arXiv subjects

Alexander Migdal

Publications and source records attributed to Alexander Migdal.

At least 19 recordsLinked to original sources

arXiv2026

Superloop Equations and Minimal Surfaces I: Confining minimal surface in $4D, N=1$ SYM

We formulate the loop equations of pure $4d, N=1$ super Yang--Mills (SYM) theory in a finite geometric form. The usual equal-point loop derivatives are singular because the order of gauge-field insertions is lost when contour points coincide. Our path-ordered operator calculus (POOC) keeps the insertions in distinct ordered slots, forms the graded commutators and Jacobi combinations, and only then takes the coincidence limit. This removes the spurious kinematical singularities without introducing a cutoff; the genuine ultraviolet contact remains a separate physical distribution. We apply POOC to the exact Itoyama--Takashino superloop hierarchy and construct a Lorentzian supersymmetric Hodge-dual (SHD) surface functional. Its Hodge-resolved area derivative is annihilated by the local chiral and anti-chiral loop operators. Exact additivity under surface-geodesic sewing shows that the exponential of this zero mode multiplies any solution of the complete finite-$N$ hierarchy without changing the equations. For planar contours the SHD functional is the geometric area. In particular, for a long rectangular Wilson loop, $W[C_{T,L}]\sim\exp[-i\sigma_kLT]$ and $E_k(L)=\sigma_kL>0$. Thus the construction gives an exact nonperturbative confining area-law factor in $ N=1$ SYM. After Euclidean continuation, the same SHD factor is an exact fixed point of the deterministic zero-noise SYM gradient flow. Its interpretation as a dynamically selected equilibrium still requires stability in the long-flow-time, large-volume, and zero-noise limits, but this does not affect the exact zero-mode and finite-$N$ dressing theorems. The complete planar Wilson-loop solution, including the undressed fluctuation factor and excitation spectrum, will be developed in the next papers of this series.

hep-th↗

arXiv2026

Euler Ensemble as Decaying Turbulence Attractor: Universality, Stability and Parity Classes

We study local Lyapunov stability of the Euler ensemble in compact rescaled momentum-loop dynamics for freely decaying Navier-Stokes turbulence. The ensemble consists of exact self-similar finite-cutoff solutions whose momentum loops are equal-step polygons on a sphere; their planar representatives define the Euler ensemble. In two dimensions, after quotienting global rotation and the time-origin shift, the odd-$N$ planar representatives have no local shape instability. The even-$N$ ensemble contains an alternating unstable mode with $\lambda=\cot^2(\pi p/q)>0$, and is excluded as a local attractor. Thus the odd Euler ensemble is the locally stable planar sector. For $d>2$, the planar ensemble lies in a continuous manifold of equal-step spherical polygons. Transverse deformations along this manifold are exact tangent zero modes. Integrating over these unconstrained spherical modes in the continuum limit gives a singular Wilson-loop functional supported only on collapsed, globally rotated planar coordinate loops. These modes are therefore projected out of the admissible homogeneous isotropic ensemble. The physical stability problem is the decay of normal perturbations, represented by local edge-length defects. In the endpoint-local sector, the planar odd representative has exact defect spectrum $\lambda_m=-\sec^2(\pi m/N)<0$. In the smooth angular continuum limit, the leading stability operator is the universal angular Laplacian. Nearby spherical zero modes do not modify this leading operator; their first effect appears only at order $N^{-4}$, through higher-derivative corrections computed symbolically. Hence the leading local stability mechanism is universal and independent of spherical zero modes.

