arXiv subjects

Sonakshi Sachdev

Publications and source records attributed to Sonakshi Sachdev.

7 recordsLinked to original sources

arXiv2024

Level crossing instabilities in inviscid isothermal compressible Couette flow

We study the linear stability of inviscid steady parallel flow of an ideal gas in a channel of finite width. Compressible isothermal two-dimensional monochromatic perturbations are considered. The eigenvalue problem governing density and velocity perturbations is a compressible version of Rayleigh's equation and involves two parameters: a flow Mach number $M$ and the perturbation wavenumber $k$. For an odd background velocity profile, there is a $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry and growth rates $γ$ come in symmetrically placed 4-tuples in the complex eigenplane. Specializing to uniform background vorticity Couette flow, we find an infinite tower of noninflectional eigenmodes and derive stability theorems and bounds on growth rates. We show that eigenmodes are neutrally stable for small $k$ and small $M$ but that they otherwise display an infinite sequence of stability transitions with increasing $k$ or $M$. Using a search algorithm based on the Fredholm alternative, we find that the transitions are associated to level crossings between neighboring eigenmodes. Repeated level crossings result in windows of instability. For a given eigenmode, they are arranged in a zebra-like striped pattern on the $k$-$M$ plane. A canonical square-root power law form for $γ(k,M)$ in the vicinity of a stability transition is identified. In addition to the discrete spectrum, we find a continuous spectrum of eigenmodes that are always neutrally stable but fail to be smooth across critical layers.

physics.flu-dyn↗

arXiv2020

Conservative regularization of neutral fluids and plasmas

Ideal systems of equations such as Euler and MHD may develop singular structures like shocks, vortex/current sheets. Among these, vortical singularities arise due to vortex stretching which can lead to unbounded growth of enstrophy. Viscosity and resistivity provide dissipative regularizations of these singularities. In analogy with the dispersive KdV regularization of the 1D inviscid Burgers' equation, we propose a local conservative regularization of ideal 3D compressible flows, MHD and 2-fluid plasmas (with potential applications to high vorticity flows with low dissipation). The regularization involves introducing a vortical `twirl' term lambda^2 w x curl w in the velocity equation. The cut-off length lambda must be inversely proportional to square root of density to ensure the conservation of a `swirl' energy. The latter includes positive kinetic, compressional, magnetic and vortical contributions, thus leading to a priori bounds on enstrophy. The extension to 2-fluid plasmas involves additionally magnetic `twirl' terms in the ion and electron velocity equations and a solenoidal addition to the current in Ampere's law. A Hamiltonian-Poisson bracket formulation is developed using the swirl energy as Hamiltonian. We also establish a minimality property of the twirl regularization. A swirl velocity field is shown to transport vortex and magnetic flux tubes as well as w/rho and B/rho, thus generalizing the Kelvin-Helmholtz and Alfven theorems. The steady regularized equations are used to model a rotating vortex, MHD pinch and vortex sheet. Our regularization could facilitate numerical simulations and a statistical treatment of vortex and current filaments in 3D. Finally, we briefly describe a conservative regularization of shock-like singularities in compressible flow generalizing both the KdV and nonlinear Schrodinger equations to the adiabatic dynamics of a gas in 3D.

physics.flu-dyn↗

arXiv2020

Nonlinear dispersive regularization of inviscid gas dynamics

Ideal gas dynamics can develop shock-like singularities with discontinuous density. Viscosity typically regularizes such singularities and leads to a shock structure. On the other hand, in 1d, singularities in the Hopf equation can be non-dissipatively smoothed via KdV dispersion. Here, we develop a minimal conservative regularization of 3d ideal adiabatic flow of a gas with polytropic exponent $γ$. It is achieved by augmenting the Hamiltonian by a capillarity energy $β(ρ) (\nabla ρ)^2$. The simplest capillarity coefficient leading to local conservation laws for mass, momentum, energy and entropy using the standard Poisson brackets is $β(ρ) = β_*/ρ$ for constant $β_*$. This leads to a Korteweg-like stress and nonlinear terms in the momentum equation with third derivatives of $ρ$, which are related to the Bohm potential and Gross quantum pressure. Just like KdV, our equations admit sound waves with a leading cubic dispersion relation, solitary and periodic traveling waves. As with KdV, there are no steady continuous shock-like solutions satisfying the Rankine-Hugoniot conditions. Nevertheless, in 1d, for $γ= 2$, numerical solutions show that the gradient catastrophe is averted through the formation of pairs of solitary waves which can display approximate phase-shift scattering. Numerics also indicate recurrent behavior in periodic domains. These observations are related to an equivalence between our regularized equations (in the special case of constant specific entropy potential flow in any dimension) and the defocussing nonlinear Schrodinger equation (cubically nonlinear for $γ= 2$), with $β_*$ playing the role of $\hbar^2$. Thus, our regularization of gas dynamics may be viewed as a generalization of both the single field KdV & NLS equations to include the adiabatic dynamics of density, velocity, pressure & entropy in any dimension.

