Gravity wave drag Parameterization of orographic related momentum fluxes in a numerical weather processing model Andrew Orr [email protected] 30 May 2012.

Gravity wave drag

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

Andrew [email protected]

30 May 2012

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)

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)

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

Information on scales by performing a spectral analysis on the data

From Rontu (2007)

Some mountain-related atmospheric processes

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

From Bougeault (1990)

Upstream / low-level blocking

Orographic turbulence (see Turbulent Orographic Form Drag)

Gravity waves and upstream blocking

Mountain waves / gravity waves / buoyancy waves / internal gravity waves

01

01

gx

p

z

ww

x

wu

t

w

x

p

z

uw

x

uu

t

u

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

with the continuity and thermodynamic equations given by

0

0)(1

zw

xu

t

wzx

u

(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

0

0 uuu

ww 0

results in the perturbation variables

0

0

01

01

0

000

00

zw

xu

t

z

w

x

u

gx

p

x

wu

t

w

x

p

x

uu

t

u

(5)

(6)

(7)

(8)

Density fluctuations due to pressure changes are small compared with those due to temperature changes, so we can write

00

(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

02

22

2

2

2

22

0

x

wN

z

w

x

w

xu

t

w

We now look for sinusoidal solutions of the general form

tmzkxi exp

220

222220

~

0

mk

Nkku

kNmkku

(11)

xk 2

zm 2

where

is the horizontal wavenumber

is the vertical wavenumber

is the wave frequency

which when substituted into (10), give the dispersion relation

~where

is the wave intrinsic frequency

kxhh m sin Lk 2Let us now restrict ourselves to stationary waves forced by sinusoidal orography with elevation h(x) given by with

hm

L

u0

The lower boundary condition (the vertical component of the wind at the surface must vanish) is

kxkhux

huzw m cos)0( 00

(12)

(i) Evanescent solution 0uNk

Solutions periodic in x are of the form

zmzmikx BeAeew Re

The condition of finite amplitude (B=0) and the lower boundary condition (12) gives

kxekhuw zmm cos0

where2/12

0

2

u

Nkm

From the continuity equation,

kxemhuu zmm sin0

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)

00 wuNotice 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

H HH

L LWind

H and L indicate positions of maximum and minimum pressure perturbation, respectively

(ii) Propagating solution

Solutions will be in the form

mzkximzkxi BeAew Re

Radiation condition implies that B=0 (ie the perturbation energy flux must be upward)

The full solution, then, is

0uNk

)cos(

cos

0

0

mzkxmhuu

mzkxkhuw

m

m

H HL

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

High and low pressures are now on the nodes, so there is a net force on the topography in the direction of the flow

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 solutionNon-dimensional length NL/U>1

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

)cos( mzkxAw kxAew zm cos

Durran, 2003

The surface pressure drag (or drag force) on the orography (per wavelength) is given by

dxx

hxpdx

x

hhxpD

kk

/2

0

/2

0

)0,(),(

Drag force

Using the lower boundary condition and the x-component of the momentum equation this can be re-expressed as

dxdxzwuDk

S

k

/2

0

/2

0

0 )0(

Note that units of stress/pressure are Pascals, with 1 Pa = 1 N m-2.

20

200 2

1mkmhuwu

0u

Nm

2000 2

1mkNhuwu

For long (hydrostatic) waves with k2<<m2 the dispersion relationship simplifies to

And so

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

i.e. for propagating modes the surface pressure drag or ‘drag force’ is non-zero.

•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)

(c)

(b)(a)

Gravity wave saturation / momentum sink

2

2

2

222

~~ c

NNkm

•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

Ucc

kU

~

~

0

Evident from radiosondes

Gravity waves observed over the Falkland Islands from radiosonde ascent

Vosper and Mobbs

Durran, 2003

Single lenticular cloud

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

Alexander and Teitelbaum, 2007

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 25.0 mSs UNhkwu

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

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

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

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-gird scale parameterization required

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

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.

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.