The zonally symmetric inertial instability of oceanic near-equatorial flows is studied through high-resolution numerical simulations. In homogeneous upper layers, the instability of surface-confined westward currents implies potentially fast downward mixing of momentum with a predictable final equilibrium. With increasing Reynolds number, latitudinal scales along the surface associated with the instability become ever smaller and initially the motions are ever more concentrated underneath the surface. The results suggest that even if the upper layer is stratified, it may still be necessary to include the full Coriolis force in the dynamics rather than use the traditional beta-plane approximation.

}, keywords = {approximation, atmosphere, beta-plane, centrifugal-type instabilities, criterion, nonlinear instability, ocean processes, pacific-ocean, rotating flows, saturation, shear flows, symmetric stability, vertical scale selection}, isbn = {0022-1120}, doi = {10.1017/jfm.2017.377}, url = {In three-dimensional flow, a vortex can become turbulent and be destroyed through a variety of instabilities. In rotating flow, however, the result of the breakup of a vortex is usually a state comprising several vortices with their axes aligned along the ambient rotation direction. This article is a review of our recent work on how the combined effect of centrifugal and barotropic instabilities can breakup a vortex and lead to its reformation in a predictable way even though an intermediate stage in the evolution is turbulent. Centrifugal instability tends to force the unstable vortex into a turbulent state that mixes absolute angular momentum in such a way as to precondition the flow for a subsequent barotropic instability. A method for predicting the redistribution of angular momentum and theresulting velocity profile is discussed. The barotropic instability horizontally redistributes the component of vorticity that is aligned along the ambient rotation vector, resulting in the final byproducts of the instability, which are stabilized by the effects of ambient rotation. A prediction scheme that puts the tendencies of these two instabilities together proves to be very reliable. (C) 2015 Elsevier Masson SAS. All rights reserved.

}, keywords = {barotropic vortices, circular, evolution, flow, inertial instability, Rotating flow, rotating fluid, saturation, stability, tripolar vortices, vortex dynamics, vortex merger, vortices}, isbn = {0997-7546}, doi = {10.1016/j.euromechflu.2015.06.007}, url = {Inertial instability in parallel shear flows and circular vorwes in a uniformly rotating system (f-plane) redistributes absolute linear momentum or absolute angular momentum in such a way as to neutralize the instability. In previous studies we showed that, in the absence of other instabilities, at high Reynolds numbers the final equilibrium can be predicted with a simple construction based on conservation of total momentum. In this paper we continue this line of research with a study of barotropic shear flows on the equatorial /3-plane, Through numerical simulations the evolution of the instability is studied in select illuminating cases: a westward flowing Gaussian jet With the flow axis exactly on the equator, a uniform shear flow and eastward and Wes t w ard flowing jets that have their flow axis shifted away from the equator. In the numerical simulations it is assumed that there are no along -stream variations. {\textquoteright}fins suppresses equatorial Rossby W a ve s and barotropic shear instabilities and allows only inertial instability to develop. We investigate whether for these flow S the equatorial t-plane the final equilibrated flow can be predicted as was possible for flows on the f-plane. For the Gaussian jet centred on the equator the prediction of the equilibrated flow is obvious by mere inspection of the initial momentum distribution and by assuming that momentum is mixed and homogenized to render the equilibrated flow inertially stable. For the uniform shear flow, however, due to the peculiar nature of the initial momentum distribution and the fact that the Coriolis parameter f varies with latitude, it appears that, unlike in our earlier studies of flows on the f-plane, additional constraints need to be considered to correctly predict the outcome of the highly nonlinear evolution of the instability, The mixing range of the linear shear flow and the value of the mixed momentum is determined numerically and this is used to predict the equilibrated flow that emerges from an eastward flowing jet that is shifted a small distance away from the equator. For shifts large enough to induce no shcar at the equator the equilibrium flow can be well predicted using the simple rccipe used in our carlicr studies of parallel shear flows on the f-plane. For thc westward flowing jet shifted a very small distance from the equator, no prcdiction appears feasible. For modcstly small shifts a prediction is possiblc by combining the empirical prcdiction for thc linear shear flow with a prediction similar to what we used in our previous studies for flows on the f-plane.

}, keywords = {barotropic vortices, geophysical and geological flows, instability, shear flows, symmetric stability, vertical scale selection, vortex, vortex dynamics}, isbn = {0022-1120}, doi = {10.1017/jfm.2015.63}, url = {The concept of a minimal flow unit (MFU) for the study of the basic physics of turbulent flows is introduced. The MFU is an initial vorticity configuration that consists of a few simple well-defined large-scale vortex structures. The form and position of these structures are chosen so that their interaction produces turbulence capturing many of the essential characteristics of isotropic homogeneous turbulence produced from random-phase initial conditions or that produced by continual random-phase forcing. The advantage of using the MFU is that the evolution of the vortex structures can be followed more clearly and the relationship between the evolving vortex structures and the various ranges in the energy spectrum can be more clearly defined. The addition of passive scalar fields to the MFU permits an investigation of passive scalar mixing that is relevant to the study of combustion. With a particular choice of the MFU, one that produces a trend to a finite-time singularity in the vorticity field, it is demonstrated that passive scalar distributed in the original large-scale vortices will develop intense gradients in the region where the vorticity is tending toward a singularity. In viscous flow, the evolution of the MFU clearly shows how the volume of the regions where originally well-separated passive scalars come into contact increases with increasing Reynolds number.

}, keywords = {constant, direct numerical simulation, direct numerical-simulation, fluid, high-symmetry, incompressible euler equations, isotropic, passive scalar turbulence, singularities, spectrum, turbulence, vortex dynamics, vorticity moments}, isbn = {1468-5248}, doi = {10.1080/14685248.2014.927066}, url = {A current in a homogeneous rotating fluid is subject to simultaneous inertial and barotropic instabilities. Inertial instability causes rapid mixing of streamwise absolute linear momentum and alters the vertically averaged velocity profile of the current. The resulting profile can be predicted by a construction based on absolute-momentum conservation. The alteration of the mean velocity profile strongly affects how barotropic instability will subsequently change the flow. If a current with a symmetric distribution of cyclonic and anticyclonic vorticity undergoes only barotropic instability, the result will be cyclones and anticyclones of the same shape and amplitude. Inertial instability breaks this symmetry. The combined effect of inertial and barotropic instability produces anticyclones that are broader and weaker than the cyclones. A two-step scheme for predicting the result of the combined inertial and barotropic instabilities is proposed and tested. This scheme uses the construction for the redistribution of streamwise absolute linear momentum to predict the mean current that results from inertial instability and then uses this equilibrated current as the initial condition for a two- dimensional simulation that predicts the result of the subsequent barotropic instability. Predictions are made for the evolution of a Gaussian jet and are compared with three-dimensional simulations for a range of Rossby numbers. It is demonstrated that the actual redistribution of absolute momentum in the three-dimensional simulations is well predicted by the construction used here. Predictions are also made for the final number and size of vortices that result from the combined inertial and barotropic instabilities.

