The Magnetic Nozzle

Consider the behaviour of a perfect gas of infinite conductivity flowing with a velocity through a channel of varying cross-sectional area A (Fig. 1). The channel has a constant width, w, a varying height, l, and a magnetic field applied in the direction.

To ensure that the magnetic field is always perpendicular to the side walls, it is assumed that the sides parallel to the x-y plane are composed of a material with infinite permeability. Such a magnetic field can ``stiffen'' the gas, so that the signal velocity of the medium is now the fast magnetosonic speed. To exploit this property, and to drive the flow, an electric field is applied in direction.

Our fundamental equations are the steady-state forms of Faraday's law, Ampere's law, plus the isentropic magnetohydrodynamic (MHD) equations and the ``frozen-in-flux'' approximation.

  
Figure 1: Channel flow with a magnetic field between high permeability pole pieces. The flow is driven by drift.

By examining the suitable one dimensional forms of these equations, one can produce (Morozov and Solov'ev 1980, Liffman and Siora 1997) a nozzle equation with the Hugoniot form:
 
where u is the x component of , is the fast magnetosonic speed (), the sound speed (), the Alfvn speed (), B - magnetic field strength, - permeability of free space, - density, p - pressure, and - the ratio of specific heats.

If we wish to accelerate the flow () then , i.e., when the flow starts, the nozzle has to converge. Similarly, , so once we are past the critical point in the flow the nozzle must diverge. Clearly, the critical speed is the magnetosonic speed and given that the Alfvn speed has the value

the potential for high speed flow is obvious.

Of course, the frozen - in - flux behaviour of magnetic fields implies that the plasma will be fixed to the magnetic field, so even though the signal velocity of the medium is the fast magnetosonic speed the actual flow speed would - at first glance - be equal to zero. This intransigence can be overcome however, by applying an electric field perpendicular to the magnetic field such that the vector is pointing in the flow direction. The plasma will then move via the mechanism of drift.

Further manipulation of the MHD equations (Morozov and Solov'ev 1980, Contopoulos 1995, Liffman and Siora 1997). gives the flow constants
 
and
 
with an MHD-Bernoulli equation
 

In the ``cold'' plasma limit () one can use the MHD-Bernoulli equation (see Liffman and Siora 1997) to show that
 
where is the exit speed of the nozzle and is the Alfvn speed at the throat of the nozzle.

One can also show

where and are the gas densities at the throat and entrance (or reservoir) of the nozzle, respectively. Similarly, and refers to the magnetic field strength at the throat and entrance of the nozzle. So, one can have a dramatic increase in the flow speed from u = 0 in the reservoir, to at the throat, but suffer only a 1/3 decrease in the magnetic field strength and gas density. This raises the possibility that such flows may be quite efficient in ejecting dust and, possibly, larger macroscopic material.

But what, if anything, does this have to do with YSO jets?

© Copyright Astronomical Society of Australia 1997