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Tort, M, Winters KB.  2018.  Poleward propagation of near-inertial waves induced by fluctuating winds over a baroclinically unstable zonal jet. Journal of Fluid Mechanics. 834:510-530.   10.1017/jfm.2017.698   AbstractWebsite

We investigate the excitation and radiation of near-inertial internal gravity waves continuously excited by a latitudinally confined temporally fluctuating wind in a numerical model of a stratified ocean on a beta-plane at mid-latitude. The surface wind forcing contains both high- and low-frequency components which excite propagating waves and a baroclinically unstable zonal jet respectively. Wentzel-Kramers-Brillouin (WKB) ray theory implies that near-inertial waves propagate strictly towards the equator. We seek to refine this view here by (i) adding the non-traditional Coriolis force (accounting for the horizontal component of the Earth's rotation) into the equations of motion, in order to allow poleward sub-inertial propagation to occur, and (ii) relaxing the conceptual constraint of no zonal variability, to allow the zonal jet to undergo instability, to meander and to sustain an active field of mesoscale eddies, potentially impacting the excitation of near-inertial waves. The key results are that, while (i) permits weakly stratified waveguides with sub-inertial poleward wave propagation to develop in accord with theory, the sub-inertial energy flux observed is very small compared with the equatorward flux. Thus, in terms of energy radiated from the storm track, non-traditional effects are small for wind-driven near-inertial waves. The consequences of (ii) are much more pronounced. Refinement (ii) produces a radiating wave field that is bidirectional, i.e. with both poleward and equatorward components. We show that the presence of regions of significant background vorticity with horizontal scales significantly smaller than the width of the storm track provides the scale selection mechanism to excite waves with sufficiently super-inertial frequencies to propagate poleward distances of the order of 1000 km.

Jagannathan, A, Winters KB, Armi L.  2017.  Stability of stratified downslope flows with an overlying stagnant isolating layer. Journal of Fluid Mechanics. 810:392-411.   10.1017/jfm.2016.683   AbstractWebsite

We investigate the dynamic stability of stratified flow configurations characteristic of hydraulically controlled downslope flow over topography. Extraction of the correct 'base state' for stability analysis from spatially and temporally evolving flows that exhibit instability is not easy since the observed flow in most cases has already been modified by nonlinear interactions between the instability modes and the mean flow. Analytical studies, however, can yield steady solutions under idealized conditions which can then be analysed for stability. Following the latter approach, we study flow profiles whose essential character is determined by recently obtained solutions of Winters & Armi (J. Fluid Mech., vol. 753, 2014, pp. 80-103) for topographically controlled stratified flows. Their condition of optimal control necessitates a streamline bifurcation which then naturally produces a stagnant isolating layer overlying an accelerating stratified jet in the lee of the topography. We show that the inclusion of the isolating layer is an essential component of the stability analysis and further clarify the nature and mechanism of the instability in light of the wave-interaction theory. The spatial stability problem is also briefly examined in order to estimate the downstream location where finite-amplitude features might be manifested in streamwise slowly varying flows over topography.

Winters, KB, Armi L.  2014.  Topographic control of stratified flows: upstream jets, blocking and isolating layers. Journal of Fluid Mechanics. 753:80-103.   10.1017/jfm.2014.363   AbstractWebsite

Optimal solutions to the nonlinear, hydrostatic, Boussinesq equations are developed for steady, density-stratified, topographically controlled flows characterized by blocking and upstream influence. These flows are jet-like upstream of an isolated obstacle and are contained within an asymmetric, thinning stream tube that is accelerated as it passes over the crest. A stagnant, nearly uniform-density isolating layer, surrounded by a bifurcated uppermost streamline, separates the accelerated flow from an uncoupled flow above. The flows are optimal in the sense that, for a given stratification, the solutions maximize the topographic rise above the blocking level required for hydraulic control while minimizing the total energy of the flow. Hydraulic control is defined mathematically by the asymmetry of the accelerated flow as it passes the crest. A subsequent analysis of the Taylor-Goldstein equation shows that these sheared, non-uniformly stratified flows are indeed subcritical upstream, critical at the crest, and supercritical downstream with respect to gravest-mode, long internal waves. The flows obtained are relevant to arrested wedge flows, selective withdrawal, stratified towing experiments, tidal flow over topography and atmospheric flows over mountains.

Winters, KB, Barkan R.  2013.  Available potential energy density for Boussinesq fluid flow. Journal of Fluid Mechanics. 714:476-488.   10.1017/jfm.2012.493   Abstract

