Two-fluid Hydrodynamic Equations Research Articles

Исследование влияния неоднородности плазмы с учётом тормозного и фоторекомбинационного излучений на поглощение альфвеновской волны

We numerically study the nonlinear absorption of a plane Alfven wave incident on a fixed boundary of a dissipative inhomogeneous plasma, which is caused by dissipative factors, including bremsstrahlung and photorecombination radiation. Density inhomogeneities specified using a Gaussian distribution are considered. The study is based on the equations of two-fluid electromagnetic hydrodynamics with full allowance for the inertia of electrons. The proposed implicit difference scheme for calculating plane-parallel flows of a two-fluid plasma made it possible to reveal a number of important absorption patterns. In particular, it is shown that absorption with allowance for bremsstrahlung and photorecombination radiation for an inhomogeneous plasma leads to blocking of the Alfven wave in a dissipative plasma. The dependences of the depth of penetration of the Alfven wave and the maximum temperatures of electrons and ions on the size of the density inhomogeneity are obtained.

Modelling multiphase gases in cosmological simulations using compressible multifluid hydrodynamics

ABSTRACT The diffuse medium in and around galaxies can exist in a multiphase state: small, cold gas clouds contributing significantly to the total mass embedded in pressure equilibrium with a hotter, more diffuse volume-filling component. Modelling this multiphase state in cosmological simulations poses a significant challenge due to the requirements to spatially resolve the clouds and consequently the interactions between the phases. In this paper, we present a novel method to model this gas state in cosmological hydrodynamical simulations. We solve the compressible two-fluid hydrodynamic equations using a moving-mesh finite-volume method and define mass, momentum, and energy exchange terms between the phases as operator-split source terms. Using a stratified flow model, our implementation is able to maintain volume fraction discontinuities in pressure equilibrium to machine precision, allowing for the treatment of both resolved and unresolved multiphase fluids. The solver remains second order accurate on smooth hydrodynamics problems. We use the source and sink terms of an existing two-phase model for the interstellar medium to demonstrate the value of this type of approach in simulations of galaxy formation, compare it to its effective equation of state implementation, and discuss its advantages in future large-scale simulations of galaxy formation.

Isaak Markovich Khalatnikov. 17 October 1919—9 January 2021

Isaak Khalatnikov made major contributions to two different branches of theoretical physics: quantum liquids, particularly liquid helium, and general relativity and cosmology. He made outstanding contributions to our theoretical understanding of superfluidity in liquid 4 He and in 3 He– 4 He mixtures. He developed the formulation of the two-fluid hydrodynamic equations, including the proper description of very complex dissipative and nonlinear effects. He also developed the idea, originally due to Landau, that the normal fluid can be described in terms of a gas of weakly-interacting excitations. His work on general relativity and cosmology began with the paper written with E. M. Lifshitz (ForMemRS 1982) in 1963. They extended Lifshitz's pioneering paper of 1946, showing how small departures from homogeneity and isotropy in the early Universe would have grown as the Universe expanded. They also tackled the problem of whether or not the equations of general relativity predict that space-time contains singularities, either in the past or in the future. They initially concluded that more general solutions would not contain singularities, but later work with Belinskiĭ not only showed that general solutions could contain singularities, but also indicated how the Universe would behave near a singularity. Later, Khalatnikov showed that nearly all cosmological solutions with a massive scalar field have a period of exponential or inflationary expansion. He became the founding director of the Landau Institute for Theoretical Physics of the USSR Academy of Sciences in 1964. He brought together a powerful group of theoreticians, creating a scientific centre that has played a major role in the development of theoretical physics.

Влияние неоднородности плазмы с учётом тормозного излучения на поглощение альфвеновской волны диссипативной плазмой

The paper investigates a mathematical model of the absorption of an Alfvén wave in an inhomogeneous incompressible dissipative plasma using the equations of two-fluid electromagnetic hydrodynamics. It is shown that a consequence of taking into account bremsstrahlung is the finiteness of the penetration depth of the Alfvén wave into an inhomogeneous plasma and a steady quasi-stationary regime of the Alfvén wave absorption. Density inhomogeneities of two types are considered – hump and hollows, which are distributed according to the Gaussian law. The dependences on the value of the hump of the penetration depth of the Alfvén wave into the inhomogeneous plasma and the maximum temperatures of electrons and ions are obtained. The study showed that an increase in the amplitude of the incident wave leads to an increase in the maximum values of the electron and ion temperatures, as well as the depth of penetration of the Alfvén wave into an inhomogeneous dissipative plasma.

