Our **first paper**
identified some persistent wave-induced forces that
are missing from the standard atmosphere-ocean
paradigm:

Oliver Bühler and Michael E. McIntyre

*J. Fluid Mech* vol. **492**, pp. 207-230 (2003).

The final published version can be downloaded from here as a .pdf (acrobat) file (0.3 Mbyte, © 2003 Cambridge University Press).

A
**followup paper**
was published in July 2005.
It is entitled

*J. Fluid Mech* **534**, pp. 67-95 (2005)

and takes a further step
toward understanding the missing-forces problem.
There are also spinoffs for understanding phonon-vortex
and roton-vortex interactions
in Bose condensates (Iordanskii forces, etc).
We rectify what might be called an `Einsteinian mismatch' between
the `force' and `vortex-interaction' viewpoints -- local thinking versus
action-at-a-distance thinking (balance, inversion, remote recoil etc).
Einstein's famous 1905 paper on the photoelectric effect began by
remarking on the profound mismatch between
physicists' then ways of thinking about
matter, quantized, and radiation, smooth.
See also
**`On the wave momentum myth'**
below.
The final published version of
**`Wave capture and wave-vortex duality'**
can be
downloaded from here
as a .pdf file (0.4Mbyte, © 2005 Cambridge University Press).
Here's the abstract:

New and unexpected results are presented regarding the nonlinear interactions between a wavepacket and a vortical mean flow, with an eye towards internal wave dynamics in the atmosphere and oceans and the problem of `missing forces' in atmospheric gravity-wave parametrizations. The present results centre around a pre-wave-breaking scenario termed `wave capture', which differs significantly from the standard such scenarios associated with critical layers or mean density decay with altitude. We focus on the peculiar wave-mean interactions that accompany wave capture. Examples of these interactions are presented for layerwise-2-dimensional, layerwise-nondivergent flows in a 3-dimensional Boussinesq system, in the strong-stratification limit.

The nature of the interactions can be summarized in the phrase `wave-vortex duality', whose key points are firstly that wavepackets behave in some respects like vortex pairs, as originally shown in the pioneering work of Bretherton (1969), and secondly that a collection of interacting wavepackets and vortices satisfies a conservation theorem for the sum of wave pseudomomentum and vortex impulse, provided that the impulse is defined appropriately. It must be defined as the rotated dipole moment of the Lagrangian-mean potential vorticity (PV). This PV differs crucially from the PV evaluated from the curl of either the Lagrangian-mean or the Eulerian-mean velocity. The results are established here in the strong-stratification limit for rotating (quasi-geostrophic) as well as for non-rotating systems. The concomitant momentum budgets can be expected to be relatively complicated, and to involve far-field recoil effects in the sense discussed in Bühler & McIntyre (2003). The results underline the three-way distinction between impulse, pseudomomentum, and momentum. While momentum involves the total velocity field, impulse and pseudomomentum involve, in different ways, only the vortical part of the velocity field.

(*J. Fluid Mech* vol. **492**, pp. 207-230)

We present a theoretical study of a fundamentally new wave-mean or wave-vortex interaction effect able to force persistent, cumulative change in mean flows in the absence of wave breaking or other kinds of wave dissipation. It is associated with the refraction of nondissipating waves by inhomogeneous mean (vortical) flows. The effect is studied in detail in the simplest relevant model, the two-dimensional compressible flow equations with a generic polytropic equation of state. This includes the usual shallow-water equations as a special case. The refraction of a narrow, slowly varying wavetrain of small-amplitude gravity or sound waves obliquely incident on a single weak (low Froude or Mach number) vortex is studied in detail. It is shown that, concomitant with the changes in the waves' pseudomomentum due to the refraction, there is an equal and opposite recoil force that is felt, in effect, by the vortex core. This force is called a ``remote recoil'' to stress that there is no need for the vortex core and wavetrain to overlap in physical space. There is an accompanying ``far-field recoil'' that is still more remote, as in classical vortex-impulse problems.

