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\newcommand{\sliver}{{\hskip0.6pt}}  %Fine control of math spacing
\newcommand{\ssliver}{{\hskip0.9pt}}
\newcommand{\Sliver}{{\hskip1.2pt}}
\newcommand{\antisliver}{{\hskip-0.6pt}}
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\newcommand{\half}{{\textstyle\frac{1}{2}}}
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\newcommand{\corioliso}{f_{\antisliver 0}}
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\newcommand{\LD}{{L_\mathrm{D}}}
\newcommand{\LDSQ}{L_\mathrm{D}^{2}}
\newcommand{\LDMT}{L_\mathrm{D}^{-2}}
\newcommand{\PV}{{Q}}
\newcommand{\PVB}{{\PV_{\mathrm{b}}}}
\newcommand{\PVA}{{\PV_{\mathrm{a}}}}
\newcommand{\pv}{{q}}
\newcommand{\pvb}{{\pv_{\mathrm{b}}}}
\newcommand{\pva}{{\pv_{\mathrm{a}}}}
\newcommand{\pvj}{{\pv_{\sliver\mathrm{j}}}}
\newcommand{\uj}{{u_{\sliver\mathrm{j}}}}
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\newcommand{\DG}{{D_{\antisliver \rm g\sliver}}}
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\newcommand{\lapl}{\nabla_{\! H}^2} %(and in Lapl. want 2nd order explicit)
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\newcommand{\BOMEGA}{{\mathbf{\Omega}}}
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\newcommand{\K}{\mbox{\Sliver K}}
\newcommand{\degree}{{$^\circ$}}  %\deg already {\rm deg}
\newcommand{\km}{\mbox{\Sliver km}}
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\newcommand{\cm}{\mbox{\Sliver cm}}
\newcommand{\ms}{\mbox{\Sliver m\Sliver s$^{-1}$}}
\newcommand{\mph}{\mbox{\sliver m\sliver p\sliver h}}
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\newcommand{\mmss}{\mbox{\Sliver m$^2$s$^{-2}$}}
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\newcommand{\PVU}{\mbox{$10^{-6}$m$^2$s$^{-1}$K\Sliver kg$^{-1}$}}

\newcommand{\ao}{atmosphere--ocean}
\newcommand{\te}{thought-experiment}

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\chapter[The atmospheric wave--turbulence jigsaw]{The atmospheric
wave--turbulence jigsaw
\label{ch1}}

\centerline{
Michael E. McIntyre
}
\vspace{0.2cm}
\centerline{\small
\textit{Dept.\ of Applied Mathematics \& Theoretical Physics,
University of Cambridge}
}
\centerline{\small
\textit{www.atm.damtp.cam.ac.uk/people/mem}
}

\vspace{0.4cm}



\vspace{+0.001cm}\section{Introduction}\label{sec1.1}\vspace{+0.001cm}

It was a huge honour to be asked to give the Marshall
Rosenbluth Memorial Lecture.
Having never worked on plasma
physics, though, I also feel some diffidence\sliver!
The closest I've ever come has been involvement
in some peculiar MHD problems that promise an improved understanding
of the solar tachocline --- more about confining
a magnetic field within a plasma than
a plasma within a magnetic field.
Before proceeding
I want to thank Drs Laur\`ene Jouve and Chris McDevitt for producing the
first draft of this chapter following my lecture.  Chris also kindly
lent assistance with the source files and graphics.
However, the final responsibility for this chapter and any errors it
may contain is mine alone.

What I do know about is the kind of fluid dynamics
that has helped us to understand the
Earth's atmosphere and oceans.  We still have an
enormous phase space,
albeit with fewer degrees of freedom than for
plasma physics.  Thanks to countless observations and to
the peculiarities of
flow heavily constrained by Coriolis effects and
stable density stratification,
great progress has been made in penetrating the nonlinear
dynamics.  Indeed, and quite surprisingly,
researchers into \ao\ dynamics have gained
insight in a way that cuts straight to strong nonlinearity,
avoiding
the standard paradigms.
And an even greater surprise, to me at least,
has been what
Pat Diamond, Paul Terry and others
have been saying in
recent years,
namely
that some insights from \ao\ dynamics
are relevant to some aspects of plasma behaviour in tokamaks
and stellarators ---  henceforth ``tokamaks'' for brevity ---
especially the self-organizing zonal
or quasi-zonal
flows that appear so important for
plasma confinement.  See, e.g.,
\cite{Diamond:2005}, \cite{Terry:2000}, \cite{Krommes:2012}
and references therein.


The standard paradigms avoided
include those of
weak nonlinearity, strong scale separation, ``cascades''
in the strict sense of being local in wavenumber space, and indeed
homogeneous turbulence theory
in all its flavours.
Ever since the 1980s when infrared remote sensing from space
began to give us global-scale views of, especially,
stratospheric fluid flow,
it has become apparent that
we are dealing with a
 highly inhomogeneous
``wave--turbulence jigsaw puzzle''
with no scale separation but with weakly and strongly nonlinear
regions closely adjacent and intimately interdependent
\cite{McIntyre:1983}, \cite{dimbo:2011}.
One of the tools that have helped us to make sense of this
has been the
finite-amplitude ``wave--mean interaction theory''
developed over many years and recently summarized in a
beautiful new book by my colleague Oliver B\"uhler \cite{Buhler:2009}.



For instance a typical phenomenon, once completely mysterious
but now well understood, and understood in a very simple way,
is the
self-organization and the peculiar persistence and quasi-elasticity of
the great \ao\ jet streams.
They can persist over surprisingly large distances.
If ordinary, domestic-scale
jets behaved similarly, you could blow out your birthday candles
from the far end of the room.
There are
``anti-frictional'' effects that
prevent the great jets from spreading out dissipatively,
tending to re-sharpen their velocity profiles if
something smears them out.
As recorded in the famous books by
Edward N. Lorenz, the father of chaos theory, and
Victor P. Starr, the pioneer of postwar global
upper-air data analysis, this behaviour used to be called
``negative viscosity''
and regarded as a profound enigma
\cite{Lorenz:1967}, \cite{Starr:1968}.
Such was the state of things when I became Jule G. Charney's postdoc
at MIT in the late 1960s.

The most conspicuous jets include
the Gulf Stream, the Kuroshio
Current, and the atmospheric jet streams that are
typically found
at airliner cruise altitudes,
fast-flowing rivers of air a a few kilometres deep
and a few hundred kilometres wide, roughly speaking.
As airline operators know very well,
these atmospheric jet streams --- which are among
the most comprehensively observed of natural phenomena ---
can persist for thousands of kilometres and
can have wind speeds sometimes exceeding
even \mbox{100\ms} or \mbox{200\knot}.
The cores of these thin jets may meander,
river-like, with large amplitudes, but nevertheless
form resilient, flexible
barriers tending to inhibit
the turbulent transport of material
across them --- by contrast with the high-speed
advective transport along them
--- making these
jets almost the
``veins and arteries of the climate system''.
Equally
spectacular, though far less well understood,
are the prograde jets and associated transport barriers
in the visible
weather layer of the planet Jupiter.

I find myself wondering whether
tokamak zonal flows are closer to the terrestrial or
Jovian cases.  Our relatively poor understanding of Jupiter
makes this a hard question to answer at present.
For Jupiter there are
relatively few observational constraints,
apart from the wind fields derived from cloud-top motions
and
the peculiar straightness of the prograde jets that makes
them, as it were, so conspicuously
unearthly.  It might be
good news if the zonal jets in big tokamaks were more
Jupiter-like than Earth-like, because less meandering
might mean better confinement.

It
cannot be too strongly emphasized that,
for all the prolific literature,
our understanding of the Jovian problem
is indeed in its infancy.
For one thing, progress has been impeded
by a tendency to forget that in the real planet,
as distinct from many models of it that have been studied,
there is no solid surface
or phase change sufficiently near the visible surface
to support the type of baroclinic instabilities
that excite terrestrial atmospheric jets.
There is a ``convective thermostat'' mechanism --- see
\cite{Rogers:1995} and references therein,
also footnote~4 of \cite{Dritschel:2008} ---
that pretty much
precludes any major role for such baroclinic instabilities.
And our understanding of the most basic aspect of all, namely the
coupling between Jupiter's weather
layer and the underlying convection zone,
is very poor indeed.
Here I think
there are some interesting paradigm changes in progress.
But let me come back to Earth and to things
we know much more about.


\vspace{+0.001cm}\section{On eddy-transport barriers}\vspace{+0.001cm}
\label{sec:barriers}

The turbulent transport inhibition or
``eddy-transport barrier'' effect at jet cores
goes hand-in-hand with the anti-frictional effects.
It depends
on the strong, self-maintaining
horizontal shears adjacent to the jet core
as well as on the \emph{potential-vorticity gradients}
--- see
below --- that are concentrated at
the jet core.
The importance of shear was pointed out in \cite{Juckes:1987}.
So although we used to speak of
``potential-vorticity barriers'', the tendency
in the \ao\ community these days is to call them
``eddy-transport barriers''.
Their dynamics involves both wave propagation and turbulence ---
strongly nonlinear and strongly inhomogeneous spatially, with
no spatial scale separation.
As already hinted, this is far beyond the reach both of
homogeneous turbulence theory and of weakly-nonlinear wave or
weak-turbulence theory, also called ``wave-turbulence'' theory.



The eddy-transport-barrier effect has
been demonstrated again and again
from observations, from laboratory experiments, and from high-resolution
numerical models.
An early and very striking observational demonstration
came from studies of the radioactive debris from atmospheric nuclear
tests published in 1968.
Using an instrumented aircraft,
two material air masses well characterized by
differing radioactive properties
were observed flowing side by side in close proximity,
without mixing, on either side of a
jet core \cite{Danielsen:1968}.

Another striking demonstration
came from a laboratory experiment in Harry Swinney's
big rotating tank at the University of Texas at
Austin \cite{Sommeria:1989b}, \cite{Sommeria:1991};
see \fref{fig:tank}. \
Dye injected on one side of a jet stayed there,
after more than 500 tank revolutions ---
almost perfectly confined despite the large-amplitude undulation
of the jet.
The jet core and velocity maximum were found to be
almost coincident with
the dye boundary.


And again, there has been a huge amount of observational and
numerical modelling work in connection with
the concern over ozone hole in the Antarctic
stratosphere, where the polar-night jet, or polar-vortex edge,
keeps itself sharp and acts as an eddy-transport barrier
within which the ozone chemistry proceeds differently
from the chemistry outside.  This has been studied
using intensive observations and a large hierarchy of models
for over two decades now; among the many landmarks we may note
a remarkable pair
of papers by
Norton \cite{Norton:1994}
and
Waugh and Plumb \cite{Waugh:1994}.
My website has a
movie from Norton's work that visualizes the barrier effect
rather spectacularly ---
websearch \texttt{"dynamics that is significant for chemistry"}.

One might ask why there should be any comparison between
the abovementioned laboratory experiment and the atmosphere.
Admittedly both are rapidly-rotating systems, in the sense that
Coriolis effects are strong.  However,
stable stratification
is very important in the atmosphere, and in the oceanic examples too,
whereas the laboratory experiment used an unstratified fluid.
The answer is, I've always thought, a rather surprising one.
The kind of dynamics involved in
all these systems has the same generic structure, explaining
the many qualitative similarities between the systems.
Even more surprisingly, the same generic structure
is found in
the tokamak models of
Hasegawa, Mima and Wakatani.
As we'll see,
it is well illustrated by the simplest such model defined by the
Hasegawa--Mima equation
(\sref{sec:inv} below).
And it is this structure that allows the insight into strong
nonlinearity.


\begin{figure}[t]
\centerline{\psfig{file=lab-expt.eps,width=6cm}}
\caption{From the laboratory study of Sommeria et al.\
\protect \cite{Sommeria:1989b}.
Courtesy Dr Jo\"el Sommeria.
}
\label{fig:tank}
\end{figure}



\vspace{+0.001cm}\section{The generic dynamics}\vspace{+0.001cm}
\label{sec:generic}

The generic dynamics is
shared by
a whole hierarchy
of models of stratified, rotating
\ao\ dynamics and their unstratified laboratory counterparts,
including some remarkably accurate stratified models
that easily explain, for instance,
the characteristic cross-sectional
structure of jet streams that has long been familiar to
observational meteorologists
\cite{Riehl:1962}, \cite{Palmen:1969};
see \fref{fig:upper-cyclone} below.
In all these models
one has a single scalar field $\PV(\bx,\sliver t)$ that is a
\emph{material invariant} for ideal fluid flow
--- that is, $\PV$ is constant on each material particle ---
expressing the advective nonlinearity in a way that
is easy to understand and to visualize.
And more than that, the evolution of the $\PV$ field
captures \emph{everything} about the advective nonlinearity
because,
to the extent that these models are accurate, the
$\PV$ field contains all the dynamical information at each instant.
So if for example
$\bu(\bx,\sliver t)$ is the velocity field,
which for definiteness we'll take relative to the rotating Earth,
then $\bu$ can be deduced at each instant
from the $\PV$ field alone.

$\PV$, to be defined shortly,
is called the \emph{potential vorticity} (PV) of the model, and
the mathematical process of deducing the $\bu$ field from the
$\PV$ field is called \emph{PV inversion}.
The hierarchy of models
arises because there is an array of different PV inversion operators.
They are inverse elliptic operators, hence nonlocal.
They differ among themselves
for two reasons.  One is simply to accommodate
the different physical systems that might be of interest,
for example the atmosphere, or the
laboratory system of \fref{fig:tank}, or
the tokamak.
The other is that, for a given physical system,
some PV inversion operators are more accurate than others.
In \ao\ dynamics, at least, there
is always a tradeoff between simplicity and accuracy.
Anyone who is curious as to what a highly accurate
inversion operator looks like --- the technicalities are
nontrivial --- may consult a
little review that I wrote
for \emph{Advances in Geosciences}
 \cite{McIntyre:2008}.
The property shared by the whole hierarchy, that
the $\PV$ field contains all the dynamical information,
is sometimes called the
``PV invertibility principle''.