nlin.CD↗

arXiv2026

Geometric QCD III: Exact transition amplitudes and the glueball spectrum

We complete the analysis of planar Makeenko--Migdal loop equations in the Lorentz-invariant continuum limit. Using our confining twistor-string representation, we compute the quantum fluctuation determinant, which in Minkowski space reduces to a discrete product of finite-dimensional matrix quadratures. The $\zeta$-regularized weight is independent of winding number $w$. Near the mass shell, the pole singularity is generated by $w \to \infty$, suppressing fluctuation variance as $1/w$. The path integral localizes on the classical trajectory, rendering the pole spectrum and transition residues parametrically exact in the large-winding WKB limit. For the open-string meson sector, we fit 40 observed states across five topological boundary sectors ($h=0, \pm 1, \pm 2$). The holonomy shift $h$ accounts for exact geometric degeneracies between parity families, reproducing mass splittings without phenomenological spin-orbit parameters. Evaluated one-loop residues yield theoretical transition cross-sections capturing heavy-mass quenching and phase-space enhancement for high-spin light states. Applying this framework to the pure Yang--Mills closed string, we demonstrate the dynamical stability of the pure-gauge minimal surface: the conformal Liouville anomaly drives the string strictly to the trigonometric minimum ($q=0$). The complex elliptic geometry analytically collapses, yielding linear Regge trajectories. The translation zero-mode measure dynamically nullifies the transition amplitude of the massless scalar ghost, providing an analytic mechanism for a purely gluonic mass gap. Anchoring parameter-free glueball trajectories to the open-string tension natively recovers the exact L"uscher intercept $\alpha(0)=1/12$, perfectly matching established PDG unassigned isoscalar candidates and large-$N_c$ lattice QCD extrapolations.

hep-th↗

arXiv2026

Decaying Turbulence and the Riemann Hypothesis: The number theory behind the infinite-time singularity

We derive a formal statistical solution of freely decaying incompressible turbulence in arbitrary dimension $d>1$ using Navier--Stokes loop equations. The loop Fourier transform maps smooth deterministic Cauchy data in infinite space to a one-dimensional momentum-loop quantum field theory, giving a geometric origin of spontaneous stochasticity. In bounded-variation calculus the nonlinear advection term becomes a closed-loop total derivative and cancels on the compact spherical target, leaving a diffusive momentum-loop evolution. The universal attractor is the planar Euler ensemble of rational star-polygon walks. Its continuum limit splits into two parity sectors, $\eta=N\bmod 2$. Both Euler ensembles are marginally Lyapunov-stable in the continuum limit and give dimension-independent energy scaling functions $H(k\sqrt{\tilde\nu t})$. Their Mellin amplitudes differ only by the odd-sector prime-$2$ Euler factor $(1-2^{-(p+17/2)})^{-1}$. Both share the Riemann-wall poles $p=-8+i\rho_n$, generated by the non-trivial zeros $1/2+i\rho_n$ of $\zeta(s)$, while the odd ensemble also contains the dyadic wall $p=-17/2+2\pi i m/\log2$, $m\ne0$. Thus the two sectors have distinct Stokes staircases, although their spectra agree with present $4096^3$ DNS within statistical uncertainty. Assuming the Riemann Hypothesis and simplicity of the zeros, the common Riemann-wall activations occur at $t_n\propto\rho_n^3$ and condense into an infinite-time essential singularity.

hep-th↗

arXiv2026

Geometric QCD II: The Confining Twistor String and Meson Spectrum

We present a local, asymptotically free solution of the planar Makeenko--Migdal loop equations in the continuum limit with full Lorentz invariance. The solution is constructed by quantizing internal Majorana fermions (referred to here as ``elves'') on a rigid Hodge-dual minimal surface. These worldsheet degrees of freedom provide the algebraic mechanism required to satisfy the unintegrated vector loop equations, with the Pauli principle enforcing planar factorization. In the local limit, the theory reduces to a confining analytic twistor-string representation. By analyzing the monodromy structure of the complexified effective action, we show that the discrete mass spectrum is organized by topological data associated with twistor singularities. The simplest sector with one branch point yields parametric Regge trajectories expressed in terms of trigonometric functions. These trajectories are non-linear but approximately linear over a broad range and are in agreement with experimental data for light mesons. The asymptotic behavior of the trajectory $J = \alpha(M^2)$ and its subleading corrections arise from the interaction between the Liouville term and the twistor monodromy, without introducing additional assumptions about string excitations. In our solution, the QCD mass spectrum follows from a generalized eigenvalue problem in complexified phase space, effectively reducing the problem to classical geometry. Within this framework, the large-$N_c$ Master Field is realized as a classical trajectory in twistor space.

