Where the inverse cascade stops

The cascade that runs backwards establishes why two-dimensional turbulence sends energy up in scale rather than down. The vortex-stretching term vanishes identically in a plane, enstrophy becomes a second inviscid invariant, and Fjørtoft’s argument — computed there on three modes, giving eighty per cent of the energy to the larger scale — forces the energy upward and the enstrophy downward.

This essay asks what happens when the energy arrives at the top, and the answer has three parts: it arrives quickly, it has nowhere to go, and the limit in which nothing removes it does not exist.

It arrives quickly, and the arithmetic is the cascade’s own #

The time an eddy of size $l$ takes to turn over is $t(l) = \varepsilon^{-1/3}l^{2/3}$, so larger eddies are slower. Summing over octaves from the forcing scale up to the box,

$$T = \sum_n \varepsilon^{-1/3}l_n^{2/3}, \qquad l_n = l_f 2^n,$$

is a geometric series with ratio $2^{-2/3} = 0.63$, and it converges. Its sum is

$$\frac{1}{1 - 2^{-2/3}} = 2.702$$

times the box’s own turnover time, whatever lies below.

Computed over ten octaves the total is 2.686 box turnovers, and adding ten more octaves below the forcing changes it by 0.6 per cent. The last octave costs more than all the rest together.

So the journey up is not slow. A two-dimensional flow forced at a small scale fills its domain with large eddies in a few turnover times of the largest eddy it can have, and nothing about the number of octaves in between matters.

The convergence of that sum is worth a sentence because it is not obvious that it should converge, and because the same arithmetic in the other direction is the reason where the energy goes can treat the forward cascade as instantaneous. Going down, the turnover times shrink geometrically and the sum is dominated by the first octave; going up, they grow geometrically and it is dominated by the last. Either way the total is a small multiple of one octave’s time, and the number of octaves in between is irrelevant.

What that means physically is that a cascade is never rate-limiting. Whatever the range of scales, the transfer takes a couple of turnover times of the slowest eddy involved, and everything else in the problem is slower.

And then there is nowhere further to go #

The upward cascade needs somewhere to deliver energy, and in an infinite domain there always is. In a box there is not: the largest eddy is the box, and once the energy is there the cascade has run out of room.

What happens then is that it accumulates. The flow develops a pair of counter-rotating vortices filling the domain — the condensate — carrying most of the energy, and growing until something stops it.

The usual something is a linear friction, which in a geophysical or laboratory two-dimensional flow is real: bottom drag in a rotating tank, Ekman friction in the atmosphere, air drag on a soap film.

The condensate is not a subtle feature. In a numerical experiment it is unmistakable — the vorticity field, which started as small-scale noise, becomes two enormous blobs of opposite sign occupying the whole domain, with the small scales reduced to filaments being wound around them.

And it changes the problem it grew out of. The condensate imposes a large-scale strain on everything below it, which shears the smaller eddies and interferes with the cascade that produced it. A two-dimensional turbulence with a condensate is not the same system as one without, which is why the friction is not a detail.

Where friction wins, and the scale it wins at #

A linear drag $-\alpha\mathbf{u}$ removes energy at a rate $\alpha$ per unit energy, independent of scale. The cascade transfers at the turnover rate $\varepsilon^{1/3}k^{2/3}$, which falls as the scale grows. So the cascade wins at small scales and the friction wins at large ones, and they change places where

$$\varepsilon^{1/3}k^{2/3} = \alpha \quad\Longrightarrow\quad k_\alpha = \sqrt{\frac{\alpha^3}{\varepsilon}}.$$

That relation is verified to $10^{-12}$ by evaluating both rates at the computed $k_\alpha$. If $k_\alpha$ is larger than the box’s wavenumber the cascade is arrested before the box and there is no condensate; if it is smaller, there is one.

The limit that does not exist #

Now weaken the friction, which is the limit a theorist would want to take.

In a statistically steady state the friction must remove exactly what the forcing supplies: $\varepsilon = 2\alpha E$. So

$$E = \frac{\varepsilon}{2\alpha}, \qquad U = \sqrt{\frac{\varepsilon}{\alpha}},$$

and the exponent measured across the sweep is $\alpha^{-0.500000000}$.

As $\alpha \to 0$ the condensate’s energy diverges. There is no limiting flow: the limit of no friction is not a two-dimensional turbulence with a weak condensate, it is a flow with no steady state at all, in which the energy grows without bound for as long as the forcing runs.

That is as clean a case of a limit leaving a residue as the collection has. The limit is perfectly natural to want — remove the friction, see the pure inverse cascade — and what it leaves behind is a divergence.