}, keywords = {2, 2-dimensional turbulence, dimensions, fluid, geophysical and geological flows, instability, isolated vortices, jets, nonlinear evolution, shear flows, turbulent flows, vortex}, isbn = {0022-1120}, doi = {10.1017/jfm.2013.191}, url = {We present high-resolution numerical simulations of the Euler and Navier-Stokes equations for a pair of colliding dipoles. We study the possible approach to a finite-time singularity for the Euler equations, and contrast it with the formation of developed turbulence for the Navier-Stokes equations. We present numerical evidence that seems to suggest the existence of a blow-up of the inviscid velocity field at a finite time (t(s)) with scaling vertical bar u vertical bar(infinity) similar to (t(s) - t)(-1/2), vertical bar omega vertical bar(infinity) similar to (t(s) - t)(-1). This blow-up is associated with the formation of a k(-3) spectral range, at least for the finite range of wavenumbers that are resolved by our computation. In the evolution toward t(s), the total enstrophy is observed to increase at a slower rate, Omega similar to (t(s) - t)(-3/4), than would naively be expected given the behaviour of the maximum vorticity, omega(infinity) similar to (t(s) - t)(-1). This indicates that the blow-up would be concentrated in narrow regions of the flow field. We show that these regions have sheet-like structure. Viscous simulations, performed at various Re, support the conclusion that any non-zero viscosity prevents blow-up in finite time and results in the formation of a dissipative exponential range in a time interval around the estimated inviscid t(s). In this case the total enstrophy saturates, and the energy spectrum becomes less steep, approaching k(-513). The simulations show that the peak value of the enstrophy scales as Re-3/2, which is in accord with Kolmogorov phenomenology. During the short time interval leading to the formation of an inertial range, the total energy dissipation rate shows a clear tendency to become independent of Re, supporting the validity of Kolmogorov{\textquoteright}s law of finite energy dissipation. At later times the kinetic energy shows a t(-1.2) decay for all Re, in agreement with experimental results for grid turbulence. Visualization of the vortical structures associated with the stages of vorticity amplification and saturation show that, prior to ts, large-scale and the small-scale vortical structures are well separated. This suggests that, during this stage, the energy transfer mechanism is non-local both in wavenumber and in physical space. On the other hand, as the spectrum becomes shallower and a k(-5/3) range appears, the energy-containing eddies and the small-scale vortices tend to be concentrated in the same regions, and structures with a wide range of sizes are observed, suggesting that the formation of an inertial range is accompanied by transfer of energy that is local in both physical and spectral space.

}, keywords = {3-dimensional turbulence, blowup, direct numerical-simulation, energy-transfer, equations, homogeneous isotropic turbulence, Navier-Stokes equations, statistics, taylor-green vortex, turbulence simulation, vortex dynamics, Vorticity}, isbn = {0022-1120}, doi = {10.1017/jfm.2011.430}, url = {A method for predicting the outcome of vortex breakup in a rotating flow is introduced. The vortices dealt with here are subject to both centrifugal and barotropic instabilities. The prediction of the aftermath of the breakup relies on knowing how both centrifugal and barotropic instabilities would equilibrate separately. A theoretical model for non-linear equilibration in centrifugal instability is wedded to two-dimensional simulation of barotropic instability to predict the final vortices that emerge from the debris of the original vortex. This prediction method is tested against three-dimensional Navier-Stokes simulations. For vortices in which a rapid centrifugal instability triggers a slower barotropic instability, the method is successful both qualitatively and quantitatively. The skill of the prediction method decreases as the time scales of the two instabilities become comparable.

}, keywords = {2-dimensional incompressible flows, atmosphere, baroclinic stability, barotropic vortices, circular vortices, evolution, fluid, generation, geophysical and geological flows, inertial, instability, tripolar vortices, turbulent flows, vortex flows}, isbn = {0022-1120}, doi = {10.1017/s0022112010004945}, url = {Geophysical turbulence is strongly affected by the variation of the Coriolis parameter with latitude. This variation results in the so-called beta-effect, which forces energy from small-scales to be transferred preferentially into zonal motions. This effect results in the formation of narrow jet-like zonal flows that dominate the dynamics and act as transport barriers. Here, laboratory experiments are used to reproduce this effect in decaying turbulent flows. An electromagnetic cell is used to generate an initial field of vorticity in a rotating tank. Under conditions of quasigeostrophic flow, the beta-effect is produced by depth variation of the flow instead of variation of the Coriolis parameter. The effects of changing the container geometry and the overall fluid depth on the production of jets are investigated. The results suggest that this laboratory configuration can be used to model jet formation in the oceans and that increasing fluid depth is a practical way to decrease viscous effects. (C) 2008 Elsevier B.V. All rights reserved.

}, keywords = {beta-plane, generation, jets, topography, turbulence, vortices}, isbn = {0924-7963}, doi = {10.1016/j.jmarsys.2008.10.015}, url = {The linear instability of a barotropic flow with uniform horizontal shear in a stratified rotating fluid is investigated with respect to perturbations invariant in the alongflow direction. The flow can be inertially unstable if there is sufficiently strong anticyclonic shear, but only for sufficiently high Reynolds numbers Re. We determine the critical Reynolds numbers required for amplification of the instability as a function of Prandtl number, strength of the stratification and magnitude of the shear. The vertical scales at the onset of the instability are calculated. For Prandtl number P \< 1.44 instability always sets in through stationary overturning motions, for P \> 1.44 it may also commence through overstable (oscillatory) motions. For Re exceeding the critical value, we determine the vertical scale of the most rapidly amplifying modes and the corresponding growth rates and how they vary with Re, P, the shear and the strength of stratification.

}, keywords = {barotropic vortices, criterion, flows, fluids, plane, stability, vortex}, isbn = {0022-1120}, doi = {10.1017/s0022112007009007}, url = {Freely decaying quasi-2D turbulence under the influence of a meridional variation of the Coriolis parameter f ( effect) is experimentally and numerically modelled. The experimental flow is generated in a rotating electromagnetic cell where the variation of f is approximated by a nearly equivalent topographical effect. In the presence of a high effect, the initial disordered vorticity field evolves to form a weak polar anticyclonic circulation surrounded by a cyclonic zonal jet demonstrating the preferential transfer of energy towards zonal motions. In agreement with theoretical predictions, the energy spectrum becomes peaked near the Rhines wave number with a steep fall-off beyond, indicating the presence of a soft barrier to the energy transfer towards larger scales. DNS substantially confirmed the experimental observations.

}, keywords = {-plane turbulence, 2-dimensional turbulence, barriers, beta-plane, cascade, flows, jets, Rhines scale, rotating sphere, spectra, topography, transport, zonal jets}, isbn = {1468-5248}, doi = {10.1080/14685240802464417}, url = {The influence of the Mid Adriatic Pit (MAP) on the general circulation of the Adriatic is explored through numerical simulations. The numerical code used is the DieCAST model specifically modified for application to the Adriatic Sea. A ten-year simulation is performed and the ability of the model to capture important features of the Adriatic circulation is demonstrated. A series of numerical experiments on the importance of the MAP on the general circulation is performed. It is demonstrated that the current over the northern flank of the MAP, which flows from the Croatian toward the Italian coast, is primarily a topographic current and that such a current would reverse direction if the gradient of the bathymetry were reversed.

