nostic equation for the rate of change of Q, supplemented, if necessary, by a

prognostic (induction) equation for any magnetic ¬elds that may be present.

The equation for Q can be DQ/Dt = 0 or an appropriate generalization,

symbolically

DQ/Dt = viscous, diabatic and MHD terms . (8.5)

The single time derivative reminds us that Rossby waves and other layer-

wise-two-dimensional motions, viewed in the rotating frame, are chiral: they

notice the direction and sense of „¦0 . The mirror-image motion is impossible.

All this is simplest to see in the limiting case of anelastic motion and

in¬nitely heavy strati¬cation, in which N 2 ’ ∞ and Ri ’ ∞. The surfaces

S become rigid and horizontal “ horizontal in the billiard-table sense, with

the sum of the gravitational and centrifugal potentials constant “ and the

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118

¬‚ow on each S becomes strictly horizontal and strictly incompressible. Then

v = ˆ — ∇Sψ for some streamfunction ψ, and

r

Q = b ’1 (f + ∇2 ψ) (8.6)

S

with b strictly constant, where f is the vertical component of 2„¦0 and ∇S

and ∇2 are the two-dimensional gradient and Laplacian on the surface S.

S

We may regard (8.6) as a Poisson equation to be solved for ψ when Q is

given. Solving it is a well de¬ned, and well behaved, operation provided that

the given Q ¬eld satis¬es the integral relation (8.4) on each S. Symbolically,

ψ = ∇’2 (bQ ’ f ) .

v = ˆ — ∇Sψ with (8.7)

r S

This expresses PV invertibility in the limiting case. Notice that the limiting

case is degenerate in that the radial coordinate r enters the problem only as

a parameter. There is no derivative ‚/‚r anywhere in the problem, either

in the Laplacian or in the material derivative D/Dt = ‚/‚t + v·∇, v

now being strictly horizontal. Not only is the motion layerwise-two-dim-

ensional, but the layers are completely decoupled. There is, therefore, an

implicit restriction on magnitudes of ‚/‚r, i.e. an implicit restriction on the

smallness of vertical scales, as the limit is taken.

More realistically, when N 2 and Ri are large but ¬nite, ‚/‚r reappears in

the problem and brings back vertical coupling. The motion remains layer-

wise-two-dimensional in the sense that the notional ˜PV particles™ move

along each strati¬cation surface S, but the surfaces themselves are no longer

quite horizontal, nor quite rigid. All the vertical coupling comes from the

PV inversion operator. The two-dimensional inverse Laplacian in (8.7) is

replaced by an inverse elliptic operator that resembles a three-dimensional

inverse Laplacian when a stretched vertical coordinate N r/f is used.

Here one has to make tradeo¬s between accuracy and simplicity. The sim-

plest though least accurate inversion operator is that arising in the standard

˜quasi-geostrophic theory™, an asymptotic theory for large Ri and f = 0,

valid away from the equator. MHD forces are still absent from the inver-

sion and, if signi¬cant at all, enter the problem only through the prognostic

equation (8.5). The operator ∇2 becomes ∇2 + ρ’1 ‚r ρf 2 N ’2 ‚r . We

S S

may expect this to be a self-consistent approximation if Alfv´n speeds are

e

of the order of |v| or less. Notice that for tachocline eddies of horizon-

tal scale 105 km, say, the vertical coupling extends over a vertical scale

∼ (f /N ) — 105 km ∼ 0.004 — 105 km at latitude 45—¦ , fairly small in com-

parison with a tachocline thickness of 0.13 — 105 km.

Some idea of what is involved in constructing more accurate inversion

operators can be gained from the recent work of Ford et al. (2000) and

Solar tachocline dynamics: eddy viscosity or anti-friction? 119

Mohebalhojeh & Dritschel (2001 & refs.) and summarized in a recent review

of mine (2001); see also the earlier discussion by Hoskins et al. (1985). Subtle

generalizations of the notions of ˜geostrophic balance™ and ˜magnetostrophic

balance™ are involved, and there are ultimate limitations on the accuracies

attainable and on good mathematical behaviour, owing to phenomena such

as Lighthill radiation, equatorial inertial instabilities, symmetric-baroclinic

or Høiland instabilities, and magneto-rotational or Chandrasekhar“Fricke“

Balbus“Hawley instabilities.

Before going further with the theory, let us take note of what layerwise-

two-dimensional motion looks like in the real-world example that has been

the most thoroughly studied, the Earth™s stratosphere. There, thanks to

today™s observing systems, we can see many of the associated phenomena

in remarkable detail, including conspicuous examples of Rossby-wave prop-

agation.

