Andrei Natarov

Beyond-all-orders instability in the equatorial Kelvin wave

Abstract The equatorial Kelvin wave is destabilized by cross-equatorial shear in a rather bizarre fashion. There is no well-defined inviscid neutral curve. Instead, using as an exemplar the linear shear U ( y )= ϵy where U is the mean zonal flow and y is non-dimensional latitude, the Kelvin wave is unstable for all ϵ , however small. We show through multiple precision numerical calculations that the imaginary part of the phase speed is proportional to the square of the unperturbed eigenfunction at the critical latitude, that is, I (c)∝ exp (−1/ϵ 2 ) for small ϵ . Such exponential dependence on a small parameter implies that the result lies “beyond-all-orders” in a perturbation series in powers of ϵ . We connect our numerical work here with our previous study of a model problem through ‘beyond-all-orders’ perturbation theory to argue that the Kelvin instability is a critical latitude phenomenon.

Three-dimensional instabilities of oscillatory equatorial zonal shear flows

In this paper, we investigate the linear stability of oscillating zonal flows on the equatorial β-plane in the presence of fully three-dimensional disturbances. To exclude inflection point effects, we focus on the simplest case of a linear meridional shear with time-mean and oscillating components. For purely oscillatory background flows we find that in addition to resonant excitation of ‘additive’ type that occurs in the zonally invariant case, resonant excitation of ‘difference’ type is also possible. For flows with an oscillatory shear superimposed on an unstable time-mean shear it is shown that while the oscillatory shear has a stabilizing influence on disturbances with a small zonal wave number k, at higher k the effect of the oscillating shear diminishes and can even be destabilizing. Overall, a small oscillatory shear tends to reduce the fastest growth rate in the system and pushes the dominant k to higher values. Calculation of dominant zonal and vertical modes shows that the zonally asymmetric modes dominate a large portion of the parameter space, especially at high time-mean background shear and low oscillatory shear. As a result, the dominant vertical mode can have a somewhat larger vertical scale than in the zonally invariant case. At intermediate values of the time-mean shear the growth rate is relatively flat with respect to the zonal mode number, with maximum growth rate occurring in bands of high and low k. We have uncovered a rich assortment of vertical and zonal modes which are likely to play a role in the nonlinear evolution of equatorial flows.

Two-dimensional instabilities of time-dependent zonal flows: linear shear

In this study, we investigate the stability of time-dependent zonal flows to two-dimensional (zonally symmetric) disturbances. While steady currents can only experience inertial instability (II) in this setting, unsteady ones may be destabilized in other ways. For example, time-periodic flows can be subject to parametric subharmonic instability (PSI). Motivated by observations of salinity interleaving patterns in the upper equatorial Pacific Ocean, our objective is to determine the basic properties of dominant instabilities (their generation mechanism, spatial and temporal characteristics, and finite-amplitude development) for background flows that are representative of those in the upper-equatorial ocean, yet still amenable to a computational sweep of parameter space. Our approach is to explore the stability of solutions to linear and nonlinear versions of a two-dimensional model for an idealized background flow with oscillating linear shear. To illustrate basic properties of the instabilities, the f -plane and equatorial β-plane scenarios are studied using a linear model. Stability regime diagrams show that on the f -plane there is a clear separation in dominant vertical scales between PSI- and II-dominated regimes, whereas on the equatorial β-plane the parameter space contains a region where dominant instability is a mixture of the two types. In general, PSI favours lower vertical modes than II. The finite-amplitude development of instabilities on the equatorial β-plane is explored using a nonlinear model, including cases illustrating the equilibration of pure II and the development of pure PSI and mixed instabilities. We find that unless the instabilities are weak enough to be equilibrated by viscosity at low amplitude, disturbances continue to grow until the vertical shear of their meridional velocity field becomes large enough to allow for Richardson numbers less than 1/4; as a consequence, PSI-favoured vertical modes are able to reach higher amplitudes than II-favoured modes before becoming susceptible to Kelvin–Helmholtz instability, and induce tracer intrusions of a considerably larger meridional extent.

Shear‐generated turbulence in the equatorial Pacific produced by small vertical scale flow features

We investigate the characteristics of shear-generated turbulence in the natural environment by considering data from a number of cruises in the western equatorial Pacific. In this region, the vertical shear of the flow is dominated by flow structures that have a relatively small vertical scale of O(10 m). Combining data from all cruises, we find a strong relationship between the turbulent dissipation rate, ϵ, vertical shear, S, and buoyancy frequency, N. Examination of ϵ at a fixed value of Richardson number, Ri = N2∕S2, shows that ϵ∝ut2N for a wide range of values of N, where ut is an appropriate velocity scale which we assume to be the horizontal velocity scale of the turbulence. The implied vertical length scale, lv = ut∕N, is consistent with theoretical and numerical studies of stratified turbulence. Such behavior is found for Ri < 0.4. The vertical diffusion coefficient then scales as κv∝ut2/N at a fixed value of Richardson number. The amplitude of ϵ is found to increase with decreasing Ri, but only modestly, and certainly less dramatically than suggested by some parameterization schemes. Provided the shear generating the turbulence is resolved, our results point to a way to parameterize the unresolved turbulence.

Persistent presence of small vertical scale velocity features during three-dimensional equilibration of equatorial inertial instability

Small vertical scale velocity features (SVSs) are ubiquitous in the upper equatorial ocean but their lifecycle and role in the large scale dynamics are only beginning to be understood. In this article, we study the development of SVSs generated through inertial instability of a prototypical equatorial zonal flow with a uniform meridional shear, U(y) = Λy. While previous studies employ a zonally symmetric setting, in which the flow is constrained to remain invariant in the streamwise direction throughout its evolution, here we use a fully three-dimensional framework. We choose a setting, in which the fastest growing linear modes are zonally symmetric and the initial perturbation is almost zonally symmetric, so that the flow remains nearly two-dimensional until the symmetric instability is fully neutralized. The secondary instabilities of the modified zonal mean flow favor zonally nonsymmetric modes, which leads to three-dimensionalization of the flow. It has been previously conjectured in the literature that the dominant secondary instability is an inflection point-type barotropic instability that favors disturbances with a large vertical scale and the necessary condition for which is the reversal of the meridional gradient of the background potential vorticity. We show that although the conditions necessary for barotropic instability indeed arise after the neutralization of the symmetric instability, the dominant secondary instabilities form a sequence of high vertical mode disturbances excited through zonally nonsymmetric inertial instability of the evolving zonal mean flow. The barotropic eddies, therefore, remain underdeveloped and do not play an important role in the dynamics. We cannot rule out the possible influence of the applied lateral hyperdiffusion (used to control near-grid point noise in the integrations) on the relative roles of barotropic and inertial instability. The competition between these two possible secondary instabilities needs further investigation with higher resolution experiments. We note, however, that when zonally asymmetric inertial instability is not possible (achieved here by shortening the zonal extent of the integration region while keeping the hyperdiffusion coefficient the same), barotropic instability fully develops and the flow is dominated by the barotropic mode.