ELECTRO-MAGNETISM1
next up previous Next: The Energy Equation Up: Fluid Equations Previous: Fluid Equations The Equation of Motion In (2.9) the external force, ${\bf F}$, consists of several terms. The importance of these terms depends on the particular situation being modelled. The dominant term for magnetised plasmas is the magnetic force, the Lorentz force, given by \begin{displaymath} {\bf j} \times {\bf B}. \end{displaymath} In addition, the gravitational force $\rho {\bf g}$ is frequently included as well as a viscous force. The actual form of the viscous term is complicated (see Braginskii, 19 ) but it may be approximated by a kinematic viscosity of the form \begin{displaymath} \rho \nu \nabla^{2}{\bf v}, \end{displaymath} for an incompressible flow. Here $\nu$ is the coefficient of kinematic viscosity which Spitzer (1962) gives as \begin{displaymath} \rho \nu = 2.21 \times 10^{-16}{T^{5/2}\over \hbox{ln} \Lambda} \hbox{kg m}^{-1}\hbox{s}^{-1}. \end{displaymath} Thus, (2.9) becomes \begin{displaymath} \rho {D{\bf v}\over Dt} = - \nabla p + {\bf j} \times {\bf B} + \rho {\bf g} + \rho \nu \nabla^{2}{\bf v}. \end{displaymath} (2.14) Note that the Lorentz force couples the fluid equations to the electromagnetic equations. Prof. Alan Hood 2000-01-11
Next: The Equation of Motion Up: MHD Equations Previous: Electromagnetic Equations
Fluid Equations
The first equation is mass continuity
and it states that matter is neither created or destroyed. Obviously if the plasma is sufficiently hot and dense that nuclear reactions occur it may be necessary to modify this equation. Next, Newton's second law of motion, mass acceleration = applied force, gives
The energy equation can be written in the form
Here is the total energy loss function, is the ratio of specific heats, and is normally taken as 5/3.
The last equation is an equation of state that is used to close the system of equations. This normally taken as the ideal gas law
In (2.11) is the plasma temperature, J Kkg is the gas constant, the factor 2 is correct for a pure hydrogen plasma but in the solar corona this should be replaced by 1.67. (2.11) can be written in terms of the number density since and so
where is the mass of a proton and is Boltzmann's constant. In (2.9) and (2.10), is the convective time derivative
that describes the time derivative as a quantity moves with the plasma.
A simple derivation of the 1-D version of (2.8)
can be obtained by considering how the density of a small region of width , centred about the general point , varies from time to time .
The initial mass in the cell, at time , is
and the final mass is approximately . Thus, the change in mass during the time interval is
Now the mass entering the cell is
whereas the the mass leaving the cell is
Therefore, equating these expressions gives the result
The final step is to let and tend to zero so that the partial derivatives are obtained.
Electromagnetic Equations
The electromagnetic equations are given in MKS units but the magnetic field is frequently quoted in terms of Gauss, where 1 tesla = Gauss. Maxwells's equations are
where the last term on the right hand side is the displacement current.
is the solenoidal condition for the magnetic induction, indicating that there are no magnetic monopoles. That is there are no sources and sinks for magnetic field lines. Next, we have Faraday's law of magnetic induction with
showing that a spatially varying electric field can induce a magnetic field. Finally, charge conservation gives
In the above equations
- B is the magnetic induction, but is usually referred to as the magnetic field,
- j is the current density,
- E is the electric field,
- is the charge density, where is the number of ions and is the number of electrons,
- is the permittivity of free space and
- is the magnetic permeability in a vacuum and is Hm.
Auxiliary variables are the electric displacement and the magnetic field . Finally, the speed of light in a vacuum is ms.
As the equation only has two terms they must be equal and so we obtain
Now in (2.1) the left hand side is approximately but the displacement current is
Hence, if the typical plasma velocities satisfy then the displacement current is much smaller than . This is the MHD approximation, so that Ampere's law simplifies to
Using similar ideas and by assuming the plasma obeys charge neutrality, , where is the total number density, we can neglect the charge density in (2.4) provided
is the charge on an electron.
The final electromagnetic equation is ohm's law,
where is the electrical conductivity in mho m. This is effectively a generalisation of the simple voltage = current times resistance to a moving conductor. is the plasma velocity and it is Ohm's law that provides the link between the electromagnetic equations and the plasma fluid equations.
Next: The Equation of Motion Up: MHD Equations Previous: Electromagnetic Equations
2000-01-11
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