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Moment equations

In fluid descriptions of plasmas (see plasma modeling and magnetohydrodynamics (MHD)) one does not consider the velocity distribution. This is achieved by replacing $f({\vec {r}},{\vec {v}},t)$ with plasma moments such as number density, $n$, mean velocity, $\mathbf {u}$ and pressure, $\mathbf {p}$ . They are named plasma moments because the nth moment of $f$ can be found by integrating $v^{n}f$ over velocity. These variables are only functions of position and time, which means that some information is lost. In multifluid theory, the different particle species are treated as different fluids with different pressures, densities and flow velocities. The equations governing the plasma moments are called the moment or fluid equations.

Below the two most used moment equations are presented (in SI units). Deriving the moment equations from the Vlasov equation requires no assumptions about the distribution function.

Continuity equation

The continuity equation describes how the density changes with time. It can be found by integration of the Vlasov equation over the entire velocity space.

$\int {\frac {\mathrm {d} }{\mathrm {d} t}}fd^{3}v=\int \left({\frac {\partial }{\partial t}}f+({\vec {v}}\cdot \nabla _{r})f+({\vec {a}}\cdot \nabla _{v})f\right)d^{3}v=0$

After some calculations, one ends up with

${\frac {\partial }{\partial t}}n+\nabla \cdot (n\mathbf {u} )=0$.

The particle density $n$, and the average velocity $\mathbf {u}$, are zeroth and first order moments:

$n=\int fd^{3}v$

$\mathbf {u} =\int {\vec {v}}fd^{3}v$

Momentum equation

The rate of change of momentum of a particle is given by the Lorentz equation:

$m{\frac {\mathrm {d} {\vec {v}}}{\mathrm {d} t}}=q({\vec {E}}+{\vec {v}}\times {\vec {B}})$

By using this equation and the Vlaslov Equation, the momentum equation for each fluid becomes

$mn{\frac {\mathrm {d} }{\mathrm {d} t}}\mathbf {u} =-\nabla \cdot \mathbf {p} +qn{\vec {E}}+qn\mathbf {u} \times {\vec {B}}$,

where $\mathbf {p}$ is the pressure tensor. The total time derivative is

${\frac {\mathrm {d} }{\mathrm {d} t}}={\frac {\partial }{\partial t}}+(\mathbf {u} \cdot \nabla )$.

The pressure tensor is defined as the mass density times the covariance matrix of the velocity:

$p_{ij}=mn\int (v_{i}-u_{i})(v_{j}-u_{j})fd^{3}v$.

The frozen-in approximation

As for ideal MHD, the plasma can be considered as tied to the magnetic field lines when certain conditions are fulfilled. One often say that the magnetic field lines are frozen into the plasma. The frozen-in conditions can be derived from Vlasov equation.

We introduce the scales $T$, $L$ and $V$ for time, distance and speed respectively. They represent magnitudes of the different parameters which give large changes in $f$. By large we mean that

${\frac {\partial f}{\partial t}}T\sim f\quad |{\frac {\partial f}{\partial {\vec {r}}}}|L\sim f\quad |{\frac {\partial f}{\partial {\vec {v}}}}|V\sim f.$

We then write

$t^{\prime }={\frac {t}{T}}\quad {\vec {r}}^{\prime }={\frac {\vec {r}}{L}}\quad {\vec {v}}^{\prime }={\frac {\vec {v}}{V}}.$

Vlasov equation can now be written

${\frac {1}{T}}{\frac {\partial f}{\partial t^{\prime }}}+{\frac {V}{L}}{\vec {v}}^{\prime }\cdot {\frac {\partial f}{\partial {\vec {r}}^{\prime }}}+{\frac {q}{mV}}({\vec {E}}+V{\vec {v}}^{\prime }\times {\vec {B}})\cdot {\frac {\partial f}{\partial {\vec {v}}^{\prime }}}=0.$

So far no approximations have been done. To be able to proceed we set $V=R\omega _{g}$, where $\omega _{g}=qB/m$ is the qyro frequency and R is the gyroradius. By dividing with $\omega _{g}$, we get

${\frac {1}{\omega _{g}T}}{\frac {\partial f}{\partial t^{\prime }}}+{\frac {R}{L}}{\vec {v}}^{\prime }\cdot {\frac {\partial f}{\partial {\vec {r}}^{\prime }}}+({\frac {\vec {E}}{VB}}+{\vec {v}}^{\prime }\times {\frac {\vec {B}}{B}})\cdot {\frac {\partial f}{\partial {\vec {v}}^{\prime }}}=0$

If $1/\omega _{g}<<T$ and $R<<L$, the two first terms will be much less than one since $\partial f/\partial t^{\prime }\sim 1$, $v^{\prime }\lesssim 1$ and $\partial f/\partial {\vec {r}}^{\prime }\sim 1$ due to the definitions of $T$, $L$ and $V$ above. Since the last term is of the order of one, we can neglect the two first terms and write

$({\frac {\vec {E}}{VB}}+{\vec {v}}^{\prime }\times {\frac {\vec {B}}{B}})\cdot {\frac {\partial f}{\partial {\vec {v}}^{\prime }}}\approx 0\Rightarrow ({\vec {E}}+{\vec {v}}\times {\vec {B}})\cdot {\frac {\partial f}{\partial {\vec {v}}}}\approx 0$

This equation can be decomposed into a field aligned and a perpendicular part:

${\vec {E}}_{||}{\frac {\partial f}{\partial {\vec {v}}_{||}}}+({\vec {E}}_{\perp }+{\vec {v}}\times {\vec {B}})\cdot {\frac {\partial f}{\partial {\vec {v}}_{\perp }}}\approx 0$

The next step is to write ${\vec {v}}={\vec {v}}_{0}+\Delta {\vec {v}}$, where

${\vec {v}}_{0}\times {\vec {B}}=-{\vec {E}}_{\perp }$

It will soon be clear why this is done. With this substitution,

${\vec {E}}_{||}{\frac {\partial f}{\partial {\vec {v}}_{||}}}+\Delta {\vec {v}}\cdot {\frac {\partial f}{\partial {\vec {v}}_{\perp }}}\approx 0$

If the parallel electric field is small,

$(\Delta {\vec {v}}_{\perp }\times {\vec {B}})\cdot {\frac {\partial f}{\partial {\vec {v}}_{\perp }}}\approx 0$

This equation means that the distribution is gyrotropic. The mean velocity of a gyrotropic distribution is zero. Hence, ${\vec {v}}_{0}$ is identical with the mean velocity, $\mathbf {u}$, and we have

${\vec {E}}+\mathbf {u} \times {\vec {B}}\approx 0$

To summarize, the gyro period and the gyro radius must be much smaller than the typical times and lengths which give large changes in the distribution function. The gyro radius is often estimated by replacing $V$ with the thermal_velocity or the Alfvén_velocity. In the latter case $R$ is often called the inertial length. The frozen-in conditions must be evaluated for each particle species separately. Because electrons have much smaller gyro period and gyro radius than ions, the frozen-in conditions will more often be satisfied.

Extensions to magnetohydrodynamics

Collisionless MHD

MHD is also often used for collisionless plasmas. In that case the MHD equations are derived from the Vlasov equation<ref name="space">.