Turbulent flows are notoriously difficult to describe and understand based on first principles. One reason is that turbulence contains highly intermittent bursts of vorticity and strain-rate with highly non-Gaussian statistics. Quantitatively, intermittency is manifested in highly elongated tails in the probability density functions of the velocity increments between pairs of points. A long-standing open issue has been to predict the origins of intermittency and non-Gaussian statistics from the Navier-Stokes equations. Here we derive, from the Navier-Stokes equations, a simple nonlinear dynamical system for the Lagrangian evolution of longitudinal and transverse velocity increments. From this system we are able to show that the ubiquitous non-Gaussian tails in turbulence have their origin in the inherent self-amplification of longitudinal velocity increments, and cross amplification of the transverse velocity increments.
The Lagrangian and Eulerian transversal velocity structure functions of fully developed fluid turbulence are found basing on the Navier-Stokes equation. The structure functions are shown to obey the scaling relations inside the inertial range. The scaling exponents are calculated analytically without using dimensional considerations. The obtained values are in a very good agreement with recent numerical and experimental data.
In this paper, we analyze the scaling of velocity structure functions of turbulent thermal convection. Using high-resolution numerical simulations, we show that the structure functions scale similar to those of hydrodynamic turbulence, with the scaling exponents in agreement with She and Leveque's predictions [Phys. Rev. Lett. 72, 336-339 (1994)]. The probability distribution functions of velocity increments are non-Gaussian with wide tails in the dissipative scales and become close to Gaussian in the inertial range. The tails of the probability distribution follow a stretched exponential. We …
Using a recent alternative form of the Kolmogorov-Monin exact relation for fully developed hydrodynamics (HD) turbulence, the incompressible energy cascade rate $\varepsilon$ is computed. Under this current theoretical framework, for three-dimensional (3D) freely decaying homogeneous turbulence, the statistical properties of the fluid velocity (${\bf u}$), vorticity ($\boldsymbol ω= \boldsymbol\nabla \times {\bf u}$) and Lamb vector ($\boldsymbol{\cal L}= \boldsymbol ω\times {\bf u})$ are numerically studied. For different spatial resolutions, the numerical results show that $\varepsilon$ can …
We propose an alternative formulation for the exact relations in three-dimensional homogeneous turbulence using two-point statistics. Our finding is illustrated with incompressible hydrodynamic, standard and Hall magnetohydrodynamic turbulence. In this formulation, the cascade rate of an inviscid invariant of turbulence can be expressed simply in terms of mixed second-order structure functions. Besides the usual variables like the velocity ${\bf u}$, vorticity $ω$, magnetic field ${\bf b}$ and the current ${\bf j}$, the vectors ${\bf u } \times {\boldsymbol ω}$, ${\bf u} \times {\bf b}$ and ${…
In this work we address the open problem of high Reynolds number limit in hydrodynamic turbulence, which we modify by considering a vanishing random (instead of deterministic) viscosity. In this formulation, a small-scale noise propagates to large scales in an inverse cascade, which can be described using qualitative arguments of the Kolmogorov-Obukhov theory. We conjecture that the limit of the resulting probability distribution exists as $\mathrm{Re} \to \infty$, and the limiting flow at finite time remains stochastic even if forcing, initial and boundary conditions are deterministic. This c…
In earlier papers we have studied the turbulent flow exponents $ζ_p$, where $\langle|Δ{\bf v}|^p\rangle\sim\ell^{ζ_p}$ is the contribution to the fluid velocity at small scale $\ell$. Using ideas of non-equilibrium statistical mechanics we have found $$ ζ_p={p\over3}-{1\over\lnκ}\lnΓ({p\over3}+1) $$ where $1/lnκ$ is experimentally $\approx 0.32\pm 0.01$. The purpose of the present note is to propose a somewhat more physical derivation of the formula for $ζ_p$. We also present an estimate $\approx 100$ for the Reynolds number at the onset of turbulence.
