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. 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
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)
Surface elevation along the latitude band 45oN, based on SRTM 3’’ data (65 m horizontal resolution)
GCM’s have horizontal scales ~ km, so not all processes related to orography are explicitly resolved km
Information on scales by performing a spectral analysis on the data km
Corresponding orography spectrum, i.e. variance of surface height as a function of horizontal scale, along the latitude band 45oN
From Rontu (2007)

4. Some mountain-related atmospheric processes
From Rontu (2007)
Stratification measured by N (Brunt-Vaisala frequency)
Stably stratified N>0 (i.e. potential temperature increases with height)
Unstably stratified N<0 (i.e. potential temperature decreases with height)

5. 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
(scales below a few km’s: turbulent form drag
Below smallest scale, turbulent surface friction: roughness length)
Parameterization required

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

7. 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.

8. Linear gravity-wave theory
Assumptions:
Consider two-dimensional airflow in an x-z plane over a ‘ridge’
Waves are linear, i.e. small amplitude
Ridge is sufficiently narrow that the Coriolis force can be neglected
WKBJ assumption: scale heights of background quantities such as density, temperature, and velocity, are longer than a gravity wave vertical wavelength.
Steady-state
Boussinesq flow: density differences are sufficiently small to be neglected, except where they appear in terms multiplied by g
Atmosphere inviscid
The linearized momentum equation can be reduced to a single equation for the vertical velocity
Scorer parameter
is the speed of the basic state flow
is the Brunt-Vaisala or buoyancy frequency
See Smith 1979;Houze 1993; Palmer et al. 1986

9. Constant wind speed and stability, sinusoidal ridges
Consider an infinite periodic ridge in which , i.e. sinusoidal terrain
is the horizontal wavenumber with L the width of the ridge
Solutions to the previous momentum equation can be written as
A and B are complex coefficients and Re denotes the real part
is the vertical wavenumber
Defining
the solution may be written as
Upper boundary condition implies B=0
Radiation condition implies B=0 (i.e., the perturbation energy flux must be upward)

10. k>l (i.e. narrow-ridge case)
(or equivalently U/L>N, i.e. high frequency)
Evanescent solution (i.e. fading away)
k<l (i.e. wider mountains)
(or equivalently U/L<N, i.e. low frequency)
Wave solution
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

11. Non-dimensional length: NL/U
U/L : intrinsic frequency of wave
(i.e. frequency based on time it takes for a fluid particle
to pass through disturbance)
See Houze 1993 and Palmer et al. 1986
intrinsic frequency (U/L) > buoyancy frequency (N), i.e. LN/U<1
wave cannot propagate
i.e. k>l: solution decays (or amplifies with height) – evanescent solution
not possible to support oscillations at frequencies greater than N
intrinsic frequency (U/L) < buoyancy frequency (N), i.e. LN/U>1
wave can propagate
i.e. k<l: sinusoidal solution – waves propagate without loss of amplitude
Real valued m for waves that transport momentum
More generally, vertical transport only possible when the frequency of the waves is bounded above by the buoyancy frequency and below by the Coriolis frequency h
LN/U>1 for L~1 km (with U=10 ms-1, N=0.01s-1)

12. Vertical variations in wind speed and stability,
isolated mountain case - - -
More realistic terrain and atmospheric profile
If the vertical variations in U and N are such that the Scorer parameter decreases significantly with height then trapped lee waves can extend downstream of an isolated ridge
Necessary condition and
Gravity wave Trapped lee wave
Above 3 km N is reduced by a factor of 0.4
N = 0.01 s-1 U = 10 ms-1
Wave propagates vertically in the lower layer and decays exponentially in the upper layer
Parametrization of trapped lee waves is not typical, see Gregory et al
Durran, 2003

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

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

15. Evidence of gravity waves in cloud formations
Trapped lee waves
Durran, 2003

16. Single lenticular cloud
Durran, 2003

17. 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

18. 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)

19. 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)

20. more non-linear From Scinocca and McFarlane (2000)
Vertical transport of momentum
Wave field becomes unstable or breaks above topography, redistribution of momentum between breaking level and topography, resulting in enhancement of surface pressure drag
Momentum not transported vertically away from surface, so deposited below height of topography
more non-linear
Schematic of surface pressure drag as a function of non-dimensional mountain height for constant N and U flow which characterises three flow regimes identified by numerical simulations. The value of surface pressure drag is nondimensionalized by the linear-theory pressure drag due to gravity waves (proportional to NUh2).

21. Low-level wave breaking
Wave breaking can occur immediately above the mountain top
Occurs if F is increased beyond Fc ~O(1)
Courtesy of Uni. of Utah

22. 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

23. Summary of flow classification
Orographic flow classified according to Nh/U (y-axis) and NL/U (x-axis).
Nh/U
From Rontu 2007
NL/U L

24. Influence of ‘gravity wave’ drag on the atmosphere
Flow over or around orography will typically cause the pressure to be higher on the upstream side of the mountain than on the downstream side
(c)
Net force on mountain in downstream direction from mean flow (a)
(b)
(a)
An equal and opposite force is exerted on the mean flow by the hill (b) L
: pressure anomaly
Pressure = force per area
p= F/A
1 Pa = 1 N m-2
However, this may be realised at high altitude owing to the vertical transport of momentum by gravity waves (c)
Momentum:
Vertical momentum flux:

25. Gravity wave saturation / momentum sink
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
Dispersion relationship for hydrostatic gravity waves

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. 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

30. 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-grid scale parameterization required
From Clark and Miller 1991

31. 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

32. 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.