Search

A phenomenological model to describe the Corona Virus(covid-19) Pandemic spread in a given population is developed. It enables the identification of the key quantities required to form adequate policies for control and mitigation in terms of observable parameters using the Landau-Stuart equation. It is intended to be complementary to detailed simulations and methods published recently by Ferguson and collaborators, March 16, (2020). The results suggest that the initial growth/spreading rate gamma-c of the disease, and the fraction of infected persons in the population p-i can be used to define a `retardation/inhibition coefficient' k-star , which is a measure of the effectiveness of the control policies adopted. The results are obtained analytically and numerically using a simple Python code. The solutions provide both qualitative and quantitative information. They substantiate and justify two basic control policies enunciated by WHO and adopted in many countries: a) Systematic and early intensive testing individuals for covid-19 and b) Sequestration policies such as `social/physical distancing' and population density reduction by strict quarantining are essential for making k-star greater than 1, necessary for suppressing the pandemic. The model indicates that relaxing such measures when the infection rate starts to decrease as a result of earlier policies could simply restart the infection rate in the non-infected population. Presently available available statistical data in WHO and other reports can be readily used to determine the the key parameters of the model. Possible extensions to the basic model to make it more realistic are indicated., Comment: 12 pages, 5 figures

We present numerical simulation results of the nonlinear evolution of the (1,1) internal kink mode in the presence of various kinds of equilibrium plasma flows. The present studies are carried out in the framework of a two fluid model to extend our past investigations done with a reduced magneto-hydrodynamic (RMHD) model. Two-fluid effects are found to significantly influence the mode dynamics in a number of different ways. In the linear regime diamagnetic effects in combination with flows provide a synergistic stabilizing influence that also carries over to the nonlinear regime. In addition one observes novel symmetry breaking phenomena in the linear growth rates as well as in the nonlinear saturated states of the mode. Our study also explores the influence of strong viscosity on the mode evolution and finds interesting modifications in the real frequency of the mode in the linear regime., Comment: submitted to Nuclear Fusion

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 $\gamma$. It is achieved by augmenting the Hamiltonian by a capillarity energy $\beta(\rho) (\nabla \rho)^2$. The simplest capillarity coefficient leading to local conservation laws for mass, momentum, energy and entropy using the standard Poisson brackets is $\beta(\rho) = \beta_*/\rho$ for constant $\beta_*$. This leads to a Korteweg-like stress and nonlinear terms in the momentum equation with third derivatives of $\rho$, 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 $\gamma = 2$, 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 $\gamma = 2$), with $\beta_*$ 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., Comment: 19 pages, 20 figure files

We report on numerical simulation studies of the dynamical behavior of edge localized modes (ELMs) under the influence of repetitive injection of pellets. In our nonlinear 2-fluid model the ELMs are excited by introducing a particle source in the confinement region and a particle sink in the edge region. The injection of pellets is simulated by periodically raising the edge density in a pulsed manner. We find that when the edge density is raised to twice the normal edge density with a duty cycle (on time:off time) of 1:2, the ELMs are generated on an average at a faster rate and with reduced amplitudes. These changes lead to significant improvements in the plasma beta indicative of an improvement in the energy confinement due to pellet injection. Concurrently, the plasma density and temperature profiles also get significantly modified. A comparative study is made of the nature of ELM dynamics for different magnitudes of edge density enhancements. We also discuss the relative impact on ELMs from resonant magnetic perturbations (RMPs) compared to pellet injection in terms of changes in the plasma temperature, density, location of the ELMs and the nonlinear spectral transfer of energies.

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 $\lambda_l^2 ({\bf w}_l + \frac{q_l}{m_l} {\bf B}) \times (\nabla \times {\bf w}_l)$ in the ion/electron velocity equations ($l = i,e$) where ${\bf w}_l = \nabla \times {\bf v}_l$ are vorticities. The cut-off lengths $\lambda_l$ must be inversely proportional to the square-roots of the number densities $(\lambda_l^2 n_l = C_l)$ 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 $\sum_l \nabla \times \nabla \times \lambda_l^2 {\bf j}_{{\rm flow},l}$. The resulting equations imply conserved linear and angular momenta and a positive definite swirl energy density ${\cal E}^*$ which includes an enstrophic contribution $\sum_l (1/2) \lambda_l^2 \rho_l {\bf w}_l^2$. It is shown that the equations admit a Hamiltonian-Poisson bracket formulation. Furthermore, singularities in $\nabla \times {\bf B}$ are conservatively regularized by adding $(\lambda_B^2/2 \mu_0) (\nabla \times {\bf B})^2$ 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., Comment: 13 pages, extended Discussion section, similar to published version

