A cascade that arrives as stripes

The inverse cascade carries energy up in scale, and the upward direction has an end: the box. Without friction to remove the energy before it arrives, it accumulates there as a pair of counter-rotating vortices filling the domain, and the limit of no friction has no steady state at all.

Both of those are calculations on a plane. A planet is not a plane, and the difference is not the curvature. It is that the component of the planet’s rotation perpendicular to the surface varies with latitude, so the vorticity a parcel starts with depends on where it starts — and the vorticity equation acquires one term it did not have.

One term, and it supports a wave #

Write the planetary vorticity as $f = f_0 + \beta y$ — the β-plane, which is the tangent plane to a rotating sphere with the latitude dependence kept to first order. Absolute vorticity is what a turning frame conserves, and conserving it then gives

$$\frac{\partial \zeta}{\partial t} + J(\psi, \zeta) + \beta\,\frac{\partial\psi}{\partial x} = 0.$$

The first two terms are the ordinary two-dimensional vorticity equation. The third is new, it is linear, and a linear term supports a wave. Dropping the nonlinearity and looking for $\psi \propto e^{i(k_x x + k_y y - \omega t)}$ gives

$$\omega = -\frac{\beta k_x}{k^2},$$

the Rossby wave. Two things about it matter here. Its phase always travels westward, whatever the wavenumber — the sign is fixed by $\beta$ and by nothing else. And its frequency falls with increasing wavenumber, which is backwards from every wave in the collection so far and is the whole of what follows.

A wave is a way of not cascading #

Turbulence transfers energy between scales because eddies strain one another, and straining takes time. A disturbance that oscillates faster than it is strained is gone before the transfer can happen: it propagates away as a wave instead of being torn up, and the nonlinear term, which needs the disturbance to sit still, has nothing to act on.

So the two rates have to be compared. The eddy turnover rate at wavenumber $k$ is $Uk$, which rises with $k$; the Rossby wave frequency is $\beta k_x/k^2$, which falls.

The cascade runs from small scales towards large ones, so it is running into the region where the waves win. That is the reverse of the usual situation, in which a cascade running downscale has to be stopped by viscosity: here it is stopped by a wave, and stopped at a scale that has nothing to do with dissipation.

The Rossby wave’s backwards dispersion is what makes that possible and it deserves its own sentence. Almost every wave in fluid mechanics travels faster at shorter wavelength or at least not slower: sound does not disperse at all, a deep-water gravity wave goes as the square root of its wavelength, and a capillary wave goes the other way but only because surface tension is doing the restoring. A Rossby wave’s restoring mechanism is the gradient of planetary vorticity, and a small parcel displaced across that gradient acquires a small anomaly, so a short wave is restored weakly and oscillates slowly.

A wave whose frequency falls with wavenumber is a wave that can only win at large scales, and a cascade that runs towards large scales is the only kind of cascade it could ever stop. The two facts have to be put together deliberately, and neither of them alone suggests the other: an inverse cascade on a plane with an ordinary wave in it would be stopped at small scales, where it never goes, and would arrive at the box exactly as before.

The barrier has a shape, and that is the result #

Setting the two rates equal gives the boundary:

$$U k = \frac{\beta\,|\!\cos\theta|}{k} \qquad\Longrightarrow\qquad k_\beta(\theta) = \sqrt{\frac{\beta\,|\!\cos\theta|}{U}},$$

with $\theta$ the angle from the east–west axis. Had the Rossby frequency been isotropic this would have been a circle and the story would have ended with a scale. It is not a circle. It vanishes identically at $\theta = \pm 90^\circ$, because a mode with $k_x = 0$ has no Rossby wave at all — the term that produced the wave is a derivative with respect to longitude, and a flow that does not vary with longitude does not feel it.

Modes with $k_x = 0$ are exactly the zonal flows: currents running east or west, varying only with latitude. The barrier is open to them and closed, at a finite scale, to everything else. So the cascade proceeds until it meets the boundary, is deflected along it, and arrives at the large scales in the one direction where nothing stops it.

It is worth being careful about what “deflected” means, because the picture invites a mechanical reading it does not deserve. Nothing steers an eddy. What happens is that transfers into the blocked region are undone — the disturbance made there oscillates away rather than being strained — while transfers into the open wedge are not, so the energy that ends up anywhere ends up there. It is a survival argument rather than a routing one, and the distinction matters because a survival argument predicts only where the energy is, and says nothing about the path it took.

