arXiv subjects

Alexander Migdal

Publications and source records attributed to Alexander Migdal.

35 records · Page 2Linked to original sources

arXiv2023

Topological vortexes, asymptotic freedom, and multifractals

We study the Kelvinons: monopole ring solutions to the Euler equations, regularized as the Burgers vortex in the viscous core. There is finite anomalous dissipation in the inviscid limit. However, in the anomalous Hamiltonian, some terms are growing as logarithms of Reynolds number; these terms come from the core of the Burgers vortex. In our theory, the turbulent multifractal phenomenon is similar to asymptotic freedom in QCD, with these logarithmic terms summed up by an RG equation. The small effective coupling does not imply small velocity; on the contrary, velocity is large compared to its fluctuations, which opens the way for a quantitative theory. In the leading order in the perturbation theory in this effective coupling constant, we compute running multifractal dimensions for high moments of velocity circulation, in good agreement with the data for quantum Turbulence and available data for classical Turbulence. The logarithmic dependence of fractal dimensions on the loop size comes from the running coupling in anomalous dimensions. This slow logarithmic drift of fractal dimensions would be barely observable at Reynolds numbers achievable at modern DNS.

physics.flu-dyn↗

arXiv2023

Statistical Equilibrium of Circulating Fluids

We are investigating the inviscid limit of the Navier-Stokes equation, and we find previously unknown anomalous terms in Hamiltonian, Dissipation, and Helicity, which survive this limit and define the turbulent statistics. We find various topologically nontrivial configurations of the confined Clebsch field responsible for vortex sheets and lines. In particular, a stable vortex sheet family is discovered, but its anomalous dissipation vanishes as $\sqrtν$. Topologically stable stationary singular flows, which we call Kelvinons, are introduced. They have a conserved velocity circulation $Γ_α$ around the loop $C$ and another one $Γ_β$ for an infinitesimal closed loop $\tilde C$ encircling $C$, leading to a finite helicity. The anomalous dissipation has a finite limit, which we computed analytically. The Kelvinon is responsible for asymptotic PDF tails of velocity circulation, \textbf{perfectly matching numerical simulations}. The loop equation for circulation PDF as functional of the loop shape is derived and studied. This equation is \textbf{exactly} equivalent to the Schrödinger equation in loop space, with viscosity $ν$ playing the role of Planck's constant. Kelvinons are fixed points of the loop equation at WKB limit $ν\rightarrow 0$. The anomalous Hamiltonian for the Kelvinons contains a large parameter $\log \frac{|Γ_β|}ν$. The leading powers of this parameter can be summed up, leading to familiar asymptotic freedom, like in QCD. In particular, the so-called multifractal scaling laws are, as in QCD, modified by the powers of the logarithm.

physics.flu-dyn↗

arXiv2021

Confined Vortex Surface and Irreversibility. 2. Hyperbolic Sheets and Turbulent statistics

We continue the study of Confined Vortex Surfaces (\CVS{}) that we introduced in the previous paper. We classify the solutions of the \CVS{} equation and find the analytical formula for the velocity field for arbitrary background strain eigenvalues in the stable region. The vortex surface cross-section has the form of four symmetric hyperbolic sheets with a simple equation $|y| |x|^μ=1$ in each quadrant of the tube cross-section ($x y $ plane). We use the dilute gas approximation for the vorticity structures in a turbulent flow, assuming their size is much smaller than the mean distance between them. We introduce the Gaussian random background strain for each vortex surface as an accumulation of a large number of small random contributions coming from other surfaces far away. We compute this self-consistent background strain, relating the variance of the strain to the energy dissipation rate. We find a universal asymmetric distribution for energy dissipation. A new phenomenon is a probability distribution of the shape of the profile of the vortex tube in the $x y$ plane. This phenomenon naturally leads to imitation of the "multi-fractal" scaling of the moments of velocity difference $v(\vec r_1) - \vec v(\vec r_2)$. These moments have a nontrivial dependence of $n, \log |r_1 - r_2|$, approximating power laws with nonlinear index $ζ(n)$. The rough estimate we provide here is not matching the observed DNS data, which may indicate necessity of the full 3D solution of the \CVS{} equations. We argue that the approximate relations for these moments suggested in a recent paper by Sreenivasan and Yakhot are consistent with the \CVS{} theory. We reinterpret their renormalization parameter $α\approx 0.95$ in the Bernoulli law $ p = - \frac{1}{2}α\vec v^2$ as a probability to find no vortex surface at a random point in space.

