2 Ideal gas dynamics
2.2 Eulerian and Lagrangian viewpoints
In the Eulerian viewpoint we consider how fluid properties vary in time at a point that is fixed in space, i.e. attached to the (usually inertial) coordinate system. The Eulerian timederivative is simply the partial differential operator
In the Lagrangian viewpoint we consider how fluid properties vary in time at a point that moves with the fluid at velocity
$\boldsymbol{u}(\boldsymbol{x},t)$
. The Lagrangian time derivative is then
2.3 Material points and structures
A material point is an idealized fluid element, a point that moves with the bulk velocity
$\boldsymbol{u}(\boldsymbol{x},t)$
of the fluid. (Note that the true particles of which the fluid is composed have in addition a random thermal motion.) Material curves, surfaces and volumes are geometrical structures composed of fluid elements; they move with the fluid flow and are distorted by it.
An infinitesimal material line element
${\it\delta}\boldsymbol{x}$
(figure 1) evolves according to
It changes its length and/or orientation in the presence of a velocity gradient. (Since
${\it\delta}\boldsymbol{x}$
is only a timedependent vector rather than a vector field, the time derivative could be written as an ordinary derivative
$\text{d}/\text{d}t$
. The notation
$\text{D}/\text{D}t$
is used here to remind us that
${\it\delta}\boldsymbol{x}$
is a material structure that moves with the fluid.)
Figure 1. Examples of material line, surface and volume elements.
Infinitesimal material surface and volume elements can be defined from two or three material line elements according to the vector product and the triple scalar product (figure 1)
They therefore evolve according to
as follows from the above equations (exercise). The second result is easier to understand: the volume element increases when the flow is divergent. These equations are most easily derived using Cartesian tensor notation. In this notation the equation for
${\it\delta}\boldsymbol{S}$
reads
2.4 Equation of mass conservation
The equation of mass conservation,
has the typical form of a conservation law:
${\it\rho}$
is the mass density (mass per unit volume) and
${\it\rho}\boldsymbol{u}$
is the mass flux density (mass flux per unit area). An alternative form of the same equation is
If
${\it\delta}m={\it\rho}{\it\delta}V$
is a material mass element, it can be seen that mass is conserved in the form
2.5 Equation of motion
The equation of motion,
derives from Newton’s second law per unit volume with gravitational and pressure forces.
${\it\Phi}(\boldsymbol{x},t)$
is the gravitational potential and
$\boldsymbol{g}=\boldsymbol{{\rm\nabla}}{\it\Phi}$
is the gravitational field. The force due to pressure acting on a volume
$V$
with bounding surface
$S$
is
Viscous forces are neglected in ideal gas dynamics.
2.6 Poisson’s equation
The gravitational potential is related to the mass density by Poisson’s equation,
where
$G$
is Newton’s constant. The solution
generally involves contributions from both the fluid region
$V$
under consideration and the exterior region
$\hat{V}$
.
A nonselfgravitating fluid is one of negligible mass for which
${\it\Phi}_{\mathit{int}}$
can be neglected. More generally, the Cowling approximation
^{1}
consists of treating
${\it\Phi}$
as being specified in advance, so that Poisson’s equation is not coupled to the other equations.
2.7 Thermal energy equation
In the absence of nonadiabatic heating (e.g. by viscous dissipation or nuclear reactions) and cooling (e.g. by radiation or conduction),
where
$s$
is the specific entropy (entropy per unit mass). Fluid elements undergo reversible thermodynamic changes and preserve their entropy.
This condition is violated in shocks (see § 6.3).
The thermal variables
$(T,s)$
can be related to the dynamical variables
$(p,{\it\rho})$
via an equation of state and standard thermodynamic identities. The most important case is that of an ideal gas together with blackbody radiation,
where
$k$
is Boltzmann’s constant,
$m_{H}$
is the mass of the hydrogen atom,
${\it\sigma}$
is Stefan’s constant and
$c$
is the speed of light.
