Progress of Theoretical Physics Supplement No. 130, 1998
137
Breaking Waves and Global-Scale Chemical Transport in the
Earth's Atmosphere, with Spinoff's for the Sun's Interior
Michael E. MciNTYRE*)
Centre for Atmospheric Science at the
Department of Applied Mathematics and Theoretical Physics
Silver Street, Cambridge CB3 9EW, U. K.
(Received May 15, 1998)
The atmosphere used to be thought of using classical ideas about turbulence that looked
back to analogies with gas kinetic theory, involving among other things an assumption that
departures from spatial homogeneity are weak. This led to problematic notions like 'negative eddy viscosity'. However, more recent advances in understanding the global-scale atmospheric circulation have shown the importance of recognizing
as essential, leading-order
features - the strong spatial inhomogeneity of atmospheric turbulence together with the
crucial role of wave propagation. For this purpose one can usefully draw a rough analogy with
an ocean beach, where (a) turbulence in the surf zone owes its existence to waves arriving
from elsewhere, and where (b) the spatial inhomogeneity of that turbulence is an essential
feature of what is called wave dissipation by breaking. There is a phase-coherent interaction
between the waves and the highly inhomogeneous turbulence. One well known consequence
is the generation of mean currents along beaches by the convergence of the radiation stress
or wave-induced momentum transport. For the global atmospheric circulation, the two most
important kinds of waves are internal gravity waves and Rossby or vorticity waves. The
chirality of Rossby waves, tied to the sense of the Earth's rotation, results in an angular momentum transport that is intrinsically one-signed and therefore ratchet-like, producing via
Coriolis effects an inexorable 'gyroscopic pumping' of air systematically poleward that dominates, for instance, the global-scale transport of chlorofluorocarbons and other long-lived
greenhouse gases in the stratosphere. The Rossby-wave counterpart to ocean-beach wave
breaking involves not 3-dimensional but 'layerwise 2-dimensional' turbulence, producing inhomogeneous mixing, quasi-horizontally along stratification surfaces, of a spin-like material
invariant called the Rossby-Ertel potential vorticity.
Some of the same considerations apply to the fluid dynamics of the Sun's stably stratified
radiative interior. Together with recent helioseismic data they are forcing us to a novel
conclusion: the Sun not merely can, but must, have in its radiative interior a poloidal
magnetic field that is strong enough (,...., 1 gauss or 10- 4 tesla by a preliminary rough estimate)
to reshape, drastically, the circulation and differential rotation in the interior. This has farreaching consequences for understanding solar spindown history and internal variability, and
for performing helioseismic inversions. It is helping to disentangle magnetic from soundspeed effects in the inversions, and should yield otherwise unobtainable information about
differential rotation in the Sun's deepest interior. It suggests a possible new resolution of
the lithium-burning enigma.
§1.
Introduction
It is a great honor to be invited to give this lecture to the Twelfth Nishinomiya
Symposium in memory of Hideki Yukawa. I understand that this is the first of the
Nishinomiya-Yukawa symposia to be devoted to such a widely interdisciplinary range
of topics. Communication between scientists in different disciplines, as well as be•) http:/ /www.atmos-dynamics.damtp.cam.ac.uk
138
M. E. Mcintyre
tween scientists and lay people, is more important than ever in today's complex and
confusing world. As a fluid dynamicist interested in the workings of the atmosphere,
I propose using the lecture to sketch something of what we know about global-scale
atmospheric circulations and what these tell us about other naturally occurring rotating, stratified bodies of fluid, including the radiative interiors of stars like our own
Sun, whose past, present, and future behavior has special human importance.
In a symposium on the dynamic organization of fluctuations, it is appropriate
to remark on the way in which order emerges from the most complicated imaginable
chaos in the Earth's atmosphere: Section 5 will give an especially striking example
of such order, the celebrated 'quasi-biennial oscillation' (QBO), of which we have a
fairly secure qualitative understanding. Reynolds numbers in the Earth's atmosphere
and oceans are of course huge, ,;(: 10 10 ; and, as expected from this, one sees motion
on a correspondingly vast range of scales all the way up from millimetres to the
scale of the planet itself- much of which motion can be described as consisting,
in one sense or another, of intermittently turbulent or chaotic fluctuations. It is
astonishing, especially from a mathematical viewpoint, that one can understand or
predict anything at all about so vast and complicated a fluid system. Vast though
the system is in physical space, it is still vaster in phase space: one feels little surprise
that the attempts to find 'weather attractors' of very low dimension ~ 6 have not
produced convincing results. 1 ), 2 ) What is more surprising is that numerical weather
prediction, for instance, with sparse data inputs and atmospheric models whose
phase-space dimensions are more like 107 or so*) - still minuscule in comparison
with reality - works as well as it does now, in a fair proportion of cases, over
timescales of a few days.
Indeed, a growing repertoire of observing techniques and strategies, involving everything from commercial aviation data to clever remote sensors, 3 ) together
with hierarchies of models - simple conceptual or intuitive through analyticalmathematical all the way to the biggest numerical models of the global atmosphereocean system- has led to very significant progress on several fronts. Not only do
we have a degree of predictive ability over timescales of days, despite all the problems posed by gaps, fuzzinesses, and malfunctions in the observing system, but, also
predictions of predictability 4 ) - of phase space sensitivity - and what currently
looks as if it might be the beginning of successful stochastic 'climate' prediction on
seasonal timescales. 5 ), 6 ) We can claim some genuine qualitative understanding of
the basic mechanisms and causal linkages that operate over a still larger range of
timescales, including some understanding of what is robust and what is sensitive.
Such understanding and predictive ability look set to improve steadily, as observ•) Today's state-of-the-art global weather prediction models extend to altitudes 30-50 km and
have equivalent mesh sizes "' 10° km vertically on average (say 30 points in the vertical) and just
a little finer than 10 2 km horizontally (say 10- 2 of the earth's radius). Thus, with about 5 main
2 2
dependent variables being time-stepped, we have a phase-space dimension"' 30 x 411" x (10 ) x 5"'
7
Weather
Range
Medium
for
Centre
European
the
at
that
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the
of
one
10 • For instance
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7
horizontal at the time of writing, early 1998, giving 31 x 3192 x 5 -:::= 1.6 x 10 • The timestep currently
used is 20 minutes.
The Earth's Atmosphere and the Sun's Interior
139
ing methods, computers, models, and data-assimilation systems improve - leading
not only to improved short-term and seasonal predictions in which people and machines both play their best roles 7 ) but also (contrary to what you might read in
some newspapers) to the possibility of answering more clearly the great questions,
involving timescales of decades or more, about global environmental change and its
predictability, or otherwise.
(Future sea-level rise is one of those great questions, and is also a simple example both of robustness and of sensitivity. The timescale of the ocean's thermal
response to a global-scale temperature increment at its surface is robustly multicentury, despite being determined in a very complicated way; s), 9 ) the magnitude is
sensitive, and highly uncertain, because the temperature increment itself is sensitive
to such things as poorly known cloud physics and dynamics. Given that atmospheric
greenhouse-gas concentrations are practically certain to stay high for a century or
more after their present buildup is stopped, IO), ll) the implication is that, once the
sea level starts to rise - if it does so significantly - the rise will be unstoppable*)
for a century or more.)
§2.
Global-scale atmospheric circulations: Some basic aspects
Here I shall concentrate mainly on the aspect of the foregoing that I am most
qualified to discuss, and where good qualitative understanding can reasonably be
claimed, viz., the highly organized, systematic global-scale circulation in what is
now usually called the 'middle atmosphere'. This is the region lying roughly between
altitudes of 10 to 100 km and consisting of the stratosphere below and the mesosphere
above (Fig. 1 and its caption). As I hope to convince you, its circulation is due
almost entirely to the dynamic organization of fluctuations over a vast range of
smaller scales. The circulation is important for moving greenhouse gases around;
in particular, it is part of why certain man-made greenhouse gases emitted in the
northern hemisphere, the chlorofluorocarbons or CFCs, can produce an ozone hole
in the southern hemisphere.
The middle-atmospheric circulation is simpler and better understood, and more
accessible to remote sensing, than the tropospheric circulation beneath. One reason is the middle atmosphere's chemical diluteness (with ozone, water vapor, and
methane mixing ratios measured in parts per 106 and most other trace chemicals in
parts per 109 ), hence lack of chemically active or passive compositional convection.
•) The multi-century timescale, stretching to millennia before a heat pulse reaches the ocean
floor 5 km down (especially if the thermohaline circulation is shut off) is largely controlled by a quasidiffusive process with vertical diffusivity ~ 10- 4 m 2 s- 1 : the random-walking of heat energy in small
vertical steps across the ocean's stable stratification, by intermittent small-scale turbulent mixing
events, implying in turn an 'underlying rate of sea level inflation' due to thermal expansion. Over
a single century, the inflation rate is roughly proportional to the global mean surface temperature
increment, if the latter is held constant. The rate is therefore closely tied to -- is almost certainly a
monotonic function of - the excess greenhouse-gas concentration. As with the ozone layer with its
natural replenishment rate of order 3 million tonnes per day, proposed technofixes have not taken
seriously the orders of magnitude, hence hard economic costs, to say nothing of the uncertainties,
that would be involved in trying to counter the greenhouse forcing itself.
