32 research outputs found
A limitation on Long's model in stratified fluid flows
The flow of a continuously stratified fluid into a contraction is examined, under the assumptions that the dynamic pressure and the density gradient are constant upstream (Long's model). It is shown that a solution to the equations exists if and only if the strength of the contraction does not exceed a certain critical value which depends on the internal Froude number. For the flow of a stratified fluid over a finite barrier in a channel, it is further shown that, if the barrier height exceeds this same critical value, lee-wave amplitudes increase without bound as the length of the barrier increases. The breakdown of the model, as implied by these arbitrarily large amplitudes, is discussed. The criterion is compared with available experimental results for both geometries
A note on the motion of surfaces
We study the motion of surfaces in an intrinsic formulation in which the
surface is described by its metric and curvature tensors. The evolution
equations for the six quantities contained in these tensors are reduced in
number in two cases: (i) for arbitrary surfaces, we use principal coordinates
to obtain two equations for the two principal curvatures, highlighting the
similarity with the equations of motion of a plane curve; and (ii) for surfaces
with spatially constant negative curvature, we use parameterization by
Tchebyshev nets to reduce to a single evolution equation. We also obtain
necessary and sufficient conditions for a surface to maintain spatially
constant negative curvature as it moves. One choice for the surface's normal
motion leads to the modified-Korteweg de Vries equation,the appearance of which
is explained by connections to the AKNS hierarchy and the motion of space
curves.Comment: 10 pages, compile with AMSTEX. Two figures available from the author
The Korteweg-de Vries equation and water waves. Part 2. Comparison with experiments
The Korteweg-de Vries (KdV) equation is tested experimentally as a model for moderate amplitude waves propagating in one direction in relatively shallow water of uniform depth. For a wide range of initial data, comparisons are made between the asymptotic wave forms observed and those predicted by the theory in terms of the number of solitons that evolve, the amplitude of the leading soliton, the asymptotic shape of the wave and other qualitative features. The KdV equation is found to predict accurately the number of evolving solitons and their shapes for initial data whose asymptotic characteristics develop in the test section of the wave tank. The accuracy of the leading-soliton amplitudes computed by the KdV equation could not be conclusively tested owing to the viscous decay of the measured wave amplitudes; however, a procedure is presented for estimating the decay in amplitude of the leading wave. Computations suggest that the KdV equation predicts the amplitude of the leading soliton to within the expected error due to viscosity (12%) when the non-decayed amplitudes are less than about a quarter of the water depth. Indeed, agreement to within about 20% is observed over the entire range of experiments examined, including those with initial data for which the non-decayed amplitudes of the leading soliton exceed half the fluid depth
Instabilities in the two-dimensional cubic nonlinear Schrodinger equation
The two-dimensional cubic nonlinear Schrodinger equation (NLS) can be used as
a model of phenomena in physical systems ranging from waves on deep water to
pulses in optical fibers. In this paper, we establish that every
one-dimensional traveling wave solution of NLS with trivial phase is unstable
with respect to some infinitesimal perturbation with two-dimensional structure.
If the coefficients of the linear dispersion terms have the same sign then the
only unstable perturbations have transverse wavelength longer than a
well-defined cut-off. If the coefficients of the linear dispersion terms have
opposite signs, then there is no such cut-off and as the wavelength decreases,
the maximum growth rate approaches a well-defined limit.Comment: 4 pages, 4 figure
Explosive instability due to 4-wave mixing
It is known that an explosive instability can occur when nonlinear waves
propagate in certain media that admit 3-wave mixing. The purpose of this paper
is to show that explosive instabilities can occur even in media that admit no
3-wave mixing. Instead, the instability is caused by 4-wave mixing: four
resonantly interacting wavetrains gain energy from a background, and all blow
up in a finite time. Unlike singularities associated with self-focussing, these
singularities can occur with no spatial structure - the waves blow up
everywhere in space, simultaneously
The nonlinear Schrödinger equation, dissipation and ocean swell
Non UBCUnreviewedAuthor affiliation: University of ColoradoFacult