Nonperturbative renormalization group study of the stochastic Navier-Stokes equation

We study the renormalization group flow of the average action of the stochastic Navier-Stokes equation with power-law forcing. Using Galilean invariance, we introduce a nonperturbative approximation adapted to the zero-frequency sector of the theory in the parametric range of the Hölder exponent $4-2\varepsilon$ of the forcing where real-space local interactions are relevant. In any spatial dimension $d$, we observe the convergence of the resulting renormalization group flow to a unique fixed point which yields a kinetic energy spectrum scaling in agreement with canonical dimension analysis. Kolmogorov's $-5/3$ law is, thus, recovered for $\varepsilon =2$ as also predicted by perturbative renormalization. At variance with the perturbative prediction, the $-5/3$ law emerges in the presence of a saturation in the $\varepsilon$ dependence of the scaling dimension of the eddy diffusivity at $\varepsilon =3/2$ when, according to perturbative renormalization, the velocity field becomes infrared relevant.

Figure 1

Result of the numerical integration of the dimensionless renormalized functions ${\gamma }^{(1,1)}\left(p\right)$ and ${\gamma }^{(0,2)}\left(p\right)$ for $d=3$ and $\varepsilon =2$. The dashed line indicates the value 1. (Inset) Convergence of ${\gamma }^{(1,1)}\left(p\right)$ from its initial (unforced limit) value ${\gamma }^{(1,1)}\left(p\right)=1$ toward stationarity (as indicated by the arrow).

Figure 2

Dependence of the fixed point $({\lambda }_0,{\lambda }_1)$ (blue dots) on $\varepsilon$ and $d=3$. The fixed point tends toward $(0,0)$ as $\varepsilon \to 0$.

Figure 3

Scaling exponents $\Lambda$ in Eq. (93) of the dimensionless renormalized functions ${\gamma }^{(1,1)}\left(p\right)$ and ${\gamma }^{(0,2)}\left(p\right)$ and of the energy spectrum $\mathcal{E}$ as a function of $\varepsilon$ (blue solid circles) and $d=3$. We also show in red open squares the dependence of the scaling exponents ${\eta }_{\kappa }$ (upper panel) and ${\eta }_F$ (middle panel) on $\varepsilon$. The solid lines correspond to the respective renormalization group scaling of Eqs. (75) and to $p^{-4\varepsilon /3}$ for the energy spectrum. The dashed lines in the upper and middle panels stand for the scalings $p^{-1}$ and $p^{1-d-4\varepsilon /3}$, respectively.

Figure 4

Basin of attraction of the fixed point in three dimensions for different values of $\varepsilon$. Each yellow (light gray) dot denotes an initial condition of $[{\lambda }_{\left(0\right)},{\lambda }_{\left(1\right)}]$ for which convergence was reached. The square with dashed sides indicates the domain in which random initial conditions were drawn. The light gray solid circles denote the trajectory of the fixed point in the ${\lambda }_{\left(0\right)}$-${\lambda }_{\left(1\right)}$ plane and the blue (dark gray) circle denotes the fixed point for the specific value of $\varepsilon$.

Figure 5

Scaling exponents $\Lambda$ in Eq. (93) of the dimensionless renormalized functions ${\gamma }_{☆}^{(1,1)}\left(p\right)$ and ${\gamma }_{☆}^{(0,2)}\left(p\right)$ and of the energy spectrum $\mathcal{E}$ as a function of $\varepsilon$ (blue solid circles) for the simplified model discussed in Sec. 6 and for $d=3$. The red open squares in the upper panel correspond to ${\eta }_{\kappa }$. The solid and dashed lines correspond to the predicted scaling of Eqs. (87), (90), and (91) for $\varepsilon <3/2$ and $\varepsilon >3/2$, respectively.

Figure 6

Dependence of the fixed point $({\lambda }_0,{\lambda }_1)$ (blue dots) on $\varepsilon$ and $d=2$. The fixed point tends toward $(0,0)$ as $\varepsilon \to 0$.

Figure 7

Scaling exponents $\Lambda$ in Eq. (93) of the dimensionless renormalized functions ${\gamma }^{(1,1)}\left(p\right)$ and ${\gamma }^{(0,2)}\left(p\right)$ and of the energy spectrum $\mathcal{E}$ as a function of $\varepsilon$ (blue solid circles) and $d=2$. We also show in red open squares the dependence of the scaling exponents ${\eta }_{\kappa }$ (upper panel) and ${\eta }_F$ (middle panel) on $\varepsilon$. The solid and dashed lines correspond to the predicted scaling of Eqs. (87), (90), and (91) for $\varepsilon <3/2$ and $\varepsilon >3/2$, respectively.

Figure 8

Basin of attraction of the fixed point in two dimensions for different values of $\varepsilon$. Each yellow (light gray) dot denotes an initial condition of $[{\lambda }_{\left(0\right)},{\lambda }_{\left(1\right)}]$ for which convergence was reached. The square with dashed sides indicates the domain in which random initial conditions were drawn. The light gray solid circles stand for the trajectory of the fixed point in the ${\lambda }_{\left(0\right)}$-${\lambda }_{\left(1\right)}$ plane and the blue (dark gray) circle denotes the fixed point for the specific value of $\varepsilon$.