}, keywords = {adjustment, circulation model, coastal current, dynamics, gradient, ocean modeling system, resolution, rossby, slope, statistical-mechanics, topography}, isbn = {1124-1896}, doi = {10.1393/ncc/i2007-10242-x}, url = {Dynamical arguments based on the structure of the Euler equations suggest the possibility of rapid amplification of vorticity in which the vorticity and the rate of strain grow proportionately. During such growth, the vorticity is expected to amplify as a (t(s)-t)(-1) power-law in time. This behavior is difficult to demonstrate numerically, in part, because initial transients tend to obscure it. Lamb dipoles are used here to construct the initial vorticity. This helps to avoid these transients and results in a flow exhibiting the expected power-law vorticity amplification for a period of time. The spatial region where the vorticity growth rate is maximal is investigated in detail using a decomposition of the vorticity along the principal axes of the rate-of-strain tensor. It is demonstrated that the vorticity and strain rate in one direction in this decomposition are proportional during the period of rapid vorticity growth. These findings suggest that, during this period, the Euler equations can be reduced to a one-dimensional model equation for vorticity in the rate-of-strain coordinate system. (C) 2007 American Institute of Physics.

}, keywords = {incompressible euler equations, instability, singularities, turbulence, vortex}, isbn = {1070-6631}, doi = {10.1063/1.2732438}, url = {The unfolding of inertial instability in intially barotropic vortices in a uniformly rotating and stratified fluid is studied through numerical simulations. The vortex dynamics during the instability is examined in detail. We demonstrate that the instability is stabilized via redistribution of angular momentum in a way that produces a new equilibrated barotropic vortex with a stable velocity profile. Based on extrapolations from the results of a series of simulations in which the Reynolds number and strength of stratification are varied, we arrive at a construction based on angular momentum mixing that predicts the infinite-Reynolds-number form of the equilibrated vortex toward which inertial instability drives an unstable vortex. The essential constraint is conservation of total absolute angular momentum. The construction can be used to predict the total energy loss during the equilibration process. It also shows that the equilibration process can result in anticyclones that are more susceptible to horizontal shear instabilities than they were initially, a phenomenon previously observed in laboratory and numerical studies.

}, keywords = {3-dimensional instability, columnar vortices, criterion, environment, f-plane, flow, laboratory experiments, nonlinear evolution, stability, vortex}, isbn = {0022-1120}, doi = {10.1017/s0022112007006325}, url = {Inertial instability in a rotating shear flow redistributes absolute linear momentum in such a way as to neutralize the instability. In the absence of other instabilities, the final equilibrium can be predicted by a simple construction based on conservation of total momentum. Numerical simulations, invariant in the along-stream direction, suppress barotropic instability and allow only inertial instability to develop. Such simulations, at high Reynolds numbers, are used to test the theoretical prediction. Four representative examples are given: a jet, a wall-bounded jet, a mixing layer and a wall-bounded shear layer.

}, isbn = {0022-1120}, doi = {10.1017/s0022112007006593}, url = {For simple parallel shear flows on the f-plane and the equatorial beta-plane we derive an energy norm for zonally invariant perturbations. It is used to derive the linear stability boundary for when these flows are inertially stable in the classical sense but may be destabilized due to unequal rates of diffusion of momentum and heat. The analysis is valid when there are arbitrary, zonally invariant, no-slip boundaries which are perfect thermal conductors.

}, keywords = {instability, stability}, isbn = {0022-1120}, doi = {10.1017/s0022112007006933}, url = {The evolution of a barotropic coastal current in the presence of a bottom ramp-shaped topography is studied by means of laboratory experiments and numerical simulations. The experiments are performed in a rectangular rotating tank filled with freshwater. The fluid depth is shallow at one side of the domain and deeper at the other side, and both regions are divided by a narrow slope, whose depth contours are perpendicular to the long sides of the tank. A current approaching the slope is produced along one of the vertical walls, having the boundary at its right. Two configurations are analyzed: when the current flows from shallow to deep water and when flowing in the opposite direction. In the first scenario, the current is divided in two parts, one of them following the coastline. The other part of the current pairs with a cell of negative relative vorticity generated at the slope due to squeezing effects, forming a dipolar structure moving offshore, back toward the shallow side. In addition, a weak current moving inshore along the slope is clearly formed. In the second configuration, when the flow goes from deep to shallow water, a part of or even the whole current might be forced to move along the contours of the topography, away from the coast. In this case there is no dipole formation. The experiments are well reproduced by means of quasigeostrophic numerical simulations, which allow a more detailed systematic study of the influence of flow parameters such as the topography height and the width of the slope.

}, keywords = {adjustment, barotropic vortices, bottom topography, circulation, currents, escarpment, evolution, rossby, shelf, statistical-mechanics, waves}, isbn = {0022-3670}, doi = {10.1175/jpo2815.1}, url = {Chandrasekhar (1961) extensively investigated the linear dynamics of Rayleigh-Benard convection in an electrically conducting fluid exposed to a uniform vertical magnetic field and enclosed by rigid, stress-free, upper and lower boundaries. He determined the marginal stability boundary and critical horizontal wavenumbers for the onset of convection as a function of the Chandrasekhar number Q or Hartmann number squared. No closed-form formulae appeared to exist and the results were tabulated numerically. We have discovered simple expressions that concisely describe the stability properties of the system. When the Prandtl number Pr is greater than or equal to the magnetic Prandtl number Pm the marginal stability boundary is described by the curve Q = pi(-2)[R - (RcR2/3)-R-1/3] where R is the Rayleigh number and R-c = (27/4)pi(4) is Rayleigh{\textquoteright}s famous critical value for the onset of stationary convection in the absence of a magnetic field (Q = 0). When Pm \> Pr the marginal stability boundary is determined by this curve until intersected by the curve Q = 1/pi(2)[Pm-2(1 + Pr)/Pr-2(1 + Pm)R - ((1 + Pr)(Pr + Pm)/Pr-2)(1/3) (Pm-2(1 + Pr)/Pr-2(1 + Pm))(2/3) (RcR2/3)-R-1/3]. An expression for the intersection point is derived and also for the critical horizontal wavenumbers for which instability sets in along the marginal stability boundary either as stationary convection or in an oscillatory fashion. A simple formula is derived for the frequency of the oscillations. Also we show that in the limit of vanishing magnetic diffusivity, or infinite electrical conductivity, the system is unstable for sufficiently large R. Instability in this limit always sets in via overstability.

