Fig. 8.1. Nitrous oxide (N2 O) mixing ratios XN2 O observed at two stratospheric al-

titudes on 11 August 1997 by the CRISTA infrared spectrometer, from Riese et al.

(2002). White areas are data gaps. On Rossby-wave timescales of days and weeks

N2 O is an accurate passive tracer, though destroyed photochemically on Brewer“

Dobson timescales of years. In the right half of each picture XN2 O values increase

equatorward nearly monotonically or stepwise monotonically (being nearly con-

stant over the large medium gray regions on the right). Polar-vortex values (dark

central regions) are close to zero, and tropical values are high, imported from the

troposphere by the Brewer“Dobson upwelling. At left and right respectively:

pressure-altitudes are 4.64 hPa and 10 hPa, roughly 37 km and 31 km; ranges of

mixing ratios in parts per billion by volume are 0“90+ and 0“150+ with contour

intervals 10 and 16.67, where ˜+™ signi¬es that maximum values may slightly over-

shoot the plotted range; the light band in the subtropics highlights the ranges 60“70

and 100“116.67 ppbv. CRISTA (CRyogenic Infrared Spectrometers and Telescopes

for the Atmosphere) detects a number of chemical species through their infrared

spectral signatures and is a large (1350 kg) helium-cooled instrument ¬‚own from

the Space Shuttle.

which must here be taken to be somewhat faster than the rotation of the

solid Earth. North“south material displacements across that gradient give

rise to a pattern of ¬‚uctuating Q anomalies on the surfaces S that alternate

in sign downstream, every 90—¦ of longitude in the case of Figure 8.1. Inver-

sion of that pattern of Q anomalies to obtain the ¬‚uctuating velocity ¬eld

produces north“south velocities that lag north“south displacements by a

quarter wavelength in longitude, implying one-way phase propagation with

¯

highest Q values on the right, i.e. retrograde phase propagation.

To check this qualitative picture in the simplest possible way using the

standard Rossby“Haurwitz wave theory, take the limiting case (8.7), lin-

earize the prognostic equation DQ/Dt = 0 for small disturbances Q , ψ , v

about a mean state of solid rotation „¦0 , regard each strati¬cation surface

S as precisely spherical and look for disturbances with complex amplitude

Q and spherical-harmonic structure Q = Re{QPn (cos θ) exp(imφ ’ iωt)}.

ˆ ˆm

McIntyre

122

The linearized prognostic equation is

¯

‚Q vθ ‚ Q

+ = 0, (8.8)

‚t r ‚θ

with ‚ Q/‚θ = ’2„¦0 b ’1 sin θ, b = constant, and „¦0 = |„¦0 |. PV inversion

¯

reduces to ψ = ∇’2 (bQ ), hence ψ = ’r 2 b Q/{n(n+1)}, with ψ the complex

ˆ ˆ

ˆ

S

ˆm

amplitude of ψ , i.e., ψ = Re{ψPn (cos θ) exp(imφ ’ iωt)}. Noting that

vθ = ’(r sin θ)’1 ‚ψ /‚φ and that ‚/‚φ = im, we have

2„¦0 m

ω=’ . (8.9)

n(n + 1)

This illustrates the qualitative picture sketched above, including the one-way

propagation associated with chirality “ the single power of ω coming from

the single time derivative. Because the angular phase velocity ω/m < 0,

the phase propagation is retrograde and the meridional disturbance velocity

ˆ

vθ , with complex amplitude ∝ ’imψ, lags the displacement, with complex

amplitude ∝ ’imψ/(’iω) = mω ’1 ψ, by a quarter wavelength in longitude.

