Notes on the basic two-fluid collisionless magnetoplasma model
Assumptions
These assumptions define the regime of validity of the model:
- Fully-ionized ion-electron plasma (we ignore neutrals).
- ∂/∂t=0: Steady-state plasma.
- λ2D=ε0Te/(ne2)≪L2: quasineutral plasma n=ni=ne.
- χH=ωce/νe≫1, ui/L≫νi: Collisionless plasma.
- ℓe≪L: electrons are fully magnetized. We only keep zeroth-order terms in ℓe/L.
- meu2e≪Te: negligible electron inertia and bulk velocity.
- miu2i∼Te: moderate bulk ion velocity.
- Ti≪Te: cold ions.
- β=2nTe/(μ0B2)≪1: negligible induced magnetic field.
Ion equations
Continuity: \nabla \cdot (n \bm u_i) = 0
Momentum: m_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 m_i\gg m_e.
- Left-hand side can be expressed as ($\bm 1{t i}, \bm 1{n i} are the tangent and normal unit vectors to ion trajectories; \kappa_i$ the ion trajectory curvature): m_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 \bm 1_{\parallel i} is integrable: \frac{1}{2}m_i u_i^2 + e\phi = H_i
- Mechanical energy equation.
- Valid along ion streamlines.
Electron equations
We write $\bm u_e = u_{\parallel e} \bm 1b + \bm u{\times e}. We will call \bm 1\times = \bm u{\times e} / u_{\times e}, and \bm 1\perp = \bm 1\times\times \bm 1b. {\bm 1_b,\bm 1\perp,\bm 1_\times}$ is a right-handed, orthonormal vector basis.
Continuity: \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_{\parallel e} is present.
State: p_e = nT_e
- Isotropic electrons.
Momentum: \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 h_e exists such that \nabla h_e = (\nabla p_e)/n, then: \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, T_e=\text{const} and \gamma=1, or polytropic, T_e\propto n^{\gamma-1}: h_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 $\bm 1b or along \bm 1\times$ yields: 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: h_e-e\phi = H_e
- Energy conservation equation.
- Valid along electron streamlines = magnetic streamlines.
- Valid in the $\bm 1\times direction too: \bm 1_b and \bm 1\times$ define nested surfaces. We can call them H_e-surfaces.
We can re-write the momentum equation as: \nabla H_e = - e\bm u_{\times e} \times \bm B
- \nabla H_e = (\nabla p_e)/n - e\nabla\phi is equal to the magnetic force per electron
Crossing with \bm B and operating: \bm u_{\times e} = \frac{\nabla H_e\times \bm B}{eB^2}
- Explicit expression for u_{\times e}, the Hall velocity, and \bm 1_\times, the Hall direction.
- \bm u_{\times e} is the sum of the \bm E\times\bm B drift and the diamagnetic drift \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
\nabla \cdot (n \bm u_i) = 0 m_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 \nabla H_e and e \bm u_i \times \bm B.
Important remarks
- Electron momentum equation is fully algebraic after applying the simplifying assumptions.
- Given \bm B and boundary conditions, H_e, u_{\times e}, \bm 1_\times are fully determined before computing anything else.
- The boundary conditions must set one and only one value of H_e on each magnetic line (else they are incomplete or incompatible).
- Once H_e is known, we can integrate the ion differential equations, finding n, \bm u_i.
- Once n is known, the electron continuity equation can be used to compute u_{\parallel e}.
- Once n and H_e are known, from closure relation and the state equation we can compute p_e, T_e.
- Finally, from the conservation of H_e we can compute \phi.
Model extensions
Assumptions can be relaxed, but the model complexity increases accordingly:
- Adding collisions allows electrons to move in the $\bm 1\perp direction. Also, u{\parallel e}$ 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.