arXiv subjects

Ritwik Mukherjee

Publications and source records attributed to Ritwik Mukherjee.

At least 19 recordsLinked to original sources

arXiv2026

Disentangling intermittent flow structure contributions to anomalous scaling and multifractality in turbulence

Intermittency in turbulence manifests as intense vortices and sharp peaks of dissipation. Causing the breakdown of Kolmogorov's simple self-similar theory, it leads to anomalous scaling, multifractality and so far remains beyond the scope of a complete theoretical description. How intermittent flow structures influence these different measurements is not known quantitatively. With a simple filtering procedure-thresholding vorticity and inverting the Biot-Savart law to generate filtered velocity fields-we show the effects of intermittent flow structures can be disentangled. As extreme vorticity contributions to the velocity field are filtered out, the energy spectrum scaling persists, while the bottleneck is flattened, and structure function scalings tend towards their Kolmogorov values. The approach is more rapid for transverse exponents, revealing the selective importance of intensely swirling flow regions. Similarly, the extent of multifractality reduces as intermittency is filtered, shrinking the range of roughness singularity exponents. The residual fields are curiously more multifractal, but their structure begins to break away from an underlying turbulence skeleton. The effects on vortex stretching and strain self-amplification are quantified. Our work shows that a Biot-Savart approach can selectively remove the effects of intermittency from turbulence, and hence from its scalings.

physics.flu-dyn↗

arXiv2026

Parity-Dependent Scaling of Velocity-Gradient Correlations in Turbulence

We investigate two-point velocity-gradient correlation functions in homogeneous isotropic turbulence using exact relations and direct numerical simulations. The second-order gradient correlation is shown to be exactly related to the Laplacian of the velocity correlation, implying inertial-range scaling $C_2^{1,1}(r)\sim r^{-4/3}$. At higher orders, we uncover a parity-dependent organization of gradient correlations: odd-odd correlations exhibit scaling close to $r^{-4/3}$ with weak dependence on order, whereas even-even correlations display systematically different exponents. We show that this distinction originates from the sign structure of the gradient field: sign decorrelation suppresses intermittent contributions in odd-odd sectors, while even-even correlations retain them and remain sensitive to the spatial organization of intense structures. The measured even-even exponents are quantitatively consistent, across two Reynolds numbers, with independently measured box-counting dimensions of intermittent gradient structures. These results identify parity under sign reversal as a fundamental organizing principle for higher-order turbulent correlations and establish a direct connection between sparse intermittent geometry and scaling exponents in turbulence.

physics.flu-dyn↗

arXiv2026

Reduction of Triadic Interactions Suppresses Intermittency and Anomalous Dissipation in Turbulence

We investigate how the defining statistical features of three-dimensional turbulence respond to systematic reductions of the Fourier-space triadic interaction network. Using direct numerical simulations of both fractally and homogeneously decimated Navier-Stokes dynamics, we show that progressive thinning of the set of active modes leads to a systematic suppression of intermittency and, most strikingly, to the vanishing of the mean dissipation rate in the large-Reynolds-number limit. Structure-function exponents collapse onto their dimensional values, the multifractal singularity spectrum contracts, and the analyticity width extracted from the exponential spectral tail increases monotonically with decimation-each indicating a substantial regularization of the velocity field. Together, these results provide direct evidence that anomalous dissipation in incompressible turbulence is not a generic property of the Navier-Stokes equations, but instead requires the full combinatorial richness of their triadic nonlinear interactions.

physics.flu-dyn↗

arXiv2025

Forcing regimes in the two-dimensional Navier-Stokes equations

In the standard theoretical setting of body-forced turbulence, the forcing that sustains the flow is concentrated in a narrow range of length scales. However, in experiments of fractal-grid turbulence and in numerical simulations inspired by the renormalization group approach, more general forcing functions have been considered. These studies have shown that the phenomenology of turbulence is sensitive to the regularity of the forcing, which raises the wider question of the sensitivity of all Navier--Stokes mathematical estimates to the regularity of body forces. To answer this question, it is necessary to convert the traditional estimates based on the Grashof number, a dimensionless measure of the magnitude of the forcing, to a form dependent on the Reynolds number, the usual dimensionless number in experimental measurements and statistical theories of turbulence. To investigate these issues we consider the two-dimensional case and employ the full range of forcing regularity allowed by the theory of weak solutions to extend available estimates not only for the energy and enstrophy dissipation rates, but also for the dimension of the global attractor. What emerges is the existence of three distinct regimes as a function of the regularity of the forcing.