The Lagrangian velocity structure functions in the inertial range of fully developed fluid turbulence are derived basing on the Navier-Stokes equations. For time $τ$ much smaller than the correlation time, the structure functions are shown to obey the scaling relations $K_n(τ)\propto τ^{ζ_n}$. The scaling exponents $ζ_n$ are calculated analytically. The obtained values are in amazing agreement with the unique experimental results of the Bodenschatz group \cite{Bod2}. New notion -- the Lagrangian position structure functions $R_n(τ)$ is introduced. All the $R_n$ of the order $n>3$ are shown to …
The appearance of sharp vorticity gradients in two-dimensional hydrodynamic turbulence and their influence on the turbulent spectra is considered. We have developed the analog of the vortex line representation as a transformation to the curvilinear system of coordinates moving together with the di-vorticity lines. Compressibility of this mapping can be considered as the main reason for the formation of the vorticity discontinuities at high Reynolds numbers. For two-dimensional turbulence in the case of strong anisotropy the vorticity discontinuities can generate spectra with the fall-off at la…
This article presents experimental and numerical studies on the effect of free stream turbulence on evolution of flow over an autonomous underwater vehicle (AUV) hull form at three Reynolds numbers with different submergence depths and angles of attack. The experiments were conducted in a recirculating water tank and the instantaneous velocity profiles were recorded along the AUV using Acoustic Doppler Velocimetry (ADV). The experimental results of stream-wise mean velocity, turbulent kinetic energy(TKE) and Reynolds stresses were used to validate the predictive capability of a Reynolds stress…
We analyze time series stemming from experiments and direct numerical simulations of hydrodynamic and magnetohydrodynamic turbulence. Simulations are done in periodic boxes, but with a volumetric forcing chosen to mimic the geometry of the flow in the experiments, the von Kármán swirling flow between two counter-rotating impellers. Parameters in the simulations are chosen to (within computational limitations) allow comparisons between the experiments and the numerical results. Conducting fluids are considered in all cases. Two different configurations are considered: a case with a weak externa…
The aim of this paper is to devise a turbulence model for the particle method Smoothed Particle Hydrodynamics (SPH) which makes few assumptions, conserves linear and angular momentum, satisfies a discrete version of Kelvin's circulation theorem, and is computationally efficient. These aims are achieved. Furthermore, the results from the model are in good agreement with the experimental and computational results of Clercx and Heijst for two dimensional turbulence inside a box with no-slip walls. The model is based on a Lagrangian similar to that used for the Lagrangian averaged Navier Stokes (L…
We present a numerical study of the statistical properties of three-dimensional dissipative turbulent flows at scales larger than the forcing scale. Our results indicate that the large scale flow can be described to a large degree by the truncated Euler equations with the predictions of the zero flux solutions given by absolute equilibrium theory, both for helical and non-helical flows. Thus, the functional shape of the large scale spectra can be predicted provided that scales sufficiently larger than the forcing length scale but also sufficiently smaller than the box size are examined. Deviat…
We study the statistical correlation functions for the three-dimensional hydrodynamic turbulence onset when the dynamics is dominated by the pancake-like high-vorticity structures. With extensive numerical simulations, we systematically examine the two-points structure functions (moments) of velocity. We observe formation of the power-law scaling for both the longitudinal and the transversal moments in the same interval of scales as for the energy spectrum. The scaling exponents for the velocity structure functions demonstrate the same key properties as for the stationary turbulence case. In p…
Pulsating flows through tubular geometries are laminar provided that velocities are moderate. This in particular is also believed to apply to cardiovascular flows where inertial forces are typically too low to sustain turbulence. On the other hand flow instabilities and fluctuating shear stresses are held responsible for a variety of cardiovascular diseases. Here we report a nonlinear instability mechanism for pulsating pipe flow that gives rise to bursts of turbulence at low flow rates. Geometrical distortions of small, yet finite amplitude are found to excite a state consisting of helical vo…
We investigate the scaling behavior of longitudinal and transverse structure functions in homogeneous and isotropic magneto-hydrodynamic (MHD) turbulence by means of an exact hierarchy of structure function equations as well as by direct numerical simulations of two- and three-dimensional MHD turbulence. In particular, rescaling relations between longitudinal and transverse structure functions are derived and utilized in order to compare different scaling behavior in the inertial range. It is found that there are no substantial differences between longitudinal and transverse structure function…
In active nematic liquid crystals activity is able to drive chaotic spatiotemporal flows referred to as active turbulence. Active turbulence has been characterized through theoretical and experimental work as a low Reynolds number phenomenon. We show that, in two-dimensions, the active forcing alone is able to trigger hydrodynamic turbulence leading to the coexistence of active and inertial turbulence. This type of flows develops for sufficiently active and extensile flow-aligning nematics. We observe that the combined effect of an extensile nematic and large values of the flow-aligning parame…
We consider the steady state statistics of turbulence in general classes of dissipative hydrodynamic equations, where the fluctuations are sustained by a random source concentrated at large scales. It is well known that in some particular cases, such as non-relativistic incompressible turbulence, a Kolmogorov-type exact scaling relation for a correlation function holds. We show that all such scaling relations follow from a general relation on the current-density correlation function. The derivation does not require an energy cascade picture and suggests that this traditional interpretation of …
We reformulate the problem of hydrodynamic turbulence as a heat flow problem. We obtain thus a prediction $$ ζ_p={p\over3}-{1\over\lnκ}\lnΓ({p\over3}+1) $$ for the exponents of the structure functions ($<|Δ_rv|^p>=r^{ζ_p}$). The meaning of the adjustable parameter $κ$ is that when an eddy of size $r$ has decayed to eddies of size $r/κ$ their energies have a thermal distribution. The above formula, with $(\lnκ)^{-1}=.32\pm.01$ is compatible with experimental data. This agreement lends supports to our physically motivated picture of turbulence.
