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

2. Scales of orography and atmospheric processes
Length scales of Earth’s orography
L ~ km
L ~ 1-10 km
L ~ 1000–10,000 km
courtesy of shaderelief.com

3. Height scales of Earth’s orography
Surface elevation along the latitude band 45oN, based on SRTM 3’’ data (65 m horizontal resolution)
From Rontu (2007)

4. Information on scales by performing a spectral analysis on the data
Planetary/synoptic scales give largest variance (most important for global weather and climate)
But variance on all scales (small and mesoscale variations influence local weather and climate)
GCM’s have horizontal scales ~ km, so not all processes related to orography are explicitly resolved km km
Corresponding orography spectrum, i.e. variance of surface height as a function of horizontal scale, along the latitude band 45oN
From Rontu (2007)

5. Some mountain-related atmospheric processes
From Rontu (2007)

6. Resolved and parametrized processes
After Emesis (1990)
Pressure drag
lift
drag
Lift
Drag
Hydrostatic drag
Form drag
Wave drag
Gravity waves
(L>5km)
Inertial waves
Rossby waves
Viscosity of the air
Orographic turbulence
(L<5km)
Upstream blocking
Explicitly represented by resolved flow
Parameterization required

7. Gravity waves and upstream blocking
Mountain waves / gravity waves / buoyancy waves / internal gravity waves
Orographic turbulence
(see Turbulent Orographic Form Drag)
Upstream / low-level blocking
From Bougeault (1990)

8. Simple properties of gravity waves
(After T. Palmer ‘Theory of linear gravity waves’, ECMWF meteorological training course, 2004)
In order to prepare for a description of the parametrization of gravity-wave drag, we examine some simple properties of gravity waves excited by two-dimensional stably stratified flow over orography.
We suppose that the horizontal scales of these waves is sufficiently small that the Rossby number is large (ie Coriolis forces can be neglected), and the equations of motion can be written as
(1)
(2)
See Smith 1979;Houze 1993; Palmer et al. 1986

9. with the continuity and thermodynamic equations given by
(3)
(4)
Using the Boussinesq approximation whereby density is treated as a constant except where it is coupled to gravity in the buoyancy term of the vertical momentum equation. Linearising (1)-(4) about a uniform hydrostatic flow u0 with constant density ρ0 and static stability N

10. results in the perturbation variables
(5)
(6)
(7)
(8)

11. Density fluctuations due to pressure changes are small compared with those due to temperature changes, so we can write
(9)
(10)
Using (9), (5)-(8) are four equations in four unknowns. After some manipulation these can be reduced to one equation with one unknown
We now look for sinusoidal solutions of the general form

12. where
is the horizontal wavenumber
is the vertical wavenumber
is the wave frequency
which when substituted into (10), give the dispersion relation
(11)
where
is the wave intrinsic frequency

13. Let us now restrict ourselves to stationary waves forced by sinusoidal orography with elevation h(x) given by with u0 hm L
The lower boundary condition (the vertical component of the wind at the surface must vanish) is
(12)

14. (i) Evanescent solution
Solutions periodic in x are of the form
The condition of finite amplitude (B=0) and the lower boundary condition (12) gives
where
From the continuity equation,

15. Hence, these evanescent solutions take the form of a sinusoidal wave field decaying without phase tilt, showing that energy is trapped near the ground (sinuous lines indicate displacement of isopycnal surfaces)
Wind L L H H H
H and L indicate positions of maximum and minimum pressure perturbation, respectively
Notice that the vertical flux of momentum for these waves. Here the overbar represents the average along the x-direction.
If we take u0~10 m s-1 and N~0.01 s-1 then with these evanescent solutions occur when L<6 km, ie small-wavelength topography / narrow-ridge case

16. (ii) Propagating solution
Solutions will be in the form
Radiation condition implies that B=0 (ie the perturbation energy flux must be upward)
The full solution, then, is

17. Now the displacement of the isopycnals is uniform with height, but wave crests move upstream with height, ie the phase lines are tilted. The group velocity relative to the air is along these phase lines.
Wind H L H
High and low pressures are now on the nodes, so there is a net force on the topography in the direction of the flow

18. k>N/U (i.e. narrow-ridge case)
(or equivalently U/L>N, i.e. high frequency)
Evanescent solution (i.e. fading away)
Non-dimensional length NL/U<1
k<N/U (i.e. wider mountains)
(or equivalently U/L<N, i.e. low frequency)
Wave solution
Non-dimensional length NL/U>1
waves decay exponentially with height
vertical phase lines
no momentum transport
energy/momentum transported upwards
waves propagate without loss of amplitude
phase lines tilt upstream as z increases
Durran, 2003

19. Drag force
The surface pressure drag (or drag force) on the orography (per wavelength) is given by
Using the lower boundary condition and the x-component of the momentum equation this can be re-expressed as
Note that units of stress/pressure are Pascals, with 1 Pa = 1 N m-2.