Using the notations 
$D/Dt
\sliver\definedas\,
\partial/\partial t + \bu\bcdot\nabla
$
for the material derivative and
$\Sliver\INV\sliver$ for the inversion operator,
we can write the
generic dynamics in great generality as the following pair of
equations:
\vspace{0.1cm}
\begin{subequations}
\begin{equation}
D\PV/Dt
\;=\;
\mbox{forcing} \,+\, \mbox{dissipation}
\;,
\label{eq:generic1}
\end{equation}
\vspace{-0.7cm}
\begin{equation}
\bu(\bx,\sliver t)
\;=\;
\INV[\Sliver\PVA(\sliver\bcdot\Sliver,\Sliver t)\Sliver]
\;,
\vspace{0.1cm}
\label{eq:generic2}
\end{equation}
where $\PVA(\bx,\sliver t)$
is the PV anomaly field relative to a background
state at relative rest in the rotating system,
$\bu\equiv0$, whose PV is
$\PVB$, say:
\begin{equation}
\PVA
\definedas
\PV \,-\, \PVB
\label{eq:generic3}
\end{equation}
\end{subequations}
The dot in \pref{eq:generic2} signals that
the inversion operator
acts nonlocally on the
$\PVA$ field.
We have $\INV[\PVA] \equiv 0$
if and only if $\PVA\equiv 0$ everywhere.
Dissipation in \pref{eq:generic1}
may include negative dissipation, i.e.\
self-excitation.

PV inversion is a diagnostic, as distinct from
a prognostic, operation.  Diagnostic means that
\pref{eq:generic2} contains neither time derivatives
nor history integrals.
The single time derivative in \pref{eq:generic1}
is the only time derivative in the problem.
The forcing and dissipation
or self-excitation
terms in \pref{eq:generic1}
will of course depend on the particular physical system and
model assumptions, but may often be considered small
for practical purposes.

The single time derivative has strange and interesting consequences.
It tells us at once that
the only possible wave propagation in this kind of system
will be
one-way propagation, in the sense that
the dispersion relation can have
only a single frequency branch.
These are the famous westward-propagating Rossby
or vorticity waves of \ao\ dynamics and the equally famous
electrostatic drift waves of tokamak dynamics,
arising from gradients in the background state $\PVB$.
The waves seen in \fref{fig:tank} are Rossby waves.
They are very different from
the classical types of waves we are all brought up on,
which have pairs of opposite-signed dispersion-relation branches
reflecting time reversibility.
Rossby waves and drift waves have their own peculiar arrow of time.

How do these
waves know which way to go?
In the \ao\ case it's in part because they notice which way the Earth,
or the laboratory tank, is rotating,
along with certain background gradients in the thermodynamic variables.
In the tokamak case, they notice which way the
azimuthal magnetic field component is pointing,
along with the facts that the
ions are positively charged and much heavier
than the electrons,
and that there are background density
and pressure
gradients.
This one-wayness has consequences reaching far
beyond small-amplitude wave theory.
The single time derivative in \pref{eq:generic1} is still there,
no matter how strongly nonlinear things become.


\begin{figure}[t]
\centerline{\psfig{file=pv-def.eps,width=7cm}}
\caption{Two stratification surfaces $\theta=$ constant in
stratified, rotating flow.
The material invariance of $\PV$ for ideal-fluid flow,
in other words the constancy of absolute Kelvin circulation
for an infinitesimal material circuit $\Gamma$ lying in a
stratification surface, exactly captures
how the component of vorticity normal to each stratification surface
changes under vortex stretching and vortex tilting.
}
\label{fig:pv-def}
\end{figure}

Of the various definitions of $\PV$
the most accurate and general,
in \ao\ dynamics, can be stated as follows.
Up to a constant normalizing factor, $\PV$ is the
absolute Kelvin circulation around an infinitesimally
small circuit $\Gamma$ lying in a
stratification surface.
That is, $\PV$ is proportional to
the loop integral
$\textstyle\oint_{\Sliver\Gamma}
 (\bu \,+\, \BOMEGA\antisliver\times\antisliver\bx)\antisliver\bcdot\antisliver
 d\bx
$
where $\BOMEGA$ is the Earth's angular velocity.
Stratification surfaces are surfaces of constant $\theta$, where
$\theta$ is a thermodynamic material invariant for ideal-fluid
flow, more generally
\vspace{0.15cm}
\begin{equation}
D\theta/Dt
\;=\;
\mbox{forcing} \,+\, \mbox{dissipation}
\,.
\label{eq:theta}
\vspace{0.15cm}
\end{equation}
For instance we can take $\theta$ to be the
specific entropy or the so-called potential temperature.
\Fref{fig:pv-def} shows two of the
stratification surfaces, which are usually close to
horizontal in practice.
Also shown are two of the small circuits $\Gamma$.
The material invariance of $\PV$ for ideal-fluid flow,
i.e.\ \pref{eq:generic1} with right-hand side exactly zero,
is an immediate corollary of Kelvin's circulation theorem.
In the ideal-fluid limit, \pref{eq:theta} also has right-hand side zero
and the stratification surfaces become material surfaces;
so the $\Gamma$'s can be taken as material circuits
and Kelvin's circulation theorem applies.

An equivalent definition of $\PV$ can be written
in terms of the mass density field $\rho$,
the absolute vorticity
$2\BOMEGA \,+\sliver \nabla\Antisliver\times\Antisliver\bu$ and
the stratification field
$\theta$
as follows:
\vspace{-0.35cm}
\begin{equation}
\PV
\definedas
\rho^{-1}
(2\BOMEGA \,+\, \nabla\antisliver\times\antisliver\bu)
\bcdot
\nabla\theta
\;.
\label{eq:rossbyertelpv}
\end{equation}
Let $\Delta\theta$ be the
\mbox{$\theta$-increment} between the two stratification surfaces,
taken infinitesimally small,
and  $\Delta\antisliver\mass$ the
mass of the infinitesimally small pillbox-shaped fluid element
defined by the pair of $\Gamma$'s in \fref{fig:pv-def}.
Using Stokes' theorem we see at once that
\pref{eq:rossbyertelpv} is the Kelvin circulation
multiplied by $\Delta\theta/\Delta\antisliver\mass$.
For ideal-fluid flow, $\Delta\theta$, $\Delta\antisliver\mass$,
the Kelvin circulation,
and hence \pref{eq:rossbyertelpv} are all
exact material invariants.
Yet another demonstration starts with the
vorticity equation, i.e.\ the curl of the momentum equation,
taking its scalar product with $\nabla\theta$
to annihilate the baroclinic vector product
$\nabla\rho\times\nabla\pressure$
involving the pressure $\pressure$, noting that
$\theta$ is a thermodynamic function of $\rho$ and $\pressure$
alone.  That is the route taken in most textbooks.

In the unstratified laboratory case, almost the same picture applies
except that the two stratification surfaces in
\fref{fig:pv-def} are replaced by the top and bottom
boundaries of the tank.
The layer now has finite thickness, a function of radius in
the case of \fref{fig:tank}.  But
because there is no stratification
the rapid rotation of the tank keeps
the flow approximately two-dimensional
--- the so-called Taylor--Proudman effect ---
causing the material circuits $\Gamma$ to move in parallel.

Historically, the central importance of
$\PV$ to \ao\ dynamics was first recognized
by Carl-Gustaf Rossby in his seminal papers of
1936, 1938 and 1940\:
\cite{Rossby:1936},
\cite{Rossby:1938},
\cite{Rossby:1940}.
Rossby's 1938 paper recognized the exact equivalence to
the Kelvin circulation and thus showed, in principle, how to define
$\PV$ exactly both for continuous stratification and for
layered systems, including the tank system of \fref{fig:tank}.
Rossby also presented
hydrostatic approximations to \pref{eq:rossbyertelpv}
for use with weather data.
The
formula \pref{eq:rossbyertelpv}
was itself first published by Hans Ertel in 1942 \cite{Ertel:1942},
after visiting Rossby at MIT in 1937.
The PV invertibility principle, implicit in Rossby's work,
was articulated with increasing explicitness
from the late 1940s onward
through the work
of Jule G. Charney
\cite{Charney:1948},
Aleksandr Mikhailovich Obukhov
\cite{Obukhov:1949},
and
Ernst Kleinschmidt
\cite{Kleinschmidt:1950}.



\vspace{-0.2cm}\section{Cyclone and jet structure}\vspace{-0.1cm}
\label{sec:cyclone-jet}

To complete the generic picture
we need a qualitative feel for what PV inversion is like.
Illustrative formulae are given in \sref{sec:inv}; but
one way to gain generic insight
is to say that
on each stratification surface
the operator $\Sliver\INV\sliver$
delivers a horizontal velocity field $\bu$
qualitatively like the electric field $\BE$ in a
horizontally two-dimensional
electrostatics problem, but rotated through a right angle,
with $\PVA$ in the role of minus the excess charge density
--- not quite the tokamak case, but qualitatively similar
especially as regards rotating the $\BE$ vector.
(``Horizontal'' corresponds to a cross-sectional
plane of the tokamak, with the background magnetic field $\BB$
nearly ``vertical''.)

In realistic, stably-stratified atmospheric models,
the electrostatic analogy is again only qualitative,
though still insightful.
I believe it was first put forward by Obukhov.
The notional electrostatics
is ``layerwise-two-dimensional'' insofar as one takes
the horizontal component of $\BE$
on each stratification surface and ignores the vertical component.
The rotation of $\BE$ about the vertical to get $\bu$ is in a
clockwise sense when viewed from above, i.e.\ compass-wise,
in the northern hemisphere, corresponding to upward $\BB$. \
However, thanks to Coriolis effects the field corresponding to
the electrostatic potential needs to be calculated
in three dimensions,
inverting an elliptic operator resembling a modified
three-dimensional rather than two-dimensional Laplacian,
e.g.\ \eref{eq:inv3}
below.
Thus the flow on one stratification
surface is influenced by the notional charge distributions
$\PVA$ on other such surfaces
over a significant depth of the atmosphere.

\begin{figure}[t]
\centerline{\psfig{file=
upper-cyclone-beij3.eps,width=5.1cm,viewport=16 10 691 1394,angle=-90}}
\caption{Vertical section through an
axisymmetric model of an upper-air cyclone; see text.
Calculation by Dr A. J. Thorpe,
from the review \protect\cite{Hoskins:1985}.
The entire structure comes from inverting a single positive
PV anomaly located in the central stippled region.  The heavy
curve is the model tropopause, and the closed contours show
the wind-speed profile, typical of atmospheric jets.  Stronger jets
go with steeper, or even reversed, tropopause slopes, but their
structure is otherwise similar.
}
\label{fig:upper-cyclone}
\end{figure}

\Fref{fig:upper-cyclone} refines this rough
picture by showing the result of an \emph{accurate} three-dimensional
inversion, taken from the 1985 review by Hoskins et al
\cite{Hoskins:1985}; q.v.\ for technical detail.
The stratification surfaces $\theta=$ constant are
the light curves that tend toward the horizontal
at the periphery.
They are more crowded above the tropopause, i.e.\ in the stratosphere
which, as its name suggests, is more stably stratified,
increasing the magnitude of $|\nabla\theta|$
in \pref{eq:rossbyertelpv}.
The heavy curve represents the tropopause,
dipping down from altitudes around 10\km\ down to 5\km,
corresponding to pressure-altitudes (right-hand scale)
ranging from around
250
to 500 millibar.
The vertical scale is stretched in the conventional way,
to make the structure visible.
In this case, which uses realistic atmospheric parameter values,
the horizontal extent of \fref{fig:upper-cyclone} is 5000\km.
So the stratification surfaces
are actually very close to being horizontal.
The exact (Rossby--Ertel)
$\PV$ field
\pref{eq:rossbyertelpv} has an axisymmetric positive
anomaly\footnote{As
explained in the review \cite{Hoskins:1985},
accurate inversion operators $\Sliver\INV\sliver$ require
the anomaly $\PVA$ --- there
referred to as an ``IPV anomaly'' --- to be defined
relative to values on the same stratification surface
$\theta=$ constant, or
``isentropic surface'' 
in atmospheric-science language.
This term arises
because $\theta$ can be
taken to be the specific entropy.  So
``IPV anomaly'' means ``isentropic anomaly of PV''.
In the example of \Fref{fig:upper-cyclone}
the positive anomaly is due mainly to
the large magnitude of the factor
$\nabla\theta$
in \pref{eq:rossbyertelpv}, i.e.\ to the presence of
stratospheric rather than tropospheric air, in the central
stippled region.
}
located in the central stippled region.
In the electrostatic analogy we can think of it as
a concentration of negative charge, making $\BE$ point inward.


The remaining light curves in \fref{fig:upper-cyclone}
are the contours of constant wind
speed, into the paper on the right and out of the paper on the left
as expected from
the inward-pointing $\BE$ vector in
the layerwise-two-dimensional electrostatic analogy.
The maximum wind speed occurs at the tropopause
around 8\km\ altitude, showing a jet structure that is
very typical.
In this case the maximum wind speed has a rather modest value,
just over 20\ms.
Stronger PV anomalies produce
stronger jets with the same structure except that the tropopause,
defined as the PV anomaly boundary, tends to
slope more steeply and indeed can become vertical or
overturned, producing what is famously called a ``tropopause fold'',
e.g.\ Fig.\ 9b of \cite{Hoskins:1985}.