${\it\mu}$
is the mean molecular weight (the average mass of the particles in units of
$m_{H}$
), equal to
$2.0$
for molecular hydrogen,
$1.0$
for atomic hydrogen,
$0.5$
for fully ionized hydrogen and approximately
$0.6$
for ionized matter of typical cosmic abundances. Radiation pressure is usually negligible except in the centres of highmass stars and in the immediate environments of neutron stars and black holes. The pressure of an ideal gas is often written in the form
${\mathcal{R}}{\it\rho}T/{\it\mu}$
, where
${\mathcal{R}}=k/m_{H}$
is a version of the universal gas constant.
We define the first adiabatic exponent
which is related to the ratio of specific heat capacities
by (exercise)
where
can be found from the equation of state. We can then rewrite the thermal energy equation as
For an ideal gas with negligible radiation pressure,
${\it\chi}_{{\it\rho}}=1$
and so
${\it\Gamma}_{1}={\it\gamma}$
. Adopting this very common assumption, we write
2.9 Microphysical basis
It is useful to understand the way in which the fluid dynamical equations are derived from microphysical considerations. The simplest model involves identical neutral particles of mass
$m$
and negligible size with no internal degrees of freedom.
2.9.1 The Boltzmann equation
Between collisions, particles follow Hamiltonian trajectories in their sixdimensional
$(\boldsymbol{x},\boldsymbol{v})$
phase space:
The distribution function
$f(\boldsymbol{x},\boldsymbol{v},t)$
specifies the number density of particles in phase space. The velocity moments of
$f$
define the number density
$n(\boldsymbol{x},t)$
in real space, the bulk velocity
$\boldsymbol{u}(\boldsymbol{x},t)$
and the velocity dispersion
$c(\boldsymbol{x},t)$
according to
Equivalently,
The relation between velocity dispersion and temperature is
$kT=mc^{2}$
.
In the absence of collisions,
$f$
is conserved following the Hamiltonian flow in phase space. This is because particles are conserved and the flow in phase space is incompressible (Liouville’s theorem). More generally,
$f$
evolves according to Boltzmann’s equation,
The collision term on the righthand side is a complicated integral operator but has three simple properties corresponding to the conservation of mass, momentum and energy in collisions:
The collision term is local in
$\boldsymbol{x}$
(not even involving derivatives) although it does involve integrals over
$\boldsymbol{v}$
. The equation
$(\partial f/\partial t)_{c}=0$
has the general solution
with parameters
$n$
,
$\boldsymbol{u}$
and
$c$
that may depend on
$\boldsymbol{x}$
. This is the Maxwellian distribution.
2.9.3 Validity of a fluid approach
The essential idea here is that deviations from the Maxwellian distribution are small when collisions are frequent compared to the characteristic time scale of the flow. In higherorder approximations these deviations can be estimated, leading to the equations of dissipative gas dynamics including transport effects (viscosity and heat conduction).
The fluid approach breaks down if the mean flight time
${\it\tau}$
is not much less than the characteristic time scale of the flow, or if the mean free path
${\it\lambda}\approx c{\it\tau}$
between collisions is not much less than the characteristic length scale of the flow.
${\it\lambda}$
can be very long (measured in astronomical units or parsecs) in very tenuous gases such as the interstellar medium, but may still be smaller than the size of the system.
Some typical orderofmagnitude estimates:
Solartype star: centre
${\it\rho}\sim 10^{2}~\text{g}~\text{cm}^{3}$
,
$T\sim 10^{7}~\text{K}$
; photosphere
${\it\rho}\sim 10^{7}~\text{g}~\text{cm}^{3}$
,
$T\sim 10^{4}~\text{K}$
; corona
${\it\rho}\sim 10^{15}~\text{g}~\text{cm}^{3}$
,
$T\sim 10^{6}~\text{K}$
.