}, isbn = {0022-1120}, doi = {10.1017/s0022112003005329}, url = {The linear dynamics of rotating Rayleigh-Benard convection with rigid stress-free boundaries has been thoroughly investigated by Chandrasekhar (1961) who determined the marginal stability boundary and critical horizontal wavenumbers for the onset of convection and overstability as a function of the Taylor number T. No closed-form formulae appeared to exist and the results were tabulated numerically. However, by taking the Rayleigh number R as independent variable we have found remarkably simple expressions. When the Prandtl number P greater than or equal to P-c = 0.67659, the marginal stability boundary is described by the curve T(R) = R[(R/R-c)(1/2) - 1] where R-c = 27/4pi(4) is Rayleigh{\textquoteright}s famous critical value for the onset of stationary convection in a non-rotating system (T = 0). For P \< P-c the marginal stability boundary is determined by this curve until it is intersected by the curve T(R, P) = R [(1+P/2(3) P-4)(1/2) (R/R-c)(1/2) - 1 + P/2P(2)]. A simple expression for the intersection point is derived and also for the critical horizontal wavenumbers for which, along the marginal stability boundary, instability sets in either as stationary convection or in an oscillatory fashion. A simple formula is derived for the frequency of the oscillations. Further, we have analytically determined critical points on the marginal stability boundary above which an increase of either viscosity or diffusivity is destabilizing. Finally, we show that if the fluid has zero viscosity the system is always unstable, in contradiction to Chandrasekhar{\textquoteright}s conclusion.

}, isbn = {0022-1120}, doi = {10.1017/s0022112002003294}, url = {It is demonstrated that the growth of the mixing zone generated by Rayleigh-Taylor instability can be greatly retarded by the application of rotation, at least for low Atwood number flows for which the Boussinesq approximation is valid. This result is analysed in terms of the effect of the Coriolis force on the vortex rings that propel the bubbles of fluid in the mixing zone.

}, isbn = {0022-1120}, doi = {10.1017/s0022112002007772}, url = {The possibility of diminishing the danger of trailing vortices through thermal forcing is investigated. It is shown that heating the vortices would have two beneficial effects. First, it would cause the vortices to descend more rapidly thus clearing the flight path more quickly. Second, it would cause the vortices to draw closer together, thus greatly increasing the growth rate of the short-wave instabilities that can ultimately destroy the vortices through cross-diffusion. (C) 2001 Editions scientifiques et medicales Elsevier SAS.

}, keywords = {3-dimensional instability, fluid, short-wave instability, strained vortices, vortex pair}, isbn = {0997-7546}, doi = {10.1016/s0997-7546(01)01131-1}, url = {The buoyancy range, which represents a transition from large-scale wave-dominated motions to small-scale turbulence in the oceans and the atmosphere, is investigated through large-eddy simulations. The model presented here uses a continual forcing based on large-scale standing internal waves and has a spectral truncation in the isotropic inertial range. Evidence is presented for a break in the energy spectra from the anisotropic k(-3) buoyancy range to the small-scale k(-5/3) isotropic inertial range. Density structures that form during wave breaking and periods of high strain rate are analysed. Elongated vertical structures produced during periods of strong straining motion are found to collapse in the subsequent vertically compressional phase of the strain resulting in a zone or patch of mixed fluid.

}, keywords = {3, dimensions, evolution, gravity-wave breaking, internal waves, isotropic turbulence, large-eddy simulation, parametric-instability, shear, spectra, temperature, vorticity dynamics}, isbn = {0022-1120}, doi = {10.1017/s002211200000241x}, url = {The applicability of the super-slip boundary condition in wind-driven quasigeostrophic ocean circulation models is reexamined. The energy and enstrophy characteristics of the super-slip condition are discussed for the equilibrium state. A model is constructed with super-slip on the western boundary and free slip on the other boundaries. Both linear and nonlinear solutions are presented. Compared to the case with all free-slip boundaries, this new model gives a more energetic and narrower western boundary current, but otherwise the differences are not very great.

}, isbn = {0022-3670}, doi = {10.1175/1520-0485(2001)031<2489:dbcr>2.0.co;2}, url = {The evolution of a coastal current as it encounters an escarpment depends strongly on whether the geometry of the coast and escarpment is right or left "handed," independent of the direction of the coastal current. Handedness is defined such that right-handed means that when looking across the escarpment from the deep to the shallow side, the coast is found on the right. The essential aspects of the difference in behavior of the current in the two geometries are captured by a simple quasigeostrophic model of coastal flow over a step. An exact analytic solution to the nonlinear stationary problem is obtained. This solution shows that, when a coastal current crosses an escarpment in the left-handed geometry, the speed of the current will increase independent of whether the flow is from shallow to deep or from deep to shallow. For the right-handed geometry, the speed of the current decreases, also independent of the direction of the coastal flow. In the left (right)-handed geometry, there is associated to the coastal flow an inshore (offshore) current along the escarpment. These results are explained in terms of linear wave theory and vortex dynamics. Numerical simulations are used to examine the evolution of the flow from the initial encounter to the establishment of a stationary flow. The relevance of this research is discussed in light of recent results from laboratory experiments and oceanic observations.

}, keywords = {circulation, eddies, evolution, flow, fluid, rossby adjustment, separation, shelf, statistical-mechanics, topographic waves}, isbn = {0022-3670}, doi = {10.1175/1520-0485(1999)029<0969:boacca>2.0.co;2}, url = {Laboratory observations and numerical experiments have shown that a variety of compound vortices can emerge in two-dimensional flow due to the instability of isolated circular vortices. The simple geometrical features of these compound vortices suggest that their description may take a simple form if an appropriately chosen set of functions is used. We employ a set which is complete on the infinite plane for vorticity distributions with finite total enstrophy. Through projection of the vorticity equation (Galerkin method) and subsequent truncation we derive a dynamical system which is used to model the observed behaviour in as simple as possible a fashion. It is found that at relatively low-order truncations the observed behaviour is qualitatively captured by the dynamical system. We determine what the necessary ingredients are for saturation of instabilities at finite amplitude in terms of wave-wave interactions and feedback between various azimuthal components of the vorticity field.

}, keywords = {2-dimensional incompressible flows, emergence, rotating fluid, stability, tripoles, turbulence, vortex}, isbn = {0022-1120}, doi = {10.1017/s0022112099004760}, url = {Laboratory experiments have shown that monopolar isolated vortices in a rotating flow undergo instabilities that result in the formation of multipolar vortex states such as dipoles and tripoles. In some cases the instability is entirely two-dimensional, with the vortices taking the form of vortex columns aligned along the direction of the ambient rotation at all times. In other cases, the vortex first passes through a highly turbulent three-dimensional state before eventually reorganizing into vortex columns. Through a series of three-dimensional numerical simulations, the roles that centrifugal instability, barotropic instability, and the bottom Ekman boundary layer play in these instabilities are investigated. Evidence is presented that the centrifugal instability can trigger the barotropic instabilities by the enhancement of vorticity gradients. It is shown that the bottom Ekman layer is not essential to these instabilities but can strongly modify their evolution.

