\thesischapter{Wave Heating of the Chromosphere}

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\begin{center}
{\em If the Sun did not have a magnetic field it would be as uninteresting a star as most night time observers believe it to be}\\
\hspace{3.5cm} - R. Leighton
\end{center}
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\label{mhdstuff}
The majority of solar structure and phenomena are a result of the interaction of the solar plasma with the magnetic field (Osterbrock 1961). Such interactions are governed by a set of equations known collectively as magnetohydrodynamics (MHD). These are a combination of the equations of electric and magnetic fields, and those of fluid dynamics. Using simple approximations and combining linearised versions of these equations leads to wave dispersion relationships which describe the behaviour of a plasma in a magnetic field. This chapter contains a brief summary of these equations and the properties of the wave modes which arise from them in both a homogeneous and inhomogeneous plasma. 

\section{Maxwell's Equations}
The set of equations describing the behaviour of the electric field, {\bf E}, magnetic field, {\bf B} (actually {\bf B} is magnetic induction but is commonly referred to as magnetic field in astrophysics), and current density, {\bf j}, were first combined by Maxwell (1873). These are as follows. Firstly, Poisson's equation, 
\begin{equation}
\nabla.{\bf E} = \frac{\rho{^c}}{\varepsilon_0} \ ,
\end{equation}  
where $\rho^c$ is charge density and $\varepsilon_0$ is the permittivity of free space. This states that a net charge will act as a source for an electric field. Secondly, Faraday's law, 
\begin{equation}
\nabla\times{\bf E} = - \frac{\partial{\bf B}}{\partial t} \ ,
\end{equation}  
which states that a change in magnetic field with time will produce an electric field. Thirdly, Ampere's law,
\begin{equation}
\label{amplaw}
\nabla\times{\bf B} = \mu_0{\bf j} + \frac{1}{c^2}\frac{\partial{\bf E}}{\partial t} \ ,
\end{equation}  
which states that either a current, or time-varying electric field will produce a magnetic field. In this equation $\mu_0$ is the magnetic permeability of free space, related to the speed of light, c, by,
\begin{equation}
c^2 = \frac{1}{\mu_0\varepsilon_0} \ .
\end{equation}
Under the MHD approximation, whereby plasma velocities, {\bf v}, are much less than the speed of light, the second term on the RHS of Eqn.~\ref{amplaw} is ignored. Finally, Gauss' law, 
\begin{equation}
\nabla.{\bf B} = 0 \ ,
\end{equation}
such that there are no magnetic monopoles. In addition to the four Maxwell's equations above, Ohm's law states that, in a frame of reference moving with the plasma, the current density is proportional to the sum of the electric field due to the movement of the plasma (${\bf v} \times {\bf B}$) plus the electric field which would act on it at rest, {\bf E}, i.e.,
\begin{equation}
{\bf j} = \sigma({\bf E} + {\bf v}\times{\bf B}) \ ,
\end{equation}
where $\sigma$ is electrical conductivity. 

\subsection{The Induction Equation}
The electric field can be eliminated between Faraday's law and Ohm's law to give,
\begin{equation}
\frac{\partial{\bf B}}{\partial t} = \nabla\times({\bf v}\times{\bf B}) - \nabla\times\frac{{\bf j}}{\sigma} \ .
\end{equation}
This reduces, by Ampere's law, to,
\begin{equation}
\frac{\partial{\bf B}}{\partial t} = \nabla\times({\bf v}\times{\bf B}) - \nabla\times(\eta\nabla\times{\bf B}) \ ,
\end{equation}
where $\eta = 1/(\mu_0\sigma)$ is magnetic diffusivity. By simple use of a vector identity, using Gauss's law, and assuming constant{\footnotemark[1]} $\eta$ this further reduces to,
\footnotetext[1]{In fact $\eta$ varies with both temperature and density, from $\sim\!10^3$~m$^2$~s$^{-1}$ in the photosphere to $\sim\!1$~m$^2$~s$^{-1}$ in the corona (Priest 1982).}
\begin{equation}
\label{induction}
\frac{\partial{\bf B}}{\partial t} = \nabla\times({\bf v}\times{\bf B}) + \eta\nabla^2{\bf B} \ ,
\end{equation}
which is known as the induction equation. This states that a local change in the magnetic field is due to both convection and diffusion. The magnetic Reynolds number is the ratio of the convective to the diffusive term, 
\begin{equation}
R_m = \frac{\nabla\times({\bf v}\times{\bf B})}{\eta\nabla^2{\bf B}} \ ,
\end{equation}
and is an indication of the coupling between the plasma flow and the magnetic field. This can simply be approximated to, 
\begin{equation}
R_m = \frac{\mathrm{v}_0 l_0}{\eta} \ ,
\end{equation}
for length scale, $l_0$, and plasma speed, $\mathrm{v}_0$.