20. Drag force of propagating modes
For the evanescent modes the drag force is zero as the vertical flux of momentum is zero.
For the propagating modes the horizontally averaged momentum flux
For long (hydrostatic) waves with k2<<m2 the dispersion relationship simplifies to
And so
i.e. for propagating modes the surface pressure drag or ‘drag force’ is non-zero.

21. Net force on mountain in downstream direction from mean flow (a)
An equal and opposite force is exerted on the mean flow by the hill (b)
(b)
(a)
However, this may be realised at high altitude owing to the vertical transport of momentum by gravity waves (c)

22. Gravity wave saturation / momentum sink
For the propagating modes the mean flow experiences this drag force where the wave activity is dissipative, which can be well above the boundary layer. This can occur because in the real atmosphere is not constant but decrease exponentially with height.
Convective instability occurs when the wave amplitude becomes large relative to the vertical wavelength. The streamlines become very steep and the wave ‘breaks’, much as waves break in the ocean.
Convective overturning can occur as the waves encounter increased static stability N or reduced wind speed U (typically upper troposphere or lower stratosphere). They also occur due to the tendency for the waves to amplify with height due to the decrease in air density.
Elimination of wave as its energy is absorbed and transferred to the mean wind. Drag exerted on flow as wave energy converted into small-scale turbulent motions acts to decelerate the mean velocity, i.e. wave drag ‘drags’ the flow velocity U to the gravity wave phase speed c (=0).
Dissipation can also occur as the waves approach a critical level (c = U)
Leads to wave breaking and turbulent dissipation of wave energy. This is termed ‘wave saturation’, and is a momentum sink

23. Evident from radiosondes
Gravity waves observed over the Falkland Islands from radiosonde ascent
Vosper and Mobbs

24. Single lenticular cloud
Durran, 2003

25. Evident from satellites (AIRS: Atmospheric Infra-red Sounder)
Alexander and Teitelbaum, 2007

26. Eliassen-Palm theorem
waves steepen leading to wave breaking and elimination of wave (i.e. λ >λsat) as its energy is absorbed and transferred to the mean wind, i.e. drag exerted on flow as wave energy converted into small-scale turbulent motions
Eliassen-Palm theorem: stress unchanged at all levels in the absence of wave breaking / dissipation (i.e. λ=λs)
i.e. amplitude of vertical displacement must increase as the density decreases upwards λ
Linear, 2d, hydrostatic surface stress
After Rontu et al. (2002)
(measured in Pa (N/m2)

27. Momentum flux observations
Stress rapidly changing; strong dissipation/wave breaking;
Stress largely unchanged; little dissipation/wave breaking;
Mean observed profile of momentum flux over Rocky mountains on 17 February 1970 (from Lilly and Kennedy 1973)

28. Gravity wave observations
steepening of waves leading to eventual wave breaking and turbulence
increasing vertical displacement as density decreases
trapped lee waves
downslope wind-storm
Potential temperature cross-section over the Rocky mountains on 17 February Solid lines are isentropes (K), dashed lines aircraft or balloon flight trajectories (from Lilly and Kennedy 1973)

29. Mountain flow regimes
Flow processes governed by horizontal and vertical scales (in absence of rotation) h
L~ km; h ~3-5 km
Non-dimensional height: Nh/U
U: upstream velocity
h: mountain height
N: Brunt-Vaisala frequency
linear/flow-over regime (Nh/U small)
L~ km; h ~1-3 km
non-linear/blocked regime (Nh/U large)
Non-dimensional mountain length: NL/U
L: mountain length
waves cannot propagate (NL/U small)
waves can propagate (NL/U large)
L~1-10 km; h ~ m

30. Non-dimensional height: Nh/U
Gravity waves
After Lott and Miller (1997)
Coriolis effect ignored
zblk h
linear/flow-over regime (Nh/U small)
non-linear/blocked regime (Nh/U large)
Blocking is likely if surface air has less kinetic energy than the potential energy barrier presented by the mountain
See Hunt and Snyder (1980)

31. Sensitivity to Nh/U Nh/U=0.5 Nh/U=1 Nh/U=1.4 Nh/U=2.2
linear
wave-breaking (some drag)
portion of flow goes over
highly non-linear
smooth gravity wave
almost entirely blocked upstream
wave-steepening
lee-vortices
blocked flow
large horizontal deviation
Cross section / near-surface horizontal flow
Dashed contour show regions of turbulent kinetic energy (ie wave breaking)
From Olafsson and Bougeault (1996)

32. Sensitivity to Nh/U: case studies
Wind vectors at a height of 2km in the Alpine region at 0300 UTC simulated by the UK Met Office UM model at 12 km.
Blocked flow (6 Nov 1999) Flow-over (20 Sep 1999)
From Smith et al. 2006

33. Sensitivity to model resolution: A finite amplitude mountain wave model
Topographic map of Carpathian mountains
Streamlines over the Carpathian profile with different resolutions: orography smoothed to 32, 10, and 3.3 km
Drag D expressed as pressure difference (unit Pa)
From Rontu 2007

