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Q.
J.
R.
Meteorol.
Sac.
(1998),
124,
pp. 2129-2148
Permeability
of
the stratospheric vortex edge: On the mean mass flux due
to
thermally dissipating, steady, non-breaking Rossby waves
By RUPING MO’, OLIVER BUHLER* and MICHAEL E. McINTYRE
Centre for Atmospheric Science?, Cambridge,
UK
‘Present
aflliation:
McGill
University,
Canada
(Received
4
November 1997; revised
29
January
1998)
SUMMARY
As
part of an assessment of the flowing-processor hypothesis of Tuck
et
al.
(1993) and references-see also
Rosenlof
et
al.
(I
997)-this paper estimates possible contributions to flow through the edge of the stratospheric
polar vortex due solely
to
distortion of the vortex by thermally dissipating Rossby waves forced from below. To
isolate such contributions in a clear-cut way, and to eliminate questions about numerical dissipation and truncation
error, an idealized model is studied analytically. It assumes steady conditions and non-breaking waves, the waves
being stationary in some rotating frame such as that of the earth. The model is studied using two approaches: first
via the generalized Lagrangian-mean formalism of Andrews and McIntyre (1978), simplified by assuming small
wave amplitude
a;
and second via a direct consideration of the three-dimensional, finite-amplitude undulations of
the vortex edge, as defined by isentropic contours of potential vorticity, avoiding the use of any mean-and-deviation
formalism. It is shown, in particular, that under quasi-geostrophic scaling the Lagrangian-mean meridional velocity
the altitude,
q’
the meridional particle displacement and
x‘
the
wave-induced fluctuation in the diabatic rate of
change of potential temperature
8.
The formula for
VL
is
shown to be consistent with the independently derived
finite-amplitude result; and the implication of
both
results is that, for disturbances dissipated by infrared radiative
relaxation in the wintertime lower stratosphere,
CL
may well be directed into rather
than
out of the vortex, though
weak outward flow is possible in some cases. There is, in addition, a vertical mean flow
uIL
controlled by eddy
dynamics above the altitude under consideration. This is usually directed downward
(EL
<
0),
and can therefore
push mass
out
of
the
vortex if
the
vortex edge has its usual upward equatorward slope. However, under typical
parameter conditions for the winter stratosphere, the magnitudes are nowhere near large enough to be consistent,
by
themselves, with Tuck
et
al.’s
statement that the vortex
is
‘flushed several times’ during a single winter.
-L
u
IS
‘
given correct to
O(a2)
by
VL
2
-(a@B/&)-’Xelt3q’/&,
where
@B
is
the basic-state potential temperature,
z
KEYWORDS:
Dynamics Eddy transport Polar vortex Stratosphere
1.
INTRODUCTION
The problem of understanding the observed midlatitude stratospheric ozone depletion
(e.g. Albritton
et
al.
1995,
and references) involves not only chemistry and radiation but
also nontrivial, and subtle, fluid-dynamical questions. There are reasons to suppose that
the edge of the stratospheric polar vortex acts as a flexible eddy-transport barrier, to some
extent leaky yet strongly inhibiting the material transport associated with layenvise-two-
dimensional turbulence on isentropic surfaces. The inhibition is due in part to Large-scale
‘Rossby-wave elasticity’ and in part to horizontal shear acting at smaller scales, especially
shear just outside the vortex edge (e.g. Juckes and McIntyre
1987;
Norton
1994).
In
the
lower stratosphere, such inhibition tends to isolate vortex air chemically ‘primed for ozone
destruction’ from extravortical, midlatitude air, most of which is not thus ‘primed’.
However, there has been controversy about exactly how effective the eddy-transport
barrier might be.
At
one extreme, flow down through the vortex and out into the mid-
latitude lower stratosphere has been suggested as a significant factor
in
the chemistry of
midlatitude ozone depletion. Tuck
et
al.
(1993)
estimate from observational data that the
lower-stratospheric part of the vortex is ‘flushed several times’ during a single winter,
*
Corresponding author: Department of Applied Mathematics and Theoretical Physics, University
of
Cambridge,
Silver Street, Cambridge CB3 9EW,
UK.
t
The Centre for Atmospheric Science is a joint initiative of the Department of Chemistry and the Department
of
Applied Mathematics and Theoretical Physics in the University of Cambridge. Further information is available on
the Internet at web page
http://www.atm.ch.cam.ac.uk/casl.
2129
2130
R.
MO,
0.
BUHLER
and
M.
E.
McINTYRE
implying, presumably, that it sustains a throughput of the order of a vortex mass or more
per month, from the vortex to the midlatitude lower stratosphere. (In Rosenlof
et
al.
(1
997)
these throughput estimates have been lowered in
the
light of more recent data analyses.
However, even these lowered estimates would still imply that the vortex is flushed several
times each winter, 70% per month, according to a referee for this paper.) If this were
correct, then the vortex could act to a significant extent as a ‘flowing processor’, or ‘flow
reactor’, priming large amounts of air for catalytic ozone destruction and exporting
it
to
sunlit middle latitudes
in
the lower stratosphere, with implications for the ozone layer over
densely populated areas.
In view of the potential seriousness of the problem there is a need to look carefully at
all possibilities, even though the large throughputs estimated by Tuck
et
al.
(1993) seem
at first sight fluid-dynamically implausible. What exactly, we need to ask, are the possible
mechanisms that could make the edge leaky, weakening the eddy-transport barrier effect,
or in any way permitting a large flow through the vortex?
One mechanism, already studied extensively, is the effect
of
Rossby-wave break-
ing. This typically takes the
form
of ‘erosion’, ‘peeling’ or ‘stripping’ of filaments of
air from a narrow neighbourhood
of
the vortex edge, and their subsequent equatorward
mixing into the midlatitude stratospheric ‘surf zone’ by layerwise-two-dimensional tur-
bulence
on
isentropic surfaces. This erosion mechanism can obviously remove some air
from the vortex, helped perhaps by the effects of inertia-gravity waves (e.g. Pierce
et
al.
1994); these can break in their own manner, contributing to small-scale vertical mixing
and to horizontal parcel dispersion, and hence to vortex-edge cross-diffusion and leakiness
(McIntyre 1995). Tuck
et
al.
(1992) had earlier suggested, from a study of aircraft data
and of isentropic maps of Rossby-Ertel potential vorticity (PV) derived from meteorologi-
cal analyses, that outward material transport
on
isentropic surfaces, presumably resulting
from some such process or processes, might be significant. On the other hand, a number
of
investigations using numerical modelling and observational data analyses indicate typical
vortex throughput rates that fall well short of a vortex mass per month. For a survey, and
critical discussion, of such work see Sobel
et
al.
(1997). There must always be doubts as
to whether numerical and data-analytic resolutions are good enough to quantify isentropic
transport processes like layerwise-two-dimensional turbulence and vortex erosion; both
encompass a large range of spatial scales, some
of
which are almost always too fine to be
well resolved (e.g. Legras and Dritschel 1993)-probably a few tens of kilometres, and
possibly even kilometres in the real lower stratosphere. Such processes must inevitably
be affected by the artificial diffusivities used in numerical models. But, regardless of this,
we may note the obvious fact that vortex erosion at a rate of a vortex mass per month
would by itself, relative to the three-month time-scale of a whole winter, amount to rapid
destruction
of,
rather than flow through, the vortex. In other words,
it
is
obvious that ero-
sion, with or without the help of inertia-gravity wave breaking, cannot by itself produce a
sustained flow of the required magnitude through an intact vortex (as contrasted with the
‘sub-vortex’
,
see below). Effects other than erosion, e.g. diabatic effects, would also have
to be important and might even be dominant.