physics.flu-dyn↗

arXiv2018

Conservative regularization of compressible dissipationless two-fluid plasmas

This paper extends our earlier approach [cf. Phys. Plasmas 17, 032503 (2010), 23, 022308 (2016)] to obtaining a priori bounds on enstrophy in neutral fluids (R-Euler) and ideal magnetohydrodynamics (R-MHD). This results in a far-reaching local, three-dimensional, non-linear, dispersive generalization of a KdV-type regularization to compressible/incompressible dissipationless two-fluid plasmas and models derived therefrom (quasi-neutral, Hall and ideal MHD). It involves the introduction of vortical and magnetic `twirl' terms $λ_l^2 ({\bf w}_l + \frac{q_l}{m_l} {\bf B}) \times (\nabla \times {\bf w}_l)$ in the ion/electron velocity equations ($l = i,e$) where ${\bf w}_l = \nabla \times {\bf v}_l$ are vorticities. The cut-off lengths $λ_l$ must be inversely proportional to the square-roots of the number densities $(λ_l^2 n_l = C_l)$ and may be taken as Debye lengths or skin-depths. A novel feature is that the `flow' current $\sum_l q_l n_l {\bf v}_l$ in Ampere's law is augmented by a solenoidal `twirl' current $\sum_l \nabla \times \nabla \times λ_l^2 {\bf j}_{{\rm flow},l}$. The resulting equations imply conserved linear and angular momenta and a positive definite swirl energy density ${\cal E}^*$ which includes an enstrophic contribution $\sum_l (1/2) λ_l^2 ρ_l {\bf w}_l^2$. It is shown that the equations admit a Hamiltonian-Poisson bracket formulation. Furthermore, singularities in $\nabla \times {\bf B}$ are conservatively regularized by adding $(λ_B^2/2 μ_0) (\nabla \times {\bf B})^2$ to ${\cal E}^*$. Finally, it is proved that among regularizations that admit a Hamiltonian formulation and preserve the continuity equations along with the symmetries of the ideal model, the twirl term is unique and minimal in non-linearity and space derivatives of velocities.

physics.plasm-ph↗

arXiv2016

Conservative regularization of compressible flow and ideal magnetohydrodynamics

Ideal systems like MHD and Euler flow may develop singularities in vorticity (w = curl v). Viscosity and resistivity are dissipative regularizations. We propose a minimal, local, conservative, nonlinear, dispersive regularization of compressible flow and ideal MHD, in analogy with the KdV regularization of the 1D Hopf equation. This work significantly extends earlier work on incompressible Euler and ideal MHD. It involves a cut-off lambda inversely proportional to square-root of density rho, which is like a position-dependent mean free path. In MHD, lambda can be taken of order the electron collisionless skin depth. The regularizing `twirl' term is - lambda w x curl w. Such a term could be important in high speed flows with vorticity and arise in an expansion of kinetic equations in Knudsen number. A magnetic analogue of the twirl term - (B x curl B)/(rho mu_0), arises as the Lorentz force in ideal MHD. Our regularization preserves symmetries of the ideal systems, and with appropriate boundary conditions, implies associated conservation laws. Energy and enstrophy are subject to a priori bounds determined by initial data. A Hamiltonian and Poisson bracket formulation is developed and used to generalize the constitutive relation to bound higher moments of w and curl w. A `swirl' velocity field is shown to transport w/rho and B/rho, generalizing the Kelvin-Helmholtz and Alfvén theorems. The steady regularized equations are used to model a rotating vortex, MHD pinch, plane vortex sheet, channel flow, plane flow and propagating spherical and cylindrical vortices; solutions are more regular than corresponding Eulerian ones. The proposed regularization could facilitate simulations of fluid/MHD equations and provide a consistent statistical mechanics of vortices/current filaments in 3D, without blowup of enstrophy. Implications for detailed analyses of fluid and plasma systems are discussed.

physics.flu-dyn↗

arXiv2016

Algebra and geometry of Hamilton's quaternions

Inspired by the relation between the algebra of complex numbers and plane geometry, William Rowan Hamilton sought an algebra of triples for application to three dimensional geometry. Unable to multiply and divide triples, he invented a non-commutative division algebra of quadruples, in what he considered his most significant work, generalizing the real and complex number systems. We give a motivated introduction to quaternions and discuss how they are related to Pauli matrices, rotations in three dimensions, the three sphere, the group SU(2) and the celebrated Hopf fibrations.

math.HO↗

arXiv2016

Local conservative regularizations of compressible magnetohydrodynamic and neutral flows

Ideal systems like MHD and Euler flow may develop singularities in vorticity (w = curl v). Viscosity and resistivity provide dissipative regularizations of the singularities. In this paper we propose a minimal, local, conservative, nonlinear, dispersive regularization of compressible flow and ideal MHD, in analogy with the KdV regularization of the 1D kinematic wave equation. This work extends and significantly generalizes earlier work on incompressible Euler and ideal MHD. It involves a micro-scale cutoff length lambda which is a function of density, unlike in the incompressible case. In MHD, it can be taken to be of order the electron collisionless skin depth c/omega_pe. Our regularization preserves the symmetries of the original systems, and with appropriate boundary conditions, leads to associated conservation laws. Energy and enstrophy are subject to a priori bounds determined by initial data in contrast to the unregularized systems. A Hamiltonian and Poisson bracket formulation is developed and applied to generalize the constitutive relation to bound higher moments of vorticity. A `swirl' velocity field is identified, and shown to transport w/rho and B/rho, generalizing the Kelvin-Helmholtz and Alfven theorems. The steady regularized equations are used to model a rotating vortex, MHD pinch and a plane vortex sheet. The proposed regularization could facilitate numerical simulations of fluid/MHD equations and provide a consistent statistical mechanics of vortices/current filaments in 3D, without blowup of enstrophy. Implications for detailed analyses of fluid and plasma dynamic systems arising from our work are briefly discussed.

physics.plasm-ph↗