M. E. Mcintyre
140
80° 70° 60° 50° 40° 30°
s
zoo
10°
0
10°
LATITUDE
zoo
30° 40° 50° 60° 70° 80°
N
Fig. 1. Mean temperature cross-section for January conditions, constructed from satellite radiome12
ter data for several successive Januaries. ) The data are averaged in longitude as well as in
time. The left-hand scale is pressure altitude p in millibars or hectopascals ( hPa); the righthand scale is the same expressed as ln(po/p) where po = 1000 hPa, a nominal sea-level pressure.
{The e-folding pressure scale height RT / g is almost exactly 7 km at temperature T = 239 K in a
hydrostatic, isothermal, perfect-gas atmosphere with specific gas constant R = 287m 2 s- 2 K- 1 ,
with gravity g = 9.8ms- 2 ; thus the right-hand scale times 7km gives geometric altitude to
a fair approximation.) The troposphere, here conspicuous via its temperature lapse rate, occupies the lowest layer with thickness about one pressure scale height in polar and just over
two in tropical regions, above which the stratosphere, the most stably stratified region, extends
up to the temperature maximum at pressure altitudes"' 10°hPa= 10- 3 p 0 , i.e., about 50km.
Above this, in turn, lies the mesosphere, extending up to the overlying temperature minimum,
at around 83 km near the summer pole but higher and more variable, often as high as 100 km
or so, at other latitudes.
Clouds and aerosols are mostly thin or non-existent (though crucial to the chemistry
itself 10 l). By contrast, compositional convection is important in the troposphere,
which has a far greater water content, producing the familiar cumulus and cumulonimbus or thunderstorm clouds. Such convection and the associated latent heat
release are especially important in parts of the deep tropics, usually between about
±10° latitude. The tropospheric circulation is driven partly by this convection, especially in the tropics, and partly by quite different mechanisms, more like those to
be discussed here. Of course the troposphere and the overlying stratosphere are not
always to be thought of in separate mental or bureaucratic compartments; 13 ) for instance extratropical 'tropospheric' cyclogenesis often depends in an important way,
dynamically speaking, upon the entrainment of lower-stratospheric air. 14 )- 16 ), 7 )
The Earth's Atmosphere and the Sun's Interior
141
Middle-atmosphere dynamics is well described by radiative heat transfer (for
which chemical composition does matter crucially) together with ordinary, classical
continuum mechanics. So far as we know, there are nothing exotic, unless you count
as 'exotic' the important Coriolis and buoyancy effects associated with rotation and
stable stratification. These effects are very strong, for present purposes. Why?
Typical buoyancy timescales, the timescales of high-frequency internal gravity waves,
are of the order of minutes- as measured by the reciprocal of the buoyancy or BruntVaisala frequency N, whose square N 2 equals gravity times the logarithmic vertical
gradient of specific entropy - and Coriolis timescales are of the order of hours. At
the poles, for instance, Coriolis or 'inertial' oscillations of a horizontal fluid layer have
period 12 hours, hence timescale, in the sense of time for significant change relevant
to estimating a time derivative, "' 12 hours/27r or about 2 hours.*) By comparison,
the timescales for significant chemical transport in the middle atmosphere are far
longer.
One relevant timescale, more aptly timespan, is the time for a nitrogen, oxygen,
or other molecule from the troposphere to circulate through the middle atmosphere
and return to the troposphere. This is typically several years. 17 ) Such circulations
feel the Coriolis and buoyancy effects as immensely strong, like stiff springs. This
implies stiffness of the fluid-dynamical equations, in the numerical-analytic sense.
Indeed it is this very stiffness that, famously, numerical weather prediction systems
must take trouble to cope with even over much shorter timespans of days.
§3. Thermal and chemical evidence
Before saying more about the circulation dynamics, let us note two simple facts
that provide important clues. One is that, at solstice, the summer pole is the sunniest
place on Earth. Diurnally averaged insolation - the solar energy arriving per unit
time per unit horizontal area · is a maximum there, because of the substantial tilt
of the Earth's axis, 23.6° or 0.41 radian. The other is that, at altitudes just above
80 km, the summer pole is observed to be the coldest place on Earth. This region is
called the polar summer mesopause. Temperatures go below 160 K or -l13°C even
in a heavily averaged picture such as Fig. 1, top left. 12 ) There are large fluctuations
about the mean shown there: rocket-borne instruments have recorded, on some
occasions, 18 ) temperatures as low as 110 K or -163°C.
Partly because of these extremes of cold, the polar summer mesopause is the
region of formation of, by far, the world's highest clouds. These are the so-called
'noctilucent' or 'night shining' clouds. 19 ) They are usually seen at altitudes near
83 km by ground-based observers at high latitudes, soon after midnight on a proportion of clear summer nights. They show a characteristic electric blue color, and
are sometimes featureless and sometimes intricately patterned. I have seen them
•l The period is indeed 12 and not, as the term 'inertial' might suggest, 24 hours. The mathematics is trivial, the same as for a particle subject only to a Coriolis force -20 xu, where 0 is the
Earth's angular velocity and u the relative velocity; but seeing why this does not correspond to a
particle at rest relative to the cosmos - why 'inertial' is a misnomer- is a good brain-teaser for
students. The key is to remember what 'horizontal' means, as with billiard tables and ice rinks.
142
M. E. Mcintyre
myself, looking north from a low hill near Cambridge, England, which at 52.2°N is
getting near the equatorward limit of observability. The electric blue color is from
sunlight coming over the pole then passing down and up again through the ozone
layer, between altitudes of order 10-50 km, before returning to 83 km and scattering off the clouds toward the observer. Ozone weakly absorbs visible light, in its
Chappuis bands, 10), 20 ) with a bias toward the yellow-red part of the spectrum.
Now noctilucent clouds are believed on good evidence, including in situ rocket
measurements, to be made of ice crystals, their formation depending not only on the
low temperature but also, crucially, on a supply of water vapor from below. 19 ) Water
vapor is photolyzed on timescales of days by the very hard solar ultraviolet that can
penetrate to 83 km, 20 ) especially Lyman alpha radiation at "' 120 nm. The necessary
supply of water vapor depends on the existence of a systematic rising motion over the
summer polar cap, as suggested schematically by the heavy dashed curve in Fig. 2,
whose different vertical scale should be noted. Air and water vapor are thus brought
up from the less fiercely irradiated, and less dry, lower layers. Water vapor mixing
ratios near 50 km altitude, for instance, are maintained rather stably at just over
6 ppmv (parts per million by volume), partly by direct supply of water and partly
by the oxidation of methane, both coming from the troposphere far below.
The rising motion in the summer polar mesosphere is part of a self-consistent
picture that checks out in other ways. For instance it is an essential part of why
summer polar mesopause temperatures are so low. There is a refrigerating action
from the expansion of the air as it rises through the stable stratification. The consequent adiabatic cooling comes into balance with net radiative heating, which has
positive contributions both from the Sun and from infrared radiation emitted by
the warmer layers below, at center left of Fig. 1. (The infrared photons have long
free paths at these altitudes, many of them escaping directly to space, but a few
are reabsorbed on their way up.) The result is that temperatures near 83 km are
pulled down, by the refrigerating action, by as much as 100 K or so below radiative equilibrium, though quantitative estimates are difficult because of uncertainties
about the detailed photochemistry, and about departures from local thermodynamic
equilibrium. 23 ), 26 )
Temperatures in certain other parts of the middle atmosphere also differ significantly from radiative equilibrium temperatures, for instance over the winter polar
cap. There air is compressed, and its temperature pushed above radiative equilibrium, by the systematic descending motion seen on the right of Fig. 2. On the
other hand the summer stratopause, the temperature maximum near center left of
Fig. 1, at pressure altitudes near 10° hPa, or about 50 km or 7 scale heights, is relatively stagnant and relatively close to radiative equilibrium, probably within 10 K or
so. 23 ), 22 ) The high temperatures at the summer stratopause are well explained by
the absorption, by ozone, of solar ultraviolet at wavelengths ,....,200-300 nm, which can
penetrate down to 50km or lower. 20 l• 23 ) Implicit in all the foregoing is, of course,
the notion of a radiative equilibrium toward which temperatures, in the absence of
any circulation, would tend to relax. This is well justified by detailed studies of
radiative transfer; 23 ), 22 ) and relaxation times are of the order of days to weeks.
