Physical Sciences

Abstract: This paper aims to reveal the systematic unification of dimensional analysis with scaling Lie group invariants and Lie algebra representation theory, proving that its essence is the symmetry reduction of physical laws under the scaling group (stretching group). By introducing the theory of Lie groups and their Lie algebras, we reinterpret dimensionless numbers as absolute invariants under group action, and the dimensional matrix as the representation matrix of the Lie algebra. On this basis, we demonstrate in detail how to solve the invariant equations via infinitesimal generators, and through the rigorous solution of two classic physics examples. We elucidate that the Lie group method can not only naturally derive the Buckingham $\Pi$ theorem but also reveal deep structural insights into the orthogonal decoupling between physical quantities.

Abstract: The multi-scale self-similarity and intermittency in incompressible turbulence fundamentally stem from scaling symmetry and its breaking. Using Lie group and infinitesimal generator theory, this paper algebraically reconstructs the Kolmogorov K41 and She-Leveque (SL) scaling laws as an equivalent mapping of existing phenomenological models rather than a first-principles derivation. We show that K41 corresponds to the invariance (zero eigenvalue) of energy flux under the scaling generator, yielding a strictly linear character, whereas anomalous scaling reflects symmetry breaking. Furthermore, SL's hierarchical recursion is demonstrated to be equivalent to an eigenvalue difference equation of the prolonged generator acting on the hierarchy. This difference eigenvalue quantifies the hierarchy's ``nonlinear sensitivity'' to scale variations---akin to weight differences in representation theory---characterizing the degree of symmetry breaking. By solving this recursion with the geometric boundary conditions of 3D vortex tubes, the SL formula is reconstructed. Our framework reveals K41 as the flat spacetime of the turbulent mean field, and SL as the curved spacetime induced by vortex singularities, with Lie algebra providing the elegant mathematical language for this transition.

Abstract: This paper develops a thermo-acoustic continuation framework for physically admissible compressible Navier–Stokes–Fourier evolution. The analysis is formulated under the assumptions of positivity of density and temperature, entropy admissibility, free-energy dissipation, finite acoustic propagation, strict hyperbolicity, uniformly subsonic evolution, constitutive smoothness, and a finite-energy weak solution framework. The admissibility conditions are treated as the physical regime of the theory. The central objective is to determine whether thermo-acoustic dissipative structure suppresses scale-critical concentration compatible with singularity formation. A localized entropy concentration quantity is introduced using the entropy-production density generated by viscous deformation and thermal diffusion. The analysis establishes localized thermo-acoustic coercivity, derives nonlinear subcriticality estimates for transport, thermal, acoustic, pressure, coefficient, and commutator remainders, and obtains higher thermo-acoustic integrability through compactness and Meyers-type arguments. Campanato iteration then yields oscillation decay, localized Hölder regularization, and thermo-acoustic ε-regularity. Within the admissible thermo-acoustic regime, persistent scale-critical concentration is excluded. Consequently, admissible thermo-acoustic evolution admits continuation beyond finite admissible evolution intervals. The continuation mechanism is generated by entropy production, thermal diffusion, free-energy dissipation, and finite-speed acoustic redistribution.The paper also studies incompressible projection of the thermo-acoustic system. Using projection fibers and conditional disintegration theory, it is shown that the entropy-generating thermo-acoustic structure is not generally reconstructible from incompressible projected variables alone. The analysis identifies a structural difference between admissible thermo-acoustic compressible evolution and mechanically projected incompressible evolution.The paper does not prove unconditional global regularity for arbitrary compressible Navier–Stokes–Fourier solutions, unconditional propagation of thermo-acoustic admissibility, or regularity or singularity formation for incompressible Navier–Stokes evolution. The continuation result is conditional on persistence of the admissible thermo-acoustic regime.

