arXiv subjects

Ahmed Farooq

Publications and source records attributed to Ahmed Farooq.

8 recordsLinked to original sources

arXiv2026

A Variational Formulation of the MHD Induction Equation

We present a variational formulation of the induction equation in magnetohydrodynamics (MHD) based on Gauss's principle of least constraint. The central result is the Euler--Lagrange equation $\mathbf{Z}_B = -\nabla h_m$, where $h_m = \mathbf{A}\cdot\mathbf{B}$ is the magnetic helicity density. This reveals that the magnetic helicity gradient $\nabla h_m$ acts as the constraint force maintaining the solenoidality of the magnetic field, exactly as the pressure gradient $\nabla p$ maintains incompressibility in Taha et al.'s pressure-gradient minimization principle. The magnetic helicity density---which, though gauge-dependent locally, yields a gauge-invariant variational principle---naturally emerges as the Lagrange multiplier enforcing $\nabla\cdot\mathbf{B}=0$. At the solution, the field minimizes the norm of the magnetic helicity gradient $\|\nabla h_m\|^2$. This establishes a structural analogy: magnetic helicity is to the magnetic field as pressure is to velocity. The variational principle connects to Woltjer's theorem, Taylor relaxation, and the Hamiltonian structure of MHD.

physics.flu-dyn↗

arXiv2026

A Variational Principle for the Vorticity Equation

We present a variational formulation of the incompressible vorticity equation based on Gauss's principle of least constraint using the Gauss constraint functional $\Zvec_ω$. The central result is the Euler--Lagrange equation $\mathbf{Z}_ω= -\nabla h$, where $h = \mathbf{u} \cdot\boldsymbol{omega}$ is the helicity density. This reveals that the helicity gradient $\nabla h$ acts as the constraint force maintaining the solenoidality of the vorticity field, exactly as the pressure gradient $\nabla p$ maintains incompressibility in Taha et al.'s pressure-gradient minimization principle. The helicity density naturally emerges as the Lagrange multiplier enforcing $\nabla \cdot \boldysmbol{omega}=0$, and at the solution the flow minimizes the norm of the helicity gradient $\|\nabla h\|^2$. This establishes the exact duality: helicity is to vorticity as pressure is to velocity. We apply the variational principle to the Burgers vortex and verify the the Euler-Lagrange equation. The variational principle connects to Moffatt's helicity conservation theorem, Arnold's geometric formulation of ideal fluid flow, and Kambe's gauge-theoretic formulation. This work provides a unified variational framework for fluid dynamics that spans classical mechanics, geometric mechanics, and topological field theory. We discuss how this work may provide a theoretical foundation for understanding the role of helicity gradients in boundary layer dynamics, with potential implications for the formation of coherent structures and the onset of transition. These applications are reserved for future work.

physics.flu-dyn↗

arXiv2026

An SO(3) Gauge Theory of Turbulence with Spontaneous Symmetry Breaking

Fully developed isotropic turbulence exhibits a dual nature: a continuous, scale-invariant energy cascade coexists with discrete, intense vortex filaments. We show that this duality arises from a spontaneously broken SO(3) gauge symmetry. By identifying the specific angular momentum $\mathbf{L} = \mathbf{r}\times\mathbf{u}$ as a non-Abelian gauge connection and the radial velocity $u_r$ as a Higgs field, the turbulent vacuum is described by the SO(3) Georgi-Glashow model. When the radial strain condenses, the symmetry breaks SO(3) $\to$ U(1), generating a topological mass gap $M_W = gv$. This gap partitions the energy into a massless U(1) sector (the solenoidal background) that sustains the Kolmogorov cascade, and a massive SO(3)/U(1) sector that is confined to vortex filaments. Using high-resolution DNS data (JHTDB, $Re_λ\approx433$), we empirically verify three key predictions: (i) the energy spectra obey a strict 1:2 equipartition over the inertial range, with a sharp divergence at $M_W \approx 40$; (ii) the radial Higgs field extracted around isolated vortex cores follows the exact BPS monopole profile $H(r)=\coth(r/η)-η/r$ with $η= 0.0093$ domain units and the VEV $v = 0.338$, identifying the ubiquitous "worms" as macroscopic 't Hooft-Polyakov monopoles; (iii) the Wilson loop computed from the velocity field exhibits a clean area law $\langle W_C \rangle \sim e^{-σA}$ with string tension $σ= 0.303 \pm 0.009$, directly confirming the confining nature of the turbulent vacuum.

