MHD Description of Plasma

The abbreviation MHD stands for magnetohydrodynamics. The MHD model is one of the simplest models for describing the interaction between a perfect conducting fluid and a magnetic field. Many feature of instabilities are predicted by the MHD model.

MHD Equations for a Plasma:

Define the total mass density tex2html_wrap_inline157 , the average flow velocity u, the current density j as following: (In fully ionized plasma)
tex2html_wrap_inline159 (1)
tex2html_wrap_inline161 (2)
tex2html_wrap_inline163 (3)
Equations of motion for ions and electrons are:
tex2html_wrap_inline165 (4)
tex2html_wrap_inline167 (5)
Equations of continuity for mass of ions and electrons are:
tex2html_wrap_inline169 (6)
tex2html_wrap_inline171 (7)
Equations of continuity for charge of ions and electrons are:
tex2html_wrap_inline173 (8)
tex2html_wrap_inline175 (9)
Where momentum transfer tex2html_wrap_inline177 and tex2html_wrap_inline179 . They are dissipation terms related to friction and thermal force. ([2], pp.37).
The set of MHD equations can be derived by playing with above equations for ions and electrons. They are:
Equation of motion: tex2html_wrap_inline181 (10)
Equation of continuity: tex2html_wrap_inline183 (11); tex2html_wrap_inline185 (12)
Generalized Ohm's law: tex2html_wrap_inline187 (13)
Equation of state (adiabatic relation): tex2html_wrap_inline189 (14)
Where tex2html_wrap_inline191 , is adiabatic index.
The system of equations is closed by including with Maxwell's equations:
Farady's law: tex2html_wrap_inline193 (15)
Ampere's law: tex2html_wrap_inline195 (16)
tex2html_wrap_inline197 (17)
Coulomb's law: U tex2html_wrap_inline199 (18)
Usually, term tex2html_wrap_inline201 is neglected.
Magnetohydrodynamics

1. The quasi-neutrality Approximation: tex2html_wrap_inline203 , but tex2html_wrap_inline205
2. The "small Larmor Radius" Approximation:[9]-120
Assume that the ion Larmor radius is very small compared to the scale-length of the fluid motion, i.e. tex2html_wrap_inline207 . This is called "finite Larmor radius" treatment. The Ohm's law becomes simple:
tex2html_wrap_inline209 (19)
3. Ideal MHD approximation:
Assume that the conductivity of the plasma is infinite, which means that the plasma is tied to the magnetic field lines. Ohm's law is simply: tex2html_wrap_inline211 (20)
4. Conservation of Magnetic Flux:
If assume plasma is a perfect conductor, the magnetic flux through any closed contour that moves with the plasma is constant:
tex2html_wrap_inline213 (21)
5. Magnetic Reynolds Number:[9]-127~128
How good must a plasma's conductivity be that ideal MHD is valid? This is decided by the magnetic Reynolds number: tex2html_wrap_inline215 (22)
where L is characteristic scale-length, and u a characteristic plasma velocity. If tex2html_wrap_inline221 is high enough, the perfect conductor assumption is valid. For fully developed magnetohydrodynamic motion, the characteristic velocities are very large, and magnetic Reynolds number in low-resistivity plasmas can range up to tex2html_wrap_inline223 , or higher.[9]