Abstract: We develop a thermodynamically closed framework for the three-dimensional compressible Navier–Stokes–Fourier system and analyze the nonlinear continuation problem through a unified thermo–acoustic formulation. The analysis begins with a Fourier–triadic decomposition of the compressible dynamics, showing that strongly nonlocal interactions remain perturbative while potentially dangerous amplification is localized near coherent same-scale High–High transfer. The compressible system is then reformulated in entropy variables, yielding a skew–dissipative thermo–acoustic structure in which the transport component becomes entropy-skew and the dissipative component remains monotone through entropy production and thermodynamic diffusion. This structure generates local thermo–acoustic coercivity directly controlling the gradient structure of the entropy variables. Using critical blow-up normalization, nonlinear thermo–acoustic ancient profiles are constructed without assuming linearization of the limiting dynamics. The entropy-variable structure is shown to survive the limiting process and induces a thermodynamic rigidity mechanism for all admissible nonlinear ancient limits. In particular, the localized gradient entropy density satisfies a parabolic subsolution inequality, leading to a Liouville-type classification theorem excluding nontrivial critical thermo–acoustic ancient profiles. Combining the rigidity theory with blow-up compactness and thermo–acoustic ε-regularity, we prove that persistent critical entropy concentration cannot occur. Consequently, finite-time continuation breakdown becomes incompatible with the thermodynamically admissible compressible Navier–Stokes–Fourier dynamics. This yields global strong regularity under the thermodynamic closure framework developed in the present paper. The later parts of the paper further establish weak–strong uniqueness, relative entropy stability, and long-time thermodynamic relaxation toward equilibrium. The overall framework connects nonlinear transfer localization, concentration exclusion, thermodynamic stability, and asymptotic relaxation within a unified thermo–acoustic dynamical structure.

Abstract: The topological properties of planetary fluids are typically analyzed by mapping classical fluid equations onto complex quantum mechanical models. Here we present a purely real, six-dimensional Stueckelberg quantum mechanical formulation of the rotating shallow water equations to demonstrate that these topological features are intrinsic to the classical kinematics itself. Operating entirely within R⁶ we decouple the complex quantum geometric tensor into an independent real Fubini-Study metric and a real antisymmetric Berry curvature. Our real-variable approach explicitly derives a topological magnetic monopole of charge C = −2 and captures the inherent scale invariance of the fluid's geometry without explicit complex coordinate representation. We suggest that continuous variations in the Coriolis parameter model the adiabatic geometric evolution of the Archean Earth, and we propose a laboratory rotating-tank experiment to physically measure this parameter sweep. Finally, we show that our real 6D formulation naturally maps to unbroken supersymmetric quantum mechanics. By identifying a purely real supercharge and calculating a fluid Witten index of W = −2, we demonstrate a strict mathematical symmetry between the topological charge of the propagating bands and the invariant of the unbroken zero-energy geostrophic vacuum. We advance the mathematically supported viewpoint that steady-state geostrophic weather patterns represent the exact supersymmetric ground states of the rotating fluid system. Consequently, the topological isolation of this vacuum naturally restricts the spectral flow across the equator, providinga theoretical explanation for the unidirectional eastward motion of equatorial boundary waves.

Abstract:

Dixit et al. proposed an asymptotic drag scaling for zero-pressure-gradient flat-plate turbulent boundary layers based on the approximation $M\sim U_{\tau}^2\delta$, where $M$ is the kinematic momentum rate through the boundary layer, $U_{\tau}$ is the friction velocity, and $\delta$ is the boundary-layer thickness. In the present paper, an explicit Reynolds-number-dependent correction to this approximation is derived from the logarithmic mean-velocity profile. Integration of the log law across the layer yields $M\sim U_{\tau}^2\delta\,f(Re_{\tau})$, where $Re_{\tau}=\delta U_{\tau}/\nu$ is the friction Reynolds number and $f(Re_{\tau})$ is given analytically. Application of the correction to the dataset compiled by Dixit et al. shows that the corrected scaling gives an exponent consistent with the asymptotic value $-1/2$ within bootstrap confidence intervals, whereas the uncorrected formulation does not. The correction should be viewed as a leading-order amendment, since the derivation uses the logarithmic law outside its strict range of validity.

Abstract: The effect of atom size on the shock wave structure in a binary monatomic gas mix with Rydberg atoms has been investigated. The problem was solved numerically using the system of hydrodynamic equations in Argon gas, for the atom size ratios between 2 and 100, T = 1500 K, and the density between 10¹⁷ and 10²⁰ m^-3. It was found that the presence of larger size atoms in the mix results in the shock front splitting that is on the order of mean free path for this component. The results can be of interest in supersonic plasma dynamics and in astrophysics studying shock waves in the environments where high-n Rydberg states are present.