We have investigated the effect of sheared equilibrium flows on the $m = 1, n = 1$ resistive internal kink mode in the framework of a reduced magnetohydrodynamic model in a periodic cylindrical geometry. Our numerical studies show that there is a significant change of the scaling dependence of the mode growth rate on the Lundquist number in the presence of axial flows compared to the no flow case. Poloidal flows do not influence the scaling. We have further found that viscosity strongly modifies the effect of flows on the (1,1) mode both in the linear and nonlinear regime. Axial flows increase the linear growth rate for low viscosity values, but they decrease the linear growth rate for higher viscosity values. In the case of poloidal flows the linear growth rate decreases in all cases. Additionally at higher viscosity, we have found strong symmetry breaking in the behaviour of linear growth rates and in the nonlinear saturation levels of the modes as a function of the helicities of the flows. For axial, poloidal and most helical flow cases, there is flow induced stabilisation of the nonlinear saturation level in the high viscosity regime and destabilisation in the low viscosity regime., Comment: submitted to Physics of Plasmas

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., Comment: 12 pages, 2 figures. Minor changes, published version. arXiv admin note: substantial text overlap with arXiv:1510.01606

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 Alfv\'en 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., Comment: 45 pages, 8 figures, significantly updated with additional material on regularization of MHD, examples, integral invariants of regularized flow, regularizations bounding higher moments of vorticity and its curl etc. This is a more detailed version of 1602.04323 published in Physics of Plasmas

Certain recent semi-classical theories of spin-half quantum plasmas are examined with regard to their internal consistency, physical applicability and relevance to fusion, astrophysical and condensed matter plasmas. It is shown that the derivations and some of the results obtained in these theories are internally inconsistent and contradict well-established principles of quantum and statistical mechanics, especially in their treatment of fermions and spin. Claims of large semi-classical effects of spin magnetic moments that could dominate the plasma dynamics are found to be invalid both for single-particles and collectively. Larmor moments dominate at high temperature while spin moments cancel due to Pauli blocking at low temperatures. Explicit numerical estimates from a variety of plasmas are provided to demonstrate that spin effects are indeed much smaller than many neglected classical effects. The analysis presented suggests that the aforementioned `Spin Quantum Hydrodynamic' theories are not relevant to conventional laboratory or astrophysical plasmas., Comment: 11 pages, To appear in Contributions to Plasma Physics. Minor correction on page 3 to electron spin magnetic moment

A comment on the Letter by S. Braun, F. A. Asenjo and S. M. Mahajan, Phys. Rev. Lett., 109, 175003 (2012). We show that recent arguments for light amplification driven by inhomogeneous quantum spin fields in low temperature electron plasmas in metals are invalid. In essence, a neglect of Pauli `blocking' led the authors to over-estimate the effects of intrinsic spin., Comment: 2 pages. Published version of Comment along with an addendum, updated references

In this Letter certain fundamental physics issues relating to recent theories of so-called `spin quantum plasmas' are examined. It is shown that the derivations and some of the results obtained in these theories contradict well-established principles of quantum mechanics, especially in their treatment of fermions and spin. The analysis presented suggests that the aforementioned theories do not apply for any range of temperatures at the stated densities and furthermore fail to make any experimentally accessible and testable predictions., Comment: 5 pages

We discuss a new non-linear PDE, u_t + (2 u_xx/u) u_x = epsilon u_xxx, invariant under scaling of dependent variable and referred to here as SIdV. It is one of the simplest such translation and space-time reflection-symmetric first order advection-dispersion equations. This PDE (with dispersion coefficient unity) was discovered in a genetic programming search for equations sharing the KdV solitary wave solution. It provides a bridge between non-linear advection, diffusion and dispersion. Special cases include the mKdV and linear dispersive equations. We identify two conservation laws, though initial investigations indicate that SIdV does not follow from a polynomial Lagrangian of the KdV sort. Nevertheless, it possesses solitary and periodic travelling waves. Moreover, numerical simulations reveal recurrence properties usually associated with integrable systems. KdV and SIdV are the simplest in an infinite dimensional family of equations sharing the KdV solitary wave. SIdV and its generalizations may serve as a testing ground for numerical and analytical techniques and be a rich source for further explorations., Comment: 15 pages, 4 figures, corrected sign typo in KdV Lagrangian above equation 31

