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
two-fluid model;
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 p
and electric current density J are:
p=pi+peJ=qiniui+qeneue
Meanwhile, the macroscopic mass density ρ
and center-of-mass flow velocity u
are as follows, although the ions dominate due to their large mass:
We will assume that electrons’ inertia
is negligible compared to the Lorentz force.
Let τchar be the characteristic timescale of the plasma’s dynamics,
i.e. nothing noticable happens in times shorter than τchar,
then this assumption can be written as:
Where we have recognized the cyclotron frequency ωc (see Lorentz force article).
In other words, our assumption is equivalent to
the electron gyration period 2π/ωce
being small compared to the macroscopic dynamics’ timescale τchar.
By construction, we can thus ignore the left-hand side
of the electron momentum equation, leaving:
Where we have used fiemini=feimene
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 lengthλD
is small compared to a “blob” dV of the fluid,
we can invoke quasi-neutrality qini+qene=0.
Using that ρ≈mini and u≈ui,
we thus arrive at the momentum equation:
ρDtDu=J×B−∇p
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:
E+ue×B−qene∇pe=qefeime(ue−ui)
Again using quasi-neutrality qini=−qene,
the current density J=qene(ue−ui),
so:
E+ue×B−qene∇pe=ηJη≡neqe2feime
Where η 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 u≈ui,
we add (u−ui)×B≈0 to the equation,
and insert J again:
Next, we want to get rid of the pressure term.
To do so, we take the curl of the equation:
∇×(ηJ)=−∂t∂B+∇×(u×B)+∇×qeneJ×B−∇×qene∇pe
Where we have used Faraday’s law.
This is the induction equation,
and is used to compute B.
The pressure term can be rewritten using the ideal gas law pe=kBTene:
It is reasonable to assume that ∇Te and ∇ne
point in roughly the same direction,
in which case the pressure term can be neglected.
Consequently, pe has no effect on the dynamics of B,
so we argue that it can be dropped from the original (non-curled) equation too, leaving:
E+u×B+qeneJ×B=ηJ
This is known as the generalized Ohm’s law,
since it contains the relation E=ηJ.
Next, consider Ampère’s law,
where we would like to neglect the last term:
∇×B=μ0J+c21∂t∂E
From Faraday’s law, we can obtain a scale estimate for E.
Recall that τchar is the characteristic timescale of the plasma,
and let λchar≫λD be its characteristic lengthscale:
∇×E=−∂t∂B⟹∣E∣∼τcharλchar∣B∣
From this, we find when we can neglect
the last term in Ampère’s law:
the characteristic velocity vchar
must be tiny compared to c,
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:
∇×B=μ0J∇×E=−∂t∂B
Finally, we revisit the thermodynamic equation of state,
for a single fluid this time.
Using the product rule of differentiation yields:
0=DtD(ργp)=DtDpρ−γ−pγρ−γ−1DtDρ
The continuity equation allows us to rewrite
the material derivativeDρ/Dt as follows:
∂t∂ρ+∇⋅(ρu)=∂t∂ρ+ρ∇⋅u+u⋅∇ρ=ρ∇⋅u+DtDρ=0
Inserting this into the equation of state
leads us to a differential equation for p:
0=DtDp+pγρ1ρ∇⋅u⟹DtDp=−pγ∇⋅u
This closes the set of 14 MHD equations for 14 unknowns.
Originally, the two-fluid model had 16 of each,
but we have merged ni and ne into ρ,
and pi and pi into p.
Ohm’s law variants
It is worth discussing the generalized Ohm’s law in more detail.
Its full form was:
E+u×B+qeneJ×B=ηJ
However, most authors neglect some of its terms:
this form is used for Hall MHD,
where J×B 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 ∇p≈0
to obtain the scale estimate J×B∼ρvchar/τchar.
In other words, if the ion gyration period is short τchar≫ωci,
and/or if the electron gyration period is long
compared to the electron-ion collision period ωce≪fei,
then we are left with this form of Ohm’s law, used in resistive MHD:
E+u×B=ηJ
Finally, we can neglect the resisitive term ηJ
if the Lorentz force is much larger.
We formalize this condition as follows,
where we have used Ampère’s law to find J∼B/μ0λchar:
Where we have defined the magnetic Reynolds numberRm as follows,
which is analogous to the fluid Reynolds numberRe:
Rm≡η/μ0vcharλchar
If Rm≪1, the plasma is “electrically viscous”,
such that resistivity needs to be accounted for,
whereas if Rm≫1, the resistivity is negligible,
in which case we have ideal MHD:
E+u×B=0
References
P.M. Bellan,
Fundamentals of plasma physics,
1st edition, Cambridge.
M. Salewski, A.H. Nielsen,
Plasma physics: lecture notes,
2021, unpublished.