Response function of turbulence computed via fluctuation-response relation of a Langevin system with vanishing noise

Abstract

For a shell model of the fully developed turbulence and the incompressible Navier-Stokes equations in the Fourier space, when a Gaussian white noise is artificially added to the equation of each mode, an expression of the mean linear response function in terms of the velocity correlation functions is derived by applying the method developed for nonequilibrium Langevin systems [Harada and Sasa, Phys. Rev. Lett. 95, 130602 (2005)]. We verify numerically for the shell-model case that the derived expression of the response function, as the noise tends to zero, converges to the response function of the noiseless shell model.

Correlation function and linear response function of homogeneous isotropic turbulence in the Eulerian and Lagrangian coordinates

“…By increasing the number of samples, the deviations becomes smaller, however. The worse agreement of H (T) has been anticipated from our previous study of the shell model (Matsumoto et al 2014) since the summations in the shell-model equivalent of (2.15) caused loss of significant digits, in particular, in the inertial range. This is also the case for the Navier-Stokes case, as we will now show.…”

“…In particular, the twofold cancellations of the Harada-Sasa FRR indicated that the deviation from the FDT is caused by both the nonlinearity and the dissipation. Whether or not the effect of the dissipation diminishes as we increase the Reynolds number, as suggested by the shell-model study (Matsumoto et al 2014), remains to be seen. If the nonlinear part of the Harada-Sasa FRR, which corresponds to the energy transfer function at equal times, becomes dominant at high Reynolds numbers, it is tempting to connect the breakdown of the FDT with the energy cascade.…”

“…In particular, it was shown that the FDT is invalid for forced Navier–Stokes turbulence in the dissipation range in Carini & Quadrio (2010) and for the forced Sabra shell model in our previous work (Matsumoto et al. 2014). The breakdown of the FDT is surely a manifestation of the out-of-equilibrium character of turbulence and of the shell model.…”

“…This recent development in non-equilibrium statistical mechanics has urged us to consider the correlation function and the linear response function of turbulence from a new perspective. This Harada-Sasa relation was the basis of our previous study (Matsumoto et al 2014) to consider a similar FRR for the shell model and the Navier-Stokes equations in the Eulerian coordinates. With random noise, there is yet another general expression of the linear response function in terms of the correlation between the random noise itself and the solution.…”

“…In this paper we call it Novikov-Carini-Quadrio FRR. Now we move to another expression of the linear response function in terms of the two-point or multi-point correlation functions ofû, which was outlined in Matsumoto et al (2014). For brevity, we write the nonlinear term and the large-scale forcing as…”