For large $R_m$ ($\gg 1$), as found in most astrophysical cases, the convective term dominates in Eqn.~\ref{induction}, and the field lines move as if they are frozen into the plasma ({\alf}, 1943) with typical timescale, $t_c = l_0 / \mathrm{v}_0$. For small $R_m$ ($\ll 1$), typically found in laboratory plasmas, the diffusion term dominates and flux `leaks' with typical ohmic diffusion timescales then given by $t_d = l^2/\eta$. For typical solar photospheric values ($\mathrm{v}_0\sim\!10$~m~s$^{-1}$, $\eta\sim\!10^3$~m$^2$~s$^{-1}$), $R_m$ becomes less than unity ($t_d\!\sim\!t_c$) for $l_0\sim\!100$~m, ohmic dissipation becomes important, and magnetic reconnection may occur.       

\section{Fluid Dynamics}
This section describes the three basic equations of fluid dynamics. In each case the equation is initially stated in its {\em material time derivative} ($D/Dt$) form, and changed into the {\em partial derivative} (${\partial}/{\partial t}$) form via,
\begin{equation}
\frac{D}{Dt} \equiv \frac{\partial}{\partial t} + {\bf v}.\nabla \ ,
\end{equation}
where {\bf v} is plasma velocity as before.

\subsection{Equation of Mass Continuity} 
The equation of mass conservation is, 
\begin{equation}
\label{fulleomc} 
\frac{D\rho}{Dt} + \rho\nabla.{\bf v} = 0 \ ,
\end{equation}
By use of a vector identity, Eqn.~\ref{fulleomc} may be expressed as, 
\begin{equation}
\label{eomc}
\frac{\partial\rho}{\partial t} + \nabla.\rho{\bf v} = 0 \ .
\end{equation}
This simply states that, at any point, the density increases if there is material flow into the region ($\partial\rho/\partial t > 0$ for $\nabla.\rho\!{\bf v} < 0$) and decreases with material flow away from the region (i.e., positive divergence). 

\subsection{Equation of Energy Conservation} 
The fundamental heat equation is given as, 
\begin{equation}
{\rho}T\frac{Ds}{Dt} = -L \ ,
\label{fund_heat}
\end{equation}
where s is entropy per unit mass, and simply states that the rate of increase of heat is due to $L$, the net effect of all sources and sinks of energy. By using the first law of thermodynamics it can be shown that Eqn.~\ref{fund_heat} can be rewritten in a concise form,
\begin{equation}
\label{par_eoec}
\frac{\rho^\gamma}{\gamma-1} \frac{D}{Dt} \left( \frac{p}{\rho^\gamma} \right) = - L \ ,
\end{equation}
where $p$ is pressure and $\gamma$ is the ratio of specific heats (polytropic index). For a thermally isolated plasma such that no heat is exchanged ($L=0$), Eqn.~\ref{par_eoec} reduces to,
\begin{equation}
\label{eoec_mtd}
\frac{D}{Dt} \left( \frac{p}{\rho^\gamma} \right) = 0 \ .
\end{equation} 
which may be expressed as,
\begin{equation}
\label{eoec}
\frac{\partial P}{\partial t} + {\bf v}.{\nabla}P -c_S^2 \left(\frac{\partial\rho}{\partial t} + {\bf v}.{\nabla}\rho\right) = 0 \ .
\end{equation} 

\subsection{Equation of Momentum Conservation}
\label{eomc_sec}
By Newton's second law the equation of motion may be written as,
\begin{equation}
\rho\frac{D{\bf v}}{Dt} = -\nabla p + {\bf j}\times{\bf B} + F_g \ ,
\end{equation}
where $F_g$ is the force due to gravity (viscosity is neglected). Thus, the force per unit volume (LHS) is equal to the summation of forces given by a pressure gradient, a Lorentz force, and gravity. This equation connects the fluid motions to Maxwell's equations via the ${\bf j}\times{\bf B}$ term. By assuming a gravitational force of the form $F_g = -\rho g ({\bf \hat{z}})$ (where ${\bf \hat{z}}$ is the outward normal from the solar surface) this becomes,
\begin{equation}
\label{eomom}
\rho\frac{D{\bf v}}{Dt} = -\nabla p + {\bf j}\times{\bf B} - \rho g {\bf \hat{z}} \ . 
\end{equation}

By simple use of Ampere's law (Eqn.~\ref{amplaw}) and a vector identity, the Lorentz force in Eqn.~\ref{eomom} reduces to,
\begin{equation}
\label{lorentz}
{\bf j}\times{\bf B}  =  \frac{({\bf B}.\nabla){\bf B}}{\mu_0} - \nabla\frac{B^2}{2\mu_0} \ .
\end{equation}
The first term on the RHS represents a force due to magnetic tension, parallel to {\bf B}, which results from the curvature of the magnetic field lines. The second term on the RHS is a scalar magnetic pressure force. The plasma $\beta$ parameter is the ratio of the the plasma pressure to this magnetic pressure and is given by,
\begin{equation}
\beta = \frac{2{\mu_0}p}{B^2} \ .
\end{equation}
So for $\beta > 1$ pressure dominates, whereas for $\beta < 1$ magnetism dominates.