A lattice Boltzmann scheme simulating the dynamics of shell models of turbulence is developed. The influence of high order kinetic modes (ghosts) on the dissipative properties of turbulence dynamics is studied. It is analytically found that when ghost fields relax on the same time scale as the hydrodynamic ones, their major effect is a net enhancement of the fluid viscosity. The bare fluid viscosity is recovered by letting ghost fields evolve on a much longer time scale. Analytical results are borne out by high-resolution numerical simulations. These simulations indicate that the hydrodynamic …
We derive a hierarchy of evolution equations for multi-point probability density functions in magneto-hydrodynamic (MHD) turbulence. We discuss the relation to the moment hierarchy in MHD turbulence derived by Chandrasekhar and derive a functional equation for a joint characteristic functional which can be considered as the analogon to the Hopf functional in hydrodynamic turbulence. Furthermore, we develop a closure method for the evolution equation of the single-point magnetic field probability density function which is based on a joint Gaussian assumption for unclosed terms. It is explicitly…
Superstatistics is a `statistics of a statistics' relevant for driven nonequilibrium systems with fluctuating intensive parameters. It contains Tsallis statistics as a special case. We show that the probability density functions of velocity differences and accelerations measured in Eulerian and Lagrangian turbulence experiments are well reproduced by simple superstatistics models. We compare fits obtained for log-normal superstatistics and chi-squared superstatistics (= Tsallis statistics).
We study formation of quasi two-dimensional (thin pancakes) vortex structures in three-dimensional flows, and quasi one-dimensional structures in two-dimensional hydrodynamics. These structures are formed at high Reynolds numbers, when their evolution is described at the leading order by the Euler equations for an ideal incompressible fluid. We show numerically and analytically that the compression of these structures and, as a consequence, the increase in their amplitudes is related to the compressibility of the frozen-in-fluid fields: the field of continuously distributed vortex lines in the…
We report a direct-numerical-simulation study of Taylor-Couette flow in the quasi-Keplerian regime at shear Reynolds numbers up to $\mathcal{O}(10^5)$. Quasi-Keplerian rotating flow has been investigated for decades as a simplified model system to study the origin of turbulence in accretion disks that is not fully understood. The flow in this study is axially periodic and thus the experimental end-wall effects on the stability of the flow are avoided. Using optimal linear perturbations as initial conditions, our simulations find no sustained turbulence: the strong initial perturbations distort…
{\bf Direct} transition from low Reynolds number "weak" Gaussian turbulence to fully developed "strong" turbulence at a critical Reynolds number $R^{tr}_λ\approx 8.91$ has recently been theoretically predicted and tested in high resolution numerical simulations of V. Yakhot \& D. A. Donzis, Phys. Rev. Lett. {\bf 119}, 044501 (2017) \& PhysicaD, {\bf 384-385}, 12 (2018) on an example of a flow excited by a Gaussian random force. The matching between the low-Reynolds number Gaussian asymptotic ($Re<<Re^{tr}$) and the multi-scaling one, dominating the high-Re limit ($Re>>Re^{tr}$), led to closed …
Compressible hydrodynamic turbulence is studied under the assumption of a polytropic closure. Following Kolmogorov, we derive an exact relation for some two-point correlation functions in the asymptotic limit of a high Reynolds number.