}, keywords = {2-dimensional incompressible flows, barotropic vortices, stability, tripolar, turbulent-flow, vortex, vortices}, isbn = {0022-1120}, doi = {10.1017/s0022112098003693}, url = {A vortex approaching a no-slip wall {\textquoteright}rebounds{\textquoteright} due to the creation of vorticity at the wall in a viscous boundary layer. Here it is demonstrated that a purely inviscid mechanism can also produce vortex rebound from a slip wall. In inviscid vortex rebound, vortex tube stretching generates the necessary vorticity to allow rebound, eliminating the need for viscous vorticity generation. This vortex stretching mechanism is demonstrated through numerical simulations and laboratory experiments on dipole-vortex rebound from a boundary. In an application to oceanography, numerical simulations of both quasi-geostrophic and shallow water dynamics are used to show that the beta-effect at an eastern boundary can produce this inviscid rebound. Through a series of numerical experiments in which the strength of the beta-effect is varied, a formula for predicting the point of separation of the vortices from the boundary in a dipole-coast collision is deduced. Through simulations, the flux of vorticity and fluid away from the boundary is measured as a function of beta and initial angle of incidence. It is found that, in contrast to viscous vortex rebound, which typically does not produce a flux of material away from the boundary farther than a distance comparable to the initial vortex radius, the beta-induced rebound does carry fluid far from the coast. Laboratory experiments in a rotating tank are used to show that a sloping bottom can also provide an inviscid mechanism for dipole-vortex rebound from the wall of the tank under certain conditions. A relation determining the conditions under which inviscid or viscous processes will dominate in the rebound of the dipole from a boundary is obtained.

}, keywords = {barotropic vortices, beta-plane, california, cold filaments, fluid, modons, propagation, rotating tank, surface, topography}, isbn = {0022-1120}, doi = {10.1017/s0022112097007155}, url = {Numerical experiments are used to study the evolution of perturbed vortex tubes in a rotating environment in order to better understand the process of two-dimensionalization of unsteady rotating flows. We specifically consider non-axisymmetric perturbations to columnar vortices aligned along the axis of rotation. The basic unperturbed vortex is chosen to have a Gaussian cross-sectional vorticity distribution. The experiments cover a parameter space in which both the strength of the initial perturbation and the Rossby number are varied. The Rossby number is defined here as the ratio of the maximum amplitude of vorticity in the Gaussian vorticity profile to twice the ambient rotation rate. For small perturbations and small Rossby numbers, both cyclones and anticyclones behave similarly, relaxing rapidly back toward two-dimensional columnar vortices. For large perturbations and small Rossby numbers, a rapid instability occurs for both cyclones and anticyclones in which antiparallel vorticity is created. The tubes break up and then re-form again into columnar vortices parallel to the rotation axis (i.e. into a quasi-two-dimensional flow) through nonlinear processes. For Rossby numbers greater than 1, even small perturbations result in the complete breakdown of the anticyclonic vortex through centrifugal instability, while cyclones remain stable. For a range of Rossby numbers greater than 1, after the breakdown of the anticyclone, a new weaker anticyclone forms, with a small-scale background vorticity of spectral shape given approximately by the -5/3 energy spectral law.

}, keywords = {barotropic vortices, evolution, fluid, homogeneous turbulence, isotropic turbulence, laboratory experiments, large-eddy simulation, merger, tank, waves}, isbn = {0022-1120}, doi = {10.1017/s0022112097005430}, url = {The formation and the evolution of axisymmetric vortex rings in a uniformly rotating fluid, with the rotation axis orthogonal to the ring vorticity, have been investigated by numerical and laboratory experiments. The flow dynamics turned out to be strongly affected by the presence of the rotation. In particular, as the background rotation increases, the translation velocity of the ring decreases, a structure with opposite circulation forms ahead of the ring and an intense axial vortex is generated on the axis of symmetry in the tail of the ring. The occurrence of these structures has been explained by the presence of a self-induced swirl flow and by inspection of the extra terms in the Navier-Stokes equations due to rotation. Although in the present case the swirl was generated by the vortex ring itself, these results are in agreement with those of Virk et al. (1994) for polarized vortex rings, in which the swirl flow was initially assigned as a {\textquoteright}degree of polarization{\textquoteright}. If the rotation rate is further increased beyond a certain value, the flow starts to be dominated by Coriolis forces. In this flow regime, the impulse imparted to the fluid no longer generates a vortex ring, but rather it excites inertial waves allowing the flow to radiate energy. Evidence of this phenomenon is shown. Finally, some three-dimensional numerical results are discussed in order to justify some asymmetries observed in flow visualizations.

}, isbn = {0022-1120}, doi = {10.1017/s0022112096000730}, url = {For an anisotropic topographic feature in a large-scale flow, the orientation of the topography with respect to the flow will affect the vorticity production that results from the topography-flow interaction. This in turn affects the amount of form drag that the ambient flow experiences. Numerical simulations and perturbation theory are used to explore these effects of change in topographic orientation. The flow is modelled as a quasi-geostrophic homogeneous fluid on anf-plane. The topography is taken to be a hill of limited extent, with an elliptical cross-section in the horizontal. It is shown that, as a result of a basic asymmetry of the quasi-geostrophic flow, the strength of the form drag depends not only on the magnitude of the angle that the topographic axis makes with the oncoming stream, but also on the sign of this angle. For sufficiently low topography, it is found that a positive angle of attack leads to a stronger form drag than that for the corresponding negative angle. For strong topography, this relation is reversed, with the negative angle then resulting in the stronger form drag.

}, isbn = {0022-1120}, doi = {10.1017/s0022112095000565}, url = {Associated with intense propagating vortices is a separatrix defining a region of fluid that is transported with the vortex. The distortion of this separatrix, under external perturbations, leads to entrainment and detrainment of fluid. The detrained fluid is shed in lobes in the wake of the vortex. Examples of this phenomenon for a propagating monopole and dipole are provided from rotating-tank experiments and numerical simulations.

}, keywords = {barotropic vortices, beta-plane, filamentation, flow, laboratory experiments, modons, stability, stratified fluid, topography, vortex}, isbn = {0167-2789}, doi = {10.1016/0167-2789(94)90256-9}, url = {Laboratory observations and numerical simulations reveal that, in addition to monopoles, dipoles and tripoles, yet another stable coherent vortex may emerge from unstable isolated circular vortices. This new vortex is the finite-amplitude result of the growth of an azimuthal wavenumber-3 perturbation. It consists of a triangular core of single-signed vorticity surrounded by three semicircular satellites of oppositely signed vorticity. The stability of this triangular vortex is analysed through a series of high-resolution numerical simulations and by an investigation of point-vortex models. This new compound vortex rotates about its centre and is stable to small perturbations. If perturbed strongly enough, it undergoes an instability in which two of the outer satellites merge, resulting in the formation of an axisymmetric tripole, which subsequently breaks down into either a pair of dipoles or a dipole plus a monopole. The growth of higher-azimuthal-wavenumber perturbations leads to the formation of more intricate compound vortices with cores in the shape of squares, pentagons, etc. However, numerical simulations show that these vortices are unstable, which agrees with results from point-vortex models.

}, keywords = {geostrophic vortices, motion, rotating fluid, vortex}, isbn = {0022-1120}, doi = {10.1017/s0022112094000157}, url = {It is shown how symmetric dipolar vortices can be formed by the action of an impulsive jet in a homogeneous single layer of fluid in a rotating tank. These dipoles are allowed to interact with a constant topographic slope, which can model a beta-plane or a continental shelf. A dipole{\textquoteright}s trajectory bends toward the right when climbing a slope and to the left when descending, as predicted by numerical simulations and analytical arguments. The maximum penetration of the dipoles over a slope, the adjustment to the slope, and formation of trailing lobes are compared with both numerical simulations and a two-point vortex model. The results suggest that Rossby wave radiation plays an important role in the interaction process.