Interstellar medium: molecular clouds
$n\sim 10^{3}~\text{cm}^{3}$
,
$T\sim 10~\text{K}$
; cold medium (neutral)
$n\sim 10100~\text{cm}^{3}$
,
$T\sim 10^{2}~K$
; warm medium (neutral/ionized)
$n\sim 0.11~\text{cm}^{3}$
,
$T\sim 10^{4}~K$
; hot medium (ionized)
$n\sim 10^{3}10^{2}~\text{cm}^{3}$
,
$T\sim 10^{6}~K$
.
The Coulomb crosssection for ‘collisions’ (i.e. largeangle scatterings) between charged particles (electrons or ions) is
${\it\sigma}\approx 1\times 10^{4}(T/\text{K})^{2}~\text{cm}^{2}$
. The mean free path is
${\it\lambda}=1/(n{\it\sigma})$
.
Related examples (see appendix A): A.1–A.4.
4 Conservation laws, symmetries and hyperbolic structure
4.1 Introduction
There are various ways in which a quantity can be said to be ‘conserved’ in fluid dynamics or MHD. If a quantity has a density (amount per unit volume)
$q(\boldsymbol{x},t)$
that satisfies an equation of the conservative form
then the vector field
$\boldsymbol{F}(\boldsymbol{x},t)$
can be identified as the flux density (flux per unit area) of the quantity. The rate of change of the total amount of the quantity in a timeindependent volume
$V$
,
is then equal to minus the flux of
$\boldsymbol{F}$
through the bounding surface
$S$
:
If the boundary conditions on
$S$
are such that this flux vanishes, then
$Q$
is constant; otherwise, changes in
$Q$
can be accounted for by the flux of
$\boldsymbol{F}$
through
$S$
. In this sense the quantity is said to be conserved. The prototype is mass, for which
$q={\it\rho}$
and
$\boldsymbol{F}={\it\rho}\boldsymbol{u}$
.
A material invariant is a scalar field
$f(\boldsymbol{x},t)$
for which
which implies that
$f$
is constant for each fluid element, and is therefore conserved following the fluid motion. A simple example is the specific entropy in ideal fluid dynamics. When combined with mass conservation, this yields an equation in conservative form,
4.2 Synthesis of the total energy equation
Starting from the ideal MHD equations, we construct the total energy equation piece by piece.
Kinetic energy:
Gravitational energy (assuming initially that the system is nonselfgravitating and that
${\it\Phi}$
is independent of
$t$
):
Internal (thermal) energy (using the fundamental thermodynamic identity
$\text{d}e=T\,\text{d}sp\,\text{d}v$
):
Sum of these three:
The last term can be rewritten as
Using mass conservation:
Magnetic energy:
Total energy:
where
$h=e+p/{\it\rho}$
is the specific enthalpy and we have used the identity
$\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }(\boldsymbol{E}\times \boldsymbol{B})=\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\times \boldsymbol{E}\boldsymbol{E}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\times \boldsymbol{B}$
. Note that
$(\boldsymbol{E}\times \boldsymbol{B})/{\it\mu}_{0}$
is the Poynting vector, the electromagnetic energy flux density. The total energy is therefore conserved.
For a selfgravitating system satisfying Poisson’s equation, the gravitational energy density can instead be regarded as
$g^{2}/8{\rm\pi}G$
:
The total energy equation is then
It is important to note that some of the gravitational and magnetic energy of an astrophysical body is stored in the exterior region, even if the mass density vanishes there.
4.3 Other conservation laws in ideal MHD
In ideal fluid dynamics there are certain invariants with a geometrical or topological interpretation. In homentropic or barotropic flow, for example, vorticity (or, equivalently, circulation) and kinetic helicity are conserved, while, in nonbarotropic flow, potential vorticity is conserved (see Example A.2). The Lorentz force breaks these conservation laws because the curl of the Lorentz force per unit mass does not vanish in general. However, some new topological invariants associated with the magnetic field appear.