physics.flu-dyn↗

arXiv2021

Confined Vortex Surface and Irreversibility. 1. Properties of Exact solution

We revise the steady vortex surface theory following the recent finding of asymmetric vortex sheets (AM,2021). These surfaces avoid the Kelvin-Helmholtz instability by adjusting their discontinuity and shape. The vorticity collapses to the sheet only in an exceptional case considered long ago by Burgers and Townsend, where it decays as a Gaussian on both sides of the sheet. In generic asymmetric vortex sheets (Shariff,2021), vorticity leaks to one side or another, making such sheets inadequate for vortex sheet statistics and anomalous dissipation. We conjecture that the vorticity in a turbulent flow collapses on a special kind of surface (confined vortex surface, or CVS), satisfying some equations involving the tangent components of the local strain tensor. The most important qualitative observation is that the inequality needed for this solution's stability breaks the Euler dynamics' time reversibility. We interpret this as dynamic irreversibility. We have also represented the enstrophy as a surface integral, conserved in the Navier-Stokes equation in the turbulent limit, with vortex stretching and viscous diffusion terms exactly canceling each other on the CVS surfaces. We have studied the CVS equations for the cylindrical vortex surface for an arbitrary constant background strain with two different eigenvalues. This equation reduces to a particular version of the stationary Birkhoff-Rott equation for the 2D flow with an extra nonanalytic term. We study some general properties of this equation and reduce its solution to a fixed point of a map on a sphere, guaranteed to exist by the Brouwer theorem.

physics.flu-dyn↗

arXiv2021

Vortex Sheet Turbulence as Solvable String Theory

We study steady vortex sheet solutions of the Navier-Stokes in the limit of vanishing viscosity at fixed energy flow. We refer to this as the turbulent limit. These steady flows correspond to a minimum of the Euler Hamiltonian as a functional of the tangent discontinuity of the local velocity parametrized as $Δ\vec v_t =\vec \nabla Γ$. This observation means that the steady flow represents the low-temperature limit of the Gibbs distribution for vortex sheet dynamics. The normal displacement $δr_\perp$ of the vortex sheet as a Hamiltonian coordinate and $Γ$ as a conjugate momentum. An infinite number of Euler conservation laws lead to a degenerate vacuum of this system, which explains the complexity of turbulence statistics and provides the relevant degrees of freedom (random surfaces). The simplest example of a steady solution of the Navier-Stokes equation in the turbulent limit is a spherical vortex sheet, which we investigate. This family of steady solutions provides an example of the Euler instanton advocated in our recent work, which is supposed to be responsible for the dissipation of the \NS{} equation in the turbulent limit. We further conclude that one can obtain turbulent statistics from the Gibbs statistics of vortex sheets by adding Lagrange multipliers for the conserved volume inside closed surfaces, the rate of energy pumping, and energy dissipation. The effective temperature in our Gibbs distribution goes to zero as $\mbox{Re}^{-\frac{1}{3}}$ with Reynolds number in the turbulent limit. \textbf{The Gibbs statistics in this limit reduces to the solvable string theory in two dimensions (so-called $c=1$ critical matrix model)}. This opens the way for non-perturbative calculations in the Vortex Sheet Turbulence, some of which we report here.

