Magnetohydrodynamics (MHD) describes the dynamics
of fluids that are electrically conductive.
Notably, it is often suitable to describe plasmas,
and can be regarded as a special case of the
we will derive it as such,
but the results are not specific to plasmas.
In the two-fluid model, we described the plasma as two separate fluids,
but in MHD we treat it as a single conductive fluid.
The macroscopic pressure
and electric current density are:
Meanwhile, the macroscopic mass density
and center-of-mass flow velocity
are as follows, although the ions dominate due to their large mass:
With these quantities in mind,
we add up the two-fluid continuity equations,
multiplied by their respective particles’ masses:
After some straightforward rearranging,
we arrive at the single-fluid continuity relation:
Next, consider the two-fluid momentum equations
for the ions and electrons, respectively:
We will assume that electrons’ inertia
is negligible compared to the Lorentz force.
Let be the characteristic timescale of the plasma’s dynamics,
i.e. nothing noticable happens in times shorter than ,
then this assumption can be written as:
Where we have recognized the cyclotron frequency (see Lorentz force article).
In other words, our assumption is equivalent to
the electron gyration period
being small compared to the macroscopic dynamics’ timescale .
By construction, we can thus ignore the left-hand side
of the electron momentum equation, leaving:
We add up these momentum equations,
recognizing the pressure and current :
Where we have used
because momentum is conserved by the underlying
Rutherford scattering process,
which is elastic.
In other words, the momentum given by ions to electrons
is equal to the momentum received by electrons from ions.
Since the two-fluid model assumes that
the Debye length
is small compared to a “blob” of the fluid,
we can invoke quasi-neutrality .
Using that and ,
we thus arrive at the momentum equation:
However, we found this by combining two equations into one,
so some information was implicitly lost;
we need a second momentum equation.
Therefore, we return to the electrons’ momentum equation,
after a bit of rearranging:
Again using quasi-neutrality ,
the current density ,
Where is the electrical resistivity of the plasma,
see Spitzer resistivity
for more information, and a rough estimate of this quantity for a plasma.
Now, using that ,
we add to the equation,
and insert again:
Next, we want to get rid of the pressure term.
To do so, we take the curl of the equation:
Where we have used Faraday’s law.
This is the induction equation,
and is used to compute .
The pressure term can be rewritten using the ideal gas law :
The curl of a gradient is always zero,
and we notice that .
Then we use the vector identity ,
It is reasonable to assume that and
point in roughly the same direction,
in which case the pressure term can be neglected.
Consequently, has no effect on the dynamics of ,
so we argue that it can be dropped from the original (non-curled) equation too, leaving:
This is known as the generalized Ohm’s law,
since it contains the relation .
Next, consider Ampère’s law,
where we would like to neglect the last term:
From Faraday’s law, we can obtain a scale estimate for .
Recall that is the characteristic timescale of the plasma,
and let be its characteristic lengthscale:
From this, we find when we can neglect
the last term in Ampère’s law:
the characteristic velocity
must be tiny compared to ,
i.e. the plasma must be non-relativistic:
We thus have the following reduced form of Ampère’s law,
in addition to Faraday’s law:
Finally, we revisit the thermodynamic equation of state,
for a single fluid this time.
Using the product rule of differentiation yields:
The continuity equation allows us to rewrite
the material derivative
Inserting this into the equation of state
leads us to a differential equation for :
This closes the set of 14 MHD equations for 14 unknowns.
Originally, the two-fluid model had 16 of each,
but we have merged and into ,
and and into .
Ohm’s law variants
It is worth discussing the generalized Ohm’s law in more detail.
Its full form was:
However, most authors neglect some of its terms:
this form is used for Hall MHD,
where is called the Hall term.
This term can be dropped in any of the following cases:
Where we have used the MHD momentum equation with
to obtain the scale estimate .
In other words, if the ion gyration period is short ,
and/or if the electron gyration period is long
compared to the electron-ion collision period ,
then we are left with this form of Ohm’s law, used in resistive MHD:
Finally, we can neglect the resisitive term
if the Lorentz force is much larger.
We formalize this condition as follows,
where we have used Ampère’s law to find :
Where we have defined the magnetic Reynolds number as follows,
which is analogous to the fluid Reynolds number :
If , the plasma is “electrically viscous”,
such that resistivity needs to be accounted for,
whereas if , the resistivity is negligible,
in which case we have ideal MHD:
- P.M. Bellan,
Fundamentals of plasma physics,
1st edition, Cambridge.
- M. Salewski, A.H. Nielsen,
Plasma physics: lecture notes,