Diabatic heating and cooling are included, explicitly or implicitly, in many of the
studies just cited. However, in order to develop insight into the significance of such effects,
and into the robustness or sensitivity of any associated flow-rate estimates, we consider
here an idealized, analytical model that tentatively assumes the dominance of diabatic
effects, and neglects vortex erosion altogether. This requires non-breaking waves, i.e. a
purely undular vortex edge; and, consistent with this, the model further assumes that
conditions are strictly steady in some rotating reference frame. Recent work described
in Biihler and Haynes (1998) generalizes this model
to
include statistically steady flows,
PERMEABILITY
OF
THE STRATOSPHERIC VORTEX EDGE
2131
which corroborates that our results derived here do not depend crucially on the strict
steadiness of the flow. The outcome is a clean thought-experiment exhibiting the diabatic
effects in pure form, i.e. a thought-experiment in which all the sideways vortex leakage is
unequivocally due to diabatic effects. Furthermore, even though sharp-edged and multiple-
edged vortices are included among a large range of possible cases, there are no lingering
concerns about problems of numerical accuracy. The same problem was considered by
McIntyre and Norton (1990), hereafter MN, (see also Nash
et
al.
1996) whose work
prepared some of the theoretical ground; however, it did not
go
far enough to be directly
applicable to the present problem of understanding vortex-edge leakage, in part because
of a mistake in interpretation (see section
3
below). This comes down
to
a tacit, and for
present purposes inappropriate, assumption that the vortex edge is vertical everywhere, an
assumption that is far too restrictive, if only because it would force us to confine attention
to altitude-independent undulations.
As expected from general considerations about wave-mean interaction, the present
analysis confirms that diabatic effects can by themselves make the vortex edge leaky. By
contrast with MNs’ analysis we allow for a fully three-dimensional distortion or undulation
of the vortex edge. We find that the leakage in this model can be regarded as the
sum
of
two contributions that can usefully be distinguished from each other. The first is a mean
inflow or outflow along each isentropic surface, which in this model will turn out to be
controlled entirely locally in altitude, in a thought-experiment in which one prescribes
the undulation
of
the vortex edge on the isentropic surface in question. This contribution
depends on the vortex being distorted away from a zonally symmetric state by sideways
undulations that are altitude-dependent (see
(1)
and (13) below). The reason is that the
PV
‘impermeability theorem’ (Haynes and McIntyre 1990) implies, under quasi-geostrophic
scaling-and this is the real significance of MNs’ result-that the diabatic leakage due to
local eddy correlations, averaged around the edge
of
an intact vortex, can be computed as
if the three-dimensional eddy diabatic circulation consisted of vertical motion only, i.e. as
if the horizontal (radial and azimuthal) components required by mass continuity did not
exist. In order to get leakage from eddy correlations involving the vertical motion alone,
the undulation (more precisely the sideways eddy displacement of the vortex edge) must
have a vertical gradient. The second contribution
is
from mean diabatic descent across
isentropes, and therefore across the vortex edge if the edge has a
mean
slope. In this
model-as in a more general class of models postulating statistically steady, ‘perpetual
winter’, conditions-such mean descent is controlled entirely by the eddy dynamics above
the isentrope of interest, in the sense discussed in Haynes
et
al.
(1991, 1996) under the
heading ‘downward control’. In the present model this corresponds to a thought-experiment
in which one prescribes the undulations of material contours at all altitudes directly above
the vortex edge on the isentrope of interest, as made precise by
(2)
below; in practice, ‘all
altitudes’ will usually mean a few scale heights,
To derive these results two independent approaches are used here, which check each
other and help to develop a feel for robustness or sensitivity. The first, presented in sec-
tion
2
and appendix
A,
considers the small-amplitude problem within the framework of
the GLM (generalized Lagrangian-mean) theory of Andrews and McIntyre (1978). This
automatically provides
a
view
of
the problem in vortex-following coordinates, subject to
the general caveats noted in McIntyre (1980a), and an easy route to the results. One result
is a simple formula for the first, locally controlled contribution to vortex leakage, which
in the GLM framework is associated with the Lagrangian-mean meridional velocity com-
ponent
TL.
(This is related to the transport velocity
ET
of Plumb and Mahlman (1987),
see appendix A.) The formula, (1) below, is derived using quasi-geostrophic theory for
the disturbance structure. It verifies the crucial importance of altitude-dependent vortex
2132
R.
MO.
0.
BUHLER
and
M.
E.
McINTYRE
undulations. The second contribution, controlled from above, is associated in the GLM
framework with the Lagrangian-mean vertical velocity component
ZL.
The second independent approach to our results, presented in sections
3
and
4,
avoids
the use of any mean-and-deviation formalism. This approach also applies straightforwardly
at finite disturbance amplitude, more
so
than the GLM approach which is applicable in prin-
ciple but only at the cost of some analytical complexity (again see the caveats in McIntyre
1980a). The cross-checks between the two approaches are given in section
5
and appendix
B.
Section
6
gives typical order-of-magnitude estimates for the two contributions to vortex
leakiness, suggesting that, as well as often being of opposite sign, each contribution has,
in any case, almost certainly much smaller magnitude than a vortex mass per month.
Before turning to details, we note that it remains fluid-dynamically possible for the
‘sub-vortex’ region below about
400
K
or
70
hPa to act as a significant flowing processor
or
flow
reactor. The sub-vortex
is
not only denser than the vortex proper, but also, in the
Antarctic at least, sufficiently cold for chemical priming to take place by processes like
chlorine activation. Fluid-dynamically speaking, the sub-vortex is far more susceptible to
mass throughput because, almost by definition,
it
is the polar stratospheric layer stirred by
the upward-evanescent disturbances associated with tropospheric weather. Even though
the large-scale wind field still looks vortex-like (as might be expected from PV inversion
of the overlying PV anomaly, e.g. Robinson 1988), the sub-vortex can have its own PV
anomaly significantly disrupted-or even, in principle, completely obliterated-by the
stirring from below; this makes a model of the present kind inappropriate below about
400
K,
and implies no fluid-dynamical difficulty in sustaining a large throughput
of
air and
chemical constituents via inflow and outflow along isentropes. One may think of this sub-
vortex transport as having the character of layerwise-two-dimensional turbulent transport,
at an opposite extreme to the purely diabatic transport through an intact vortex studied,
in idealized form, in the present paper. The sub-vortex problem will not be pursued here,
but some further discussion from a fluid-dynamical viewpoint may be found in McIntyre
(19954, and from the viewpoint of chemical evidence in Jones and Kilbane-Dawe (personal
communication).
2.
LAGRANGIAN-MEAN
MERIDIONAL
CIRCULATION
INDUCED
BY
SMALL-AMPLITUDE
Returning to the steady, purely diabatic vortex model, consider the first contribution
uL
in the idealized situation in which the dissipation of the wave-mean system is assumed
to be purely thermal and the system has settled down as a whole to an exactly steady state,
i.e. not only is the wave amplitude steady, but also the mean circulation. We further assume,
in this first approach, that the waves involved are of small amplitude and that they approxi-
mately satisfy geostrophic and hydrostatic balance (as in quasi-geostrophic theory, though
not assuming latitude-independent static stability). Under such circumstances, it follows
easily from established GLM results (details in appendix
A),
and also from the alternative
approach via transformed Eulerian-mean theory (Mo and McIntyre 1998,
(5.17)),
that,
away from the pole, at given co-latitudinal distance
r,
say
(I
being defined as co-latitude
times the earth’s radius), and at given altitude
z,
DISTURBANCES
-
where the sign convention for
gL(t-,
z)
is
the same as that
of
$(I,
z),
the latitudinal particle
displacement. We shall take both to be positive when poleward, i.e. positive for inflow and
PERMEABILITY
OF
THE STRATOSPHERIC VORTEX EDGE
2133
inward displacement, in deference to the standard sign convention but in the sense opposite
to
Y.
