George Miloshevich

Concomitant Hamiltonian and topological structures of extended magnetohydrodynamics

Abstract The paper describes the unique geometric properties of ideal magnetohydrodynamics (MHD), and demonstrates how such features are inherited by extended MHD, viz. models that incorporate two-fluid effects (the Hall term and electron inertia). The generalized helicities, and other geometric expressions for these models are presented in a topological context, emphasizing their universal facets. Some of the results presented include: the generalized Kelvin circulation theorems; the existence of two Lie-dragged 2-forms; and two concomitant helicities that can be studied via the Jones polynomial, which is widely utilized in Chern–Simons theory. The ensuing commonality is traced to the existence of an underlying Hamiltonian structure for all the extended MHD models, exemplified by the presence of a unique noncanonical Poisson bracket, and its associated energy.

Remarkable connections between extended magnetohydrodynamics models

Through the use of suitable variable transformations, the commonality of all extended magnetohydrodynamics (MHD) models is established. Remarkable correspondences between the Poisson brackets of inertialess Hall MHD and inertial MHD (which has electron inertia, but not the Hall drift) and extended MHD (which has both effects) are established. The helicities (two in all) for each of these models are obtained through these correspondences. The commonality of all the extended MHD models is traced to the existence of two Lie-dragged 2-forms, which are closely associated with the canonical momenta of the two underlying species. The Lie-dragging of these 2-forms by suitable velocities also leads to the correct equations of motion. The Hall MHD Poisson bracket is analyzed in detail, the Jacobi identity is verified through a detailed proof, and this proof ensures the Jacobi identity for the Poisson brackets of all the models.

Traveling solitons in long-range oscillator chains

We investigate the existence and propagation of solitons in a long-range extension of the quartic Fermi-Pasta-Ulam (FPU) chain of anharmonic oscillators. The coupling in the linear term decays as a power-law with an exponent greater than 1 and less than 3. We obtain an analytic perturbative expression of traveling envelope solitons by introducing a Non Linear Schrodinger (NLS) equation for the slowly varying amplitude of short wavelength modes. Due to the non analytic properties of the dispersion relation, it is crucial to develop the theory using discrete difference operators. Those properties are also the ultimate reason why kink-solitons may exist but are unstable, at variance with the short-range FPU model. We successfully compare these approximate analytic results with numerical simulations.

Action principles for relativistic extended magnetohydrodynamics: A unified theory of magnetofluid models

Two types of Eulerian action principles for relativistic extended magnetohydrodynamics (MHD) are formulated. With the first, the action is extremized under the constraints of density, entropy, and Lagrangian label conservation, which leads to a Clebsch representation for a generalized momentum and a generalized vector potential. The second action arises upon transformation to physical field variables, giving rise to a covariant bracket action principle, i.e., a variational principle in which constrained variations are generated by a degenerate Poisson bracket. Upon taking appropriate limits, the action principles lead to relativistic Hall MHD and well-known relativistic ideal MHD. For the first time, the Hamiltonian formulation of relativistic Hall MHD with electron thermal inertia (akin to [Comisso \textit{et al.}, Phys. Rev. Lett. {\bf 113}, 045001 (2014)] for the electron--positron plasma) is introduced. This thermal inertia effect allows for violation of the frozen-in magnetic flux condition in marked contrast to nonrelativistic Hall MHD that does satisfy the frozen-in condition. We also find violation of the frozen-in condition is accompanied by freezing-in of an alternative flux determined by a generalized vector potential. Finally, we derive a more general 3+1 Poisson bracket for nonrelativistic extended MHD, one that does not assume smallness of the electron ion mass ratio.

