deletions | additions       diff --git a/Local Analysis.tex b/Local Analysis.tex  index 035281b..e101336 100644  --- a/Local Analysis.tex  +++ b/Local Analysis.tex 
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Many of the properties of magnetohydrodynamic waves can be understood from a local analysis for high wavenumbers ${\bf k}$, in which $k r \gg 1$ and $k H \gg 1$. This analysis is technically only valid for large horizontal wavenumbers, $k_\perp r \gg 1$, whereas dipole oscillation modes have $k_\perp r = \sqrt{2}$. We must therefore be careful in extrapolating to global scale waves, especially for arbitrary magnetic field geometries.   In the adiabatic, anelastic, and ideal MHD approximations, the local dispersion relation for MHD waves is \cite{1989nos..book.....U} \citep{1989nos..book.....U}  \label{eqn:disp}  \bigg( \omega^2 - \omega_A^2 \bigg) \bigg( \omega^2 - \frac{k_\perp^2}{k^2}N^2 - \omega_A^2 \bigg) = 0.  \end{equation}