Folding in fluids and MHD

Evgenii Kuznetsov
P.N. Lebedev Physical Institute of RAS, Moscow, Russia; L.D. Landau Institute for Theoretical Physics of RAS, Chernogolovka, Moscow region, Russia; Skolkovo Institute of Science and Technology, Skolkovo, Russia

The formation of the coherent vortical structures in the form of thin pancakes for three-dimensional flows and quasi-shocks of the vorticity in two-dimensional turbulence is studied at the high Reynolds regime when, in the leading order, the development of such structures can be described within the Euler equations for ideal incompressible fluids. Numerically and analytically on the base of the vortex line representation we show that compression of such structures and respectively increase of their amplitudes are possible due to the compressibility of the vorticity $\mathbf{\omega}$ in the 3D case and of the di-vorticity field ${\bf B}=\mbox{rot}\,\mathbf{\omega}$ for 2D geometry. It is demonstrated that, in both cases, this growth has an exponential behavior and can be considered as folding (analog of breaking) for the divergence-free fields of both vorticity and di-vorticity. At high amplitudes this process in 3D has a self-similar behavior connected the maximal vorticity and the pancake width by the relation of the universal type [1]: $ \omega_M\propto l^{-2/3} $ . For the 2D turbulence numerically it is shown that $B_M (t)$ depends on the quasi-shock thickness according to the same power law: $B_M(t)\sim \ell^{-\alpha}(t)$, where the exponent $\alpha\approx 2/3$, that indicates also in favor of folding for the di-vorticity field [2]. Appearance of the $2/3$-law in fluids is a consequence of frozenness for both vorticity and di-vorticity fields. In this talk we consider also the problem of generation of strong magnetic fields in MHD due to the folding mechanism predicted in [3]. On our opinion, the formation of magnetic filaments in the convective zone of the Sun can be explained by this mechanism. At the end of this talk we discuss the role of folding structures in the formation of the Kolmogorov spectrum in 3D and the Kraichnan spectrum for two-dimensional turbulence.

[1] D.S. Agafontsev, E.A. Kuznetsov and A.A. Mailybaev, Development of high vorticity structures and geometrical properties of the vortex line representation, Phys. Fluids 30, 095104-13 (2018).

[2] E.A. Kuznetsov and E.V. Sereshchenko, Folding in two-dimensional hydrodynamic turbulence, Pis'ma ZhETF, 109, 231 – 235 (2019) [JETP Letters, 109, issue 4 (2019), DOI 10.1134/S0021364019040039].

[3] E.A. Kuznetsov, T. Passot and P.L. Sulem, Compressible dynamics of magnetic field lines for incompressible MHD flows, Physics of Plasmas, 11, 1410-1415 (2004).