Processing math: 15%

Notes on the basic two-fluid collisionless magnetoplasma model

Assumptions

These assumptions define the regime of validity of the model:

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

bg right:35% 90%


Projection along \bm 1_{\parallel i} is integrable: \frac{1}{2}m_i u_i^2 + e\phi = H_i

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

State: p_e = nT_e


Momentum: \bm 0 = -\nabla p_e + en \nabla \phi - en \bm u_{\times e} \times \bm B

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


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

Integrating: h_e-e\phi = H_e


We can re-write the momentum equation as: \nabla H_e = - e\bm u_{\times e} \times \bm B

Crossing with \bm B and operating: \bm u_{\times e} = \frac{\nabla H_e\times \bm B}{eB^2}

bg right:25% 80%

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

Important remarks

Model extensions

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