I should explain that in order
to do the inversion in this kind of problem
one has to prescribe the mass under each
stratification surface.
That is how one tells the model to have
a stratosphere with larger values of $|\nabla\theta|$
than the troposphere beneath \cite{Hoskins:1985}.
One also has to impose what is called
a \emph{balance condition}.  In this case it is enough to say that
the flow is in hydrostatic and cyclostrophic balance.  Cyclostrophic
means that horizontal pressure gradients are in balance with the
Coriolis force plus the centrifugal force $|\bu|^2/r$ of the relative
motion where $r$ is horizontal distance from the symmetry axis.
Thus pressures $\pressure$ are low at the centre of the structure.
Although it is left implicit in \pref{eq:generic2},
inversion delivers the $\pressure$, $\rho$ and $\theta$ fields
as well as the $\bu$ field,
as the invertibility principle says it must.
In more complicated, non-axisymmetric cases,
accurate balance conditions can still
be imposed over a surprisingly large range of parameter values,
but at the cost of becoming technically much more complicated; see
\cite{McIntyre:2008} and references therein.

Notice the power of the invertibility principle.
The entire structure in \fref{fig:upper-cyclone},
long familiar and easily recognizable
from the
zoological
annals of observational meteorology ---
under such names as
``upper-air cutoff cyclone'' or ``cutoff low'',
e.g.\ Fig.\ 10.8 of \cite{Palmen:1969} or
Fig.\ 8 of \cite{Hoskins:1985} ---
follows from
having just a single positive PV anomaly
on stratification surfaces intersecting the tropopause.
``Cutoff'' refers to the way in which such
PV anomalies are formed in reality, by a
mass of high-$\PV$ stratospheric air being advected into lower-$\PV$
surroundings and then wrapping itself up into a cyclonic vortex
(e.g.\ the coloured air over the Balkans in
\fref{fig:balkans} below,
also examples in \cite{Hoskins:1985}).
The idea that a PV anomaly can wrap \emph{itself} up makes
perfect sense if
the invertibility principle holds.

As already suggested,
the jet structure illustrated in
\fref{fig:upper-cyclone} is typical and very robust,
over a large range of jet speeds and tropopause steepnesses.
Although the
example in  \fref{fig:upper-cyclone}
is idealized as being
axisymmetric, one gets the same
jet structure in more complicated cases
whenever there is a concentrated gradient of
$\PV$ on stratification surfaces, with high values adjacent to
low values.
The way in which such ``isentropic'' gradients arise --- and
observations repeatedly show that they are are commonplace ---
is precisely through the strong nonlinearity I have been hinting at.
Indeed, the way in which the concentrated gradients
arise can be viewed as a rather simple kind of
strongly-nonlinear \emph{saturation}.
The oft-observed jet structure is telling us that, in reality,
saturated states are often approached.


It is worth pointing out for later reference
that even without
relative motion $\bu$
there may well be pre-existing \emph{large-scale} gradients
in $\PVB$, the background PV distribution.
For the tokamak they come from the
radial
density and pressure gradients.
For realistic \ao\ dynamics
they come from the fact that, in the formula \pref{eq:rossbyertelpv},
horizontal
stratification surfaces pick out the
vertical
component, $f$ say, of the
planetary vorticity $2\BOMEGA$.
At latitude $\lambda$ we have to good accuracy
\vspace{0.16cm}
\begin{equation}
\coriolis
\;=\;
2|\BOMEGA| \sin\lambda
\,;
\label{eq:coriolis}
\vspace{0.16cm}
\end{equation}
 $\coriolis$ is called the Coriolis parameter.  Its
northward gradient
is conventionally denoted by $\beta$ and
also by the phrase ``beta effect'',
not to be confused with plasma beta.


Before leaving this topic I should point out that if one wants to
cover a wider range of significant cases
then one has to count as part of
the $\PV$ field the distribution of $\theta$ at the Earth's surface.
This last behaves somewhat like a Dirac delta function in
the vertical distribution of $\PV$,
and is important in some dynamical processes.  A prime example is
the baroclinic instability and the associated wrapping-up and
``frontogenesis'' in the \mbox{surface-$\theta$} field
that's so important in the terrestrial
atmosphere
\cite{Thorncroft:1993}, even though absent in the Jovian.
The Earth's large-scale \mbox{surface-$\theta$} gradient across latitudes
provides an effective \emph{southward} PV gradient,
opposite to that of the beta effect.  The opposing gradients
mediate a powerful shear instability, the baroclinic
instability in question.
See \S\ref{sec:waves} below.  The resulting
frontogenesis in the surface $\theta$ distribution
is another variation on the theme of saturation.
This baroclinic instability is the \ao\ counterpart to
plasma drift wave self-excitation by resistive instability,
negative dissipation in \pref{eq:generic1}.


\begin{figure}[t]
\centerline{\psfig{file=
cont-advec-wan-STE.ps,width=11cm}}
\caption{Estimated map of $\PV$,
the exact PV defined by \eref{eq:rossbyertelpv},
on a stratification
surface near 10\km\ altitude.  From \protect\cite{Appenzeller:1996}.
The computation assumes that material invariance of $\PV$
is a good approximation over a 4-day time interval, and
uses a state-of-the-art advection algorithm and weather-forecasting data
to trace the flow of high-$\PV$ stratospheric air (coloured)
and low-$\PV$ tropospheric air (clear).
The main boundary between stratospheric and tropospheric air
marks a jet core showing large-amplitude meandering,
from Greenland toward Spain and then back to northern Norway.
The leakage of stratospheric air into the troposphere
signals
intermittent attrition of
the eddy-transport barrier at the jet core.
The different chemical signatures of the
stratospheric and tropospheric air are easily detectable
and have been demonstrated in meaurement campaigns,
even for fine filamentary structures like those shown
\protect\cite{Waugh:1994}.
The high-$\PV$ anomaly over the
Balkans illustrates the ``cutoff'' or self-wrapping-up
process that occurs when
sufficiently large masses of stratospheric air
overcome the barrier.  The wrapping-up produces structures
like that in \fref{fig:upper-cyclone}.
}
\label{fig:balkans}
\end{figure}


   \vspace{-0.1cm}\section{Strong nonlinearity is ubiquitous}\vspace{-0.1cm}
\label{sec:nonlinear}

Why should concentrated isentropic gradients of PV
be so commonplace
along with their inversion signatures, the great jet streams,
and indeed surface fronts as well,
and what justifies
associating them with
a saturation process?
What produces the accompanying anti-frictional or
``negative viscosity'' effects that used to be thought so mysterious?
The answer lies in the idea of
\emph{\mbox{inhomogeneous} PV mixing} ---
more precisely, in the idea of
spatially \mbox{inhomogeneous},
layerwise-two-dimensional,
turbulent PV mixing,
the mixing of PV along stratification surfaces
in some regions but not in others.
Mixing is a strongly nonlinear
process because it involves not weak distortions or
resonant-triad interactions but, rather, drastic advective
rearrangements of large-scale into smaller-scale PV fields.
It has the potential to weaken PV gradients
in some places on stratification surfaces,
and to strengthen them in others to form jets and
eddy-transport barriers; see \srefs{sec:waves}{sec:phillips}
below.

Neither mixing nor eddy-transport barriers need
be perfect.  Indeed in many cases the vortex
dynamics produces new coherent structures on different spatial
scales, as illustrated in \fref{fig:balkans}.
On the other hand,
whenever perfect or near-perfect mixing is achieved
within some region,
we have a rather simple kind of strongly nonlinear saturation
because, once we have a well-mixed region, further mixing
has little further effect.


\Eref{eq:generic1} with its advective nonlinearity
tells us that we can, indeed, reasonably
regard $\PV$ as a mixable quantity,
to that extent like a chemical tracer
consisting of notional charged particles.
(Moreover,
there is an exact ``impermeability theorem'' stating that
the notional particles behave as if trapped on each
stratification surface \cite{Haynes:1990b}
even if air is crossing the surface diabatically.)

Today the reality of PV mixing, and the tendency toward
piecewise-saturated,
piecewise-well-mixed states in many cases, in the real atmosphere,
has extremely strong observational support.
We can routinely
map real atmospheric $\PV$ fields
through the highly sophisticated (spacetime Bayesian)
observational data-assimilation technology that underpins operational
weather forecasting.
The results support
\eref{eq:generic1} with small right-hand side
as a key
to understanding the ubiquity
of real jets, upper-air cyclones, surface fronts
and many other meteorological phenomena.

Examples like
that of \fref{fig:balkans} are
scrutinized in \cite{Appenzeller:1996} and \cite{Martius:2010}.
Other examples
come from higher altitudes,
in the winter stratosphere.
They have been
intensively studied
in connection with the ozone-hole problem already mentioned.
Again and again, we see large regions on
stratification surfaces within which $\PV$ is roughly constant as
a result of mixing, bordered by concentrated gradients
marking the eddy-transport barriers at jet cores.
We can see the mixing taking place in more and more detail, over
an increasingly large range of spatial scales as computer power
increases.  Reference \cite{dimbo:2011} includes a
movie from state-of-the-art data-assimilation,
for a much-studied stratospheric case.
These situations are at an opposite extreme from those assumed in
classical homogeneous turbulence theory.
In a nutshell, reality is
highly inhomogeneous.

Numerical experiments tell the same story in a variety of
cases, e.g.\ \cite{Esler:2008} and \cite{Scott:2012}, as does
the laboratory experiment of \fref{fig:tank}.
In the latter case
the laboratory data were good enough to enable
the experimenters to map the PV field.
The resulting PV maps \cite{Sommeria:1989b}, \cite{Sommeria:1991}
closely resemble
the dye pattern in \fref{fig:tank}, with well-mixed regions
on either side of the jet core.
Even though the visible wavy pattern might
suggest the validity of
linearized or weakly-nonlinear Rossby-wave theory,
the suggestion is misleading because
the fluid motion has already
nonlinearly rearranged its PV distribution
to be close to piecewise uniform,
uniform to either side of the jet core,
with a concentrated PV gradient
at the core.
Strong nonlinearity did most of its work
before the dye was injected.
At the instant shown in \fref{fig:tank}, piecewise regional
mixing continues on either side of the jet but has become invisible,
since there were no further dye injections.
As far as the dynamics is concerned,
the mixing process has saturated.


\Fref{fig:tank} well illustrates what I mean by the
inhomogeneous ``wave--turbulence jigsaw''.
We have a wavy, weakly nonlinear jet-core region
adjacent to
strongly nonlinear mixing regions
on either side of the jet core.
The different regions are coupled dynamically to each other,
in a manner to be further analysed
in \sref{sec:toymodel} below.
The jigsaw has to fit together dynamically as well as geometrically.



A recent numerical model study \cite{Dritschel:2011}
finesses these
inhomogeneous PV-mixing ideas through numerical experiments showing
the co-development of jets, well-mixed areas and coherent vortices,
with the vortices actively contributing to the mixing,
an idealized version of \fref{fig:balkans}.
The discussion of strongly-nonlinear coherent structures in
Diamond et al.\ (this volume) hints that we might end up
with a similar picture for
real drift-wave turbulence in big tokamaks.


\vspace{+0.001cm}\section{Rossby waves and drift waves}\vspace{+0.001cm}
\label{sec:waves}

\Fref{fig:rossby-wave}
shows the generic propagation mechanism for
Rossby and drift waves.
The background rotation $\BOMEGA$ and azimuthal magnetic field $\BB$
point toward the viewer.
Propagation occurs whenever
the $\PV$ field has
a background gradient.
We consider a background $\PV$ field $\PVB = \PVB(y)$,
where in \ao\ cases we take
$y$ as pointing northward,
in the direction of increasing $\PVB$,
and in tokamak cases as pointing radially toward
decreasing background pressure and mass density.
The $y$ direction is toward the top of the figure
and the $x$ direction toward the right;
please
note that this is not the usual coordinate convention for tokamaks.
For the stratified atmosphere
$d\PVB/dy>0$ is the isentropic gradient, i.e.,
the gradient
along a stratification surface as already mentioned.


\begin{figure}[t]
\centerline{\psfig
 {file=rossby-sharp-gimped.eps,width=9.5cm}}
\vspace{+0.1cm}
\caption{
Plan view of an
$x$-periodic Rossby wave, where $x$ points
toward the right in the figure.  The
plus and minus signs respectively indicate the centres of the cyclonic and
anticyclonic PV anomalies due to southward and northward air-parcel
displacements across a basic northward PV gradient.
For a tokamak drift wave,
``cyclonic'' means a vorticity vector pointing in the $+\BB$
direction, toward the viewer in this case.
The plus signs then correspond to
\emph{negative} net charge, and
minimal density,
and in the background gradients
high $\PV$ corresponds to low background pressure and density.
}
\label{fig:rossby-wave}
\end{figure}

With no disturbance the constant-$\PV$ contours would be
straight and parallel to the $x$ axis.  We imagine that
a small disturbance
makes them wavy, as shown.
We assume ideal-fluid motion, i.e., zero on the right of
\eref{eq:generic1}.
Then the wavy
$\PV$ contours
are also material contours.
Linearized wave theory
requires the
undulations
to be gentle: their sideways slopes must be
much smaller than unity.  We have
a row of PV anomalies $\PVA = \PV - \PVB$ of alternating sign, as
suggested
by the
plus and minus signs enclosed by circular arrows.
In the electrostatic analogy --- be careful ---
plus means a \emph{negative} charge and vice versa.
(If only the world were made of antimatter, it would be easier to
be lucid here.)

