7th Asia-Pacific Conference on Plasma Physics, 12-17 Nov, 2023 at Port Messe Nagoya

Linear response function of turbulence and its time scale

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Takeshi Matsumoto1, Michio Otsuki2, Takeshi Ooshida3, Susumu Goto2
Dep. Phys. Kyoto Univ., 2 Dep. Eng. Sci. Osaka Univ, 3Dep. Eng. Tottori Univ.
e-mail (speaker):takeshi@kyoryu.scphys.kyoto-u.ac.jp

Two-point correlation function, such as the energy
spectrum and the Reynolds stress, is the central object inthe turbulence research. We can write the equation of a
two-point correlation function by using the governing
equation, for example, the Navier-Stokes equations.
However the resultant equation involves three-point
correlation functions. In other words, it is not closed withthe two-point correlation functions, known as the closure
problem. Therefore we need to model higher ordercorrelation functions somehow in terms of the two-point
correlation functions.
One systematic method of such a modeling is arguably
to use linear response function. For the closure problem
on the energy spectrum of homogeneous isotropic
turbulence (HIT), the linear response function was firstused by Kraichnan in 1950s [1]. His closure
approximation, called direct interaction approximation(DIA), has opened up a number of new research avenues
while overcoming its initial failure in predicting theKolmogorov
law of the energy spectrum ( iswavenumber). Our first focus here is on one of theavenues, which concerns time scales of the two-point
correlation function and, in particular, of the linear
response function and how they differ depending on theEulerian or Lagrangian coordinates. Our second focus isan exact formula of the linear response function, which
can be obtained with a recent technique of nonequilibrium statistical mechanics [2].
Regarding the first focus, we perform a direct numericalsimulation (DNS) of HIT of a neutral fluid with a forcing
in a periodic cube using the spectral method. We thencalculate the temporal correlation functions of theFourier coefficients of the velocity and the linear
response function of the same Fourier coefficients. We do
this both in the Eulerian and Lagrangian coordinates (forthe latter we use so-called Lagrangian history velocity).
We find in the inertial range that their time scales in theEulerian coordinates are proportional toand that thetime scales in the Lagrangian coordinates areproportional toin agreement with the classicalpredictions. See Fig.1 for the measured time scale of thelinear response function as a function of the wavenumber
in both the Eulerian and Lagrangian cases.
Regarding the second focus, we adapt the theory for
nonlinear Langevin equations in [2] to HIT by adding aGaussian random noise term to the Navier-Stokes

equation in the Fourier space in addition to the largescale forcing. We obtain an exact formula of the linear
response function of the Fourier coefficients of thevelocity in the Eulerian coordinates. It involves two-point
and three-point correlation functions and the amplitudeof the Gaussian noise. We show that, for sufficiently
small noise, the formula agrees with the linear response
function without the noise. Also the formula shed somelight on the discrepancy between the two-point
correlation function and the linear response function(breakdown of the fluctuation dissipation relation).
Further details of this presentation can be found in [3].

Figure 1. Time scales of the linear response function as
a function of wavenumber . Here the time scale isdefined as the halving time of the linear response
function (the time at which the linear response function
is equal to 0.5). Herecorresponds to the time scaleof the linear response function of the velocity Fourier
coefficients in the Eulerian coordinates. And
corresponds to the time scale of the linear response
function in the Lagrangian coordinates.
References[1] R.H. Kraichnan, J. Fluid Mech., 5 497 (1959).
[2] T. Harada and S. Sasa, Phys.Rev.Lett., 95 130602(2005).
[3] T. Matsumoto, M. Otsuki, T. Ooshida and S. Goto, J.
Fluid Mech., 919 A9 (2021)