Emergent dynamical scaling in the inviscid limit of 3D stochastic Navier-Stokes equation with thermal noise

TL;DR

The paper investigates the stochastic 3D Navier–Stokes equation with thermal noise (model A) using functional renormalization group (FRG) and direct numerical simulations (DNS). It identifies a nonperturbative inviscid fixed point (IFNS) governing the large-momentum dynamics, with a dynamical exponent $z=1$ and exact large-momentum closures from Ward identities, alongside an IR fixed point (FNS) with $z=2$. The results show a crossover between UV $z=1$ and IR $z=2$ regimes, corroborated by DNS which reveal two scaling regimes in temporal correlations and a clear UV/IR separation. The findings link fluctuating hydrodynamics to Burgers–KPZ universality, providing a framework to understand high-wavenumber temporal correlations in both near-equilibrium and turbulent settings, though the UV $z=1$ scaling is unlikely to be observed in equilibrium molecular fluids. Overall, the work advances nonperturbative analyses of stochastic fluid dynamics and elucidates how symmetry constraints drive universal temporal scaling at large $k$.

Abstract

In this work, we investigate the Navier-Stokes equation in the presence of thermal noise, both at finite viscosity (revisiting the seminal work by Forster-Nelson-Stephen) and in the inviscid limit, which has not yet been explored. We determine the space-time velocity correlations in this dynamics, using functional renormalisation group and direct numerical simulations. While spectrally truncated three-dimensional Euler flows reach a stationary equilibrium state, they exhibit non-trivial temporal correlations. We show that these non-trivial correlations persist for small but finite viscosity, yielding an emergent $τ\sim k^{-1}$ dynamical scaling, where $τ$ is the decorrelation time. We characterise the crossover from the scaling $τ\sim 1/(νk^2)$, expected at large viscosity, to the scaling $τ\sim 1/(u_{\rm rms}k)$ found in the inviscid limit.

Figures (6)

• Figure 1: FRG flows of the non-linear coupling $\hat{w}_\kappa$ for (a) the stochastic 1D Burgers and (b) the stochastic 3D Navier-Stokes equations: in red flows from $s=\ln(\kappa/\Lambda)=0$ to $s\to-\infty$ ($\kappa\to0$) towards the IR fixed points for different initial conditions $\hat{w}_\Lambda$; in blue flows from small initial $\kappa_0$ (large negative $s\lesssim -20$) to $s\to0$ ($\kappa\to\Lambda$) towards the UV fixed points for different initial conditions $\hat{w}_{\kappa_0}$.
• Figure 2: Sketch of the fixed point structure of the stochastic 1D Burgers and stochastic 3D Navier-Stokes equations in the presence of thermal noise. The red points represent attractive (IR) fixed points and the blue points repulsive (UV) ones. The red-shaded (resp. blue-shaded) area hence symbolises the IR (resp. UV) modes controlled by the IR (resp. UV) fixed points.
• Figure 3: Half decay frequency for different initial values of the viscosity. The IR modes show a diffusive FNS scaling $\omega_{1/2}\sim p^2$. The IFNS scaling, characterised by $\omega_{1/2}\sim p$, appears for UV modes when the viscosity decreases. In this figure, the momentum $p$ is measured in units of $\Lambda$ and the viscosity $\nu$ in units of $(\Lambda\chi)^{1/2}$.
• Figure 4: Contour plots of the correlation function $\bar{C}(t,k)/\bar{C}(0,k)$ represented in the natural logarithm $(\log(k),\log(t))$ plane; (left): inviscid scaling obtained at $\nu=0$; (right): viscous scaling obtained at $\nu= 0.016$. $\bar{C}(t,k)/\bar{C}(0,k)$ contour levels are drawn from $0.0$ to $0.9$ and spaced by $0.1$. The dashed black lines indicate (left) $t\sim k^{-1}$ and $t\sim k^{-2}$; (right) $t\sim k^{-2}$. The points $\tau_{1/2} (k)$, computed independently using $\bar{C}(\tau_{1/2},k)={\frac{1}{2}}\bar{C}(0,k)$ are indicated by red circles.
• Figure 5: Contour plots of $\omega S(\omega,k)$, where $S(\omega,k)$ is the power spectrum, i.e the Fourier transform of $\bar{C}(t,k)/\bar{C}(0,k)$, represented in the natural logarithm $(\log(k),\log(\omega))$ plane; (left): inviscid scaling obtained at $\nu=0$; (right): viscous scaling obtained at $\nu= 0.016$. The dashed black lines indicate (left) $\omega\sim k^{1}$ and $\omega\sim k^{2}$; (right) $\omega\sim k^{2}$. The contour levels of $\omega S(k,\omega)$ are: (left) from $0.0$ to $0.09$ (spaced by $0.01$) and (right) $0.0$ to $0.9$ (spaced by $0.1$).