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 (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
of a particle with mass and charge is as follows,
when subjected only to the Lorentz force:
From here, the derivation is similar to that of the
We replace with a
material derivative ,
and define as the blob’s center-of-mass velocity:
Where we have multiplied by the number density of the particles.
Due to particle collisions in the fluid,
stresses become important. Therefore, we include
the Cauchy stress tensor ,
leading to the following two equations:
Where the subscripts and 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:
Where is the mean frequency at which an ion collides with electrons,
and vice versa for .
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 of a scalar pressure :
Next, we demand that matter is conserved.
In other words, the rate at which particles enter/leave a volume
must be equal to the flux through the enclosing surface :
Where we have used the divergence theorem.
Since is arbitrary, we can remove the integrals,
leading to the following continuity equations:
These are 8 equations (2 scalar continuity, 2 vector momentum),
but 16 unknowns , , , , , , and .
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 redundant; see the article on Maxwell’s equations):
Now we have 14 equations, so we need 2 more, for the pressures and .
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:
Where is the heat capacity ratio,
and can be calculated from the number of degrees of freedom
of each particle in the gas.
In a fully ionized plasma, .
The density ,
so since is constant in time,
for some constant :
In the two-fluid model, we thus have the following two equations of state,
giving us a set of 16 equations for 16 unknowns:
Note that from the relation ,
we can calculate the term in the momentum equation,
using simple differentiation and the ideal gas law:
Note that the ideal gas law was not used immediately,
to allow for .
The momentum equations reduce to the following
if we assume the flow is steady ,
and neglect electron-ion momentum transfer on the right:
We take the cross product with ,
which leaves only the component of
perpendicular to in the Lorentz term:
Isolating for tells us
that the fluids drifts perpendicularly to ,
with velocity :
The last term is often neglected,
which turns out to be a valid approximation if ,
or if is parallel to .
The first term is the familiar drift
from guiding center theory,
and the second term is called the diamagnetic drift :
It is called diamagnetic because
it creates a current that induces
a magnetic field opposite to the original .
In a quasi-neutral plasma ,
the current density is given by:
Using the ideal gas law ,
this can be rewritten as follows:
Curiously, 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.
- F.F. Chen,
Introduction to plasma physics and controlled fusion,
3rd edition, Springer.
- M. Salewski, A.H. Nielsen,
Plasma physics: lecture notes,