hep-th↗

arXiv2025

Geometric QCD I: The Hodge-Dual Surface and Quark Confinement

This is the first of two papers presenting a geometric framework for Planar QCD ($N_c \to \infty$). In this part, we establish the kinematic foundation of the theory by constructing the unique stable vacuum of the loop equation. We demonstrate that the Makeenko-Migdal loop equation admits a solution of the form $W[C] = W_{pert}[C] \exp{-\kappa S[C]}$, provided $S[C]$ is a specific minimal surface possessing a self-dual area derivative. We prove that such a surface exists and corresponds to the Hodge-dual projection of a minimal surface in $\mathbb{R}^3 \otimes \mathbb{R}^4$. Crucially, this confinement mechanism relies on the self-duality of the area derivative -- a property that exists exclusively in four dimensions. This geometric constraint ensures stability only in $D=4$, distinguishing the resulting theory from standard string models which require higher critical dimensions. We relate the string tension parameter $\kappa$ to the gluon condensate via the Operator Product Expansion. The dynamical quantization of the Fermi string on this rigid surface and the resulting meson spectrum are derived in the companion paper \cite{Migdal2026GeometricQCDII}.

hep-th↗

arXiv2025

Geometric Solution of Turbulence as Diffusion in Loop Space

Strongly nonlinear dynamics, from fluid turbulence to quantum chromodynamics, have long constituted some of the most challenging problems in theoretical physics. This review describes a unified theoretical framework, the loop space calculus, which offers an analytical approach to these problems. The central idea is a shift in perspective from pointwise fields to integrated loop observables, a transformation that recasts the governing nonlinear equations into a universal linear diffusion equation in the space of loops. This framework, supported by recent mathematical analysis, is analytically solvable and yields an exact, parameter-free solution for decaying hydrodynamic turbulence--the Euler ensemble--which is shown to be dual to a solvable string theory. The theory's predictions include: (i) the unification of spatial and temporal scaling laws, governed by two related, infinite spectra of intermittency and decay exponents derived from the nontrivial zeros of the Riemann zeta function; (ii) a first-order phase transition in magnetohydrodynamic (MHD) turbulence; and (iii) the formation of quantized, concentric shells in passive scalar mixing. The appearance of identical mathematical structures as solutions to the turbulent regime of Yang-Mills gradient flow points to the broad applicability of this approach. The framework also yields a new type of analytic Hodge-dual matrix surface that solves the Yang-Mills fixed-point loop equation by harmonic map, opening the way for a geometric formulation of QCD string theory.

physics.flu-dyn↗

arXiv2025

Exact Confining Solution of the Planar QCD Loop Equation via a Matrix Ensemble

We construct an exact solution to the planar QCD loop equation in four-dimensional Euclidean space using a novel matrix-valued momentum loop formalism. Central to this construction is a new loop calculus, in which functional derivatives act directly on the loop velocity $ \dot{C}(\theta) $. This framework yields finite, well-defined expressions for point and area derivatives in loop space and reveals the role of the loop-space Bianchi identity in ensuring gauge consistency. The Wilson loop is expressed as an average over matrix-valued momentum loops $ P(\theta) $ tracing closed paths in a compact complex manifold, constrained by self-duality and boundary conditions. Self-duality dynamically nullifies the classical Yang--Mills term in the loop equation, while the boundary constraints eliminate the contact terms in the planar ($ N \to \infty$)) limit. Thus, the full planar QCD loop equation is satisfied exactly and without modification. We further argue that these random walks define a self-dual minimal surface in an extended matrix space, and \textbf{prove} that the area of this matrix minimal surface scales proportionally to the Euclidean minimal area bounded by the same loop $ C $, thereby providing a sufficient condition for quark confinement. In the example of an circular loop, we construct the exact surface and compute its area analytically, finding that it equals the Euclidean area multiplied by a universal factor of $ 6 \sqrt{2}$. These results provide the first known explicit construction of a quark-confining minimal surface that solves the planar QCD loop equation exactly, and open a new path toward understanding gauge--string duality from first principles.