The
8/32
signals the importance
of
altitude-dependent vortex undulations already
mentioned;
z
can be expressed in any units, for instance pressure or log pressure; note that
the units cancel on the right-hand side of (1). The other symbols
are
t!?B
(r,
z),
the basic-state
potential temperature, and
X’,
the wave-induced heating rate expressed as the fluctuation
in the diabatic rate of change of potential temperature
-9,
not ordinary temperature
T.
That
is,
X’
is
eB/
TB
times the wave-induced fluctuation in the diabatic contribution to the rate
of change
of
T,
with
TB(r,
t)
the basic-state temperature; note that
&/TB
-
2
to
3
in the
lower stratosphere.
As the form
of
(1)
suggests, it does not depend on any quasi-Cartesian or channel
approximation but applies equally well to a polar vortex having finite radius,
r
=
R(z)
say; indeed it applies at any finite radius
r
%
q’
at
or
away from the vortex edge, though
we shall be interested mainly in applying it with
Y
taken to be the undisturbed vortex-edge
position
R.
When the formula is thus applied, the overbar signifies the corresponding zonal
average.
Equation
(1)
shows that
VL
at a given altitude can be related entirely locally to the cor-
relation between the disturbance diabatic heating rate
X’
and the vertical derivative
of
the
meridional particle displacement
q’.
In particular,
CL
is
zero for disturbances that are either
adiabatic
(X’
=
0)
or produce altitude-independent vortex-wall undulations
aq’/dz
=
0.
Apart from the restriction to small amplitude, this last is the particular case to which the
analysis of MN applies, for which they found, at finite as well as small amplitude, that
‘there is no diabatic mass transport’ across the vortex edge. This will be further clarified
in section
3.
The second contribution
EL,
controlled from above in the sense already explained,
is given in this model to the same (quasi-geostrophic) accuracy as
(1)
by the associated
‘downward control integral’ (Haynes
et
al.,
op.
cit.),
namely:
where
4
is latitude and
pB
(r,
z)
is
the basic-state mass density. The units of
EL
are units
of
z
per unit time. Note the possibility that vortex leakage could be exactly zero
if
the vortex
edge slope \dR/dz\
=
\EL/TBLI
with appropriate signs, for instance inflow and descent, in
the usual case of vortex radius increasing with altitude.
In deriving
(1)
and
(2)
we assume that the displacements
q’
are small;
GLM
theory
can be used at finite amplitude, but
the
details are complicated and not worth pursuing
here. The second approach, given in section
3,
yields finite amplitude results more simply,
taking direct advantage of the model assumption of steady, non-breaking waves. The upshot
will be that, judiciously interpreted,
(1)
and
(2)
are qualitatively correct even at realistic
amplitudes, provided always that wave breaking remains unimportant. (When it comes
to order-of-magnitude estimates, however, we shall avoid using
(2)
beyond its role as a
reminder of the downward control principle. Thus, the actual estimates will, in the end,
rely only on
(1)
and not on
(2);
hence, they will rely on neglecting wave breaking only in
the altitude range, say 15-25
km,
to which the flowing-processor hypothesis is considered
To obtain a more specific idea of the likely implications
of
(l),
consider a case in
to apply.)
which
X’
is modelled by weak Newtonian cooling, i.e.
2134
R. MO,
0.
BUHLER
and
M.
E.
McINTYRE
where
a
is the Newtonian radiative cooling coefficient,
8’
the isobaric potential-temperature
fluctuation,
a’
the isobaric geopotential fluctuation, and
g
the magnitude of the acceleration
due to gravity; see
(A.4).
For steady, quasi-geostrophic disturbances, it is readily shown
(see
A.7)
that
where
f
is the Coriolis parameter and
u(r,
z)
the basic zonal flow. Substituting
(3)
and
(4)
into
(I),
we immediately obtain
rl’
=
Wfm,
(4)
(5)
In this expression
0
is a zonal average as before. The first term within braces is always
positive, corresponding to inflow, i.e. to a poleward rather than equatorward wave-induced
mass flux. This positive definiteness, corresponding to the usual sense of the ‘gyroscopic
pumping’ of mean circulations by thermally dissipating Rossby waves, can be seen from
the foregoing to be characteristic of any small amplitude quasi-geostrophic disturbance if
the basic vertical shear
ag/az
is
zero or negligible. In all such cases the correlation in
(1)
is
robustly negative, as is obvious from inspection of
(3)
and
(4),
when
u
is independent
of altitude
z.
Included among such cases are the standard Rossby-wave solutions familiar
from slow-modulation or group-velocity theory; however, there is no restriction on the
horizontal shear
au/ar.
The results apply, in particular, to waves on a sharp-edged vortex.
If
u
is altitude-dependent, then the second term of
(5)
might not be negligible.
Under normal winter stratospheric conditions,
a
>
0
(in a frame of reference following
the wave) and
au/az
>
0.
Then the two terms reinforce, again giving inflow, whenever
the wave has a sufficiently diffractive vertical structure, such that decreases upward,
as with synoptic-scale weather systems penetrating the stratosphere. Otherwise, with
increasing upward, as often happens with disturbances of larger horizontal scale, we may
expect some cancellation between the two terms. In all cases, an upper bound on possible
outflow
rates
--VL
is evidently
This bound is unlikely to be sharp. Outflow, corresponding to positive
-TL,
requires the
second term in
(5)
to overcome the first. The ratio of the two terms satisfies the inequality
This is easy to prove, for instance by writing
W(r,
z),
without loss of generality, as ampli-
tude
IO’(r,
z)l
times a phase factor cos{nx
-
x
(r,
z)},
for any zonal wavenumber
n
and
function
~(r,
z).
The essential point is that, for any given amplitude profile
I@’@,
z)l,
increasing the vertical phase tilt will increase the magnitude of the first term in
(5)
but not
that
of
the second.
The possibility that the ratio of the two terms in
(5)
exceeds unity and weak outflow
occurs
is
known to be realized, in at least one case of a theoretical Rossby-wave solu-
tion with sufficiently strong positive vertical shear
au/az
(Mo
and McIntyre
1998).
The
anomalous outflow was, however, confined to a rather narrow altitude range. The well
PERMEABILITY
OF
THE STRATOSPHERIC VORTEX
EDGE
2135
known and well observed case of
20-22
January
1992,
which occurred during the Euro-
pean Arctic Stratospheric Ozone Experiment, is of interest here because the vortex edge
was nearly vertical during a major disturbance induced by a large blocking anticyclone
over Scandinavia (e.g. Norton and Carver
1994);
this suggests relatively small values of
aq’/az
and, to the extent to which the Newtonian model applies, near-cancellation
of
the
two terms in
(5).
Further discussion is postponed to section
6,
where some numerical
estimates are given.
3.