When the plasma flow speed is less than the sound speed, {\alf} speed and free-fall speed, magnetohydrostatic equilibrium exists and the forces in Eqn.~\ref{eomom} balance. In a direction perpendicular to the field lines, {\bf n}, and substituting for the Lorentz force from Eqn.~\ref{lorentz}, Eqn.~\ref{eomom} reduces to,
\begin{equation}
\label{presbal}
\frac{\partial}{\partial n} \left(p + \frac{B^2}{2\mu_0}\right) = 0 \ .
\end{equation} 
Define a thin flux tube such that the internal and external pressures are $p_i$ and $p_e$ respectively, such that the {\em total} pressure inside and outside the tube is the same, i.e., 
\begin{equation}
p_e = p_i + \frac{B^2}{2\mu_0} \ .
\end{equation}
Hence $p_i$ will be less than $p_e$, the internal mass density will also be lower than the external mass density, and (assuming thermal balance) the flux tube will rise. Thus magnetic buoyancy will cause magnetic flux tubes in the convection zone to rise to the surface.    

\section{Waves}
Waves in a medium are the direct result of periodic perturbations. The basic modes of waves are classified by their restoring forces, which in a non-viscous, perfectly conducting plasma are gravity, pressure, and magnetism. As well as acting individually, these may act in three sets of pairs or all at once. This section contains a brief summary of the general dispersion relation for each mode of wave, followed by a qualitative description of the multi-force modes. For a more detailed description the reader is referred to any MHD textbook (e.g., Priest 1982, Choudhuri 1998) or Stein \& Leibacher (1974).

\subsection{Acoustic Velocity \& the Klein Gordon Equation}
\label{avkge}
Initially I will neglect the magnetic field, and concentrate on the effects of sound waves in a stratified atmosphere (i.e., set {\bf B}=0). Assume a simple adiabatic expansion of the form, 
\begin{equation}
p = A \rho^{\gamma} ,
\end{equation}
where A is a constant. Taking the gradient of both sides of this equation leads to,
\begin{equation}
\label{speedeqn}
\nabla p = c_S^2 \nabla\rho ,
\end{equation}
where the acoustic velocity, $c_S$, is given by,
\begin{equation}
c_S^2  = \frac{\gamma p}{\rho} = \frac{{\gamma}{K_B}T}{{\mu}m_p} \ ,
\end{equation}
where $K_B$ is the Boltzmann constant, ${\mu}$ is the mean molecular weight, and $m_p$ is the proton mass. Hence $c_S~{\propto}~T^{0.5}$ and varies from around 9~km~s$^{-1}$ in the photosphere to over 200~km~s$^{-1}$ in the corona. 

Under hydrostatic equilibrium conditions, Eqn.~\ref{eomom} becomes,
\begin{equation}
\label{eomom_b}
\nabla p = - \rho g {\bf \hat{z}} \ .
\end{equation}
Using the ideal gas equation, and for an isothermal medium stratified in the z axis, this can simply be integrated to obtain (Roberts 2003),
\begin{equation}
p(z) = p(0) e^{-\frac{z}{H}} \ ,
\end{equation}
\begin{equation}
\rho(z) = \rho(0) \frac{H(0)}{H(z)} e^{-\frac{z}{H}} \ ,
\end{equation}
where H is the pressure scale height,
\begin{equation}
H = \frac{p_0}{{\rho_o}g} = \frac{RT}{g} \ , 
\end{equation}
typically around 150~km in the photosphere. Hence pressure and density both fall off exponentially with height. Furthermore, by considering small temporal perturbations in the z axis, it can be shown that acoustic type waves will obey the Klein Gordon equation (Lamb 1909; Roberts 2003) whereby, 
\begin{equation}
\frac{{\partial}^{2}Q}{{\partial}t^{2}} - c_S^2 \frac{{\partial}^{2}Q}{{\partial}z^{2}} + \Omega^2 Q = 0 \ ,
\label{kg} 
\end{equation}
where $\Omega$ has frequency dimensions, and the initial perturbation, u, is directly proportional to the new term, Q, such that $|u|$~$\propto$~p$^{-0.5}|Q|$. Hence a typical photospheric velocity of around 1~km~s$^{-1}$ will rapidly increase to a value approaching the sound speed within 600~km (4 scale heights approximates to a 2$e$ increase), leading to the possibility of damping mechanisms (e.g., wave shocks (Section~\ref{heat_theor})).