physics.flu-dyn↗

arXiv2025

Geometric Intermittency in Turbulence

Equal-time scaling exponents in fully developed turbulence typically exhibit non anomalous scaling in the inverse cascade of two-dimensional (2D) turbulence and anomalous scaling in three dimensions. We demonstrate that multiscaling is not confined to longitudinal, scalar velocity increments, but also emerges in increments associated with the magnitude and orientation of the velocity vector. This decomposition uncovers a multiscaling in the 2D inverse cascade, which remains obscured when using conventional structure functions. Our results highlight a decoupling between velocity amplitude and flow geometry, offering new insight into the statistical structure of turbulent cascades as well as showing how different classes of multiscaling emerge.

physics.flu-dyn↗

arXiv2025

Extreme dynamics and relaxation of quantum gases: A hydrodynamic approach

The evolution of quantum gases, released from traps, are studied through hydrodynamics, both analytically and numerically, in one and two dimensions. In particular, we demonstrate the existence of long time self-similar solutions of the Euler equations, for the density and velocity fields, and derive the scaling exponents as well as the scaling functions. We find that the expanding gas develops a shock front and the size of the cloud grows in time as a powerlaw. We relate the associated exponent to that appearing in the corresponding equation of state of the quantum gas. Furthermore, we study the relaxation dynamics of a trapped quantum gas and show that the resulting steady state is in excellent agreement with that derived analytically. Our hydrodynamic approach is versatile and can be used to unravel several other far-from-equilibrium collective phenomenon of extreme nature, relevant to the growing experimental interests in quantum gases.

cond-mat.quant-gas↗

arXiv2025

Intermittent fluctuations determine the nature of chaos in turbulence

We adapt recent ideas for many-body chaos in nonlinear, Hamiltonian fluids [Murugan \textit{et al.}, Phys. Rev. Lett. 127, 124501 (2021)] to revisit the question of the Reynolds number Re dependence of the Lyapunov exponent $\lambda\propto{\rm Re}^\alpha$ in fully developed turbulence. The use of such decorrelators allow us to investigate the interplay of the competing effects of viscous dissipation and nonlinearity. We obtain a precise value of $\alpha = 0.59 \pm 0.04$ and show that departure from the Kolmogorov mean field result $\lambda \propto \sqrt{{\rm Re}}$ is a consequence of the intermittent fluctuations in the velocity-gradient tensor. The robustness of our results are further confirmed in a local, dynamical systems model for turbulence.

physics.flu-dyn↗

arXiv2024

Enumeration of Rational Cuspidal Curves via the WDVV equation

We give a conjectural formula for the characteristic number of rational cuspidal curves in the projective plane by extending the idea of Kontsevich's recursion formula (namely, pulling back the equality of two divisors in the four pointed moduli space). The key geometric input that is needed here is that in the closure of rational cuspidal curves, there are two component rational curves which are tangent to each other at the nodal point. While this fact is geometrically quite believable, we haven't as yet proved it; hence our formula is for the moment conjectural. The answers that we obtain agree with what has been computed earlier Ran, Pandharipande, Zinger and Ernstrom and Kennedy. We extend this technique (modulo another conjecture) to obtain the characteristic number of rational quartics with an E6 singularity.

math.AG↗

arXiv2024

On a fibre bundle version of the Caporaso-Harris formula

The Caporaso-Harris formula gives a recursive algorithm to enumerate delta nodal degree d curves in P^2. The recursion is obtained in terms of curves of lower degree that are tangent to a given divisor. This paper presents two generalizations of this method. The first result is on enumeration of one cuspidal curves on P^2, and the second result is an extension to the fiber bundle setting. We solve the question of counting the characteristic number planar nodal cubics in P^3 by extending the idea of Caporaso-Harris.