The comparison with the decaying case is instructive. What decay never forgets has a three-dimensional turbulence whose large scales are conserved and whose energy falls, and the conserved large-scale quantity sets the rate. Here the large scales are conserved in the same sense — nothing above them takes anything away — and the energy rises, because there is a forcing below feeding them. Same structural fact, opposite sign, and in one case it produces a decay exponent and in the other a divergence.

What that means for a computation, which is not academic #

The divergence has a practical face and every two-dimensional simulation meets it.

Run a forced two-dimensional simulation in a periodic box with no large-scale drag and the energy grows linearly with time for ever, because the forcing puts it in and nothing takes it out. The spectrum develops a spike at the box wavenumber that keeps rising, and the “statistically steady state” the simulation was supposed to reach does not exist.

So every such computation has a friction in it, and the friction is a choice. Its value sets the condensate’s amplitude, which sets the shear the small scales feel, which sets very nearly everything else. A reader comparing two two-dimensional simulations is often comparing two frictions.

That is a different kind of arbitrariness from the one in the three-dimensional problem, where the dissipation is set by the cascade and the viscosity merely decides where it happens — the limit that is not the value is that essay. Here the removal mechanism is not downstream of anything; it is a boundary condition on the whole flow.

The other range, which is never the power law it is called #

The downward, enstrophy-carrying cascade has its own problem and it is worth the second half of this essay.

Kraichnan’s dimensional argument gives $E(k) = C\eta^{2/3}k^{-3}$ in the enstrophy range, by the same reasoning that gives $-5/3$ for energy. But $-3$ is a special exponent: a $k^{-3}$ spectrum makes the strain rate contributed by each octave the same, so the strain at any scale is a sum over all the octaves above it, and the cascade is non-local.

Kraichnan’s own correction accounts for it and makes the transfer time pick up a logarithm:

$$E(k) = C\,\eta^{2/3}k^{-3}\big[\ln(k/k_f)\big]^{-1/3}.$$

Differentiate and the local slope is $-3 - 1/(3\ln(k/k_f))$, checked here to $10^{-5}$ against the computed derivative. It is $-3.240$ two octaves above the forcing, $-3.060$ at eight, $-3.015$ at thirty-two and $-3.0075$ at sixty-four.

Forty-eight octaves #

The number worth carrying is what it costs to reach a given accuracy.

$|\text{slope} + 3| = \text{tol}$ needs $\ln(k/k_f) = 1/(3\,\text{tol})$, so the separation required is the exponential of the reciprocal of the tolerance. Reaching $-3.01$ needs 48 octaves; reaching $-3.002$ needs 240.

Forty-eight octaves is a scale separation of $3\times10^{14}$. The atmosphere does not have it, a laboratory does not have it, and no computation ever will. So the $k^{-3}$ range is a limit that is not merely unreached in practice — it is unreachable in principle by anything anybody could build.

That is worse than the three-dimensional case. The range a real Reynolds number does not have records that a clean $-5/3$ needs a few decades of scale separation, which the atmosphere supplies. Here the approach is logarithmic in the logarithm, and the numbers are not of the same kind.

What is actually measured, and how the two failures interact #

Measured two-dimensional spectra in the enstrophy range cluster between $-3$ and $-4$, and the literature has argued about whether the steepening is Kraichnan’s logarithm or something else — coherent vortices, which are known to steepen the spectrum by taking the enstrophy out of the cascade and holding it.

Nothing here settles that. What the arithmetic establishes is that a measured slope steeper than $-3$ is expected even with no vortices at all, and by an amount that is computable: at a laboratory scale separation of eight octaves the expected departure is 0.06, and at three octaves it is 0.16.

So a measurement of $-3.2$ over three octaves is consistent with pure Kraichnan and is not evidence for anything else. That is a useful null hypothesis to have, and it is what the logarithm is for.

Why this collection’s two cascades fail differently #

Putting the three-dimensional and two-dimensional cases side by side is the cleanest way to remember both.

Three dimensions. Energy goes down, arrives at the viscous scale, and is destroyed. The mechanism that removes it is inside the fluid, it is set by the cascade rather than by the viscosity, and the limit that is not the value shows the removal rate surviving the limit $\nu \to 0$ untouched. Nothing has to be supplied from outside.

Two dimensions. Energy goes up, arrives at the box, and stays. The mechanism that removes it is outside the fluid — a drag on a boundary, a bottom, a surface — and it does not survive being taken to zero. Something has to be supplied from outside.

That asymmetry is not an accident of the models. It follows from the direction of the cascade: a cascade towards small scales always finds a dissipation mechanism, because viscosity acts most strongly on the smallest scales, and a cascade towards large scales finds nothing at all.

What a physical two-dimensional flow actually is #

A note on where any of this applies, because strictly two-dimensional flows do not exist.

