Gravity wave drag

Parameterization of orographic related momentum fluxes in a numerical weather processing model

Andrew Orranmcr@bas.ac.uk

Lecture 1: Atmospheric processes associated with orography

Lecture 2: Parameterization of subgrid-scale orography

Scales of orography and atmospheric processes

L ~ 1000–10,000 km

L ~ 100-1000 kmL ~ 1-10 km

courtesy of shaderelief.com

Length scales of Earth’s orography

From Rontu (2007)

Surface elevation along the latitude band 45oN, based on SRTM 3’’ data (65 m horizontal resolution)

•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)

Height scales of Earth’s orography

Information on scales by performing a spectral analysis on the data

1000-10000 km100-1000 km

Corresponding orography spectrum, i.e. variance of surface height as a function of horizontal scale, along the latitude band 45oN

GCM’s have horizontal scales ~ 10-100 km, so not all processes related to orography are explicitly resolved

Some mountain-related atmospheric processes

z

gN

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)

From Rontu (2007)

Resolved and parametrized processes

Inertial waves

Hydrostatic drag

After Emesis (1990)

Wave dragForm drag

Rossby waves

Pressure drag

Explicitly represented by resolved flowParameterization required

Viscosity of the air

Upstream blocking

Orographic turbulence

(L<5km)

Gravity waves

(L>5km)

DragLiftdrag

lift

(scales below a few km’s: turbulent form dragBelow smallest scale, turbulent surface friction: roughness length)

From Bougeault (1990)

Upstream / low-level blocking

Orographic turbulence (see Turbulent Orographic Form Drag)

Gravity waves and low-level blocking

Mountain waves = gravity waves = buoyancy waves

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.

Linear gravity-wave theory

022

2

2

2

wlz

w

x

w

See Smith 1979;Houze 1993; Palmer et al. 1986

Scorer parameterU

Nl

z

U

UU

Nl

1

2

2

2

22

The linearized momentum equation can be reduced to a single equation for the vertical velocity

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

)(zU is the speed of the basic state flow

)(zN is the Brunt-Vaisala or buoyancy frequency

kxhh sin0Lk /2

222 klm

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

)(Re mzkximzkxi BeAew 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

iklm 22 NLUlkBeAeew zzikx /or Re

NLUlkBeAew mzkximzkxi /or Re

Defining the solution may be written as

kxAew z sin

)sin( mzkxAw

Upper boundary condition implies B=0

Radiation condition implies B=0 (i.e., the perturbation energy flux must be upward)

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

0wu

•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

)sin( mzkxAw kxAew z sin

Durran, 2003

intrinsic frequency (U/L) > buoyancy frequency (N), i.e. LN/U<1

intrinsic frequency (U/L) < buoyancy frequency (N), i.e. LN/U>1

See Houze 1993 and Palmer et al. 1986U/L : intrinsic frequency of wave (i.e. frequency based on time it takes for a fluid particle to pass through disturbance)

wave cannot propagate i.e. k>l: solution decays (or amplifies with height) – evanescent solutionnot possible to support oscillations at frequencies greater than N

wave can propagate i.e. k<l: sinusoidal solution – waves propagate without loss of amplitude

Non-dimensional length: NL/U

222 klm

LN/U>1 for L~1 km (with U=10 ms-1, N=0.01s-1)

2

22

U

Nk Real valued m for waves that transport momentum

2222 / NLUf 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

Vertical variations in wind speed and stability, isolated mountain case

N = 0.01 s-1 U = 10 ms-1

More realistic terrain and atmospheric profileIf 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

Above 3 km N is reduced by a factor of 0.4

---

Wave propagates vertically in the lower layer and decays exponentially in the upper layer

Gravity wave Trapped lee wave

Durran, 2003

Necessary condition and222

4hll UL

Ul

Ll

UL lkl

Parametrization of trapped lee waves is not typical, see Gregory et al. 1998.

Evident from radiosondes

Gravity waves observed over the Falkland Islands from radiosonde ascent

Vosper and Mobbs

Evident from satellites (AIRS: Atmospheric Infra-red Sounder)

Alexander and Teitelbaum, 2007

Durran, 2003

Evidence of gravity waves in cloud formations

Trapped lee waves

Durran, 2003

Single lenticular cloud

Mountain flow regimes

•linear/flow-over regime (Nh/U small)

Non-dimensional height: Nh/U U: upstream velocityh: mountain heightN: Brunt-Vaisala frequency

L~1000-10000 km; h ~3-5 km

L~100-1000 km; h ~1-3 km

L~1-10 km; h ~100-500 m

•non-linear/blocked regime (Nh/U large)

Non-dimensional mountain length: NL/UL: mountain length

h

•waves cannot propagate (NL/U small)•waves can propagate (NL/U large)