Abstract: During atmospheric reentry, a spacecraft is enveloped by a turbulent plasma sheath that induces severe signal degradation and communication blackout. Conventional mitigation strategies primarily focus on reducing average attenuation but fail to address the dynamic fluctuations in plasma density (typically 20%–40%), which cause significant group velocity dispersion (GVD), pulse broadening, and intersymbol interference. To overcome this limitation, this paper proposes an active decoupling framework that dynamically tunes an external magnetic field to suppress turbulence-induced signal distortion in the reentry plasma sheath. By establishing a wave propagation model for right-hand circularly polarized (RCP) waves in magnetized collisional plasma and introducing a sensitivity analysis of propagation parameters with respect to plasma density fluctuations, we derive the condition under which the first-order sensitivity of GVD vanishes. Under this condition, a dynamic balance between collisional effects and frequency detuning renders the system immune to density perturbations, effectively decoupling signal transmission from plasma turbulence. Numerical simulations demonstrate that, under optimal parameter matching, pulse broadening is suppressed by several orders of magnitude, and the broadening factor remains near unity over extended propagation distances. Furthermore, reentry trajectory analysis reveals that static matching is insufficient in dynamically evolving environments, motivating the necessity of adaptive magnetic field control. This work provides a novel physical-layer paradigm for mitigating reentry blackout by actively decoupling signals from turbulence via dynamically tuned magnetic fields.

Abstract: In this study, we have investigated the existence and properties of solitons in an unmagnetized plasma composed of positive ions, negative ions, negatively charged dust grains, non-thermal electrons and non-extensive positrons. We have conducted our study on this complex plasma model because it moves away from simplistic and idealized plasma models. Also, study of solitons has not been conducted previously on this complex plasma model. Through the Sagdeev potential method we have derived the energy integral and investigated the variation of the Sagdeev potential for different values of the parameters that are involved in our plasma model. We have found that the non-thermal parameter (β) and the non-extensive parameter(q) significantly influence the features of the solitons. The features of the solitons are also found to be influenced by the Mach number (M), the negative ion to positive ion mass ratio (Ω), the positron to positive ion density ratio (δp) , the electron to positron temperature ratio (σp), the dust charge density ratio (δd ) and the negative ion to positive ion density ratio(δ_ ). The results from our study can be useful in investigating plasma in astrophysical environments, such as cometary tails and interstellar clouds.

Abstract: We investigate the continuation problem for the three-dimensional incompressible Navier–Stokes equations from a structural, assumption-free perspective. Using the exact Fourier–helical representation and a dyadic shell decomposition, the nonlinear term is reformulated in terms of triadic interactions, allowing a scale-resolved analysis of energy transfer. Within this framework, we establish a complete structural reduction of the nonlinear dynamics. All cross-scale and non-coherent interactions are shown to be perturbatively controlled on every finite time interval and cannot produce non-integrable accumulation in weighted Sobolev norms on compact subintervals. As a result, any potential finite-time blow-up must be supported by a sharply restricted class of residual mechanisms. More precisely, we show that non-integrable accumulation of positive Sobolev-weighted transfer can occur only through either large-transfer same-scale interactions or endpoint accumulation of perturbative remainder contributions. All other interaction channels are excluded as possible sources of divergence by structural and energetic arguments. The analysis is entirely assumption-free and does not rely on any phase closure, temporal localization, or statistical modeling. It therefore provides a complete obstruction formulation of the continuation problem: blow-up is reduced to the viability of a minimal set of explicitly identified mechanisms. We further show that these residual mechanisms persist because the incompressible Navier–Stokes equations do not constitute a thermodynamically complete system. Interpreting the incompressible equations as a singular limit of the compressible formulation, we identify the loss of entropy-based dissipation as the structural origin of the missing control on positive nonlinear transfer. Motivated by this observation, we introduce a minimal ε-retained thermodynamic extension that restores a remnant of the free-energy dissipation mechanism. Under this extension, we show that the positive transfer becomes integrable and that both residual blow-up mechanisms are eliminated under the stated closure condition. This yields a precise conditional closure of the continuation problem. The results clarify the exact scope and limitation of Navier–Stokes-based analysis and reduce the global regularity problem to the question of whether a thermodynamic-type dissipation principle can be rigorously derived within, or as a limit of, the governing equations.