}, keywords = {california current, coastal current, couples, eddies, evolution, fluid, laboratory experiments, meanders, modons, vortex, vortices}, isbn = {0377-0265}, doi = {10.1016/0377-0265(93)90032-3}, url = {In this paper, we will describe what are (in our view) the newest and most exciting trends in current research on language development; trends that are likely to predominate in the few years that remain until the millennium. The paper is organized into six sections: (1) advances in data sharing (including the Child Language Data Exchange System), (2) improved description and quantification of the linguistic data to which children are exposed and the data that they produce (with implications for theories of language learning); (3) new theories of learning in neural networks that challenge old assumptions about the "learnability" (or unlearnability) of language, (4) increased understanding of the nonlinear dynamics that may underlie behavioral change, (5) research on the neural correlates of language learning, and (6) an increased understanding of the social factors that influence normal and abnormal language development.

}, keywords = {acquisition, child language, cortex, focal brain injury, late talkers, lexical development, models, negative evidence, positron emission tomography, speech sounds}, isbn = {0273-2297}, doi = {10.1006/drev.1993.1020}, url = {The second variation of a linear combination of energy and angular momentum is used to investigate the formal stability of circular vortices. The analysis proceeds entirely in terms of Lagrangian displacements to overcome problems that otherwise arise when one attempts to use Arnol{\textquoteright}d{\textquoteright}s Eulerian formalism. Specific attention is paid to the simplest possible model of an isolated vortex consisting of a core of constant vorticity surrounded by a ring of oppositely signed vorticity. We prove that the linear stability regime for this vortex coincides with the formal stability regime. The fact that there are formally stable isolated vortices could imply that there are provable nonlinearly stable isolated vortices. The method can be applied to more complicated vortices consisting of many nested rings of piecewise-constant vorticity. The equivalent expressions for continuous vorticity distributions are also derived.

}, keywords = {flows, nonlinear stability, Parallel, theorem}, isbn = {0022-1120}, doi = {10.1017/s0022112092002362}, url = {A recently proposed scaling theory of two-dimensional turbulent decay, based on the evolutionary pathway of successive mergers of coherent vortices, is used to predict the rate and end state of the evolution. These predictions differ from those based on the selective-decay hypothesis and traditional ideas of spectrum evolution, and they are in substantially better agreement with numerical solutions at large Reynolds number.

}, isbn = {0899-8213}, doi = {10.1063/1.858251}, author = {Carnevale, G. F. and McWilliams, J. C. and Pomeau, Y. and Weiss, J. B. and Young, W. R.} } @article {25321, title = {Fluctuation-response relations in systems with chaotic behavior}, journal = {Physics of Fluids a-Fluid Dynamics}, volume = {3}, number = {9}, year = {1991}, note = {n/a}, month = {Sep}, pages = {2247-2254}, type = {Article}, abstract = {The statistics of systems with good chaotic properties obey a formal fluctuation-response relation which gives the average linear response of a dynamical system to an external perturbation in terms of two-time correlation functions. Unfortunately, except for particularly simple cases, the appropriate form of correlation function is unknown because an analytic expression for the invariant density is lacking. The simplest situation is that in which the probability distribution is Gaussian. In that case, the fluctuation-response relation is a linear relation between the response matrix and the two-time two-point correlation matrix. Some numerical computations have been carried out in low-dimensional models of hydrodynamic systems. The results show that fluctuation-response relation for Gaussian distributions is not a useful approximation. Nevertheless, these calculations show that, even for non-Gaussian statistics, the response function and the two-time correlations can have similar qualitative features, which may be attributed to the existence of the more general fluctuation-response relation.

}, keywords = {dissipation, dynamics, linear response, model, statistical-mechanics, turbulence}, isbn = {0899-8213}, doi = {10.1063/1.857905}, url = {Freely evolving two-dimensional turbulence is dominated by coherent vortices. The density of these vortices decays in time as rho approximately t^-ɛ with ɛ almost-equal-to 0.75. A new scaling theory is proposed which expresses all statistical properties in terms of ɛ. Thus the average circulation of the vortices increases as t^ɛ/2 and their average radius as t^ɛ/4. The total energy is constant, the enstrophy decreases as t^ɛ/2, and the vorticity kurtosis increases as t^ɛ/2. These results are supported both by numerical simulations of the fluid equations and by solutions of a modified point-vortex model.

}, keywords = {2 dimensions, dimensional decaying turbulence, dynamics, merger, vortices}, isbn = {0031-9007}, doi = {10.1103/PhysRevLett.66.2735}, url = {n/a

}, month = {May}, pages = {1411-1415}, type = {Article; Proceedings Paper}, abstract = {Theory and simulations based on the two-dimensional Euler equation predict a critical distance of separation for the merger of two like-signed vortices. By the symmetry of the equation, this separation must be the same for both cyclone and anticyclone pairs. In rotating-tank experiments, the observed critical separation distance for anticyclone merger is in accord with predictions; however, pairs of cyclones have been found to merge in all cases examined, even with separations substantially greater than the predicted critical separation. The hypothesis that this discrepancy is due to the presence of Ekman volume fluxes, which are not incorporated in the two-dimensional theory, is examined and found not quantitatively supportable. A second hypothesis is that the parabolic curvature of the free upper surface of the fluid in the rotating tank induces motion of the cyclones toward the center of the tank and hence promotes the cyclone pair merger. Quasigeostrophic simulations which capture this "topography effect" show good agreement with the rotating-tank experiments.

}, keywords = {2 dimensions, vortices}, isbn = {0899-8213}, doi = {10.1063/1.858019}, url = {A small-scale cyclonic vortex in a relatively broad valley tends to climb up and out of the valley in a cyclonic spiral about the centre, and when over a relatively broad hill it tends to climb toward the top in an anticyclonic spiral around the peak. This phenomenon is examined here through two-dimensional numerical simulations and rotating-tank experiments. The basic mechanism involved is shown to be the same as that which accounts for the northwest propagation of cyclones on a beta-plane. This inviscid nonlinear effect is also shown to be responsible for the observed translationary motion of barotropic vortices in a free-surface rotating tank. The behaviour of isolated vortices is contrasted with that of vortices with non-vanishing circulation.

}, keywords = {beta-plane, eddies, eddy, evolution, flow, gulf-stream rings, modons, motion, statistical-mechanics, turbulence}, isbn = {0022-1120}, doi = {10.1017/s0022112091000411}, url = {Analytic expressions are given for statistical mechanical equilibrium solutions of two-field turbulence model equations that are used in describing plasma drift waves and for passive scalar advection in a neutral fluid. These are compared with those previously proposed [Phys. Fluids B 1, 1331 (1989)], in particular regarding the role of the cross correlations between fields. Implications for two-point closure calculations are discussed.

