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The absolute frequency is the generally complex angular frequency at the physically selected absolute wavenumber. Its imaginary part is the fixed-position exponential growth rate under the convention $e^{-i\omega t}$. Marginal zero exponential growth may still carry an algebraic prefactor.

An absolute wavenumber is a complex wavenumber at the selected zero-group velocity saddle of a dispersion relation. For the linear complex Ginzburg-Landau equation, $k_0=-U/[2(c_d-i)]$; the positive real part of the diffusion coefficient makes the Gaussian saddle accessible from the real Fourier contour.

A normal mode $e^{i(kx-\omega t)}$ has dispersion relation $\omega =Uk+(c_d-i)k^2+i\mu$. The absolute wavenumber is the saddle where $d\omega /dk=0$. The Green function of the linear complex Ginzburg-Landau equation distinguishes convective hydrodynamic instability from absolute hydrodynamic instability.

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.

At the absolute wavenumber, the group velocity $d\omega /dk$ is zero. The corresponding accessible saddle therefore governs the disturbance seen at a fixed position. Its exponential growth rate isSaddle accessibility can be checked directly here. With $D=1+i{c}_d$, whose real part is positive, Fourier inversion gives the Green function of the linear complex Ginzburg-Landau equation,At fixed $x$ its exponential rate is $\mathrm{Im}{\omega }_0$, while along the packet centre $x=Ut$ it is $\mu$. For real wavenumbers of Fourier modes, the temporal growth rate is $\mathrm{Im}\omega (k)=\mu -k^2$, so the flow is temporally unstable exactly when $\mu >0$.

Consequently the classifications for a localized disturbance in this laboratory frame areFor convective hydrodynamic instability, a travelling wave packet amplifies but the response at each fixed position decays. For absolute hydrodynamic instability, that fixed-position response amplifies. The equality is the marginal absolute threshold: its exponential rate is zero and the impulse response has a $t^{-1/2}$ prefactor. For $\mu <0$ there is no temporal growth; $\mu =0$ is temporally marginal. When $U=0$ the convective window is empty. These conclusions use the infinite-line impulse problem; a general dispersion relation requires its own spatial-branch or saddle selection.

Take the physical coefficients $U,\mu ,c_d$ real. Substitution of $e^{i(kx-\omega t)}$ into the linear complex Ginzburg-Landau equation gives $-i\omega +ikU-\mu +(1+i{c}_d)k^2=0$, henceLet $A=c_d-i$. Completing the square yields $\omega =A(k+U/(2A){)}^2+i\mu -U^2/(4A)$. Therefore the absolute wavenumber and absolute frequency areTheir explicitly separated parts areThus $d\omega /dk=U+2(c_d-i)k$ vanishes at $k_0$. The complex saddle describes the localized Green function response; it is different from selecting a real wavenumber for a Fourier mode for temporal amplification.