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Lecture #3:Gravity Waves in GCMs

Charles McLandress (Banff Summer School 7-13 May 2005)

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Outline of Lecture

1. Role of GWs in the middle atmosphere2. Background theory3. Resolved GWs in GCMs4. Parameterized GWs in GCMs

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Part 1: The Role of GravityWaves in the Middle

Atmosphere• keep mesosphere far from radiative equilibrium

(e.g., cold summer mesopause and reversal ofthe mesospheric jets).

• alleviate cold bias in Southern Hemispherewinter polar stratosphere.

• help drive quasi-biennial oscillation (QBO) intropical lower stratosphere.

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Gravity waves seen in noctilucent clouds in mesosphere

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Observed Temperature (January)

“Radiative” Temperature (January)

Cold Summer Mesopause

150K

220K

Cooling is due to adiabaticascent in the summer polethat results from aneastward force exerted bybreaking gravity waves:

- the eastward forcegenerates aneastward wind that isdeflected toward thewinter pole by theCoriolis force.

- conservation of massthen implies ascent atthe summer pole.

from Fels (1987)

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Observed zonal mean zonal winds in mesosphere

40S 40N

Reversal of summer mesospheric winds

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Alleviation of cold bias is SH winter polar stratosphere

• downwelling produced by mesosphericgravity wave drag extends into thestratosphere as a consequence of‘downward control’ and the long radiativetimescales in the cold winter pole.

• this downwelling produces adiabaticwarming as a result of compression of airparcels.

• this effect is more important in the SHwhere the effects of other waves (planetaryRossby waves and topographicallygenerated gravity waves) are weaker.

• GCMs typically have a cold bias in the SHwinter polar stratosphere, which is alsoreferred to as the ‘cold pole problem’.

(from Garcia & Boville, 1994)

without GWD

with GWD

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Quasi-biennial oscillation

• an oscillation of the zonal mean zonalwinds in the equatorial lower stratospherewith a period varying from 22 to 34 months.

(Radiosonde observations from 1953-2003)

• the QBO is a wave-drivenphenomenon.

• the original theory proposedby Holton and Lindzen (1972)assumes that equatorialplanetary waves provide thewave driving.

• Dunkerton (1997) showedthat in the presence of tropicalupwelling planetary waveswere not enough; he proposedthat gravity waves providedthe additional wave driving.

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Part 2: Background Theory

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Buoyancy oscillations(Newton’s 2nd Law)

(hydrostatic basic state)

(Ideal gas law and definitionof potential temperature)

(potential temperatureof parcel is conserved)

(for small verticaldisplacement)

(buoyancy force)

Parcel oscillates in the vertical with the buoyancy frequency N.

displacedparcel

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Internal gravity waves

k

m

Acceleration of parcel up the slope is

Component of buoyancy force alongthe slope is

Newton’s 2nd law gives

Now consider the parcel of air displacedupward along a sloping surface:

t = 0

which is the dispersion relation for a hydrostaticgravity wave in the Boussinesq approximation.

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Linear gravity wave equationsEquations governing hydrostatic gravity waves without rotation are:

Consider plane wave solutions:Resulting dispersion relation is:

(intrinsic frequency)

(zonal momentum equation)

(thermodynamic equation)

(mass continuity equation)

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Nonhydrostatic and rotational effects

• the previous example was for linear hydrostatic GWswithout the effects of the Earth’s rotation.

• a more general dispersion relation which accountsfor nonhydrostatic and rotational effects is:

where f is the Coriolis parameter.

• a vertically propagating wave (m real) now requires that

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Satellite image of vertically trapped mountain waves (courtesy of Sam Shen, U of Alta)

Mean winddirection (U)

Vertically trapped mountain waves

• For a stationarymountain wave with

• Dispersion relationbecomes

• Short horizontalwavelengths (k > N/U)are vertically trapped.

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Momentum flux

• vertically propagating gravitywaves “transport” momentumover large height ranges.

• consider the continuity equationfor the hydrostatic gravity in thecase where m << 2H:

!

"

!

