The scalar has its own cascade

A dye stirred into a turbulent flow has a spectrum of its own, and it is not the velocity’s.

That is not obvious. The dye is carried by the velocity field, it has no dynamics of its own, and it does nothing to the flow. It would be reasonable to expect its statistics to be the velocity’s, inherited. Over one range of scales they are, and over another they are not, and the second range is the one worth an essay.

Two exponents rather than one #

The inertial-convective range is Obukhov’s and Corrsin’s, and it is the inherited part. Where the velocity is carrying the scalar and neither viscosity nor diffusivity matters, dimensional reasoning on the scalar dissipation and the energy dissipation gives

$$E_\theta(k) = \beta\,\varepsilon_\theta\,\varepsilon^{-1/3}\,k^{-5/3},$$

the same five-thirds the velocity has, for the same reason.

The viscous-convective range is Batchelor’s, and it is the interesting one. Below the Kolmogorov scale there is no turbulence left: the velocity field there is smooth, and its spectrum is falling away exponentially. The scalar cascades anyway, on

$$E_\theta(k) = q\,\varepsilon_\theta\sqrt{\nu/\varepsilon}\;k^{-1},$$

an exponent of minus one, with no velocity spectrum in the amplitude at all.

Measured off a model spectrum built to be self-consistent — normalised so that its own dissipation integral returns the scalar dissipation it was given — the two slopes come out at −1.659 and −1.018.

It is worth being clear about what “the scalar has no dynamics” buys and what it does not. The dye does not push the fluid, so the velocity field is whatever it would have been; that is what makes the problem tractable. But the scalar’s equation has a diffusivity in it that is not the viscosity, and that single independent parameter is enough to give the scalar a smallest scale of its own — and with it a range of behaviour the velocity field does not have.

So a passive scalar is not a diagnostic of the flow. It is a second field with its own physics, sharing a velocity field with the first, and the two agree only where neither of their diffusivities matters. That range — the inertial-convective one — is the only place where a dye picture can be read as a picture of the turbulence.

Why the second range exists at all #

The condition is that the scalar diffuse more slowly than momentum, which is the Schmidt number being greater than one.

The velocity’s smallest scale is Kolmogorov’s, set by the balance between inertia and viscosity. The scalar’s smallest scale is set by the balance between the strain that is stretching it and the diffusivity that is smoothing it, and since the strain below the Kolmogorov scale is constant, that balance gives

$$\eta_B = \frac{\eta}{\sqrt{\mathrm{Sc}}}$$

exactly. At a Schmidt number of one the two are one scale; at two thousand the scalar’s structure is forty-five times finer than the velocity’s; below one it is coarser, and there is no second range.

And the exponent follows from the same constancy. Below the Kolmogorov scale the velocity field is smooth, so a blob of dye is stretched by a strain rate that is the same at every scale — the Kolmogorov strain, the square root of the dissipation over the viscosity. A quantity stretched at a constant rate has its variance spread over wavenumber as the reciprocal of the wavenumber, because each doubling of wavenumber takes the same time to reach and carries the same variance. That is the whole derivation of $k^{-1}$, and there is nothing about turbulence in it beyond the smoothness.

There is a striking consequence of the strain being constant that is worth extracting. In the inertial range each eddy is stretched by the strain of eddies of its own size, and those strains get faster as the scale shrinks — which is why the cascade accelerates downwards and why the time spent at each scale falls. Below the Kolmogorov scale that stops: every scale is stretched at the same rate, so every octave takes the same time, and the cascade proceeds at a uniform pace all the way to the Batchelor scale.

That uniformity is why the range has an exponent at all, and why the exponent is so simple. It also gives the timescale on which mixing at those scales finishes: the reciprocal of the Kolmogorov strain, regardless of how many decades of Batchelor range there are. Adding Schmidt number adds range without adding time, which is the reason a high-Schmidt-number scalar mixes to molecular contact almost as quickly as a low one despite having so much further to go.

What each fluid actually has #

The range spans half a decade for every hundredfold of Schmidt number, so the answer varies enormously between things a reader would call the same problem.

Heat in air has a Schmidt number of 0.7 and therefore no second range at all — its Batchelor scale is larger than its Kolmogorov scale, so the scalar is smoothed before the velocity is. A gas mixing into another gas is much the same. Heat in water is at 7, giving four tenths of a decade; salt in water at 700 gives one and a half; dye at 2,000 gives one and seven tenths; a polymer at 10⁵ gives two and a half.

So “mixing in a turbulent flow” is not one problem. In a gas the scalar field ends where the velocity field ends; in a liquid it goes on for one to three decades below, in a regime whose physics is a smooth straining flow rather than turbulence.

Where the variance is, and where it is destroyed #

The two ranges do different jobs, and the split is stark.