FINITE-AMPLITUDE
DISTURBANCES
We now derive a finite-amplitude expression for the mass flux across a three-dimen-
sional vortex edge due to steady, non-breaking, thermally dissipating Rossby waves. The
finite-amplitude vortex-edge definition is now based on isentropic contours of
PV,
rather
than material contours*. The derivation corrects and extends the first attempt on the prob-
lem discussed in MN; they did not take the vertical structure of the vortex edge properly into
account. It also: (a) allows an independent check of the small-amplitude results obtained
in the previous sections, bypassing the
GLM
formalism; and (b) allows greater confidence
in our order-of-magnitude estimates for realistically large vortex-edge distortions. Indeed,
the expression to be obtained is quantitatively exact in principle, though to apply it quan-
titatively we would need to know the precise three-dimensional shape of the vortex edge,
which in turn would require elaborate numerical calculations not worth pursuing here.
For clarity
it
is easiest to carry out the derivation first in a polar-tangent plane ap-
proximation, and afterwards generalize to spherical geometry in section
4.
We use the
hydrostatic primitive equations in isentropic coordinates, in which
6
is
used as a vertical
coordinate and
x,
y
denote arbitrarily oriented Cartesian horizontal coordinates, though
plane polars could equally well be used. The general time-dependent form of the gov-
erning hydrostatic primitive equations in isentropic coordinates is well known, see for
example Andrews
et al.
(1987)
or
Haynes and McIntyre
(1
987, 1990).
Important aspects
of these equations include that in
xye
space isentropes appear as flat ‘horizontal’ planes,
and that the material velocity becomes
(u,
v,
X),
where
u
=
Dx/Dt,
v
=
Dy/Dt
(not to
be confused with the poleward-pointing
u
of section
2)
are the true horizontal velocity
components and not components along isentropes, and where as before
X
=
DB/Dt
is a
measure of the diabatic heating.?
Observational and numerical-modelling evidence indicate that the polar vortex tends
to be bounded radially by strong isentropic gradients
of
PV,
which together with horizontal
shear (Juckes and McIntyre
1987)
form an ‘eddy-transport barrier’, strongly inhibiting
turbulent eddy transport along isentropes. This suggests the use of appropriately chosen
PV
contours,
r(6)
say, to mark the barrier region on isentropes and with it the vortex
edge$. The
PV
values associated with these
PV
contours are denoted by
Qr(0),
and may
vary continuously with
O
as appropriate. The result
is
a curved surface marking the vortex
edge, and for convenience we consider its three-dimensional geometry
as
seen in an
xye
space, with origin at the pole (see Fig.
1).
We now consider the mass flux,
h?
say, into the vortex through a band
B
of the vortex
edge bounded by two isentropes
Ol
and
0,.
For steady flow as
is
assumed here,
A?
is given
*
The recent work
in
Biihler and Haynes
(1998)
generalizes the expression derived here to statistically steady flows,
in which the circulation around the
PV
contour bounding the vortex edge
is
constant on average.
t
It
should also be noted that steady flow in
xyz
space implies steady flow in
xy8
space, because the former then
implies that the variable transformation to isentropic coordinates
is
time-independent.
$
This assumes that the vortex edge can be marked approximately on a given isentrope by a single
PV
contour.
We do not wish to suggest that such an edge definition would apply more generally than in the restricted problem
considered here; cf. e.g.
Tao
and Tuck
(I
994).
2136
R.
MO,
0.
BUHLER
and
M.
E.
McINTYRE
X
Figure
1.
A
schematic perspective view
of
the model
vortex
edge. Although the horizontal coordinates
are
labelled
x,
y
for
convenience, they need not be Cartesian (see
also
section
4).
by a surface integral over
B
as
=
-
/l
afu,
u,
%)
.
dB,
where
(T
is the mass ‘density’ in isentropic-coordinate space such that
(T
dx
dy d8 is the
mass element, and where
dB
=
(dy d0, d6’
dx,
-d~
dy)
is an oriented surface element of
B
in
xy0
space, pointing outwards from the vortex by
definition. The dot product between the velocity vector and the surface element is simply
the sum of the products
of
the corresponding three components (see section
4
for further
discussion of this flux formula). This expression for
M
can be substantially simplified,
because the ‘horizontal’ contribution to the mass flux in
(8)
is in fact approximately zero for
steady flows in the usual quasi-geostrophic scaling regime (for large Richardson number,
in a frame rotating approximately with the vortex edge).
This
can be shown, following MN, by considering the
PV
evolution equation for
steady flow in isentropic coordinates, i.e.
Here
Q
is the Rossby-Ertel
PV,
and the symbol
V
stands for
(3/3x,
a/dy,
3/80).
Note
that in (9) we have again used the assumption of purely thermal dissipation. In the assumed
scaling regime, namely standard quasi-geostrophic scaling as also assumed in section
2,
the second term in (9) is negligible (e.g. Haynes and McIntyre 1987); therefore
v
*
[cTQ(u,
u,
O)]
0.
(10)
PERMEABILITY
OF
THE STRATOSPHERIC VORTEX EDGE
2137
On a given isentrope
(10)
can be integrated over the vortex area on that isentrope, and an
application of the two-dimensional divergence theorem then yields
~Q(u,
V,
0)
*Sds
=
Qr(8)
O(U,
~,0)
.S~S
EO,
(1
1)
kel
4F,,)
where ds is the line element along the PV contour marking the vortex edge, and where
S=
(g,
-%
ds
0)
is the unit vector normal to the contour in xy8 space, pointing outwards by definition. In
(1
1)
Qr
(8)
could be moved outside the contour integral, because by assumption
r
(Q)
is
a contour of constant PV. Note that only the integral around the closed contour vanishes,
not the integrand itself.
On the other hand, the horizontal contribution to the mass flux M in
(8)
is given by
and comparison with
(1
1)
now shows that the inner integral is approximately zero on any
given isentrope.
Therefore, in the
xyQ
view, only the non-horizontal part
of
the scalar product in
(8)
contributes to the mass flux into the vortex; i.e.
(8)
is replaced by
where
dx
dy stands for the horizontally projected area
of
the edge band element dB. (Note
that
crX
is evaluated
on
the vortex edge band
B,
and not on its horizontal, i.e. xy-plane,
projection,) Each area element
dx
dy would be zero for a vortex edge that does not tilt
away from the vertical in xy8 space, and the sign of
dx
dy conforms to the convention
dxdY
{
<
0
:
edge tilts
inwards
from the vortex with increasing
8.
This convention expresses the fact that diabatic heating
%
>
0
produces a mass flux into
the vortex wherever the vortex edge is tilting outwards with increasing
8,
and
vice versa.
Note that the result (13) has a highly nontrivial aspect; it tells us that the total diabatic
mass flux into or out of the vortex through the band
B
can be computed as if the horizontal
particle velocity components at the band could be ignored. In the assumed circumstances
all such horizontal contributions, though not locally zero, sum to zero around the band.
That is the significance of the partial result previously obtained in
MN,
described there
in terms of the motion
of
‘quasi-material’ contours on isentropic surfaces, which ‘do not
drift systematically northwards or southwards’. MNs’ interpretation of this to mean that
there is no diabatic mass transport across the vortex edge was wrong, as (13) shows, except
when the vortex edge is vertical everywhere, making the right hand side of
(13)
zero by
making the projected area elements
dx
dy all zero. In other words, MNs’ interpretation
conflated what they called quasi-material transport with true material transport. Recall that
the ‘assumed circumstances’ both here and in
MN
are: first, that quasi-geostrophic scaling
holds, permitting neglect of the
8/88
terms in
(9);
and second, that all dissipative effects
are diabatic. This excludes, for instance, forces attributable to gravity-wave drag.