The magnetic helicity in a volume
$V$
with bounding surface
$S$
is defined as
where
$\boldsymbol{A}$
is the magnetic vector potential, such that
$\boldsymbol{B}=\boldsymbol{{\rm\nabla}}\times \boldsymbol{A}$
. Now
where
${\it\Phi}_{e}$
is the electrostatic potential. This can be thought of as the ‘uncurl’ of the induction equation. Thus
In ideal MHD, therefore, magnetic helicity is conserved:
However, care is needed because
$\boldsymbol{A}$
is not uniquely defined. Under a gauge transformation
$\boldsymbol{A}\mapsto \boldsymbol{A}+\boldsymbol{{\rm\nabla}}{\it\chi}$
,
${\it\Phi}_{e}\mapsto {\it\Phi}_{e}\partial {\it\chi}/\partial t$
, where
${\it\chi}(\boldsymbol{x},t)$
is a scalar field,
$\boldsymbol{E}$
and
$\boldsymbol{B}$
are invariant, but
$H_{m}$
changes by an amount
Therefore
$H_{m}$
is not uniquely defined unless
$\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{n}=0$
on the surface
$S$
.
Magnetic helicity is a pseudoscalar quantity: it changes sign under a reflection of the spatial coordinates. Indeed, it is nonzero only when the magnetic field lacks reflectional symmetry. It can also be interpreted topologically in terms of the twistedness and knottedness of the magnetic field (see Example A.10). Since the field is ‘frozen in’ to the fluid and deformed continuously by it, the topological properties of the field are conserved. The equivalent conserved quantity in homentropic or barotropic ideal gas dynamics (without a magnetic field) is the kinetic helicity
The crosshelicity in a volume
$V$
is
It is helpful here to write the equation of motion in ideal MHD in the form
using the relation
$\text{d}h=T\,\text{d}s+v\,\text{d}p$
. Thus
and so crosshelicity is conserved in ideal MHD in homentropic or barotropic flow.
Bernoulli’s theorem follows from the inner product of (4.25) with
$\boldsymbol{u}$
. In steady flow
which implies that the Bernoulli function
$(1/2)u^{2}+{\it\Phi}+h$
is constant along streamlines, but only if
$\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{F}_{m}=0$
(e.g. if
$\boldsymbol{u}\,\Vert \,\boldsymbol{B}$
), i.e. if the magnetic field does no work on the flow.
Related examples: A.10, A.11.
4.4 Symmetries of the equations
The equations of ideal gas dynamics and MHD have numerous symmetries. In the case of an isolated, selfgravitating system, these include:

(i) Translations of time and space, and rotations of space: related (via Noether’s theorem) to the conservation of energy, momentum and angular momentum.

(ii) Reversal of time: related to the absence of dissipation.

(iii) Reflections of space (but note that
$\boldsymbol{B}$
is a pseudovector and behaves oppositely to
$\boldsymbol{u}$
under a reflection).

(iv) Galilean transformations.

(v) Reversal of the sign of
$\boldsymbol{B}$
.

(vi) Similarity transformations (exercise): if space and time are rescaled by independent factors
${\it\lambda}$
and
${\it\mu}$
, i.e.
then (This symmetry requires a perfect gas so that the thermodynamic relations are scale free.)
In the case of a nonisolated system with an external potential
${\it\Phi}_{\mathit{ext}}$
, these symmetries (other than
$\boldsymbol{B}\mapsto \boldsymbol{B}$
) apply only if
${\it\Phi}_{\mathit{ext}}$
has them. However, in the approximation of a nonselfgravitating system, the mass can be rescaled by any factor
${\it\lambda}$
such that
(This symmetry also requires a perfect gas.)
