Notes on the basic two-fluid collisionless magnetoplasma model


These assumptions define the regime of validity of the model:

Ion equations

Continuity: \(\nabla \cdot (n \bm u_i) = 0\)

Momentum: \(m_i \bm u_i \cdot \nabla \bm u_i = -e \nabla \phi + e \bm u_i \times \bm B\)

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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: \(\bm u_{\times e} = \frac{\nabla H_e\times \bm B}{eB^2}\)

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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: