Notes on the basic Two-fluid collisionless magnetoplasma model

                Equipo de Propulsión Espacial y Plasmas, Mario Merino, 2021

Assumptions

These assumptions define the regime of validity of the model:

  • Fully-ionized ion-electron plasma (we ignore neutrals).
  • ∂/∂t=0\partial/\partial t = 0: Steady-state plasma.
  • λD2=ε0Te/(ne2)≪L2\lambda_D^2 = \varepsilon_0 T_e/(ne^2)\ll L^2: quasineutral plasma n=ni=nen=n_i=n_e.
  • χH=ωce/νe≫1\chi_H = \omega_{ce}/\nu_e \gg 1, ui/L≫νiu_i/L\gg\nu_i: Collisionless plasma.
  • ℓe≪L\ell_e \ll L: electrons are fully magnetized. We only keep zeroth-order terms in ℓe/L\ell_e/L.
  • meue2≪Tem_e u_e^2 \ll T_e: negligible electron inertia and bulk velocity.
  • miui2∼Tem_i u_i^2 \sim T_e: moderate bulk ion velocity.
  • Ti≪TeT_i \ll T_e: cold ions.
  • β=2nTe/(μ0B2)≪1\beta = 2nT_e/(\mu_0 B^2)\ll 1: negligible induced magnetic field.

Ion equations

Continuity:

∇⋅(nui)=0\nabla \cdot (n \bm u_i) = 0

Momentum:

miui⋅∇ui=−e∇ϕ+eui×Bm_i \bm u_i \cdot \nabla \bm u_i = -e \nabla \phi + e \bm u_i \times \bm B

  • Note this is the "non-conservative form."
  • Ion magnetic force is often neglected (unmagnetized ions). Note that mi≫mem_i\gg m_e.
  • Left-hand side can be expressed as (1ti\bm 1_{t i}, 1ni\bm 1_{n i} are the tangent and normal unit vectors to ion trajectories; κi\kappa_i the ion trajectory curvature):

miui⋅∇ui=12mi∂ui2∂1ti1ti+miui2κi1nim_i \bm u_i \cdot \nabla \bm u_i = \frac{1}{2}m_i \frac{\partial u_i^2}{\partial \bm 1_{t i}} \bm 1_{t i} + m_i u_i^2 \kappa_i\bm 1_{n i}

Projection along 1∥i\bm 1_{\parallel i} is integrable:

12miui2+eϕ=Hi\frac{1}{2}m_i u_i^2 + e\phi = H_i

  • Mechanical energy equation.
  • Valid along ion streamlines.

Electron equations

We write ue=u∥e1b+u×e\bm u_e = u_{\parallel e} \bm 1_b + \bm u_{\times e}. We will call 1×=u×e/u×e\bm 1_\times = \bm u_{\times e} / u_{\times e}, and 1⊥=1××1b\bm 1_\perp = \bm 1_\times\times \bm 1_b.
{1b,1⊥,1×}\{\bm 1_b,\bm 1_\perp,\bm 1_\times\} is a right-handed, orthonormal vector basis.

Continuity:

∇⋅(nue)=0∇⋅(nu∥e1b+nu×e)=0\nabla \cdot (n \bm u_e) = 0 \\ \nabla \cdot(nu_{\parallel e} \bm 1_b + n\bm u_{\times e}) = 0

  • This is the only electron equation where u∥eu_{\parallel e} is present.

State:

pe=nTep_e = nT_e

  • Isotropic electrons.

Momentum:

0=−∇pe+en∇ϕ−enu×e×B\bm 0 = -\nabla p_e + en \nabla \phi - en \bm u_{\times e} \times \bm B

  • Balance of pressure, electric, and magnetic forces.