4.5 Hyperbolic structure
Analysing the socalled hyperbolic structure of the equations of AFD is one way of understanding the wave modes of the system and the way in which information propagates in the fluid. It is fundamental to the construction of some types of numerical method for solving the equations. We temporarily neglect the gravitational force here, because in a Newtonian theory it involves instantaneous action at a distance and is not associated with a finite wave speed.
In ideal gas dynamics, the equation of mass conservation, the thermal energy equation and the equation of motion (omitting gravity) can be written as
and then combined into the form
where
is a fivedimensional ‘state vector’ and
$\unicode[STIX]{x1D63C}_{x}$
,
$\unicode[STIX]{x1D63C}_{y}$
and
$\unicode[STIX]{x1D63C}_{z}$
are the three
$5\times 5$
matrices
This works because every term in the equations involves a first derivative with respect to either time or space.
The system of equations is said to be hyperbolic if the eigenvalues of
$\unicode[STIX]{x1D63C}_{i}n_{i}$
are real for any unit vector
$\boldsymbol{n}$
and if the eigenvectors span the fivedimensional space. As will be seen in § 6.2, the eigenvalues can be identified as wave speeds, and the eigenvectors as wave modes, with
$\boldsymbol{n}$
being the unit wavevector, locally normal to the wavefronts.
Taking
$\boldsymbol{n}=\boldsymbol{e}_{x}$
without loss of generality, we find (exercise)
where
is the adiabatic sound speed. The wave speeds
$v$
are real and the system is indeed hyperbolic.
Two of the wave modes are sound waves (acoustic waves), which have wave speeds
$v=u_{x}\pm v_{s}$
and therefore propagate at the sound speed relative to the moving fluid. Their eigenvectors are
and involve perturbations of density, pressure and longitudinal velocity.
The remaining three wave modes have wave speed
$v=u_{x}$
and do not propagate relative to the fluid. Their eigenvectors are
The first is the entropy wave, which involves only a density perturbation but no pressure perturbation. Since the entropy can be considered as a function of the density and pressure, this wave involves an entropy perturbation. It must therefore propagate at the fluid velocity because the entropy is a material invariant. The other two modes with
$v=u_{x}$
are vortical waves, which involve perturbations of the transverse velocity components, and therefore of the vorticity. Conservation of vorticity implies that these waves propagate with the fluid velocity.
To extend the analysis to ideal MHD, we may consider the induction equation in the form
and include the Lorentz force in the equation of motion. Every new term involves a first derivative. So the equation of mass conservation, the thermal energy equation, the equation of motion and the induction equation can be written in the combined form
where
is now an eightdimensional ‘state vector’ and the
$\unicode[STIX]{x1D63C}_{i}$
are three
$8\times 8$
matrices, e.g.
We now find, after some algebra,
The wave speeds
$v$
are real and the system is indeed hyperbolic. The various MHD wave modes will be examined later (§ 5).
In this representation, there are two modes that have
$v=u_{x}$
and do not propagate relative to the fluid. One is still the entropy wave, which is physical and involves only a density perturbation. The other is the ‘
$\text{div}\boldsymbol{B}$
’ mode, which is unphysical and involves a perturbation of
$\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{B}$
(i.e. of
$B_{x}$
, in the case
$\boldsymbol{n}=\boldsymbol{e}_{x}$
). This must be eliminated by imposing the constraint
$\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{B}=0$
. (In fact the equations in the form we have written them imply that
$(\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{B})/{\it\rho}$
is a material invariant and could be nonzero unless the initial condition
$\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{B}=0$
is imposed.) The vortical waves are replaced by Alfvén waves with speeds
$u_{x}\pm v_{ax}$
.
4.6 Stress tensor and virial theorem
In the absence of external forces, the equation of motion of a fluid can usually be written in the form
where
$\unicode[STIX]{x1D64F}$
is the stress tensor, a symmetric secondrank tensor field. Using the equation of mass conservation, we can relate this to the conservative form of the momentum equation,
which shows that
$\unicode[STIX]{x1D64F}$
is the momentum flux density excluding the advective flux of momentum.