math.AG↗

arXiv2024

Weyls's law for Compact Rank One Symmetric Spaces

Weyls law is a fundamental result governing the asymptotic behaviour of the eigenvalues of teh Laplacian. It states that for a compact d dimensional manifold M (without boundary), the eigenvalue counting function has an asymptotic growth, whose leading term is of the order of d and the error term is no worse than order d-1. A natural question is: when is the error term sharp and when can it be improved? It has been known for a long time that the error term is sharp for the round sphere (since 1968). In contrast, it has only recently been shown (in 2019) by Iosevich and Wyman that for the product of spheres, the error term can be polynomially improved. They conjecture that a polynomial improvement should be true for products in general. In this paper we extend both these results to Compact Rank One Symmetric Spaces (CROSSes). We show that for CROSSes, the error term is sharp. Furthermore, we show that for a product of CROSSes, the error term can be polynomially improved. This gives further evidence to the conjecture made by Iosevich and Wyman.

math.DG↗

arXiv2024

Hydrodynamics of a hard-core active lattice gas

We present a fluctuating hydrodynamic description of an active lattice gas model with excluded volume interactions that exhibits motility-induced phase separation under appropriate conditions. For quasi-one dimension and higher, stability analysis of the noiseless hydrodynamics gives quantitative bounds on the phase boundary of the motility-induced phase separation in terms of spinodal and binodal. Inclusion of the multiplicative noise in the fluctuating hydrodynamics describes the exponentially decaying two-point correlations in the stationary-state homogeneous phase. Our hydrodynamic description and theoretical predictions based on it are in excellent agreement with our Monte Carlo simulations and pseudospectral iteration of the hydrodynamics equations. Our construction of hydrodynamics for this model is not suitable in strictly one-dimension with single-file constraints, and we argue that this breakdown is associated with micro-phase separation.

cond-mat.stat-mech↗

arXiv2024

Kac's Central Limit Theorem by Stein's Method

In $1946$, Mark Kac proved a Central Limit type theorem for a sequence of random variables that were not independent. The random variables under consideration were obtained from the angle-doubling map. The idea behind Kac's proof was to show that although the random variables under consideration were not independent, they were what he calls \textit{statistically independent} (in modern terminology, this concept is called long range independence). The final conclusion of his paper was that the sample averages of the random variables, suitably normalized converges to the standard normal distribution. We describe a new proof of Mark Kac's result by applying Stein's method and show that the normalized sample averages converge to the standard normal distribution in the Wasserstein metric, which is stronger than the convergence in distribution.

math.PR↗

arXiv2023

Counting Curves with Tangencies

Interpreting tangency as a limit of two transverse intersections, we obtain a concrete formula to enumerate smooth degree $d$ plane curves tangent to a given line at multiple points with arbitrary order of tangency. Extending that idea, we then enumerate curves with one node with multiple tangencies to a given line of any order. Subsequently, we enumerate curves with one cusp, that are tangent to first order to a given line at multiple points. We also present a new way to enumerate curves with one node; it is interpreted as a degeneration of a curve tangent to a given line. That method is extended to enumerate curves with two nodes, and also curves with one tacnode are enumerated. In the final part of the paper, it is shown how this idea can be applied in the setting of stable maps and perform a concrete computation to enumerate rational curves with first-order tangency. A large number of low degree cases have been worked out explicitly.

math.AG↗

arXiv2023

Turbulent flows are not uniformly multifractal

The Frisch-Parisi multifractal formalism remains the most compelling rationalisation for anomalous scaling in fully developed turbulence. We now show that this formalism can be adapted locally to reveal the spatial distribution of generalized dimensions and of how multifractal the energy dissipation field is. In particular, we show that most regions of the flow are close to being mono-fractal and these are interspersed with islands of multifractality corresponding to the most singular structures in the flow. By defining a suitable measure $\Phi ({\bf x})$ of the spatial variation of multifractality, we show that this grows logarithmically with the extent to which the energy dissipation varies locally around ${\bf x}$. These results suggest ways to understand how singularities could arise in disparate regions of a flow and provides new directions in understanding anomalous dissipation and intermittency. We then employ the same technique to a non-intermittent, model turbulent flow to check the robustness of our conclusions.

physics.flu-dyn↗

arXiv2022

Counting rational curves with an $m$-fold point

We obtain a recursive formula for the number of rational degree $d$ curves in $\mathbb{CP}^2$ that pass through $3d+1-m$ generic points and that have an $m$-fold singular point. The special case of counting curves with a triple point was solved earlier by other authors. We obtain the formula by considering a family version of Kontsevich's recursion formula, in contrast to the excess intersection theoretic approach of others. A large number of low degree cases have been worked out explicitly.

math.AG↗

arXiv2020

Counting planar curves in $\mathbb{P}^3$ with degenerate singularities

In this paper, we consider the following question: how many degree $d$ curves are there in $\mathbb{P}^3$ (passing through the right number of generic lines and points), whose image lies inside a $\mathbb{P}^2$, having $\delta$ nodes and one singularity of codimension $k$. We obtain an explicit formula for this number when $\delta+k \leq 4$ (i.e. the total codimension of the singularities is not more than four). We use a topological method to compute the degenerate contribution to the Euler class; it is an extension of the method that originates in a paper by A. Zinger and which is further pursued by S. Basu and the second author. Using this method, we have obtained formulas when the singularities present are more degenerate than nodes (such as cusps, tacnodes and triple points). When the singularities are only nodes, we have verified that our answers are consistent with those obtained by by S. Kleiman and R. Piene and by T. Laarakker. We also verify that our answer for the characteristic number of planar cubics with a cusp and the number of planar quartics with two nodes and one cusp is consistent with the answer obtained by R. Singh and the second author, where they compute the characteristic number of rational planar curves in $\mathbb{P}^3$ with a cusp. We also verify some of the numbers predicted by the conjecture made by Pandharipande, regarding the enumerativity of BPS numbers for $\mathbb{P}^3$.

math.AG↗

arXiv2020

Rational Cuspidal Curves in a moving family of $\mathbb{P}^2$

In this paper we obtain a formula for the number of rational degree d curves in $\mathbb{P}^3$ having a cusp, whose image lies in a $\mathbb{P}^2$ and that passes through $r$ lines and $s$ points (where $r + 2s = 3d + 1$). This problem can be viewed as a family version of the classical question of counting rational cuspidal curves in $\mathbb{P}^2$, which has been studied earlier by Z. Ran, R. Pandharipande and A. Zinger. We obtain this number by computing the Euler class of a relevant bundle and then finding out the corresponding degenerate contribution to the Euler class. The method we use is closely based on the method followed by A. Zinger and I. Biswas, S. D'Mello, R. Mukherjee and V. Pingali. We also verify that our answer for the characteristic numbers of rational cuspidal planar cubics and quartics is consistent with the answer obtained by N. Das and the first author, where they compute the characteristic number of $\delta$-nodal planar curves in $\mathbb{P}^3$ with one cusp (for $\delta \leq 2$).

math.AG↗

arXiv2019

Counting curves in a linear system with upto eight singular points

In this paper, we develop a systematic approach to enumerate curves with a certain number of nodes and one further singularity which maybe more degenerate. As a result, we obtain an explicit formula for the number of curves in a sufficiently ample linear system, passing through the right number of generic points, that have $δ$ nodes and one singularity of codimension $k$, for all $δ+k \leq 8$. In particular, we recover the formulas for curves with upto six nodal points obtained by Vainsencher. Moreover, all the codimension seven numbers we have obtained agree with the formulas obtained by Kazarian. Finally, in codimension eight, we recover the formula of A.Weber, M.Mikosz and P.Pragacz for curves with one singular point and we also recover the formula of Kleiman and Piene for eight nodal curves. All the other codimension eight numbers we have obtained are new.

math.AG↗