34. Sensitivity to model resolution
Sensitivity of resolved pressure drag (i.e. no SSO parameterization scheme) over the Alps to horizontal resolution
No GWD scheme
Convergence of drag with resolution,
i.e. good estimate of drag at high resolution
large underestimation of drag at coarse resolution
i.e. sub-gird scale parameterization required
From Clark and Miller 1991

35. Fundamentals of gravity waves
Basic forces that give rise to gravity waves are buoyancy restoring forces.
If a stably stratified air parcel is displaced vertically (i.e., as it ascends a mountain barrier) the buoyancy difference between the parcel and its environment will produce a restoring force and accelerate the parcel back to its equilibrium position.
The energy associated with the buoyancy perturbation is carried away from the mountain by gravity waves.
Gravity waves forced by mountains often ‘breakdown’ due to convective overturning in the upper levels of the atmosphere, in doing so exerting a decelerating force on the large-scale atmospheric circulation, i.e., a drag.
The basic structure of a gravity wave is determined by the size and shape of the mountain and by vertical profiles of wind speed and temperature.
A physical understanding of gravity waves can be got using linear theory, i.e., the gravity waves are assumed to small-amplitude.
Gravity waves that do not break down before reaching the mesosphere are dissipated by ‘radiative damping’, i.e., via the transfer of infra-red radiation between the warm and cool regions of the wave and the surrounding environment.

36. Summary: ‘gravity wave’ drag and ‘blocking’ drag
When atmosphere stably stratified (N>0)
Create obstacles, i.e. blocking drag
Generation of vertically propagating waves
→ transport momentum between their source regions where they are dissipated or absorbed, i.e. gravity wave drag
This can be of sufficient magnitude and horizontal extent to substantially modify the large scale mean flow
Coarse resolution models requires parameterization of these processes on the sub-grid scale
→sub-grid scale orography (SSO) parametrization
Fine-scale models can mostly explicitly resolve these processes

37. References
Allexander, M. J., and H. Teitelbaum, 2007: Observation and analysis of a large amplitude mountain wave event over the Antarctic Peninsula, J. Geophys. Res., 112.
Bougeault, P., B. Benech, B. Carissimo, J. Pelon, and E. Richard, 1990: Momentum budget over the Pyrenees: The PYREX experiment. Bull. Amer. Meteor. Soc., 71,
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,
Durran, D. R., 1990: Mountain waves and downslope winds. Atmospheric processes over complex terrain, American Meteorological Society Meteorological Monographs, 23,
Durran, D. R., 2003: Lee waves and mountain waves, Encylopedia of Atmospheric Sciences, Holton, Pyle, and Curry Eds., Elsevier Science Ltd.
Emesis, S., 1990: Surface pressure distribution and pressure drag on mountains. International Conference of Mountain Meteorology and ALPEX, Garmish-Partenkirchen, 5-9 June, 1989,
Gregory, D., G. J. Shutts, and J. R. Mitchell, 1998: A new gravity-wave-drag scheme incorporating anisotropic orography and low-level breaking: Impact upon the climate of the UK Meteorological Office Unified Model, Quart. J. R. Met. Soc., 124,
Houze, R. A., 1993: Cloud Dynamics, International Geophysics Series, Academic Press, Inc., 53.
Hunt, J. C. R., and W. H. Snyder, 1980: Experiments on stably and neutrally stratified flow over a model three-dimensional hill, J. Fluid Mech., 96,
Lilly, D. K., and P. J. Kennedy, 1973: Observations of stationary mountain wave and its associated momentum flux and energy dissipation. Ibid, 30, Lott, F. and M. J. Miller, 1997: A new subgrid-scale drag parameterization: Its formulation and testing, Quart. J. R. Met. Soc., 123,
Olafsson, H., and P. Bougeault, 1996: Nonlinear flows past an elliptic mountain ridge, J. Atmos. Sci., 53,
Olafsson, H., and P. Bougeault, 1997: The effect of rotation and surface friction on orographic drag, J. Atmos. Sci., 54,
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,
Rontu, L., K. Sattler, R. Sigg, 2002: Parameterization of subgrid-scale orography effects in HIRLAM, HIRLAM technical report, no. 56, 59 pp.
Rontu, L., 2007, Studies on orographic effects in a numerical weather prediction model, Finish Meteorological Institute, No. 63.
Scinocca, J. F., and N. A. McFarlane, 2000, :The parameterization of drag induced by stratified flow over anisotropic orography, Quart. J. R. Met. Soc., 126,
Smith, R. B., 1989: Hydrostatic airflow over mountains. Advances in Geophysics, 31, Academic Press,
Smith, R. B., 1979: The influence of mountains on the atmosphere. Adv. in Geophys., 21,
Smith, R. B., S. Skubis, J. D. Doyle, A. S. Broad, C. Christoph, and H. Volkert, 2002: Mountain waves over Mont Blacn: Influence of a stagnant boundary layer. J. Atmos. Sci., 59,
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,
Vosper, S., and S. Mobbs: Numerical simulations of lee-wave rotors.