The $\Omega$ term in the Klein Gordon equation (\ref{kg}) introduces a timescale onto the perturbations. Assume a solution to Eqn.~\ref{kg} of the form, 
\begin{equation}
Q(z,t) = Q e^{i(k_{z}z - {\omega}t)} \ ,
\end{equation}
where $\omega$ is the angular frequency ($=2\pi\nu$) and $k_z$ is the wavenumber ($=2\pi/\lambda$). Hence $\frac{\partial}{\partial t} \rightarrow -i\omega$ and $\nabla \rightarrow i{\bf k}$ in Eqn.~\ref{kg}, which becomes,
\begin{equation}
\label{ac_cutoff}
\omega^2 = k_z^2c_S^2 + \Omega^2 \ .
\end{equation}   
From Eqn.~\ref{ac_cutoff}, it is clear that real solutions only exist for values of $\omega$ greater than $\Omega$, otherwise waves are evanescent (i.e., $k_z < 0$). This {\em acoustic cut-off frequency} is defined as, 
\begin{equation}
\Omega = \frac{c_S}{2H} \ = 2\pi\nu_{ac} \,
\end{equation}
which leads to $\nu_{ac}~\sim~5$~mHz around temperature minimum.

The increased temperature in the solar interior forms a refractive boundary for acoustic waves. A second boundary is formed at the photosphere due to a sharp density gradient. Waves with a frequency above the acoustic cutoff can escape the photospheric boundary, otherwise they will be reflected back toward the resonant cavity and form a standing wave (and studied as p-modes in helioseismology).
 
It can be shown (e.g., Lamb 1932; Erd\'{e}lyi 1954) that the impulsive generation of a wave will lead to a wavefront propagating with velocity $c_S$. Ahead of this wavefront the medium will be at rest, but the trailing wake of the wavefront will be oscillating at the characteristic frequency, $\Omega$. In addition the amplitude of this oscillation will decay in time as the wavefront moves through the medium. This is directly analogy with the inhomogeneous case (Section~\ref{inhomo}).

\subsection{Acoustic, Gravity, \& Acoustic-Gravity Waves}
Assume a plasma with no magnetic field ({\bf B}=0) at rest with constant density, $\rho_0$, undergoes a small perturbation (subscripts 1) resulting in a plasma flow such that, \\
\indent$\rho$ = $\rho_0$ + $\rho_1$, \\
\indent{\bf v} = {\bf v}$_1$, \\
Differentiating Eqn.~\ref{eomom} with respect to time, substituting linearised forms of Eqn.~\ref{eomc} and Eqn.~\ref{eoec}, and assuming an isothermal, gravitationally stratified atmosphere as before leads to a generalised wave equation, 
\begin{equation}
\label{acos_grav}
\frac{\partial^2{\bf v}_1}{\partial t^2} - c_S^2\nabla(\nabla.{\bf v}_1) + (\gamma -1)g{\bf{\hat{z}}}(\nabla.{\bf v}_1) + g\nabla{\bf v}_{1z} = 0 \ .
\end{equation}
Once again plane wave solutions for {\bf v}$_1$ exist of of the form,
\begin{equation}
{\bf v}_1 ({\bf r},t) = {\bf v}_1 e^{i({\bf k}{\bf .}{\bf r} - {\omega}t)} \ ,
\end{equation}
where propagation is along the wavevector {\bf k} ($=k{\bf{\hat{k}}}$). Hence, as before,  $\frac{\partial}{\partial t} \rightarrow -i\omega$ and $\nabla \rightarrow i{\bf k}$ in Eqn.~\ref{acos_grav}, which becomes,
\begin{equation}
\label{disrel_aco_grav}
\omega^2{\bf v}_1 - c_S^2{\bf k}({\bf k}.{\bf v}_1) - i(\gamma-1)g{\bf{\hat{z}}}({\bf k}.{\bf v}_1) + ig{\bf k}{\bf v}_{1z} = 0 \ , 
\end{equation}
which forms the basis for discussing acoustic and gravity waves.

\subsubsection{Acoustic waves}
When pressure is the only restoring force (i.e., g=0), Eqn.~\ref{disrel_aco_grav} becomes,
\begin{equation}
\omega^2{\bf v}_1 - c_S^2{\bf k}({\bf k}.{\bf v}_1) = 0 \ , 
\end{equation}
and assuming a compressible gas (i.e., $\nabla . {\bf v}_1 \neq 0$), this simplifies by taking the square root of the dot product with {\bf k} to,
\begin{equation}
\omega= kc_S ,
\end{equation} 
the dispersion relation for acoustic waves. Hence acoustic waves are longitudinal ({\bf v}$_1$ is along {\bf k}), non-dispersive (phase velocity v$_p$ = $\frac{\omega}{k}$ = $c_S$, and group velocity v$_g$ = $\frac{d\omega}{dk}$ = $c_S$ are equal), and isotropic (phase speed does not vary with {\bf k}). 

