Andrew-Wave Mean-slide 2009-04-09 EP CD

8/11/2019 Andrew-Wave Mean-slide 2009-04-09 EP CD

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Eliassen-Palm, Charney-Drazin,

and the development of wave,

mean-flow interaction theories

in atmospheric dynamics

David Andrews


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My personal memories

Ill describe some of the work I didwith Michael McIntyre from 1971-78.

After ~ 30 years, some aspects mayhave been erased from my memory!

But Ill try to explain how our ideas

developed and set them in context.


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Earlier history

Since the work of Starr (MIT) andothers in the 1950s-60s,

meteorologists had been analysingatmospheric data in terms of zonalmeans and waves or eddies, e.g.

Zonal mean Wave


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Cancellation

It is sometimes found that theeddy and mean terms are nearlyequal, suggesting that they are

somehow related:


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The full QG set of equations is

Subscripts =partial

derivatives

Eddyheatflux


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Looking at only one equation (e.g.

the zonal momentum equation)can be misleading!

Eddy fluxes also appear in the

zonal-mean thermodynamicequation.


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By the early 1970s several theoreticalstudies had looked at wave-mean

interaction in the stratosphere:

Matsuno (1971): stratospheric sudden

warmings, mean-flow accelerationdriven by Rossby waves.

Lindzen & Holton (1968), Holton &

Lindzen (1972): quasi-biennialoscillation (QBO), mean-flowacceleration driven by equatorial Kelvinand Rossby-gravity (Yanai) waves.


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How did I get involved?

In 1971, I started a PhD atCambridge with Michael McIntyre

Me in1974


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Michael was (among other things) very interestedin some wave, mean problems, and had (I think)recognised the importance of wave transience andwave dissipation in driving mean-flow changes.

He suggested I should look at the O(amplitude2)effect of various waves on mean flows, using atwo-timing technique, etc.

I also looked at Lagrangian means, proposed by F.P. Bretherton (1971).

All this was entirely analytical no computerswere used!

The most interesting application was to theinteraction of Kelvin and RG waves to the QBO(later published inJAS 33, 2049-53, 1976)


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Another important influence on mewas Jim Holton, who had a years

sabbatical in Cambridge while Iwas doing my PhD.

James R Holton (1938-2004)


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Eliassen and Palm (1961)

I read thisfamouspaper while

I was astudent

Arnt Eliassen, 1915-2000


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but I didnt fully understand it at

the time! Near the end of this paper was a

section on general steady, non-

dissipated waves in a zonal meanflow.

It gave some mysterious relationsbetween energy fluxes,

momentum fluxes and heatfluxes associated with the waves


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I dont think anyone (except possiblyEliassen) really understood at the timewhat these meant!


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Charney and Drazin (1961)

I also read this important paper:

Philip Drazin, 1934-2002

Jule Charney, 1917-81


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It was most famousfor working out the

mean-flowconditions underwhich linearplanetary (Rossby)waves canpropagate into theupper atmosphere.

But it also had asection on thenonlinear effects ofthese waves on themean flow.


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It showed (following a suggestionby Eliassen) that the steady, non-

dissipated waves they considered,had no effect on the mean flow.

Later this came to be called a

non-acceleration theorem.


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Uryu (1973):clarified EP using

Lagrangian particledisplacements.

Uryu (1974a,1974b, 1975..):several papers onO(amplitude2)

mean motionsinduced by wavepackets


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After finishing my PhD

In 1975 I went to work on otherproblems with Raymond Hide at the UKMet Office and Brian Hoskins at ReadingUniversity.

However, I kept up my interest inwave-mean theory, in particular

wondering whether a general theorycould be developed that took wavetransience and dissipation into account.


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Generalisation of EP and CD

After much algebra, McIntyre and Ifound that we could generalise theresults of Eliassen & Palm andCharney & Drazin.

EPs mysterious eddy relation(10.8) was shown to be a specialcase of a conservation law for

wave properties, valid when thewaves are steady and non-dissipated.


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Transformed Eulerian-mean formulation


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So the eddy heat and momentumfluxes do not act separately, but inthe combination

the EP flux divergence


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Similar to the omega equation


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Reduction to EP and CD


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Further generalisations

McIntyre and I originally did this for theBoussinesq primitive equations on abeta-plane, and applied it to equatorialwaves and the QBO. (JAS 1976.)

We also generalised it to other equationsets and spherical geometry. (JAS1978.)

At the same time, John Boyd (JAS 33,2285-2291, 1976) had similar ideas.


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Most other people found this paper mysterious, too!


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Lagrangian means

Suggested by Bretherton in 1971, extendingStokes (1847) for water waves, and `acousticstreaming ideas for sound waves.

Take time-average following a fluid particle

(Lagrangian mean), not at a fixed point (Eulerianmean).


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Generalised Lagrangian Mean (1978)

For finite-amplitude waves,in principle [not restricted toO(amplitude2)] and includesother averages.

Conservation laws: Wave action (average over

phase)

Pseudo-momentum

(x-average) Pseudo-energy (t-average)

Finite-amplitude GLM isdifficult to use in practice!


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A&Ms interpretation

Matsunos interpretation

Interpreting particle displacements and Lagrangianmean velocity when anx-average is used


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This takes us up to 1978. Whathas happened in 30 years since? Many researchers (especially

Japanese!) have used the transformedEulerian mean / EP fluxes fordiagnosing atmospheric waves inmodels and data.

The EP flux vector F can give an idea ofdirection of wave propagation

(generalisation of group velocity). Its divergence gives a force per unit

mass acting on the mean flow.

Some early examples


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Introduced byEdmon et al.

(JAS 1980)

EP cross-sections


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Interpretation of model

sudden warmings

Dunkerton et al. (JAS 1981)


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Tracer transport Variants of the TEM and GLM formalisms

have been used (e.g. Dunkerton,JAS 1978)to diagnose wave-driven tracer transport instratospheric models (e.g. Brewer-Dobsoncirculation, upper mesospheric circulation).

LagrangianEulerian


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An unusual application:orthogonality of modes in shear flow

Held,JAS 1985: linear modes in shearare not orthogonal in `energy sense,i.e. for 2 modes the total energy sum

of energies of separate modes. However, they are orthogonal in the

pseudo-energy or pseudo-momentumsense.


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More recently

There have been many otherapplications of the theory.

I have not done much work in this

area for many years, and I am notfamiliar with them all!

However, recently I have beencollaborating with researchers inthe UK Met Office, to help set upEP diagnostics suitable for their

non-hydrostatic GCM.


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2 weeks ago I was asked to reviewyet another paper on a variant ofthe Generalised LagrangianMean!


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Limitations of the approach

EP diagnostics may not work wellfor large-amplitude disturbances

(e.g. breaking Rossby waves in thestratosphere, baroclinic waves inthe troposphere). Wave, mean separation may not be

appropriate then. Potential vorticity diagnostics may

be more useful in these cases.


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Final point

This theory has shown that thereis no unique way of defining waveand mean quantities.

Formulations such as the TEM andGLM may be better than theEulerian mean for interpretingsome processes.

But the Eulerian mean may still bethe best for other purposes.


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The end

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Andrew-Wave Mean-slide 2009-04-09 EP CD