The candidates are a soap film, a thin layer of fluid on a rotating table, a magnetically forced layer of electrolyte, and — much more importantly — the large scales of the atmosphere and the ocean, which are quasi-two-dimensional because rotation and stratification suppress vertical motion.

The number that stops the mixing is about the stratification half of that suppression and five numbers one name about how many distinct Richardson numbers are hiding inside one symbol. What matters here is that in every one of those systems there is a real friction — bottom drag, Ekman pumping, air drag on a film — and it is what stops the condensate.

So the divergence computed above is not a pathology that real flows escape. It is the reason every real quasi-two-dimensional flow has a large-scale drag, and the reason removing it in a model produces nonsense rather than an idealisation.

What a condensate does to the flow that made it #

The last thing worth saying is that the condensate is not a passive accumulation sitting at the top of the spectrum. It changes the flow underneath it.

A pair of domain-scale vortices imposes a large-scale strain and shear on everything below. Small eddies in that field are stretched into filaments and wound around the large ones, which is exactly the mechanism longer with nothing pulling it describes for a material line, applied to vorticity.

The consequence for the cascade is that the transfer becomes non-local: energy at small scales is being sheared by the condensate rather than interacting with its neighbours, and the local cascade picture that produced the inverse-cascade prediction in the first place is being undermined by its own output.

That is why the friction is not a detail and why the arbitrariness noted above matters. A simulation with a weak drag is not a simulation of a purer inverse cascade; it is a simulation of a different flow, in which a large coherent structure dominates the dynamics that produced it.

The number to carry #

$k_\alpha = \sqrt{\alpha^3/\varepsilon}$ is the whole of the practical content, and it is worth converting.

The arrest scale is $2\pi/k_\alpha$, so a friction time $1/\alpha$ of an hour with an energy input of $10^{-8}$ square metres per cubic second — atmospheric values, roughly — puts the arrest at a few thousand kilometres, which is the scale of the largest atmospheric eddies.

That is the right order of magnitude and it is one of the reasons the quasi-two-dimensional picture of large-scale atmospheric flow is taken seriously: the arrest scale computed from a friction and an energy input lands where the observed spectral peak is, without anything having been fitted.

Halve the friction and the arrest scale grows by $2^{3/2}$, which is a factor of nearly three. That sensitivity is the same divergence as before, read at a finite value rather than in the limit.

Limits recorded rather than smoothed over #

No two-dimensional turbulence is computed here. Everything above is arithmetic on the standard scaling arguments — turnover times, an energy balance, and Kraichnan’s corrected spectrum. There is no simulation, no spectrum measured off a field, and no vortices.

The condensate’s structure is not addressed. The energy balance gives its amplitude and says nothing about its shape, and the shape — usually a pair of counter-rotating vortices, sometimes a jet, depending on the domain’s aspect ratio — matters for what the condensate then does to the small scales.

The friction is linear. Real bottom drag is quadratic at high Reynolds number, and the exponents above are the linear ones.

And the enstrophy-range correction is Kraichnan’s, which is itself a scaling argument. It is a correction to a dimensional prediction, obtained by the same kind of reasoning as the prediction, and it is not derived from the equations. What is exact is the arithmetic of what it implies.

The asymmetry, stated once more #

The clearest way to hold the two-dimensional case in mind is against the three-dimensional one, and the difference comes down to a single question: does the cascade run towards a place where something removes energy?

In three dimensions it does, always, because viscosity acts most strongly on the smallest scales and the cascade goes there. The removal mechanism is inside the fluid, it is found automatically, and — as the limit that is not the value establishes — its rate is independent of how strong the viscosity is.

In two dimensions it does not. The cascade goes to the largest scales, where nothing in the fluid acts preferentially, and it stops only when it meets a boundary or an externally imposed drag. There is no internal mechanism to find.

Everything in this essay is a consequence of that one sentence: the divergence of the condensate, the arbitrariness of the friction, the difficulty of defining a statistically steady state, and the fact that a two-dimensional turbulence in a box is a different problem from a two-dimensional turbulence in an unbounded plane.

The residue #

Two limits, and neither of them has the answer a reader would expect.

The friction $\alpha \to 0$: no steady state, a condensate diverging as $\alpha^{-1/2}$, and a flow whose energy grows for ever. The residue is the whole existence of the solution.

The scale separation $\to\infty$: an enstrophy range whose exponent is $-3$ in the limit and never within a hundredth of it in anything achievable. The residue is a logarithm, and a logarithm’s approach to its own limit is the slowest thing in this subject.

What the two have in common is the direction. A cascade that runs upward has no natural end, and both residues are consequences of the same missing thing — a sink at the top of the ladder.