Eliassen-Palm, Charney-Drazin,
and the development of wave,
mean-flow interaction theories
in atmospheric dynamics
David Andrews
1

My personal memories…
• I’ll describe some of the work I did
with Michael McIntyre from 1971-78.
• After ~ 30 years, some aspects may
have been erased from my memory!
• But I’ll try to explain how our ideas
developed…
• … and set them in context.
2

Earlier history• Since the work of Starr (MIT) andothers in the 1950s-60s,
meteorologists had been analysing
atmospheric data in terms of zonalmeans and ‘waves’ or ‘eddies’, e.g.

Zonal mean

Wave

3

• We can write dynamical equationsin terms of mean and wave terms.
• Take quasi-geostrophic zonal-mean
momentum equation, for simplicity:
Convergenceof ‘eddy
momentum
flux’
Zonal-mean
zonalacceleration

Coriolis term associatedwith mean meridional
circulation
4

Interpretation?
•We seem to have a nice physicalinterpretation: mean acceleration is“due to”a) mean Coriolis termb) eddy (or wave) fluxes.

••

BUT the mean meridional circulation
is not independent of the eddies/waves. It may even be forced bythem! (See later.)Direction of causality is not clear!
5‘Cancellation’• It is sometimes found that the‘eddy’ and ‘mean’ terms are nearly
equal, suggesting that they aresomehow related:6• The full QG set of equations isEddy
heatflux
Subscripts =partialderivatives7• Looking at only one equation (e.g.
the zonal momentum equation)
can be misleading!
• Eddy fluxes also appear in thezonal-mean thermodynamic
equation.

8

By the early 1970s several theoreticalstudies had looked at wave-mean
interaction in the stratosphere:• Matsuno (1971): stratospheric sudden
warmings, mean-flow acceleration
driven by Rossby waves.
• Lindzen & Holton (1968), Holton &Lindzen (1972): quasi-biennial
oscillation (QBO), mean-flow
acceleration driven by equatorial Kelvinand Rossby-gravity (Yanai) waves.
9How did I get involved?
• In 1971, I started a PhD atCambridge with Michael McIntyre

Me in197410• Michael was (among other things) very interested
in 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 a“two-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 in JAS 33, 2049-53, 1976)11• Another important influence on me
was Jim Holton, who had a year’ssabbatical in Cambridge while I
was 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

13• … but I didn’t fully understand it atthe time!
• Near the end of this paper was asection on general steady, nondissipated waves in a zonal meanflow.
• It gave some mysterious relations
between “energy fluxes”,“momentum fluxes” and “heatfluxes” associated with the waves…
14• I don’t think anyone (except possiblyEliassen) really understood at the time
what these meant!
15

Charney and Drazin (1961)• I also read this important paper:

Jule Charney, 1917-81

Philip Drazin, 1934-2002

• It was most famous
for working out themean-flow
conditions underwhich linear
planetary (Rossby)
waves canpropagate into theupper atmosphere.
• But it also had asection on the
nonlinear effects ofthese waves on themean flow.

17• It showed (following a suggestionby Eliassen) that the steady, nondissipated waves they considered,
had no effect on the mean flow.
• Later this came to be called a
non-acceleration theorem.

18• Uryu (1973):
clarified EP using
Lagrangian particle
displacements.
• Uryu (1974a,1974b, 1975..):several papers onO(amplitude2)
mean motions
induced by wavepackets
19After finishing my PhD…
• In 1975 I went to work on otherproblems with Raymond Hide at the UK
Met Office and Brian Hoskins at Reading
University.
• However, I kept up my interest in
wave-mean theory, in particularwondering whether a general theory
could be developed that took wave
transience and dissipation into account.
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Generalisation of EP and CD• After much algebra, McIntyre and I
found that we could generalise theresults of Eliassen & Palm andCharney & Drazin.
• EP’s mysterious eddy relation(10.8) was shown to be a special
case of a “conservation law” forwave properties, valid when thewaves are steady and nondissipated.
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Transformed Eulerian-mean formulation23

• So the eddy heat and momentum
fluxes do not act separately, but inthe combination

 the EP flux divergence24

Similar to the ‘omega equation’25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 equatorial
waves and the QBO. (JAS 1976.)• We also generalised it to other equation
sets 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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29

Lagrangian means• Suggested by Bretherton in 1971, extending
Stokes (1847) for water waves, and `acoustic
streaming’ 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 includes
other averages.
• Conservation laws:– Wave action (average overphase)– Pseudo-momentum(x-average)– Pseudo-energy (t-average)• Finite-amplitude GLM isdifficult to use in practice!
31

Interpreting particle displacements and Lagrangianmean velocity when an x-average is used
A&M’s interpretation

Matsuno’s interpretation
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This takes us up to 1978. Whathas happened in 30 years since?
• Many researchers (especiallyJapanese!) have used the transformed
Eulerian mean / EP fluxes fordiagnosing atmospheric waves inmodels and data.
• The EP flux vector F can give an idea of
direction 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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EP cross-sections
Introduced byEdmon et al.
(JAS 1980)34

Interpretation of model
sudden warmings
Dunkerton et al. (JAS 1981)35

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-Dobson
circulation, upper mesospheric circulation).

Lagrangian

Eulerian36

An unusual application:orthogonality of modes in shear flow
• Held, JAS 1985: linear modes in shear
are not orthogonal in `energy’ sense,
i.e. for 2 modes the total energy ≠ sumof energies of separate modes.
• However, they are orthogonal in thepseudo-energy or pseudo-momentum
sense.

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More recently…
• There have been many otherapplications of the theory.
• I have not done much work in thisarea for many years, and I am notfamiliar with them all!
• However, recently I have beencollaborating with researchers in
the UK Met Office, to help set up
EP diagnostics suitable for their‘non-hydrostatic’ GCM.
38• 2 weeks ago I was asked to review
yet another paper on a variant of
the 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 beappropriate then.

• Potential vorticity diagnostics may
be more useful in these cases.
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Final point• This theory has shown that there
is no unique way of defining ‘wave’and ‘mean’ quantities.
• Formulations such as the TEM andGLM may be better than theEulerian mean for interpretingsome processes.
• But the Eulerian mean may still be
the best for other purposes.
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The end42