hep-th↗

arXiv2025

Spontaneous quantization of the Yang--Mills gradient flow

We formulate a nonsingular loop-space calculus for Yang-Mills (YM) gradient flow directly in terms of Wilson loops. Variations act within the manifold of smooth loops via finite, reparametrization-invariant "dot derivatives," eliminating cusp/backtracking singularities. This yields a closed linear diffusion equation in loop space for Wilson loops. The associated loop operator is gauge invariant and universal; the construction applies to any (Abelian or non-Abelian) gauge group. We then exhibit two classes of exact solutions. (i) A self-dual (Hodge-dual) minimal surface whose exponentiated dual area solves the fixed-point loop equation without contact terms or ambiguities. For planar loops the dual area equals 2*sqrt(2) times the Euclidean minimal area, providing a geometric confinement mechanism. We also show that the ordinary minimal surface in R^4 does not solve the fixed-point equation because the loop operator produces a singular nonzero contribution. (ii) A decaying-flow solution in which the momentum loop performs a periodic random walk on regular star polygons (the "Euler ensemble" known from Navier-Stokes turbulence). Both solutions realize spontaneous quantization: the self-dual one gives a stationary quantized state (a fixed manifold of the flow), while the decaying one yields a quantized trajectory that relaxes to a pure-gauge vacuum along a universal path. Thus we obtain exact solutions for Wilson-loop evolution in YM gradient flow and a concrete route by which quantum-like Wilson-loop statistics emerge from deterministic classical dynamics. We outline implications for confinement in QCD and the role of the above manifolds/trajectories as attractors of the flow.

hep-th↗

arXiv2025

Geometric Solution of Turbulent Mixing

We derive an analytical solution for the one-point distribution of a passive scalar in decaying homogeneous turbulence, in the limit of strong turbulence (high Re, fixed Schmidt number). Velocity statistics are governed by the Euler ensemble, a spontaneously stochastic exact solution of the loop equation from the Navier-Stokes equations in the strongly turbulent regime. The scalar's advection-diffusion problem is also recast as a solvable linear loop equation. For a localized initial condition, the solution consists of expanding concentric shells: the radial scalar profile is quantized and piecewise parabolic, with gaps organized by Euler totients - an arithmetic structure distinct from conventional scaling. This shell pattern is the unique solution in the Euler ensemble, smoothed by any finite diffusivity. The result provides the underlying geometric structure for scalar transport in decaying strong turbulence, relevant in astrophysical or quantum-fluid regimes where dissipation is negligible. This may explain the "ramp-cliff" structures observed in turbulent mixing half a century ago. While this shell structure is hard to resolve in DNS, its statistical signature is robustly captured by the volume-averaged scalar density, a measurable quantity.

physics.flu-dyn↗

arXiv2025

Dual Theory of MHD Turbulence

We present an exact analytic solution for decaying incompressible magnetohydrodynamic (MHD) turbulence. Our solution reveals a dual formulation in terms of two interacting Euler ensembles--one for hydrodynamic and another for magnetic circulation. This replaces empirical scaling laws with an infinite set of power terms with calculable decay exponents, some of which appear as complex-conjugate pairs related to the Riemann zeta function. A key result of our analysis is the explicit dependence of the solution on the Prandtl number ($\mathrm{Pr} = \nu/\eta$), leading to a phase transition at $\mathrm{Pr} = 1$. In the $\mathrm{Pr}<1$ regime, turbulence is dominated by hydrodynamic fluctuations, while for $\mathrm{Pr}>1$, two distinct solutions emerge: a metastable one in which magnetic fluctuations grow with $\mathrm{Pr}$ and a stable one where they remain balanced with hydrodynamic fluctuations. We compare our theoretical predictions with recent direct numerical simulations (DNS) and discuss their implications for astrophysical plasmas, fusion devices, and laboratory MHD experiments. Our results provide a rigorous mathematical framework for understanding MHD turbulence and its dependence on fundamental parameters, offering a new perspective on turbulence in highly conducting fluids.