\subsubsection{Gravity Waves}
When gravity is the only restoring force, Eqn.~\ref{disrel_aco_grav}, after taking the dot product with {\bf k} and with ${\bf{\hat{z}}}$, and some rearrangement becomes, 
\begin{equation}
\omega^2c_S^2 = (\gamma - 1)g^2(1-\frac{k_z^2}{k^2}) \ ,
\end{equation}
where $k_z$ is the component of {\bf k} along {\bf z}, and which further simplifies to, 
\begin{equation}
\label{gravityeqn}
\omega = \omega_{BV} sin \theta_z \ ,
\end{equation}
where $\theta_z$ is the angle between {\bf k} and {\bf z}, and $\omega_{BV}$ is the Brunt-V\"ais\"al\"a frequency, 
\begin{equation}
\label{gravityeqn2}
\omega_{BV}^2 = \frac{(\gamma - 1)g^2}{c_S^2} \ .
\end{equation}
It is clear from Eqn.~\ref{gravityeqn} that gravity waves are dispersive (v$_p$ varies with wavelength), and anisotropic (v$_p$ varies with {\bf k}). In addition gravity waves can only propagate with angular frequencies less than $\nu_{BV}$ and cannot propagate vertically ($\theta~\neq~0$). The group velocity is also perpendicular to the phase velocity such that when waves are upwardly-propagating, energy is downwardly-propagating.   

\subsubsection{Acoustic-Gravity Waves}    

\begin{figure}
\begin{center}
\epsfig{figure=propdiag.eps,width=5in}
\caption[Acoustic-gravity dispersion diagram]{Acoustic-gravity dispersion diagram (courtesy of Paul Cally). Waves are propagating in the acoustic and gravity dominated regimes. Between these, waves are evanescent.}
\label{AG}
\end{center}
\end{figure}

Under the combined forces of pressure and gravity, the two modes remain distinct. Figure~\ref{AG} shows the dispersion relation for waves propagating in the vertical direction (i.e., along the z axis). Acoustic waves propagate for $\omega > \omega_{ac}$, with group and phase velocity greater and less than the sound speed respectively, and approaching zero and infinity at $\omega_{ac}$ respectively. At high frequencies, the waves are purely acoustic. Gravity waves are propagating for $\omega < \omega_{BV}$, with phase and group velocity less than the sound speed. For large $k_x$, the waves are purely gravitational. The acoustic cut-off is larger than the Brunt-V\"ais\"al\"a frequency by a factor,
\begin{equation}
\frac{\omega_{ac}}{\omega_{BV}} = \frac{c_S}{2H} \times \frac{c_S}{(\gamma - 1)^{1/2}g} = \frac{\gamma}{2(\gamma - 1)^{1/2}} \ ,
\end{equation}
which is $\sim 1.02$ for $\gamma = 5/3$. Between the acoustic and gravity dominated regimes, waves are evanescent.

\subsection{Magnetic Waves}
By neglecting gravity, but now including magnetism, a plasma at rest with constant density, $\rho_o$, and uniform magnetic field, ${\bf B}_0$, undergoes a small perturbation (subscripts 1) resulting in a plasma flow such that \\
\indent{\bf B} = ${\bf B}_0$ + ${\bf B}_1$ \\
\indent$\rho$ = $\rho_0$ + $\rho_1$ \\
\indent{\bf v} = {\bf v}$_1$ \\
Differentiating Eqn.~\ref{eomom} with respect to time, and substituting linearised forms of Eqns.~\ref{induction} (with a large $R_m$), ~\ref{eomc}~\&~\ref{eoec} leads to a second generalised wave equation, 
\begin{equation}
\label{genwaveeqn2}
\frac{\partial^2{\bf v}_1}{\partial t^2} - c_S^2\nabla(\nabla.{\bf v}_1) -  (\nabla\!\times\!(\nabla\!\times({\bf v}_1\!\times\!{\bf B_0})))\!\times\!\frac{\bf B_0}{\mu\rho_0} = 0 \ . 
\end{equation}
Assuming a plane wave solution, as for Eqn.~\ref{acos_grav}, leads to,
\begin{equation}
\label{disrel}
\omega^2{\bf v}_1 - c_S^2{\bf k}({\bf k}.{\bf v}_1) - ({\bf k}\!\times\!({\bf k}\!\times\!({\bf v}_1\!\times\!{\bf B_0})))\!\times\!\frac{\bf B_0}{\mu\rho_0} = 0 \ .
\end{equation}