Drift towards isotropization during the 3D hydrodynamic turbulence onset
Shirshov Institute of Oceanology of RAS（俄罗斯科学院谢尔肖夫海洋学研究所）；Skolkovo Institute of Science and Technology（斯科尔科沃科学技术学院）；P.N. Lebedev Physical Institute of RAS（俄罗斯科学院列别杰夫物理研究所）；National Research University Higher School of Economics（国立高等经济大学）
The study is devoted to the development of new effective tools and methods of ana-lytical hydrodynamics, including problems of existence, smoothness and structure of laminar and turbulent flows. The main problem is complex Navier-Stokes equations and turbulence in complex spaces. The necessity of introducing complex spaces in hydrodynamics is deter-mined by the mechanism of transition of a laminar flow into a turbulent flow. The author pro-poses a non-traditional scenario of the transition: the cause of turbulence is in destruction (cessation of existence) of a laminar flow. The article contai…
A stochastic Euler equation is proposed, describing the motion of a particle density, forced by the random action of virtual photons in vacuum. After time averaging, the Euler equation is reduced to the Reynolds equation, well studied in turbulent hydrodynamics. It is shown that the average pressure is nonlocal and the magnitude of the turbulent flow obeys the Fick law. Using the Madelung transformation, the Schrodinger equation is derived without any other assumptions.
We introduce a novel approach to scale-decomposition of the fluid kinetic energy (or other quadratic integrals) into band-pass contributions from a series of length-scales. Our decomposition is based on a multiscale generalization of the ``Germano identity'' for smooth, graded filter kernels. We employ this method to derive a budget equation that describes the transfers of turbulent kinetic energy both in space and in scale. It is shown that the inter-scale energy transfer is dominated by local triadic interactions, assuming only the scaling properties expected in a turbulent inertial-range. W…
We derive exact scaling relations for two-dimensional relativistic hydrodynamic turbulence in the inertial range of scales. We consider both the energy cascade towards large scales and the enstrophy cascade towards small scales. We illustrate these relations by numerical simulations of turbulent weakly compressible flows. Intriguingly, the fluid-gravity correspondence implies that the gravitational field in spacetimes with anti-de Sitter asymptotics should exhibit similar scaling relations.
Using numerical simulations, we show that the asymptotic states of two-dimensional (2D) Euler turbulence exhibit large-scale flow structures due to nonzero energy transfers among small wavenumber modes. These asymptotic states, which depend on the initial conditions, are out of equilibrium, and they are different from the predictions of Onsager and Kraichnan. We propose ``hydrodynamic entropy'' to quantify order in 2D Euler turbulence; we show that this entropy decreases with time, even though the system is isolated with no dissipation and no contact with a heat bath.
We analyze the data stemming from a forced incompressible hydrodynamic simulation on a grid of 2048^3 regularly spaced points, with a Taylor Reynolds number of Re~1300. The forcing is given by the Taylor-Green flow, which shares similarities with the flow in several laboratory experiments, and the computation is run for ten turnover times in the turbulent steady state. At this Reynolds number the anisotropic large scale flow pattern, the inertial range, the bottleneck, and the dissipative range are clearly visible, thus providing a good test case for the study of turbulence as it appears in na…
A $p$-adic hydrodynamic type equation with two integrals of motion is proposed. It can be considered as a model cascade equation for energy dissipation in fully developed turbulence. Some of special cases of the proposed equation are detailed and it is shown that for a specific choice of parametrs they have stationary solutions that correspond to the 2/3 Kolmogorov-Obukhov law. Possible further studies of the proposed model are discussed.
