37. Eye on PRL - On statistical theory of developed turbulence
DmitryAlthough 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
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!
| 2.5 |
Related posts:
36. Eye on ArXiv: 6 May 2008 - Where does the cosmological perturbation theory actually break down?
DmitryI am going to briefly discuss today the paper by Cristian Armendariz-Picon, Michele Fontanini, Riccardo Penco, and Mark Trodden “Where does cosmological perturbation theory break down?”
The issue of possibility to describe inflation in terms of effective QFT keeps bothering people’s minds :) Let me remind you that two weeks ago we had a paper by Steven Weinberg discussing the similar subject. What is the essence of the problem that Christian et al. found? Naively, it seems that as long as 
, dynamics of inflaton + metric perturbations can be described in terms of QFT (canonically normalized scalar field with some potential). Well, not quite so, as the authors say.
If you start to take effective QFT tree level corrections into account, the correlator for Mukhanov variable will have the form
(1)
etc., so the actual smallness parameter in the effective QFT expansion should not be 
, but something like 
instead.
I believe this fact by itself is not new. Take a correlation function for curvature perturbation 
. Apart from the standard 
term there are corrections to this correlator small with respect to the parameter
(2)
where 
is the number of efoldings (see our paper for example as well as recent works by David Seery, David Lyth etc.) These corrections grow while the number of efoldings grows and the expansion breaks down sooner or later. Now, if we take a look at the formula (1), we see that the smallness parameter there looks remarkably similar to (2). Indeed, the number of efoldings is defined as
where 
is the slow roll parameter.
What is the physical meaning of the effective QFT breakdown the authors found? The point is that the number of efoldings cannot be very large, and if it is, then inflation enters eternal self-reproducing regime. There the bare notion of homogeneous isotropic FRW backround breaks down (eternally inflating Universe consists of different Hubble patches, and the distribution of 
among them is highly inhomogeneous). Still, the theory is under control even in this regime (instead of effective QFT, we can describe dynamics of the inflaton field by stochastic formalism).
By the way, authors say on the p. 7 that by in-vacuum they mean Bunch-Davies. This is probably a misprint: in-vacuum corresponds to the absence of particles at 
, while Bunch-Davies is Euclidean vacuum corresponding to the linear combination of in- and out-modes.
P.S. There is also very nice description of the Schwinger-Keldysh formalism in the paper.
| 2.5 |
Related posts:
