, HIRLAM, etc z blk hz/z blk h ef f h

Evaluation of blocking height Characterise incident (low-level) flow passing over the mountain top by ρ H, U H, N H (averaged between μ and 2μ) Define non-dimensional mountain height H n = hN H /U H In ECMWF model assume h=3μ Blocking height z blk satisfies: Where H ncrit ≈1 tunes the depth of the blocked layer (uses wind speed U p calculated by resolving the wind U in the direction of U H )

Evaluation of blocked-flow drag Assume sub-grid scale orography has elliptical shape For z<z blk flow streamlines divide around mountain. Drag exerted by the obstacle on the flow at these levels can be written as l(z): horizontal width of the obstacle as seen by the flow at an upstream height z ( assumes each layer below z blk is raised by a factor H/z blk, i.e. reduction of obstacle width ) r: aspect ratio of the obstacle as seen by the incident flow C d (~1): form drag coefficient (proportional to ψ) B,C: constants Summing over number of consecutive ridges in a grid point gives the drag This equation is applied quasi-implicitly level by level below z blk See Lott and Miller 1997

Evaluation of gravity wave surface stress Consider again an elliptical mountain Gravity wave stress can be written as (Phillips 1984) G (~1): constant (tunes amplitude of waves) Typically L 2 /4ab ellipsoidal hills inside a grid point. Summing all forces we find the stress per unit area (using a=μ/σ)

Evaluation of stress profile Gravity wave breaking only active above z blk (i.e. λ=λ s for 0<z< z blk ) Above z blk stress constant until waves break (i.e. convective overturning) This occurs when the local Richardson number Ri min < Ri crit (=0.25), i.e. saturation hypothesis (Lindzen 1981) :amplitude of wave :mean Richardson number Values of the wave stress are defined progressively from the top of the blocked layer upwards

Evaluation of stress profile Set λ=λ s and Ri min =0.25 at model level representing top of blocked layer Assume stress at any level U k-1,T k-1 U k-3,T k-3 U k-2,T k-2  k-2  k-1 z=0; λ= λ s z k =z blk ; λ k = λ s Height Calculate Ri at next level Set λ k-1 =λ k to estimate δh using Calculate Ri min If Ri min >=Ri crit estimate  h set  k-1 =  k go to next level If Ri min <Ri crit set Ri min =Ri crit estimate  h=  h sat estimate  =  sat go to next level Repeat

Gravity wave stress profile U  Deceleration Wave breaking 10km Weak winds at low-level can result in low-level wave breaking. Corresponding drag distributed linearly over a depth Δz (above the blocked flow) Note, trapped lee waves not represented in Lott and Miller scheme. However, accounted for in UK Met Office UM model (see Gregory et al. 1998)

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 Strong interaction/compensation between drag contributions

From ECMWF T511 operational model Parameterized surface stresses

Sensitivity of resolved orographic drag to model resolution From Smith et al drag converging parameterization still required at high-resolution Weak flow: most drag produced by flow splitting Strong flow: short- scale trapped lee waves produce significant fraction of drag (Georgelin and Lott, 2001

Orographic form drag due to scales <5000m Effective roughness concept (Taylor et al. 1989) Enhancement of roughness length above its vegetative value in areas of orography Disadvantages: Can reach 100’s of meters Roughness lengths for heat and moisture have to be reduced New scheme: Directly parameterises TOFD and distributes it vertically (Beljaars et al. 2004) Vegetative roughness treated independently Requires filtering of orography field to have clear separation of horizontal scales Spectrum of orography represented by piecewise empirical power law Integrates over the spectral orography to represent all relevant scales Wind forcing level of the drag scheme depends on horizontal scale of orography

Enhancement of convection by orography: Simulation of mid-afternoon precipitation maximum

July 2003 mean operational T511 cross-sections of wind (m/s) and specific humidity (g/kg) morning afternoon evening night

References Baines, P. G., and T. N. Palmer, 1990: Rationale for a new physically based parameterization of sub-grid scale orographic effects. Tech Memo 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, 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, Eliassen, A. and E., Palm, 1961: On the transfer of energy in stationary mountain waves, Geofys. Publ., 22, 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, 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, Lilly. D. K., 1978: A severe downslope windstorm and aircraft turbulence event induced by a mountain wave, J. Atmos. Sci., 35, Lindzen, R. S., 1981: Turbulence and stress due to gravity wave and tidal breakdown. J. Geophys. Res., 86, Lott, F. and M. J. Miller, 1997: A new subgrid-scale drag parameterization: Its formulation and testing, Quart. J. R. Met. Soc., 123, Queney, P., 1948: The problem of airflow over mountains. A summary of theoretical studies, Bull. Amer. Meteor. Soc., 29, 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, Phillips, D. S., 1984: Analytical surface pressure and drag for linear hydrostatic flow over three-dimensional elliptical mountains. J. Atmos. Sci., 41, Smith, S., J. Doyle., A. Brown, and S. Webster, 2006: Sensitivity of resolved mountain drag to model resolution for MAP case studies. Submitted to Quart. J. R. Met. Soc.. 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,