"

u’

• GWs with eastward (westward)intrinsic frequencies have positive(negative) momentum flux.

X

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Momentum flux deposition

• gravity waves interact with the mean flow throughthe deposition of momentum:

• for a steady, linear & undamped GW, the momentumflux is independent of height GWD = 0.

• GWD arises when the momentum flux changes withheight, which will occur if:

1. the GW approaches a critical level (c = U)

2. the GW ‘breaks’ and undergoes turbulent dissipation.

GWD

!

"

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Cartoon depicting critical level ‘filtering’ andnonlinear breakdown of a monochromatic GW

Mean wind

Momentum fluxGWD

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Mesoscale model simulations ofconvectively generated GWs

• to simulate small-scale GWs generated by convection a2D or 3D mesoscale model must be used.

• compared to a GCM a mesoscale model employs a veryfine horizontal resolution (1km) and a short timestep (5 s).

• here we will examine results from a 3D simulation of atropical squall line.

• the model equations are nonlinear, compressible,nonhydrostatic and nonrotating; cloud microphysicsparameterization is used.

19X (km) X (km) EW

Hei

ght (

km)

Hei

ght

(km

)

550 km

40km

40km

QBO east

QBO west

Y (k

m)

Y (k

m)

Eastward propagatingGWs filtered out byeastward QBO winds

Westward propagatingGWs filtered out bywestward QBO winds

From Piani and Durran (JAS 2001)3D mesoscale model results

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Satellite observations of convectively generatedsmall-scale gravity waves

McLandress et al (2000)

Temperature variance strongestover convective regions

Microwave Limb Sounder geometry

Single day of temperature data

Seasonal average (NH summer)

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Part 3: ResolvedGravity Waves in GCMs

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Kinetic energy spectra in middle atmosphere GCMs

Koshyk et al (1999)

CMAM

CMAM

Rotational KE

Divergent KE

• resolved GWs are included inthe divergent part of the KE.

• all models show that divergentKE grows faster with height thanthe rotational KE.

• there are large differences inthe amplitude of the spectrabetween models.

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Wavenumber-frequencyspectra of precipitation and thevertical component of the EPflux for different GCMs

precipitation Fz (70 hPa) Fz (1 hPa)

Horinouchi et al (JAS 2003)

• 2D spectra of precipitationand vertically propagatingwaves in the tropics is closelytied to the type of convectiveparameterization that is usedin the GCM.

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Impact of model resolution on mesospheric mean winds

from Hamilton (JATP 1996)

• zonal mean zonal windsfor June-August from theSKYHI model for differenthorizontal resolutions.

• As the model resolutionincreases the speed of themesospheric jets decrease:

- higher resolutionmeans more GWs areresolved.

- more GWs meanmore drag andweaker winds.120 m/s

160 m/s

-60 m/s

-40 m/s

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Resolved mesospheric gravity wave drag

• Eliassen-Palm fluxdivergence for June-August from theSKYHI model fordifferent horizontalresolutions

• the magnitude ofthe EPFD increasesas model resolutionincreases.

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Part 4: ParameterizedGravity Waves in GCMs

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• the SKYHI model results demonstrated that a GCMwith extremely high resolution is required to simulatethe small-scale GWs that produce the required GWD inthe mesosphere.

• this is further demonstratedby comparing the vertical fluxof horizontal momentum fromthe low and high resolutionsimulations of SKYHI

- the shallow slope meansthat very high horizontalresolution is required.

• since such high resolution is currently not possible for aGCM the effects of these ‘missing’ gravity waves must beparameterized.

Hamilton (1996)

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Overview of the GWD parameterization problem

• the basic idea is to include the effects of GWD on the meancirculation without actually solving for the gravity waves.

• a GWD parameterization is built on ideas from linear gravity wavetheory in conjunction with a mechanism for the nonlinear breakdown(i.e., dissipation) of GWs.

• gross simplifications are required in order that the parameterizationbe computationally efficient.