Inversion gives a velocity field $\bu$
($\propto\BE\times\BB$ for the tokamak)
whose north-south component
is a quarter wavelength out of phase
with the north-south displacement field, as suggested by
the big dark arrows in Fig. \ref{fig:rossby-wave}.
(In the meteorological literature the velocity field resulting from a
PV inversion is sometimes called the wind field ``induced'' by the PV
anomalies.\footnote{Here
meteorologists are following the language of
aerodynamics dating from the pioneering days of
Frederick Lanchester and Ludwig Prandtl,
where three-dimensional \emph{vorticity inversion}
using a Biot--Savart integral has long
been a basic conceptual tool.  As an aerodynamicist would put it,
aeroplanes stay up thanks to the downward velocity ``induced'' by the
trailing wingtip vortices.})

So to understand Rossby-wave and drift-wave propagation ---
generically, and not just in textbook cases --- one need only use
one's visual imagination to turn \fref{fig:rossby-wave} into
a movie.  With velocity a
quarter wavelength out of phase with displacement,
the pattern will start to propagate to the left.  But invertibility,
\eref{eq:generic2}, says that the velocity pattern must remain
phase-locked to the displacement pattern!\;  So
the propagation continues indefinitely.
With the signs shown, the propagation is
in the negative $x$ direction ---
the famous westward, or quasi-westward, or high-$\PV$-on-the-right
phase propagation, relative of course to any mean flow.
We may think of the $\PV$ contours as possessing a peculiar
Rossby-wave ``quasi-elasticity''.
This shows up spectacularly in Norton's movie mentioned in
\sref{sec:barriers}.

If we do a \te\ in which all the signs are changed in
\fref{fig:rossby-wave}, replacing westward phase propagation
 by eastward,
then we are describing the effect of the
large-scale gradient in surface $\theta$ noted at the end of
\sref{sec:cyclone-jet}.  The simplest and most powerful
baroclinic-instability mode can be thought of as a pair of
counterpropagating Rossby waves on the opposing interior and surface
PV gradients, phase-locked with the help of the mean vertical shear
\cite{Hoskins:1985}.

The propagation mechanism
works similarly for uneven $\PV$ contour spacing,
including the case of a jet with most of the
$\PV$ contours bunched up at the jet core.
This means that jets can act as Rossby waveguides,
with quasi-elastic cores.
\Fref{fig:tank} is an example.
Some atmospheric examples are discussed in
\cite{Martius:2010}.
Explicit toy-model solutions will illustrate the same point in
\sref{sec:toymodel}.



If the $\PV$ contours in \fref{fig:rossby-wave} were
deforming irreversibly rather than gently undulating,
then we would say that the Rossby waves are \emph{breaking}.
Indeed,
for strong reasons grounded in wave--mean interaction theory
we may regard such irreversible deformation as the
defining property of wave breaking
\cite{Buhler:2009}, \cite{McIntyre:1985}.
In the atmosphere, at least, there is no doubt that
the breaking of Rossby waves is Nature's principal way of causing
PV mixing and its typical consequences, anti-frictional
jet sharpening and eddy-transport-barrier reinforcement.
In many cases
the associated radiation stress or wave-induced momentum transport
is an essential part of how the wave--turbulence jigsaw fits together
\cite{McIntyre:2008},
\cite{Dritschel:2008},
\cite{Wood:2010}.
One way of seeing more precisely how it fits together
is through what is called the \emph{Taylor identity};
see \pref{eq:taylor-id}
and \pref{eq:tay-id-model3}ff.\ below.

In \sref{sec:toymodel} we'll see that jet-guided
Rossby waves have a strong tendency to break on one or
both sides of the jet, leaving the jet core intact.
That is, there is a
systematic tendency for PV mixing to occur preferentially
on the flanks of a jet ---
sharpening the jet anti-frictionally, reinforcing the
eddy-transport-barrier effect,
and keeping the wave--turbulence jigsaw highly inhomogeneous.
We may think of jets almost as \emph{self-sustaining}
elastic structures; the only help
they need is for their Rossby waves to be excited now and again,
for instance by baroclinic instabilities.


   \vspace{-0.1cm}\section{The PV Phillips effect}\vspace{-0.1cm}
\label{sec:phillips}

There is an even simpler, and independent,
argument suggesting that the spatial
inhomogeneity or regionality
I keep talking about is generically likely.
Imagine a large-scale, initially uniform
PV gradient subject to random disturbances.
Suppose that these disturbances produce PV mixing
along stratification surfaces, the mixing being
slightly stronger in some regions than others.
The regions where the mixing is stronger will have their
overall PV gradients weakened.  But then those regions
will have weaker Rossby-wave quasi-elasticity, and will be
even easier to mix.  Other things being equal, there is
a positive feedback
that tends to push PV contours apart in some regions and
bunch them together in others.

I like to call this the ``PV Phillips effect'', after
Owen M. Phillips'
original suggestion in 1972 that the same thing happens with
vertical gradients of $\theta$, i.e.\ with stable stratification
\cite{Phillips:1972}; see also
\cite{Dritschel:2008}, \cite{Haynes:2007}
and references therein.
In that case the suggestion was beautifully
verified in a non-rotating, stratified laboratory tank experiment
\cite{Ruddick:1989}.  One starts with the uniform stratification
created by a smooth vertical gradient of salinity.  Salinity is a
convenient laboratory counterpart to $\theta$,
or rather to $-\sliver\theta$ because more salinity means
more density and less buoyancy.
One stirs the tank with
smooth vertical rods.  This imposes
no vertical scale of motion smaller than the depth of the tank.
Nevertheless the stratification rearranges itself into horizontal
layers.  Layers with
weak stratification, relatively small $|\nabla\theta|$, are
sandwiched between horizontal interfaces with strong stratification,
relatively large $|\nabla\theta|$.
The vertical scale depends on how vigorously one
stirs.  With sufficiently vigorous stirring one can of course
homogenize the whole tank.  Otherwise, we have
 --- guess what ---
yet another example of the tendency for real turbulence to be
highly inhomogeneous and for
eddy-transport barriers to form.

\begin{figure}[t]
\centerline{\psfig{file=PV_phillips_2.eps,width=9cm}}
\caption{Schematic of the positive feedback loops for the 
original Phillips effect and its PV counterpart.
Courtesy of Drs Jouve and McDevitt.  In the PV case the
positive feedback is reinforced by jet shear, leading to
the formation of eddy-transport barriers such as that
illustrated in \fref{fig:tank}.  Further reinforcement can come
from the preferred phase speeds of disturbances, as discussed
in \sref{sec:toymodel}.
} 
\label{fig:phillips}
\end{figure}

The original Phillips effect and its PV counterpart are summarized in
\fref{fig:phillips}, courtesy of Drs Jouve and McDevitt.  In the
original Phillips effect the relevant wave elasticity is
the  elasticity associated with internal gravity waves --- the waves
that owe their existence to the
stable stratification $\nabla\theta$.

Notice by the way that the positive-feedback argument does not
depend on whether the mixing can be described as Fickian eddy
diffusion, i.e.\ as a random walk with \emph{short} steps.
The argument transcends
any such restrictive modelling assumptions.

In the case of PV
the positive feedback tends to be reinforced by the
jet shear effects
\cite{Dritschel:2008},
\cite{Juckes:1987},
contributing to eddy-transport-barrier formation as
suggested in \sref{sec:barriers}.
Wherever PV contours bunch together, inversion gives jetlike
velocity profiles hence shear.
It turns out that the shearing of small-scale
disturbances is just as important as the Rossby-wave quasi-elasticity
felt by larger-scale disturbances.
There is a smoothing effect or \emph{scale effect},
coming from the inverse-Laplacian-like character of
the inversion operator,
that weakens the $\bu$ field
of the smallest-scale PV anomalies.
So the small-scale behaviour tends to be
passive-tracer-like,
as indeed was suggested
by the filamentary structures in \fref{fig:balkans}.





\vspace{+0.001cm}\section{Some simple inversion operators}\vspace{+0.001cm}
\label{sec:inv}


To check our insights we often use
models with simplified but qualitatively reasonable
inversion operators  $\Sliver\INV\sliver$.
The most important of these models are the so-called
\emph{quasi-geostrophic} models.  They are
not quantitatively accurate but are conceptually important
because their dynamics still has the generic form
\prefs{eq:generic1}{eq:generic3}
and, at a good qualitative level --- better than that of
the crude electrostatic analogy of \sref{sec:cyclone-jet} ---
they describe phenomena such as
the scale effect,
Rossby-wave propagation, and
Rossby-wave breaking and
other strongly nonlinear phenomena such as
vortex interactions and so-called ``cascades''\!.\;


PV inversion, which generically
is a mildly nonlinear operation,
albeit a smoothing operation because of the scale effect,
becomes strictly linear in these models.  This allows free use of
the superposition principle and
helps to expand the repertoire of
mathematically precise
illustrative solutions.
The advective nonlinearity is the only nonlinearity.
The quasi-geostrophic models come in a number of versions,
including single-layer, multi-layer, and continuously stratified.
The standard single-layer or ``shallow-water'' version is
isomorphic to
the standard Hasagawa--Mima tokamak model.

The term quasi-geostrophic comes from the
balance condition used to contruct the inversion operator
$\Sliver\INV\sliver$,
geostrophic balance, in which we entirely neglect
the relative centrifugal and other small terms
in the momentum equation.
Geostrophic balance means simply a balance between
Coriolis forces and horizontal pressure gradients.
It is valid as an asymptotic approximation
in the limit of small Rossby number
$\Ro:=
\coriolis^{-1}
\!\parallel\!
  \bzhat\bcdot\nabla\antisliver\times\antisliver\bu
\!\parallel
$
where $\coriolis$ is the Coriolis parameter as before,
$\bzhat$ is a unit vertical vector,
and $\parallel\!~\!\parallel$ denotes a typical magnitude.

In these models it is convenient to
use modified definitions of the PV, to be denoted
by $\pv$,
with background $\pvb$.\;
Within the asymptotic approximation schemes that lead to the models,
which originated in the independent pioneering work of
Charney \cite{Charney:1948} and Obukhov \cite{Obukhov:1949}
and are described in many textbooks,
we may regard the velocity field as purely horizontal and
nondivergent to leading order.  An $O(\Ro)$ correction is
implicit, allowing weak vertical motion
and horizontal divergence.
So to leading order
we can introduce a streamfunction
$\phi(\bx,t)$,
which is a suitably Coriolis-scaled pressure anomaly
such that, to leading order,
$
\bu
\,=\,
\bug
\sliver\definedas\,
\bzhat\times\nabla\phi
\,=\,
(-\phi_y,\,\phi_x,\,0)
$,
expressing geostrophic balance.
Suffixes $x$ and $y$ denote partial differentiation.
In terms of the corresponding
``geostrophic material derivative''
$\DG/Dt
\sliver\definedas\,
\partial/\partial t
\,+\,
\bug\bcdot\nabla
$
the dynamics
takes the form
\vspace{0.3cm}
\begin{subequations}
\begin{equation}
\DG\pv/Dt
\;=\;
\mbox{forcing} \,+\, \mbox{dissipation}
\;,
\label{eq:generic1qg}
\end{equation}
\begin{equation}
\bug(\bx,\sliver t)
\;=\;
\INV[\Sliver\pva(\sliver\bcdot\Sliver,\Sliver t)\Sliver]
\,\definedas\Sliver
\bzhat\times\antisliver\nabla\Sliver\LL^{-1}\antisliver\pva
\;,
\label{eq:generic2qg}
\vspace{-0.1cm}
\end{equation}
\begin{equation}
\pva
\sliver\definedas\,
\pv \,-\, \pvb
\;,
\label{eq:generic3qg}
\vspace{0.2cm}
\end{equation}
\end{subequations}
where again
``dissipation'' includes negative dissipation, i.e.\
self-excitation,
and where $\LL$ is a
linear elliptic operator
given by the horizontal Laplacian
$\lapl
\,=\,
\partial_{xx} \,+\, \partial_{yy}
$\;
plus extra terms that vary from model to model.
These operat\-ors have well-behaved inverses $\LL^{-1}$
if reasonable boundary conditions are given.
Of course \prefs{eq:generic2qg}{eq:generic3qg}
amount to saying that in each case the
definition of~\;$\pv$\; is $\pvb + \LL\phi$. \
However, saying it
via \prefs{eq:generic2qg}{eq:generic3qg}
emphasizes that these models
are indeed examples of the generic dynamics.

Examples include
the following three.
The first two are single-layer, two-dimensional models with
$\phi=\phi(x,y,t)$, \Sliver
involving a fixed lengthscale $\LD$
to be specified shortly.
The third is three-dimensional with
$\phi=\phi(x,y,z,t)$, \Sliver
taking account of continuous background stratification:
\vspace{0.1cm}
\begin{subequations}
\begin{equation}
\mbox{Model~1:}\qquad
\LL\phi
\,\definedas\sliver
\lapl\phi
\,-\,
\LDMT\phi
\,,\qquad\qquad
\label{eq:inv1}
\vspace{0.15cm}
\end{equation}
\begin{equation}
\mbox{Model~2:}\qquad
\LL\phi
\,\definedas\sliver
\lapl\phi
\,-\,
\LDMT\tilde\phi
\,,\qquad\qquad
\label{eq:inv2}
\end{equation}
\begin{equation}
\hspace{2.3cm} \mbox{Model~3:\qquad}
\LL\phi
\,\definedas\sliver
\lapl\phi
\;+\,
\frac{1}{\rho_0}
\frac{\partial}{\partial z}
\left(
    \frac{\rho_0\corioliso^2}{N^2}
    \frac{\partial\phi}{\partial z}
\right)
.
\label{eq:inv3}
\vspace{0.05cm}
\end{equation}
\end{subequations}
Model~2 is exclusive to the tokamak, having no \ao\ counterpart
beyond its conformity to the generic dynamics, whose
most important implication, for our purposes,
is PV mixability.
The tilde in model~2 denotes the departure from a zonal
or $x$-average: $\tilde\phi\definedas\phi-\langle\phi\rangle$.