(14)
>
0
:
edge tilts
outwards
from the vortex with increasing
8,
21
38
R.
MO,
0.
BUHLER
and
M.
E.
McINTYRE
4.
GENERALIZATION
TO
SPHERICAL
GEOMETRY
AND
ARBITRARY
COORDINATES
Equation (13) for the mass flux is now generalized to spherical geometry, using
the straightforward spherical generalization of the tangent-plane primitive equations in
isentropic coordinates (e.g. Andrews
et
al.
1987,
section 3.8). This allows us to check,
for instance, that the curvature of geopotentials in spherical geometry does not affect our
results. It is also shown how the mass flux in spherical geometry can be expressed in
terms of arbitrary coordinates
(a,
b,
c),
where
(a,
b)
are quasi-horizontal (e.g. longitude
and latitude) and
c
is a quasi-vertical coordinate (e.g. pressure altitude or
6).
This allows
us to check that the polar-tangent plane of the previous section is in fact a surprisingly
accurate approximation, even for a very large polar vortex, and to do
so
in a succinct and
economical way.
To
begin with, we note that the derivation of (13) (i.e.
(8)
and
(10)
to
(12))
relies
only on two things: first, on defining
cr
such that
cr
dx
dy d0
is
the mass element; and
second, on defining
u
and
u
as material rates of change of coordinate values,
u
=
Dx/Dt
and
u
=
Dy/Dt. With such definitions, and with
(x,
y,
6)
replaced by
(a,
b,
6)
(but
not
yet by
(a,
b,
c)-see below), fluxes through surfaces (i.e. the dot product in
(8)
and below)
and the divergence theorem can always be expressed
as
$
the coordinates were Cartesian;
to put it another way, the relevant equations can then all be expressed in the language of
exterior calculus, without having to use the possibly complicated metric coefficients of
the chosen coordinates.* This also establishes that the curvature of geopotentials does not
affect our previous derivation. Hence, with the choice of
cr
da db d6 as the mass element
and with
(u,
u)
=
(Da/Dt, Db/Dt) as quasi-horizontal velocity components, the above
derivation is valid for any pair of quasi-horizontal coordinates
a
and b.
We finally generalize to an arbitrary quasi-vertical coordinate
c
by noting that the
projection of the surface element in (13) is in fact independent
of
the choice of this quasi-
vertical coordinate. This means that (13) is transformed into arbitrary coordinates
(a,
b,
c)
on the sphere by simply replacing
dx
dy with
du
db, and by replacing
cr
with an expression
derived from
dm
=cr
da dbd6
=
pdx
dy dz
+
o
=
where
p
is the mass density in ordinary, geometric space, and
a
and b. Therefore
itk
is
re-expressed in arbitrary coordinates
(a,
b,
c)
on the sphere as
is evaluated at constant
where
B
stands for the image of the vortex band in the chosen coordinates and where
da
db
conforms to the sign convention in
(14).
In local Cartesian coordinates
(a,
b,
c)
=
(x,
y,
z),
which are suitable for a tangent-
plane approximation, this is simply
Alternatively,
if
longitude
A,
co-latitudinal distance
r,
and altitude
z
are used as coordinates
*
In
this notation the diabatic
flux
in the full
PV
evolution
(9)
looks more complicated on the sphere than on a
tangent plane (cf. Andrews
etal.
1987,
Eq.
(3.8.8)).
However, as argued before, the diabatic
flux
is negligible in the
assumed scaling regime.
PERMEABILITY
OF
THE STRATOSPHERIC VORTEX EDGE
2139
(a,
b,
c),
then
k
is given by
where
aE
is the earth’s radius such that the ratio
r/aE
is equal to co-latitude. The second,
approximate form comes from noting that
z
4
UE
and that
UE
sin(r/aE)
%
r
for realistic
values of co-latitudinal distance for the vortex edge, i.e. for values of
r
not much larger
than aE/2, say. (See section
6
for numerical estimates of
r.)
The second form shows that
for realistic values of
r
the mass flux
&Z
can, to a very good approximation, be calculated
using cylindrical polar coordinates based on a polar-tangent plane,
in
which co-latitudinal
distance
r
acts as radial distance. This polar-tangent-plane approximation can equally well
be described using the local Cartesian coordinates introduced in
(17),
provided that
x
and
y
are related to the co-latitudinal distance r by
r2
=
x2
+
y2.
This very useful description
of the mass
flux
M
will be used repeatedly below.
5.
CONNECTION
TO
SMALL-AMPLITUDE
THEORY
To derive a leading-order estimate for
(17)
and check that it agrees with the previous
small-amplitude result, we assume that the flow consists of three components: first, a
zonally symmetric basic state whose meridional velocity components are zero; second,
a field of small-amplitude linear Rossby waves with non-dimensional amplitude
a
<
1;
and third, an
0(a2)
mean-flow response to these linear Rossby waves. This implies that
the entire mean meridional circulation is wave-driven and
0(a2).
It
also implies that
the undisturbed, basic-state polar vortex is zonally symmetric, i.e. that its bounding
PV
contours are latitude circles. The shape of the basic-state vortex edge,
Bo
say, can then
be described by an altitude-dependent radius R
(2)
in cylindrical polar coordinates with
origin at the pole-either in tangent-plane or in spherical polar coordinates, see
(17)
and
(18).
The slope of the basic-state vortex edge is given by dR/dz, which in the real winter
stratosphere is often observed to be a modest fraction of
N/f
*
1, which latter
2
lo2.
Such a substantial slope of the basic-state vortex edge cannot be neglected
a
priori
in its
contributions to the projected surface elements
dx
dy in
(17).
This zonally-symmetric basic-state vortex edge
Bo
corresponds to the leading-order
mean vortex edge in the
GLM
formalism. This implies that the shape of the actual vortex
edge, which is undulated by the Rossby waves, is given to leading order by the map
x
I+
x
+
g’(x,
t),
for allx on
Bo.
Here g’is the
0
(a)
linearized
GLM
particle displacement
field used previously. This map from
Bo
to
B
allows us to write the integral in
(17)
as an
integral over
Bo
whilst evaluating the integrand on
B.
The leading-order result of this
procedure, whose details are given in appendix
B,
is
where
z
=
z1
and
z
=
z2
are the horizontal planes bounding the edge band
Bo.
and where
eL
is
evaluated correct to
0(a2);
q’
is,
as in previous sections, the
poleward
particle
displacement correct to
O(a),
which accounts for the minus sign in the integrand. (The
sign convention is consistent when
TTL
is poleward velocity and
k
inward mass flux, as
before.) Note that
p
and
M/az
have been replaced by their zonally symmetric basic-state
values
p~
and
MB/az.
2140
R.
MO,
0.
BUHLER
and
M. E. McINTYRE
The two terms in the square brackets are the leading-order expressions for the two
contributions to the mean transport discussed earlier. The first contribution
-
(X’aq’/az)
describes the mean transport across the vortex edge along isentropes, which is given by the
eddy structure in the local altitude range
z1
<
z
<
z2, specifically, by the local correlation
between disturbance-induced diabatic heating and vortex-edge tilt. The form of this first
term gives an independent check on the small-amplitude expression for
DL
presented in
(l),
in particular checking the previous assertion that the validity of the expression does not
depend on any quasi-Cartesian or beta-channel approximation; there are
no
extra terms in
1/R,
for instance. The second contribution gLdR/dz describes the mean transport across
isentropes, corresponding to
?IrL
in
(2).