A circulation like that in Fig. 2 and its seasonal variants is consistent with the
The Earth's Atmosphere and the Sun's Interior
143
,.~'S'{#ft~-~
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Fig. 2. Mass transport streamlines of the global-scale mean circulation for January 1979, solid
curves, from Ref. 21). This was estimated from satellite radiometer data in conjunction with
detailed (and elaborate 21 )- 23 >) calculations of radiative heating and cooling rates. The picture gives the typical latitude and height dependence of the longitude and time averaged mass
circulation of the stratosphere, defined in a quasi-Lagrangian sense 24 ), 25 ) giving a simplified,
but roughly correct, indication of the vertical advective transport of chemical constituents. The
heavy dashed streamline indicates the qualitative sense of the mesospheric branch of the circulation, deduced from other observational and theoretical evidence. 23 ) The lower pair of circulation
cells is often collectively referred to as the 'Brewer-Dobson circulation'. The altitude scale is
nominal log-pressure altitude, as in the right-hand scale in Fig. 1 times 7 km; thus the vertical
domain shown corresponds roughly to the middle third of Fig. 1. The right-hand cell of the
circulation is a little stronger than usual in this picture; the wintertime stratosphere was dynamically very active in January 1979. The northward mean velocities at top right (within the
frame, not counting the heavy dashed streamline) are of the order of 2 or 3 ms - 1 . On the heavy
dashed streamline they would be more like 10 ms - 1 .
144
M. E. Mcintyre
observed behavior not only of water vapor and methane but also of a number of other
chemical constituents (nearly all of which, incidentally, having molecules larger than
diatomic, are greenhouse gases).
For instance, man-made CFCs are chemically inert under everyday conditions.
Together with their low water-solubility and small rate of uptake by the oceans, this
means that, as is well verified by observations, 10 ) they are mixed fairly uniformly
throughout the troposphere. The troposphere can therefore be thought of as a big
tank or reservoir a few percent of whose contents are recirculated through the middle atmosphere at rates of the order suggested by Fig. 2. This accounts rather well
for the observed destruction rates of CFCs, which correspond to e-folding atmospheric lifetimes of order a century, actually somewhat shorter for CFCll (CFC1 3 )
and somewhat longer for CFC12 (CF2Cb), the latter being a bit more stable because
of its extra fluorine atom. 10 l It is mainly at altitudes ;G 25 km that the CFCs are destroyed, again mainly through ultraviolet photolysis; and so rates of CFC destruction
are closely tied to rates of circulation.. In terms of mass flow that reaches altitudes
;G 25km, these rates amount to a few percent of the tropospheric mass per year.
This is consistent with circulation times of order several years and CFC e-folding
lifetimes of order a century, because of the falloff of mass density with altitude roughly in proportion to pressure, i.e., as already mentioned, by a factor e- 1 every
7 km or so. Thus mass densities at, e.g., 30 km, are 1% of those at sea level.
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Fig. 3. The tropical stratosphere as a 'tape recorder', from Ref. 28); q.v. for a careful discussion
of the data analysis and interpretation. The time-altitude plot is from HALOE, an occultation
limb sounder on the Upper Atmosphere Research Satellite (UARS), showing the signal from
the annual cycle in water vapor being carried up with the large-scale mean upward velocities
;G 0.2 mm s- 1 . Soundings between l4°N and 14°S were used. The quantity plotted is water
vapor plus twice methane in ppmv (parts per million by volume), a notional mixing ratio
proportional to the ratio of hydrogen atoms to air molecules. Because of methane oxidation
chemistry this is to excellent approximation a material invariant, a passive tracer, in the altitude
range shown. It is strongly modulated by the water-vapor signal of the annual cycle at the
tropopause, 27 ) which thus acts as the 'recording head' of the notional tape recorder. The raw
data have been fitted to a suitable set of empirical orthogonal functions to extract the annual
cycle in an objective way; but the tape-recorder signal is still easily visible without any such
processing ··see Plate 1 of Ref. 28) ~and in data from other instruments, notably the (passive)
Microwave Limb Sounder on UARS. 29 )
The Earth's Atmosphere and the Sun's Interior
145
The orders of magnitude in this picture turn out to require upward velocities at
20 km in the tropical lower stratosphere, at bottom center of Fig. 2, of the order of
27 13
0.2 mm s- 1 or 6 km per year, or a little more, depending on the time of year. ), )
This sort of velocity is far too small to observe directly- or so it was thought until
two or three years ago. Figure 3, from Ref. 28), comes about as close as we are
likely to get to a direct observation, giving a powerful independent check on the
whole picture. It was derived from satellite data obtained recently, long after results
like Fig. 2 had been obtained with the help of an earlier generation of satellites.
Figure 3 shows the water-vapor signature of the annual cycle being carried up in
the rising flow just like a signal recorded on a moving magnetic tape. At altitudes
around 20 km, the velocity apparent in the picture is about 7 km or one scale height
per year; for a careful estimate and discussion, verifying that upward advection is
indeed the main effect near 20 km, Ref. 28) may be consulted.
But what, then, is driving this global-scale circulation? As with ordinary mechanical refrigerators or compressors, something has to pump the air around. It will
not move across the stable stratification by itself. To make noctilucent clouds you
have to pull air upward, and to warm the winter polar stratosphere you have to push
air downward. So what is doing the pulling and pushing? It is here that the dynamical organization of fluctuations comes in. The key is the irreversible transport of
angular momentum by certain wave motions generated in one place and dissipated
in another, the dissipation mechanisms including various kinds of wave breaking.
§4.
Wave-induced momentum transport
It is well known of course that wave motion of small amplitude O(a) tends to
transport momentum at rate O(a 2 ), photons in a vacuum being the simplest case,
where we can think of them just like bullets having a certain momentum and carrying
that momentum from one place to another. Waves in fluid media are often described
in these terms, but it is important to understand that the picture is not literally
true any more; the analogy with photons in a vacuum is only an analogy. This is
because the material medium is present and can, and does, transport momentum
2
via O(a 2 ) mean stresses. Careful analyses of problems of this kind correct to O(a ),
with careful attention to boundary conditions as well as to the nonlinearities in the
hydrodynamic equations, show that the notion of a wave packet possessing a definite
momentum no longer makes sense except in some very restricted circumstances.
But the notion of a wave-induced momentum flux, or angular momentum flux, does
generally make sense, and furthermore is the relevant quantity for our purpose. To
make the literature on atmospheric waves make sense, you sometimes have to insert
the word 'flux' or 'transport' after the word 'momentum'.*)
•) Essentially this point was made long ago by Leon Brillouin, and, following him, I have discussed it at some length in earlier publications; see Refs. 30), 31), and their bibliographies, tracing
the problem back to the 'Abraham-Minkowski controversy' about light waves in refractive media
and to a rare mistake by Lord Rayleigh. There is undoubtedly a source of confusion insofar as two
different conservable quantities enter the theory, momentum on the one hand and quasimomentum
or pseudomomentum, often called 'wave momentum', on the other. They each have.associated trans-
146
M. E. Mcintyre
Fig. 4. Simple experiments with water waves, illustrating the wave-induced momentum transport
- here manifested by the appearance of a strong mean flow driven by it - that results from
the generation of waves in one place and their dissipation in another (see text, § 4). The mean
flow can be made visible by sprinkling a little powder such as chalk dust on the surface of
the water. I have often done experiment (a) as a lecture demonstration in a glass tray on an
overhead projector, using a cylinder about 10 em long and 4 em in diameter oscillated vertically,
and experiment (b) using a curved wave maker that is rather bigger, about 60 em in radius and
arc length. Good results are easily obtained with capillary-gravity waves having frequencies
"'5Hz for (a) and "'3Hz for (b). From Ref. 32).
In practical terms you can demonstrate the essential effect for yourself, with
no special apparatus. Capillary-gravity waves on a water surface have typical frequencies of a few hertz, and can easily be generated by oscillating an obstacle or by
jiggling your hand in the water. Figure 4(a) shows the version demonstrated at the
symposium, in which a small cylinder is oscillated vertically and radiates capillarygravity waves anisotropically to either side. If chalk dust or other floating powder
is sprinkled on the surface, where most of the fluid motion is concentrated, one sees
that the water flows away from the wavemaker in a persistent and conspicuous mean
motion that sweeps the surface dust along with it, easily visible through an overhead
projector. The experiment is robust. It always works well, and I have often shown
it in lectures.
The observed flow depends on strong dissipation of the waves. This results in
an irreversible flux or transport of momentum from the wave maker to the fluid on
either side of it, where the waves are dissipating. The wave dissipation is greatly
enhanced by the dirty water surface, which gives rise to a strong viscous boundary
layer just under the surface, which may be turbulent. The fact that irreversible
momentum transport is involved can be seen by stopping the wavemaker, and observing that the mean flow persists much longer than does the outgoing wave field.