The other half of the same caution is that the barrier is not a wall. The comparison is between two rates of the same order on either side of it, so there is no sharp line in any real flow: the transition occupies a factor of two or three in scale, and a spectrum measured across it shows a bend rather than a cut-off. A criterion drawn as a curve is a statement about where a smooth thing changes, and the sharpness of the drawing is the figure’s rather than the flow’s.

At half the barrier’s largest radius, 84 per cent of directions are blocked and 16 per cent are not. That fraction, rather than any single scale, is the anisotropy the picture is about.

Reading that fraction is worth a moment because it is the quantitative form of the whole result. A disturbance at that wavenumber, pointed in a randomly chosen direction, has about a one-in-six chance of being in the open wedge. Energy arriving at that scale is therefore not stopped and not passed; it is filtered, with five-sixths of it turned back and a sixth going through — and the sixth that goes through accumulates, because the turned-back five-sixths has nowhere else to be.

That is a different kind of arrest from the one a plane produces. There, the cascade reaches the box and stops because the box is the largest thing there is; the arrest is a boundary condition. Here it is a filter, it acts at a scale set by the fluid rather than by the container, and what it leaves is not a state of rest but a one-dimensional flow. A filter that passes one direction turns a two-dimensional cascade into a one-dimensional one, and a one-dimensional flow varying only with latitude is what the word “band” means.

The spacing, and what it is worth #

The scale at which the barrier sits is the Rhines scale, and in the convention used here — with the factor that makes it a half-wavelength rather than a wavenumber — it is

$$L_\beta = \pi\sqrt{\frac{2U}{\beta}}.$$

That square root is the reason the prediction is worth making. A planet’s band spacing is a robust quantity and its jet speeds are not: the speeds vary by a factor of several between one jet and the next and between one decade and another, and the spacing does not move, because it depends on the square root of a speed that is itself an average.

There is a second reading of the same scale that is worth carrying, because it is the one that makes the number memorable. At the Rhines scale the eddy turnover time equals the Rossby wave period, so the scale is the size at which a turbulent structure lasts about as long as it takes a planetary wave to cross it. Larger than that, a structure is a wave; smaller, it is an eddy. The Rhines scale is the size at which turbulence stops being turbulence, and a planet’s bands are the debris left where the two descriptions meet.

Why a square root is the right amount of dependence #

The Rhines scale is the geometric mean of two lengths that have nothing to do with each other, and that is worth a paragraph because it explains why the prediction survives being fed poor inputs.

Write it as $L_\beta^2 \sim U/\beta$. The numerator is a speed and the denominator is a vorticity gradient, so the combination has dimensions of length squared and nothing else could. What follows is that a factor of four of ignorance about the stirring speed is a factor of two of ignorance about the spacing, and a factor of a hundred is a factor of ten. For a quantity that has to be estimated from cloud-tracked winds averaged over a hemisphere, that insensitivity is the difference between a prediction and an arithmetic exercise.

The same square root explains why the criterion reaches so far in latitude. β varies as the cosine of latitude, so it falls by a factor of two between the equator and sixty degrees; the Rhines scale therefore widens by only forty per cent over that range. Bands on a planet should be roughly uniformly spaced in latitude rather than crowded at the equator, which is what Jupiter shows and which a linear dependence would not have predicted.

Four rotating fluids #

Jupiter’s weather layer, stirred at forty metres a second, gives a Rhines scale of 13,500 kilometres and sixteen bands from pole to pole. Jupiter has twenty to thirty, depending on how faint a jet is counted. That is not a fit; it is a rotation rate, a radius and one velocity, and it lands within a factor of two of a number anybody can count off a photograph.

The Earth’s atmosphere gives 3,200 kilometres and six bands, and has two or three. The disagreement is in the informative direction: the Earth is too small and too gently stirred to fit many Rhines scales in, so its jets are set by the size of the planet rather than by the turbulence, and the calculation is being asked about a regime it does not describe.

The ocean is the interesting case. Stirred at five centimetres a second, it gives 223 kilometres — and satellite altimetry finds faint alternating zonal striations in the open ocean at 200 to 300 kilometres, which were not expected before they were looked for and which are now among the better pieces of evidence that this argument is about something real.

Saturn is the awkward one and it is worth not hiding. Its jets are faster than Jupiter’s — a hundred metres a second against forty — and the arithmetic therefore gives a wider Rhines scale and fewer bands, nine against Jupiter’s sixteen, where Saturn is observed to have about as many as Jupiter. Something is wrong, and the likeliest candidate is the stirring speed: the criterion wants the speed of the turbulent eddies that feed the cascade, and what is measured on a giant planet is the speed of the jets the cascade has already made. Those are the same order and they are not the same quantity, and on Saturn the difference between them is apparently larger.

