37. Eye on PRL – On statistical theory of developed turbulence

Although I am not an expert in the field of turbulence, I do remember a couple of tricks from hydrodynamics (even had a couple of papers on ideal hydrodynamics in the far past together with my friend Vitya Ruban). So when I found the following intriguely looking paper on PRL

K. P. Zybin, V. A. Sirota, A. S. Ilyin, and A. V. Gurevich, “Lagrangian Statistical Theory of Fully Developed Hydrodynamical Turbulence”,

I decided to write a couple of words about it.

When one says “fully developed turbulence”, one usually means behavior of the liquid flow driven by the Navier-Stokes equation

(1)

with all the nonlinearities taken into account. From the experiment we see that correlation functions of behave awfully; namely, all higher order (multipoint) irreducable correlators of are not negligible.

The problem of 3d developed turbulence is horrible. Needless to say, it is even unclear whether the flow develops singularity in finite time (i.e., equal time but multipoint corr. functions of diverge at some for finite separations of points) or a smooth general solution of the Naiver-Stockes equation exists for arbitrary initial conditions. In this respect, let me mention that this problem is really close to the one for the solution of which Clay Institute offers a banch of money.

One can try to simplify probem a bit by, say, discussing turbulence in 1D, or forgetting about the vector nature of the velocity (the latter approach was pursued by Kraichnan and is called “passive scalar turbulence”). As it turns out, statistics of the turbulent scalar field is such that rare events dominate and lead to strong nongaussianities. This is quite a subject by itself, but we want to be smart and to learn how to deal with developed turbulence of the vector field in 3D, don’t we?

Is there any way to understand what happens with in the regime of developed turbulence? The answer was given by Kolmogorov back in 1941 – one needs to consider physics in the inertial interval, when the viscosity in the Eq. (1) can be neglected. The reason is that viscosity is only important at relatively small scales (in Fourier, viscosity term is proportional to ) where energy dissipates.

To develop turbulence, one needs to pump energy into the system at large scales. In the regime of developed turbulence (when energy pumped into the system per unit time is equal to energy dissipated per unit time due to viscosity) Navier-Stokes equation admits an approximate scaling solution, that is known as Kolmogorov cascade. The meaning is that energy is pumped in the IR, cascades down to the UV and then dissipates when viscosity term becomes important.

Unfortunately, it turns out that it was hard to say much more compared to what Kolmogorov said – for example, behavior of the multipoint correlation functions of velocity field is very hard to understand, etc.

Let me go at this point to the paper I wanted to discuss. What is new in what these people say? Let us again focus on physics in inertial interval. When viscosity is neglected, trivially, one has

,

(i.e., Navier-Stockes equation becomes Euler equation) and

which is just continuity equation (the density is taken to be constant).

The main interesting property of the Euler equation is that vorticity field

is frozen into the motion of the fluid according to the Thompson’s circulation theorem. Each vortex line is indenendent degree of freedom (in a sense, it is a string ), and these degrees of freedom interact with each other through Coulomb interaction. So, if we learn how to deal with vortex lines, we will understand pretty much all the nontrivial physics in the inertial interval.

Second thing to understand is that vorticity is not conserved for complete problem (with viscosity and pumping taken into account). Vortex filaments appear due to pumping in the IR, then get stretched due to the interaction between them and finally the horrible mess happens with them in the UV where viscosity gets important. Only the stretching phase is under control (it proceeds in the inertial interval), while physics at pumping scales can be modelled by stochastics (saying, that filaments are generated according to a certain statistics).

Introducing Gaussian statistics, the authors were able to actually derive Fokker-Planck equation for the vorticity field, which, I think, is really cool.

My congrats to Lebedev Institute’s team!