\subsubsection{{\alf} Waves}
As discussed in Section~\ref{eomc_sec} the Lorentz force can be interpreted as the sum of a magnetic tension ($\mathrm{B}^2 / \mu_0$) and a magnetic pressure ($\mathrm{B}^2/2\mu_0$). By analogy with a string under tension, this will permit transverse waves with an {\alf} velocity, v$_A$, given by,
\begin{equation}
\label{alfspeed}
\mathrm{v}_A = \left(\frac{tension}{density}\right)^{1/2} = \left(\frac{\mathrm{B}^2}{\mu_0\rho}\right)^{1/2} \ .
\end{equation}
When the magnetic field is the only restoring force, Eqn.~\ref{disrel} reduces to,
\begin{equation}
\label{alfeqn}
\omega^2{\bf v}_1 = ({\bf k}\!\times\!({\bf k}\!\times\!({\bf v}_1\!\times\!{\bf \hat{B}_0})))\!\times\!\mathrm{v}_A^2\! {\bf\hat{B}_0} \ ,
\end{equation} 
which can be reduced by vector identity to,
\begin{equation}
({\bf k}.{\bf v}_1)(\omega^2 - k^2\mathrm{v}_A^2) = 0 \ ,
\end{equation}
which has two distinct solutions. For {\bf k}.{\bf v}$_1$ = 0, Eqn.~\ref{alfeqn} (after multiplying out the vector products) becomes
\begin{equation}
\label{alfeqn2}
\omega = k\mathrm{v}_Acos\theta_B \ ,
\end{equation}
where $\theta_B$ is the angle between {\bf k} and ${\bf B_0}$. This is the shear {\alf} wave, with a phase velocity which is just the {\alf} velocity along the magnetic field and zero perpendicular to it, and a group velocity which is always directed along the magnetic field. The {\bf k}.{\bf v}$_1$ = 0 property means that the velocity is perpendicular to the direction of propagation, hence the waves are transverse. The second solution, $\omega$ = $k$v$_A$, is the compressional {\alf} wave, which propagates isotropically.

\subsubsection{Magneto-Acoustic Waves}    
By taking the scale product of Eqn.~\ref{disrel} with {\bf k} and {\bf B}$_0$ in turn leads to a pair of equations which combine to give, 
\begin{equation}
\label{magaco}
\omega^4 - \omega^2(c_S^2 + \mathrm{v}_A^2){\bf k}^2 + c_S^2\mathrm{v}_A^2{\bf k}^4 cos^2\theta_B = 0 \ ,
\end{equation}
and (for {\bf k}.{\bf v}$_1$ = 0) the {\alf} wave of Eqn.~\ref{alfeqn2}. As Eqn.~\ref{magaco} is quadratic in ($\omega^2/k^2$) the solutions are simply, 
\begin{equation}
\label{magaco2}
\left({\frac{\omega}{k}}\right)^2 = \frac{1}{2}(c_S^2 + \mathrm{v}_A^2) \pm \frac{1}{2} [ (c_S^2 + \mathrm{v}_A^2)^2 - 4 c_S^2\mathrm{v}_A^2cos^2\theta_B ]^{1/2} \ ,
\end{equation}
yielding two outwardly propagating waves termed the fast-mode and the slow-mode (defined by their phase velocity) in addition to the {\alf} wave. The characteristics of each type of wave in a strong and weak magnetic field are summarised in Table~\ref{mag_aco}. Essentially, the fast- and slow-mode waves may be regarded as a sound wave modified by the magnetic field, and an {\alf} wave modified by the pressure. In the limits of {\bf B} = 0 or $\rho$ = 0, the fast-mode becomes a sound wave or {\alf} wave respectively and the slow-mode vanishes. 

\begin{table}
\begin{center}
\caption[Magneto-acoustic wave modes]{Summary of magneto-acoustic modes. Direction of propagation, and dominant restoring force and velocity of waves with strong and weak magnetic fields are given. (after Stein \& Leibacher 1974)}
\label{mag_aco}
\begin{tabular}{llcc} 
\hline
\hline
Mode		& Propagation 	& \multicolumn{2}{c}{Magnetic Field} \\
of Wave		& Direction	& Strong & Weak \\	
\hline
{\alf}		& along {\bf B}	& \multicolumn{2}{c}{Magnetic Tension, v$_A$} \\ \\
Fast		& Isotropic	& Magnetic Pressure	& Gas Pressure	\\
		&		& $\sim$v$_A$		& $\sim$c$_S$	\\
Slow		& approximately	& Gas Pressure		& Magnetic Tension	\\
		& along {\bf B}	& $\sim$c$_S$		& $\sim$v$_A$ \\
\hline
\end{tabular}
\end{center}   
\end{table}

\subsubsection{Magneto-Gravity Waves}    
Under the combined forces of magnetism and gravity the two modes again remain distinct. Gravity waves can propagate at frequencies less than $\omega_{BV}$ at small horizontal wavevector (essentially the Lorentz force opposes the transverse variation), and at frequencies greater than v$_A^2{\bf k}_x^2$ (Lou 1995). There are also a pair of upward- and downward- propagating waves at frequencies greater and less than $\omega_{BV}$ respectively.