}, isbn = {0899-8221}, doi = {10.1063/1.859822}, author = {Koniges, A. E. and Crotinger, J. A. and Dannevik, W. P. and Carnevale, G. F. and Diamond, P. H.} } @article {25317, title = {Statistics of ballistic agglomeration}, journal = {Physical Review Letters}, volume = {64}, number = {24}, year = {1990}, note = {n/a}, month = {Jun}, pages = {2913-2916}, type = {Article}, abstract = {We consider a {\textquoteleft}{\textquoteleft}sticky gas{\textquoteright}{\textquoteright} in which collisions between spherical particles are perfectly inelastic. Thus the two colliding particles conserve mass and momentum, but merge to form a single more massive sphere. A scaling argument suggests that the average mass of a particle grows as t^2D/(2+D), where D is the spatial dimension. In the case D=1 this result is confirmed by numerical simulation.

}, isbn = {0031-9007}, doi = {10.1103/PhysRevLett.64.2913}, url = {The continuous transformation of one flow into another of higher or lower energy while preserving the potential vorticity of all particles can be accomplished by advection with an artificial velocity field. Since isolated extremal energy states are stable states, this method can be used to find stable stationary flows on a prescribed isovortical sheet. A series of numerical simulations of this method for two-dimensional fluids that demonstrates its feasibility and utility is presented. Additionally, a corollary to Arnol{\textquoteright}d{\textquoteright}s nonlinear stability theorems is discussed, which shows that there can be at most two Arnol{\textquoteright}d stable states per isovortical sheet.

}, isbn = {0022-1120}, doi = {10.1017/s0022112090002440}, url = {Andrews (1984) has shown that any flow satisfying Arnol{\textquoteright}d{\textquoteright}s (1965, 1966) sufficient conditions for stability must be zonally-symmetric if the boundary conditions on the flow are zonally-symmetric. This result appears to place very strong restrictions on the kinds of flows that can be proved to be stable by Arnol{\textquoteright}d{\textquoteright}s theorems. In this paper, Andrews{\textquoteright} theorem is re-examined, paying special attention to the case of an unbounded domain. It is shown that, in that case, Andrews{\textquoteright} theorem generally fails to apply, and Arnol{\textquoteright}d-stable flows do exist that are not zonally-symmetric. The example of a circular vortex with a monotonic vorticity profile is a case in point. A proof of the finite-amplitude version of the Rayleigh stability theorem for circular vortices is also established; despite its similarity to the Arnol{\textquoteright}d theorems it seems not to have been put on record before.

}, isbn = {0309-1929}, doi = {10.1080/03091929008219847}, url = {Certain modifications of the Euler equations of fluid motion lead to systems in which the energy decays or grows monotonically, yet which preserve other dynamically important characteristics of the field. In particular, all topological invariants associated with the vorticity field are preserved. In cases where isolated energy extrema exist, a stable steady flow can be found. In two dimensions, highly constrained by vorticity invariants, it is shown that the modified dynamics will lead to at least one non-trivial stationary, generally stable, solution of the equations of motion from any initial conditions. Numerical implementation of the altered dynamics is straightforward, and thus provides a practical method for finding stable flows. The method is sufficiently general to be of use in other dynamical systems.Insofar as three-dimensional turbulence is characterized by a cascade of energy, but not topological invariants, from large to small scales, the procedure has direct physical significance. It may be useful as a parameterization of the effects of small unresolved scales on those explicitly resolved in a calculation of turbulent flow.

}, isbn = {0022-1120}, doi = {10.1017/s0022112089002533}, url = {In this paper we suggest that the longevity of the enhanced predictability periods often observed in the weather and general circulation models can he quantified by a study of the statistical moments of error growth rates as has been demonstrated for dynamical systems. As an illustration, it is shown how this approach can he pursued in simple cases. For the Lorenz model, the probability density distribution of error growth is close to log-normal and the average growth rate is two times shorter than the most probable. In general, we argue that the ratio of the average growth rate to the most probable is a measure of enhanced predictability.

}, isbn = {0022-4928}, doi = {10.1175/1520-0469(1989)046<3595:apmolp>2.0.co;2}, url = {A summary of a numerical study of the stability of modons to topographic perturbation is presented. Previous studies have suggested a monotonic relationship between the horizontal scale of the perturbation and the amplitude needed to destroy a modon{\textemdash}as the scale of the perturbation increases the strength needed for destruction decreases. The results presented here show that this relationship does not hold for scales larger than the modon radius. For large-scale perturbations, the strength needed for destruction again increases. The modon is most stable to perturbations of horizontal scale either much larger or much smaller than the modon radius. Stability graphs are presented for three types of perturbations; ridges, hills, and irregular terrain.

}, isbn = {1070-6631}, doi = {10.1063/1.866533}, url = {Lyapunov stability arguments may be used to show that an otherwise unstable flow can be stabilized by restriction of the class of possible perturbations. It is shown that, in general, such a restriction applied only to the initial perturbation does not imply stability for dynamics on the entire phase space nor does it necessarily imply a delay of the onset of instability. As a result, proofs of linear stability based on a restriction of the initial perturbation actually are not valid. In particular, certain criteria for the stability of modons given by Pierini [Dyn. Atmos. Oceans 9, 273 (1985)] and Swaters [Phys. Fluids 29, 1419 (1986)] and synthesized by Flierl [Annu. Rev. Fluid Mech. 19, 493 (1987)] do not, in fact, ensure stability. A model is used to demonstrate that these stability criteria do not preclude instantaneous onset of linear instability. The model also demonstrates that, although conservation of energy and enstrophy implies that the transfer of energy in an instability must be to scales both larger and smaller than the modon scale, the principal direction of transfer remains undetermined.

}, isbn = {1070-6631}, doi = {10.1063/1.866534}, url = {This is a broad survey of the interaction of modons with topography in a one-layer, quasigeostrophic model. Numerical simulations of modons interacting with ridges, hills, random topography and other obstacles are presented. The behavior of the modon is compared to numerical simulations of a two-point-vortex model, which proves a useful guide to the basic trajectory deflection mechanism. Under sufficiently strong but quasigeostrophically valid topographic perturbations, the modon is shown to fission into two essentially independent, oppositely-signed vortices. In the breakup of a modon near a hill it is found that the positive vortex migrates to the top of the hill. The resulting correlation between the positive vorticity trapped over the hill and the topography is in sharp contrast with the theories of turbulent flow over topography and generation of flow over topography by large scale forcing, both of which describe the development of vorticity anticorrelated with topography. A heuristic explanation of this new behavior is provided in terms of the dynamics of β bT-plane vortices. Further, it is found that a modon travelling over rough topography homogenizes the field of potential vorticity in its vicinity. This behavior is explained in terms of the induced eddy activity near the modon.

}, isbn = {0309-1929}, doi = {10.1080/03091928808208831}, url = {The stability properties and stationary statistics of inviscid barotropic flow over topography are examined. Minimum enstrophy states have potential vorticity proportional to the streamfunction and are nonlinearly stable; correspondingly, canonical equilibrium based on energy and enstrophy conservation predicts mean potential vorticity is proportional to the mean streamfunction. It is demonstrated that in the limit of infinite resolution the canonical mean state is statistically sharp, that is, without any eddy energy on any scale, and is identical to the nonlinearly stable minimum enstrophy state. Special attention is given to the interaction between small scales and a dynamically evolving large-scale flow. On the β-plane, these stable flows have a westward large-scale component. Possibilities for a general relation between inviscid statistical equilibrium and nonlinear stability theory are examined.

