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

2. Equation of horizontal hydrostatic motion
Drag effects acceleration through tendencies of the horizontal wind
Dyn: dynamics (resolved), e.g. advection, pressure gradient and Coriolis force
Param: physical parameterizations of sub-grid scale processes:
e.g. radiation, convection, boundary layer, orography
(linearise around
mean state)
(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)
Tendency’s due to sub-grid scale orography (SSO)
Equation for resolved part of the horizontal momentum

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

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

5. 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)

6. 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

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

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

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

10. Compute gravity wave surface stress
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
Assume sub-grid scale orography has elliptical shape
Gravity 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)
See Lott and Miller 1997

11. 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. λ equals 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)

12. 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
:amplitude of wave
:mean Richardson number

13. Gravity wave breaking only active above zblk (i. e
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 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)

14. Calculate Ri at 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
Assume stress at any level:
Calculate Ri at next level
Uk-2,Tk-2
λk-2
Set λk-1=λk to estimate δh
using
Repeat
Uk-1,Tk-1
λk-1
Calculate Rimin
Height
z=0; λ= λs
zk=zblk; λk= λs
If Rimin<Ricrit
set Rimin=Ricrit
estimate h=hsat
Estimate λ= λsat
go to next level
set λk-1= λk
go to next level
If Rimin>=Ricrit

15. Compute 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)
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. Compute blocked-flow drag
Consider again an elliptical mountain
For z<zblk flow streamlines divide around mountain
r: aspect ratio of the obstacle as seen by the incident flow; 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 zblk
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 zblk is raised by a factor H/zblk, i.e. reduction of obstacle width)
Cd (~1): form drag coefficient (proportional to ψ)

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

18. 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.

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

20. Alleviation of systematic westerly bias
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

21. less impact in southern-hemisphere
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

22. Sensitivity of resolved orographic drag to model resolution
Flow-blocking case
Flow-over 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

23. Parameterization schemes in NWP models
Blocking
Gravity wave
ECMWF √
HIRLAM
DWD
Meteofrance
ECHAM
UK UM

24. With GWD scheme - control
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

25. 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)

26. Effect of rotation on drag
Drag normalised by that predicted analytically for linear 2d non-rotating frictionless flow
High-drag hydraulic state discussed previously
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))

27. 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

28. 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, 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,
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,
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,
Olafsson, H., and P. Bougeault, 1997: The effect of rotation and surface friction on orographic drag, J. Atmos. Sci., 54,
Queney, P., 1948: The problem of airflow over mountains. A summary of theoretical studies, Bull. Amer. Meteor. Soc., 29,
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,
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,
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,
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,