Parameterization of gravity waves: my perspective…

Timothy J Dunkerton

NorthWest Research Associates

…and my regrets

• No opportunity to bid good-bye to Byron• Since Waikaloa, same questions but few answers• Funding & time scarce after an extraordinary 15 years 1984-98

• Is gravity wave parameterization a dying discipline?

– The realm of unresolved dynamics is shrinking.– Contrast 3d turbulence, which will always be out of reach, and

microphysical processes outside the realm of classical mechanics.

• Quote of Dunkerton (1997 JGR):

What makes a parameterization respectable

Based on the laws of physics Incorporates relevant processes (many to select from)

Realistic parameter settings yield realistic results Agrees with a fully resolved process simulation Correctly predicts sensitivity & response to changes

Anecdotal evidence that implementation of gw param’s in GCMs is a labor-intensive tuning exercise guided by a posteriori results.

Little effort has been devoted to explicit modeling of unresolved processes.

We are left wondering whether sensitivity studies can be done at all.

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Dunkerton & Fritts (1984 JAS)Quasi-monochromatic gravity wave incident on critical level, model vs theory

Contemporary version: wave packet in 3d model incident on critical levelwith fully developed gravity-wave spectrum included in initial condition,perhaps an ensemble of many azimuthal orientations, rotation, β…purposewould be to calculate force rather than mean flow response (constant)

CL

Kitchen and McIntyre (1980 JMSJ)

• Critical level behavior is likely only if waves (i) literally have nowhere else to go or (ii) are entirely dissipated (e.g., obliterated by breaking) prior to their escape

• Caustics can act as nonsingular turning points, causing horizontal or vertical reflection w/o wave amplification

• Equatorial Kelvin wave in vertical shear of zonal mean zonal flow

• Horizontal reflection forms modal structure in latitude

• Latitudinal extent of waveguide shrinks approaching the critical level

• Ray is effectively contained, wave breaking and/or absorption presumed

critical level

EQJones C

L

Booker-Bretherton CL:usually, but not always, an absorber

Jones CL:usually, but not always, a reflector

f^

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Broutman (1986 JPO)

• Time dependence of basic state (e.g., low-frequency inertia-gravity wave) may allow a relatively fast wave to escape absorption, at least in part

• Transmission to higher levels• Reflection to lower levels• Relevant to oceanic or other

context in which mean flow is relatively weak

• A strongly sheared mean flow may preclude vertical propagation notwithstanding the Broutman mechanism: (i) absorption (w/ CL), (ii) reflection (w/o CL)

• Stratospheric QBO: strong mean shear, large equatorial waves embedded

Dunkerton (1984 JAS)

• Extension of Dunkerton and Butchart (1984) ray tracing to low-frequency inertia-gravity waves

• Horizontal propagation and refraction are more important than for high-frequency waves

• Combination of caustic and critical-level behavior

• A never-land between rays and modes

• Force paradox: wave orientation and radiation stress are rotated by horizontal shear of mean flow, but net force is applied in the original direction of propagation; evidently a horizontal pressure gradient force is required

• Remote recoil idea of Buhler and McIntyre (2003) may be relevant

meridional wavenumber = 0 initially

ray timein hours

Kawatani et al. (2003 GRL)

Alexander et al. (2002 JAS)

For a given equivalent depth, higher modes of equatorial IGWs have identical latitudinal scale of Gaussian envelope but the Hermite polynomial is of higher

order, so the eigenfunction is wider in latitude.

Kawatani et al. (2004 GRL)

mess

• Develop a general gravity-wave parameterization that (i) accounts for horizontal propagation: exchange information between columns*; and (ii) accounts for horizontal refraction: stretching and/or rotation of horizontal component of wavevector due to horizontal gradients of mean flow: alter horizontal wavenumber and azimuth

A fork in the road

• Continue to rely on a single-column parameterization of gravity waves without horizontal propagation and refraction but require that (i) the parameterization be uniformly valid for unresolved waves and (ii) that all other waves for which this requirement is not met be explicitly resolved by the large-scale model (“hybrid” approach)

How to deal with horizontal propagation & refraction

Unresolved waves must disappear before violating this requirement

*Wave action density at caustic may be problematic (spurious over-concentration)

• Combination of (i) continuous spectrum in single-column parameterization and (ii) discrete representation of parameterized modes: e.g., Dunkerton (1997 JGR)

• Speaking from experience, somewhat pedantic & clumsy; requires simplifications for effects of shear on modes

• Good constraints for convectively forced modes (OLR), none yet for laterally forced modes

Two possible “hybrid” approaches

• Combination of (i) continuous spectrum in single-column parameterization and (ii) explicit numerical modeling of resolved spectrum: e.g., Giorgetta et al. (2005 JGR)

• Spectacular success achieved in modeling the gw spectrum (e.g., Sato et al., 1999 JAS)

• However, horizontal resolution of caustics is uncertain, and similarly…

• Vertical resolution of critical layers & momentum deposition

Where to draw the line between resolved & unresolved?

O’Sullivan & Dunkerton, 1995 JAS

130 hPa DIV

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ω / f ratio

0.0, 0.5,1.0,…3.0

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Pathological nature of the IGW manifoldGravity waves cannot be properly modeled by any (non-LES) model currently in use.

Unified mathematical framework for spectral parameterization

source

saturated

obliterated

We are working on a unified scheme: Lindzen, AD, W-McI & Hines

Mean flow

Phase velocity

Hines’ critical circle

Applies to any model of wave force that dependson the local intrinsic frequency of the wave.

Possibly a novel use for equatorial IGW modes

• It is uncertain to what extent inertia-gravity waves trapped in the equatorial waveguide are able to form modes resembling equatorially-trapped waves.

• Horizontal refraction of these slow IGW rays cannot be neglected, however.

• In the spirit of gw param, we are not concerned with individual IGW events, only their ensemble average. It seems silly to project individual events onto EQW modes!

• A better approach is calculate the ensemble-average projection of all events onto a single EQW mode, for each mode n=-1,0,1,2,3…

• In this manner a gravity-wave parameterization may account for horizontal refraction in a simplified framework of EQW modes which are known from (exact or approximate) theory.

Idea #1

Tangent linear gravity wave parameterization

• Perhaps we are asking our gw param’s to do too much.• Sensitivity issues suggest an alternative approach:

• Constrain the large-scale model to resemble observations using relaxation to the observed state (“control” run; D’97 JGR).

• Save the required body force for future model integrations under slightly different external forcings (ibid.).

• Use the gravity-wave parameterization to calculate the difference between the original gravity-wave induced force (not actually used in the control run) and the modified gw force in altered state.

• Calculate the model sensitivity to change in external forcing by combining the original “perfect” missing force and gw force difference as determined by the parameterization.

• This idea is well-suited to assimilation models where O-F is known.

• Optional: inverse model of “saved” force for gw parameters

Idea #2

Kawatani et al. (2005 GRL)

Award winning QBO?

Parameterization? What parameterization?

1st Workshop on Spontaneous Imbalance

University of WashingtonSeattle WA

August 7-10, 2006

current stats: ~42 oral presentations

http://www.nwra.com/siw1/