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The fixed observer is the ray $V=0$. The physically selected zero-group velocity saddle defines the absolute wavenumber $k_0$ and absolute frequency ${\omega }_0$:For a smooth frequency branch with $D_{\omega }\ne 0$, the saddle condition is equivalently $D_k(k_0,{\omega }_0)=0$. The absolute growth rate isHere $k_0$ and ${\omega }_0$ can be complex. A physical saddle point is selected by the localized impulse response and deformation of its Fourier transform contour; an arbitrary algebraic stationary point need not determine absolute hydrodynamic instability.

The dispersion relation gives the zero-group velocity saddleThus the absolute growth rate is $\sigma (0)=\mu -U^2/[4(1+c_d^2)]$. More generally the saddle growth rate along a ray isso a frame travelling at $V=U$ sees the maximal rate $\mu$.

For $U>0$, the three open regions areThe line $\mu =0$ is the temporal marginal boundary; the parabola $\mu =U^2/[4(1+c_d^2)]$ is the convective-to-absolute boundary. The Green function of the linear complex Ginzburg-Landau equation has the prefactor $t^{-1/2}$, so zero exponential rate on the latter boundary still allows algebraic decay of the impulse response. This is the stability diagram of the linear complex Ginzburg-Landau equation.

On a ray $x/t=V$, a localized wave packet is governed by a physically selected saddle point satisfying $d\omega /dk=V$ in its analytic dispersion relation. The exponent's real growth is $\mathrm{Im}({\omega }_{\star }-V{k}_{\star })$. The fixed-frame case $V=0$ gives the absolute growth rate. This ray formulation makes the frame dependence of convective hydrodynamic instability explicit.

For the linear complex Ginzburg-Landau equation with real $U,c_d,\mu$ and positive real diffusion coefficient scaled to one, the maximum temporal growth rate is $\mu$ and the absolute growth rate is $\mu -U^2/[4(1+c_d^2)]$. For $U>0$, $\mu <0$ is stable, $0<\mu <{\mu }_a$ has convective hydrodynamic instability, and $\mu >{\mu }_a$ has absolute hydrodynamic instability. Both equality boundaries are marginal in exponential rate.