Rieutord M., Georgeot B., and Valdettaro L., 2001, JFM, 435, 103

8

Solar tachocline dynamics: eddy viscosity,

anti-friction, or something in between?

MICHAEL E. McINTYRE

Centre for Atmospheric Science at the

Department of Applied Mathematics and Theoretical Physics,

Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK,

http://www.atm.damtp.cam.ac.uk/people/mem/

The tachocline has values of the strati¬cation or buoyancy frequency N two

or more orders of magnitude greater than the Coriolis frequency. In this

and other respects it is very like the Earth™s atmosphere, viewed globally,

except that the Earth™s solid surface is replaced by an abrupt, magnetically-

constrained ˜tachopause™(Gough & McIntyre 1998). The tachocline is heli-

um-poor through fast ventilation from above, down to the tachopause, on

timescales of only a few million years. The corresponding sound-speed an-

omaly ¬ts helioseismic data with a tachocline thickness (0.019 ± 0.001)R ,

about 0.13 — 105 km (Elliott & Gough 1999), implying large values of the

gradient Richardson number such that strati¬cation dominates vertical shear

even more strongly than in the Earth™s stratosphere, as earlier postulated by

Spiegel & Zahn (1992). Therefore the tachocline ventilation circulation can-

not be driven by vertically-transmitted frictional torques, any more than the

ozone-transporting circulation and di¬erential rotation of the Earth™s strato-

sphere can thus be driven. Rather, the tachocline circulation must be driven

mainly by the Reynolds and Maxwell stresses interior to the convection zone,

through a gyroscopic pumping action and the downward-burrowing response

to it. If layerwise-two-dimensional turbulence is important, then because of

its potential-vorticity-transporting properties the e¬ect will be anti-frictional

rather than eddy-viscosity-like. In order to correctly predict the di¬erential

rotation of the Sun™s convection zone, even qualitatively, a convection-zone

model must be fully coupled to a tachocline model.

8.1 Introduction

In the quintessential Douglas Gough manner I am going to be provocative

straight o¬ and say, in answer to the question in the title, that ˜anti-friction™

is closer to the mark “ ¬‚ying in the face of classical turbulence theories.

111

McIntyre

112

How can I make such an outrageous assertion? I can do so because in

signi¬cant respects the Sun™s interior is very like the Earth™s atmosphere,

and we observe the Earth™s atmosphere doing it all the time, that is, showing

anti-frictional behaviour. By ˜anti-frictional™ I mean that if we describe the

¬‚uid system in terms of a di¬erentially-rotating mean state with angular

¯ ¯

velocity „¦(r, θ, t) and azimuthal velocity vφ = r sin θ „¦(r, θ, t), where r, θ, φ

¯

are radius, colatitude, and longitude and t is time, plus chaotic ¬‚uctuations

v about that state, then the averaged e¬ect of the ¬‚uctuations is to drive

the system away from solid rotation.

This of course contradicts the classical idea, enshrined in the term ˜eddy

viscosity™, that chaotic ¬‚uctuations by themselves should drive, or rather

relax, the system toward solid rotation. The attractiveness of that classical

idea illustrates the perils of con¬‚ating ˜chaos™ with ˜turbulence™. The idea

would be correct if another classical idea were correct, namely that turbu-

lence theory should be like gas-kinetic theory, with the turbulent ¬‚uctuations

acting like molecular-scale ¬‚uctuations about a nearly homogeneous mean

state. Thus the gas-kinetic mean free path is replaced by some ˜mixing

length™, ˜Austausch length™, or other lengthscale representative of the irre-

versible ¬‚uctuating displacements of ¬‚uid elements. That lengthscale may

or may not be hidden from view within the complexities of a turbulence

theory based on ˜closure™. If momentum is transported by the ¬‚uctuating

displacements, and if typical displacements are much smaller than the scales

of variation of the mean state “ as implied by the stipulation ˜nearly homo-

geneous™ “ then the e¬ect of the ¬‚uctuations on the mean state is like that of

a viscosity, relaxing the system toward solid rotation, essentially because of

the scale separation just mentioned and the implied ¬‚ux“gradient relations.

