Expert: Philip A. Stahl - 5/18/2011

Question
I'm not a student but just interested in plasma physics and have done lots of work on it, mostly self- study. Anyway, can you clarify where two fluid theory fits in with general plasma schemes and also what makes it up? You know, the sort of equations that apply and so on? Thanks!

Answer
Hello,

Classification of differing fluid regimes is far beyond the scope of this answer, but you can find any number of good introductory plasma physics text that delve into details (e.g. 'Introduction to Plasma Physics' by Francis F. Chen).

The bottom line overview is that one proceeds by taking moments of the Boltzmann equation.

E.g. the Boltmann eqn. is:  @f/ @t + v*grad f + F/m*@f/@t = (@f/@t)_C

where @ denotes partial derivative, and (@f/@t)_C  is the time rate of change in f due to collisions.

The first moment, which yields a 'two-fluid' (e.g. electron-ion) medium is obtained by integrating the above eqn. with F = q/m (E + v X B). If one then assumes a sufficiently hot plasma so it's collisionless, the term on the RHS, (@f/@t)_C ->  0.

This is the Vlasov equation:

@f/@t + v*grad f + q/m (E + v X B)*@f/@t = 0

The 2nd moment is obtained by multiplying the original eqn. (Boltzmann) by m v then integrating it over dv.

Anyway, the progression by using this procedure is that one gets in succession:


Two -fluid theory (e.g. ions and electrons treated as a separate fluids)

!
!
!
V

One fluid theory (introducing low frequency, long wave length and quasi -neutral approximations, e.g. n_e ~ n_i
!
!
!
V

MHD Theory (proceeds from 1-fluid theory with further assumptions, simplifications)

In this development, we can see that the two-fluid paradigm embodies much more detail and accuracy than the cruder 1-fluid and MHD approaches, though it is or can be more difficult to apply.  Basically given an “ion fluid” and an “electron fluid” there are two essential equations which apply to describe the properties:

1)   The Continuity equation:

Which is a statement of conservation of mass to the effect that the density of the fluid increases (decreases) with time at a point that is equal to the net fluid influx (outflux) of the fluid in a volume element at that point in space. This can be expressed mathematically as:

@_t  n_s(x,t) + del (n_s V_s) = 0

Where @_t denotes the partial derivative with respect to time, V_s is the fluid velocity of the sth species (ion or electron) and ‘del’ is an operator, denoting the gradient in this instance, i.e.

If f = f(x,y,z) then grad f = @f/ @x + @f/@y + @f/ @z

We can shorten this to represent:

grad = (@_x, @_y, @_z) which means the same thing, so grad f=  = (@_x, @_y, @_z)f

IN the fluid case cited grad(‘flux’) = grad (n_s V_s)  = (@_x, @_y, @_z)(n_s V_s)

Which  is positive or negative depending on whether flux enters or leaves the volume element.

2) The force equation. Newton’s equation for the motion of a volume element may be expressed:

n_s m_s d_t V_s(x,t)= F_s(x,t)

Where F_s(x,t) is the “body force” consisting of pressure, electric and magnetic contributions, viz.

F = - grad p + qn E + qn V X B = -grad p +qn(E + V X B)

Where I’ve dropped the subscript ‘s’ for greater clarity but we understand the equation above refers to the electron and ion fluids in turn. Note that q is the electronic charge or q = 1.6 x 10^-19 C. E is the electric intensity and B the magnetic induction.

Now, the quantity d_t V_s(x,t) is defined:

d_t V_s(x,t)= @_t V_s(x,t) + (V_s*del)V_s

and since we already know F (see above) we can write in toto:

@_t V(x,t) + (V*del)V  = - grad p +qn(E + V X B)

Where again, I’ve omitted the subscript ‘s’ throughout but we do understand it applies and take it into account, e.g. in the respective computations for ions, elections, for all the pertinent quantities (e.g. n, V, p, q).

As you can see, the quantitative intricacies are one reason why so many plasma physicists gravitate toward the ultra-simplifying paradigm that represents MHD, as opposed to two-fluid theory!