physics.flu-dyn↗

arXiv2021

Asymmetric Vortex Sheet

We present a steady analytical solution of the incompressible Navier-Stokes equation for arbitrary viscosity in an arbitrary dimension $d$ of space. It represents a $d-1$ dimensional vortex "sheet" with an asymmetric profile of vorticity as a function of the normal coordinate $z$. This profile is related to the Hermite polynomials $H_μ(z)$ which are analytically continued to the negative fractional index $μ= -\frac{d}{d-1}$. In $d=2$ dimensions, the solution degenerates to a constant vorticity flow. In $ d \ge 3$ dimensions, the vorticity is confined to the thin layer around the hyperplane with Gaussian decay on one side of the hyperplane and the power decay on another side. One can adjust the common scale of velocity so that the dissipation will stay finite at vanishing viscosity. In this limit, the width $w$ of the viscous lawyer will shrink to zero as $ν^{\frac{3}{5}}$ for arbitrary dimension $d>3$. In $d=3$ dimensions, this power law is also accompanied by powers of the logarithm.

physics.flu-dyn↗

arXiv2020

Towards Field Theory of Turbulence

We revisit the problem of stationary distribution of vorticity in three-dimensional turbulence. Using Clebsch variables we construct an explicit invariant measure on stationary solutions of Euler equations with the extra condition of fixed energy flow/dissipation. The asymptotic solution for large circulation around large loops is studied as a WKB limit (instanton). The Clebsch fields are discontinuous across minimal surface bounded by the loop, with normal vorticity staying continuous. There is also a singular tangential vorticity component proportional to $δ(z)$ where $z$ is the normal direction. Resulting flow has nontrivial topology. This singular tangent vorticity component drops from the flux but dominates the energy dissipation as well as the Biot-Savart integral for velocity field. This leads us to a modified equation for vorticity distribution along the minimal surface compared to that assumed in a loop equations, where the singular terms were not noticed. In addition to describing vorticity distribution over the minimal surface, this approach provides formula for the circulation PDF, which was elusive in the Loop Equations.

hep-th↗

arXiv2020

Probability Distribution of Velocity Circulation in Three Dimensional Turbulence

We elaborate the statistical field theory of Turbulence suggested in the previous paper \cite{M20a}. We clarify and simplify the basic Energy pumping equation of that theory and study mathematical properties of singular field configuration (instanton) which determine the tails of PDF for the velocity circulation around large loop $C$ in isotropic turbulence at highest Reynolds numbers. Explicit analytic solution is found for the Clebsch instanton in an Euler equation for a planar loop circulation problem. This solution for vorticity is has a term proportional to a delta function in normal direction to the minimal surface bounded by the loop. The smoothing of $δ$ functions in the vorticity in the full Navier-Stokes equations is investigated and exponential profile of smoothed singularity is found. The PDF for circulation is now an infinite sum of decreasing exponential terms $\EXP{- n |w|}\sqrt{\frac{n}{|w|}}$, with $ w = \fracΓ{Γ_0[C]}$, and $ Γ_0[C] \sim \sqrt{A_C} $ with minimal area $A_C$. The leading term fits with adjusted $R^2 = 0.9999$ the PDF tail found in DNS over more than six orders of magnitude. The area dependence of the ratio of the circulation moments $M_8/M_6$ fits with adjusted $R^2=0.9996$ the DNS in inertial range of square loop sizes from $100 $ to $500$ Kolmogorov scales. Thus, our theory explains DNS with high degree or confidence. For a flat loop we derive two-dimensional integral equation for the dependence of the scale $Γ_0[C] $ of circulation as a function of the shape of the loop (aspect ratio for rectangular loop

