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On the zeroth law of turbulence for the stochastically forced Navier-Stokes equations

We consider three-dimensional stochastically forced Navier-Stokes equations subjected to white-in-time (colored-in-space) forcing in the absence of boundaries. Upper and lower bounds of the mean value of the time-averaged energy dissipation rate, $\mathbb{E} [\langle\varepsilon \rangle] $, are derived directly from the equations. First, we show that for a weak (martingale) solution to the stochastically forced Navier-Stokes equations, \[ \mathbb{E} [\langle\varepsilon \rangle] \leq G^2 + (2+ \frac{1}{Re})\frac{U^3}{L},\] where $G^2$ is the total energy rate supplied by the random force, $U$ is the root-mean-square velocity, $L$ is the longest length scale in the applied forcing function, and $Re$ is the Reynolds number. Under an additional assumption of energy equality, we also derive a lower bound if the energy rate given by the random force dominates the deterministic behavior of the flow in the sense that $G^2 > 2 F U$, where $F$ is the amplitude of the deterministic force. We obtain, \[\frac{1}{3} G^2 - \frac{1}{3} (2+ \frac{1}{Re})\frac{U^3}{L} \leq \mathbb{E} [\langle\varepsilon \rangle] \leq G^2 + (2+ \frac{1}{Re})\frac{U^3}{L}\,.\] In particular, under such assumptions, we obtain the zeroth law of turbulence in the absence of the deterministic force as, \[\mathbb{E} [\langle\varepsilon \rangle] = \frac{1}{2} G^2.\] Besides, we also obtain variance estimates of the dissipation rate for the model.