}, isbn = {0022-1120}, doi = {10.1017/s002211208700034x}, url = {The nonlinear stability properties of stationary exact nonzonal solutions for inviscid flow over topography are examined within a barotropic model in spherical geometry. For stationary solutions, such that the potential vorticity is proportional to the streamfunction, necessary and sufficient conditions for nonlinear stability are established. For a truncated system with rhomboidal truncation wave number J these are that the solid body rotation component of the zonal wind u(i) be negative, corresponding to westward flow, as J -\>infinity. These results are established by using the methods of statistical mechanics. The sufficient condition for stability is also established by applying Arnol{\textquoteright}d{\textquoteright}s method. The results are illustrated by numerical calculations. The stationary solutions are perturbed by disturbances in the streamfunction fields or by small changes in the topographic height; the climatic states for the system are obtained directly using statistical mechanics methods. The nonlinear stability properties of the stationary solutions are obtained by comparing the stationary solution with the climate, which for inviscid flow is shown to be unique. Stationary flows for which u(i) is eastward, are found to be unstable even in the limit as the streamfunction perturbation or change in the topographic height vanish. Large amplitude transient waves are generated which break the time invariance symmetry of the initial stationary flows. In contrast, for stationary flows with westward u(i), the climate is identical to the initial flow in the limit as the initial streamfuncton perturbation or the change in the topographic height vanishes. The linear instability characteristics of the stationary solutions are also obtained by solving a linear eigenvalue problem. The difficulties in establishing the stability properties of more general exact solutions, where the streamfunction is a general differentiable function of the potential vorticity, within numerical spectral models are discussed.

}, isbn = {0309-1929}, doi = {10.1080/03091928608245892}, url = {n/a

}, pages = {289-303}, type = {Article}, abstract = {Approximations in statistical turbulence theory often rely on modelling the decay in time of velocity correlations with a simple exponential decay. The decay rate is viewed as a renormalized viscosity. The three simplest implementations of this approximation scheme were originally given independently by Kraichnan, Edwards and Leslie. Each of these investigators used a different formalism and each achieved different renormalization prescriptions. These three different results are reexamined here entirely in terms of direct-interaction theory. The difference in the prescriptions of Kraichnan and Leslie is shown to be the product of different definitions of renormalized viscosity. Edwards{\textquoteright} prescription is shown to result from an inconsistent identification of the non-stationary energy-spectrum relaxation rate with the viscosity. An assessment of the validity of the Markovian closure approximation, and a prescription for non-stationary renormalized viscosity are provided.

}, isbn = {0022-1120}, doi = {10.1017/s0022112083001330}, url = {In this paper we define the Painlev{\'e} property for partial differential equations and show how it determines, in a remarkably simple manner, the integrability, the B{\"a}cklund transforms, the linearizing transforms, and the Lax pairs of three well-known partial differential equations (Burgers{\textquoteright} equation, KdV equation, and the modified KdV equation). This indicates that the Painlev{\'e} property may provide a unified description of integrable behavior in dynamical systems (ordinary and partial differential equations), while, at the same time, providing an efficient method for determining the integrability of particular systems.

}, isbn = {0022-2488}, doi = {10.1063/1.525721}, url = {A statistical dynamical closure theory describing the interaction of strongly (and weakly) nonlinear two-dimensional internal waves in the presence of viscous dissipation and thermal conduction is derived. By applying renormalization methods originally formulated for quantum and classical statistical field theory, closures similar to the Direct Interaction and eddy-damped quasi-normal procedures of turbulence are derived. These methods are applied directly to the strongly nonlinear primitive field equations in Eulerian variables, thus avoiding the small amplitude assumptions inherent in the resonant interaction formalism. Propagator renormalization techniques provide formulas for the nonlinear internal wave frequency and spectral width in terms of the energy spectrum. The commonly used multiple time and space scale analysis is replaced by an analysis of the two-point correlation functions in terms of sum and difference variables. This permits the systematic development of a Landau equation. This generalization of the Boltzmann equation incorporates spatial variation of the group velocity and scattering due to spatial inhomogeneity. In the limit of weakly interacting waves and zero viscosity, the closures reduce to the resonant interaction approximation formalism. It is shown that the inviscid resonant interaction limit is singular in the sense that the quilibrium spectrum differs from that of the general inviscid nonlinear off-resonant case. This is due to the fact that in the resonant interaction limit there is an additional constant of motion, viz. {\textquotedblleft}z-momentum{\textquotedblright}. The implications of these results are discussed.

}, isbn = {0309-1929}, doi = {10.1080/03091928308209042}, url = {A derivation of two-point Markovian closure is presented in classical statistical field theory formalism. It is emphasized that the procedures used in this derivation are equivalent to those employed in the quantum statistical field theory derivation of the Boltzmann equation. Application of these techniques to the study of two-dimensional flow on a β-plane yields a quasi-homogeneous, quasi-stationary transport equation and a renormalized dispersion relation for Rossby waves

}, isbn = {0309-1929}, doi = {10.1080/03091928208209002}, url = {A measure of predictability that has many superior features compared to currently used measures is introduced. Through statistical theory it is demonstrated that in inviscid truncated flow this new predictability measure increases monotonically in time while all initial information about the system decays. Under the influence of forcing and viscosity the behaviour of this measure is shown always to satisfy intuitive expectations.

}, isbn = {0022-1120}, doi = {10.1017/s0022112082000391}, url = {Liouville{\textquoteright}s equation for randomly forced two-dimensional flow with Rayleigh friction is examined. An exact nonstationary solution is presented for a special form of the forcing and zero energy initial condition. This solution is such that the fluctuation-dissipation relation is valid at all times.

}, isbn = {1070-6631}, doi = {10.1063/1.863942}, url = {Statistical fluid dynamics identifies a functional of the fluid energy spectrum that plays the role of Boltzmann{\textquoteright}s entropy for fluids. Through a series of two-dimensional flow simulations we confirm the theoretical predictions for the behaviour of this entropy functional. This includes a demonstration of Loschmidt{\textquoteright}s paradox and an examination of the effects of Rossby waves and viscosity on the behaviour of the entropy.

}, isbn = {0022-1120}, doi = {10.1017/s0022112082002134}, url = {It is demonstrated that the second-order Markovian closures frequently used in turbulence theory imply an H theorem for inviscid flow with an ultraviolet spectral cut-off. That is, from the inviscid closure equations, it follows that a certain functional of the energy spectrum (namely entropy) increases monotonically in time to a maximum value at absolute equilibrium. This is shown explicitly for isotropic homogeneous flow in dimensions d\>or=2, and then a generalised theorem which covers a wide class of systems of current interest is presented. It is shown that the H theorem for closure can be derived from a Gibbs-type H theorem for the exact non-dissipative dynamics.

}, isbn = {0305-4470}, doi = {10.1088/0305-4470/14/7/026}, url = {