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

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.

Figure 1

Second-order moment of the absolute value of the shell-model variable $u_j\left(t\right)$ with and without the Gaussian white noise, as a function of the shell index. Here ${\sigma }_j^2=1$ in Eq. (2). Inset: The energy flux function $\Pi \left(k_j\right)$ showing that the constant-energy-flux structure is preserved for $T\le {10}^{-3}$.

Figure 2

Directly calculated response function of the shell model $G_{jj}^{\left(0\right)}$ (zero temperature), $G_{jj}^{\left(T\right)}$ with $T={10}^{-4}$, and the FRR expression of the response function $H_{jj}^{\left(T\right)}$ with $T={10}^{-4}$, the right-hand side of Eq. (5), for the shell indices $j=9,10,11$, and 12 (from top to bottom). The gray curve is $C_{jj}^{\left(0\right)}(t-s)/C_{jj}^{\left(0\right)}\left(0\right)$ for $j=12$. Inset: Approach of $G_{jj}^{\left(T\right)}$ to $G_{jj}^{\left(0\right)}$ as $T\to 0$ for $j=9$, plotted with $H_{jj}^{\left(T\right)}$.

Figure 3

Numerical difficulty of the FRR. (a) Discrepancy between $H_{jj}^{\left(T\right)}$ and $G_{jj}^{\left(0\right)}$ for $j=4,6$, and 8 (from top to bottom) with $T={10}^{-4}$; The error bars of $G_{jj}^{\left(0\right)}$ are shown for the index $j=6$ up to $t-s=1.5$. Inset: Same as the outset but for $j=2,4,6$, and 8 with $T={10}^{-3}$. (b) Cancellation between the real parts of $\langle {\Lambda }_j^{*}\left(s\right)u_j\left(t\right)\rangle$ and $\langle {\Lambda }_j^{*}\left(t\right)u_j\left(s\right)\rangle$ with $T={10}^{-4}$. (c) Symmetry of the real parts of $\langle {\Lambda }_j^{*}\left(s\right)u_j\left(t\right)\rangle$ and $\langle {\Lambda }_j^{*}\left(t\right)u_j\left(s\right)\rangle$ for $j=8$ (solid) and 11 (dashed) with $T={10}^{-4}$.