Abstract: Neoclassical tearing modes (NTMs) are magnetohydrodynamic instabilities that generate magnetic islands in tokamak plasmas, degrading confinement and potentially limiting high-performance operation. Their stabilization typically requires precise alignment and appropriate injection of electron cyclotron (EC) power beams, making real-time control a challenging task. In this work, we present a proof-of-principle study aimed at investigating the potential role of neural networks in the control of plasma instabilities. The objective is not to develop a device-specific controller, but rather to explore, within a synthetic environment, how a learning-based agent can autonomously discover effective stabilization strategies. To this end, a neural network controller is trained using reinforcement learning techniques, resulting in an intelligent and agnostic control system. The controller is defined as intelligent in the sense that it learns the optimal strategy directly from interaction with the environment, without being explicitly programmed or guided by a predefined control law. It is agnostic because it does not rely on equilibrium reconstruction or explicit knowledge of the deposition location relative to the island. Instead, it operates solely on feedback derived from a representation of the magnetic island width, using this information to adapt its actions. Two control tasks are considered: pure angular alignment and combined angular alignment with power control. This exploratory study establishes a framework for assessing the potential advantages of data-driven approaches in magnetic island control and provides a basis for future investigations aimed at improving alignment and suppression strategies in fusion plasmas.

Abstract: This article reviews the research of Fedor Vasilievich Shugaev, Doctor of Physical and Mathematical Sciences and Professor at the Faculty of Physics of M. V. Lomonosov Moscow State University. Over a career at MSU spanning more than six decades, Professor Shugaev has published 146 journal articles, 4 monographs, 46 conference papers and 83 invited talks, and supervised twelve Candidate-of-Sciences dissertations and twenty diploma theses. The main research lines covered below are the propagation and reflection of shock waves; shock-wave interaction with vortices, acoustic disturbances and turbulent fluctuations; shock-wave dynamics in low-temperature and discharge plasmas; the geometry and stability of magnetised and astrophysical bow shocks; Navier–Stokes-based methods for vortex acoustics; laser-beam propagation through the turbulent atmosphere; and a recent cycle of work in theoretical astrophysics.

Abstract: The composition of partially ionised plasmas is investigated for densities and temperatures at which the free electrons are degenerate. Based on a quantum statistical approach, the effect of Pauli blocking is addressed. Specifically, one- and two-electron ions are studied. New results regarding the degree of ionisation and the Mott effect are presented. Standard codes for plasma properties do not take Pauli blocking effects into account and are therefore unable to explain the experiments in the high-density regime, where the electrons are degenerate.

Abstract: A numerical bifurcation analysis is presented for inductively coupled plasmas and wall stabilized arcs for argon and hydrogen. Because of the non–linear transport and radiative properties both problems admit multiple solutions, up to three for argon and up to four for hydrogen. The multiplicity structure primarily dependents on the non–linear and especially the non–monotonic relationship between thermal conductivity and temperature. As a result of the non-monotonicity a multipoint energy equilibrium between Joule heating (heat generation) and heat dissipation by conduction and radiation exists, giving rise to the multiplicity which is a characteristic feature of both radiating and non–radiating arcs. Despite the relatively simple one–dimensional model employed the agreement with the experimental data is good.