8.4 A glimpse of the Earth™s stratosphere

Figure 8.1 presents two snapshots of the stratosphere, showing at a spatial

resolution of a few degrees latitude the e¬ects of layerwise-two-dimensional

motion on two strati¬cation surfaces S. These surfaces lie at altitudes of

about 31 and 37 km. An animated version can be seen on my website.† The

¬gure is reproduced by courtesy of Dirk O¬ermann, Martin Riese, and the

other scientists involved in the CRISTA space-based remote-sensing project;

see Riese et al. (2002). The quantity shown is the mixing ratio XN2 O of a

biogenic chemical tracer, nitrous oxide, that is destroyed photochemically on

a timescale of years but resupplied, across the strati¬cation surfaces S from

the troposphere below, on the same timescale of years, by a global-scale

circulation called the Brewer“Dobson circulation. This is a stratospheric

counterpart of the tachocline ventilation circulation. In the stratosphere

the upwelling branch of the circulation is in the tropics; therefore XN2 O

values are highest there. White areas are data gaps.

The layerwise-two-dimensional motion has far greater horizontal velocities

than the Brewer“Dobson circulation, and far shorter timescales of days to

weeks. On such timescales XN2 O is a near-perfect passive tracer, indeed

material invariant, DXN2 O /Dt = 0 to good approximation. Thus, apart

from the overall pole-to-equator gradient, the patterns seen in Figure 8.1

are shaped almost exclusively by the layerwise-two-dimensional motion.‡

† In colour, at www.atm.damtp.cam.ac.uk/people/mem/papers/LIM/index.html#crista-movie

‡ The observational resolution is enough for our purposes, though there must in reality be invis-

ible ¬ne-grain detail, such as the ¬lamentary, cream-on-co¬ee patterns found in recent high-

resolution observational and modelling studies of stratospheric ¬‚ows at lower altitudes (e.g.

Norton 1994, Waugh & Plumb 1994, Waugh et al. 1994, Appenzeller et al. 1996).

McIntyre

120

Each snapshot shows similar features, notably the well-mixed region (med-

ium gray) on the right, with nearly uniform tracer values, sandwiched be-

tween relatively isolated polar and tropical airmasses having very di¬erent

tracer values, with steep gradients in transition zones between. It is clear

from the animated version and from numerical model simulations, which

produce generically similar tracer distributions (e.g. Norton 1994), that the

layerwise-two-dimensional motion is causing strong mixing on each strati-

¬cation surface S in an extensive midlatitude region sandwiched between

the polar and tropical airmasses. A long tongue of tropical air is being

drawn eastward past the tip of South America (light gray, inner band on

the left) and marks the early stages of a typical mixing event, in which air

is visibly recirculating within the midlatitude region at the instant shown.

This horizontal recirculation is conspicuous in the animation. Because of

the strong mixing, it is reasonable to regard the motion as fully turbulent,

in the layerwise-two-dimensional sense, in middle latitudes.

However, the motion as a whole has not only its turbulent aspect but

also the wavelike aspect anticipated theoretically. This too is conspicuous

in the animated version of Figure 8.1, which shows the long axis of the

central, elongated dark region rotating clockwise through an angle of about

70—¦ longitude in 5 days, 10“15 August 1997, relative to the Earth. The

central region marks the core of the ˜polar vortex™, characterized by large

negative values of Q. Because of the approximate material invariance of Q,

it behaves like an advected tracer on the short timescales of the layerwise-

two-dimensional motion, and has a distribution somewhat like that of XN2 O

apart from an additive constant.

The rate at which the long axis rotates is determined by a competition

between the mean winds “ which broadly speaking blow clockwise, at speeds

of the order of 80 m s’1 , about nine times faster than 70—¦ in 5 days “ and

a wave propagation mechanism that powerfully rotates the long axis an-

ticlockwise relative to the air. This is the Rossby-wave or vorticity-wave

mechanism.† The phase progression is necessarily one-way (here anticlock-

wise, or retrograde, relative to the air), as a consequence of the chirality

associated with the single time derivative in equation (8.5).

As is well known, the Rossby-wave mechanism operates whenever Q has a

mean gradient ‚ Q/‚θ on strati¬cation surfaces S, such as the gradient asso-

¯

ciated with the global-scale polarization “ the positive-to-negative, pole-to-

pole variation in Q values due to the rotation of the whole system, „¦0 say,

† As usual, terminology contradicts historical precedent. Carl-Gustaf Rossby was one of the

greatest pioneers in atmospheric science, and his memory deserves special honour, but the

wave mechanism was noted decades earlier by Kelvin and Kirchho¬, in special cases at least.