In models~1 and~3, the extra terms added to the Laplacian
represent
hydrostatic balance together
with vortex stretching by the implicit, $O(\Ro)$ vertical motion,
essentially the ballerina effect from the accompanying horizontal
convergence.
The continuous background stratification
in model~3 is represented approximately in terms of
background profiles $\rho=\rho_0(z)$ and
$\theta=\theta_0(z)$, with
the buoyancy frequency $N(z)$
defined in terms of the gravity acceleration $g$ by
$N^2\definedas \sliver g\sliver d\antisliver\ln\theta_0/dz$.
\ Coriolis effects are represented by a
domain-average Coriolis parameter $\coriolis=\corioliso=$ constant.
In model~1 the lengthscale $\LD$, called the
Rossby deformation length, is $\corioliso^{-1}$ times
the gravity-wave speed for the layer, which has a free top surface.
More precisely, one uses the notional gravity-wave speed that
would apply if $\corioliso$ were zero,
measuring
the hydrostatic free-surface gravitational elasticity.


Since model~1 is often called the Hasegawa--Mima model
or sometimes, for the sake of historical justice, the
Charney--Obukhov--Hasegawa--Mima model,
it is reasonable to call model~2 with the tilde a
``generalized'', or ``modified'', or ``extended'' Hasegawa--Mima model, e.g.\
\cite{Krommes:2000}, \cite{Gallagher:2012}. 
Model~1 in its \ao\ applications is given the self-explanatory
name ``shallow-water quasi-geostrophic model'' and
sometimes, less transparently, ``equivalent barotropic model''.
Notice that we can turn model~3 into model~1
by assuming a fixed vertical structure with
a single vertical scale $H$.  Then 
a scale analysis
applied to the extra term
with the vertical derivatives gives $\LD = NH/\corioliso$,
related to the notional
internal-gravity-wave speed $NH$ for $\corioliso=0$.
Typical extratropical $\LD$ values
range between
\mbox{$\sim\!10^3$\km\Sliver--$10^2$\km} for the atmosphere
and $\sim\!10^2$\km\Sliver--$10^1$\km\ for the~oceans,
with $H$ in the ballpark of say 5--10\km.


Model~3 requires a boundary condition $\phi_z=0$
at a flat lower boundary, say $z=0$, idealizing the Earth's surface.
(In otherwise-unbounded domains we take evanescent boundary
conditions.)  For model~3 one can show from hydrostatic balance that
$\phi_z=g/\corioliso$ times
the anomaly in \,$\ln\theta$ \
($\phi_z$ having dimensions of velocity, like $\phi_x$ and $\phi_y$).
So the nonvanishing $\theta$ anomalies at $z=0$,
which as already mentioned are
critical to baroclinic instability and frontogenesis,
correspond to nonvanishing $\phi_z$ at $z=0$.
As hinted at the end of \sref{sec:cyclone-jet}, we can regard
this situation as equivalent to $\phi_z=0$ at $z=0$
together with a compensating delta function in the last term of
\pref{eq:inv3}, coming from a jump discontinuity in $\phi_z$
\cite{Bretherton:1966c}.

In model~2
the ballerina effect is replaced by
spinup via the magnetic Lorentz force $\bu\times\BB$
when ions converge, $\bu$ being the ion flow component
normal to a
background magnetic field $\BB=|\BB|\bzhat$.
Geostrophic balance is replaced by the leading-order force balance
$\bu\times\BB\approx-\BE$,
implying that the disturbance streamfunction
$\tilde\phi$ is now $|\BB|^{-1}$ times the
disturbance to the electrostatic potential.
In other words
the pressure-gradient force is replaced
by the electrostatic force
and the Coriolis parameter
by $|\BB|$, suitably normalized ---
most conveniently as the
ion gyrofrequency $\omega_{ci}$ given by $|\BB|$
times the ions' charge-to-mass ratio.
Model~2 assumes that
ion temperatures are much lower than electron temperatures
so that
the ion pressure gradient
is relatively unimportant in the ion flow dynamics.


Also implicit in the dynamics of model~2
are assumptions that
a typical disturbance $\tilde\phi$, while treated as
two-dimensional in the cross-sectional plane of the tokamak,
actually has a
finite wavenumber component
$k_{\parallel}$ parallel to $\BB$,
allowing
the electrons
to adjust quasi-statically
along the $\BB$ lines
encircling the tokamak.  More precisely,
with thermalized, hence Boltzmann-distributed,
electrons at fixed temperature $T_e$
there is a mutual adjustment
between $\tilde\phi$ and
the electron density anomaly $\tilde n_e$
such that
$\tilde n_e
\propto
\tilde\phi$.
For an insightful discussion see \cite{Stoltzfus:2013}.
Together with quasineutrality
this mutual adjustment gives
rise to a notional isothermal sound speed $c_{\rm s}$
based on electron temperature but ion mass,
corresponding to the notional
shallow-water gravity-wave speed in model~1; \
$c_{\rm s}^2$ is
of the order of
the electron-to-ion mass ratio times
the square of the electrons' thermal speed $v_e$.
The corresponding $\LD$ value
$c_{\rm s}/\omega_{ci}$ is
conventionally denoted by $\rho_{\rm s}$
and is
small in comparison with tokamak dimensions.
It would be equal to the ions' gyroradius if the ions were heated
up to
temperature $T_e$,
as indeed happens in models more realistic than model~2.
Also implicit
in model~2 is an assumption that the
zonal-mean flow $\langle\phi\rangle$ does, by contrast,
have $k_{\parallel}=0$, \
suppressing the
mutual adjustment between $\tilde\phi$ and $\tilde n_e$
and
giving rise to the tilde in \eref{eq:inv2}.
In more realistic tokamak models
$\langle~~\rangle$ becomes an average over an entire
torus-shaped magnetic flux surface encircling the tokamak.


The Rossby number is replaced by the small parameter
$\omega_{ci}^{-1}
\!\parallel\!
  \bzhat\bcdot\nabla\antisliver\times\antisliver\bu
\!\parallel\Antisliver.$
For the background PV gradient the \ao\ $\beta$,
with dimensions vorticity/length,
is replaced by
$\omega_{ci}/L$ where $L$ is the lengthscale of the background
electron-pressure
gradient, with signs as in \fref{fig:rossby-wave}.
Also, ``dissipation'' in \eref{eq:generic1qg}
includes the negative dissipation from resistive
self-excitation of drift waves \cite{Wakatani:1984},
predominantly at scales of the order of
$\LD = \rho_{\rm s} = c_{\rm s}/\omega_{ci}$
and coming from a slight time delay in the mutual adjustment
between $\tilde\phi$ and $\tilde n_e$.
 

Omitted from the list above are
various two-layer and higher multi-layer \ao\
models that
also conform to the generic dynamics,
and for which the PV-mixing paradigm is also robust.
They are essentially stacks of
shallow water quasi-geostrophic models
and are popular because they capture some aspects of continuous
stratification, model~3, but with a reduced computational burden.
There is a rigidly-bounded
two-layer model that might, however, be worth exploration as
a more consistent version of model~2 for the tokamak.  In its \ao\
interpretation it represents a physically
consistent thought-experiment with two $\LD$ values, one finite
and the other infinite, that might be put into correspondence
with the tokamak's finite-$k_\parallel$ and zero-$k_\parallel$ modes.
It has been intensively studied, e.g.\ \cite{Thompson:2007}
\&~refs.

Such a two-layer model might well, on the other hand,
fail to improve on
model~2 especially if PV mixing turns out to be important in
real tokamaks.  This is because mixing due to
advection by a chaotic disturbance velocity field
$\tilde\bu$
is insensitive to sign changes
$\tilde\bu\rightarrow-\tilde\bu$.
So mixing by finite-$k_\parallel$ disturbances in a real tokamak
should be well able to robustly generate and maintain
jets in the flux-surface-averaged, zero-$k_\parallel$
velocity field $\langle\tilde\bu\rangle$,
a scenario that is implicit in
model~2 despite its being heavily idealized.


Taking
$\pvb = \beta y \;+$ constant as the background PV,
with constant gradient $\beta$,
we can easily check
that
\mbox{models~1--3}
possess
waves that propagate in
the manner sketched
in \fref{fig:rossby-wave}. \ 
For instance both
model~1 and model~2 linearized about relative rest
have the same elementary wave solutions
$\tilde\phi\propto\exp(ikx+ily-i\omega t)$
with the same, single-branched
dispersion relation
\vspace{-0.21cm}
\begin{equation}
\omega
\;=\;
\frac{-\beta k}{k^2 + l^2 + \LDMT}
\;,
\label{eq:plane-rossby-wave-disp}
\end{equation}
where the denominator comes from
$\;\pva=\tilde\pv=\LL\tilde\phi
= -(k^2 + l^2 + \LDMT)\tilde\phi
$
in the~linearized \pref{eq:generic1qg} with right-hand side zero,
$
\partial\tilde\pv/\partial t + \DG\pvb/Dt
=
\partial(\LL\tilde\phi)/\partial t + \beta\sliver\partial\tilde\phi/\partial x
= 0
$.
Notice the scale effect:
wave propagation is weakened as scales become smaller and
$k^2 + l^2$ larger.
In the long-wave limit $(k^2 + l^2)\to0$ the phase
velocity asymptotes to $-\beta\LDSQ$, which
in the tokamak case
coincides with the background diamagnetic drift velocity,
not of ions but of electrons.

The laboratory flow in \fref{fig:tank} has a rigid lid and
is well described by model~1
with $\LD=\infty$.  This is ordinary (Euler, inertial)
two-dimensional vortex dynamics except that there is a nontrivial
background
$\pvb = \beta y \;+$ constant. \
This comes from a gently sloping cone-shaped
tank bottom,  where now $y$ is radial distance
toward the tank centre.

For model~1 in an unbounded domain with finite $\LD$
one has the explicit inversion formula
\begin{equation}
\phi(\bx,\,t)
\;=\;
\LL^{-1}\antisliver\pva
\definedas\antisliver
-\frac{1}{2\pi}
\int\!\!\int
K_0
\left(
    \frac{|\bx-\bx'|}
    {\LD}
\right)
\pva(\bx',\,t)\Sliver dx'dy'
\vspace{0.1cm}
\label{eq:greensintegral}
\end{equation}
where $|\bx-\bx'|^2 = (x-x')^2 + (y-y')^2$ and where $K_0$ is the
modified Bessel function \cite{Obukhov:1949}.  Its exponential decay
shows
that both linear and nonlinear interactions become very weak at
distances significantly greater than $\LD$.
Aphoristically speaking, we have action at a distance but not too
great a distance, which helps to explain the surprising robustness
of the generic dynamics
\cite{McIntyre:2008}
which, in effect, through the balance
condition, assumes that gravity-wave propagation is instantaneous.
In model~2
this corresponds to instantaneous acoustic propagation
in the thermalized electron gas.



\vspace{+0.001cm}\section{Pseudomomentum and the Taylor identity}\vspace{+0.001cm}
\label{sec:pmomtaylor}

Models~1--3 and their multi-layer counterparts all
possess what are called
pseudomomentum theorems and Taylor
identities.
All these theorems and identities stem from the seminal
1915 work of
Sir Geoffrey Ingram Taylor \cite{Taylor:1915} and its
further development in the 1960s by
Jule G. Charney and Melvin E. Stern \cite{Charney:1962} and by
Francis P. Bretherton \cite{Bretherton:1966b}.
(I'm almost
a direct intellectual
descendant because Taylor was Bretherton's PhD advisor and
Bretherton was mine.)  Taylor's results apply to models~1 and~2.
Charney, Stern and Bretherton
extended them to model~3,
leading in turn to a milestone 1969 paper by Robert E. Dickinson
\cite{Dickinson:1969}.  Dickinson's paper took what I regard as the
decisive step toward solving the
``negative viscosity'' enigma for the real atmosphere,
though further work was needed to see how simply
it could be understood
using the concept of Rossby-wave breaking.
For~more history see \cite{Dritschel:2008}.


In their usual forms the
pseudomomentum theorems depend on linearization and so
apply to small-amplitude waves only.
The Taylor
identity, by contrast, is
valid at any amplitude, and so applies to the whole
wave--turbulence jigsaw.


