Si. Badulin et Vi. Shrira, ON THE IRREVERSIBILITY OF INTERNAL-WAVE DYNAMICS DUE TO WAVE-TRAPPINGBY MEAN FLOW INHOMOGENEITIES .1. LOCAL ANALYSIS, Journal of Fluid Mechanics, 251, 1993, pp. 21-53
The propagation of guided internal waves on non-uniform large-scale fl
ows of arbitrary geometry is studied within the framework of linear in
viscid theory in the WKB-approximation. Our study is based on a set of
Hamiltonian ray equations, with the Hamiltonian being determined from
the Taylor-Goldstein boundary-value problem for a stratified shear fl
ow. Attention is focused on the fundamental fact that the generic smoo
th non-uniformities of the large-scale flow result in specific singula
rities of the Hamiltonian. Interpreting wave packets as particles with
momenta equal to their wave vectors moving in a certain force field,
one can consider these singularities as infinitely deep potential hole
s acting quite similarly to the 'black holes' of astrophysics. It is s
hown that the particles fall for infinitely long time, each into its o
wn 'black hole'. In terms of a particular wave packet this falling imp
lies infinite growth with time of the wavenumber and the amplitude, as
well as wave motion focusing at a certain depth. For internal-wave-fi
eld dynamics this provides a robust mechanism of a very specific conse
rvative and moreover Hamiltonian irreversibility. This phenomenon was
previously studied for the simplest model of the flow non-uniformity,
parallel shear flow (Badulin, Shrira & Tsimring 1985), where the term
'trapping' for it was introduced and the basic features were establish
ed. In the present paper we study the case of arbitrary flow geometry.
Our main conclusion is that although the wave dynamics in the general
case is incomparably more complicated, the phenomenon persists and re
tains its most fundamental features. Qualitatively new features appear
as well, namely, the possibility of three-dimensional wave focusing a
nd of 'non-dispersive' focusing. In terms of the particle analogy, the
latter means that a certain group of particles fall into the same hol
e. These results indicate a robust tendency of the wave field towards
an irreversible transformation into small spatial scales, due to the p
resence of large-scale flows and towards considerable wave energy conc
entration in narrow spatial zones.