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Consider the equation

$latex dz+\nu A^{\alpha}z=C^{1/2}dW, z(0)=0$

where for $latex x:=\sum_{k\in Z^{2}\setminus \{(0,0)\}} x_{k}e_{k}$, with $latex \sum |x_{k}|^{2}<\infty$,

we set $latex A^{\alpha}(x):=\sum_{k\in Z^{2}_{*}}|k|^{\alpha} x_{k}e_{k}$ for $latex \alpha>1$

and $latex C(x):=\sum_{k\in Z^{2}_{*}}\sigma_{k}^{2}x_{k}e_{k}$ such that for $latex \delta\in (1-\alpha,2-\alpha)$ and $latex \frac{c_{2}}{|k|^{\delta}}\leq |\sigma_{k}|\leq \frac{c_{3}}{|k|^{\delta}}$.

RO: The dimension of the zero level set is $latex dim(\{x\in T_{2}: z_{t}(x)=0\})=3-\alpha-\delta$.

Reference: Hausdorff dimension of the level sets of some stochastic PDEs from fluid dynamics

Conformal invariance (CI)

In “Conformal invariance in two-dimensional turbulence” by Bernard etal, it was guessed that numerically the zero level set (ZLS) for the vorticity field has the same properties as an SLE curve.

• Since NSE and Euler equations are volume preserving, the equations themselves are not conformally invariant. So one has to study the equations satisfied by the countour lines themselves.
•  The zero level set is in the intersection of the support of all invariant measures. Recall that this holds, because there is a small probability that the random forces are close to zero for any given time interval so the fluid flow slows down due to viscosity.  Moreover, the invariant measures seem to possess strong scale invariance properties, at least in the inverse cascade regime. So the zero level set has some special statistical properties that other level sets do not.
•  Since the linearization is stochastic heat equation (SHE) whose stationary solution is the Gaussian free field (GFF), it is reasonable that in the long run the ZLS of 2DNSE is close to the ZLS of SHE which is in turn close to the ZLS of GFF aka SLE(4).

 

Dimension of the zero level set

What is the dimension of zero vorticity level set?