Ninety-five per cent of the scalar’s variance sits above the Kolmogorov scale, in the range a simulation resolving the velocity can see. That is the part a reader would look at: the visible structure, the blobs and filaments of a dye photograph, the concentration fluctuations a probe measures.

Seventy-seven per cent of the destruction of that variance happens below it, where such a simulation has nothing at all.

The dissipation integrand peaks at the Batchelor scale rather than at the Kolmogorov one, three decades above where the variance is. The scalar’s variance is made at large scales, carried down by the velocity to the Kolmogorov scale, carried further by a smooth strain to the Batchelor scale, and destroyed there.

That separation is the practical content. A calculation that gets the variance right and the dissipation wrong will look correct in every visualisation and be wrong about the thing mixing is for — because what a reaction, a combustion or a dissolution needs is molecular contact, and molecular contact is the dissipation rather than the variance.

What it costs #

The extra range costs the three-halves power of the Schmidt number in grid points, because it is a factor of $\sqrt{\mathrm{Sc}}$ in each of three directions.

For heat in air that is a factor of 0.6 — nothing. For heat in water, 19. For salt, 19,000. For dye in water, 89,000, and that is the number that decides the practice: a simulation that resolves the velocity of a laboratory water experiment cannot resolve the dye in it by a factor of ninety thousand in cost, at any Reynolds number, on any machine anybody is likely to have.

So the scalar field of such a simulation is under-resolved, and it is under-resolved in the range where the dissipation is. That is not a criticism of anybody’s practice; it is a statement of what the arithmetic permits, and the reason for saying it is that the consequence — a scalar dissipation that is set by the grid rather than by the fluid — is invisible in every plot of the scalar field.

What under-resolution actually does #

It is worth saying what goes wrong rather than only that something does, because the failure has a signature.

A grid that stops at the Kolmogorov scale supplies its own smallest scale for the scalar, and the scalar’s dissipation then happens at that scale rather than at the Batchelor one. The rate is set by the cascade, which is resolved, so the total dissipation is roughly right — the variance still has to go somewhere and it goes there. What is wrong is where it happened, and therefore how much of the scalar has actually reached molecular contact.

The visible consequence is that the scalar field looks too smooth. Its filaments terminate at the grid rather than at the Batchelor scale, its gradients are under-estimated by whatever factor the missing decades represent, and any quantity built from a gradient — a reaction rate limited by mixing, a scalar dissipation rate, a flame surface density — is under-estimated with them.

The invisible consequence is worse: the fields look perfectly plausible. A dye picture from an under-resolved simulation has the same large-scale structure as a resolved one because ninety-five per cent of the variance is at large scales, and the missing part is the part nobody photographs.

Where this fits #

The pattern is one this collection keeps meeting. The energy cascade is a flux through scales whose rate is set at the large scales and whose destruction happens at the small ones, and the same is true of the scalar’s variance one range further down. The other layer, and the number that separates it from the first, is the laminar version of the same statement: a thermal layer and a momentum layer of different thickness, separated by a ratio of diffusivities.

What is new here is the exponent. Every other range in this subject inherits its scaling from the velocity field; the viscous-convective range does not, because there is no velocity spectrum left to inherit from. The strain rate does the work, it is constant, and the constant is what fixes the slope.

Where the two ranges are actually used #

Three places where the distinction is not academic.

Combustion. A non-premixed flame burns where fuel and oxidiser are in molecular contact, so its rate is controlled by the scalar dissipation rather than by the scalar variance. Gaseous combustion is at a Schmidt number near one and so has almost no Batchelor range, which is a considerable mercy; liquid-phase reaction is not, and the mixing-limited rate there depends on scales nothing resolves.

Oceanography. Heat and salt in seawater have Schmidt numbers of about seven and seven hundred, so salt has one and a half more decades of range than heat does. That difference is the origin of double-diffusive convection, in which a stably stratified column overturns because one scalar diffuses away faster than the other, and it is a direct consequence of the two Batchelor scales being different.

And laboratory measurement. A dye visualisation is a picture of the scalar at whatever scale the optics resolve, which is usually far above the Batchelor scale. The filaments in such a picture are optically thin structures whose true width is smaller than the image can show, which is why measured scalar gradients depend on the resolution of the instrument in a way that measured velocities usually do not.

In each case the thing that matters is the same: the gradient rather than the field, and the gradient lives in the range that is hardest to see.

How this sits beside the rest of the cascade #

It is worth placing the second range among the things this collection has already established about cascades, because it is both an extension of them and an exception to one of them.

The energy cascade is a flux of energy through scales at a rate set at the top and dissipated at the bottom, and the grid nobody can build prices the range of scales that implies. The scalar’s variance cascade is the same statement about a different quantity: a flux, set at the top, destroyed at the bottom.

The range a real Reynolds number does not have is the reminder that inertial ranges in practice are short. The viscous-convective range is the one piece of good news in that direction: it exists at any Reynolds number, because it does not need an inertial range above it.