• all GWD parameterizations currently used in GCMs consist of threecomponents:

1. ‘Launch’ spectrum in the lower atmosphere

2. Linear propagation upward from the launch level

3. Nonlinear dissipation mechanism when waves attain largeamplitudes.

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1. Launch spectrum

• must specify a discrete or continuous spectrum of GWs at somelevel in the lower atmosphere which is assumed to lie above theactual gravity source regions.

• spectrum is a function of horizontal wavenumber and frequency.

• amplitude of waves in spectrum determine the amount ofmomentum flux they carry.

• great uncertainty in how to specify the launch spectrum sincethere are insufficient observations.

• a further complication is that waves with long verticalwavelengths (i.e., high intrinsic frequencies) are impossible tomeasure in the lower atmosphere:

• BUT these are the waves that have the greatest impact onthe mesosphere.

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2. Linear propagation

• current GWD parameterizations assume that GW propagationis straight up (i.e., no propagation into adjacent grid boxes).

• a linear dispersion relation is used to relate the various waveparameters (e.g., the dispersion relation for hydrostaticnonrotating GWs is often employed).

• a GW is assumed to propagate linearly, conserving itsmomentum flux until either:

1. a critical level is reached where c=U or

2. its amplitude exceeds a specified threshold.

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3. Nonlinear dissipation mechanisms

1. Convective instability and saturation:

• Lindzen (1981)

• amplitude of GW increases to point where the totaltemperature lapse rate is convectively unstable.

• Small-scale turbulence damps the GW and maintains it atan amplitude of neutral stability.

2. Convective instability and obliteration:

• Alexander and Dunkerton (1999)

• Uses Lindzen’s convective instability criterion but depositsall of the GWs momentum flux at the breaking height.

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3. Doppler spreading:

• Hines (1991, 1993, 1997)

• a GW is dissipated as it reaches a critical level givenby the background wind and the wind associated withthe parameterized spectrum (i.e., c = U + u’).

• the term ‘spreading’ is used since the Doppler shiftingof the vertical wavenumber is defined only in astatistical sense.

4. Nonlinear diffusion:

• Weinstock (1982); Medvedev and Klaassen (1995)

• A complicated and virtually incomprehensible (to me atleast) mechanism that we will discuss no more.

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5. Empirical saturation condition:

• Warner and McIntyre (1986)

• The observed m-3 form of the observed spectrum isused to limit wave amplitudes

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Simulations using the Canadian MiddleAtmosphere Model (CMAM)

• the CMAM is a middle atmosphere GCM that extends fromthe ground to 100 km.

• horizontal resolution is T32 (about 5-6 deg).

• comprehensive set of physical parameterizations relevantto the troposphere and middle atmosphere.

• a single GWD parameterization that contains differentnonlinear dissipation mechanisms.

• experiments all use identical launch spectrum for theparameterized GWs.

• McLandress and Scinocca (JAS, in press).

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Filtering of GW spectrum by tropospheric winds

• Simulations using Hines Doppler spread parameterization:Strongreversal

Weakreversal

Strongwesterlies

Weakwesterlies

CL filtering bytroposphericwesterlies

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Simulations using different nonlinear dissipationmechanisms in the GWD parameterization

Warner &McIntyre(1996)

HinesDopplerspread

Modified Warner &McIntyre (1996)

With onlycritical levelfiltering

• important message here is that the different dissipationmechanisms can be made to produce nearly same GCM responseby a simple scaling of the height at which momentum is deposited.

• this means that details of dissipation are not that important.

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1D-QBO model results

prescribedupwelling

wave momentum fluxPlanetary waves (Holton& Lindzen, 1972)

Small-scale gravity wavesparameterized usingHines (1997)

QBO is suppressed when upwelling isincreased (only planetary waves).

QBO reappears when gravity wave momentumflux is increased (both PWs and GWs).

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Simulating the QBOin the CMAM

98 levels

71 levels

Pres

sure

(hPa

)Pr

essu

re (h

Pa)

Month

Month

• T47 horizontal resolution

• Warner and McIntyre GWDparameterization:

- momentum fluxenhanced by factor offour in tropics.

96 km

73 km

30 km

30 km

Zonal mean zonal windsat equator

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The End