There is a significant contribution to the mean flow that is not a mere Stokes drift
or other reversible mean-flow contribution, dependent on the continued presence of
lational symmetries, and so tend to look similar within the mathematical formalisms of Hamiltonian
and Lagrangian dynamics, a simple example being Hamilton's equations for ray tracing. But the
two translational symmetries are physically different, in the present context of waves in a material
medium. One is translational invariance of the physics, giving rise to conservation of momentum.
The other is translational invariance, i.e., homogeneity, of the medium, giving rise to conservation
of quasimomentum.
The Earth's Atmosphere and the Sun's Interior
147
the outgoing waves (and on the choice of averaging operator) and returning to zero
32 33
as soon as the waves propagate out of the region of interest. ), )
The fact that the flow is due to wave-induced momentum transport and not to
the oscillatory boundary layer on the wavemaker itself is also clear in the experiment - if one uses vertical oscillations of the wavemaker. The Rayleigh-Schlichting
boundary-laye r streaming from the surface of the wavemaker is then toward the
wavemaker, i.e., in a sense opposite to that observed. Moreover, with a larger tank,
one can do versions of the experiment in which the waves are made by a larger, concave wavemaker that focuses them at a more distant spot, as suggested in Fig. 4(b).
Sure enough, the momentum is transported, then, over the greater distance. The
strongest mean flow is seen where the waves are focused.
A spectacular case of wave-induced momentum transport over long distances was
demonstrated by Walter Munk and colleagues 34 ) in a famous observational study of
surface gravity waves propagating from the Southern Ocean across the entire Pacific,
all the way to beaches on Alaska, where the waves generate longshore mean currents
in fundamentally the same way as in my lecture demonstration . In other words,
forces exerted on the Southern Ocean, by winds in the latitudes of the 'roaring forties'
and 'fifties', were shown to drive mean currents on the coast of Alaska. Even more
than Fig. 4(b) this underlines a crucial point of principle, namely that momentum
transport by wave propagation can take place over vastly greater distances than the
mixing lengths associated with turbulent eddies.*)
§5. The quasi-bienni al oscillation (QBO)
Fundamentall y similar effects occur with, for example, internal gravity waves
in a stably stratified fluid. Figure 5 shows the celebrated Plumb-McEw an experiment, 35 ) in which the effects manifest themselves in an especially interesting form
that I think must surprise anyone encountering them for the first time. A flexible
membrane is oscillated in a standing wave, at the bottom of an annular container
filled with a stably stratified salt solution. The imposed conditions are close to
mirror-symme tric. There is no externally-imp osed difference between the clockwise
and counterclockwise directions around the annulus; in this experiment, Coriolis effects are negligible. When the bottom boundary is oscillated, with a suitable period
> 2rr / N, the buoyancy period of the stratification, internal gravity waves are generated. They can be regarded as a superposition of two progressive modes, clockwise
and counterclockwise. They propagate upward and dissipate mainly by viscosity,
salt diffusivity being negligible for this purpose.
Naive symmetry arguments suggest that nothing interesting will happen; but,
when the experiment is run with sufficient wave amplitude, the mirror symmetry is
broken spontaneously and a conspicuous mean flow around the annulus is generated.
This goes robustly into a quasi-periodic sequence of states, in which the mean flow
keeps reversing and the long time average state is approximately mirror-symme tric.
Sufficient amplitude means sufficient to overcome viscous drag on the mean flow.
•) For a careful discussion of why the photon analogy is still only an analogy, see Ref. 31).
148
M. E. Mcintyre
Statically
stable
Densitystratified
fluid
The camera
looks into
material
the bottom
circuit r
half of the
(consider
annulus
fru. ds)
Fig. 5. Schematic of the Plumb-McEwan experiment, 35 ) showing how a reversing mean flow, first
clockwise then counterclockwise around the annulus, can be generated by nothing more than
standing oscillations of a flexible membrane at the bottom of an annular container filled with
a stably stratified salt solution. The standing oscillations generate internal gravity waves at
periods around 20 s > 27r / N, the buoyancy period of the stable stratification. The camera
position shown corresponds to the speeded-up video of the experiment shown at the Symposium.
The annular gap width in the original apparatus was 12 em, which is smaller than depicted here,
relative to the outer radius, 30 em. Also, the original apparatus had twice the number of flexible
segments, giving 8 full wavelengths around the annulus. With gap widths significantly less than
12 em, and water as the working fluid, the experiment fails because of viscous drag on the mean
flow.
The mechanism operating in the experiment is simple and well understood. 35 ), 23 )
It involves irreversible wave-induced transport of angular momentum, dependent on
the viscous dissipation of the waves and on wave refraction and Doppler shifting by
the resulting shear flow, operating in a feedback loop. The Doppler shifting leads to
enhanced dissipation, reduced vertical group velocity, and reduced vertical penetration of the clockwise and counterclockwise progressive wave components alternately.
This gives rise to a characteristic spacetime signature in which features such as zero
crossings in the mean velocity profile propagate inexorably downward.
Just such a spacetime signature, shown in Fig. 6, is also observed in the real
atmosphere. This is the famous quasi-biennial oscillation (QBO) of the zonal or
east-west winds in the equatorial lower to middle stratosphere. The overwhelming
weight of evidence, from observation and modeling, says that this flow in the real
atmosphere is, indeed, a wave-driven mean flow, like the laboratory experiment, 36 )
even though there are still, even today, large uncertainties as to exactly which wave
types are the most important. 37 )- 39 ), 36 ) As it happens, the whole oscillation, with
features descending at about a scale height per year, takes place in air rising at about
the same rate, as illustrated in Fig. 3. The wave-induced angular momentum flux
therefore has to be about twice what would otherwise be estimated. The important
The Earth's Atmosphere and the Sun's Interior
149
~~~~~~~~~=~~~~~~~~
~~~--+----+---J""'
-~~~~~~~r=~~~~~~~
~~-+-~~
Fig. 6. Time-height cross-section showing the QBO (quasi-biennial oscillation) of the east-west
wind in the real equatorial stratosphere between about 15 km and 30 km log-pressure altitude.
This is a reversing mean flow fundamentally similar to that in the Plumb-McEwan experiment
shown in Fig. 5. From radiosonde data at Canton Island, Can/Maldives, and Singapore (since
1976), by kind courtesy of Dr. Barbara :'-Jaujokat of the Free University of Berlin. Contour values
are in ms- 1 , with shaded regions marked 'W' representing 'westerly' flow, meaning eastward
flow (by a perverse but ineradicable meteorological convention). Unshaded regions marked 'E'
similarly represent 'easterly', meaning westward, flow.
150
M. E. Mcintyre
wave types include certain vertically propagating waves trapped in an equatorial
waveguide 40 ), 41 ), 23 ) by the 'potential well' in the Coriolis parameter
f
= 2101 sin(latitude) ,
(5·1)
which is the vertical component of 20 picked out by the atmosphere's stable stratification, where 0 is the Earth's angular velocity. Prominent among these equatorially
guided waves is a so-called Kelvin wave, which looks like an internal gravity wave
when viewed in the equatorial plane and whose disturbance fields have Gaussian
profiles in latitude (the latitudinal scale being frequency-dependent but typically
"'±10°).
Two further differences between the laboratory experiment and the real atmosphere can be noted. First, rather than simple viscous dissipation, the dominant
wave dissipation mechanisms are relaxational thermal damping, by radiative heat
transfer, and various kinds of wave breaking. 42 ) Second, none of the candidate-waves
have well-understood sources. About all one can safely say is that the wave sources
seem to involve highly nonlinear, chaotic fluid dynamics in the troposphere on a
vast range of scales. The source mechanisms may even include rotating, stratified
counterparts of 'Lighthill radiation' or aeroacoustic radiation from chaotic vortex
motion. 43 ) Along with detailed numerical studies of wave breaking, too numerous to
cite here, this is very much at the tough edge of the research frontier.
Despite all this, the QBO itself is still, by its mere persistence, the most predictable atmospheric phenomenon on timescales greater than a few days, discounting
the annual cycle. (The recent improvements in El Nino predictability 5 ) have not, to
my knowledge, yet claimed great skill for timespans much beyond 6 months.) The
QBO is indeed a remarkable illustration of order emerging out of chaos. To get a
robust QBO-type mean flow oscillation, the theoretical models suggest 44 ) that we
need only have a wave field that, in some long-term average sense, has sufficient
amplitude and a degree of long-term stability in its statistics, and a sufficient spread
of east-west phase speeds, tens of meters per second each way, spanning the range
of mean wind speeds seen in Fig. 6.