Dynamic Kosterlitz-Thouless theory for two-dimensional ultracold atomic gases

In this letter we develop a theory for the first and second sound in a two-dimensional atomic superfluid across the superfluid transition based on the dynamic Koterlitz-Thouless theory. We employ a set of modified two-fluid hydrodynamic equations which incorporate the dynamics of the quantised vortices, rather than the conventional ones for a three-dimensional superfluid. As far as the sound dispersion equation is concerned, the modification is essentially equivalent to replacing the static superfluid density with a frequency dependent one, renormalised by the frequency dependent "dielectric constant" of the vortices. This theory has two direct consequences. First, because the renormalised superfluid density at finite frequencies does not display discontinuity across the superfluid transition, in contrast to the static superfluid density, the sound velocities vary smoothly across the transition. Second, the theory includes dissipation due to free vortices, and thus naturally describes the sound-to-diffusion crossover for the second sound in the normal phase. With only one fitting parameter, our theory gives a perfect agreement with the experimental measurements of sound velocities across the transition, as well as the quality factor in the vicinity of the transition. The predictions from this theory can be further verified by future experiments.

Influence of Synchrotron and Photorecombinational Radiation on Alfven Wave Absorption by Dissipative Plasma

A mathematical model of the Alfven wave absorption is proposed, the study of which is based on the equations of two-fluid electromagnetic hydrodynamics (EMHD) having taken the electron inertia and bremsstrahlung into account. Special attention is paid to the research of the additional effects of photorecombinational and synchrotron radiation. The investigation revealed, in particular, the finite depth of the Alfven wave penetration into the plasma. Another important effect is the establishment of the parameters of the absorbed Alfven wave and their convergence to the quasi-stationary regime.

Acoustic modes in He I and He II in the presence of an alternating electric field

The vibrational modes in isotropic nonpolar dielectrics He I and He II are studied in the presence of an alternating electric field E = E0izsin(k0z–ω0t), by solving the equations of ordinary and two-fluid hydrodynamics. There is a “coupling” between the electric field and the density fluctuations, since the density gradient leads to the spontaneous polarization Ps, and the electric force contains the term (Ps∇) E. Analysis shows that the wave velocities of the first- and second-sounds propagating along E change according to the formula uj ≈ cj + χjE02 (where j = 1, 2; cj is the speed of the jth sound at E0 = 0, and χj is a constant). It is found that the field E, together with the wave of the first- (second) sound (ω, k), should create in He II hybrid acoustoelectric (thermoelectric) density waves (ω + lω0, k + lk0), where l = ± 1, ± 2,… The amplitudes of the acoustoelectric waves and the quantity |u1−c1| are negligibly small, but at certain ω and ω0 they should increase resonantly. The first resonance seems to correspond to the decay of a photon into two photons with the recoil momentum being transferred to the liquid as a whole. Therefore, the electromagnetic signal spectrum should have a narrow absorption line, similar to the Mössbauer effect.

Influence of an External AC Electric Field on Plasma Turbulence in the Tokamak Near-Wall Layer

Braginskii reduced equations of two-fluid hydrodynamics are modified to take into account the presence of an external ac electric field localized in the tokamak near-wall layer. Numerical simulations show that, after reaching certain amplitude, such a field oscillating with the frequency ω ≈ ωGAM is capable of suppressing turbulent processes. The turbulence suppression mechanism consists in a sharp decrease in the growth rate of drift-resistive ballooning instability due to the appearance of additional nonlinear terms related to the external field in the equation for the vorticity.