6.
NUMERICAL
ORDERS
OF
MAGNITUDE
We now use (19) to obtain some typical order-of-magnitude estimates for the two
contributions
to
the mass flux, with realistic values for the diabatic heating and wave
amplitude. Although finite-amplitude analogues of (19) can be derived in principle from
(17),
by using suitably defined finite-amplitude formulae for the edge projections
dx
dy
and
so
on, it is clear from the analysis just sketched (given in detail in appendix
B)
and
from the associated geometric picture, that (1
9)
is robust enough to be used for order-of-
magnitude estimates.
The ratio between the vortex mass
M
and the mass transport rate
M
into or out of the
vortex may be called the vortex flushing time,
tF
say, i.e.
which is positive for outflow if
kl
<
0
corresponds to outflow. In a steady state
(M/M
constant),
tF
is the time required for a throughput of just
1
vortex mass. Note also that
tF
allows us to define unambiguously the relative vortex-mass throughput per unit time in
our thought-experiment. We assume that the vortex edge is sufficiently well-defined, and
undular, from
8
=
400
K
upward to use the corresponding altitude
z1
as the lower boundary
of the vortex edge in our model. The vortex itself is assumed to extend over several
(3-5,
say) density-scale heights
H
-
7
km in the vertical. The rapid decay of the basic state
mass density
h(r,
z)
with altitude, which tends to make high-altitude contributions to
M
unimportant, then allows us to treat the vortex height as effectively infinite for the purpose
of calculating
M.
For the purpose of estimating the first contribution to (19), that involving
aq’/az,
the geopotential disturbance
a’
is assumed to have a simple exponential vertical
structure without phase tilt. The no-phase-tilt assumption is not essential, but is made here
because non-zero phase tilt always increases the inflow component of this contribution
(noted already below
(7)).
The second contribution, that involving
gL,
will be estimated
without direct reference to the types of disturbances that might be acting to pump air down
from above, but simply by using observational estimates of typical descent and cooling
rates. We look at this next.
Consider, then, the second contribution
in
(19),
i.e. the transport due
to
mean diabatic
heating
ZL
in the presence of
a
sloping mean vortex edge. We assume that
‘%
is
constant
and equal to a typical negative value, compatible with studies of mean descent in the polar
vortex (e.g. Manney
et
al.
1994). The mean vortex-edge slope dR/dz
is
assumed to
be
a
positive constant,
y
say, whose value is a moderate fraction of
N/f.
This is consistent
with typical observations. Together these assumptions imply that the second contribution
-L
.
PERMEABILITY
OF THE
STRATOSPHERIC VORTEX
EDGE
2141
corresponds to
hi
<
0,
i.e. to mean outflow. The vortex mass is estimated as
where
p1
and
RI
are the density and vortex radius at the lower boundary,
p~
=
pi
exp(-
(z
-
zl)/H), and R
=
R1
+
y(z
-
21).
The second orgL contribution to (19) is found to be
where
QB
%
Q1
exp(lc(z
-
zl)/H)
has been assumed, with
K
=
(gas
pressure specific heat)
=
0.286. Combining (22) and (21) we obtain
(22)
constant)/( constant-
Using
%‘
=
-0.6
K
day-’ in 6
(-
-0.3
K
day-’ in
T;
e.g. Manney,
op.
cit.),
Q1
=
400
K,
y
=
0.5N/f
%
80,
H
=
7 km, and
RI
=
3000
km
(which corresponds to a mean vortex
edge bounded on
8
=
400
K
by 63” latitude) then gives
tF
x
850
days. This shows that the
mean outflow due to diabatic descent across the sloping vortex edge is weak; weaker even
than the mean outflow through the vortex bottom into the ‘sub-vortex’ below
8
=
400
K
(cf.
McIntyre 1995,
$6;
Jones and Kilbane-Dawe personal communication), which by itself can
be estimated as
&f
X
n
RfplgL H/(K&), yielding a shorter flushing time
tF
x
270 days.
Consider finally the first or
X’
contribution to (19), i.e. the along-isentrope transport
due to a correlation between fluctuations
aq’/az
in vortex-edge tilt and fluctuations
X’
in
diabatic heating rate, corresponding to
EL
in
(1).
This equation was rewritten in terms of
geopotential disturbance
@’
in
(5)
for the case of small-amplitude Rossby waves dissipated
by Newtonian cooling in the wintertime lower stratosphere. It is reasonable to expect that
such a relation also applies approximately to finite-amplitude undulations of the vortex
edge whose temperature anomalies suffer radiative relaxation, and whose phase relations
are qualitatively those implied by geostrophic and hydrostatic balance. Hence, for the
purpose of order-of-magnitude estimates, we assume that the relations between
X’,
q’
and
a’
implied by the small-amplitude
(3)
and
(4),
namely
also hold approximately at finite amplitude. Here
a
is
a suitably chosen Newtonian radiative
cooling coefficient.
We assume a simple form of
@’
on the vortex edge without vertical phase tilt, namely
@’
=
@’,
cos(kx) exp(-(z
-
zl)/h),
(25)
where
@’,
is
the
geopotential-disturbance
amplitude at the vortex bottom,
k
is a zonal
wavenumber, and
h
is the exponential envelope scale. Both decaying
(h
>
0)
and growing
(h
<
0)
disturbances are considered. We now restrict attention to the lower part of the
vortex, say the lowest 10
km
or
so,
i.e.
D
=
z2
-
zl
=
10 km. This avoids non-essential
2142
R. MO,
0.
BUHLER
and
M.
E.
McINTYRE
convergence problems of the integral in (19) if the disturbances are growing very strongly
with altitude; see also the Concluding Remarks. Non-zero vertical shear must be explicitly
recognized, as in
(3,
and the along-isentrope mass flux in (19) is then estimated from (24)
and
(25)
as
f2
vp
h
au
D
h
x
napl
R~Ro--
N2
h2
[
1
+
5x1
1
exp(-z/A) (1
+
z)
dz (26)
where the effective exponential envelope scale
A,
which includes density decay as well
as
the amplitude envelope of
W2,
is given by
h
=
(;
+
f)'
.
This uses the same assumptions for
p~
etc. as before,
Ro
=
ul/(f
R,)
is the Rossby number
at the vortex bottom,
q;
is the vortex undulation amplitude at the bottom, and
u(z)
has
been approximated with
u,
except in the explicit shear term, which is itself approximated
as a constant. Note that the square bracket determines the sign of
h,
and that this bracket
exhibits the previously mentioned competition between inflow and outflow contributions
in the case of
h
<
0
and
u-'au/az
>
0.
The integral in (26) can be evaluated as
Y
(
{
R1
I"
exp(-z/h)
1
+
-
dz
=
h
1
+
A-
-
exp(-D/A)
(28)
and hence the corresponding flushing time is estimated as the ratio
of M
from (21)
and
-h
from (26), (27) and (20). Let
a!
=0.05
day-', Ro=0.1 (which corresponds
to
u1
x
40 m
s-I),
D
=
10 km,
qi
=
1500
km, and let the remaining parameters take
the values previously used. Consider first a case without vertical shear (i.e.
a'il/az
=
0)
and with a decaying
geopotential-disturbance
structure described by
h
=
10
km.
This
yields
tF
x
-700
days, where it should be noted that the sense of this contribution
is
into
the vortex. Second, still without vertical shear, let the geopotential disturbance be
growing with
h
=
-10
km.
