The phenomenon of shelf generation by long nonlinear internal waves
in stratified flows is investigated. The problem of primary interest
is the case of a uniformly
stratified Boussinesq fluid of finite depth. In analysing the
transient evolution of a
finite-amplitude long-wave disturbance, the expansion procedure of Grimshaw
& Yi
(1991) breaks down far downstream, and it proves expedient to follow a
matched-asymptotics procedure: the main disturbance is governed by the
nonlinear theory of
Grimshaw & Yi (1991) in the ‘inner’ region, while the
‘outer’ region comprises multiple
small-amplitude fronts, or shelves, that propagate downstream and
carry O(1) mass.
This picture is consistent with numerical simulations of uniformly
stratified flow past
an obstacle (Lamb 1994). The case of weakly nonlinear long waves in a fluid
layer
with general stratification is also examined, where it is found that
shelves of fourth
order in wave amplitude are generated. Moreover, these shelves may extend
both
upstream and downstream in general, and could thus lead to an upstream
influence
of a type that has not been previously considered. In all cases, transience
of the main
nonlinear wave disturbance is a necessary condition for the formation of
shelves.
Liquid holdup optical measurements for horizontal stratified flows with an opaque fluid layer