A new global two-fluid electromagnetic turbulence code, CENTORI, has been developed for the purpose of studying magnetically-confined fusion plasmas on energy confinement timescales. This code is used to evolve the combined system of electron and ion fluid equations and Maxwell equations in toroidal configurations with axisymmetric equilibria. Uniquely, the equilibrium is co-evolved with the turbulence, and is thus modified by it. CENTORI is applicable to tokamaks of arbitrary aspect ratio and high plasma beta. A predictor-corrector, semi-implicit finite difference scheme is used to compute the time evolution of fluid quantities and fields. Vector operations and the evaluation of flux surface averages are speeded up by choosing the Jacobian of the transformation from laboratory to plasma coordinates to be a function of the equilibrium poloidal magnetic flux. A subroutine, GRASS, is used to co-evolve the plasma equilibrium by computing the steady-state solutions of a diffusion equation with a pseudo-time derivative. The code is written in Fortran 95 and is efficiently parallelized using Message Passing Interface (MPI). Illustrative examples of output from simulations of a tearing mode in a large aspect ratio tokamak plasma and of turbulence in an elongated conventional aspect ratio tokamak plasma are provided., Comment: 9 figures

The evolution of non-potential perturbations to a current-free magnetic X-point configuration is studied, taking into account electron inertial effects as well as resistivity. Electron inertia is shown to have a negligible effect on the evolution of the system whenever the collisionless skin depth is less than the resistive scale length. Non-potential magnetic field energy in this resistive MHD limit initially reaches equipartition with flow energy, in accordance with ideal MHD, and is then dissipated extremely rapidly, on an Alfvenic timescale that is essentially independent of Lundquist number. In agreement with resistive MHD results obtained by previous authors, the magnetic field energy and kinetic energy are then observed to decay on a longer timescale and exhibit oscillatory behavior, reflecting the existence of discrete normal modes with finite real frequency. When the collisionless skin depth exceeds the resistive scale length, the system again evolves initially according to ideal MHD. At the end of this ideal phase, the field energy decays typically on an Alfvenic timescale, while the kinetic energy (which is equally partitioned between ions and electrons in this case) is dissipated on the electron collision timescale. The oscillatory decay in the energy observed in the resistive case is absent, but short wavelength structures appear in the field and velocity profiles, suggesting the possibility of particle acceleration in oppositely-directed current channels. The model provides a possible framework for interpreting observations of energy release and particle acceleration on timescales down to less than a second in the impulsive phase of solar flares., Comment: 30 pages, 8 figures

An explicitly energy-conserving full orbit code CUEBIT, developed originally to describe energetic particle effects in laboratory fusion experiments, has been applied to the problem of proton acceleration in solar flares. The model fields are obtained from solutions of the linearised MHD equations for reconnecting modes at an X-type neutral point, with the additional ingredient of a longitudinal magnetic field component. To accelerate protons to the highest observed energies on flare timescales, it is necessary to invoke anomalous resistivity in the MHD solution. It is shown that the addition of a longitudinal field component greatly increases the efficiency of ion acceleration, essentially because it greatly reduces the magnitude of drift motions away from the vicinity of the X-point, where the accelerating component of the electric field is largest. Using plasma parameters consistent with flare observations, we obtain proton distributions extending up to gamma-ray-emitting energies (>1MeV). In some cases the energy distributions exhibit a bump-on-tail in the MeV range. In general, the shape of the distribution is sensitive to the model parameters., Comment: 14 pages, 4 figures, accepted for publication in Solar Physics

Magnetohydrodynamic (MHD) and two-fluid quasi-neutral equilibria with azimuthal symmetry, gravity and arbitrary ratios of (nonrelativistic) flow speed to acoustic and Alfven speeds are investigated. In the two-fluid case, the mass ratio of the two species is arbitrary, and the analysis is therefore applicable to electron-positron plasmas. The methods of derivation can be extended in an obvious manner to several charged species. Generalized Grad-Shafranov equations, describing the equilibrium magnetic field, are derived. Flux function equations and Bernoulli relations for each species, together with Poisson's equation for the gravitational potential, complete the set of equations required to determine the equilibrium. These are straightforward to solve numerically. The two-fluid system, unlike the MHD system, is shown to be free of singularities. It is demonstrated analytically that there exists a class of incompressible MHD equilibria with magnetic field-aligned flow. A special sub--class first identified by S. Chandrasekhar, in which the flow speed is everywhere equal to the local Alfven speed, is compatible with virtually any azimuthally symmetric magnetic configuration. Potential applications of this analysis include extragalactic and stellar jets, and accretion disks., Comment: 18 pages, 0 figures