MHD equations

Magnetohydrodynamic theory involves a further simplification of fluid theory, where the proton and electron fluids are combined and assumed to possess a common flow velocity U. We also assume that the relevant time scales are long in comparison with microscopic particle motion time scales ( $\tau > \Omega_{e}^{-1}, \, \omega_{p}^{-1}, \, \nu_{e}^{-1}$) and that spatial scale lengths are long in comparison with the Debye length and the thermal ion gyroradius. The equation of motion for the MHD fluid is derived by adding electron and proton forms of (), to give

$$\frac{\partial} {\partial t}(\eta U_{s}) + \frac{\partial} {\... ...ho E_{s} - \varepsilon_{sjk} j_{j} B_{k} - \eta g_{s} = 0 \, ,$$ (3.31)

where p_is = p_p,is + p_e,is = p; i.e., we assume that the distribution of particle velocities is sufficiently random such that p_ij can be approximated by a scalar. In vector form,

$$\eta \left[\frac{\partial {\bf U}} {\partial t} + ({\bf U} {\... ... \nabla p + \rho {\bf E} + {\bf J \times B} + \eta {\bf g}\, .$$ (3.32)

The fluid velocity ${\bf U({\bf x}, t)}$ is an Eulerian velocity, which refers to the velocity of a fluid element, and not the the velocity of individual particles that constitute that fluid element at any one time. This is to be contrasted with a Lagrangian velocity, which is the time derivative of the position vector of a particle, and is thus only a function of time; e.g., Newton's equation of motion for a single particle is Lagrangian. The term $({\bf U}$ $\mbox{\boldmath$\cdot \nabla$ }){\bf U}$ is called the convective derivative.

A further relation, linking ${\bf J}$ and the fields, is obtained by multiplying the proton form of () by - e/m_e and the electron form of () by e/m_p (where e is the charge of an electron). Terms quadratic in velocity are ignored ensuring that the resulting expression will be linear in ${\bf J}$. Adding the two equations,

$$\frac{\partial J_{s}} {\partial t} = - \frac{e}{m_{p}} \frac{... ..._{p}} + \frac{n_{e}}{m_{e}}\right) E_{s} \mbox{\hspace{1.5cm}}$$

$$\mbox{\hspace{1.0cm}} + \frac{e^{2}}{c} \varepsilon_{sjk} \le... ... + e \left(\frac{1}{m_{p}} + \frac{1}{m_{e}}\right) P_{s} \, .$$ (3.33)

The following simplifying approximations are made (given that $m_{e} \ll m_{p}$):

$$n_{e} \approx n_{p} \approx \frac{\eta}{m_{p}} \,, \,\,\,\,\,... ...\,\, u_{e,s} \approx U_{s} - \frac{m_{p} c}{\eta e} J_{s} \,.$$

Further, we assume that the momentum exchanged between electrons and ions is proportional to the relative velocity of the two types of particles, with

$$P_{s} = - \frac{\eta e J_{s}}{m_{p} \sigma} \, ,$$ (3.34)

where $\sigma$ is the conductivity coefficient, with $\sigma = \varepsilon_{0} \omega_{p}^{2}/\nu_{e}$. This gives the generalized form of Ohm's law:

$${\bf J} + \frac{\sigma m_{p} m_{e}}{\eta e^{2}} \frac{\partia... ...{\bf U \times B} + \frac{m_{p}}{\eta e} \nabla p \right) \, .$$ (3.35)

For low-frequency disturbances, with characteristic frequency $\omega \ll \nu_{e}$, the second term on the left-hand side may be dropped. In situations where the electron cyclotron frequency $\Omega_{e} \ll \nu_{e}$, the third term on the left-hand side may also be dropped. If the pressure gradient term is also insignificant, then () reduces to

$${\bf J} = \sigma ({\bf E} + {\bf U \times B}) \, .$$ (3.36)

In the perfectly conducting limit ( $\sigma = \infty$), () further simplifies to

$${\bf E} + {\bf U \times B} = 0 \, .$$ (3.37)

This means that a highly conducting plasma with zero current must set up an electric field ${\bf E}=-{\bf U \times B}$. This is a convection electric field that leads to ${\bf E \times B}$ drift of the plasma perpendicular to ${\bf B}$. See Lecture 2 for examples. Equation () also leads to the condition of frozen-in magnetic flux (not proven here) in which a plasma carries a magnetic field along with it. This condition may be stated formally as the magnetic flux through a closed loop that moves with the fluid is constant in time, where the magnetic flux $\Phi := \int {\bf B \cdot {\hat n}} {\rm d}S$, where ${\bf {\hat n}}$ is the unit normal to a surface ${\bf S}$. This is illustrated in Figure for a closed loop at two consecutive times t₁ and t₂, where the loop is stretched out as the fluid locally expands. The density of magnetic field lines enclosed by the loop decreases so as to conserve magnetic flux. The frozen-in flux condition $d \Phi/dt =0$ can be proven by substituting () into Faraday's Law () to give

$$\frac{\partial {\bf B}}{\partial t} = {\mbox{\boldmath$\nabla \times$ }} ({\bf U \times B}) \, ,$$ (3.38)

and using Gauss's law and Stoke's theorem.

We define a magnetic flux tube to be the surface generated by moving any closed loop parallel to the magnetic field lines it intersects at any given time. This surface encloses a constant amount of magnetic flux. As a consequence of flux conservation, the same fluid elements constitute a flux tube at different times. The fluid and magnetic field lines move together. A further consequence of the frozen-in flux condition is that all particles initially in a flux tube will remain in the same flux tube at later times.

  

Figure: The closed loop S embedded in the fluid is stretched out at a later time t₂>t₁ by a non-uniform fluid velocity profile. The magnetic flux through S remains constant and the field lines are tied to the fluid.

An equation of energy continuity is derived by taking the second order moment of Boltzmann's equation, to give

$$\frac{\partial}{\partial t} \left( \frac{1}{2} \eta U^{2} +\f... ... {\bf U} + \frac{1}{\mu_{0}} {\bf E \times B} \right) = 0 \, .$$ (3.39)

where $\Gamma$ is the adiabatic index which takes the value 5/3 for a monatomic gas. Equation () assumes the adiabatic equation of state, for which there is no change in internal energy of a fluid element as it propagates, with

$$p \propto \eta^{\Gamma} \, .$$ (3.40)