An exact expression E-a for available potential energy density in Boussinesq fluid flows (Roullet & Klein, J. Fluid Mech., vol. 624, 2009, pp. 45-55; Holliday & McIntyre, J. Fluid Mech., vol. 107, 1981, pp. 221-225) is shown explicitly to integrate to the available potential energy E-a of Winters et al. (J. Fluid Mech., vol. 289, 1995, pp. 115-128). E-a is a positive definite function of position and time consisting of two terms. The first, which is simply the indefinitely signed integrand in the Winters et al. definition of E-a, quantifies the expenditure or release of potential energy in the relocation of individual fluid parcels to their equilibrium height. When integrated over all parcels, this term yields the total available potential energy E-a. The second term describes the energetic consequences of the compensatory displacements necessary under the Boussinesq approximation to conserve vertical volume flux with each parcel relocation. On a pointwise basis, this term adds to the first in such a way that a positive definite contribution to E-a is guaranteed. Globally, however, the second term vanishes when integrated over all fluid parcels and therefore contributes nothing to E-a. In effect, it filters the components of the first term that cancel upon integration, isolating the positive definite residuals. E-a can be used to construct spatial maps of local contributions to E-a for direct numerical simulations of density stratified flows. Because E-a integrates to E-a, these maps are explicitly connected to known, exact, temporal evolution equations for kinetic, available and background potential energies.

Winters, KB, Armi L.  2012.  Hydraulic control of continuously stratified flow over an obstacle. Journal of Fluid Mechanics. 700:502-513.   10.1017/jfm.2012.157   AbstractWebsite

Motivated by the laboratory experiments of Browand & Winant (Geophys. Fluid Dyn., vol. 4, 1972, pp. 29-53), a series of two-dimensional numerical simulations of flow past a cylinder of diameter d are run for different values of the approach Froude number Fr-0 = U/Nd between 0.02 and 0.2 at Re = O(100). The observed flow is characterized by blocking and upstream influence in front of the cylinder and by relatively thin, fast jets over the top and bottom of the cylinder. This continuously stratified flow can be understood in terms of an inviscid non-diffusive integral inertia-buoyancy balance reminiscent of reduced-gravity single-layer hydraulics, but one where the reduced gravity is coupled to the thickness of the jets. The proposed theoretical framework describes the flow upstream of the obstacle and at its crest. The most important elements of the theory are the inclusion of upstream influence in the form of blocked flow within an energetically constrained depth range and the recognition that the flow well above and well below the active, accelerated layers is dynamically uncoupled. These constraints determine, through continuity, the transport in the accelerated layers. Combining these results with the observation that the flow is asymmetric around the cylinder, i.e. hydraulically controlled, allows us to determine the active layer thicknesses, the effective reduced gravity and thus all of the integral flow properties of the fast layers in good agreement with the numerically computed flows.

Winters, KB, Bouruet-Aubertot P, Gerkema T.  2011.  Critical reflection and abyssal trapping of near-inertial waves on a β-plane. Journal of Fluid Mechanics. 684:111-136.   10.1017/jfm.2011.280   AbstractWebsite

We consider near-inertial waves continuously excited by a localized source and their subsequent radiation and evolution on a two-dimensional beta-plane. Numerical simulations are used to quantify the wave propagation and the energy flux in a realistically stratified ocean basin. We focus on the dynamics near and poleward of the inertial latitude where the local value of the Coriolis parameter f matches the forcing frequency sigma, contrasting the behaviour of waves under the traditional approximation (TA), where only the component of the Earth's rotation aligned with gravity is retained in the dynamics, with that obtained under the non-traditional approach (non-TA) in which the horizontal component of rotation is retained. Under the TA, assuming inviscid linear wave propagation in the WKB limit, all energy radiated from the source eventually propagates toward the equator, with the initially poleward propagation being internally reflected at the inertial latitude. Under the non-TA however, these waves propagate sub-inertially beyond their inertial latitude, exhibiting multiple reflections between internal turning points that lie poleward of the inertial latitude and the bottom. The numerical experiments complement and extend existing theory by relaxing the linearity and WKB approximations, and by illustrating the time development of the steadily forced flow and the spatial patterns of energy flux and flux divergence. The flux divergence of the flow at both the forcing frequency and its first harmonic reveal the spatial patterns of nonlinear energy transfer and highlight the importance of nonlinearity in the vicinity of near-critical bottom reflection at the inertial latitude of the forced waves.

Winters, KB.  2008.  Growth of inertia-gravity waves in sheared inertial currents. Journal of Fluid Mechanics. 601:85-100.   10.1017/s0022112008000621   AbstractWebsite

The linear stability of inviscid non-diffusive density-stratified shear flow in a rotating frame is considered. A temporally periodic base flow, characterized by vertical shear S, buoyancy frequency N and rotation frequency f, is perturbed by infinitesimal inertia-gravity waves. The temporal evolution and stability characteristics of the disturbances are analysed using Floquet theory and the growth rates of unstable solutions are computed numerically. The global structure of solutions is addressed in the dimensionless parameter space (N/f, S/f, phi) where phi is the wavenumber inclination angle from the horizontal for the wave-like perturbations. Both weakly stratified rapidly rotating flows (N < f) and strongly stratified slowly rotating flows (N > f) are examined. Distinct families of unstable modes are found, each of which can be associated with nearby stable solutions of periodicity T or 2T where T is the inertial frequency 2 pi/f. Rotation is found to be a destabilizing factor in the sense that stable non-rotating shear flows with N N(1)/S(2) > 1/4 can be unstable in a rotating frame. Morever, instabilities by parametric resonance are found associated with free oscillations at half and integer multiples of the inertial frequency.