= Weak hyperbolicity of the unsteady top-hat plume
{title2=$A_t+wA_z+Aw_z=\cdots,\quad w_t+ww_z=\cdots$}
For plume area $A$ and reduced gravity $g'$, the principal matrix in variables $(A,w,g')$ is $\begin{pmatrix}w&A&0\\0&w&0\\0&0&w\end{pmatrix}$. At $A>0$ it has a triple real speed $w$ but only two independent eigenvectors. Thus the model is weakly hyperbolic, not strongly or strictly hyperbolic. Real speeds alone do not provide a complete set of characteristic fields or a uniform high-frequency estimate. Shape factors representing nonuniform radial profiles can alter this degeneracy.
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