That is the standing weakness of every Rhines-scale comparison and it is not a detail. The criterion is written in terms of the eddy velocity and evaluated with the jet velocity, because the jet velocity is what a telescope sees. Where the two agree the prediction looks good; where they do not, the prediction is being tested with the wrong number rather than being falsified.

What the picture cannot show #

The flow is barotropic — one layer, no stratification. A real atmosphere or ocean is stratified, and the same argument run on a stratified fluid brings a second deformation radius and a second barrier, which is why the literature on this is a great deal longer than one criterion.

The criterion has no forcing scale in it. It compares a turnover rate with a wave frequency and says nothing about how the turbulence was made — unlike the enstrophy budget that sends the energy upscale in the first place, which does care. A cascade fed at a scale already larger than the Rhines scale never meets the barrier and makes no bands at all.

And it is a rate comparison rather than a solution. Nothing above solves the nonlinear equation; it compares two terms and says which should dominate where. That is the standard way this subject establishes a scale, it is right far more often than it has any right to be, and it does not say how sharp the transition is or what the jets’ profiles look like.

The waves and the turbulence are treated as separate things and they are not. The argument computes a wave frequency from the linear term and a turnover rate from the nonlinear one, and compares them as though a disturbance were one or the other. A disturbance at the barrier is both: its own nonlinearity is exactly as strong as its wave behaviour, which is the definition of where the barrier is, and no description that treats it as a wave or as an eddy can be right there. The same difficulty appears wherever a balance is its own error term.

And β is a fiction of the tangent plane, which is the same approximation a rotating bath is usually argued away with. A sphere’s planetary vorticity varies as the sine of latitude, not linearly, and a band spacing computed from a local β at thirty degrees is being applied to a hemisphere. The correction is of order one over the number of bands, so it is small where the prediction is most confident and large where it is not.

Who found it, and when #

Rossby derived the wave in 1939, from conservation of absolute vorticity on a β-plane, to explain the westward propagation of pressure patterns in the upper atmosphere. Rhines gave the arrest argument and the scale in 1975, in a paper that also reported the numerical experiments showing the bands forming. Vallis and Maltrud worked out the shape of the barrier in the wavenumber plane in 1993, and it is their dumbbell that is drawn here.

The order of those dates is worth noticing. The wave came from meteorology and the arrest argument from turbulence theory, thirty-six years apart, and the connection between them was not obvious to either field: Rossby was explaining why weather patterns drift west, and Rhines was asking what stops an inverse cascade. The bands on Jupiter had been catalogued for three centuries by the time anybody proposed that they were the answer to that question.

The surprising connection is with a mechanism that has nothing rotating in it. A cascade is arrested whenever the transfer rate falls below the rate at which something else removes the disturbance, and the something else need not be dissipation. Here it is a wave. In a stratified fluid it is also a wave, at the Ozmidov scale. In a conducting fluid with a magnetic field it is an Alfvén wave, and the cascade there becomes anisotropic about the field direction for exactly the reason it becomes anisotropic about the zonal axis here. Three subjects, one criterion, and in all three the shape of the anisotropy is the shape of the wave’s own dispersion relation rather than anything about the turbulence.

Still open: whether the jets are made by the cascade or merely fitted by it #

The Rhines scale is a criterion, and a criterion that lands within a factor of two of four different systems is doing something right. What it is not is a mechanism for a jet.

The arrest argument says the cascade cannot proceed except along the zonal axis. It does not say that energy does proceed along it, nor how a flow that is being blocked in most directions organises itself into a small number of sharp, persistent, alternating currents rather than into a broad-band zonal mess. Several mechanisms have been proposed — a secondary instability of the Rossby-wave field, potential-vorticity mixing between sharp gradients, and a direct upscale transfer into the zonal mode — and they predict different jet profiles while agreeing on the spacing.

The measurement that would separate them is the one thing a spacing does not test: the shape. A potential-vorticity staircase predicts jets whose curvature reaches the β threshold exactly, with mixed plateaux between; a secondary-instability mechanism does not. Jupiter’s jets have been profiled well enough by Cassini to ask the question, and the answer has been argued both ways for twenty years — which suggests the discriminating quantity has not yet been chosen carefully enough rather than that the data are inadequate.