physics.flu-dyn↗

arXiv2024

Duality of Navier-Stokes to a one-dimensional system

The Navier--Stokes (NS) equations describe fluid dynamics through a high-dimensional, nonlinear system of partial differential equations (PDEs). Despite their fundamental importance, their behavior in turbulent regimes remains incompletely understood, and their global regularity is still an open problem. Here, we reformulate the NS equations as a nonlinear equation for the momentum loop $\vec{P}(\theta, t)$, effectively reducing the original three-dimensional PDE to a one-dimensional problem. We present an explicit analytical solution -- the Euler ensemble -- which describes the universal asymptotic state of decaying turbulence and is supported by numerical simulations and experimental validation. This Euler ensemble is equivalent to a string theory with discrete target space given by a set of regular star polygons, with additional Ising (Fermi) degrees of freedom at the vertices. This string theory can also be interpreted as a random walk on regular star polygons. The Wilson loop for turbulence, \[ \left\langle \exp\left( \imath \oint d\theta\, \vec{C}'(\theta) \cdot \vec{v}(\vec{C}(\theta, t)) \right) \right\rangle, \] reduces to a dual amplitude of this string theory with distributed external momentum proportional to $\vec{C}'(\theta)/\sqrt{t}$.

math-ph↗

arXiv2023

Dual theory of decaying turbulence. 2. Numerical simulation and continuum limit

This is the second paper in a cycle investigating the exact solution of loop equations in decaying turbulence. We perform numerical simulations of the Euler ensemble, suggested in the previous work, as a solution to the loop equations. We designed novel algorithms for simulation, which take a small amount of computer RAM so that only the CPU time grows linearly with the size of the system and its statistics. This algorithm allows us to simulate systems with billions of discontinuity points on our fractal curve, dual to the decaying turbulence. For the vorticity correlation function, we obtain \textbf{quantum fractal laws} with regime changes between universal discrete values of indexes. The traditional description of turbulence with fractal or multifractal scaling laws does not apply here. The measured conditional probabilities of fluctuating variables are smooth functions of the logarithm of scale, with statistical errors being negligible at our large number of random samples $T = 167,772,160$ with $N = 200,000,000$ points on a fractal curve. The quantum jumps arise from the analytical distribution of scaling variable $X = \frac{\cot^2(\pi p/q)}{q^2}$ where $\frac{p}{q}$ is a random fraction. This distribution is related to the totient summatory function and is a discontinuous function of $X$. In particular, the energy spectrum is a devil's staircase with tilted uneven steps with slopes between $-1$ and $-2.03$. The logarithmic derivative of energy decay $n(t)$ as a function of time is jumping down the stairs of universal levels between $1.50$ and $1.53$. The quantitative verification of this quantization would require more precise experimental data.

physics.flu-dyn↗

arXiv2023

Quantum Solution of Classical Turbulence. Decaying Energy Spectrum

This paper presents a recent advancement that transforms the problem of decaying turbulence in the Navier-Stokes equations in $3+1$ dimensions into a Number Theory challenge: finding the statistical limit of the Euler ensemble. We redefine this ensemble as a Markov chain, establishing its equivalence to the quantum statistical theory of $N$ fermions on a ring, interacting with an external field associated with random fractions of $\pi$. Analyzing this theory in the turbulent limit, where $N \to \infty$ and $\nu \to 0$, we discover the solution as a complex trajectory (instanton) that acts as a saddle point in the path integral over the density of these fermions. By computing the contribution of this instanton to the vorticity correlation function, we obtain an analytic formula for the observable energy spectrum -- a complete solution of decaying turbulence derived entirely from first principles without the need for approximations or fitted dimensionless parameters. Our analysis reveals the full spectrum of critical indices in the velocity correlation function in coordinate space, determined by the poles of the Mellin transform, which we prove to be a meromorphic function. Real and complex poles are identified, with the complex poles reflecting dissipation and uniquely determined by the famous complex zeros of the Riemann zeta function. Universal functions of the scaling variables supersede the traditional turbulent scaling laws (K41, Heisenberg, and multifractal). These functions for the energy spectrum, energy decay rate, and velocity correlation significantly deviate from power laws but closely match the results from grid turbulence experiments \cite{GridTurbulence_1966, Comte\_Bellot\_Corrsin\_1971} and recent DNS data \cite{SreeniDecaying} within experimental error margins.