Consider an arbitrary zonal-mean state
$\langle\pv\rangle
=
\pvb + \langle\pva\rangle$.
The non-background part $\langle\pva\rangle$ might for instance
represent a jet flow.
Take the disturbance part $\tilde\pv=\LL\tilde\phi$ of
\pref{eq:inv1} or \pref{eq:inv2}.
Multiply it by $\tilde\phi_x$
and take the zonal mean.  The $\LD$ terms disappear, because
$
\big\langle\tilde\phi_x\tilde\phi\big\rangle
=
\big\langle\half(\tilde\phi^2)_x\big\rangle
=0
$.
From the horizontal Laplacian we have
$
\big\langle\tilde\phi_x\tilde\phi_{xx}\big\rangle
=
\big\langle\half(\tilde\phi_x^2)_x\big\rangle
=0
$
and so, noting also that
$
\big\langle\tilde\phi_{xy}\tilde\phi_{y}\big\rangle
=
\big\langle\half(\tilde\phi_y^2)_x\big\rangle
=0
$
and writing
$\tilde v = \vg = \tilde\phi_x$,
$\tilde u = \ug = -\tilde\phi_y$,
dropping the suffix $\mathrm g$ from now on, we have
\vspace{0.1cm}
\begin{equation}
\big\langle \tilde v \tilde \pv \big\rangle
\;=\;
-\partial\big\langle \tilde u \tilde v \big\rangle/\partial y
\;.
\label{eq:taylor-id}
\vspace{0.05cm}
\end{equation}
This is the Taylor identity for models~1 and~2.  It is
valid at any amplitude and is indifferent to
whether the dynamics is ideal-fluid or not:
\eref{eq:generic1qg} was never used.  It
was derived by Taylor for the nondivergent barotropic
case $\LD=\infty$.  As just shown, however, it extends
trivially to any value of $\LD$.
It tells us that in these models
the eddy flux of PV, including
any contributions due to strongly nonlinear processes like wave breaking
and PV mixing, is directly
tied to the eddy flux of momentum and hence to the
self-sharpening, anti-frictional jet dynamics.
The negative-viscosity enigma has
vanished in a puff of insight!
It is the mixable quantity PV that
tends to go down its mean gradient ---
not momentum, which there is no reason to suppose is mixable.


For model~3
one replaces the momentum-flux convergence on
the right of \pref{eq:taylor-id}
by its counterpart in the $yz$ plane, the convergence of
an effective momentum flux whose vertical component is
minus what oceanographers call the \emph{form stress}
across an undulating stratification surface
due to correlations between pressure fluctuations and
stratification-surface slopes.  This gives
the
Taylor identity for model~3:
\vspace{0.1cm}
\begin{equation}
\big\langle \tilde v \tilde \pv \big\rangle
\;=\;
-\frac{\partial\langle\tilde u \tilde v\rangle}{\partial y}
\;+\,
\frac{1}{\rho_0}
\frac{\partial}{\partial z}
\left(
    \frac{\rho_0\corioliso^2}{N^2}
\left\langle
    \frac{\partial\phi}{\partial x}
    \frac{\partial\phi}{\partial z}
\right\rangle
\right)
.
\label{eq:tay-id-model3}
\vspace{0.1cm}
\end{equation}
To see that the last term contains the form stress,
within the round brackets, note that
$\langle\phi_x\phi_z\rangle
\;=\;
-\langle\phi\phi_{zx}\rangle
$
and that
$\rho_0\corioliso\phi$ is the pressure fluctuation
while, thanks to hydrostatic balance,
$-\corioliso N^{-2}\phi_z$ is the stratification-surface displacement
and $-\corioliso N^{-2}\phi_{zx}$
its slope.
For historical reasons this effective momentum flux
has often been
defined with a perverse sign convention
and labelled
the \emph{Eliassen--Palm flux}.
By the usual conventions,
$-\langle\tilde u \tilde v\rangle$
ought to be minus the momentum flux or plus the stress,
and similarly for the vertical, form-stress term.
Like
\pref{eq:taylor-id},
the
identity
\pref{eq:tay-id-model3} holds
whether the dynamics is ideal-fluid or not and
for disturbances of any amplitude whatever,
wavelike, or turbulent, or both.

The small-amplitude pseudomomentum
theorems
were also derived by Taylor and
Bretherton
albeit in slightly disguised form.
As things panned out historically,
full clarity (in the \ao\ community)
had to await introduction of what
are usually called the ``transformed Eulerian-mean equations''
\!\cite{Andrews:1976},
shortly after which the conceptual
connection with theoretical-physics principles ---
translational invariance, quasi-particle gases and so on ---
was finally made, helped by a correspondence I had with
Sir Rudolf Peierls.  An in-depth discussion is given
in~\cite{Buhler:2009}.


Here we define the pseudomomentum $P$ per unit mass for small-amplitude
fluctuations $\tilde\pv$ about the translationally-invariant mean state
$\langle\pv\rangle$, for all three models, as
\vspace{-0.1cm}
\begin{equation}
P
\,\definedas
-\half
\langle \tilde \pv^2 \rangle
/
\langle \pv \rangle_y
\;.
\label{eq:pmomdef}
\end{equation}
The sign convention is chosen to make $P$
agree with the usual ray-theoretic
pseudomomentum or quasimomentum, i.e.\
wave action times wave vector.  We expect a minus sign
precisely because of the one-wayness of Rossby and drift waves.


Linearizing \pref{eq:generic1qg} with right-hand side zero, about
the mean state $\langle\pv\rangle$,
we easily find, on multiplication by $\tilde\pv$ and
taking the zonal mean, the disturbance potential-enstrophy equation
$\;
\partial_t
    \half\langle \tilde \pv^2 \rangle
=
-\langle \pv \rangle_y
\langle \tilde v \tilde \pv \rangle
$\;
for all three models.
The Taylor
identity
says that we can turn the potential-enstrophy equation
into a conservation theorem if we
divide it by
$-\langle \pv \rangle_y$
and use the fact that, at small amplitude, we may consistently
neglect the rate of mean-state evolution
$\langle \pv \rangle_{yt}$.
Thus for models~1 and~2, for instance,
\vspace{0.2cm}
\begin{equation}
\frac{\partial P}{\partial t}
\;+\;
\frac{\partial\langle \tilde u \tilde v \rangle}
{\partial y}
\;=\;
0
\label{eq:pmom-cons-12}
\vspace{0.2cm}
\end{equation}
which is indeed in conservation form, as is the corresponding result
for model~3 for which we need only replace the
second term of \pref{eq:pmom-cons-12} by
minus the right-hand side of \pref{eq:tay-id-model3}:
\vspace{0.2cm}
\begin{equation}
\frac{\partial P}{\partial t}
\;+\;
\frac{\partial\langle\tilde u \tilde v\rangle}{\partial y}
\;-\,
\frac{1}{\rho_0}
\frac{\partial}{\partial z}
\left(
    \frac{\rho_0\corioliso^2}{N^2}
\left\langle
    \frac{\partial\phi}{\partial x}
    \frac{\partial\phi}{\partial z}
\right\rangle
\right)
\;=\;
0
.
\label{eq:pmom-cons-3}
\vspace{0.2cm}
\end{equation}
Of course one can usefully
repeat these derivations with the
forcing or dissipation terms explicitly included
on the right of \pref{eq:generic1qg}
whenever one has a particular model for those terms,
such as a viscous term,
or infrared radiative damping,
or drift-wave self-excitation.
Then one has sources and sinks of pseudomomentum $P$. \
Such sources and sinks do not necessarily require external forces to be
exerted.  They require only that waves be generated or dissipated
somehow.  This
underlines the fact that pseudomomentum is not the same thing
as momentum, \emph{despite} sharing the same flux terms.



The small-amplitude pseudomomentum conservation theorems
are sometimes called Charney--Drazin
theorems,
even though they originated in the 1915 work of
Taylor \cite{Taylor:1915}.  The 1961 work of
Charney and Drazin \cite{Charney:1961}
restricted attention to
steady, nondissipating waves on
a special mean flow $\langle u\sliver\rangle(z)$ with vertical
shear only, and vanishing horizontal shear
$\langle u\sliver\rangle_y$ and vanishing Reynolds stress
$\langle\tilde u \tilde v\rangle$, in model~3.
Their theorem, Eqs.\ (8.13)ff.\
of \cite{Charney:1961},
though influential in its time,
said only that, in this special steady-waves case,
the $\partial/\partial z$ term of \pref{eq:pmom-cons-3} vanishes ---
the counterpart to saying that
$\partial\langle\tilde u \tilde v\rangle/\partial y$ vanishes
for the steady-waves case of
\pref{eq:pmom-cons-12}.
Unlike Taylor,
they did not consider time-dependent waves
and so did not find any results
like 
\pref{eq:pmom-cons-12} and \pref{eq:pmom-cons-3}
involving
$\partial P/\partial t$.


The reader interested in deeper aspects of the theory is
recommended to read the penetrating account in B\"uhler's book
\cite{Buhler:2009}.
For instance
the assumption $\langle \pv \rangle_{yt}=0$
used above,
in order to derive
\pref{eq:pmom-cons-12} and \pref{eq:pmom-cons-3},
is consistent with
the mean state being not only translationally but also
temporally invariant.  Then
pseudoenergy as well as pseudomomentum
is conserved for ideal-fluid flow.
Generically, the pseudoenergy $E$
in a particular frame of reference can be defined as
$\langle u\sliver\rangle P$ plus the positive-definite wave-energy
of the linearized disturbance
equations.
This holds for arbitrarily-sheared mean flows $\langle u\sliver\rangle$.

Domain-integrated $P$ and $E$ conservation
immediately give us
the well known Rayleigh--Kuo--Charney--Stern
and Fj\o rtoft shear stability theorems.
The first implies stability whenever $P$ is inherently one-signed,
i.e.\ whenever
$\langle\pv\rangle_y$ is one-signed, and the second
whenever a translating frame of reference can be found that makes
$\langle u\sliver\rangle P$ positive definite,
i.e.\ $\langle u\sliver\rangle\langle\pv\rangle_y$ negative definite,
even if $\langle\pv\rangle_y$ is two-signed.  This last
points to the role of counterpropagating Rossby waves
in shear instabilities mentioned in \sref{sec:waves}. \
States in which $\langle\pv\rangle_y$ is only just one-signed
(e.g.\ \fref{fig:tank}, also ``PV staircases'', next section)
are sometimes called
``states of Rayleigh--Kuo--Charney--Stern marginal stability'',
or for brevity
``states of Rayleigh--Kuo marginal stability''.
Beautiful generalizations and extensions of all the
stability theorems were discovered by
V. I. Arnol'd; see e.g.\
\cite{Wood:2010}, \cite{McIntyre:1987b}, and references therein.


B\"uhler's book \cite{Buhler:2009} also makes
clear to what
extent one can generalize small-amplitude
results like \pref{eq:pmom-cons-12} and
\pref{eq:pmom-cons-3}
to finite amplitude, for \ao\ models at least.
The finite-amplitude
counterparts,
though conceptually important, are
computationally impractical in the cases that interest us here
because they require
retention of Lagrangian flow information --- more precisely,
the shapes of originally-zonal material contours
entering Kelvin's circulation theorem.  That is no great
problem for waves that are not breaking,
like the waves in \fref{fig:rossby-wave},
but becomes
hopelessly complicated in turbulent zones
where the waves are breaking, meaning that
the material contours deforming irreversibly.
It is perhaps worth noting that this difficulty has, nevertheless,
been circumvented in a recent proof of a finite-amplitude
version of the Rayleigh--Kuo--Charney--Stern stability theorem
that is even more general than Arnold's version, allowing
fully-turbulent wave breaking and PV mixing
\cite{Wood:2010}, in models~1 and~3. \
(The Fj\o rtoft theorem then fails: it is easy to find counter\-examples.)

It is Kelvin's circulation theorem applied to
originally-zonal material contours
that accounts for the otherwise mysterious fact that,
even though momentum and pseudomomentum
are different physical quantities related to different
translational symmetry operations, they
share the same off-diagonal flux terms.
Kelvin's circulation theorem is exactly the ideal-flow
constraint preventing
what Diamond et al.\ (this volume) call
``the slippage of a quasi-particle gas'' of
drift waves relative to the zonal flow.

We may think of the momentum-flux convergences that appear in
\pref{eq:pmom-cons-12}, \pref{eq:pmom-cons-3}
and their generalizations, including the generalizations to finite
disturbance amplitude,
as effective wave-induced forces felt by the mean state.
Such effective forces may or may not be equal to the actual
mean-flow acceleration $\partial\langle u\sliver\rangle/\partial t$.
If they are, then
we have simply
\vspace{0.01cm}
\begin{equation}
    \partial_t
\left(
          \langle u \rangle -  P
\right)
 = 0
\label{eq:gas-slippage}
\end{equation}
for ideal-fluid flow.
It can be shown that this always holds in model~2,
and in model~1 for $\LD=\infty$.
It is not true more generally because the response to
the effective mean force normally includes implicit
$O(\Ro)$ mean motions and mass fluxes in the $y$ direction,
whose Coriolis forces contribute to
$\partial\langle u\sliver\rangle/\partial t$.
In the case $\LD=\infty$,
and in model~2 for all $\LD$,
such mean mass fluxes are shut down
by the kinematic constraints. \
$\LD=\infty$ models are
two-dimensionally nondivergent
to sufficient accuracy
since with gravity $g=\infty$
we have replaced the free upper surface with a rigid lid.
Otherwise
it is simplest
to think directly in terms of eddy fluxes of PV,
focusing on the
left-hand sides of the Taylor
identity \pref{eq:taylor-id} or \pref{eq:tay-id-model3},
and using
PV inversion to calculate $\partial\langle u\sliver\rangle/\partial t$.
Inversion implicitly takes full account of the
implicit $O(\Ro)$ mean motions and mass fluxes.

For \mbox{models~1--3}
there are actually four
momentum-like quantities that
are liable to be conflated but need to be
distinguished, namely
(1) momentum,
(2) Eulerian pseudomomentum
    (of which $P$ is a simple
    example),
(3) Lagrangian pseudomomentum (definable at finite amplitude), and
(4) Kelvin impulse.
Adding to the potential for confusion,
the word impulse often
\emph{means} momentum in some European languages.
B\"uhler's book \cite{Buhler:2009}
keeps these
distinctions clear at all stages.

It also lays out in full detail exactly how \pref{eq:gas-slippage}
generalizes
to finite amplitude, and to
\ao\ models more accurate than quasi-geostrophic, and spells out
the price to be paid for such generalizations in terms of retaining the
accurate Lagrangian flow information required
to keep track of the shapes of material contours.