Abstract: The early turbulence phenomenon has been observed in pipe flow of very dilute polymer solutions [1–4], and the full chord laminar flow can be achieved on various laminar suction wings at high Reynolds numbers (Re) up to approximately 2 000 000 [5–7]. Their transition conditions deviate significantly from the traditional criteria, critical Re about 2 000~2 300, which is quoted in most contemporary textbooks for pipe flow [8–13]. In this paper, a new force model with a virtual fluid layer, which is of a hemispherical shell shape and with a constant thickness inside a laminar pipe flow is established, on the basis of the membrane force model of a spherical shell under uniformly distributed load conditions in structural mechanics. In laminar flow state with a lower Re and a lower pressure gradient, the curvature radius of the virtual spherical liquid layer is inversely proportional to the pressure gradient. As Re increases, pressure gradient also increases, while the curvature radius decreases. When the curvature radius decreases to be equal to and starting less than the pipe radius, the stable liquid layer structure collapses, and the laminar flow becomes turbulent. This is a transition state with a critical tensile force flow defined as twice the product of the viscosity of the fluid and the maximum velocity in pipe, divided by the pipe radius. In laminar flow situation, the shear stress at the pipe wall can be interpreted as a horizontal component of the critical tensile force flow, and the direction is against the flow. Only when the flow achieving the transition condition, the shear stress at the wall become the critical tensile force flow itself, which had already been observed in early turbulence [1–4,14]. The second case, which can be explained by the concept of critical tensile force flow, is high Re laminar pipe flow [5–7], for example, the pipe with surface suction can be considered as a part of a virtual, larger pipe with a no slip wall at where the shear stress coincides with the critical tensile force flow, the shear stress at the real pipe is smaller, with a weakening factor related to the ratio of the average velocity in the real pipe to its maximum velocity.

Abstract: In this article, the different types of self-coagulation discovered in the fluid simulations of inductively coupled plasma (abbreviated as ICP) at both the electronegative and electropositive cases are presented. Among these, the electronegative plasma sources include the Ar/O₂, Ar/Cl₂, and Ar/SF₆ ones, and the electropositive plasma source is the inertial argon plasma itself. The fluid simulation versions are not the same. Concretely, the Comsol software is used to simulate the Ar/O₂, Ar/Cl₂, and Ar/SF₆, and the pure argon ICPs, and the self-written code of fluid model is used to simulate the pure argon ICP as well, but in a different framework of fluid design. The types of self-coagulation refined from these fluid simulations are the physically ambi-polar self-coagulation of ions, the chemically ambi-polar self-coagulation of ions, the mono-polar self-coagulation of electrons, and the non-polar self-coagulation of argon metastable atoms. It is noted these self-coagulations are based on the mass and found in the Comsol fluid simulations, and moreover the self-coagulation of thermal energy of electrons is also given and found in the self-written fluid code simulation. The self-coagulations of mass and energy found in the laboratory plasmas have significant implications on ambi-polar diffusion, the wave-particle duality, application of Schrodinger equation, the positive and reverse species pair, the β and β⁺ decay, the spin orientations of neutrino and anti-neutrino, the symmetry and asymmetry, and the photon model. It is believed this interdisciplinary work of plasma physics with the quantum mechanics, the particle physics, the nuclear physics, and the optics are useful for us better understanding the mass and energy general dynamics. The self-coagulation definition constructed herein is reliable since it is validated in many circumstances, such as in the different discharging plasma species and in the thermal energy, whether the Comsol software or the self-written fluid model is used.

Abstract: This is the second article on the mechanical mechanism of laminar turbulent transition in pipe Poiseuille flow, which is one of the most important topics in turbulence research [1,2], as a representative of a large category of wall-bounded flows [3]. Traditional fluid mechanics stability research focuses on the effects of different disturbances and pays less attention to the mechanical properties of flow structures [4]. In this paper, the tensile energy flux, which is renamed from the viscous energy flux vector [5], and its divergence are deduced and visualized in pipe Poiseuille flow. The tensile energy flux vector is both zero at wall and in the center, and at a critical position 0.707R, the divergence of tensile energy flux vector is zero. Once the tensile force flow reaches its critical value [4], the critical position 0.707R is just the local position where onset of turbulence occurs, consistent with some experimental results [6]. This predicted position has a zenith angle of 45° if membrane theory of spherical shell is applied on the fluid [4], and this angle may be analogous to the cracks angle in the uniaxial compressive strength experiment of rock specimen subjected to uniaxial compression [7]. This article also proves that the critical Reynolds number during laminar turbulent transition in a circular tube is not a constant, but the ratio of critical tensile energy flux to average kinetic energy flux inside the tube is inversely proportional to the Reynolds number, similar to the inverse relationship between laminar flow resistance coefficient and Reynolds number.