Alfven wave absorption in dissipative plasma

We consider nonlinear absorption of Alfven waves due to dissipative effects in plasma and relaxation of temperatures of electrons and ions. This study is based on an exact solution of the equations of two-fluid electromagnetic hydrodynamics (EMHD) of plasma. It is shown that in order to study the decay of Alfven waves, it suffices to examine the behavior of their amplitudes whose evolution is described by a system of ordinary differential equations (ODEs) obtained in this paper. On finite time intervals, the system of equations on the amplitudes is studied numerically, while asymptotic integration (the Hartman-Grobman theorem) is used to examine its large-time behavior.

Solitary fast magnetosonic waves propagating obliquely to the magnetic field in cold collisionless plasma

Solutions describing solitary fast magnetosonic (FMS) waves (FMS solitons) in cold magnetized plasma are obtained by numerically solving two-fluid hydrodynamic equations. The parameter domain within which steady-state solitary waves can propagate is determined. It is established that the Mach number for rarefaction FMS solitons is always less than unity. The restriction on the propagation velocity leads to the limitation on the amplitudes of the magnetic field components of rarefaction solitons. It is shown that, as the soliton propagates in plasma, the transverse component of its magnetic field rotates and makes a complete turn around the axis along which the soliton propagates.

First and second sound in a two-dimensional harmonically trapped Bose gas across the Berezinskii–Kosterlitz–Thouless transition

We theoretically investigate first and second sound of a two-dimensional (2D) atomic Bose gas in harmonic traps by solving Landau’s two-fluid hydrodynamic equations. For an isotropic trap, we find that first and second sound modes become degenerate at certain temperatures and exhibit typical avoided crossings in mode frequencies. At these temperatures, second sound has significant density fluctuation due to its hybridization with first sound and has a divergent mode frequency towards the Berezinskii–Kosterlitz–Thouless (BKT) transition. For a highly anisotropic trap, we derive the simplified one-dimensional hydrodynamic equations and discuss the sound-wave propagation along the weakly confined direction. Due to the universal jump of the superfluid density inherent to the BKT transition, we show that the first sound velocity exhibits a kink across the transition. These predictions might be readily examined in current experimental setups for 2D dilute Bose gases with a sufficiently large number of atoms, where the finite-size effect due to harmonic traps is relatively weak.

First and second sound of a unitary Fermi gas in highly elongated harmonic traps

Using a variational approach, we present the full solutions of the simplified one-dimensional two-fluid hydrodynamic equations for a unitary Fermi gas trapped in a highly elongated harmonic potential, which was recently derived by G. Bertaina and coworkers [Phys. Rev. Lett. 105, 150402 (2010)]. We calculate the discretized mode frequencies of first and second sound along the weak axial trapping potential, as a function of the temperature and the form of superfluid density. We show that the density fluctuations in second-sound modes, due to their coupling to first-sound modes, are large enough to be measured in current experimental setups such as that exploited by M. K. Tey et al. at the University of Innsbruck [Phys. Rev. Lett. 110, 055303 (2013)]. Owing to the sensitivity of second sounds on the form of superfluid density, the high precision of the measured second-sound frequencies may provide us a promising way to accurately determine the superfluid density of a unitary Fermi gas, which so far has remained elusive.

First and second sound of a unitary Fermi gas in highly oblate harmonic traps

We theoretically investigate first and second sound modes of a unitary Fermi gas trapped in a highly oblate harmonic trap at finite temperatures. Following the idea by Stringari and co-workers (2010 Phys. Rev. Lett. 105 150402), we argue that these modes can be described by the simplified two-dimensional two-fluid hydrodynamic equations. Two possible schemes—sound wave propagation and breathing mode excitation—are considered. We calculate the sound wave velocities and discretized sound mode frequencies, as a function of temperature. We find that in both schemes, the coupling between first and second sound modes is large enough to induce significant density fluctuations, suggesting that second sound can be directly observed by measuring in situ density profiles. The frequency of the second sound breathing mode is found to be highly sensitive to the superfluid density.

