11  EMHD

Electron Magnetohydrodynamics (EMHD) is a specialized branch of plasma physics that focuses on the behavior of plasmas at very small spatial scales and very short timescales, where the motion of electrons becomes dominant over the heavier ions. Since electrons are much lighter and more agile than ions, they can exhibit unique behaviors in the presence of magnetic fields, leading to phenomena not captured by standard magnetohydrodynamics (MHD).

11.1 Key Concepts in EMHD

  • Fast Electron Motion: Electrons move much faster than ions due to their lower mass. This translates to a separation of timescales in EMHD; you study rapid electron dynamics while assuming the ions form a relatively stationary background.

  • Magnetic Field “Freezing”: In standard MHD, the magnetic field lines are considered to be “frozen” into the plasma. In EMHD, this concept breaks down at small scales as electrons can move independently of the magnetic field lines.

  • Whistler Waves: EMHD supports a unique type of plasma wave called a whistler wave. These waves are electromagnetic oscillations that propagate along magnetic field lines, driven by the motion of electrons.

  • Hall Effect: Electrons, due to their small mass, are readily deflected by magnetic fields. This deflection, the basis of the Hall effect, leads to the generation of electric currents perpendicular to both the magnetic field and the direction of electron flow. These currents play a role in phenomena like magnetic reconnection.

11.2 Equations of EMHD

The core equations of EMHD differ from standard MHD.

  • Generalized Ohm’s Law: Describes electron flow, including the Hall effect and pressure gradient terms absent in ideal MHD Ohm’s Law.

  • Momentum Equation for Electrons: Governs electron motion under the influence of electric and magnetic fields, as well as pressure gradients.

  • Maxwell’s Equations: Describe the evolution of the electric and magnetic fields, just as in standard MHD.