ˆ ˆ

More realistic models of stratospheric Rossby waves must take account of

the turbulent mixing in middle latitudes. The mixing has an obvious qual-

¯

itative e¬ect: it weakens the PV gradient ‚ Q/‚θ in middle latitudes and

strengthens it at the subtropical edge of the midlatitude mixing region (out-

ermost light band, clearest on the right of Figure 8.1) and at the polar edge

¯

bounding the vortex core. This characteristic reshaping of the Q(θ) pro¬le is

suggested schematically by the cartoon on the left of Figure 8.2, in which y

denotes northward distance in arbitrary units, y ∝ ’θ, in a midlatitude slab

¯

model. The dashed and heavy lines represent the Q(y) pro¬les before and

¯

after mixing. The middle graph presents the corresponding Q(y) pro¬les in

an actual numerical experiment, to be referred to shortly. Thus, in more

realistic models, the quasi-elastic resilience associated with the Rossby-wave

mechanism tends to be concentrated in transition zones of steep Q gradients,

also marked by steep XN2 O gradients, lying between the tropical, midlati-

tude and polar airmasses. The same quasi-elastic resilience is part of why

the three airmasses are chemically so distinct, with little mixing between

them, a phenomenon seen again and again by stratospheric researchers and

much studied because of its signi¬cance for ozone-layer chemistry. ˜Shear

sheltering™ is also involved (Juckes & McIntyre 1987; Hunt & Durbin 1999).

For our purposes, however, the most important point of all is that the

layerwise-two-dimensional mixing in middle latitudes owes its existence to

the Rossby waves. In this respect the situation illustrated in Figure 8.1 is

fundamentally similar to the ocean-beach situation, in which the turbulence

Solar tachocline dynamics: eddy viscosity or anti-friction? 123

in the ocean-beach surf zone owes its existence to surface gravity waves.

That is part of what I meant by the assertion that in the dynamical regimes

under discussion ˜there is no such thing as turbulence without waves™.

The midlatitude mixing occurs for well-understood reasons associated

with ¬‚ow unsteadiness, hyperbolic points, and so on “ a chaotic-advection

kinematics very much tied, in this case, to the wave propagation, as ana-

lysed in detail by, for instance, Polvani & Plumb (1992). We may say that

the turbulent mixing is intimately, and inseparably, part of the wavemotion.

It is therefore reasonable to consider these stratospheric Rossby waves to

be breaking waves. For this reason, the midlatitude mixing region is often

called the ˜stratospheric surf zone™.

Numerical experiments in which the initial condition is axisymmetric, and

in which Rossby waves are then excited somehow, commonly produce surf

zones like that seen in Figure 8.1 (e.g. Norton 1994). The formation of

surf zones is a very robust feature of such experiments, almost regardless

of how chaotic or regular the waves, as such, happen to be. In the Earth™s

stratosphere the Rossby-wave ¬elds can on occasion be fairly regular, as in

the case of Figure 8.1, or, more typically in the northern-hemispheric winter,

rather more chaotic.

A fundamentally similar phenomenon of surf-zone formation was demon-

strated long ago in the idealized numerical experiments of Rhines, in a clas-

sic paper entitled ˜Waves and turbulence on a beta-plane™ (Rhines 1975).

The designations ˜Rossby-wave breaking™ and ˜stratospheric surf zone™ can

be justi¬ed in a very general way, from wave“mean interaction theory (e.g.

McIntyre & Palmer 1985), having regard to Kelvin™s circulation theorem.

This has application to most if not all non-acoustic wave types.

In the Rossby-wave case the whole conceptual picture is illustrated by a

speci¬c model of wave breaking in a certain parameter limit, known as the

Stewartson“Warn“Warn model, in which the surf zone is narrow and the

interplay between the wavelike and turbulent dynamics can be precisely and

comprehensively described using matched asymptotic expansions (Haynes

1989 & refs.). This is based on the midlatitude slab model in the limiting

case (8.7), and has provided a set of detailed examples including that from

which Figure 8.2b is derived. The interplay works both ways, at leading

order: not only do the waves create the turbulence “ again justifying the

idea of ˜wave breaking™ “ but the turbulence, in turn, strongly in¬‚uences

the wave¬eld, and in particular the systematic correlations between vθ and

vφ that are signi¬cant for horizontal momentum transport. The wave¬eld,

through the PV inversion operator, senses the horizontal rearrangement of

PV substance by the turbulence within the surf zone.

McIntyre

124

8.5 Turbulence requires waves

There is an alternative, independent justi¬cation for the assertion that in

the dynamical regimes under discussion ˜there is no such thing as turbu-

lence without waves™. The justi¬cation follows simply and directly from

PV invertibility, involving no restriction to special parameter regimes, and

no reliance on particular mathematical techniques such as that of matched

asymptotic expansions.

We assume the existence of turbulence without waves, and show that this

leads to a contradiction. More precisely, consider a layerwise-two-dimen-

sional PV mixing event like those depicted in Figure 8.2a,b, in which the