Categories: Physics, Plasma physics.

Two-fluid equations

The two-fluid model describes a plasma as two separate but overlapping fluids, one for ions and one for electrons. Instead of tracking individual particles, it gives the dynamics of fluid elements \(\dd{V}\) (i.e. small “blobs”). These blobs are assumed to be much larger than the Debye length, such that electromagnetic interactions between nearby blobs can be ignored.

From Newton’s second law, we know that the velocity \(\vb{v}\) of a particle with mass \(m\) and charge \(q\) is as follows, when subjected only to the Lorentz force:

\[\begin{aligned} m \dv{\vb{v}}{t} = q (\vb{E} + \vb{v} \cross \vb{B}) \end{aligned}\]

From here, the derivation is similar to that of the Navier-Stokes equations. We replace \(\dv*{t}\) with a material derivative \(\mathrm{D}/\mathrm{D}t\), and define \(\vb{u}\) as the blob’s center-of-mass velocity:

\[\begin{aligned} m n \frac{\mathrm{D} \vb{u}}{\mathrm{D} t} = q n (\vb{E} + \vb{u} \cross \vb{B}) \end{aligned}\]

Where we have multiplied by the number density \(n\) of the particles. Due to particle collisions in the fluid, stresses become important. Therefore, we include the Cauchy stress tensor \(\hat{P}\), leading to the following two equations:

\[\begin{aligned} m_i n_i \frac{\mathrm{D} \vb{u}_i}{\mathrm{D} t} &= q_i n_i (\vb{E} + \vb{u}_i \cross \vb{B}) + \nabla \cdot \hat{P}_i{}^\top \\ m_e n_e \frac{\mathrm{D} \vb{u}_e}{\mathrm{D} t} &= q_e n_e (\vb{E} + \vb{u}_e \cross \vb{B}) + \nabla \cdot \hat{P}_e{}^\top \end{aligned}\]

Where the subscripts \(i\) and \(e\) refer to ions and electrons, respectively. Finally, we also account for momentum transfer between ions and electrons due to Rutherford scattering, leading to these two-fluid momentum equations:

\[\begin{aligned} \boxed{ \begin{aligned} m_i n_i \frac{\mathrm{D} \vb{u}_i}{\mathrm{D} t} &= q_i n_i (\vb{E} + \vb{u}_i \cross \vb{B}) + \nabla \cdot \hat{P}_i{}^\top - f_{ie} m_i n_i (\vb{u}_i - \vb{u}_e) \\ m_e n_e \frac{\mathrm{D} \vb{u}_e}{\mathrm{D} t} &= q_e n_e (\vb{E} + \vb{u}_e \cross \vb{B}) + \nabla \cdot \hat{P}_e{}^\top - f_{ei} m_e n_e (\vb{u}_e - \vb{u}_i) \end{aligned} } \end{aligned}\]

Where \(f_{ie}\) is the mean frequency at which an ion collides with electrons, and vice versa for \(f_{ei}\). For simplicity, we assume that the plasma is isotropic and that shear stresses are negligible, in which case the stress term can be replaced by the gradient \(- \nabla p\) of a scalar pressure \(p\):

\[\begin{aligned} m_i n_i \frac{\mathrm{D} \vb{u}_i}{\mathrm{D} t} &= q_i n_i (\vb{E} + \vb{u}_i \cross \vb{B}) - \nabla p_i - f_{ie} m_i n_i (\vb{u}_i - \vb{u}_e) \\ m_e n_e \frac{\mathrm{D} \vb{u}_e}{\mathrm{D} t} &= q_e n_e (\vb{E} + \vb{u}_e \cross \vb{B}) - \nabla p_e - f_{ei} m_e n_e (\vb{u}_e - \vb{u}_i) \end{aligned}\]

Next, we demand that matter is conserved. In other words, the rate at which particles enter/leave a volume \(V\) must be equal to the flux through the enclosing surface \(S\):

\[\begin{aligned} 0 &= \pdv{t} \int_V n \dd{V} + \oint_S n \vb{u} \cdot \dd{\vb{S}} = \int_V \Big( \pdv{n}{t} + \nabla \cdot (n \vb{u}) \Big) \dd{V} \end{aligned}\]

Where we have used the divergence theorem. Since \(V\) is arbitrary, we can remove the integrals, leading to the following continuity equations:

\[\begin{aligned} \boxed{ \pdv{n_i}{t} + \nabla \cdot (n_i \vb{u}_i) = 0 \qquad \quad \pdv{n_e}{t} + \nabla \cdot (n_e \vb{u}_e) = 0 } \end{aligned}\]

Currently, we have 8 equations (2 scalar continuity, 2 vector momentum), but 16 unknowns \(\vb{u}_i\), \(\vb{u}_e\), \(\vb{E}\), \(\vb{B}\), \(n_i\), \(n_e\), \(p_i\) and \(p_e\). We would like to close this system, so we need 8 more. An obvious choice is Maxwell’s equations, in particular Faraday’s and Ampère’s law (since Gauss’ laws are in fact redundant; see the article on Maxwell’s equations):

\[\begin{aligned} \boxed{ \nabla \cross \vb{E} = - \pdv{\vb{B}}{t} \qquad \quad \nabla \cross \vb{B} = \mu_0 \Big( n_i q_i \vb{u}_i + n_e q_e \vb{u}_e + \varepsilon_0 \pdv{\vb{E}}{t} \Big) } \end{aligned}\]

Now we have 14 equations, so we need 2 more, for the pressures \(p_i\) and \(p_e\). This turns out to be the thermodynamic equation of state: for quasistatic, reversible, adiabatic compression of a gas with constant heat capacity (i.e. a calorically perfect gas), it turns out that:

\[\begin{aligned} \frac{\mathrm{D}}{\mathrm{D} t} \big( p V^\gamma \big) = 0 \qquad \quad \gamma \equiv \frac{C_P}{C_V} = \frac{N + 2}{N} \end{aligned}\]

Where \(\gamma\) is the heat capacity ratio, and can be calculated from the number of degrees of freedom \(N\) of each particle in the gas. In a fully ionized plasma, \(N = 3\).