The theoretical modeling of the feedback loop or 'wave-mean interaction' giving
rise to the QBO is usually based on averaging around latitude circles. One may then
picture the dynamics in terms of a zonally symmetric fluid motion in which notional
rings of fluid, aligned with latitude circles, are pushed eastward or westward by the
wave-induced zonal forces. The forces, in turn, are due to wave-induced fluctuations
about the zonally symmetric mean state, with characteristic correlations. Here I am
glossing over some important distinctions between the Eulerian, Lagrangian, and
other kinds of averaging operator that can be used in these models. One can think
of the wave-induced forces as something like Reynolds stress divergences- indeed,
this is technically correct for internal gravity waves with no Coriolis effects, as in the
Plumb-McEwan experiment- but when the Coriolis parameter f is not negligible
the technicalities become more complicated. 45 ), 2 3), 46), 33), 53)
The QBO is a very clear example of what used to be called 'negative eddy
viscosity'. 47 ) The wave-induced momentum fluxes are against the mean angularvelocity gradient in substantial parts of the flow. Were it otherwise, the oscillating
The Earth's Atmosphere and the Sun's Interior
151
mean flow could not arise spontaneously. At each instant, moreover, the mean
angular-velocity gradient has zeros at certain altitudes where the momentum flux is
nonzero, and so we can have 'infinite eddy viscosity' as well. Of course to speak of a
negative or infinite diffusivity of any kind- a hopelessly, so to speak, ill-conditioned
idea - is no more than an epigrammatic way of signalling that one is dealing with
interesting phenomena that are not at all understood. 48 ) The 'negative viscosity'
idea was part of early pioneering and is not used these days. Recognition that
wave propagation has a key role makes it clear that the observed momentum fluxes
associated with fluctuations about a mean are often controlled in a very nonlocal
way, as with storms in the Southern Ocean and beaches in Alaska - quite unlike,
for instance, gradient-related momentum fluxes in gas kinetic theory or in mixinglength models of turbulence. A better epigram for the Plumb-McEwan experiment
and its real-atmospheric counterpart, the QBO, might be simply to say that waveinduced angular momentum transport tends to drive a mean flow away from, not
toward, solid rotation.
§6.
Gyroscopic pumping
With wave-induced angular momentum transport so conspicuously important
near the equator, as just illustrated, it is perhaps no surprise that it has a role at
higher latitudes as well - though just how central a role was not recognized until
13
recently, outside a small community of specialists in atmospheric dynamics ) who
gradually developed the insights involved. Those developments can be traced back
to the 1969 work of R. E. Dickinson. 49 )
The atmosphere is full of waves of various kinds, with significant amplitudes at
most latitudes, as observations repeatedly confirm. But, away from the equator, the
response to wave-induced forces, or to any other forces, depends on Coriolis effects.
These are significant, indeed immensely strong for present purposes, as hinted earlier.
The equator is exceptional because everything is dominated there by the still stronger
stable stratification as reflected (a) in the sine-latitude dependence of the Coriolis
2
parameter f defined in Eq. (5·1) above, and (b) in the large values of N/j20j,....., 10 ,
with N the buoyancy frequency as before.
The Coriolis effects give rise, therefore, to a different kind of response to waveinduced zonal forces away from the equator. It is sometimes called 'gyroscopic
pumping'. Because of the strong Coriolis effects, gyroscopic pumping is a robust
mechanism outside the tropics, under typical parameter conditions - further from
the equator than, say, about 20° latitude. If a ring of fluid feels a persistent westward
push, for instance, then Coriolis effects tend to drive it persistently poleward. This
is the essence of what is happening at most altitudes and latitudes in Fig. 2. The
main exception is at the top left, above the frame of the figure, near the summer
50
mesopause, where the push is eastward and the air is driven equatorward. )- 52 )
This pulls air upward in the polar cap, producing the refrigeration already noted.
To check that these ideas make sense and to gain more insight, one can again
make use of theoretical models based on averaging around latitude circles, with
notional rings of fluid being pushed eastward or westward by the wave-induced forces.
152
M. E. Mcintyre
Again there are nontrivial technicalities regarding choices of averaging operator -especially if one wants 'mean circulation' to have the Lagrangian significance for
chemical transport needed in §3 - which I shall again gloss over. The interested
reader may consult Refs. 53), 46), 24) and 25) on this point. The essence of the
matter is, however, very clear. In a thought-experiment in which one switches on a
given, zonally symmetric westward force in some region occupying a finite altitude
and latitude range away from the equator, and asks how the model responds, 54 ) one
sees an initial transient after which the motion settles down to a steady poleward flow
whose mass streamlines close downward. 54 ) The response looks like the lower half of
Fig. 2, if the westward force is applied wherever the streamlines indicate poleward
flow. The Coriolis force of the poleward flow comes into balance with the applied
westward force as the steady state is approached. The streamline pattern burrows
downward until it reaches the Earth's surface, where it closes off in a frictional
boundary layer.
The approach to the steady state and the downward closure of the streamlines
is related to the finiteness of the pressure and density scale heights, i.e., to the
exponential falloff of pressure and density with altitude. Again glossing over some
technicalities, 54 ) we have a situation in which the finite mass of atmosphere above
the forcing region is incapable of absorbing infinite amounts of angular momentum.
The system solves this problem, so to speak, by sending all the angular momentum
downward, and ultimately into the solid Earth, whose moment of inertia is regarded
as infinite for this purpose. That the steady state is, indeed, robustly approached, if
the force is applied far enough from the equator - weaker forces can be applied closer
- has been well demonstrated both in analytical-mathematical and in numerical
versions of the thought-experiment. Here I am talking about realistic values of the
force, for which 'far enough' means further than the abovementioned 20° latitude or
so. In such cases the response problem is linear to good approximation, and stable,
hence the robustness. For a more thorough analysis and discussion, the reader may
consult Refs. 54) and 55). The timescales of the initial transient are proportional
to thermal relaxation times and depend also on spatial scales. The adjustment is
quickest and most robust at the largest, or near-global scales, on which the finiteness
of scale heights is most strongly felt.
If one gets too close to the equator, then the response problem for a steadily
applied force (as distinct from the oscillating force involved in the QBO 55 )) becomes
nonlinear, and robustness may be lost. This is at the research frontier and is not
yet well understood, though progress is being made (P. H. Haynes, R. A. Plumb,
D. Sankey, R. K. Scott, personal communication; also Ref. 56)). There are even
some possibilities for hysteretical behavior (multiple equilibria, regime-flipping) in
the path taken by the rising, tropical branch of the circulation. But mass continuity
still says, of course, that the circulation must close somehow, as illustrated by the
rising branch of the circulation in Fig. 2.
What is known about the wave types doing the pumping in the real extratropical
middle atmosphere? Here we think we know more than in the case of the tropics
and the QBO. Below 50 km or so, extratropical pumping is believed to be produced
mainly by so-called Rossby waves, to be discussed in the next section. Rossby waves
The Earth's Atmosphere and the Sun's Interior
153
have much larger spatial scales than gravity waves, and are relatively well defined
observationally. They have ratchet-like properties that tend to produce persistently
westward, or retrograde, mean forces whenever the waves are dissipated. Observational studies show that these forces are of the right order to drive the poleward
flows in the lower part of Fig. 2, 13 ) though internal gravity waves may make a secondary contribution, especially in the summer hemisphere. 57 ) Above 50 km or so,
by contrast, the most important wave type is internal gravity. 51 ), 5 2 ) Being classical
oscillations with negligible Coriolis effects (radian wave frequencies » f), internal
gravity waves show time-reversal symmetry and can produce mean forces that are
either prograde or retrograde, depending on circumstances, just as they do in the
Plumb-McEwan experiment. Observations of internal gravity waves in the mesosphere, mostly by radar and lidar techniques, supplemented by a few rocket soundings, show that they are well able to drive the circulation suggested by the heavy
dashed curve, including the pumping required to make noctilucent clouds. 58 ) These
waves are observed to dissipate mainly by breaking, due to convective overturning of
heavy air over light, and clearly seen in the mixing of certain chemical tracers such
as atomic oxygen. 59 )
§7.
Rossby waves and Rossby-wave critical layer theory
Rossby waves have timescales of days and are tightly constrained by the atmosphere's strong stable stratification, giving them a quasi-horizontal or 'layerwise
2-dimensional' character. The essential dynamics is captured in an extremely simple
and elegant dynamical model, of which our understanding is rather complete. This
is the variant of 2-dimensional incompressible vortex dynamics known as Rossby's
'beta plane' or 'approximately flat Earth' model. It is worth indicating briefly how
this model illustrates the two properties of Ross by waves, propagation and breaking,
that are most essential to their ratchet-like pumping action.
As with ordinary incompressible vortex dynamics, all the dynamical information resides in a single scalar field, the 2-dimensional absolute vorticity Q(x, y, t),
from which a stream function 1/J(x, y, t) describing the incompressible velocity field
u(x, y, t) can be derived by inverting a Poisson equation, if we are given suitable
boundary conditions such as evanescence at infinity:
(7·1)
Here
f
is again the Coriolis parameter (5·1), and
u = (u, v) ,
u ==
-a'lj;jay'
v = a'lj;jax,
(7·2)
where (x, y) are eastward and northward Cartesian coordinates and (u, v) the corresponding components of the velocity vector u(x, y, t). There is just one evolution
equation
(7·3)
DQIDt = 0,
where D I Dt is the 2-dimensional material derivative, defined by
D 1Dt =
a1at + u . v
=
a1at + ua 1ax + va 1oy .
(7·4)
154
M. E. Mcintyre
If we wish, we can model dissipation by adding suitable 'frictional' terms to the
right of Eq. (7·3). Note incidentally that in this system we have non-locality, i.e.,
instantaneous action-at-a-distance, because acoustic degrees of freedom have been
wholly suppressed and the sound speed is infinite.
The Rossby-wave critical layer theory is the simplest example in which one has
both wavelike and, in a certain sense that is relevant, quasi-turbulent motion or, more precisely, in a sense appropriate to Rossby waves, wave breaking. 42 ) The
wavelike part owes its existence to a background gradient of absolute vorticity, which
corresponds in the real case to the latitudinal gradient of the Coriolis parameter
f, which is picked out because, as already mentioned, it is the only component
whose effects are not overwhelmed by buoyancy forces associated with strong stable
stratification N :» f. So in the 2-dimensional model we have a background gradient
(3 = df jdy of absolute vorticity where y corresponds to latitude.
Figure 7 shows the characteristic pattern of a Rossby-wave disturbance, in which
material contours, which are also contours of constant absolute vorticity, are undulated in the ±y-direction, corresponding to northward or southward on the real
Earth, implying a pattern of vorticity anomalies from which inversion, Eq. (7·1),
yields a velocity field qualitatively as shown by the solid arrows. 14 ) If one now makes
a mental movie of the resulting motion, one sees that the undulations will propagate
westward because the velocity field is half a wavelength out of phase with the displacement field. This is the Rossby-wave propagation mechanism, sometimes called
'quasi-elasticity' to emphasize that it is a restoring effect, tending to return parcels
----------------
y
L~x
(±}
~
___v.~~ Q=constant
---------------1
Fig. 7. The restoring mechanism or chiral 'quasi-elasticity' to which Rossby waves owe their existence, diagram taken from Ref. 14). The mechanism is fundamental to all large-scale dynamical
processes in the atmosphere and oceans. The contours are isopleths of the relevant material
invariant, the absolute vorticity or the PV (potential vorticity) as appropriate. Because of
material invariance, the contours are also material contours. The + and - signs respectively
indicate the centers of the cyclonic and anticyclonic vorticity or PV anomalies due to horizontal
air parcel displacements, across a basic positive isentropic y-gradient of PV, i.e., gradient on
stratification surfaces. Anomalies mean whatever is relevant to PV inversion, in the simplest
case the right-hand side of Eq. (7·1). The heavy, dashed arrows indicate the sense and relative
phase of the induced velocity field (see text), i.e., the field obtained from inversion. This advects
the contours in just such a way as to causes leftward propagation of the phase of the pattern
~ more generally, propagation with high PV on the right ~ with individual parcels always
accelerating against their displacements, hence the term 'quasi-elasticity'.
The Earth's Atmosphere and the Sun's Interior
155
to their equilibrium latitudes.
The standard Rossby wave theory - linearize Eq. (7·3) about rest, take f3 =
df jdy = const., look for solutions ex exp(ikx+ily- iwt) so that '\7-2 = -(k2 + z2)-1,
and deduce w = -f3k/(k 2 + l 2 ) ~ presumes that the sideways displacements and
slopes of the material contours are small and that the undulations are reversible.
We say that the Rossby waves 'break' when this condition of reversible undulation
is violated, and material contours deform irreversibly; the relevant arrow-of-time
'paradox' is discussed in Ref. 60). This is fundamental to the irreversibility of the
wave-induced angular momentum transport for a simple reason, namely the validity
of Kelvin's circulation theorem ~ which governs the absolute circulation integral
along just such material contours~ in the frictionless dynamical system (7·1) and
(7·3).
Now if the Ross by wave undulations take place in a shear flow (u(y), 0), and have
a longitudinal (x) phase speed that coincides with the background flow velocity at
some value of y ~this location is often called a 'critical line'~ then there is always
some surrounding region in which Rossby-wave breaking, in the foregoing sense, is
important. This region is sometimes called a 'Rossby-wave surf zone' to remind us of
its significance for irreversible momentum and angular momentum transport, which
is fundamentally the same as for the ordinary surf zone of an ocean beach, despite
a very different geometry and wave-breaking dynamics. In the case of Rossby-wave
surf zones, they can be broad or narrow according as the wave amplitude is large or
small.
Figure 8 shows the simplest known illustrative example, due to Stewartson, Warn
and Warn, hereafter 'SWW', an elegant analytic solution of the foregoing equations
describing a simple case of Ross by wave breaking. A small-amplitude Ross by wave is
excited by an undulating boundary (outside the domain of the figure) and encounters
a critical line in basic flow having constant shear. A narrow surf zone forms in which
material contours deform irreversibly in a recirculating flow, a so-called Kelvin's
cats-eye pattern, straddling the critical line. As the material contours wrap around,
the vorticity distribution changes and, through the action-at-a-distance implied by
the inversion, the '\7- 2 operator in Eq. (7·1), induces a change in the flow outside
the critical layer. This affects phase gradients with respect to y in the outer flow in
such a way that the surf zone appears as an absorber to the Rossby waves outside
it during the early stages of wave breaking, but later becomes a reflector owing to
the finite scope for vorticity rearrangement within the surf zone or critical layer.
This possibility is another important difference vis-a-vis ocean beaches. The way in
which the surf zone interacts with its surroundings is precisely expressed, in the SWW
solution, by the use of matched asympotic expansions. Full details of the solution
may be found in Ref. 61), which also establishes a general theorem showing that the
absorption-reflection behavior just described is generic, and not dependent on the
particularly simple circumstances that permit the problem to be solved analytically.
If there is some forcing effect that tends restore the background vorticity gradient
(which requires some nonzero term on the right of Eq. (7·3)), then absorption can
be continuous. Wave absorption can be quantified by a suitable conservable measure
of wave activity and its flux, most conveniently one of the Eulerian forms of the
M. E. Mcintyre
156
(a)
y
-5
r----
5F===~~===~====~~==~
(b)
y
-5~~~~~~~~~~~==~
5~~~~~9
(c)
y
(d)
y
-lt
1t
Fig. 8. Material (and absolute-vorticit y) contours in the analytically simplest model of a breaking
Rossby wave, the Stewartson-War n-Warn 'critical-layer' solution, see text. Plot taken from Ref.
61).
The Earth's Atmosphere and the Sun's Interior
157
pseudomomentum, which is used in proving the theorem just mentioned. (It is also
related to Arnol'd's hydrodynamic stability theorems, but that is another story.)
Such measures are useful in making precise the qualitative statements about how
the wave field and surf zone interact.
A feature that is robust, and appears to carry over to larger amplitude cases, is
that the flow is highly inhomogeneous, being more wavelike in some places (the outer
flow) and more 'turbulent' in others (the surf zone) - meaning that the material
contours deform irreversibly- with each region affecting the other. More realistic
cases have larger wave amplitude but typically show the same generic features, the
main difference being that the flow in the surf zone is more, intuitively speaking,
'turbulent', i.e., more chaotic-looking. A video shown at the conference illustrated
such a large amplitude case.
Figure 9 summarizes the relation between vorticity rearrangement and angular
momentum transport in the simple dynamical system (7·3) and (7·1). The graph
second from the left shows the background absolute vorticity profile Q(y) superposed
on the result of rearranging it in such as way as to homogenize it over some finite
y-interval. This mimics in idealized form what happens in the surf zone. The middle
graph is the x-averaged Q(y) profile from an actual solution, in another Ross by wave
critical layer case that is less simple ·~· the detailed flow is more chaotic, because
of the onset of secondary instabilities ~· but shows generically the same features
(P. H. Haynes, Ref. 62) and personal communication). To get the corresponding
zonally averaged momentum cha,nges, the inversion is trivial: one merely has to
integrate the vorticity change with respect to y, giving the right hand graph. The
momentum change corresponding to the more idealized, left-hand Q(y) profile is a
simple parabolic shape (not shown) qualitatively similar to the right-hand graph.
The essential point, then, is that whenever vorticity is thus rearranged, a mo-
(c)
(ll)
oo(y)
Q(y)
----~-4''------
//,
-5
Fig. 9. The relation between mean wave-induced force and PV (potential-vorticity) rearrangement
due to a breaking Rossby wave, in the simplest relevant model system, the dynamical system
(7·3) and (7·1). Courtesy P. H. Haynes; for mathematical details see Ref. 62). Plot (a) shows
idealized PV distributions before and after mixing the PV in some y-interval or latitude band;
(b) shows the x-averaged PV, or rather absolute vorticity in this case, in an actual model
simulation using Eqs. (7·3) and (7·1); (c) shows the resulting momentum deficit, whose profile
would take on a simple parabolic shape in the idealized case corresponding to (a). See Ref. 61)
for a more detailed discussion, including mathematically rigorous bounds on the momentum
transport into or out of the wave-breaking region or 'surf zone'.