For a selfgravitating system in ideal MHD, the stress tensor is
or, in Cartesian components,
We have already identified the Maxwell stress tensor associated with the magnetic field. The idea of a gravitational stress tensor works for a selfgravitating system in which the gravitational field
$\boldsymbol{g}=\boldsymbol{{\rm\nabla}}{\it\Phi}$
and the density
${\it\rho}$
are related through Poisson’s equation
$\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{g}={\rm\nabla}^{2}{\it\Phi}=4{\rm\pi}G{\it\rho}$
. In fact, for a general vector field
$\boldsymbol{v}$
, it can be shown that (exercise)
In the magnetic case (
$\boldsymbol{v}=\boldsymbol{B}$
) the first term in the final expression vanishes, while in the gravitational case (
$\boldsymbol{v}=\boldsymbol{g}$
) the second term vanishes, leaving
$4{\rm\pi}G{\it\rho}\boldsymbol{g}$
, which becomes the force per unit volume,
${\it\rho}\boldsymbol{g}$
, when divided by
$4{\rm\pi}G$
.
The virial equations are the spatial moments of the equation of motion, and provide integral measures of the balance of forces acting on the fluid. The first moments are generally the most useful. Consider
Integrate this equation over a material volume
$V$
bounded by a surface
$S$
(with material invariant mass element
$\text{d}m={\it\rho}\,\text{d}V$
):
where we have integrated by parts using the divergence theorem. In the case of an isolated system with no external sources of gravity or magnetic field,
$\boldsymbol{g}$
decays proportional to
$\boldsymbol{x}^{2}$
at large distance, and
$\boldsymbol{B}$
decays faster. Therefore
$T_{ij}$
decays proportional to
$\boldsymbol{x}^{4}$
and the surface integral can be eliminated if we let
$V$
occupy the whole of space. We then obtain (after division by
$2$
) the tensor virial theorem
where
is related to the inertia tensor of the system,
is a kinetic energy tensor and
is the integrated stress tensor. (If the conditions above are not satisfied, there will be an additional contribution from the surface integral.)
The scalar virial theorem is the trace of this tensor equation, which we write as
Note that
$K$
is the total kinetic energy. Now
for a perfect gas with no external gravitational field, where
$U$
,
$W$
and
$M$
are the total internal, gravitational and magnetic energies. Thus
On the righthand side, only
$W$
is negative. For the system to be bound (i.e. not fly apart) the kinetic, internal and magnetic energies are limited by
In fact equality must hold, at least on average, unless the system is collapsing or contracting.
The tensor virial theorem provides more specific information relating to the energies associated with individual directions, and is particularly relevant in cases where anisotropy is introduced by rotation or a magnetic field. It has been used in estimating the conditions required for gravitational collapse in molecular clouds. A higherorder tensor virial method was used by Chandrasekhar and Lebovitz to study the equilibrium and stability of rotating ellipsoidal bodies (Chandrasekhar 1969).
5 Linear waves in homogeneous media
In ideal MHD the density, pressure and magnetic field evolve according to
Consider a magnetostatic equilibrium in which the density, pressure and magnetic field are
${\it\rho}_{0}(\boldsymbol{x})$
,
$p_{0}(\boldsymbol{x})$
and
$\boldsymbol{B}_{0}(\boldsymbol{x})$
. The above equations are exactly satisfied in this basic state because
$\boldsymbol{u}=\mathbf{0}$
and the time derivatives vanish. Now consider small perturbations from equilibrium, such that
${\it\rho}(\boldsymbol{x},t)={\it\rho}_{0}(\boldsymbol{x})+{\it\delta}{\it\rho}(\boldsymbol{x},t)$
with
${\it\delta}{\it\rho}\ll {\it\rho}_{0}$
, etc. The linearized equations are
By introducing the displacement
${\bf\xi}(\boldsymbol{x},t)$
such that
${\it\delta}\boldsymbol{u}=\partial {\bf\xi}/\partial t$
, we can integrate these equations to obtain
We have now dropped the subscript ‘0’ without danger of confusion.