It is a nontrivial question whether or not model~2
admits any
corresponding
finite-amplitude results.
The reason is that, in the \ao\ cases, the application of
Kelvin's circulation theorem to originally-zonal material contours
requires the
implicit $O(\Ro)$ mean motions and mass fluxes
to be taken into account --- that is, it
requires consideration of the $O(\Ro)$ corrections to the
leading-order velocity field $\bzhat\times\nabla\phi$.
My current conjecture is that, since the zonal-mean kinematics of
model~2 would appear to constrain such corrections to be zero,
at least in the zonal mean,
there may well be a
finite-amplitude counterpart to \pref{eq:gas-slippage}
though, if so, there is still the question of how far the
Lagrangian information required might limit its usefulness
in practice.
However, all this would need to be checked in detail and preferably by
someone who understands
the plasma physics
better than I do.




\vspace{+0.001cm}\section{PV staircases}\vspace{+0.001cm}
\label{sec:staircases}

\begin{figure}[t]
\centerline{\psfig{file=staircase-jets.eps,width=8.5cm}}
\caption{Profiles
of $\phi$ and 
$u$ for perfect, zonally-symmetric PV staircases in model~1.
Tick marks are at intervals of $y = \half L = \LD$.
From left to right,
the first profile is that of $\phi$
for a single step and the second is the corresponding
$u$ profile.
The remaining profiles are those of $u=-\phi_y$
for two, three and the limiting case of
an infinite number of perfect steps.
From \protect\cite{Dritschel:2008}; q.v.\ for mathematical details of
the inversions.
}
\label{fig:staircases}
\end{figure}

Focusing again on
our main theme of strong nonlinearity,
we note that
the PV Phillips effect,
\fref{fig:phillips},
suggests the possible relevance of
idealized saturated
states consisting of perfectly mixed zones
cut out of a background $y$-profile $\pvb(y)$
having a constant gradient $\beta$
(again, not to be confused with plasma beta).
We take $\pvb =
\beta y$ as before.
The graph of $\pv$ against $y$ then looks like a staircase
cut out of a sloping hillside.  So these
idealized saturated, marginally Rayleigh--Kuo--Charney--Stern stable
states are often called ``PV staircases''.


The corresponding $\pva$ profile is a zigzag.  Inverting this
we get an array of zonally symmetric jets whose profiles depend on
the PV jump $\pvj$ at each step, as well as on the
step size $L$, i.e.\ the jet spacing, in units of $\LD$.
Some examples are shown in
\fref{fig:staircases} above,
for model~1 with $L/\LD=2$
and for staircases of one, two, and three steps along
with the limiting case of an infinite number of steps.
The two profiles on the left are the
$\phi$ and $u$ profiles
for the case of one step, an idealized version of
the terrestrial winter stratosphere in its
usual wintertime state.
The others are all $u$ profiles.
Recall that $u =-\phi_y$.
The solutions are taken from
\cite{Dritschel:2008},
q.v.\ for mathematical details
as well as for caveats regarding the
conflicting uses of the term
``Rhines scale'', whose relation, if any, to the jet spacing $L$
is
not as straightforward as is sometimes
assumed.\footnote{One
problem, sometimes forgotten,
is that Rhines'
original concept came from a variant of homogeneous turbulence theory
in which
an upscale energy ``cascade'' is all-important, yet
the Rossby waves feel only the background PV gradient
$\beta$, and therefore obey
the dispersion relation \pref{eq:plane-rossby-wave-disp}. \
Also, $\LD=\infty$ is often assumed, as in Rhines'
original work \cite{Rhines:1975}. \
We shall see shortly that Rossby waves on PV
staircases can have dispersion properties that
differ substantially from \pref{eq:plane-rossby-wave-disp},
and in a way that
can drastically reshape the
wave--turbulence interactions.
}

Some researchers think that the PV-staircase idealization
provides a good model for Jupiter's
weather layer in extratropical latitudes
 --- a persuasive case is made in the review article
by Marcus \cite{Marcus:1993} --- though, hardly surprisingly, in
view of our generally poor understanding of Jovian fluid dynamics,
this is controversial.  Alternatives have been put forward,
notably by Dowling \cite{Dowling:1993}.
What is not in doubt is that staircase-like structures
of one, two and sometimes
three steps are well observed and well documented
in the Earth's atmosphere
\cite{Martius:2010}.
The observed states are not zonally symmetric; what they share with
the simple staircase idealization are large, zonally-extensive regions
of fairly well mixed PV on stratification surfaces.
Those regions are bordered by narrow bands of
concentrated PV gradients,
the meandering jets already noted.
If PV contours on a stratification surface are splayed out
by mixing in some
places, then they must be bunched up in others.
(The only simple alternative consistent with \pref{eq:rossbyertelpv}
and its conservation properties \cite{Haynes:1990b} is
an overwhelmingly improbable alternative, namely,
to wipe out the entire pole-to-pole planetary PV gradient
and thus kill
the atmosphere's rotation
$\BOMEGA$ altogether.)
The simplest and clearest example of the splaying-out and bunching-up
that I'm talking about is the winter stratosphere
\cite{dimbo:2011},
\cite{Riese:2002},
often resembling a single-step staircase idealizable as the
left-hand case in \fref{fig:staircases}.


\begin{figure}[t]
\centerline{\psfig{file=critical_lines_5.eps,width=4cm,viewport=134 34 394 370}}
\caption{Sketch of PV discontinuity
and the corresponding velocity profile 
obtained by PV inversion.
The step size or jet spacing $L$
is much larger than the jet scale $\LD$,
and we can ignore $\beta$ and equate $\pva$ and $\pv$. \
The solid velocity profile
is from \eref{eq:single-jet}.  The angle between the dashed lines
is governed by the strength of the delta function $\pvj\sliver\delta(y)$.
}
\label{fig:single-jet}
\end{figure}


As $L/\LD$ varies, the jets retain their peaked profiles,
corresponding to the infinitely tight bunching of PV contours
in this idealization.
They change their shapes in other ways.
For large $L/\LD$ we get isolated jets,
still Rayleigh--Kuo--Charney--Stern marginally stable.
\Fref{fig:single-jet} \Sliver
zooms in on one such jet,
near which the $\pva$ zigzag looks like a simple jump
discontinuity, at $y=0$, say, as shown in the right-hand graph.
The zonally-symmetric inversion problem for $u$ then simplifies to
$
\LL u
=
u_{yy} - \LDMT u
= -
\langle\pva\rangle_y
= -
\langle\pv\rangle_y
= -
\pvj\sliver\delta(y)
$
and delivers the velocity profile shown in the left-hand graph,
\vspace{0.01cm}
\begin{equation}
u
~=~
\langle u\sliver\rangle
~=~
\uj \exp
\left(
   - |y|/\Sliver\LD
\right)
\label{eq:single-jet}
\vspace{0.15cm}
\end{equation}
where
$\uj \definedas
\half\LD\pvj$.
The dashed lines in \fref{fig:single-jet} mark the jump
in the slope $u_y$ at the jet peak,
determined by the strength of the
delta function $\pvj\sliver\delta(y)$
in the inversion problem.
Integration of
$
u_{yy} - \LDMT u
= -
\pvj\sliver\delta(y)
$
across $y=0$ gives
$
\big[
  u_y
\big]^{y=0+}_{y=0-}
= -
\pvj
$.


Staircase models are
no more than an idealization, for one thing
being much more deterministic than current models of the
fluctuating zonal flows in tokamaks.  However,
it may be interesting to speculate whether,
for strong nonlinearity,
the staircase idealization has some
relevance to the saturation problem in real tokamaks.

There is an ill-understood dependence on how the turbulence
is excited.
In the vast literature on numerical forced-dissipative versions of
model~1 (the ``beta-turbulence'' literature,
mostly on model~1 with $\pvb=\beta y$ and $\LD=\infty$), it seems that
some
of the numerical experiments produce sharp staircases,
e.g.\ \cite{Danilov:2004},
while others produce only a faint ``washboard''
(e.g.\ B. Galperin, personal communication).
This could be partly due to the fact that
differing choices of artificial forcing will
disrupt PV mixing to differing extents.

Most of these studies use a
prescribed stochastic external force field,
rather than self-excitation.
We have the same difficulty in the Jupiter problem ---
a lack of clarity as to
what kind of forcing or self-excitation best mimicks reality.
There is also great uncertainty as to when or whether upscale energy
``cascades'' are involved and when, by contrast, the
passive shearing of repeated PV-anomaly injections is more important
\cite{Thomson:1887}, \cite{Farrell:2007}, \cite{Srinivasan:2012}.
In ``cascade'' language
this last produces an enstrophy cascade but not an energy cascade.
It also disrupts simple PV mixing.

The laboratory flows exemplified by
\fref{fig:tank}
are forced in several ways that all
contrive to
preserve the material invariance of PV
to good approximation, giving PV mixing a chance to predominate.
In the experiments that
produced \fref{fig:tank} and similar flows, the
forcings are non-stochastic, though spatially variable.
In a variety of runs
the outcome, in these experiments, always seems to be
robustly a flow
like that in \fref{fig:tank},
a prograde jet that is
``staircase-like''
in the same sense as the flow
in \fref{fig:single-jet}.

In these experiments,
forcing and dissipation are kept far weaker
than in most laboratory and numerical experiments.
The tank in \fref{fig:tank} is unusually large
 --- 86.4\cm\ in diameter --- and can be rotated
unusually fast without disintegrating, up to 4~revolutions per second.
\Fref{fig:tank} used 3 per second,
$|\BOMEGA|=18.8$\smone.
The wave--turbulence interactions that shape the flow are almost
free of forcing and dissipation,
and strongly nonlinear.
See \cite{Sommeria:1989b} and \cite{Sommeria:1991} for more details
of these remarkable experiments.

The only other laboratory experiments
with comparable weakness of forcing and dissipation,
of which I'm aware, are those
of Read et al.\ \cite{Read:2007}.
They use an ingenious quasi-stochastic forcing method
in an extremely large tank, 13m diameter, though
at much smaller $|\BOMEGA|$.
Meandering jets are obtained, and temporary local reversals of PV
gradients ``indicative of Rossby wave breaking'', though in the
absence of PV maps like those in
\cite{Sommeria:1989b} and \cite{Sommeria:1991}
it is hard to make a closer assessment.  It
appears, however, that in these experiments the turbulence is too weak to
bring the system close to a well developed staircase-like regime.
In this connection some
recent numerical experiments by
R. K. Scott and D. G. Dritschel are noteworthy,
in that
with increased computing power they can reach further into
the weakly-excited, weakly-dissipative,
strongly-nonlinear
parameter ranges conducive to staircasing \cite{Scott:2012}.



T. E. Dowling and co-workers have argued that Jupiter may be
different again, possibly close to a state that is marginal not
by the Rayleigh--Kuo--Charney--Stern stability criterion but,
rather, by that of Arnol'd's Second Theorem.
The $\pv$ profiles would then be
``hyperstaircase-like''
to the extent that
they have reversed PV gradients.
There is some indirect, but I think persuasive,
observational evidence
that Jupiter's weather layer may be close to such a state
\cite{Dowling:1993}.
But the question of
what forcing or self-excitation mechanisms might lead to it
does not yet
seem to have been clearly answered ---
though an extremely promising candidate, direct external forcing by
moist (thunderstorm) convection, has been put forward
from time to time and is now under intensive investigation,
e.g.\ 
\cite{Ingersoll:2000},
\cite{Lian:2010}, and references therein.
At first sight
this
would seem to suggest
a leading role for
the passive shearing of repeated PV-anomaly injections
\cite{Thomson:1887}--\cite{Srinivasan:2012},
the simplest
mechanism
able to overcome PV-mixing tendencies and produce a hyperstaircase.




\vspace{+0.001cm}\section{Jet self-sharpening and meandering: a toy model}\vspace{+0.001cm}
\label{sec:toymodel}

The single isolated jet in \fref{fig:single-jet}
is the simplest
of all Rossby waveguides.  Its
dispersion relation differs substantially
from \pref{eq:plane-rossby-wave-disp}, and in a way that will prove
interesting.  It signals a kind of wave--turbulence,
linear--nonlinear coupling
that differs greatly from the
standard homogeneous-turbulent Rhines picture.


