¯

tion is not trivial to demonstrate, when „¦ = 0, but it can be demonstrated

from the so-called ˜nonacceleration theorem™ of wave“mean interaction the-

ory (e.g. McIntyre 2000 & refs.), essentially a consequence Kelvin™s circula-

tion theorem applied around all latitude circles.

Long-range momentum transports of the kind in question are sometimes

called radiation stresses (e.g. Brillouin 1925, on ˜tensions de radiation™).

They are usually anisotropic, contrary to what might be suggested by the

older term ˜radiation pressure™ still found in the literature. They are related

to mean gradients in ways that are anything but local, the global eigenmodes

being an extreme case. It is crucial to consider large-scale wave¬elds and

the processes of generation, dissipation, refraction, Doppler-shifting, internal

re¬‚ection, focusing and defocusing that shape the wave¬elds.

Classical turbulence theories “ all the way from simplistic mixing-length

theories to complicated closure theories “ take no account of such long-range

momentum-transport mechanisms. As already emphasized, the only mo-

mentum transport they consider is, by assumption, that arising from local,

short-range, Austausch or material-exchange types of process. It is exactly

that short-range character, and the implied or hoped-for scale separation,

that give rise to ˜turbulent stresses™ involving ¬‚ux“gradient relations and

eddy viscosities. We may summarize what happens in the Earth™s atmo-

sphere by saying that, on a global scale, radiation stresses dominate turbu-

lent stresses. We shall see nevertheless that turbulence can be important in

another way, namely through its contribution to shaping the wave¬elds, as

with surf near ocean beaches.

8.3 Potential vorticity

Anti-frictional behaviour is not inevitable when radiation stresses dominate

turbulent stresses, but experience has shown it to be commonplace. For

instance such behaviour is often produced by broadband internal gravity

Solar tachocline dynamics: eddy viscosity or anti-friction? 115

wave ¬elds “ broadband in the sense of having a range of horizontal phase

speeds “ like those generated by the Sun™s convection zone or by the Earth™s

tropical thunderstorms. In fact the Plumb“McEwan experiment, in which

the signi¬cant waves are internal gravity waves, shows that even two distinct

horizontal phase speeds can be enough.

Anti-frictional behaviour is commonplace, too, in the case of Rossby-

wave ¬elds, whether broadband or not, for quite di¬erent reasons connected

with the properties of the Rossby“Ertel potential vorticity, hereafter ˜PV™.

Anti-frictional behaviour is especially characteristic of the stresses exerted

horizontally by ¬‚uctuating layerwise-two-dimensional motion. That is why

Gough & McIntyre (1998 & refs., hereafter GM) argued against horizontal

eddy viscosity as explaining the thinness of the tachocline.

The PV, denoted here by the symbol Q, is a quantity central to the dynam-

ics of heavily strati¬ed ¬‚uid systems, including the dynamics of Rossby waves

and other nearly-horizontal, layerwise-two-dimensional motions. Such other

motions include layerwise-two-dimensional turbulence, also loosely called

˜geostrophic™ turbulence despite its possible existence near the equator. The

properties of Q will expose the fact that such turbulence is itself intimately

bound up with the Rossby-wave mechanism. This will be demonstrated in

Sections 8.4 and 8.5. In the dynamical regimes under discussion there is no

such thing as turbulence without waves.

In a reference frame rotating with angular velocity „¦0 the PV, Q, is

de¬ned as

Q = ρ’1 (2„¦0 + ∇ — v) · ∇‘ , (8.1)

where ρ is mass density and ‘ is potential temperature (materially invariant,

D‘/Dt = 0, for adiabatic motion; in place of ‘ one may equally well use

speci¬c entropy, or any other monotonic function of ‘ alone). For de¬nite-

ness we identify „¦0 with the angular velocity of the Sun™s interior just below

the tachocline, |„¦0 | ≈ 0.27 — 10’5 rad s’1 or 430 nHz, and take the axis of

coordinates parallel to „¦0 . Heavy strati¬cation means that ∇‘ is nearly

vertical, ∇‘ ≈ ˆ ‚‘/‚r, where ˆ is a unit vertical (radial) vector. Heavy

r r

strati¬cation also means that the associated buoyancy frequency N greatly

exceeds the other reciprocal timescales of interest, including |„¦0 | and the

¯

vertical shear r sin θ ‚ „¦/‚r. We recall that N is de¬ned by

N 2 = g ‘’1 ‚‘/‚r = g ‚(ln ‘)/‚r , (8.2)

g being the local gravitational acceleration, and that the value of N is of the

order of 10’3 rad s’1 near the base of the tachocline. The standard measure

of strati¬cation against vertical shear, the gradient Richardson number, is

McIntyre

116

de¬ned by

Ri = N 2 (r sin θ ‚ „¦/‚r)’2 .

