= Anisotropic absolute-instability threshold for positive diffusion
{title2=$\mu_a=\tfrac14\boldsymbol c^T\Gamma^{-1}\boldsymbol c$}
For $\eta_t+\boldsymbol c\cdot\nabla\eta=\mu\eta+\nabla\cdot(\Gamma\nabla\eta)$ with symmetric positive definite $\Gamma$, the <dispersion relation> is $\omega=\boldsymbol c\cdot\boldsymbol k+i(\mu-\boldsymbol k^T\Gamma\boldsymbol k)$. Its zero-<group velocity> saddle is $\boldsymbol k_0=-i\Gamma^{-1}\boldsymbol c/2$. The anisotropic <heat kernel> has fixed-position exponential rate $\mu-\boldsymbol c^T\Gamma^{-1}\boldsymbol c/4$, so this accessible saddle gives the actual threshold. At fixed speed $q$, the threshold ranges from $q^2/(4d_{\max})$ to $q^2/(4d_{\min})$, where $d_{\min},d_{\max}$ are the <eigenvalues> of $\Gamma$. Absence of <absolute hydrodynamic instability> in every direction requires the upper bound $\mu\le q^2/(4d_{\max})$.
Back to article page