Scaling solutions of the two-fluid hydrodynamic equations in a harmonically trapped gas at unitarity

We prove that the two fluid Landau hydrodynamic equations, when applied to a gas interacting with infinite scattering length (unitary gas) in the presence of harmonic trapping, admit exact scaling solutions of mixed compressional and surface nature. These solutions are characterized by a linear dependence of the velocity field on the spatial coordinates and a temperature independent frequency which is calculated in terms of the parameters of the trap. Our results are derived in the regime of small amplitude oscillations and hold both below and above the superfluid phase transition. They apply to isotropic as well as to deformed configurations, thereby providing a generalization of Castin's theorem (Y. Castin, C. R. Phys. \textbf{5}, 407 (2004)) holding for isotropic trapping. Our predictions agree with the experimental findings in resonantly interacting atomic Fermi gases. The breathing scaling solution, in the presence of isotropic trapping, is also used to prove the vanishing of two bulk viscosity coefficients in the superfluid phase.

Equations of two-fluid hydrodynamics in the Hubbard model

A set of equations is obtained using stationary and nonstationary superconductivity theories in terms of the Hubbard model. The equations near Tc and for a weak magnetic field have the general form of superconductivity equations with a strongly anisotropic Fermi surface. Calculations are performed using a kinetic equation for quasiparticles. The low-temperature collective excitations in a superconductor are studied. An explicit relationship for the temperature dependence of second sound is obtained.

First and second sound in a strongly interacting Fermi gas

Using a variational approach, we solve the equations of two-fluid hydrodynamics for a uniform and trapped Fermi gas at unitarity. In the uniform case, we find that the first and second sound modes are remarkably similar to those in superfluid helium, a consequence of strong interactions. In the presence of harmonic trapping, first and second sound become degenerate at certain temperatures. At these points, second sound hybridizes with first sound and is strongly coupled with density fluctuations, giving a promising way of observing second sound. We also discuss the possibility of exciting second sound by generating local heat perturbations.

Two-fluid hydrodynamic modes in a trapped superfluid gas

In the collisional region at finite temperatures, the collective modes of superfluids are described by the Landau two-fluid hydrodynamic equations. This region can now be probed over the entire BCS-Bose-Einstein-condensate crossover in trapped Fermi superfluids with a Feshbach resonance, including the unitarity region. Building on the approach initiated by Zaremba, Nikuni, and Griffin in 1999 for trapped atomic Bose gases, we present a variational formulation of two-fluid hydrodynamic collective modes based on the work of Zilsel in 1950 developed for superfluid helium. Assuming a simple variational Ansatz for the superfluid and normal fluid velocities, the frequencies of the hydrodynamic modes are given by solutions of coupled algebraic equations, with constants only involving spatial integrals over various equilibrium thermodynamic derivatives. This variational approach is both simpler and more physical than a direct attempt to solve the Landau two-fluid differential equations. Our two-fluid results are shown to reduce to those of Pitaevskii and Stringari for a pure superfluid at T=0.

Mechanism of “rigid-body” rotation of the superfluid and normal components during phase separation of a supersaturated He3–4He solution

It is shown that unstable hydrodynamic vortices can form inside subcritical domains in the normal component of a supersaturated decomposing He3–4He solution. A mechanism for the entrainment of the superfluid component by the normal component of the He3–4He solution into “rigid-body” rotation due to Hall–Vinen–Bekarevich–Khalatnikov forces in the equations of two-fluid hydrodynamics, which leads to the creation of quantized vortices, is considered. An increase of the average density of quantized vortices can increase the rate of heterogeneous decomposition of the He3–4He solution.

Rayleigh instability in Hall thrusters

Gradient-driven Rayleigh-type instabilities in Hall plasma thrusters are analyzed using linearized two-fluid hydrodynamic equations. Necessary instability conditions and a general criterion for stability of azimuthally propagating perturbations are derived. For a simplified model of the axial distribution of parameters inside the thruster channel, the growth rate of an unstable wave, resonant with the azimuthal electron flow, is obtained. The frequency and phase relations are related to the results of experimental investigations of high-frequency oscillations in Hall thrusters.

Transverse sound in aerogel with liquid [formula omitted

An experiment was performed to measure transverse sound resonances in a square slab of aerogel filled with liquid 4 He . Resonances have been observed both in the superfluid and normal phase. The dynamics of the system was modelled by combining the equations of two-fluid hydrodynamics of helium with those of elasticity of aerogel.