¯ (8.3)

If we use the re¬ned estimate of tachocline depth ∆r obtained by Elliott

& Gough (1999), (0.019 ± 0.001)R , about 0.13 — 105 km, then typical

10’5 s’1 , not much greater than |„¦0 |. Thus

vertical shears ∆¯φ /∆r

v

Ri 10’6 /10’10 ∼ 104 1 near the base of the tachocline. Even when N

is taken to be an order of magnitude smaller, 10’4 rad s’1 , as near the top

of the tachocline, we still have Ri 102 1. This says that the tachocline

is even more heavily strati¬ed than the most heavily strati¬ed portion of

the Earth™s stratosphere, where typically Ri 10 in a coarse-grain view.

Such Ri values are high enough to enforce layerwise-two-dimensional mo-

tion, everywhere including the equator, as pointed out by Spiegel & Zahn

(1992, hereafter SZ). A key property of Q during such motion is that not

only ‘ but also Q itself is materially invariant, DQ/Dt ≈ 0, if the motion

can be considered inviscid as well as adiabatic and if MHD (Lorentz) forces

can be neglected within the tachocline. Approximately inviscid motion is

consistent with large Ri values.†

A second key property of Q, which holds for any motion whatever “ even

a motion that feels MHD forces “ is the integral relation

Q b dA = 0 , (8.4)

S

where dA is the surface area element and where the integral is taken glob-

ally over a strati¬cation or isentropic surface S, on which ‘ is constant by

de¬nition. The weighting factor b = ρ/|∇‘|, a positive-de¬nite quantity. It

is a strati¬cation-related mass density in the sense that b d‘ is the mass per

unit area between neighbouring strati¬cation surfaces S; that is, b dA d‘ is

the mass element. The relation (8.4) is an immediate consequence of Stokes™

theorem, the de¬nition of Q, and the fact that each surface S is topologically

spherical and has no boundary. For present purposes both the Sun and the

Earth are rapidly rotating bodies, with strongly polarized Q ¬elds: except

near the equator, 2„¦0 dominates ∇ — v in (8.1). So (8.4) is satis¬ed through

† Note also that tachocline thermal di¬usion times estimated as (π ’1 ∆r)2 /κ, where the thermal

di¬usivity κ ∼ 107 cm2 s’1 , come out at about 500y. This is well in excess of the likely

timescales of months to years for any layerwise-two-dimensional motion that might occur in

the tachocline. Viscous and magnetic di¬usion times are far longer still. The inviscid, adiabatic

material invariance of Q (Ertel™s theorem) is easy to verify from ∇(D‘/Dt) = 0 together with

the scalar product of ∇‘ with the inviscid, adiabatic vorticity equation, or alternatively (e.g.

McIntyre 2000, Section 9) as a corollary of mass conservation together with Kelvin™s circulation

theorem applied to small constant-‘ circuits. The neglect of MHD forces is much more of an

open question, but, for what it is worth, the arguments of GM strongly justify such neglect in

the downwelling branches of the tachocline ventilation circulation.

Solar tachocline dynamics: eddy viscosity or anti-friction? 117

cancellation of strong positive and negative contributions from the northern

and southern hemispheres respectively.

Owing to the positive-de¬niteness of the weighting factor b, the relation

(8.4) imposes a severe constraint on the possible evolution of the global-scale

Q distribution on each surface S. We shall see that (8.4) is almost enough,

by itself, to guarantee that layerwise-two-dimensional ¬‚uctuations about a

mean state of solid rotation will behave anti-frictionally. Consistently with

(8.4), one may picture Q as the amount per unit mass of a ¬ctitious ˜PV sub-

stance™ composed of charged particles to which the strati¬cation surfaces S

are completely impermeable. They are impermeable even if the motion is

not adiabatic. Even if mass leaks across a surface S, through thermal di¬u-

sion, the notional particles of ˜PV substance™ remain trapped on that surface

(Haynes & McIntyre 1990). The ˜PV charge™ is conserved in the same way

as electric charge. That is, pair production and annihilation are allowed,

but no net charge creation or destruction. Just as b dA d‘ is the mass ele-

ment, Q b dA d‘ is the charge element. The picture is consistent with (8.4)

because the value, zero, of Q b dA cannot be changed by pair production

and annihilation. Nor can it be changed by the advective rearrangement of

the notional particles on each surface S by any layerwise-two-dimensional

motion.

A third key property of Q is its ˜invertibility™. This says that the isen-

tropic distributions of Q, i.e. the distributions of PV values on the surfaces

S, contain nearly all the kinematical information about the layerwise-two-

dimensional motion “ whether or not MHD forces are signi¬cant. At each

instant, to good approximation, one can ˜invert™ the PV ¬eld to get the

velocity, pressure and density ¬elds. The dynamical system is then com-