158
M. E. Mcintyre
mentum deficit appears, signalling the presence of an effective retrograde force. This
is the ratchet-like character of the process to which I referred earlier: the momentum change where Rossby waves dissipate by breaking is always retrograde, i.e.,
against the background rotation, westward in the case of the Earth. For the idealized schematic vorticity rearrangement shown in the panel (a) at the left of the
figure, the corresponding velocity-deficit profile is a simple parabola.
§8. Generalization to realistic stratified flow
The whole picture just summarized generalizes to layerwise 2-dimensional motion in a strongly stratified atmosphere provided that we reinterpret Q as the RossbyErtel potential vorticity, hereafter 'PV', and replace yr- 2 in Eq. (7·1) by a more
complicated inversion operator. 14 ), 7 ) Qualitatively, everything is much the same as
before: Rossby quasi-elasticity still works as depicted in Fig. 7, and the rearrangement of PV on stratification surfaces is still robustly associated with an angular
momentum deficit. 63 )
The standard definition of the PV in three dimensions can be given in terms of
the mass density p and potential temperature 0, the latter proportional to specific
entropy and defined as actual temperature after adiabatic compression to sea-level
pressure PO· The potential temperature() is thus, by definition, a material invariant
when the motion is adiabatic, as is approximately correct for timescales « thermal
radiative relaxation times. The standard definition is
(8·1)
This Q is a material invariant for frictionless, adiabatic motion (Ertel's theorem).
Such invariance is qualitatively valid, and qualitatively important, for Rossby-wave
undulation and breaking in the real stratosphere.
The video shown at the conference illustrated a large amplitude case very similar
to the real wintertime stratosphere at altitudes of 25 km or thereabouts. Conspicuously evident was the elastic behavior of the region of crowded PV contours at the
poleward edge of the surf zone, which latter is spread over middle latitudes, over a
zone a few tens of degrees of latitude in breadth. There is a similar but less tight
crowding of potential vorticity contours at the sub-tropical edge of this surf zone;
and one of the results is a partial chemical isolation of the tropical region (Ref. 28)
and references therein). This is why the 'tape recorder' signal in Fig. 3 is able to
persist for so long in the rising tropical stratospheric air. The polar region, the socalled polar stratospheric vortex region, is even better isolated chemically, though
only during winter. This near-isolation is significant for ozone-hole chemistry. 10 )
A basic point to note about this kind of situation - the situation idealized in
Figs. 8 and 9 but including the generalization to stratified flow, large wave amplitude,
and broad surf zones - is the tendency for the basic inhomogeneity to be selfreinforcing, the tendency toward more wavelike versus more turbulent motion in
different zones or regions. Where Q contours are crowded together, typically at the
edges of surf zones - idealized as the 'corners' in the (Q) profile in Fig. 9(a) one has a strengthening of the Rossby quasi-elasticity, and conversely a weakening
The Earth's Atmosphere and the Sun's Interior
159
within surf zones. Thus mixing becomes easier in surf zones once they begin to
be established, and vice versa. Shear effects tend to reinforce this. 64 ) Other things
being equal, then, the inhomogeneity tends to perpetuate itself. The structure can
of course be changed by sufficiently drastic changes in circumstance, such that the
introduction of strong Rossby-wave motion with a different phase speed from the
original one, tending to carve out a surf zone near a different y value. 65 )
But the result is still highly inhomogeneous. As I have argued more carefully
elsewhere, 36 ) this is almost inevitable. The reason is the fundamental integral constraint on, more aptly property of, PV rearrangement on a (topologically spherical)
stratification surface S enclosing the Earth:
(8·2)
where dAis the area element and bdAd(} is by definition the mass element. That
is, b = pfl'\701. This is just a thinly disguised form of Stokes' theorem on the closed
stratification surface S. It tells us that the only way to mix Q to homogeneity on
the surfaceS is to have zero Q everywhere, a highly improbable state on a planet as
rapidly rotating as the Earth. The upshot is, again -- though for somewhat different
reasons- a marked tendency of the associated wave-induced angular momentum
transport to drive the system away from, not toward, solid rotation.
§9.
The Sun's interior
When put together with new helioseismic data, our understanding of stratospheric circulations has just recently enabled us to learn something new and significant about the Sun's deep interior. Here I mean the radiative interior, at radii
< 5 x 105 km, as distinct from the overlying convection zone, whose upper surface
is the visible surface at about 7 x 105 km. We can now answer in the affirmative,
with high confidence, an age-old question: does a dynamically significant magnetic
field pervade the interior? The mere existence of such a field has immediate and
far-reaching consequences, as will be explained, for our ability to sharpen theoretical
hypotheses about other aspects of the Sun's interior and to make inferences from
helioseismic inversion.
Of course the possibility that there could be such an interior field has long been
recognized, if only because magnetic diffusion times ('"" 10 10 y) are comparable to, or
somewhat longer than, the age of the Sun, 66 ) the latter currently estimated as close to
4 x 109 y. Thus the interior needs no dynamo action: it could contain a magnetic field
left over from what might be called the Sun's formative years. Indeed, when the Sun
began to form as a protostar, the interstellar magnetic field must have been greatly
concentrated as material collapsed inward (e.g., Ref. 67) and references therein),
dragging the field with it and spinning up to high rotation rates. On the other
hand it is also possible, and has often been hypothesized, that any such interior field
was expelled or annihilated long ago · perhaps by some kind of turbulent motion
within the early Sun - leaving only the rapidly oscillating field associated with the
convection zone and its 22-year cycle, for which dynamo action is plausible. Models
160
M. E. Mcintyre
of solar spindown- the Sun's evolution away from its early, rapidly-rotating state,
exporting angular momentum through the solar wind - have tended, therefore, to
assume that the interior was free of dynamically signficant interior magnetic fields
for most of the relevant timespan of 4 x 109 y; see for instance Ref. 68) and references
therein.
The argument now to be sketched - · for more detail, see Refs. 36) and 69)
- depends crucially on recent helioseismic results concerning the Sun's differential
rotation. 70 ) These strongly indicate, for one thing, that classical laminar spindown
models will not fit observations. The convection zone is in quite strong differential rotation, as suggested in Fig. 10 and its caption. But the radiative interior, by
contrast, turns out to be approximately in solid body rotation except possibly near
the poles. (The helioseismic technique, which depends on observing the rotational
splitting of acoustic vibrational modes of the Sun, is insensitive to details near the
rotation axis.) Now this near-solid interior rotation, of which the first indications
began to emerge about a decade ago, is incompatible not only with classical laminar spindown but also with practically any other purely fluid-dynamical picture.
How can anyone claim such a thing? The basis is our observational and theoretical
knowledge of terrestrial middle-atmospheric dynamics.
The Sun's interior is strongly stratified, and, for present purposes, is a fluiddynamical system very like the terrestrial stratosphere, with short buoyancy and
Coriolis timescales. Even the ratio of those timescales is similar: throughout most
of the interior, N /21012:: 102 . So, in a laminar spindown model for instance, the
dynamics is qualitatively like that in a zonally symmetric terrestrial model stratosphere. The main difference is one of detail: because photon mean free paths are
small, radiative heat transfer is diffusive, and also rather weak for fluid-dynamical
purposes (diffusivities rv 103 m 2 s-1 ).
Now if laminar spindown were the only thing happening, it would drive the interior away from solid rotation toward some differentially rotating state that matches
the differential rotation of the convection zone. If, on the other hand, layerwise
2-dimensional turbulence were present in the interior, perhaps through the breaking
of Rossby waves emitted from the convection zone, then the effect would be to drive
the interior away from solid rotation in, probably, some other way, 36 ) as actually
happens in the terrestrial stratosphere. The Sun's interior is not exempt from the
integral constraint (8·2). Broadband gravity waves emitted from the convection zone
would tend to do the same thing in yet another way, as in the terrestrial QBO. One
can get all kinds of shapes and forms of departures from solid rotation, but one
cannot get solid rotation out of the fluid dynamics except by some hardly credible
accident; recall the spontaneous symmetry-breaking in the Plumb-McEwan experiment, the remarks at the end of §5, and, again, the remarks concerning the constraint
(8·2).
I am claiming, therefore, that there appears to be no purely fluid-dynamical
process that can make the interior rotate nearly solidly as observed. This is the
basis for concluding that there must be an interior poloidal magnetic field. Such a
field, sketched in the right half of Fig. 10, can do the job very easily, through Alfvenic
torques. Moreover, it is the only known way to do the job - to stop the interior
The Earth's Atmosphere and the Sun's Interior
161
~011€..­
,....__..__
r~s--..111-2---l
-;:;-( tu c. hocfi,e.)