\subsubsection{Magneto-Acoustic-Gravity Waves}
When all three forces act together the behaviour is much more complicated. In general, at high frequencies ($\omega > \omega_{BV}$) the modes are similar to the fast and slow mode magneto-acoustic waves, with the frequency cut-off dependant on the magnetic field. At low frequencies, the gravity mode is cut-off at large $k_x$ by the magnetic field (as in magneto-gravity), and at small $k_x$ by pressure (as in acoustic-gravity waves) (Stein \& Leibacher 1974).

\section{Inhomogeneous Plasmas}
\label{inhomo}
For a homogeneous plasma, the velocity, $\mathrm{v_M}$, of magneto-acoustic wave propagation normal to the magnetic field, is obtained by analogy with Eqn.~\ref{presbal} and Eqn.~\ref{speedeqn}, as,
\begin{equation}
\nabla\left( p + \frac{\mathrm{B}_0^2}{2\mu_0}\right) = \mathrm{v}_M^2 \nabla\!\rho \ ,
\end{equation}
which reduces to (Narain \& Agarwal 1994),
\begin{equation}
\mathrm{v}_M^2 = c_S^2 + \mathrm{v}_A^2 \ .
\end{equation}
In general, MHD waves in a homogeneous plasma may be anisotropic, but will always be non-dispersive.

Although MHD waves are extremely complex in the chromosphere (Rosenthal {\etal} 2002), much progress has been made in the study of thin flux tubes in this region of the atmosphere (e.g., Spruit 1981a; 1981b). This is achieved by assuming magnetic field lines are confined in thin tubes creating an inhomogeneous plasma in which various waves modes may exist. These wave modes are dispersive, which will allow for formation of wave packets and dissipation. 
\begin{figure}
\begin{center}
\epsfig{figure=modes.eps,width=5in}
\caption[Wave modes in a flux tube]{The three modes of waves in a thin flux tube. Left to Right: The sausage (longitudinal) mode, the kink (transverse) mode and the {\alf} (rotational) mode. In each case the magnetic field is directed along the z axis (Narain \& Agarwal 1994)}
\label{tube_modes}
\end{center}
\end{figure}

The three basic modes for waves in a flux tube are described in Figure~\ref{tube_modes}. The sausage-mode wave is a longitudinal wave whereby the tube cross-section expands and contracts due to compressions and rarefactions of the gas. These, acoustic-like, waves propagate along the tube with characteristic speed, $c_{\lambda}$, given by (Roberts 2000),
\begin{equation}
\frac{1}{c_{\lambda}^2} = \frac{1}{c_S^2} + \frac{1}{\mathrm{v}_A^2} \ , 
\end{equation}   
which can be also be written as (Kalkofen 1997), 
\begin{equation}
\label{longwave2}
c_{\lambda}^2 = c_S^2 \frac{2}{2+\gamma\beta} \ .
\end{equation}   

The kink-mode wave is a transverse wave whereby the field lines in the tube are displaced laterally. It can be thought of as an {\alf} wave which is reduced in speed by the inertia of the tube surroundings. It has a characteristic speed, $c_K$, given by (Roberts 2000), 
\begin{equation}
\label{kinkwave1}
c_K^2 = \frac{\rho_0\mathrm{v}_A^2 + \rho_e\mathrm{v}_{Ae}^2}{\rho_0 + \rho_e} \ ,
\end{equation}
where subscript $e$ refers to external parameters (i.e., outside the flux tube). For an isolated flux tube such as found under photospheric  conditions, $\rho_0\mathrm{v}_A^2~\gg~ \rho_e\mathrm{v}_{Ae}^2$, and Eqn.~\ref{kinkwave1} can be expressed as (Kalkofen 1998),
\begin{equation}
\label{kinkwave2}
c_K^2 = c_S^2 \frac{2}{\gamma(1+2\beta)} \ .
\end{equation}

Both modes follow a dispersion relation similar to Eqn.~\ref{ac_cutoff}, of the form (Spruit \& Roberts 1983),
\begin{equation}
\omega^2 = c_P^2 {\bf k}^2 + \Omega^2 \ ,
\end{equation}
where $c_P$ is the mode's speed of propagation ($c_K$ or $c_{\lambda}$), and $\Omega$ is the mode cut-off frequency. For each mode the wave will be propagating for $\omega > \Omega$ (i.e., ${\bf k}^2 > 0$) and evanescent for $\omega < \Omega$ (i.e., ${\bf k}^2 < 0$). The cut-off frequencies are given in Kalkofen (1998) as, 
\begin{equation}
{\nu_{\lambda}} = {\nu_{ac}}\sqrt{\frac{63+48\beta}{60+50\beta}} \ ,
\end{equation} 
\begin{equation}
{\nu_{k}} = {\nu_{ac}}\sqrt{\frac{1}{2\gamma(1+2\beta)}} \ . 
\end{equation} 
As in Section~\ref{avkge}, impulsive wave generation will result in a wave front moving at the respective tube speed, with a trailing wake which will oscillate at the respective cut-off frequency. The third mode is the torsional {\alf} wave, which is a rotational oscillation of the tube. 