The recognition that ¬‚uctuations in the Earth™s atmosphere often do the

opposite, i.e. drive the system away from solid rotation (though not, of

course, arbitrarily far away), was a major paradigm shift within the terres-

trial atmospheric sciences over the past century. That paradigm shift had its

beginnings in the work of Harold Je¬reys in the second and third decades of

the century (e.g. Je¬reys 1933†). It gathered pace in the late 1960s, stimu-

lated by an increasing wealth of observational evidence. It was fundamental

† This conference paper, originally from Proc`s-Verbaux de l™Assoc. de M´t´orol., UGGI, Lisbon,

e ee

Part II (M´moires), lucidly and cogently summarizes Je¬reys™ classic argument, developed

e

over the preceding decade or more, that observed surface winds imply the existence of what

Victor Starr later called the ˜negative viscosity™ due to the large-scale eddies, the cyclones and

anticyclones, appearing on weather maps. The reported conference discussion (Je¬reys, op.

cit., pp. 210“11) illustrates that in 1933 no-one, not even Je¬reys, had the faintest idea of

what kind of ¬‚uid dynamics might be involved. The ˜negative viscosity™ phenomenon was still

¬‚agged as a major enigma in the closing pages of the landmark review by E. N. Lorenz (1967).

Solar tachocline dynamics: eddy viscosity or anti-friction? 113

to solving some of the greatest enigmas with which the atmospheric sciences

were confronted in the 1960s.

¯

One of those enigmas was the behaviour of „¦(r, θ, t) in the tropical strato-

¯

sphere between 15“30 km, in which the sign of ‚ „¦/‚r reverses quasi-periodic-

ally with a mean period around 27 months. This surprising phenomenon was

¬rst revealed by radiosonde balloon observations, which had become routine

after the second world war in support of operational weather forecasting.

The phenomenon is known today as the quasi-biennial oscillation or QBO.

Its cause was wholly mysterious in the 1960s. Today, however, the QBO is

recognized as one of the clearest illustrations of the point I am emphasizing,

the tendency of chaotic ¬‚uctuations to drive a strati¬ed ¬‚uid system, very

often, away from solid rotation; and a further and even clearer illustration

can be found in the beautiful laboratory experiment devised and carried out

by Plumb & McEwan (1978). A strati¬ed ¬‚uid in a large annulus is driven

away from solid rotation, „¦ ≡ 0 in this case, by nothing but the imposition

¯

of ¬‚uctuations via an oscillating boundary. On a timescale of many bound-

¯

ary oscillations, „¦ evolves away from zero, and then develops a pattern of

reversals very like that of the QBO.

Together with appropriate conceptual and numerical modelling, the re-

sults from the Plumb“McEwan experiment have greatly illuminated our

thinking about the QBO, and enriched our repertoire of models of it. The

reader interested in the observed phenomena and in today™s understanding

of them, which is secure, at least qualitatively “ and in the history of ideas

leading to that understanding “ may consult my recent reviews (2000, 2002)

together with a major review of research on the QBO by Baldwin et al.

(2001), which latter includes an extensive discussion of the observational

evidence. Movies of the Plumb“McEwan experiment plus ˜technical tips™ on

how to repeat it are available on the Internet.†

8.2 Long-range and short-range momentum transport

How can the classical turbulence theories be so completely wrong, not just

quantitatively but also qualitatively? The answer is not only clear with hind-

sight but also simple. Because the Earth™s atmosphere and the Sun™s interior

are heavily-strati¬ed, rotating ¬‚uid systems, the ¬‚uctuations, chaotic though

they may be, inevitably feel the wave propagation mechanisms associated

with rotation and strati¬cation.

These include the propagation mechanisms of internal gravity waves, Cori-

olis or ˜inertial™ (epicyclic) waves, and layerwise-two-dimensional Rossby or

† at http://www.gfd-dennou.org/library/gfd exp/exp e/exp/bo/

McIntyre

114

vorticity waves. By its very nature, any wave propagation mechanism pro-

motes systematic correlations among the ¬‚uctuating ¬elds v , etc. Almost

inevitably, the upshot is that momentum and angular momentum are trans-

ported over distances far greater than mixing lengths, limited only by the

distances over which waves can propagate. Internal gravity waves provide

a well known example, in which the most signi¬cant correlations are those

between the horizontal and vertical components of v .

Of course there are exceptional cases in which the momentum transports

exactly cancel, such as perfect g modes and p modes, in the strict sense