If a barotropy function heh_e exists such that ∇he=(∇pe)/n\nabla h_e = (\nabla p_e)/n, then:

0=−∇he+e∇ϕ−eu×e×B\bm 0 = -\nabla h_e + e\nabla \phi - e\bm u_{\times e} \times \bm B

  • A barotropy function exists if the electrons are isothermal, Te=constT_e=\text{const} and γ=1\gamma=1, or polytropic, Te∝nγ−1T_e\propto n^{\gamma-1}:

he=γTeln⁡nh_e = \gamma T_e \ln n

  • This is known as a closure relation at the temperature level. Other models exist that use the energy equation and a closure relation at the heat flux level (they may not have a barotropy function).

Projection along 1b\bm 1_b or along 1×\bm 1_\times yields:

0=1⋅[−∇he+e∇ϕ],where: 1=1b,1×0 = \bm 1 \cdot[-\nabla h_e + e\nabla \phi], \quad \text{where: } \bm 1= \bm 1_b,\bm 1_\times

  • Balance of pressure and electric forces.

Integrating:

he−eϕ=Heh_e-e\phi = H_e

  • Energy conservation equation.
  • Valid along electron streamlines == magnetic streamlines.
  • Valid in the 1×\bm 1_\times direction too: 1b\bm 1_b and 1×\bm 1_\times define nested surfaces. We can call them HeH_e-surfaces.

We can re-write the momentum equation as:

∇He=−eu×e×B\nabla H_e = - e\bm u_{\times e} \times \bm B

  • ∇He=(∇pe)/n−e∇ϕ\nabla H_e = (\nabla p_e)/n - e\nabla\phi is equal to the magnetic force per electron

Crossing with B\bm B and operating:

u×e=∇He×BeB2\bm u_{\times e} = \frac{\nabla H_e\times \bm B}{eB^2}

  • Explicit expression for u×eu_{\times e}, the Hall velocity, and 1×\bm 1_\times, the Hall direction.
  • u×e\bm u_{\times e} is the sum of the E×B\bm E\times\bm B drift and the diamagnetic drift ∇pe×B/(enB2\nabla p_e \times \bm B / (enB^2). This is not a particle drift, but a fluid drift due to the collective motion of the electrons.

Ion equations, revisited

∇⋅(nui)=0\nabla \cdot (n \bm u_i) = 0

miui⋅∇ui+γTe∇ln⁡n=∇He+eui×Bm_i \bm u_i \cdot \nabla \bm u_i + \gamma T_e \nabla \ln n = \nabla H_e + e \bm u_i \times \bm B

  • These are analogous to Euler gasdynamics equations for a polytropic species, with the extra force terms ∇He\nabla H_e and eui×Be \bm u_i \times \bm B.

Important remarks

  • Electron momentum equation is fully algebraic after applying the simplifying assumptions.
  • Given B\bm B and boundary conditions, HeH_e, u×eu_{\times e}, 1×\bm 1_\times are fully determined before computing anything else.
  • The boundary conditions must set one and only one value of HeH_e on each magnetic line (else they are incomplete or incompatible).
  • Once HeH_e is known, we can integrate the ion differential equations, finding nn, ui\bm u_i.
  • Once nn is known, the electron continuity equation can be used to compute u∥eu_{\parallel e}.
  • Once nn and HeH_e are known, from closure relation and the state equation we can compute pep_e, TeT_e.
  • Finally, from the conservation of HeH_e we can compute ϕ\phi.

Model extensions

Assumptions can be relaxed, but the model complexity increases accordingly:

  • Adding collisions allows electrons to move in the 1⊥\bm 1_\perp direction. Also, u∥eu_{\parallel e} now appears in the momentum equation.
  • Adding ionization/recombination: source terms in the ion and electron equations.
  • Interactions with walls: non-neutral sheath models can be employed.
  • Time dependence: all differential equations become fully hyperbolic then.
  • Replace isothermal/polytropic closure with electron energy equation plus a higher-order closure.
  • Include heating term in energy equation (e.g. EM power deposition).
  • Extend the temperature model to include electron anisotropy.
  • Include ion temperature.
  • Include the plasma-induced magnetic field through Ampère's equation.