hep-th↗

arXiv2020

Clebsch Confinement and Instantons in Turbulence

We introduce a concept of Clebsch confinement related to unbroken gauge invariance and study Clebsch instantons: singular vorticity sheets with nontrivial helicity. This is realization of the "Instantons and intermittency" program we started back in the 90ties\cite{FKLM}. These singular solutions are involved in enhancing infinitesimal random forces at remote boundary leading to critical phenomena. In the Euler equation vorticity is concentrated along the random self-avoiding surface, with tangent components proportional to the delta function of normal distance. Viscosity in Navier-Stokes equation smears this delta function to the Gaussian with width $h \propto ν^{\nicefrac{3}{5}}$ at $ν\ra 0$ with fixed energy flow. These instantons dominate the enstrophy in dissipation as well as the PDF for velocity circulation $Γ_C$ around fixed loop $C$ in space. At large loops, the resulting symmetric exponential distribution perfectly fits the numerical simulations\cite{IBS20} including pre-exponential factor $1/\sqrt{|Γ|}$. At small loops, we advocate relation of resulting random self-avoiding surface theory with multi-fractal scaling laws observed in numerical simulations. These laws are explained as a result of fluctuating internal metric (Liouville field). The curve of anomalous dimensions $ζ(n)$ can be fitted at small $n$ to the parabola, coming from the Liouville theory with two parameters $α, Q$. At large $n$ the ratios of the subsequent moments in our theory grow linearly with the size of the loop, which corresponds to finite value of $ζ(\infty)$ in agreement with DNS.

hep-th↗

arXiv2019

Analytic and Numerical Study of Navier-Stokes Loop Equation in Turbulence

We developed analytic approach to the non-planar loop equation, which we derived in previous papers \cite{M19a},\cite{M19b},\cite{M19c}. We found quadratic integral equation for the vorticity distribution $Ω(r)$ we introduced on a minimal surface. There are no corrections to the minimal surface though: it is still defined by mean external curvature equal to zero, for arbitrary non-planar loop. We also analyzed the loop equations with viscosity term in Navier-Stokes equations. This term creates boundary condition for $Ω(r\in C)$. The leading viscosity correction term mixes the moments $\left< Γ^p \right>$ with $\left< Γ^{p-1} \right>$ resembling the bi-fractal behavior observed in \cite{S19} and explicitly breaking the time reversal symmetry. We also develop numerical approach to the loop equation with arbitrary curved loop and present \Mathematica notebook building triangulated minimal surface and then numerically solving these equations. As a result we obtain predictions for future numerical experiments which will compute vorticity distribution along the loop.

hep-th↗

arXiv2019

Exact Area Law for Planar Loops in Turbulence in Two and Three Dimensions

We study properties of the minimal surface in the Area Law Solution \cite{M93}, \cite{M19a}, \cite{M19b}. We find out that Area Law holds exactly for 2D turbulence as well as for arbitrary planar loop in higher dimensions. This relies on our previous result $α= \frac{1}{2}$ in which case the second moment of circulation can be proven to reduce to the area inside the planar loop. In $d=3$, we demonstrate how the Stokes condition $\partial_i ω_i(r)=0$ is exactly satisfied for the minimal surface solution in virtue of vanishing mean curvature at the minimal surface. In order to satisfy Loop Equation beyond planar loops, we introduce self-consistent conformal metric on the surface designed to preserve Stokes condition but to compensate the terms in the loop equation. We derive nonlinear integral equation for this conformal metric as a function of a point on a surface.

hep-th↗

arXiv2019

Scaling Index $α= \frac{1}{2}$ In Turbulent Area Law

We analyze the Minimal Area solution to the Loop Equations in turbulence \cite{M93}. As it follows from the new derivation in the recent paper \cite{M19}, the vorticity is represented as a normal vector to the minimal surface not just at the edge, like it was assumed before, but all over the surface. As it was pointed in that paper, the self-consistency relation for mean vorticity leads to $α=\frac{1}{2}$, however the similar conditions for product of two and more vorticities cannot be satisfied without extra terms, which were left undetermined in that paper. In this paper we find these missing terms -- they are delta functions at coinciding points which must be taken into considerations in surface integrals. We compare this value of $α$ with new measurements of the same team which confirmed the area law \cite{S19} and we find that asymptotic formula $λ(p) \approx 2 αp + β\ln p$, with $α=0.49 \pm 0.02, β= 0.92 \pm 0.01 $, fits all data at $p=3,...10$ within error bars.

