Viscosity-dependent inertial spectra of the Burgers and Korteweg–deVries–Burgers equations

The Inertial Range of the Burgers Equation

As is well known, the Burgers equation where x is a physical space variable, t is the time, ε is a viscosity and subscripts denote differentiation, has in the limit ε → 0 a Fourier transform û(k) with an inertial range spectrum E(k) ∼ k–², as can be readily seen from the fact that the solution u develops shocks whose spectral signature is k ^–2. But what happens to that spectrum when ε is small but finite?

To get a stationary solution of Eq. 1 we force the equation at infinity in physical space (at zero in wave number space) by imposing the boundary conditions u(–∞) = u ₀, u(+∞) = –u ₀, with u ₀ a positive constant. The corresponding solution of the equation is

It is quite satisfactory to consider a single smoothed shock (10, 11). There is no need to add a random forcing term as is done elsewhere, because this would violate the analogy to turbulence, where the focus of interest is on the intrinsic stochasticity produced by the equations and not on the response to extrinsic randomness. The problem now is simply to evaluate the Fourier transform û(k) of the solution (Eq. 2).

Define the energy spectrum E(k) by E(k) = û(k)û*(k), where the asterisk denotes a complex conjugate. Dimensional analysis (see, e.g., ref. 12) shows that the dimensionless spectrum must be a function Φ of the dimensionless variable η = kε/u ₀. When Φ = 1 the asymptotic k ^–2 spectrum ensues. We now determine Φ.

One can show that if (1 + |x|)u′(x) is in L ₁(–∞, +∞) then, in the sense of distributions, where the prime denotes a derivative and δ(k) is the delta function (see, e.g., ref. 13). One expects a delta function in an analysis of the spectrum in the inertial range, where it is the signature of the energy containing range (see ref. 14); with our boundary conditions this term is zero anyway. For the solution (Eq. 2) above the integrations can be carried out analytically and yield: Note that Φ(0) = 1 but Φ < 1 for all values of η > 0 with Φ″ < 0, so that there is a positive viscous correction to the inertial exponent –2 for any value of k and for any positive ε. This is exactly analogous to what we claim for turbulence. It is essentially different from what is claimed by others; the “intermittency corrections” derived by several authors for this very equation on the basis of tenuous analogies with field theory (see, e.g., ref. 15) do not decrease with viscosity.

In Fig. 1 we plot Φ as a function of k for several values of ε. We plot it this way, rather than showing the single curve Φ = Φ(η), to highlight the fact that, as a result of viscous effects, the inertial range spectrum is not a single function but that there is a distinct spectrum for each value of ε.

This calculation is, of course, elementary; the point is that so elementary a calculation is sufficient to display the phenomenology of the turbulence scaling models we described in the references quoted above.

To ease comparisons with the calculation in the next section, we want to point out that the spectrum of the steady traveling wave solution of Eq. 1 with boundary conditions u(–∞) = u ₀, u(+∞) = 0 is also given by Eq. 4 with 4, η replaced by 1, 2η respectively.

The Inertial Range of the KdVB Equation

We now carry out an analogous analysis for the KdVB equation where β > 0 is a dispersion coefficient of order 1. Some comments on the use of dispersive models in turbulence can be found in ref. 16. We create a steady traveling wave at long times by the boundary conditions u(–∞) = u ₀, u′(–∞) = 0, u(+∞) = 0 (see analyses in refs. 17–19). The translation of the wave does not affect the spectrum.

The wave can be found numerically after converting Eq. 5 into a second-order ordinary differential equation by first considering it in a frame of reference moving right with velocity 1 (the velocity of the steady wave) and then integrating once. This ordinary differential equation, written as a first-order system for u and w = u′, has a saddle at x = +∞ and a spiral point at x = –∞. The steady solution is the heteroclinic orbit that connects these points. The only problem is to find a point on that orbit that is not one of the end points. One can pick for that point values of x and u, say x*, u*, arbitrarily as long as u* in the range of the solution, for example x* = 0, u* = 1, and then solve for w* = u′(0) so that the orbit through (u*, w*) enter the saddle as x → +∞. Once this is done, one can march backward in x from the point (x*, w*) to the spiral point. A formula analogous to Eq. 3 holds for the Fourier transform and the integrals there can be evaluated numerically.

Dimensional analysis as above shows that where and are dimensionless variables and Ψ is a dimensionless function. The role of R bears some resemblance to that of a Reynolds number, hence the symbol (see refs. 17 and 18). Eq. 5 can be reduced by suitable changes of variables to the dimensionless form u_t + 2uu_x = R ^–1 u_xx – u_xxx with the boundary conditions u(–∞) = 1, u′(–∞) = 0, u(+∞) = 0. In Fig. 2 we display the steady wave form for this equation with R = 40.

In Fig. 3 we display the corresponding as a function of ζ for several values of R; we take the square root in Fig. 3 so that the large maximum not mask all of the detail away from the peak. In Fig. 4 we enlarge the representation of the lower ζ range. The large spike in the spectrum of Fig. 3 corresponds to the frequency of rotation around the spiral point on the left and therefore to the frequency of oscillation in Fig. 2.

In ref. 18 we showed that the averaged solution satisfies, approximately in the L ₂ sense and for l large enough, an equation of the form , where the parameter ε_eff, the effective dimensionless viscosity, satisfies the equation ε_eff = L(ℓ)R ^ν with ν > 0 a constant exponent and L a smooth function of its argument; we estimated there ν ∼ 0.75. Note that this scaling relation is singular, in the sense that as the dimensionless viscosity R ^–1 tends to zero, the effective viscosity ε_eff grows; the reason is that as the R → ∞, the wave train (see Fig. 2) becomes longer and the dissipation increases. Thus if one focuses only on the larger scales, the main effect of the dispersion is to enhance the diffusion.

Averaging damps the large ζ (above 1/ℓ) part of the spectrum. This suggests that if Ψ of Fig. 4 is plotted as function, not of ζ, but of R ^νζ, the curves would collapse onto a single curve, the spectral function Φ of the effective Burgers equation. In Fig. 5 we display some spectra in this representation with ν = 0.9 and indeed observe the collapse. With ν = 0.75, the value we obtained in ref. 18, the collapse is slightly less exact; the difference in exponents is due to (i) the difference between the L ₂ approximation in ref. 18 and the approximation in L ₁ implied here by the pointwise comparison of spectra, and (ii) the fact that we cannot access by the methods of the present article the coefficient L(ℓ) in the power law for ε_eff and had to set L = 1 arbitrarily.

The relation between R and ε_eff was produced in ref. 18 by actually solving the full equation, averaging it, and only then looking for an effective equation whose solution matched the average. We see here that one can indeed approximate the low-k part of even a highly oscillatory solution by a suitably renormalized diffusive equation, as is assumed in most numerical treatments of turbulence. However, the coefficients in that diffusive equation have to be determined by an appropriate scaling/renormalization argument and do not follow from a straightforward averaging.

Conclusions

We have used simple examples as positive evidence for two propositions, for which we have argued elsewhere in more complex settings: (i) The intermediate asymptotics of turbulence is viscosity-dependent, and (ii) averaging equations requires a more careful consideration of the properties of averaged solutions than is usually granted.