physics.flu-dyn↗

arXiv2023

To the theory of decaying turbulence

We have found an infinite dimensional manifold of exact solutions of the Navier-Stokes loop equation for the Wilson loop in decaying Turbulence in arbitrary dimension $d >2$. This solution family is equivalent to a fractal curve in complex space $\mathbb C^d$ with random steps parametrized by $N$ Ising variables $\sigma_i=\pm 1$, in addition to a rational number $\frac{p}{q}$ and an integer winding number $r$, related by $\sum \sigma_i = q r$. This equivalence provides a dual theory describing a strong turbulent phase of the Navier-Stokes flow in $\mathbb R_d$ space as a random geometry in a different space, like ADS/CFT correspondence in gauge theory. From a mathematical point of view, this theory implements a stochastic solution of the unforced Navier-Stokes equations. For a theoretical physicist, this is a quantum statistical system with integer-valued parameters, satisfying some number theory constraints. Its long-range interaction leads to critical phenomena when its size $N \rightarrow \infty$ or its chemical potential $\mu \rightarrow 0$. The system with fixed $N$ has different asymptotics at odd and even $N\rightarrow \infty$, but the limit $\mu \rightarrow 0$ is well defined. The energy dissipation rate is analytically calculated as a function of $\mu$ using methods of number theory. It grows as $\nu/\mu^2$ in the continuum limit $\mu \rightarrow 0$, leading to anomalous dissipation at $\mu \propto \sqrt{\nu} \to 0$. The same method is used to compute all the local vorticity distribution, which has no continuum limit but is renormalizable in the sense that infinities can be absorbed into the redefinition of the parameters. The small perturbation of the fixed manifold satisfies the linear equation we solved in a general form. This perturbation decays as $t^{-\lambda}$, with a continuous spectrum of indexes $\lambda$ in the local limit $\mu \to 0$.

physics.flu-dyn↗

arXiv2022

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↗

arXiv2022

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 $\sqrt{\nu}$. Topologically stable stationary singular flows, which we call Kelvinons, are introduced. They have a conserved velocity circulation $\Gamma_\alpha$ around the loop $C$ and another one $\Gamma_\beta$ 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 Schr\"odinger equation in loop space, with viscosity $\nu$ playing the role of Planck's constant. Kelvinons are fixed points of the loop equation at WKB limit $\nu \rightarrow 0$. The anomalous Hamiltonian for the Kelvinons contains a large parameter $\log \frac{|\Gamma_\beta|}{\nu}$. 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. 3. Nested Tubes and Energy Cascade

We find a new family of exact solutions of the Confined Vortex Surface equations (The Euler equations with extra boundary conditions coming from the stability of the Navier-Stokes equations in the local tangent plane). This family of solutions has an infinite number of nested tubes of varying diameters. The shape of the boundary cross-section is the same up to a scale. This Russian doll implements in physical space the scenario of the energy cascade from an eddy to a smaller eddy. This hierarchy of vortex shells is not wishful thinking but rather an exact solution of the Euler (CVS) equations. The spectrum of the size of the shells is determined from the minimization of the effective Hamiltonian of our turbulent statistics. This effective Hamiltonian is given by a surface dissipation integral, conserved in the \NS{} dynamics in virtue of the \CVS{} conditions. The thickness of each tube goes to zero as a power of Reynolds number $\R^{-\frac{3}{4}}$, compared to the average distance between tubes in the turbulent flow. Thus, at finite viscosity, there will be a logarithmic number of inner tubes nested inside the external one.

physics.flu-dyn↗