Oscillations, Waves, and Interactions - GWDG

96 D. Ronneberger et al.
powers of δα = α − αvgr0. With δω = ω − ωvgr0 one obtains
d 2 Ω n(αvgr0)
dα 2
Ω n(αvgr0 + δα) = ωvgr0 + a2δα 2 + O(δα 3 ) with a2 = 1
2
⇒ δω = a2δα 2 + O(δα 3 ) . (12)
Hence the inverse function α = inv{Ω n}(ω) has two branches in the neighborhood of
ωvgr0, namely δα(ω) = ±(δω/a2) 1/2 which have a common point if and only if ωvgr0
is exactly on the path through the complex ω-plane. Otherwise the two branches are
supposed to be the dispersion relations of two different modes. These however are
mixed when a parameter of the eigenvalue problem is continuously varied such that
ωvgr0 crosses over the path through the ω-plane (also the path through the ω-plane
may be subject to deformation); then the four parts of branches that emanate from
the branch point are differently combined before and after the crossing. In the case
considered in Fig. 13 the path through the ω-plane is the real axis and the varied
parameter is the flow velocity. So, with ℑ{ωvgr0} = 0 the flow is marginally stable
for U/c = 0.1155 but theoretically becomes absolutely unstable (ℑ{ωvgr0} > 0) with
U/c > 0.1155. The dashed curves in Fig. 13 illustrate the four possible dependencies
of ℑ{α} on the frequency: each of the two low-frequency branches can combine with
each of the two high-frequency branches. Only two of these possibilities are shown
with the dispersion relations plotted in the Figure. The absolute instability that is
expected above a critical flow velocity has not been observed in the experiments, and
before we further search for indications of an absolute instability we consider the
convective instabilities and the causality problem.
4.1.4 Convective instability and causality problem
As mentioned earlier the direction of spatial growth −ℑ{α} of a mode has to agree
with its causality direction νcaus(= ±1), i. e. νcaus · ℑ{α} < 0, if the mode is to
be classified as convective unstable. A trivial but laborious way to determine νcaus
is the computation of the total field which is caused by a local source switched on
at t = 0. To avoid this effort Briggs [37] has developed a criterion which allows to
determine the causality direction on the basis of the dispersion relation. He replaces
the switching-on by exponential growth of the source amplitude, and the imitation
of the discontinuous switching-on is the better the faster the amplitude grows, i. e.
if ℑ{ω} → ∞. It is to be expected that the excited field then concentrates at the
location of the source because no time for the spreading of the field has been left. So
each mode decays in its direction of causality: νcaus · ℑ{α} > 0. Now, according to
the Briggs criterion, ℑ{ω} is gradually reduced to zero and νcaus ·ℑ{α} is checked for
changing the sign which indicates that the respective mode finally (at real frequencies)
grows in the causality direction and thus is subject to convective instability.
However two conditions have to be fulfilled before the Briggs criterion can be applied:
(i) The result must be independent of the path through the ω-plane on which ℑ{ω}
is reduced to zero.
(ii) An upper bound ω ′′ max must exist such that νcaus · ℑ{α} > 0 for all ω with
ℑ{ω} > ω ′′ max; an equivalent and actually verifiable condition is the existence of an
upper bound ω ′′ max such that ℑ{Ωn(α)} ≤ ω ′′ max for all real α.