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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


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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 $\times$ acceleration = applied force, gives

The energy equation can be written in the form

Here ${\textsc{L}}$ is the total energy loss function, $\gamma$ is the ratio of specific heats, $c_p/c_v$ 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) $T$ is the plasma temperature, $ \textsc{R} = 8.3 \times 10^3$J K$^{-1}$kg$^{-1}$ 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 $\rho = n m_p$ and so

where $m_p$ is the mass of a proton and $k_B$ is Boltzmann's constant. In (2.9) and (2.10), $D/Dt$ 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)

\begin{displaymath} {\partial \rho \over \partial t} + {\partial \over \partial x}(\rho v) = 0, \end{displaymath}

can be obtained by considering how the density of a small region of width $\delta x$, centred about the general point $x$, varies from time $t$ to time $t + \delta t$.

The initial mass in the cell, at time $t$, is

\begin{displaymath} \int_{x-\delta x}^{x + \delta x}\rho (u,t) du \approx \rho (x,t) \delta x , \end{displaymath}

and the final mass is approximately $\rho (x, t+\delta t) \delta x$. Thus, the change in mass during the time interval $\delta t$ is

\begin{displaymath} \biggl [ \rho(x, t + \delta t) - \rho (x,t) \biggr ] \delta... ...g in time }\delta t - \hbox { mass leaving in time }\delta t \end{displaymath}

Now the mass entering the cell is

\begin{displaymath} \rho (x - \delta x, t) v (x - \delta x, t) \delta t , \end{displaymath}

whereas the the mass leaving the cell is

\begin{displaymath} \rho (x + \delta x, t) v (x + \delta x, t) \delta t . \end{displaymath}

Therefore, equating these expressions gives the result

\begin{displaymath} \biggl [ \rho(x, t + \delta t) - \rho (x,t) \biggr ] \delta... ...\rho (x + \delta x, t) v (x + \delta x, t) \biggr ] \delta t , \end{displaymath}

\begin{displaymath} {\biggl [ \rho(x, t + \delta t) - \rho (x,t) \biggr ] \over... ... - \delta x, t) v (x - \delta x, t) \biggr ] \over \delta x} . \end{displaymath}

The final step is to let $\delta t$ and $\delta x$ 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 = $10^4$ 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,


  • $\rho^* = e(n^+ - n^-)$ is the charge density, where $n^+$ is the number of ions and $n^-$ is the number of electrons,


  • $\epsilon$ is the permittivity of free space and


  • $\mu$ is the magnetic permeability in a vacuum and is $4\pi \times 10^{-7}$ Hm$^{-1}$.

Auxiliary variables are the electric displacement ${\bf D} = \epsilon {\bf E}$ and the magnetic field ${\bf H} = {\bf B}/\mu$. Finally, the speed of light in a vacuum is $c = (\mu \epsilon)^{-1/2} \approx 3 \times 10^8$ms$^{-1}$.



Example 2.1.1   The displacement current in Ampere's law may be neglected if the typical plasma velocities are much less than the speed of light, $c$. To see this we assume that the typical lengthscale for plasma variations is $L$ and that typical timescales are of order $T$. This simply means that $L$ is the spatial distance over which quantities vary. Thus, if a quantity is given by $y = e^{-x/a}$, then $y$ varies appreciably over a distance $x=a$. Thus, we would choose $L= a$ in this case. In a similar manner $T$ is an estimate of the time needed for the plasma to change. These two quantities can be used to define a typical plasma velocity as $V = L/T$. This kind of approximation allows us to estimate the typical size of the terms in (2.3). Thus,
\begin{displaymath} \nabla \times {\bf E} \approx {E \over L} \hbox{ and }{\partial {\bf B} \over \partial t} \approx {B\over T}. \end{displaymath}

As the equation only has two terms they must be equal and so we obtain
\begin{displaymath} E = {L \over T} B = VB. \end{displaymath}

Now in (2.1) the left hand side is approximately $B/L$ but the displacement current is
\begin{displaymath} {1\over c^2}{\partial {\bf E}\over \partial t} \approx {1\o... ...over L} {V\over c^2} {L\over T} = {B\over L} {V^2\over c^2}. \end{displaymath}

Hence, if the typical plasma velocities satisfy $V^2 \ll c^2$ then the displacement current is much smaller than $\nabla \times {\bf B}$. This is the MHD approximation, so that Ampere's law simplifies to

Using similar ideas and by assuming the plasma obeys charge neutrality, $n^+ - n^- \ll n$, where $n$ is the total number density, we can neglect the charge density in (2.4) provided

$e$ is the charge on an electron.

The final electromagnetic equation is ohm's law,

where $\sigma$ is the electrical conductivity in mho m$^{-1}$. This is effectively a generalisation of the simple voltage = current times resistance to a moving conductor. ${\bf v}$ is the plasma velocity and it is Ohm's law that provides the link between the electromagnetic equations and the plasma fluid equations.


next up previous
Next: The Equation of Motion Up: MHD Equations Previous: Electromagnetic Equations

Prof. Alan Hood
\begin{displaymath} {\bf j} = \sigma ({\bf E} + {\bf v} \times {\bf B}), \end{displaymath} (2.7)
\begin{displaymath} n \gg {\epsilon B V \over e L}. \end{displaymath} (2.6)
\begin{displaymath} \nabla \times {\bf B} = \mu {\bf j}. \end{displaymath} (2.5)
\begin{displaymath} \nabla \cdot {\bf E} = {1\over \epsilon} \rho^*. \end{displaymath} (2.4)
\begin{displaymath} \nabla \times {\bf E} = - {\partial {\bf B}\over \partial t}, \end{displaymath} (2.3)
\begin{displaymath} \nabla \cdot {\bf B} = 0, \end{displaymath} (2.2)
\begin{displaymath} \nabla \times {\bf B} = \mu {\bf j} + {1\over c^2}{\partial {\bf E}\over \partial t}, \end{displaymath} (2.1)
Figure 2.1: Derivation of mass continuity
\begin{displaymath} {D\over Dt} = {\partial \over \partial t} + {\bf v} \cdot \nabla, \end{displaymath} (2.13)
\begin{displaymath} p = 2 n k_B T, \end{displaymath} (2.12)
\begin{displaymath} p = 2 \rho \textsc{R}T. \end{displaymath} (2.11)
\begin{displaymath} {\rho^\gamma \over \gamma - 1} {D\over Dt}\biggl ({p\over \rho^\gamma}\biggr ) = - {\textsc{L}}. \end{displaymath} (2.10)
\begin{displaymath} \rho {D{\bf v}\over Dt} = - \nabla p + {\bf F}. \end{displaymath} (2.9)
\begin{displaymath} {\partial \rho \over \partial t} + \nabla \cdot (\rho {\bf v}) = 0, \end{displaymath} (2.8)

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