## On the dynamic excitation of internal gravity waves in the equatorial oceans

#### B. R. Sutherland,

J. Phys. Oceanogr.,

**26**, pp 2398--2419 (1996).-
It is proposed that shear instability of the upper flank of the equatorial
undercurrent may generate, under a broad range of conditions,
downward propagating internal gravity waves (IGW) of large amplitude.
The generation mechanism is shown to require only that the
background stratification is weak where the shear is large
(
*i.e.*in the mixing region) and that the stratification is sufficiently large in the far field (*i.e.*near the thermocline). In a series of studies, the generation of IGW from unstable shear flows is examined. Linear theory is used to predict under what circumstances the generation of IGW may be large and fully nonlinear simulations restricted to two dimensions are employed to provide estimates of the degree of vertical mixing and of the vertical transport of horizontal momentum by IGW. In particular, the simulations demonstrate that when large amplitude IGW are generated by shear instability the mean flow itself is significantly decelerated in the mixing region. The momentum flux associated with the radiating IGW is large and it is proposed that these may act in part as a momentum source to the deep equatorial countercurrents.

Internal gravity wave radiation into weakly stratified fluid#### B. R. Sutherland,

Phys. Fluids,

**8**, pp 430--441 (1996).- It is shown by way of nonlinear numerical simulations of flow restricted to two dimensions that a compact wavepacket of large-amplitude internal gravity waves incident upon a weakly stratified region in which the buoyancy frequency is less than the frequency of the wavepacket may partially transmit energy into this region through the generation of a wavepacket of lower frequency. In part, the transmission of waves occurs due to the transient nature of the forcing by the incident wavepacket, but if the amplitude of the wavepacket is moderately large, weakly nonlinear effects may act to significantly increase the proportion of the wavepacket that is transmitted. For a range of simulations initialised with wavepackets of different amplitude and vertical extent, the characteristics of the reflected and transmitted waves are analysed and reflection coefficients are calculated. An explanation for how the nonlinear transmission mechanism operates is given by demonstrating that the wave induced mean-flow, which is shown to be approximately equal to the horizontal wave pseudomomentum expressed in Eulerian variables, acts to adjust the frequency of the incident waves.

Self-organisation and scaling in a lattice predator-prey model#### B. R. Sutherland and A. E. Jacobs,

Complex Systems,

**8**(6), pp 385--405 (1994).-
We propose
that self-organization may provide a mechanism
by which power-law cluster distributions of mobile prey
(
*e.g.*fish, phytoplankton) may develop; in contrast, such distributions have often been attributed to scaling of the background environment. Evidence supporting our proposal is provided by examining the dynamics of a cellular automaton-like model of a predator-prey system. The model, which is discrete in space and time, is robust and evolves to a state with oscillatory, phase-shifted populations for a large range of parameter values, namely the predator and prey breeding times and the predator starvation time. The distribution function*D(s)*of the prey cluster sizes*s*has roughly power-law form,*s^(-alpha)*, over a range of moderately large sizes but is cut off at large*s*. The exponent*alpha ~= 1.35 +/- 0.10*depends only weakly on the parameters of the model.

Internal wave generation and hydrodynamic instability#### B. R. Sutherland, C. P. Caulfield, and W. R. Peltier,

J. Atmos. Sci,

**51**, pp 3261--3280 (1994).-
Two mechanisms are proposed whereby
internal gravity waves (IGW) may radiate from a
linearly unstable region of Boussinesq parallel flow that is
characterized in the far field by constant horizontal velocity and
Brunt-Vaisala frequency.
Through what is herein referred to as ``primary generation'',
IGW may be directly
excited by linear instability of the initial state parallel shear flow.
Characteristically, these waves propagate with horizontal phase speed and
wave number equal to that of the most unstable mode of linear stability
theory.
Through the second mechanism, referred to as ``secondary generation'',
IGW may be excited
*via*nonlinear modification of the initial instability into a form that couples strongly to a large amplitude outgoing internal wave field. We propose that the primary generation of IGW may occur provided a penetration condition, which we derive on the basis on linear theory, is satisfied. The penetration condition provides a limit on the growth rate of a disturbance of any particular frequency that is capable of propagating into the far field. This hypothesis is supported by a sequence of representative nonlinear numerical simulations in two spatial dimensions for both free mixing layer and jet flows with horizontal velocity profiles U(z)=tanh(z) and U(z)=sech^2(z), respectively. For the purpose of these analyses, the fluid density is taken to be such that the square of the Brunt-Vaisala frequency is given by N^2(z)=J tanh^2(z/R). Such stratification allows both for the development of large-scale eddies in the region of low static stability and, in the far field where N^2 ~= J is positive and approximately constant, for the radiation of a broad frequency spectrum of IGW.

Turbulence transition and internal wave generation in density stratified jets#### B. R. Sutherland and W. R. Peltier,

Phys. Fluids,

**6**(3), pp 1267--1284 (1994).-
The nonlinear evolution of an unstable symmetric
jet in incompressible, density stratified fluid is simulated
numerically.
When N^2 is constant and near zero, like-signed
vortices pair by way of an instability of the mean flow to a subharmonic
disturbance with wavelength twice that of the most unstable
mode of linear theory.
For small but finite and constant values of N^2,
however, the individual vortex cores are
strained and vorticity is generated at small scales
by the action of baroclinic
torques. In this case, the mean flow of the fully evolved jet
is stable to subharmonic disturbances.
The linear stability of the two-dimensional nonlinear basic
states to three dimensional perturbations is examined in detail.
From this
stability analysis it is inferred that jet flow with stratification
characterized by constant N^2 is a poor
candidate for IGW generation.
However,
the existence of an efficient mechanism whereby
IGW may be radiated to infinity from the jet core is demonstrated
*via*simulations initialized with a density profile such that N^2=J tanh^2(z/R). This mechanism is expected to be an important contributor to the wave field observed in a variety of geophysical circumstances.

The stability of stratified jets#### B. R. Sutherland and W. R. Peltier,

Geophys. Astrophys. Fluid Dyn.,

**66**, pp 101--131 (1992).- We study numerically the two dimensional linear instabilities of an incompressible, inviscid, density stratified, symmetric jet as a function of a width parameter, D. In the limit of infinite D, the maximum growth rate of temporal instability exponentially approaches that of the shear flow on either flank of the jet. The growth rate need not approach this limit monotonically, however. For some stratified flows it is possible that the odd (varicose) mode of a jet with sufficiently large width may grow more rapidly than the even (sinuous) mode. We also examine spatiotemporal instabilities of the Bickley profile, focusing specifically upon the identification of the regime of parameter space in which the flow is absolutely or convectively unstable. Finally, our methods of spatiotemporal linear stability analysis are applied to a more realistic asymmetric jet which mimics the internal wave related flow that develops in the lee of a topographic obstacle when ``breaking'' occurs. This analysis appears to bear strongly upon the interaction that occurs subsequent to wave breaking that leads to intense wave, mean-flow interaction.

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Bruce R. Sutherland, Jan 96,