The flux tubes will spread horizontally with height, due to the pressure stratification, so that somewhere in the high chromosphere they fill out to occupy the whole space and create the magnetic canopy. In the canopy the field can once again be thought of as homogeneous, the torsional wave goes over into the {\alf} wave and the sausage and kink waves go over into the slow and fast waves. Furthermore non-linear mode conversion (mode-coupling) is expected to occur in the canopy.

%\section{Standing Waves Versus Travelling waves}
%In general wave-based chromosphere heating theories consist of waves generated in the photosphere (or sub-photospheric) which travel up through the chromosphere before dissipating their energy. However some debate exists as to the existence of standing waves or travelling waves. The temperature minimum and onset of the transition region may 

%According to linear theory (Mihalis \& Mihalas, 1984), waves excited in the low chromosphere will produce a continuum intensity signal which precedes both the transition region intensity and doppler shift signals by amound which depends on both frequency of the wave, and the fraction of reflection. For total reflection at the transition wave (i.e., a standing wave), the doppler shift signal for each ion will be 90$\deg$ out of phase with the intensity signal. The signals between ions will be either in phase or 180$\deg$ out of phase. For only partial reflection, the phase shift between the doppler and intensity signals will vary locations of signal and amount of reflection. For a propagating wave, the phase shift between different ion intensity signals will be directly proportional to the frequency,
%\begin{equation}
%v_p = \frac{\delta z}{\delta \phi}\nu \,
%\end{equation}
%where $\delta z$ is the differnce in height of formation of the two lines, and $\delta \phi$ is the phase difference at frequency $\nu$. 
%Hence a plot of phase difference against frequency will yield a straight line, and with a knowledge of the difference in height of formation of the two ions, the gradient of the line yields the phase velocity. 
       
%MENTION PROPAGATING / STANDING WAVE PAPERS

\section{Heating Theories}
\label{heat_theor}
The most important factors in any heating theory involving waves are initially creating a wave and then transferring the energy to the upper chromosphere where the energy has to be deposited. These factors are generally discussed as follows.
\newline$\bullet$ Generation  \newline
Flux tube waves can be generated at the photosphere in a number of ways: transverse buffeting by granules can create kink waves; pressure changes can create sausage waves; twists in the tube may create torsional waves.
\newline$\bullet$ Propagation \newline
Only oscillations with frequencies above the respective cut-off will propagate, otherwise waves will be evanescent.
\newline$\bullet$ Dissipation \newline
When waves reach the required height in the atmosphere the mechanical energy has to be transferred to heat energy in the surrounding plasma. This can be achieved in a variety of ways, the most common of which is by the wave shocking.

\begin{figure}
\begin{center}
\epsfig{figure=shock.eps,width=5in}
\caption[Formation of a shock wave]{A sinusoidal wave changes shape with increasing height in a stratified atmosphere. As the compressed regions travel faster, a wave shock front is gradually formed. (Bird 1964)}
\label{shock}
\end{center}
\end{figure}

For longitudinal modes, the waves cannot directly heat the surrounding gas, instead they cause the plasma to vibrate. As they move up through the atmosphere $\rho$ decreases and the wave amplitude, V, increases so the total wave energy, $\rho$V$^2$, remains constant. In effect, the sinusoidal form of the original wave is distorted such that compressed regions travel faster than rarefied regions. Eventually, when the amplitude reaches the sound velocity a shock front is formed (Figure~\ref{shock}). The rise in both density and temperature behind the shock wave increases the rate of radiation, hence creating an important dissipation mechanism. This form of heating is found to be efficient in the lower chromosphere for acoustic waves (Phillips 1992). In the presence of a magnetic field shocks are much more complicated but, similar to acoustic waves, MHD waves in general shock when any disturbance within the wave moves faster than the characteristic wave speed. In recent years further theories (e.g., phase mixing, resonant absorbtion) have also been proposed as possible heating mechanisms.

Throughout Chapters~4 -- 6 I discuss various AC chromospheric heating theories based on the equations and wave modes described above. The generation, propagation and dissipation of different wave modes lead to different oscillations. The search for these oscillations gives an indication of which wave mode, and hence which heating mechanism, may be present in different parts of the chromosphere. The observations used to search for these oscillations are described in the next chapter. 