hep-th↗

arXiv2019

Universal Area Law in Turbulence

We re-visit the Area Law in Turbulence discovered many years ago \cite{M93} and verified recently in numerical experiments\cite{S19}. We derive this law in a simpler way, at the same time outlining the limits of its applicability. Using the PDF for velocity circulation as a functional of the loop in coordinate space, we obtain explicit formulas for vorticity correlations in presence of velocity circulation. These functions are related to the shape of the scaling function of the PDF as well as the shape of the minimal surface inside the loop. The background of velocity circulation does not eliminate turbulence but makes observable quantities in inertial range \textbf{calculable}. The scaling dimension of velocity circulation as a function of large area remains unknown. Numerical experiments \cite{S19} suggest transition for log-log derivative of circulation moments $\left<Γ^p\right>$ by the loop area from Kolmogorov index $\frac{2p}{3}$ at $p <4$ down to approximately $0.58 p$ for $4 \leq p \leq 10$ within available Reynolds numbers. We argue that Area Law applies to these moments only in the limit $p\rightarrow \infty$ when they are dominated by the tails of the PDF. So, these numerical experiments suggest that the scaling index in Area law is less then $\frac{2}{3}$.

hep-th↗

arXiv2011

Meromorphization of Large N QFT

We find general relations between RG equations and planar unitarity-analyticity. These relations are summarized in meromorphization procedure, generalizing the Padé approximation in the limit of infinite order. We also investigate confinement conditions for the mass spectrum in asymptoticaly free QFT and lay down systematic framework for α expansion suggested in previous papers. The new relations for meromorphization of symmetric conformal tensors are found and studied. Explicit intergal representation for the triple string vertex Γ is found. This corresponds to resonanse theory with infinite number of masses and Lagrangean \PhiQΦ+ΓΦ^3.

hep-th↗

arXiv2011

Integral Equation for CFT/String Duality

We reinterpret and extend some old work on CFT/string duality. We consider some asymptotically conformal field theory in large N limit, with conformal symmetry broken by VEV's of infinite number of operators. Assuming that this theory confines (i.e. is dual to infinite number of free composite particles) we derive explicit equation for the mass spectrum operator Q of the theory, relating this operator to terms OPE expansion of CFT. Under some general assumptions about growth of OPE coefficients (less than double factorial growth) the resulting expansion for the mass spectrum is convergent. This method applies to confining CFT of ADS family as well as any asymptotically CFT with confinement. This includes the ordinary QCD. In the latter case the first terms of our perturbation expansion have good correspondence with experimental Regge trajectories at low angular momentum.

hep-th↗

arXiv1996

Hidden Symmetries of Large N QCD

The local SUSY symmetry of the loop dynamics of QCD is found. The remarkable thing is, there is no einbein-gravitino on this theory, which makes it a 1D topological supergravity, or locally SUSY quantum mechanics. Using this symmetry, we derive the large $N_c$ loop equation in momentum superloop space. Introducing as before the position operator $\Xμ$ we argue that the superloop equation is equivalent to invariance of correlation functions of products of these operators with respect to certain quadrilinear transformation. The applications to meson and glueball sectors as well as the chiral symmetry breaking are discussed. The 1D field theory with Quark propagating around the loop in superspace is constructed.

hep-th↗

arXiv1992

Ground State of 2D Quantum Gravity and Spectral Density of Random Matrices

We compute the exact spectral density of random matrices in the ground state of the quantum hamiltonian corresponding to the matrix model whose double scaling limit describes pure gravity in 2D. We show that the non-perturbative effects are very large and in certain cases dominate the semi-classical WKB contribution studied in the earlier literature. The physical observables in this model are the loop averages with respect to the spectral density. We compute their exact ground-state expectation values and show that they differ significantly from the values obtained in the WKB approximation. Unlike the alternative regularizations of the nonperturbative 2D quantum gravity, based on analytic continuation of the Painlevé transcendent, our solution shows no pathologies.

hep-th↗