The inverse cascade is the case where a flux runs the other way, which happens because two-dimensional flow has a second invariant. Nothing of the sort happens to a scalar: its variance has only one direction to go, and what decays never comes back.

And the one exact result in turbulence has a scalar counterpart — Yaglom’s four-thirds law, which relates the mixed third moment of a velocity increment and a scalar increment to the scalar dissipation, exactly, in the inertial-convective range. It is derived the same way, it is as exact, and it holds only in the range the scalar shares with the velocity. Below the Kolmogorov scale there is no such law, because there is no cascade of velocity to write it about.

The Péclet number is doing two jobs #

There is a tidy way to hold all of this, and it is worth setting out because the usual account collapses two different statements into one number.

The Péclet number compares advection with diffusion, and this collection has an essay on what it separates. Written at the large scale it is $UL/D$, and it says whether the scalar is carried or spread — which is a statement about the outer problem, and it is large in every case considered here.

Written at the smallest scale it is one, by definition: the Batchelor scale is where advection and diffusion balance. So the interesting quantity is not the Péclet number at all but the ratio of the two smallest scales, which is the square root of the Schmidt number, and that is a property of the pair of materials rather than of the flow.

That separation explains a fact that would otherwise be puzzling. Raising the Reynolds number extends the inertial range downwards and moves the Kolmogorov scale, and it drags the Batchelor scale with it by the same factor — so the number of decades of viscous-convective range does not change. The second range’s extent is fixed by the fluid pair and not by how hard the flow is stirred. Dye in water has one and seven tenths of a decade of it in a stirred beaker and one and seven tenths in the ocean.

Which is the last thing worth carrying. Most difficulties in turbulence get worse with Reynolds number and can be deferred by working at a modest one. This one cannot: it is the same at every Reynolds number, and a low-Reynolds-number experiment with dye in water has all of it.

The one thing that does resolve it #

There is a way to compute a high-Schmidt-number scalar field without paying the full factor, and it is worth naming because it follows directly from what the second range is.

Below the Kolmogorov scale the velocity field is smooth, so it does not need resolving there at all — it can be evaluated by interpolating from the velocity grid, whose smallest structure is the Kolmogorov scale by construction. Only the scalar needs the fine grid, and only for advection by a field that is already known.

That halves the exponent in the cost: a scalar-only refinement in three dimensions costs $\mathrm{Sc}^{3/2}$ in scalar grid points and nothing extra in velocity points, and the velocity solve is usually the expensive part of a simulation. Methods built that way — a coarse velocity grid and a fine scalar grid, or a Lagrangian scalar carried on particles — exploit exactly the smoothness that gives the range its exponent.

The general observation is worth carrying. A range whose physics is smooth is a range that can be represented cheaply, and identifying which parts of a multiscale problem are smooth is often worth more than resolving all of it uniformly. The Batchelor range is the clearest example in this subject: three decades of cascading in which the flow doing the cascading has no structure at all.

A closing note on why the range is called viscous-convective. Both words are doing work. Viscous because it lies below the Kolmogorov scale, where the viscosity has already smoothed the velocity field into a smooth strain. Convective because the scalar is still being carried rather than diffused — the diffusivity has not caught up, and will not until the Batchelor scale. A range that is viscous for one field and convective for another is exactly what a Schmidt number greater than one produces, and the name records it.

And a note on Schmidt against Prandtl. The two numbers are the same ratio for two different scalars — momentum diffusivity over mass diffusivity, and over thermal diffusivity — so everything on this page applies to heat with the Prandtl number in place of the Schmidt number. Liquid metals, whose Prandtl numbers are hundredths, have a Batchelor scale far larger than their Kolmogorov scale, which is the opposite regime and has its own literature.

What is not claimed #

The spectrum is a model spectrum. It is a product of two power laws and two cut-offs, normalised so that its own dissipation integral returns the scalar dissipation it was built from, which is the only self-consistency a model spectrum can have. No turbulent flow was simulated and no data were fitted.

The exponents come out of the model because they were put into it. What is measured is that the model’s local slopes are the intended ones in the intended ranges, which is a check on the construction and on the cut-offs rather than a derivation of Obukhov–Corrsin or Batchelor scaling. The cut-off shape had to be corrected once for exactly that reason: an exponential cut borrowed from the velocity spectrum contaminated the viscous-convective slope, reading −1.59 where the model was built to give −1.

Batchelor’s picture is not universally accepted. Kraichnan’s variant gives a different form near the cut-off, and measurements of the $k^{-1}$ range are difficult and have been argued about for decades. What is not in dispute is the Batchelor scale and the separation of the two ranges.

And the cost estimate assumes the same numerical requirement per scale. A method that resolves the scalar with a different scheme, or on a separate grid, or with an explicit subgrid model for the viscous-convective range, does not pay the full factor — and building such methods is exactly why the factor is worth knowing.