This implies much stronger flow yielding
rF
x
-140
days,
but this is still a contribution into the vortex. Finally, consider strong vertical shear such
that
u(au/az)-'
=
10
km.
(This is consistent with observed values of this vertical shear
length-scale, which typically range from 10-30
km
in the lower stratosphere.) It is then
found that maximal outflow (i.e. minimal positive
tF)
is achieved when
h
x
-16 km,
yielding
tF
x
880
days.
It can be noted that the positive mean slope of the vortex
y
reduces the effectiveness of
the along-isentrope contributions, because the mass of the vortex increases as
R2
whereas
the strength of the flow along isentropes increases only as
R.
We have experimented
considerably with different choices for the various parameters that enter these estimates,
but have found no indication that the flushing times can be brought close to the times
required for the flowing-processor hypothesis mentioned before.
7.
CONCLUDING
REMARKS
Analytical formulae, and order-of-magnitude estimates, for the sideways
(FL
associ-
ated) and vertical
(EL
associated) contributions to the mean mass
flux
through the edge
PERMEABILITY
OF
THE STRATOSPHERIC VORTEX EDGE
2143
of an intact polar stratospheric vortex have been presented. The notion of intactness is
expressed theoretically by the basic assumption made here, that vortex erosion and other
effects of Rossby-wave breaking can be neglected in estimating
FL
and the associated side-
ways mass flux through the lower-stratospheric part of the vortex, say a
10
km-thick layer
above 15-16 km. The
EL
contribution, by contrast, is estimated on the assumption that
it
is
pumped from above by eddy activity, which might, and probably does, include significant
wave breaking higher up.
All
the mass-flux contributions appear to be too small to support
the version of the flowing-processor hypothesis that postulates large flows of the order of
a vortex mass per month through the polar vortex, as distinct from the sub-vortex below
about
400
K,
to which the notion of intactness does not apply and through which large
flows could much more easily take place.
Contrary
to
what the simplest wave-mean theories suggest, the sense of the
FL
con-
tribution can be either into or out of the vortex when the background zonal wind has
sufficient vertical shear, as illustrated by
(5).
There are then intermediate cases in which
the
FL
contribution vanishes even though
aW/az
and
X’
do not, of which the most obvious
are cases in which the vortex wall is distorted yet vertical everywhere-as was approxi-
mately true, for instance, in a case analysed by Norton and Carver (1994). Indeed, in such
cases there must be zero net mass flux through the edge, as can be seen from the finite-
amplitude formula
(1
3);
note further that refinements to the theory, such as going beyond
quasi-geostrophic accuracy, will not qualitatively change the picture beyond changing the
stipulation ‘vertical’ to ‘slightly sloping’ (by a small fraction of Prandtl’s ratio
N/f).
As
discussed for instance in MN and in
Mo
and McIntyre (1998), this does not, of course,
imply that other measures of mean circulation need vanish.
The assumption of negligible wave breaking is critical only as regards conditions in
the vortex edge, where material contours undulate more or less reversibly and quantities
like
EL
are well defined. The assumption probably represents a good approximation if, as
seems typical in the lower stratosphere, the actual wave breaking gives rise to no more
than the erosion of fine filaments or sheets from the vortex edge. Such erosion will not
substantially change typical
8’
anomalies and hence will not produce substantial diabatic
effects, because PV inversion tends to have the character of a smoothing operator and hardly
sees the finer-scale structures. In other words, it takes
a
large-scale distortion of the vortex
to generate substantial
8’
anomalies and eddy diabatic effects. For order-of-magnitude
purposes, such large-scale distortion is probably described well enough by a theory of
the kind used above, in which the theoretical displacement field
q’
is interpreted as, in
effect, a measure of the large-scale distortion only, and not as the exact, and irreversible,
material distortion including wave breaking effects in the form of fine-scale erosion. In this
sense, linearized wave-mean theory is ‘better’, for present purposes, than nonlinear theory.
Better still, however, is the finite-amplitude formula
(1
8),
which avoids the problems of
finite-amplitude wave-mean theory and is valid for any intact vortex. This latter would be
the best of the foregoing results to use if the theory were to be applied in an observational
case study.
One might ask about the fate of outflow from the side of the vortex, as distinct
from outflow from the bottom, as well as asking about horizontal flow through the sub-
vortex. On present evidence concerning vertical, cross-isentropic diffusivities in the lower
stratosphere (tending to confirm that they can be taken to be of the order of
0.2
m2s-l as
usually supposed, e.g. most recently in Sparling
et
al.
1998 and references) we can argue
that vertical diffusion heights would be little more than a kilometre, over the time-scale
of one winter, implying that vortex and sub-vortex outflows would tend to have separate
midlatitude destinations at separate altitudes within the layer of interest for midlatitude
ozone depletion chemistry, which we have been assuming to lie between about
15
km
and
2144
R.
MO,
0.
BUHLER
and
M.
E. McINTYRE
25
km. This suggests that flow out from the bottom of the vortex should be considered as
part of the sub-vortex problem, significant for chemistry in the midlatitude stratosphere
below about
15
km
as long as mean descent does not take it out of the stratosphere
altogether; while flow out of, or into, the sides of the vortex should be considered separately,
and
is
significant for chemistry in the midlatitude stratosphere between about
15
km and
25
km.
Although, as stated at the outset, the importance of the ozone-depletion problem
requires us to leave
no
stone unturned, the analysis has brought no surprises beyond find-
ing that weak outflow
EL
is
a
possibility, in some rather restricted circumstances. This
possibility was not revealed by earlier simplistic arguments in terms of angular momen-
tum (McIntyre 1995), which fail to allow for the diabatic distortion of material contours;
though such arguments, as it has turned out, still give a correct idea of the likely order of
magnitude of
EL.
Consistent with our present arguments about robustness, the estimates
made here are not much larger than the similarly modest vortex throughput rates derived
from the numerical-model studies; again see the critical discussion in Sobel
et
al.
(1
997)
and references. We conclude that it is difficult to find any fluid-dynamical justification
for the suggestion of high throughputs of the order of a vortex mass per month. We think
our estimates are robust enough to permit some confidence that the only way to get such
throughputs would be for cooling time-scales in the lower stratosphere to be more like
5
days than
20
days, which seems unlikely.
ACKNOWLEDGEMENTS
We thank P.
H.
Haynes, R.
L.
Jones,
I.
Kilbane-Dawe, W.
A.
Norton, J.
A.
Pyle and
T.
G. Shepherd for stimulating correspondence and conversations, and two referees for
constructive and encouraging comments. The result (1
1)
was first pointed out to
us
by
P.
H.
Haynes. RM gratefully acknowledges support by the British Council and the State
Education Commission of the People's Republic
of
China in form of a 4-year Sino-British
Friendship Scholarship. OB thanks the Gottlieb Daimler and Karl Benz foundation in
Germany and the Natural Environment Research Council for research studentships. MEM
thanks the Engineering and Physical Sciences Research Council for generous support in
the form of a senior research fellowship.
APPENDIX A
Derivation
of
(I)
and
remarks
on
ZT
Equation
(1)
can be derived from theorem
I
and corollary
I1
of Andrews and McIntyre
(1978); see also McIntyre (1980b),
Eqs.
(4.10b), (5.4a),
(5.5)
and(5.7) whichareessentially
the same results
in
a slightly more convenient form for our purpose. It is easiest to begin
by
thinking in terms of a quasi-Cartesian
or
beta-channel approximation, though it can be
shown from a version of corollary
I1
(and can be independently seen from section
5
above)
that the result, in the end, depends on no such approximation.