Linearizing \pref{eq:generic1qg} with right-hand side zero
about the profiles in \fref{fig:single-jet},
with $\LL$ defined as in \pref{eq:inv1},
and continuing to drop the suffix $\mathrm g$, we have
\begin{equation}
\tilde\pv_t
\,+\,
\langle u\sliver\rangle\sliver \tilde\pv_x
\,+\,
\langle\pv\rangle_y\tilde\phi_x
\;=\;
0
\label{eq:jet-disturbance}
\end{equation}
where $\langle u\sliver\rangle$ is given by \pref{eq:single-jet},
and $\langle\pv\rangle_y=\pvj\sliver\delta(y)$ as before.
For a disturbance
of the standard form
$\tilde\phi
\sliver\propto\sliver
\hat\phi(y)\sliver
\exp\!
\left\{
   ik\sliver(x-ct)
\right\}
$,
with phase velocity $c$ and wavenumber $k$ along the jet,
we have from \pref{eq:inv1}
\begin{equation}
\tilde\pv
\;=\;
\LL\tilde\phi
\;=\;
\tilde\phi_{yy}
\,-\,
(k^2 + \LDMT)\tilde\phi
\;.
\label{eq:wave-pv-psi}
\end{equation}
So, with
$
\hat\pv
\definedas
\hat\phi_{yy}
\,-\,
(k^2 + \LDMT)\hat\phi
$,
\pref{eq:jet-disturbance} implies when $k\ne0$ that
\begin{equation}
\left(
   \langle u\sliver\rangle - c \Sliver
\right)
\hat\pv
\,+\,
\pvj\sliver\delta(y)\hat\phi
\;=\;
0
\;,
\label{eq:jet-disturbance-mode}
\end{equation}
integration of which across $y=0$
gives
\begin{equation}
\left(
   \uj - c \Sliver
\right)
\left[
   \hat\phi_y
\right]^{y=0+}_{y=0-}
\,+\,
\pvj\sliver\hat\phi\big|_{y=0}
\;=\;
0
\label{eq:jet-disturbance-jump}
\end{equation}
for the jump in $\hat\phi_y$ across $y=0$.
Away from $y=0$, we have $\langle\pv\sliver\rangle_y=0$ hence
$\hat\pv=0$ and therefore
\begin{equation}
\hat\phi(y)
\sliver
\;\propto\;
\sliver
\exp\!
\big\{\Antisliver\antisliver
   -\Antisliver\antisliver(k^2+\LDMT)^{1/2}|y|
    \phantom{|}\!\antisliver\antisliver
\big\}
\;.
\label{eq:jet-disturbance-structure}
\end{equation}
For \pref{eq:jet-disturbance-jump} and
\pref{eq:jet-disturbance-structure}
to be compatible we must have, with $\uj=\half\LD\pvj$,
\begin{equation}
c
=
\uj
\big\{\Antisliver
   1
   -
   \left(
      {1+\hbox{$\LDSQ\sliver k^2$}}
   \right)^{\!-1/2}\!\antisliver
\big\}
\;,
\label{eq:jet-disp-rel}
\end{equation}
which may be compared with
\pref{eq:plane-rossby-wave-disp}, which has no square root in
its denominator $\LDMT + k^2 + l^2$. \
The square root in \pref{eq:jet-disp-rel}
arises from the infinite bunching of PV contours at
the jet core, giving dispersion properties
distinctly different from
those where the background PV contours are evenly spread out, as
assumed in \pref{eq:plane-rossby-wave-disp} and in its limiting case
$\LDSQ\rightarrow\infty$ used in the standard Rhines-scale argument.

In \pref{eq:jet-disp-rel} notice especially
that, as $k$ varies monotonically between $0$ and $\infty$,
$c$ varies monotonically between
$0$ and $\uj$.
That is,
for this jet there are always ``critical lines''
--- that is, $y$ values such that $\langle u\sliver\rangle = c$
--- on each flank of the jet as indicated in \fref{fig:single-jet}.
Thus,
whenever the jet is disturbed, the undulations of its core
will inevitably give rise to PV mixing
on either side.  In a frame of reference moving with the wave at
speed $c$, one sees
Kelvin-cat's-eye flow patterns in the jet flanks, which
must twist up any stray PV contours like spaghetti on a fork.
Whenever the
undulations increase in amplitude, the
widths of the mixing regions expand,
tending on average to keep the PV contours bunched up in the
core and maintaining the eddy-transport barrier there,
in the typical way.

The presence of more than one zonal wavenumber $k$
complicates the kinematical details, going over to
Kelvin sheared-disturbance kinematics \cite{Thomson:1887}
for broad spectra.  But none of this can
suppress the PV mixing in the jet flanks.
This is the typical anti-frictional jet self-sharpening process and,
broadly speaking, is
the kind of kinematics already encountered in
\fref{fig:tank} and in the other examples mentioned.  It is
typical of the real, highly inhomogeneous wave--turbulence
interactions observed in the atmosphere and oceans
--- see also \cite{Hughes:1996} for a good discussion of
some oceanic examples ---
with linear, wavelike regions closely adjacent to
fully nonlinear, wave-breaking regions.
It is well verified in fully nonlinear numerical experiments,
e.g.\ \cite{Greenslade:2008}, \cite{Esler:2008}.
As already emphasized, it is at an opposite extreme from a
homogeneous-turbulence scenario using
\pref{eq:plane-rossby-wave-disp} as the
Rossby-wave dispersion relation.
Further discussion and further
illustrations can be found for instance in
\cite{McIntyre:2008},
\cite{Dritschel:2011},
and
\cite{Wood:2010}.

The
long-wave behaviour is relevant to the issue of meandering.
If we take $k^2\ll\LDMT$ in \pref{eq:jet-disp-rel} we find
phase and group velocities
\vspace{0.1cm}
\begin{equation}
c\approx \half\uj \LDSQ k^2
\;, \qquad
c_{\rm group}\approx \threehalves\uj \LDSQ k^2
\vspace{0.1cm}
\label{eq:phase-group}
\end{equation}
that match the jet's surrounding flow velocity
$\langle u\sliver\rangle \approx 0$ for sufficiently small
$k^2$. \  This says that a
meandering jet
flows through its
surroundings almost like a river.  Large-scale meanders
have very little propensity to propagate upstream or downstream.
The same thing holds
even for large-amplitude
meanders, and for
generalizations of model~1 beyond quasi-geostrophic theory,
as shown in
\cite{Nycander:1993}.  The meandering behaviour
of a nonlinear version of our isolated-jet model
is strikingly reminiscent of the large-scale, large-amplitude
meandering of real
\ao\ jets.  The nonlinear model even succeeds in mimicking
the cutoff behaviour that sheds Gulf-stream rings.
Again, further discussion is in \cite{McIntyre:2008}.

It is worth remarking that the long-wave
dispersion properties just described
do not depend on having an idealized, perfectly sharp jet,
with infinitely close bunching of its PV
contours --- at least not within linear theory.  As long as
$\langle\pv\rangle_y
=
\langle u\sliver\rangle_{yy} - \LDMT \langle u\sliver\rangle$
to sufficient accuracy --- which means continuing to neglect the
planetary-scale background contribution $\beta$ ---
and as long as
the jet velocity profile $\langle u\sliver\rangle$
approaches zero
on either side of the jet core, we always find
phase speeds $c$ that also approach
zero as $k^2\rightarrow 0$.
This can be seen by inspection of
\pref{eq:jet-disturbance}--\pref{eq:wave-pv-psi}.
As long as $k^2$ can be neglected in \pref{eq:wave-pv-psi},
the disturbance problem \pref{eq:jet-disturbance}
evidently has a solution
$\hat\phi\propto\langle u\sliver\rangle$
with $c = 0$.
This fact
was pointed out in \cite{Drazin:1962} and extensively exploited
for planetary-scale Rossby waves on a strongly-sheared
stratospheric polar-night jet in
\cite{Simmons:1974}.
The physical reason
is that the long-wavelength
meanders tend to take place with all the jet's
PV contours
moving as one,
as long as they are sufficiently bunched to avoid
further mixing by the wave breaking on either side.


How much of this carries over to model~2\sliver?  The general
long-wave solution just described fails; but in the idealized
sharp-jet case
we get
precisely the same dispersion properties,
\pref{eq:jet-disp-rel}--\pref{eq:phase-group}, as in model~1,
despite having a very different jet velocity profile.
In model~2 the sharp-jet velocity profile is given by
the pair of dashed lines on the left of \fref{fig:single-jet}.
(Model~2, with the same PV jump $\pvj$,
requires us to set $\LD=\infty$ in the
local inversion problem for the jet profile,
i.e.\ to set $\LD=\infty$ in
$
\langle u\sliver\rangle_{yy} - \LDMT \langle u\sliver\rangle
=
\langle u\sliver\rangle_{yy}
= -
\pvj\sliver\delta(y)
$,
which is
satisfied by
the pair of dashed lines.)

The derivation
\pref{eq:jet-disturbance}--\pref{eq:phase-group}
of the linear wave theory
applies word for word and symbol for symbol,
provided that $\uj$ still denotes the maximum jet velocity.
With $\hat\pv=0$ everywhere except for the delta function
at $y=0$,
advection  $\langle u\sliver\rangle\partial_x$ by
the the mean flow $\langle u\sliver\rangle$ has no role away from
$y=0$.


The dashed lines in \fref{fig:single-jet}
are of course only a local approximation
within a full staircase with large step size $L$;
it is easy to show by restoring the $\beta$ contribution to
$\pv_y$  that the dashed lines are really part of
a set of larger-scale
parabolic profiles, qualitatively like the profiles in
\fref{fig:staircases} though shifted to the left.
Again, details are given in
\cite{Dritschel:2008}.
All that matters for the present discussion is that the range
of $\langle u\sliver\rangle$ values
expands when we go from model~1
to model~2, and in particular that model~2 still has the property
noted above that critical lines
$\langle u\sliver\rangle = c$
will always be present.




\vspace{+0.001cm}\section{Concluding remarks}\vspace{+0.001cm}

The most important idea to have emerged
from
our excursion into \ao\ dynamics
is, I'd argue, the apparent robustness of the
inhomogeneous PV-mixing paradigm
in strongly nonlinear
two-dimensional and
layerwise-two-dimensional turbulence,
whose
weakly-excited, weakly-dissipative
parameter regimes exhibiting PV mixing
are now beginning to be reached thanks to
today's computing power (e.g.\ \cite{Scott:2012}).
It is perhaps worth saying that
model~2, the extended Hasegawa--Mima model,
seems to point toward a similar conclusion
for real tokamaks.
Model~2 is heavily idealized, to be sure ---
for instance in assuming unrealistically low ion temperatures ---
but a process as robust as inhomogeneous PV mixing might well
have counterparts in
more realistic plasma models.


At first sight model~2 presents us with a
strangely inconsistent-looking
m\'elange of
the real tokamak's
zero-$k_\parallel$ and nonzero-$k_\parallel$ modes of motion.
Zonal averages $\langle\phi\rangle$ in model~2
are associated with zero $k_\parallel$
because
$\langle\,\cdot\,\rangle$
is a surrogate for
averaging over one of the magnetic flux surfaces
within the real tokamak --- these toroidal
surfaces being
approximately fixed in space,
providing a peculiar spiral scaffolding
that constrains, among other things,
the mutual adjustment
between
$\tilde n_e$ and $\tilde\phi$
in the nonzero-$k_\parallel$ modes.
The zonal average $\langle\phi\rangle$
within a single two-dimensional
cross-section of the torus is close to
the zonal average within another such cross-section, both being
close to the full flux-surface average
especially in cases where each spiral field line
intersects most neighbourhoods on the
flux surface.
By contrast, the fluctuating velocity fields
$\tilde\bu(\bx,\sliver t)$
about such a zonal average
tend to~have nonzero values of $k_\parallel$,
subjecting them to the mutual adjustment between
$\tilde n_e$ and $\tilde\phi$
and making them different in different cross-sections.

However, as already suggested in Sec.\ \ref{sec:inv},
a key point about advective \emph{mixing}
processes is that they are insensitive to details in the
fluctuating velocity fields responsible for the mixing.
Thus, for instance, 
if in model~2 a two-dimensional fluctuating velocity field
$\tilde\bu(\bx,\sliver t)$ were to be replaced by
$-\tilde\bu(\bx,\sliver t)$,
then mixing could still be expected to occur.
A random walk remains a random walk when all the
displacements are multiplied by $-1$.
Thus if model~2 exhibits inhomogeneous PV mixing, then
its entire behaviour could be a surprisingly good
surrogate
for the real, three-dimensional behaviour.

Of course the PV field
is not a passively-advected scalar, even though the
scale effect (\sref{sec:phillips})
tends to make the PV almost
passive and the velocity field effective at mixing,
to a remarkable extent
in many observed \ao\ cases.
These often have the character of
breaking Rossby waves,
involving quasi-passive mixing
by a $\bu$ field whose scale is generally larger than the
rapidly-shrinking scale of the advected PV features.
It seems possible, then, that similar ideas might help toward
understanding the dynamics
of strongly nonlinear drift-wave turbulence.



In summary, we might reasonably expect that the zonally-averaged
--- in reality flux-surface-averaged --- zonal flows in tokamaks
will be strongly affected by inhomogeneous PV mixing
even though the
details of the mixing process must change
as we go around the torus.
It would be of great interest to see in detail what model~2's PV fields
actually look like in the strongly nonlinear,
weakly-excited, weakly-dissipative cases
accessible to today's computers.
To my knowledge there are no such fields published,
though zonal-flow generation in model~2
has been clearly demonstrated in
reference \cite{Gallagher:2012}.
If we were to zoom in to look at the fine details,
would the PV look anything remotely like \fref{fig:balkans} above?

\medskip

\noindent
\emph{Note added May 2013:} \
Jupiter now looks like a different case again.
From current work with my graduate student Stephen Thomson
an impression is growing
that Jupiter has an altogether stronger ``scaffolding''
in the form of deep, relatively steady, zonally symmetric
jets generated in the massive cloud-free convection layer
that underlies the weather layer.
It now seems likely that such steady deep jets
exist in reality and guide the upper,
weather-layer jets,
suppressing
what would otherwise be their tendency to meander.
This is despite Jupiter's weather layer being in a regime
similar to that discussed in
\sref{sec:toymodel}, in which long-wavelength meandering
would be very easy to excite, in the absence of the deep jets.



\bigskip

\noindent
\textbf{Acknowledgements}

\smallskip

\noindent
I am very grateful to Laur\`ene Jouve and Chris McDevitt for producing the
first draft and to
Steve Cowley, Pat Diamond, Philippe Ghendrih, Alexey Mishchenko,
Gabriel Plunk, and Timothy Stoltzfus-Dueck
for patiently
explaining some aspects of
tokamak and stellarator dynamics, and for help with the
plasma literature and its notational conventions.
Tim Dowling, Peter Read, Richard Wood, and Stephen Thomson
have helped me with recent
developents in Jovian dynamics, real and idealized.
Huw Davies helped with recent terrestrial
developments,
and Steven Balbus and Toby Wood with solar.