The most efficient energy sources known in the Universe are accretion disks. Those around black holes convert 5 -- 40 per cent of rest-mass energy to radiation. Like water circling a drain, inflowing mass must lose angular momentum, presumably by vigorous turbulence in disks, which are essentially inviscid. The origin of the turbulence is unclear. Hot disks of electrically conducting plasma can become turbulent by way of the linear magnetorotational instability. Cool disks, such as the planet-forming disks of protostars, may be too poorly ionized for the magnetorotational instability to occur,…
We investigate experimentally three-dimensional (3D) hydrodynamic turbulence at scales larger than the forcing scale. We manage to perform a scale separation between the forcing scale and the container size by injecting energy into the fluid using centimetric magnetic particles. We measure the statistics of the fluid velocity field at scales larger than the forcing scale (energy spectra, velocity distributions, and energy flux spectrum). In particular, we show that the large-scale dynamics are in statistical equilibrium and can be described with an effective temperature, although not isolated …
Generalization of the viscous stress tensor to the case of non-small gradients of hydrodynamic velocity: a path to numerical modeling of turbulence non-locality
National Research Center “Kurchatov Institute”（库尔恰托夫国家研究中心）；National Research Nuclear University MEPhI（莫斯科国立核研究大学MEPhI）
We investigate the critical transition from an inverse cascade of energy to a forward energy cascade in a two-dimensional magneto-hydrodynamic flow as the ratio of magnetic to mechanical forcing amplitude is varied. It is found that the critical transition is the result of two competing processes. The first process is due to hydrodynamic interactions, cascades the energy to the large scales. The second process couples small scale magnetic fields to large scale flows transferring the energy back to the small scales via a non-local mechanism. At marginality the two cascades are both present and …
The statistics of 2-dimensional turbulence exhibit a riddle: the scaling exponents in the regime of inverse energy cascade agree with the K41 theory of turbulence far from equilibrium, but the probability distribution functions are close to Gaussian like in equilibrium. The skewness $\C S \equiv S_3(R)/S^{3/2}_2(R)$ was measured as $\C S_{\text{exp}}\approx 0.03$. This contradiction is lifted by understanding that 2-dimensional turbulence is not far from a situation with equi-partition of enstrophy, which exist as true thermodynamic equilibrium with K41 exponents in space dimension of $d=4/3$.…
We present results of numerical simulation of the direct cascade in two-dimensional hydrodynamic turbulence (with spatial resolution up to $16384 \times 16384$). If at the earlier stage (at the time of order of the inverse pumping growth rate $τ\simΓ_{max}^{-1}$), the turbulence develops according to the same scenario as in the case of a freely decaying turbulence \cite{KNNR-07, KKS}: quasi-singular distributions of di-vorticity are formed, which in $k$-space correspond to jets, leading to a strong turbulence anisotropy, then for times of the order of $10τ$ turbulence becomes almost isotropic.…
The classical statistics of turbulence are shown to be not specific to turbulence and can be derived from a solution for recurring unsteady state viscous flow. Care must be exercised in using them to make deductions about turbulence structures and mechanisms. The conditionally averaged statistics, particularly involving the velocities of the ejections in the burst phase, are more distinctive of turbulence. Key words: Turbulence statistics, classical, conditional average, probability density function, energy spectrum
Origin of hydrodynamic turbulence in rotating shear flow, e.g. plane Couette flow including the Coriolis force, is a big puzzle. While the flow often exhibits turbulence in laboratory experiments, according to the linear perturbation theory it should always be stable. We demonstrate that the secondary disturbance to the primarily perturbed flow triggers the elliptical instability in such a system and hence an exponential growing eigenmode. This result has an immediate application to astrophysics and geophysics which often exhibit a turbulent flow with the Keplerian angular momentum profile. We…
The effect of precession in a rotating sphere filled with fluid was studied with direct numerical simulations, both in the incompressible hydrodynamics (HD) and magnetohydrodynamics (MHD) scenarios. In both cases the asymptotic state and its dependence with both rotating and precession frequency was analyzed. For the MHD case no self-sustaining dynamos were found for the prograde precession case, whereas on the other hand a critical retrograde precession frequency was found above which dynamo action is self-sustained. It was also found that these correspond to small-scale dynamos with a develo…
We report on simulations of laboratory experiments in which magnetized supersonic flows are driven through a wire mesh. The goal of the study was to investigate the ability of such a configuration to generate supersonic, MHD turbulence. We first report on the morphological structures that develop in both magnetized and non-magnetized cases. We then analyze the flow using a variety of statistical measures, including power spectra and probability distribution functions of the density. Using these results we estimate the sonic mach number in the flows downstream of the wire mesh. We find the init…
