The global solvability of the Hall-magnetohydrodynamics system in critical Sobolev spaces

We are concerned with the 3D incompressible Hall-magnetohydrodynamic system (Hall-MHD). Our first aim is to provide the reader with an elementary proof of a global well-posedness result for small data with critical Sobolev regularity, in the spirit of Fujita–Kato’s theorem [On the Navier–Stokes initial value problem I, Arch. Ration. Mech. Anal. 16 (1964) 269–315] for the Navier–Stokes equations. Next, we investigate the long-time asymptotics of global solutions of the Hall-MHD system that are in the Fujita–Kato regularity class. A weak-strong uniqueness statement is also proven. Finally, we consider the so-called 2[Formula: see text]D flows for the Hall-MHD system (that is, 3D flows independent of the vertical variable), and establish the global existence of strong solutions, assuming only that the initial magnetic field is small. Our proofs strongly rely on the use of an extended formulation involving the so-called velocity of electron [Formula: see text] and as regards [Formula: see text]D flows, of the auxiliary vector-field [Formula: see text] that comes into play in the total magneto-helicity balance for the Hall-MHD system.

Properties of Hall-MHD Turbulence at Sub-Ion Scales: Spectral Transfer Analysis

We present results of a multiscale study of Hall-magnetohydrodynamic (MHD) turbulence, carried out on a dataset of compressible nonlinear 2D Hall-MHD numerical simulations of decaying Alfvénic turbulence. For the first time, we identify two distinct regimes of fully developed turbulence. In the first one, the power spectrum of the turbulent magnetic fluctuations at sub-ion scales exhibits a power law with a slope of ∼−2.9, typically observed both in solar wind and in magnetosheath turbulence. The second regime, instead, shows a slope of −7/3, in agreement with classical theoretical models of Hall-MHD turbulence. A spectral-transfer analysis reveals that the latter regime occurs when the energy transfer rate at sub-ion scales is dominated by the Hall term, whereas in the former regime, the governing process is the dissipation (and the system exhibits large intermittency). Results of this work are relevant to the space plasma community, as they may potentially reconcile predictions from theoretical models with results from numerical simulations and spacecraft observations.

Magnetotail reconnection asymmetries in an ion-scale, Earth-like magnetosphere

We use a newly developed global Hall magnetohydrodynamic (MHD) code to investigate how reconnection drives magnetotail asymmetries in small, ion-scale magnetospheres. Here, we consider a magnetosphere with a similar aspect ratio to Earth but with the ion inertial length (δi) artificially inflated by a factor of 70: δi is set to the length of the planetary radius. This results in a magnetotail width on the order of 30 δi, slightly smaller than Mercury's tail and much smaller than Earth's with respect to δi. At this small size, we find that the Hall effect has significant impact on the global flow pattern, changing from a symmetric, Dungey-like convection under resistive MHD to an asymmetric pattern similar to that found in previous Hall MHD simulations of Ganymede's subsonic magnetosphere as well as other simulations of Mercury's using multi-fluid or embedded kinetic physics. We demonstrate that the Hall effect is sufficient to induce a dawnward asymmetry in observed dipolarization front locations and find quasi-periodic global-scale dipolarizations under steady, southward solar wind conditions. On average, we find a thinner current sheet dawnward; however, the measured thickness oscillates with the dipolarization cycle. During the flux-pileup stage, the dawnward current sheet can be thicker than the duskward sheet. This could be an explanation for recent observations that suggest Mercury's current sheet is actually thicker on the duskside: a sampling bias due to a longer lasting “thick” state in the sheet.

Hamiltonian kinetic-Hall magnetohydrodynamics with fluid and kinetic ions in the current and pressure coupling schemes

We present two generalized hybrid kinetic-Hall magnetohydrodynamics (MHD) models describing the interaction of a two-fluid bulk plasma, which consists of thermal ions and electrons, with energetic, suprathermal ion populations described by Vlasov dynamics. The dynamics of the thermal components are governed by standard fluid equations in the Hall MHD limit with the electron momentum equation providing an Ohm's law with Hall and electron pressure terms involving a gyrotropic electron pressure tensor. The coupling of the bulk, low-energy plasma with the energetic particle dynamics is accomplished through the current density (current coupling scheme; CCS) and the ion pressure tensor appearing in the momentum equation (pressure coupling scheme; PCS) in the first and the second model, respectively. The CCS is a generalization of two well-known models, because in the limit of vanishing energetic and thermal ion densities, we recover the standard Hall MHD and the hybrid kinetic-ions/fluid-electron model, respectively. This provides us with the capability to study in a continuous manner, the global impact of the energetic particles in a regime extending from vanishing to dominant energetic particle densities. The noncanonical Hamiltonian structures of the CCS and PCS, which can be exploited to study equilibrium and stability properties through the energy-Casimir variational principle, are identified. As a first application here, we derive a generalized Hall MHD Grad–Shafranov–Bernoulli system for translationally symmetric equilibria with anisotropic electron pressure and kinetic effects owing to the presence of energetic particles using the PCS.

On the blow-up criterion for the Hall-MHD problem with partial dissipation in R^3

We deal with the incompressible 3D Hall-magnetohydrodynamics with partial dissipation, a new blow-up criterion is obtained, based on which we also prove a new global solutions with small data.