Manikandan Mathur

We present a local stability analysis to investigate the effects of differential diffusion between momentum and density (quantified by the Schmidt number ) on the three-dimensional, short-wavelength instabilities in planar vortices with a uniform stable stratification along the vorticity axis. Assuming small diffusion in both momentum and density, but arbitrary values for , we present a detailed analytical/numerical analysis for three different classes of base flows: (i) an axisymmetric vortex, (ii) an elliptical vortex and (iii) the flow in the neighbourhood of a hyperbolic stagnation point. While a centrifugally stable axisymmetric vortex remains stable for any , it is shown that can have significant effects in a centrifugally unstable axisymmetric vortex: the range of unstable perturbations increases when is taken away from unity, with the extent of increase being larger for than for . Additionally, for 1$]]>, we report the possibility of oscillatory instability. In an elliptical vortex with a stable stratification, is shown to non-trivially influence the three different inviscid instabilities (subharmonic, fundamental and superharmonic) that have been previously reported, and also introduce a new branch of oscillatory instability that is not present at . The unstable parameter space for the subharmonic (instability IA) and fundamental (instability IB) inviscid instabilities are shown to be significantly increased for , respectively. Importantly, for sufficiently small and large and 1$]], respectively, the maximum growth rate for instabilities IA and IB occurs away from the inviscid limit. The new oscillatory instability (instability III) is shown to occur only for sufficiently small

In this paper, we use asymptotic theory and numerical methods to study resonant triad interactions among discrete internal wave modes at a fixed frequency in a two-dimensional, uniformly stratified shear flow. Motivated by linear internal wave generation mechanisms in the ocean, we assume the primary wave field as a linear superposition of various horizontally propagating vertical modes at a fixed frequency. The weakly nonlinear solution associated with the primary wave field is shown to comprise superharmonic (frequency) and zero frequency wave fields, with the focus of this study being on the former. When two interacting primary modes and are in triadic resonance with a superharmonic mode, it results in the divergence of the corresponding superharmonic secondary wave amplitude. For a given modal interaction, the superharmonic wave amplitude is plotted on the plane of primary wave frequency and Richardson number, and the locus of divergence locations shows how the resonance locations are influenced by background shear. In the limit of weak background shear , using an asymptotic theory, we show that the horizontal wavenumber condition is sufficient for triadic resonance, in contrast to the requirement of an additional vertical mode number condition in the case of no shear. As a result, the number of resonances for an arbitrarily weak shear is significantly larger than that for no shear. The new resonances that occur in the presence of shear include the possibilities of resonance due to self-interaction and resonances that occur at the seemingly trivial limit of, both of which are not possible in the no shear limit. Our weak shear limit conclusions are relevant for other inhomogeneities such as non-uniformity in stratification as well, thus providing an understanding of several recent studies that have highlighted superharmonic generation in non-uniform stratifications. Extending our study to finite shear (finite) in an ocean-like exponential shear flow profile, we show that for cograde-cograde interactions, a significant number of divergence curves that start at will not extend below a cutoff. In contrast, for retrograde-retrograde interactions, the divergence curves extend all the way from to. For mixed interactions, new divergence curves appear at for and extend to other primary wave frequencies for smaller. Consequently, the total number of resonant triads is of the same order for small as in the limit of weak shear , although it attains a maximum at.

We present a theoretical study of nonlinear effects that result from modal interactions in internal waves in a non-uniformly stratified finite-depth fluid with background rotation. A linear wave field containing modes m and n (of horizontal wavenumbers km and kn) at a fixed frequency ω results in two different terms in the steady-state weakly nonlinear solution: (i) a superharmonic wave of frequency 2ω, horizontal wavenumber km and kn and a vertical structure hmn(z) and (ii) a time-independent term (Eulerian mean flow) with horizontal wavenumber km and kn. For some (m, n), hmn(z) is infinitely large along specific curves on the .!=N0; f =!/plane, where N0 and f are the deep ocean stratification and the Coriolis frequency, respectively; these curves are referred to as divergence curves in the rest of this paper. In uniform stratifications, a unique divergence curve occurs on the (ω/N0, f/ω) plane for those (m, n ≠ m) that satisfy (m/3) < n < (3m). In the presence of a pycnocline (whose strength is quantified by the maximum stratification Nmax), divergence curves occur for several more modal interactions than those for a uniform stratification; furthermore, a given .m; n/interaction can result in multiple divergence curves on the (ω/N0, f/ω) plane for a fixed Nmax=N0. Nearby high-mode interactions in a uniform stratification and any modal interaction in a non-uniform stratification with a sufficiently strong pycnocline are shown to result in near-horizontal divergence curves around f/ω ≈ 1, thus implying that strong nonlinear effects often occur as a result of interaction within triads containing two different modes at the near-inertial frequency. Notably, self-interaction of certain modes in a non-uniform stratification results in one or more divergence curves on the (ω/N0,f/ω) plane, thus suggesting that even arbitrarily small-amplitude individual modes cannot remain linear in a non-uniform stratification. We show that internal wave resonant triads containing modes m and n at frequency ! occur along the divergence curves, and their existence is guaranteed upon the satisfaction of two different criteria: (i) the horizontal component of the standard triadic resonance criterion k1 + k2 + k3 = 0 and (ii) a non-orthogonality criterion. For uniform stratifications, criterion (ii) reduces to the vertical component of the standard triadic resonance criterion. For non-uniform stratifications, criterion (ii) seems to be always satisfied whenever criterion (i) is satisfied, thus significantly increasing the number of modal interactions that result in strong nonlinear effects irrespective of the wave amplitudes. We then adapt our theoretical framework to identify resonant triads and hence provide insights into the generation of higher harmonics in two different oceanic scenarios: (i) low-mode internal tide propagating over small-or large-scale topography and (ii) an internal wave beam incident on a pycnocline in the upper ocean, for which our results are in qualitative agreement with the numerical study of Diamessis et al. (Dynam. Atmos. Oceans., vol. 66, 2014, pp. 110-137).