The remote-recoil effects are studied perturbatively using the
wave amplitude and vortex weakness as small parameters. The
nature of the remote recoil is demonstrated in various
set-ups with wavetrains of finite or infinite length. The
effective recoil force R_{V}
on the vortex core is given by an expression
resembling the classical Magnus force felt by moving cylinders
with circulation. In the case of wavetrains of
infinite length, an explicit formula for the scattering angle
theta of
waves passing a vortex at a distance is derived correct to second order
in Froude or Mach number.
To this order R_{V} is proportional to theta.
The formula is cross-checked against numerical
integrations of the ray-tracing equations.

This work is part of an ongoing study of internal-gravity-wave dynamics in the atmosphere and may be important for the development of future gravity-wave parametrization schemes in numerical models of the global atmospheric circulation. At present, all such schemes neglect remote-recoil effects caused by horizontally inhomogeneous mean flows. Taking these effects into account should make the parametrization schemes significantly more accurate.

The analysis uses classical Eulerian averaging throughout, in order
to take advantage of the irrotationality of the wavemotion and to
illustrate the remote-recoil effects in the simplest possible way.
However, the extension to continuous stratification
and atmospheric gravity waves is most easily
done using Lagrangian averaging, as shown in the 2005 paper
`Wave capture and wave-vortex duality' --
more precisely, it is most easily done
using the `generalized Lagrangian-mean theory'.
Some earlier developments in this area were reported in another
paper with Bühler
on non-dissipative wave-mean interactions,
*J. Fluid Mech.,* **354**, 301-343 (1998),
which can be
downloaded from here
as a .pdf file
(0.7 Mbyte, © 1998 Cambridge University Press).

Lagrangian-mean velocity fields and their divergence --
generally nonzero even for incompressible fluid motion --
are discussed in
my 1988 paper **`A note on the divergence effect and the
Lagrangian-mean surface elevation in water waves'**,
*J. Fluid Mech.,* **189**, 235-242.
Here's a
scanned reprint in pdf format
(0.8 Mbyte, © 1988 Cambridge University Press).
Here's a precursor to all this,
**`On the wave momentum myth'**,
*J. Fluid Mech.,* **106**, 331-347 (1981), as another
scanned reprint in pdf format
(2.4 Mbyte, © 1981 Cambridge University Press).
The 2005 wave-capture work presents striking new examples of
the wave-momentum myth's power to mislead.

Here is a pdf scan of the 1985 McIntyre-Palmer paper
**`A note on the general concept of wave breaking for
Rossby and gravity waves'**
justifying, via wave-mean interaction theory,
our **fundamental definition of wave breaking**
(pdf, 0.9Mbyte, © 1980 Birkhäuser Verlag).

The wave-mean theory that underpins the definition
is set out in a pair of papers I published in 1978 with
David Andrews,
**An exact theory of nonlinear waves on a
Lagrangian-mean flow**,
*J. Fluid Mech.*,
**89**, 609-646
and
**On wave-action and its relatives**,
*J. Fluid Mech.*,
**89**, 647-664
(pdfs, 3.7 and 2.4 Mbyte respectively, © 1978 Cambridge University Press).
This theory is often referred to as
the `GLM' or generalized Lagrangian-mean theory.
(Corrigendum at
**95**, 796
and in the
*Myth* paper
above.)
A tutorial paper illustrating how it works came out in 1980, entitled
An introduction to the generalized Lagrangian-mean description of
wave, mean-flow interaction,
*Pure Appl. Geophys.*, **118**, 152-176
(pdf, 1.6Mbyte, © 1980 Birkhäuser Verlag),
and a major connected account is now available in
Bühler's landmark book
*Waves and Mean Flows*
(Cambridge University Press, 2009).

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Michael Edgeworth McIntyre (mem at damtp.cam.ac.uk), DAMTP, University of Cambridge, Silver Street, Cambridge CB3 9EW