~
IC /0 fJ k_fiM
69
Fig. 10. Schematic of proposed new model for the Sun's interior. ) The strong differential rota70
tion of the convection zone, now well known from helioseismic inversion, ) is roughly depthindependent and is indicated schematically by the words 'PROGRADE' and 'RETROGRADE '.
6
Angular velocities range over about ±10% of an intermediate value, 101 ~ 2.7 x 10- s-I, close
to the typical interior value. Interior rotation rates are much more nearly uniform. The poloidal
magnetic field occupying the radiative interior is a necessary dynamical ingredient, crucial to
explaining the observed depth of the tachocline, the thin shear layer separating the base of
5
the convection zone from the interior, whose depth .1 is roughly of the order 0.1 x 10 km; cf.
4
5
is an
tesla
10,
.
.
,
the Sun's radius, 7 x 10 km. The estimate of magnetic field strength IBI
nonlinear
intricate
on
depend
will
which
extremely rough, preliminary estimate, refinement of
modeling not yet done, and on the fullest possible use of helioseismic data now being accumu69
lated. Because of the expected approximate scaling, ) which turns out to involve a ninth power
9
law, IBI oc .1- , it will be especially crucial to work toward refined estimates of .1. The latest
5
5
72
attempt at a quantitative estimate ) tentatively replaces 0.1 x 10 km by 0.13 x 10 km. The
next refinement will be to make .1 a function of latitude.
differential rotation that fluid-dynamic al processes would otherwise bring about.
Recent refinements to the helioseismic results, from the MDI (Michelson Doppler
Interferometer ) instrument on the SOHO spacecraft (Solar and Heliospheric Obser70
vatory) have strongly confirmed the near solid interior rotation ) and have furthermore begun to resolve the detailed vertical structure at the top of the interior, just
162
M. E. Mcintyre
under the base of the convection zone. There is strong shear in a layer of small
but now just-resolvable thickness, ;S 2% of the solar radius; but the magnitudes
involved (Richardson numbers N 2 /lshearl 2 » 1) imply still stronger control by the
stable stratification, leading to layerwise 2-dimensional motion. Older ideas that
the region might be shear-unstable, and 3-dimensionally turbulent, appear to be
definitely ruled out. Consequently, the shear layer, which solar physicists call the
'tachocline', must have a dynamical character like the rest of the interior, i.e., like
the terrestrial stratosphere.
On this basis, and from the helioseismic results, one can estimate the circulation
of the tachocline in much the same way as with the circulation shown in Fig. 2.
It turns out that the circulation or ventilation time is about 106 times longer: not
several years but more like several million years. 69 ) But this is a mere instant in
comparison with 4 x 109 y, the age of the Sun and the timescale of its spindown.
Rates of change due to spindown can therefore be neglected in theories of tachocline
structure and dynamics: the latter can be treated as quasi-steady. Furthermore,
wave-induced forces, which are hard to estimate but are generally thought to be
significant, if at all, on longer timescales"' 108 y to 109 y- see Ref. 71) and references
therein - can likewise be neglected, almost certainly, in the tachocline dynamics
with its 106 y circulation timescale.
The only remotely plausible candidate for driving a circulation that is so fast,
relative to the other timescales involved, is gyroscopic pumping by the Reynolds
stresses within the dynamically vigorous convection zone above. As in the terrestrial
stratosphere, this must produce a downward-burrowi ng circulation. The burrowing
has to be stopped, however, not by a frictional boundary layer at a solid surface but
by a magnetic diffusion layer capping the interior magnetic field. If the burrowing
is not stopped somehow, then we are back to a classical laminar spindown scenario,
which would produce a tachocline much deeper than observed.
An idealized boundary-layer theory for the magnetic diffusion layer 69 ) confirms
that the magnetic field is, indeed, capable of stopping the burrowing, and furthermore that the boundary layer is thin, more than an order of magnitude thinner
than the tachocline itself. This is turn implies that the tachocline must end in a
relatively sharp 'tachopause', information that can be fed back into helioseismic inversion models. In particular, the fast tachocline circulation implies that there has to
be a near-discontinuity in helium abundance, hence sound speed, at the tachopause;
such a structure has recently been shown to fit the helioseismic data very well and
to lead to a refined estimate of the tachocline thickness; 72 ) see caption to Fig. 10.
The boundary layer theory also gives us a first, rather crude, estimate, "'
10- 4 tesla, of typical magnetic field strengths just below the tachopause. Stronger
fields would imply smaller tachocline depths L1, and vice versa. However, this estimate is likely to need revision, partly because it turns out that L1 is only weakly
dependent on field strength (caption to Fig. 10), and partly because a quantitative
estimate will require solution of a rather complicated, and strongly nonlinear, magnetohydrodynamic problem for the upwelling branch of the tachocline circulation.
This is work in progress. In the upwelling branch, which has to be in middle lati-
The Earth's Atmosphere and the Sun's Interior
163
tudes to be consistent with the observed pattern of differential rotation, we expect
the interior field to be skimmed up into the convection zone, where it will be subject to reconnection. There will probably be significant zonal Lorentz forces in this
upwelling region.
The implications are, however, already far-reaching in at least three senses. First,
if the interior is constrained by the simplest possible magnetic field, well diffused
from its primordial state and aligned with the rotation axis, hence in the form of a
poloidal field with field lines configured as in Fig. 10, as if wrapped around a torus
or donut, then what is known as Ferraro's law of isorotation applies. This says that
the angular velocity on the surface of each torus containing closed field lines must
be constant, which is equivalent to saying that no Alfvenic torsional oscillations are
excited. (There is an argument saying that such torsional oscillations will be damped
rather quickly relative to the Sun's age, essentially because of the near-degeneracy
of the problem with singular eigenfunctions in the form of a delta function confined
to the surface of one torus. 73 )• 36 )) Because some of the surfaces penetrate deep into
the interior, this offers for the first time a chance of obtaining information about
differential rotation near the Sun's core- information that is unobtainable directly
from helioseismic inversion.
The second implication is a better understanding of spindown itself. The spindown problem under the constraint of Ferraro's law (with or without wave-induced
forces) becomes very different from the classic spindown problem, again bringing
new understanding to what we observe of the present differential rotation.
The third is a new possibility for solving the notorious lithium-burning problem. 71 ) To explain observed lithium abundances at the surfaces of various populations of stars, one needs constituent-transp orting mixing mechanisms or circulations
that reach well below the tachocline. The strong gyroscopic pumping by the convection zone might have such a role. On the simplest picture sketched in Fig. 10, there
are just two places where the circulation might be able to burrow to sufficient depth
for the purpose: they are the two poles, where the magnetic field must vanish. The
downward-burrowi ng circulation might therefore be able to 'dig' a kind of 'polar pit'
in the magnetic field, at each pole, 69 ) penetrating much more deeply than elsewhere.
9
It can probably afford to take its time over this, say ;S 10 y. This could still be
enough to burn the lithium. Even if the interior magnetic field were more complicated, the hairy-sphere theorem tells us that weak points vulnerable to 'pit digging'
must exist. Detailed predictions will again depend on numerical solution of a highly
nonlinear magnetohydrodyn amic problem, yet to be attempted.
I shall end on a much more speculative note. It is conceivable that, despite their
generally high damping rates, interior Alfvenic torsional oscillations might nevertheless be significant especially if the field somehow configures itself to minimize
the damping. Could there be, one wonders, a QBO-like torsional oscillation in the
Sun's interior excited by broadband gravity waves from the convection zone, despite
the likely weakness of such waves? This would be a bit like the balance wheel of a
36
watch. My present feeling is that it is less likely than when I argued for it in 1994 )
- but then again, people who study isotopes in paleoclimatic records tell us that
there is evidence for solar oscillations on many timescales, centuries to millennia, far
164
M. E. M clntyre
slower than the sunspot cycle. Perhaps all these fluctuations on various timescales
originate, without exception, in the convection zone. This is entirely possible in such
a strongly nonlinear dynamical subsystem. But then again, perhaps some of these
timescales come from the radiative interior.
Professor Yukawa, had he been here, might well have asked me at this point
about the solar neutrino problem. But I shall resist the temptation to say anything
about this. It would be piling speculation too high! Let us wait and see what good
modeling and the wealth of new helioseismic data can tell us.
Acknowledgements
I thank K. Browning, D. Dritschel, P. Haynes, T. Hollingsworth, B. Kerridge,
P. Mote, B. Naujokat, T. Palmer, A. Plumb, C. Rodgers, K. Rosenlof, D. Sankey,
R. Scott, A. Simmons and K. Shine for helpful comments and correspondence. This
work received generous support from the UK Natural Environment Research Council
through the UK Universities' Global Atmospheric Modelling Programme, from the
Isaac Newton Institute for Mathematical Sciences, and from a SERC/EPSRC Senior
Research Fellowship.
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