The density \(n \propto 1/V\), so since \(p V^\gamma\) is constant in time, for some constant \(C\):

\[\begin{aligned} \frac{\mathrm{D}}{\mathrm{D} t} \Big( \frac{p}{n^\gamma} \Big) = 0 \quad \implies \quad p = C n^\gamma \end{aligned}\]

In the two-fluid model, we thus have the following two equations of state, giving us a set of 16 equations for 16 unknowns:

\[\begin{aligned} \boxed{ \frac{\mathrm{D}}{\mathrm{D} t} \Big( \frac{p_i}{n_i^\gamma} \Big) = 0 \qquad \quad \frac{\mathrm{D}}{\mathrm{D} t} \Big( \frac{p_e}{n_e^\gamma} \Big) = 0 } \end{aligned}\]

Note that from the relation \(p = C n^\gamma\), we can calculate the \(\nabla p\) term in the momentum equation, using simple differentiation and the ideal gas law:

\[\begin{aligned} p = C n^\gamma \quad \implies \quad \nabla p = \gamma \frac{C n^{\gamma}}{n} \nabla n = \gamma p \frac{\nabla n}{n} = \gamma k_B T \nabla n \end{aligned}\]

Note that the ideal gas law was not used immediately, to allow for \(\gamma \neq 1\).

Fluid drifts

The momentum equations reduce to the following if we assume the flow is steady \(\pdv*{\vb{u}}{t} = 0\), and neglect electron-ion momentum transfer on the right:

\[\begin{aligned} m_i n_i (\vb{u}_i \cdot \nabla) \vb{u}_i &\approx q_i n_i (\vb{E} + \vb{u}_i \cross \vb{B}) - \nabla p_i \\ m_e n_e (\vb{u}_e \cdot \nabla) \vb{u}_e &\approx q_e n_e (\vb{E} + \vb{u}_e \cross \vb{B}) - \nabla p_e \end{aligned}\]

We take the cross product with \(\vb{B}\), which leaves only the component \(\vb{u}_\perp\) of \(\vb{u}\) perpendicular to \(\vb{B}\) in the Lorentz term:

\[\begin{aligned} 0 &= q n (\vb{E} + \vb{u}_\perp \cross \vb{B}) \cross \vb{B} - \nabla p \cross \vb{B} - m n \big( (\vb{u} \cdot \nabla) \vb{u} \big) \cross \vb{B} \\ &= q n (\vb{E} \cross \vb{B} - \vb{u}_\perp B^2) - \nabla p \cross \vb{B} - m n \big( (\vb{u} \cdot \nabla) \vb{u} \big) \cross \vb{B} \end{aligned}\]

Isolating for \(\vb{u}_\perp\) tells us that the fluids drifts perpendicularly to \(\vb{B}\), with velocity \(\vb{u}_\perp\):

\[\begin{aligned} \vb{u}_\perp = \frac{\vb{E} \cross \vb{B}}{B^2} - \frac{\nabla p \cross \vb{B}}{q n B^2} - \frac{m \big( (\vb{u} \cdot \nabla) \vb{u} \big) \cross \vb{B}}{q B^2} \end{aligned}\]

The last term is often neglected, which turns out to be a valid approximation if \(\vb{E} = 0\), or if \(\vb{E}\) is parallel to \(\nabla p\). The first term is the familiar \(\vb{E} \cross \vb{B}\) drift \(\vb{v}_E\) from guiding center theory, and the second term is called the diamagnetic drift \(\vb{v}_D\):

\[\begin{aligned} \boxed{ \vb{v}_E = \frac{\vb{E} \cross \vb{B}}{B^2} } \qquad \quad \boxed{ \vb{v}_D = - \frac{\nabla p \cross \vb{B}}{q n B^2} } \end{aligned}\]

It is called diamagnetic because it creates a current that induces a magnetic field opposite to the original \(\vb{B}\). In a quasi-neutral plasma \(q_e n_e = - q_i n_i\), the current density \(\vb{J}\) is given by:

\[\begin{aligned} \vb{J} = q_e n_e (\vb{v}_{De} - \vb{v}_{Di}) = q_e n_e \Big( \frac{\nabla p_i \cross \vb{B}}{q_i n_i B^2} - \frac{\nabla p_e \cross \vb{B}}{q_e n_e B^2} \Big) = \frac{\vb{B} \cross \nabla (p_i + p_e)}{B^2} \end{aligned}\]

Using the ideal gas law \(p = k_B T n\), this can be rewritten as follows:

\[\begin{aligned} \vb{J} = k_B \frac{\vb{B} \cross \nabla (T_i n_i + T_e n_e)}{B^2} \end{aligned}\]

Curiously, \(\vb{v}_D\) does not involve any net movement of particles, because a pressure gradient does not necessarily cause particles to move. Instead, there is a higher density of gyration paths in the high-pressure region, so that the particle flux through a reference plane is higher. This causes the fluid elements to drift, but not the guiding centers.

References

  1. F.F. Chen, Introduction to plasma physics and controlled fusion, 3rd edition, Springer.
  2. M. Salewski, A.H. Nielsen, Plasma physics: lecture notes, 2021, unpublished.

© Marcus R.A. Newman, a.k.a. "Prefetch". Available under CC BY-SA 4.0.
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