physics.flu-dyn↗

arXiv2026

Beyond Vorticity: An Angular Momentum Perspective on Fluid Flow

While vorticity is the classical tool for analyzing rotational fluid kinematics, it inherently focuses on local, differential spin. This paper introduces a complementary framework based on the angular momentum density field, $\mathbf{L} = \mathbf{r} \times \mathbf{u}$, deriving generalized transport equations that explicitly balance macroscopic torque and rotational momentum. This $\mathbf{L}$ perspective offers several distinct theoretical advantages over traditional velocity/vorticity formulations. Specifically, this approach: (i) provides a novel decomposition of the viscous torque into a diffusive component and a local spin dissipative term; (ii) shows the mechanism by which lift is generated in viscous boundary layers by vorticity acting as a source of angular momentum; it also explains stall (iii) reformulates the hydrodynamic impulse to yield a remarkably clean separation of terms into dilatational, volumetric, and rotational flux components; The $\mathbf{L}$ formalism provides the kinematic closure necessary to unify non-circulatory added mass and circulatory lift within a single, dimensionally consistent budget. (iv) enables the direct calculation of the viscous added mass force, accounting for the inertial resistance of boundary layers and separated wakes; (v) simplifies geophysical fluid dynamics by absorbing the planet's rotation, traditionally treated as an artificial virtual vorticity term which directly gets absorbed into the conserved axial angular momentum $m$, revealing the fundamental physics of global circulation through explicit torque balances; (vi) identifies the rotlet as a fundamental Green's function for the $\mathbf{L}$ transport equation in the Stokes regime; and (vii) demonstrates that both oblique shocks and vortex sheets act as singular sources of $\mathbf{L}$ that turn the macroscopic flow.

physics.flu-dyn↗

arXiv2026

A Statistical Field Theory for Isotropic Turbulence

This article establishes a first-principles statistical field theory of fully developed isotropic turbulence. Applying an exact Helmholtz decomposition to the local angular momentum field ($\Lvec = \rvec \times \uvec$) reveals a segregation into two orthogonally distinct topological phases: a longitudinal condensate of macroscopic coherent structures ($\PhiL$) and a volume-filling, transverse thermal bath ($\AL$). Constructing a Hamiltonian and evaluating the partition function of these decoupled fields demonstrates that their ergodic exploration of phase space is topologically quantized, mandating a strict $1:2$ equipartition of degrees of freedom. Inverting this topological projection back to the velocity domain isolates the radial velocity field ($\uvec_r$) (which strictly resides in the null space of the $\Lvec$ framework) revealing a recursive partitioning scheme across the cascade into a precise $1/3 : 2/9 : 4/9$ fractional hierarchy. This geometric constraint forces the turbulent steady state into a rigorous canonical equilibrium governed by the equalization of phase chemical potentials ($μ_Φ= μ_A$). The radial component acts as a non-equilibrium mechanical piston, continuously injecting energy into the tangential modes to sustain the canonical equilibrium -- a mechanism that mathematically formalizes the classical phenomenology of vortex stretching. Spectral evaluations from direct numerical simulation strongly corroborate this thermodynamic framework, establishing the universality of the partition ratios $1:2$ and $1/3 : 2/9 : 4/9$ as a fundamental signature of three-dimensional isotropic turbulence.

physics.flu-dyn↗

arXiv2025

A Hybrid Multi-Well Hopfield-CNN with Feature Extraction and K-Means for MNIST Classification

This study presents a hybrid model for classifying handwritten digits in the MNIST dataset, combining convolutional neural networks (CNNs) with a multi-well Hopfield network. The approach employs a CNN to extract high-dimensional features from input images, which are then clustered into class-specific prototypes using k-means clustering. These prototypes serve as attractors in a multi-well energy landscape, where a Hopfield network performs classification by minimizing an energy function that balances feature similarity and class assignment.The model's design enables robust handling of intraclass variability, such as diverse handwriting styles, while providing an interpretable framework through its energy-based decision process. Through systematic optimization of the CNN architecture and the number of wells, the model achieves a high test accuracy of 99.2% on 10,000 MNIST images, demonstrating its effectiveness for image classification tasks. The findings highlight the critical role of deep feature extraction and sufficient prototype coverage in achieving high performance, with potential for broader applications in pattern recognition.

cs.CV↗

arXiv2025

Long-Sequence Memory with Temporal Kernels and Dense Hopfield Functionals

In this study we introduce a novel energy functional for long-sequence memory, building upon the framework of dense Hopfield networks which achieves exponential storage capacity through higher-order interactions. Building upon earlier work on long-sequence Hopfield memory models, we propose a temporal kernal $K(m, k)$ to incorporate temporal dependencies, enabling efficient sequential retrieval of patterns over extended sequences. We demonstrate the successful application of this technique for the storage and sequential retrieval of movies frames which are well suited for this because of the high dimensional vectors that make up each frame creating enough variation between even sequential frames in the high dimensional space. The technique has applications in modern transformer architectures, including efficient long-sequence modeling, memory augmentation, improved attention with temporal bias, and enhanced handling of long-term dependencies in time-series data. Our model offers a promising approach to address the limitations of transformers in long-context tasks, with potential implications for natural language processing, forecasting, and beyond.

cs.LG↗

arXiv2025

A Framework for Non-Linear Attention via Modern Hopfield Networks

In this work we propose an energy functional along the lines of Modern Hopfield Networks (MNH), the stationary points of which correspond to the attention due to Vaswani et al. [12], thus unifying both frameworks. The minima of this landscape form "context wells" - stable configurations that encapsulate the contextual relationships among tokens. A compelling picture emerges: across $n$ token embeddings an energy landscape is defined whose gradient corresponds to the attention computation. Non-linear attention mechanisms offer a means to enhance the capabilities of transformer models for various sequence modeling tasks by improving the model's understanding of complex relationships, learning of representations, and overall efficiency and performance. A rough analogy can be seen via cubic splines which offer a richer representation of non-linear data where a simpler linear model may be inadequate. This approach can be used for the introduction of non-linear heads in transformer based models such as BERT, [6], etc.

stat.ML↗