This paper explores the application of SPH to a Direct Numerical Simulation (DNS) of decaying turbulence in a two-dimensional no-slip wall-bounded domain. In this bounded domain, the inverse energy cascade, and a net torque exerted by the boundary, result in a spontaneous spin up of the fluid, leading to a typical end state of a large monopole vortex that fills the domain. The SPH simulations were compared against published results using a high accuracy pseudo-spectral code. Ensemble averages of the kinetic energy, enstrophy and average vortex wavenumber compared well against the pseudo-spectr…
We investigate properties of the scale dependence and cross-scale transfer of kinetic energy in compressible three-dimensional hydrodynamic turbulence, by means of two direct numerical simulations of decaying turbulence with initial Mach numbers M = 1/3 and M = 1, and with moderate Reynolds numbers, R_lambda ~ 100. The turbulent dynamics is analyzed using compressible and incompressible versions of the dynamic spectral transfer (ST) and the Karman-Howarth-Monin (KHM) equations. We find that the nonlinear coupling leads to a flux of the kinetic energy to small scales where it is dissipated; at …
Statistical characteristics of freely decaying two-dimensional hydrodynamic turbulence at high Reynolds numbers are numerically studied. In particular, numerical experiments (with resolution up to $8192\times 8192$) provide a Kraichnan-type turbulence spectrum $E_k\sim k^{-3}$. By means of spatial filtration, it is found that the main contribution to the spectrum comes from the sharp vorticity gradients in the form of quasi-shocks. Such quasi-singularities are responsible for a strong angular dependence of the spectrum owing to well-localized (in terms of the angle) jets with minor and/or larg…
Using fluctuating hydrodynamics we investigate the effect of thermal fluctuations in the dissipation range of homogeneous, isotropic turbulence. Simulations confirm theoretical predictions that the energy spectrum is dominated by these fluctuations at length scales comparable to the Kolmogorov length. We also find that the extreme intermittency in the far-dissipation range predicted by Kraichnan is replaced by Gaussian thermal equipartition.
We study the possible origin of hydrodynamic turbulence in cold accretion disks such as those in star-forming systems and quiescent cataclysmic variables. As these systems are expected to have neutral gas, the turbulent viscosity is likely to be hydrodynamic in origin, not magnetohydrodynamic. Therefore MRI will be sluggish or even absent in such disks. Although there are no exponentially growing eigenmodes in a hydrodynamic disk, because of the non-normal nature of the eigenmodes, a large transient growth in the energy is still possible, which may enable the system to switch to a turbulent st…
Properties of the turbulent cascade of kinetic energy are studied using direct numerical simulations of three-dimensional hydrodynamic decaying turbulence with a moderate Reynolds number and the initial Mach number $M=1$. Compressible and incompressible versions of the Karman-Howarth-Monin (KHM) and low-pass filtering/coarse-graining approaches are compared. In the simulation the total energy is well conserved; the scale dependent KHM and coarse-grained energy equations are also well conserved; the two approaches show similar results, the system does not have an inertial range for the cascade …
In hydrodynamic turbulence, the kinetic energy injected at large scales cascades to the inertial range, leading to a constant kinetic energy flux. In contrast, in magnetohydrodynamic (MHD) turbulence, a fraction of kinetic energy is transferred to the magnetic energy. Consequently, for the same kinetic energy injection rate, the kinetic energy flux in MHD turbulence is reduced compared to its hydrodynamic counterpart. This leads to relative weakening of the nonlinear term ($\la | ({\bf u}\cdot \nabla) {\bf u} | \ra$, where ${\bf u}$ is the velocity field) and turbulent drag, but strengthening …
Statistical characteristics of the Kraichnan direct cascade for two-dimensional hydrodynamic turbulence are numerically studied (with spatial resolution $8192\times 8192$) in the presence of pumping and viscous-like damping. It is shown that quasi-shocks of vorticity and their Fourier partnerships in the form of jets introduce an essential influence in turbulence leading to strong angular dependencies for correlation functions. The energy distribution as a function of modulus $k$ for each angle in the inertial interval has the Kraichnan behavior, $\sim k^{-4}$, and simultaneously a strong depe…
The vorticity rotor field ${\bf B}=\mbox{rot}\,\mathbfω$ (di-vorticity) for freely decaying two-dimensional hydrodynamic turbulence due to a tendency to breaking is concentrated in the vicinity of the lines corresponding to the position of the vorticity quasi-shocks. The maximum value of the di-vorticity $B_{max}$ at the stage of quasi-shocks formation increases exponentially in time, while the thickness $\ell(t)$ of the maximum area in the transverse direction to the vector ${\bf B}$ decreases in time also exponentially. It is numerically shown that $B_{max} (t)$ depends on the thickness acco…