Flow processes governed by horizontal and vertical scales (in absence of rotation)

•linear/flow-over regime (Nh/U small)

blkeff

blk

zhh

NhUhz

)/(1,0max

Blocking is likely if surface air has less kinetic energy than the potential energy barrier presented by the mountain

•non-linear/blocked regime (Nh/U large)

Coriolis effect ignored

effh

zblk

h

hzblk

Gravity waves

See Hunt and Snyder (1980)

After Lott and Miller (1997)

Non-dimensional height: Nh/U

Sensitivity to Nh/U

Nh/U=0.5 Nh/U=1

From Olafsson and Bougeault (1996)

wave-breaking (some drag)smooth gravity wave wave-steepening

Nh/U=1.4 Nh/U=2.2

almost entirely blocked upstream

large horizontal deviationlee-vorticesblocked flow

portion of flow goes over

Cross section / near-surface horizontal flow

Dashed contour show regions of turbulent kinetic energy (ie wave breaking)

linear highly non-linear

From Scinocca and McFarlane (2000)

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).

more non-linear

a) Vertical transport of momentum

b) Wave field becomes unstable or breaks above topography, redistribution of momentum between breaking level and topography, resulting in enhancement of surface pressure drag

c) Momentum not transported vertically away from surface, so deposited below height of topography

•Wave breaking can occur immediately above the mountain top•Occurs if F is increased beyond Fc ~O(1)

Low-level wave breaking

Courtesy of Uni. of Utah

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

From Rontu 2007L

Summary of flow classification

Orographic flow classified according to Nh/U (y-axis) and NL/U (x-axis).

Nh/U

NL/U

Influence of ‘gravity wave’ drag on the atmosphere

dxx

hpFp

Lp: pressure anomaly

•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

•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)

•However, this may be realised at high altitude owing to the vertical transport of momentum by gravity waves (c)

Pressure = force per area

p= F/A

1 Pa = 1 N m-2

Momentum:Vertical momentum flux: wu

u

(a)(b)

(c)

Gravity wave saturation / momentum sink

2

2

2

222

~~ c

NNkm

•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 Ucc

kU

~

~

Eliassen-Palm theorem

λ

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

Linear, 2d, hydrostatic surface stress 205.0 UNhkwus

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

25.0 hUNk

After Rontu et al. (2002) (measured in Pa (N/m2)

N

Uksat

3

5.0

Momentum flux observations

wuzt

u

1

Mean observed profile of momentum flux over Rocky mountains on 17 February 1970 (from Lilly and Kennedy 1973)

Momentum flux: wu

Stress largely unchanged; little dissipation/wave breaking; 0/ tu

Stress rapidly changing; strong dissipation/wave breaking; 0/ tu

Gravity wave observations

Potential temperature cross-section over the Rocky mountains on 17 February 1970. Solid lines are isentropes (K), dashed lines aircraft or balloon flight trajectories (from Lilly and Kennedy 1973)

increasing vertical displacement as density decreases

steepening of waves leading to eventual wave breaking and turbulence

trapped lee waves

downslope wind-storm

Sensitivity to model resolution: A finite amplitude mountain wave model

From Rontu 2007

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)

Sensitivity to model resolution

From Clark and Miller 1991

Sensitivity of resolved pressure drag (i.e. no SSO parameterization scheme) over the Alps to horizontal resolution

No GWD scheme

large underestimation of drag at coarse resolution

i.e. sub-grid scale parameterization required

Convergence of drag with resolution,i.e. good estimate of drag at high resolution

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

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, 806-818.•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.•Durran, D. R., 1990: Mountain waves and downslope winds. Atmospheric processes over complex terrain, American Meteorological Society Meteorological Monographs, 23, 59-81.•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, 20-22.•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, 463-493.•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, 671-704.•Lilly, D. K., and P. J. Kennedy, 1973: Observations of stationary mountain wave and its associated momentum flux and energy dissipation. Ibid, 30, 1135-1152. 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.•Olafsson, H., and P. Bougeault, 1996: Nonlinear flows past an elliptic mountain ridge, J. Atmos. Sci., 53, 2465-2489•Olafsson, H., and P. Bougeault, 1997: The effect of rotation and surface friction on orographic drag, J. Atmos. Sci., 54, 193-210.•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.•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, 2353-2393 •Smith, R. B., 1989: Hydrostatic airflow over mountains. Advances in Geophysics, 31, Academic Press, 59-81.•Smith, R. B., 1979: The influence of mountains on the atmosphere. Adv. in Geophys., 21, 87-230.•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, 2073-2092. •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.•Vosper, S., and S. Mobbs: Numerical simulations of lee-wave rotors.