In the present steady-state problem, with small wave amplitude and purely thermal
wave dissipation and no significant wave breaking, the results just mentioned imply at
where
0
is a zonal average as before,
DL
is the GLM meridional velocity,
f
is the Cori-
olis parameter,
g
is the magnitude
of
gravity acceleration,
OB
is the basic state potential
PERMEABILITY
OF
THE STRATOSPHERIC VORTEX EDGE
2145
temperature,
{’
is the upward particle displacement,
a
is
a measure of the eddy amplitude,
and
Be
is the Lagrangian potential-temperature fluctuation given to leading order by
As before,
qf
is the northward particle displacement. (A.2) may be rearranged as
(A.3)
From hydrostatic balance we have, with
@’
the disturbance and
5
the mean geopotential,
=-f--,
(A.4)
ai7
gel
aw
g
aeB a25
oB az
’
eB
ay ayaz az
-
-
_-
-
because
Now invoking geostrophic balance, steadiness, and small wave amplitude again, we see
that, to leading order,
Q>’
is related to
r]’
through
whence
equivalent to (4). Similarly
where
X’
is
the wave-induced fluctuation in the diabatic rate of change
of
0,
and is the
basic zonal flow. Substituting (A.3) into (A.l) and using (A.7) and then (A.4) and (A.8)
gives
(1).
We remark that, as shown by Plumb (1979) within the quasi-Cartesian or beta-channel
approximation,
(EL,
EL)
=
(VT,
GT)
where
(gT,
ET)
denotes the effective transport cir-
culation in the sense of Plumb (1979) and Plumb and Mahlman (1987). An alternative
derivation is given in
Mo
and McIntyre (1998), also within the channel approximation.
Both derivations depend on the assumption
of
steady, non-breaking waves. Whether the
result, unlike
(l),
depends on the channel approximation in any essential way is not clear
at present, because the theory of the effective transport circulation does not seem
to
have
been fully worked out in curvilinear geometry. For instance, the Stokes drift contribution to
(VL,
SL)
in polar-cap
or
other curvilinear geometry includes a term involving the second
spatial derivatives of the basic zonal velocity field. These derivatives form a third-rank
tensor some of whose components are
O(1)
as distinct from
O(a2),
hence not negligible,
for disturbances of low zonal wavenumber when latitude circles are significantly curved,
and when the zonal velocity
is
O(
1).
For theoretical detail the reader may consult Andrews
2146
R.
MO,
0.
BUHLER
and
M.
E.
McINTYRE
and McIntyre (1978). For instance the second term in their (2.27), with the factor
q,i,
in-
terpreted as representing the second derivatives of the basic zonal velocity field, with one
of
the three indices implicit, contains a significant contribution proportional to the cone-
lation
im
where
(’
is the zonal disturbance displacement. To see whether similar terms
arise when generalizing the formulae for
(ET,
ZT)
to a significantly curved geometry, one
would have to develop a model for the associated three-dimensional diffusion tensor that is
sensitive to the presence of
0
(1) mean shear and mean-flow curvature, as might possibly
happen through shear-dispersion effects.
We recall finally that, even in the channel approximation,
(EL,
iZL)
and
(UT,
gT)
differ from the transformed Eulerian-mean meridional velocity
(IT*,
E*).
Mo and McIntyre
(1998) present a specific example in which they have
the
opposite sign.
APPENDIX
B
Derivation
of
(19)
The task is to derive (19) as the
0(a2)
approximation of (17) in the case of an
axisymmetric basic-state polar vortex perturbed by
0
(a)
Rossby waves. The integral over
the actual vortex edge
B
in (17) is replaced by an integral over the leading-order mean
vortex edge, i.e. the axisymmetric basic-state vortex edge
Bo.
The diabatic heating
‘X
then
has to be evaluated at the displaced position
x
+
l’,
and this gives
X(x
+
6’)
=
X’+R+
6’
*
VX’+
o(U3)
(B.1)
correct to
0
(a2),
where the terms on the right-hand side are all evaluated at
x.
Specifically,
X
and
‘X’
denote the Eulerian mean and disturbance parts of the diabatic heating
X,
where
the mean is a zonal average as before (cf. (B.7) below). Because the diabatic heating
3Y
is
itself
O(a),
all remaining terms in the integrand of (17) need only be evaluated correct to
0
(a)
in order to yield all contributions at
0
(a2).
The GLM formalism provides a general relation between
a
disturbed surface area
element dB and the corresponding mean surface area element dBo (Andrews and McIntyre
1978, section
A.3).
The leading-order form
of
this general relation
is
03.2)
where dBo
.
(V6’)T
is equal to
[dBO]jei,i
when written in Cartesian coordinates. For the
axisymmetric basic-state vortex edge the surface area element and the
O(a)
displacement
field
6’
are given by, respectively,
-
dB
=
dBo(1
+
V
.
6’)
-
dBo.
(V&’)T
+
0(a2),
dBo
=
r
-
-z
ds dz and
(-
3
The function
R(z)
gives
the
radius of the basic-state vortex edge in cylindrical polar co-
ordinates and
@,
T,
9
are local unit vectors in these coordinates, pointing in the radial,
azimuthal, and vertical directions respectively. The increment ds is taken along latitude
circles. Note that the components of the displacement vector have been chosen according
to the convention that
((’,
q’,
5’)
are the Cartesian components of the displacement vector
with respect to a local tangent plane oriented such that
(’
points
in
the zonal direction and
17’
points poleward.
PERMEABILITY
OF
THE
STRATOSPHERIC VORTEX
EDGE
2147
Using
(B.3)
and
(B.4)
in
(B.2)
allows the horizontal projections
of
dB to be evaluated
as
dx
dy
=
-dB
-2
(by chosen sign convention for
dx
dy in
(14))
dR
dz
=
-(1
+
V
.
5’
-
<:)
ds dz
-
‘7:
ds
dz
+
O(a2)
dR
=
(z
-
0:)
ds
dz
+
O(a2),
where suffixes denote differentiation. The horizontal divergence of
e’,
which
is
small for
the quasi-geostrophic motion under consideration, has been neglected in the last step.
Finally, due to the assumed quasi-geostrophic scaling, the deviations of
p
and
a8/az
from their zonally symmetric basic-state values
pB
and
i3&,/az
can be neglected completely.
The leading-order
form
of
(17)
can then be written as
Using the definition of zonal averaging
2nR
2nRn=l
()ds
03.7)
in
(B.6),
neglecting further
O(a3)
terms, and noting that
(5’
.
VX’)
is
the Stokes correction
to
%‘
such that
%
+
(5’
.
OX’)
=
%’,
now yields
(19).
Albritton, D. L., Watson, R.
T.
and
Aucam, P.
J.
(eds.)
Andrews, D.
G.
and McIntyre, M.
E.
Andrews, D.
G.,
Holton,
J.
R.
and
Buhler,
0.
and Haynes,
P.
H.
Haynes,
P.
H. and McIntyre,
M.
E.
Leovy,
C.
B.
Haynes,
P.
H., Marks,
C.
I.,
McIntyre, M.
E.,
Shepherd,
T.
G.
and
Shine,
K.
P.
Shepherd,
T.
G.
Haynes, P.
H.,
McIntyre,
M.
E.
and
Juckes,
M.
N.
and McIntyre, M.
E.
Legras,
B.
and Dritschel, D.
1995
1978
1987
1998
1987
1990
1991
1996
1987
1993
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