Abstract: We develop a unified dynamical framework for the three-dimensional incompressible Navier–Stokes equations in which global regularity and turbulent inertial-range structure emerge from a common underlying mechanism. Building on a recent result establishing global regularity via coherent-core reduction and phase non-persistence, we reformulate the nonlinear dynamics in terms of triadic interactions and their associated phase evolution. We show that nonlinear amplification is confined to a High–High interaction channel, which can be further localized to a coherent core characterized by low phase drift. The phase dynamics within this core exhibits a curvature-driven instability, implying that persistent phase coherence is dynamically impossible. As a consequence, nonlinear transfer is temporally localized, preventing cumulative growth and ensuring global regularity. Using this structure, we derive the inertial-range energy cascade directly from deterministic dynamics. The combination of time-localized interactions and scale-dependent triadic multiplicity yields a constant energy flux across scales without invoking statistical assumptions or closure models, leading to a first-principles derivation of the Kolmogorov −5/3 scaling law. Furthermore, we show that the Kolmogorov constant is not an empirical parameter but a dynamically determined quantity arising from phase-averaged triadic interactions. At the continuum level, the theory yields a structural formula together with a finite admissible interval. This remaining indeterminacy is resolved by extracting the coherent-phase quantities from a GOY shell model, used as a dynamically consistent reduced system that preserves local triadic interactions. The resulting value is thereby obtained without introducing phenomenological closure assumptions. These results establish that Navier–Stokes regularity, inertial-range cascade, and the determination of the Kolmogorov constant are not independent phenomena, but three manifestations of a single triadic phase dynamic. The mechanism that suppresses finite-time blow-up is identical to the mechanism that generates energy transfer across scales and fixes the Kolmogorov constant, providing a unified deterministic foundation for fluid dynamics.

Abstract: We propose a thermodynamically consistent framework for computational fluid dynamics based on a master equation formulation that precedes the continuum description. Instead of discretizing the Navier–Stokes equations, the fluid is modeled as a network of interacting elements whose dynamics are governed by antisymmetric conservative interactions and entropy-generating dissipative interactions. This construction embeds conservation laws and the second law of thermodynamics directly at the discrete level. A unified graph-based discretization is introduced, in which finite volume and meshless methods arise as special cases of a common interaction structure. Reversible fluxes are constructed to be entropy-neutral, while irreversible fluxes are derived from an entropy gradient flow, yielding a systematic decomposition of transport and dissipation. An implicit–explicit (IMEX) time integration scheme is then designed to preserve conservation and ensure entropy monotonicity. We establish convergence of the resulting scheme to entropy weak solutions of the Navier–Stokes equations through uniform a priori estimates and compactness arguments. Numerical experiments on compressible flow demonstrate that the proposed method maintains stability comparable to classical schemes while significantly reducing numerical dissipation, particularly in the resolution of contact discontinuities. These results suggest a shift in perspective: fluid dynamics can be understood not primarily as a system of partial differential equations, but as a thermodynamically constrained interaction system from which continuum equations emerge as effective limits. This viewpoint provides a unified foundation for the design of stable, accurate, and physically consistent numerical methods.

Abstract: We present a unified and thermodynamically consistent framework for the derivation and analysis of the two-dimensional incompressible Navier–Stokes equations based on a network-type master equation. The proposed formulation originates from a discrete, conservative interaction system endowed with a dissipative structure, and is designed to satisfy fundamental physical principles including conservation laws and, in its extended form, the second law of thermodynamics. Starting from this master equation, we construct a finite-volume discretization that preserves the antisymmetric structure of nonlinear interactions and ensures discrete energy stability. We then rigorously establish the convergence of the discrete system to a continuous limit, showing that the incompressible Navier–Stokes equations arise naturally as a singular limit of the underlying thermodynamic dynamics.Using compactness arguments of Aubin–Lions type, we prove the existence of global weak solutions in two dimensions. Furthermore, by exploiting the vorticity formulation and enstrophy estimates specific to two-dimensional flows, we demonstrate global regularity and uniqueness of solutions. These results are obtained within a single, coherent framework that connects microscopic interaction models, discrete numerical structures, and continuum fluid equations. Although the global well-posedness of the two-dimensional Navier–Stokes equations are classical, the present work provides a novel perspective by deriving these results from a physically grounded master equation, thereby offering a structurally consistent bridge between discrete thermodynamic systems and continuum fluid mechanics. This approach not only clarifies the origin of the Navier–Stokes equations but also establishes a robust foundation for future extensions to more complex systems, including compressible flows and higher-dimensional turbulence.