(The above relations allow some freedom to add arbitrary functions of
$\boldsymbol{x}$
. At least when studying wavelike solutions in which all variables have the same harmonic time dependence, such additional terms can be discarded.)
The linearized equation of motion is
where the perturbation of total pressure is
The gravitational potential perturbation satisfies the linearized Poisson equation
We consider a basic state of uniform density, pressure and magnetic field, in the absence of gravity. Such a system is homogeneous but anisotropic, because the uniform field distinguishes a particular direction. The problem simplifies to
with
Owing to the symmetries of the basic state, planewave solutions exist, of the form
where
${\it\omega}$
and
$\boldsymbol{k}$
are the frequency and wavevector, and
$\tilde{{\bf\xi}}$
is a constant vector representing the amplitude of the wave. For such solutions, (5.7) gives
where we have changed the sign and omitted the tilde.
For transverse displacements that are orthogonal to both the wavevector and the magnetic field, i.e.
$\boldsymbol{k}\boldsymbol{\cdot }{\bf\xi}=\boldsymbol{B}\boldsymbol{\cdot }{\bf\xi}=0$
, this equation simplifies to
Such solutions are called Alfvén waves. Their dispersion relation is
Given the dispersion relation
${\it\omega}(\boldsymbol{k})$
of any wave mode, the phase and group velocities of the wave can be identified as
where
$\hat{\boldsymbol{k}}=\boldsymbol{k}/k$
. The phase velocity is that with which the phase of the wave travels, while the group velocity is that which the energy of the wave (or the centre of a wavepacket) is transported.
For Alfvén waves, therefore,
where
${\it\theta}$
is the angle between
$\boldsymbol{k}$
and
$\boldsymbol{B}$
.
To find the other solutions, we take the inner product of (5.10) with
$\boldsymbol{k}$
and then with
$\boldsymbol{B}$
to obtain first
and then
These equations can be written in the form
The ‘trivial solution’
$\boldsymbol{k}\boldsymbol{\cdot }{\bf\xi}=\boldsymbol{B}\boldsymbol{\cdot }{\bf\xi}=0$
corresponds to the Alfvén wave that we have already identified. The other solutions satisfy
which simplifies to
The two solutions
are called fast and slow magnetoacoustic (or magnetosonic) waves, respectively.
In the special case
${\it\theta}=0$
(
$\boldsymbol{k}\Vert \boldsymbol{B}$
), we have
together with
$v_{p}^{2}=v_{a}^{2}$
for the Alfvén wave. Note that the fast wave could be either
$v_{p}^{2}=v_{s}^{2}$
or
$v_{p}^{2}=v_{a}^{2}$
, whichever is greater.
In the special case
${\it\theta}={\rm\pi}/2$
(
$\boldsymbol{k}\bot \boldsymbol{B}$
), we have
together with
$v_{p}^{2}=0$
for the Alfvén wave.
The effects of the magnetic field on wave propagation can be understood as resulting from the two aspects of the Lorentz force. The magnetic tension gives rise to Alfvén waves, which are similar to waves on an elastic string, and are trivial in the absence of the magnetic field. In addition, the magnetic pressure affects the response of the fluid to compression, and therefore modifies the propagation of acoustic waves.
The phase and group velocity for the full range of
${\it\theta}$
are usually exhibited in Friedrichs diagrams
^{6}
(figure 4).
We can interpret:

(i) the fast wave as a quasiisotropic acoustictype wave in which both gas and magnetic pressure contribute;

(ii) the slow wave as an acoustictype wave that is strongly guided by the magnetic field;

(iii) the Alfvén wave as analogous to a wave on an elastic string, propagating by means of magnetic tension and perfectly guided by the magnetic field.
Related example: A.12.