On the structure and statistical theory of turbulence of extended magnetohydrodynamics

Recent progress regarding the noncanonical Hamiltonian formulation of extended magnetohydrodynamics (XMHD), a model with Hall drift and electron inertia, is summarized. The advantages of the Hamiltonian approach are invoked to study some general properties of XMHD turbulence, and to compare them against their ideal MHD counterparts. For instance, the helicity flux transfer rates for XMHD are computed, and Liouvilles theorem for this model is also verified. The latter is used, in conjunction with the absolute equilibrium states, to arrive at the spectra for the invariants, and to determine the direction of the cascades, e.g., generalizations of the well-known ideal MHD inverse cascade of magnetic helicity. After a similar analysis is conducted for XMHD by inspecting second order structure functions and absolute equilibrium states, a couple of interesting results emerge. When cross helicity is taken to be ignorable, the inverse cascade of injected magnetic helicity also occurs in the Hall MHD range - this is shown to be consistent with previous results in the literature. In contrast, in the inertial MHD range, viz. at scales smaller than the electron skin depth, all spectral quantities are expected to undergo direct cascading. The consequences and relevance of our results in space and astrophysical plasmas are also briefly discussed.

Stochastic resonance in the Fermi-Pasta-Ulam chain.

We consider a damped beta-Fermi-Pasta-Ulam chain, driven at one boundary subjected to stochastic noise. It is shown that, for a fixed driving amplitude and frequency, increasing the noise intensity, the systems energy resonantly responds to the modulating frequency of the forcing signal. Multiple peaks appear in the signal-to-noise ratio, signaling the phenomenon of stochastic resonance. The presence of multiple peaks is explained by the existence of many stable and metastable states that are found when solving this boundary value problem for a semicontinuum approximation of the model. Stochastic resonance is shown to be generated by transitions between these states.

Instabilities and relaxation to equilibrium in long-range oscillator chains.

We study instabilities and relaxation to equilibrium in a long-range extension of the Fermi-Pasta-Ulam-Tsingou (FPU) oscillator chain by exciting initially the lowest Fourier mode. Localization in mode space is stronger for the long-range FPU model. This allows us to uncover the sporadic nature of instabilities, i.e., by varying initially the excitation amplitude of the lowest mode, which is the control parameter, instabilities occur in narrow amplitude intervals. Only for sufficiently large values of the amplitude, the system enters a permanently unstable regime. These findings also clarify the long-standing problem of the relaxation to equilibrium in the short-range FPU model. Because of the weaker localization in mode space of this latter model, the transfer of energy is retarded and relaxation occurs on a much longer timescale.

Direction of cascades in a magnetofluid model with electron skin depth and ion sound Larmor radius scales

The direction of cascades in a two-dimensional model that takes electron inertia and ion sound Larmor radius into account is studied, resulting in analytical expressions for the absolute equilibrium states of the energy and helicities. It is found that typically both the energy and magnetic helicity at scales shorter than electron skin depth have direct cascade, while at large scales the helicity has an inverse cascade as established earlier for reduced magnetohydrodynamics (MHD). It is also found that the introduction of gyro-effects allows for the existence of negative temperature (conjugate to energy) states and the condensation of energy to the large scales. Comparisons between two- and three-dimensional extended MHD models (MHD with two-fluid effects) show qualitative agreement between the two.

Dipolar needles in the microcanonical ensemble: Evidence of spontaneous magnetization and ergodicity breaking

We have studied needle-shaped three-dimensional classical spin systems with purely dipolar interactions in the microcanonical ensemble, using both numerical simulations and analytical approximations. We have observed spontaneous magnetization for different finite cubic lattices. The transition from the paramagnetic to the ferromagnetic phase is shown to be of first order. For two lattice types we have observed magnetization flips in the phase transition region. In some cases, gaps in the accessible values of magnetization appear, a signature of the ergodicity breaking found for systems with long-range interactions. We analytically explain these effects by performing a nontrivial mapping of the model Hamiltonian onto a one-dimensional Ising model with competing antiferromagnetic nearest-neighbor and ferromagnetic mean-field interactions. These results hint at performing experiments on isolated dipolar needles in order to verify some of the exotic properties of systems with long-range interactions in the microcanonical ensemble.