Abstract:We present a local stability analysis to investigate the effects of differential diffusion between momentum and density (quantified by the Schmidt number $Sc$) 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 $Sc$, we present a detailed analytical/numerical analysis for three different classes of base flows: (i) an axisymmetric vortex, (ii) an… Show more
“…In other words, a finite () or an infinite () range of becomes susceptible to oscillatory instability as soon as becomes different from unity. In the centrifugally unstable regime in the limit of no radial stratification (, ), oscillatory instability was previously reported only for (Singh & Mathur 2019), which is consistent with the limit in figure 8( a ). Figure 8.( a ) Neutral stability boundaries on the (, ()) plane for the overall instability (blue dashed line) and the oscillatory instability (red dashed and solid lines).…”
Section: Resultssupporting
confidence: 87%
“…, the instability criterion in (3.10) reduces to , with the maximum growth rate being given by occurring at . This no-radial-stratification inviscid limit was highlighted using the local stability approach by Singh & Mathur (2019), and is consistent with the large axial wavenumber limit of Billant & Gallaire (2005). In this unstable regime of , the range of unstable reduces from for to for a finite .…”
Section: Resultssupporting
confidence: 53%
“…For example, (2.13)–(2.14) without background rotation and buoyancy were considered by Landman & Saffman (1987), who showed that viscous terms suppress the growth rates associated with elliptic instability in an unstratified, planar elliptical vortex. In addition to elucidating viscous damping of inviscid instabilities, the scaling of momentum and mass diffusivities have helped the local approach unearth double-diffusive instabilities in stratified vortices as well (Kirillov & Mutabazi 2017; Singh & Mathur 2019). Finally, the scaling for temperature and salinity diffusivities also leads to the reduction of local stability equations to the classical salt fingering instability equation for a quiescent base flow (Appendix A).…”
Section: Theorymentioning
confidence: 99%
“…Here, we use the local stability approach (Lifschitz & Hameiri 1991) with small diffusivities in momentum and density to investigate short-wavelength instabilities in stratified eddies in thermal wind balance. Previous studies using the local stability approach have captured the effects of differential diffusion (in mass and momentum) on local instabilities in radially stratified circular Couette flow (Kirillov & Mutabazi 2017) and vertically stratified planar vortices (Singh & Mathur 2019).…”
We present a local stability analysis of an idealized model of the stratified vortices that appear in geophysical settings. The base flow comprises an axisymmetric vortex with background rotation and an out-of-plane stable stratification, and a radial stratification in the thermal wind balance with the out-of-plane momentum gradient. Solving the local stability equations along fluid particle trajectories in the base flow, the dependence of short-wavelength instabilities on the Schmidt number
$Sc$
(ratio between momentum and mass diffusivities) is studied, in the presence of curvature effects. In the diffusion-free limit, the well-known symmetric instability is recovered. In the viscous, double-diffusive regime, instability characteristics are shown to depend on three non-dimensional parameters (including
$Sc$
), and two different instabilities are identified: (i) a monotonic instability (same as symmetric instability at
$Sc = 1$
), and (ii) an oscillatory instability (absent at
$Sc = 1$
). Separating the base flow and perturbation characteristics, two each of base flow and perturbation parameters (apart from
$Sc$
) are identified, and the entire parameter space is explored for the aforementioned instabilities. In comparison with
$Sc = 1$
, monotonic and oscillatory instabilities are shown to significantly expand the instability region in the space of base flow parameters as
$Sc$
moves away from unity. Neutral stability boundaries on the plane of
$Sc$
and a modified gradient Richardson number are then identified for both these instabilities. In the absence of curvature effects, our results are shown to be consistent with previous studies based on normal mode analysis, thus establishing that the local stability approach is well suited to capturing symmetric and double-diffusive instabilities. The paper concludes with a discussion of curvature effects, and the likelihood of monotonic and oscillatory instabilities in typical oceanic settings.
“…In other words, a finite () or an infinite () range of becomes susceptible to oscillatory instability as soon as becomes different from unity. In the centrifugally unstable regime in the limit of no radial stratification (, ), oscillatory instability was previously reported only for (Singh & Mathur 2019), which is consistent with the limit in figure 8( a ). Figure 8.( a ) Neutral stability boundaries on the (, ()) plane for the overall instability (blue dashed line) and the oscillatory instability (red dashed and solid lines).…”
Section: Resultssupporting
confidence: 87%
“…, the instability criterion in (3.10) reduces to , with the maximum growth rate being given by occurring at . This no-radial-stratification inviscid limit was highlighted using the local stability approach by Singh & Mathur (2019), and is consistent with the large axial wavenumber limit of Billant & Gallaire (2005). In this unstable regime of , the range of unstable reduces from for to for a finite .…”
Section: Resultssupporting
confidence: 53%
“…For example, (2.13)–(2.14) without background rotation and buoyancy were considered by Landman & Saffman (1987), who showed that viscous terms suppress the growth rates associated with elliptic instability in an unstratified, planar elliptical vortex. In addition to elucidating viscous damping of inviscid instabilities, the scaling of momentum and mass diffusivities have helped the local approach unearth double-diffusive instabilities in stratified vortices as well (Kirillov & Mutabazi 2017; Singh & Mathur 2019). Finally, the scaling for temperature and salinity diffusivities also leads to the reduction of local stability equations to the classical salt fingering instability equation for a quiescent base flow (Appendix A).…”
Section: Theorymentioning
confidence: 99%
“…Here, we use the local stability approach (Lifschitz & Hameiri 1991) with small diffusivities in momentum and density to investigate short-wavelength instabilities in stratified eddies in thermal wind balance. Previous studies using the local stability approach have captured the effects of differential diffusion (in mass and momentum) on local instabilities in radially stratified circular Couette flow (Kirillov & Mutabazi 2017) and vertically stratified planar vortices (Singh & Mathur 2019).…”
We present a local stability analysis of an idealized model of the stratified vortices that appear in geophysical settings. The base flow comprises an axisymmetric vortex with background rotation and an out-of-plane stable stratification, and a radial stratification in the thermal wind balance with the out-of-plane momentum gradient. Solving the local stability equations along fluid particle trajectories in the base flow, the dependence of short-wavelength instabilities on the Schmidt number
$Sc$
(ratio between momentum and mass diffusivities) is studied, in the presence of curvature effects. In the diffusion-free limit, the well-known symmetric instability is recovered. In the viscous, double-diffusive regime, instability characteristics are shown to depend on three non-dimensional parameters (including
$Sc$
), and two different instabilities are identified: (i) a monotonic instability (same as symmetric instability at
$Sc = 1$
), and (ii) an oscillatory instability (absent at
$Sc = 1$
). Separating the base flow and perturbation characteristics, two each of base flow and perturbation parameters (apart from
$Sc$
) are identified, and the entire parameter space is explored for the aforementioned instabilities. In comparison with
$Sc = 1$
, monotonic and oscillatory instabilities are shown to significantly expand the instability region in the space of base flow parameters as
$Sc$
moves away from unity. Neutral stability boundaries on the plane of
$Sc$
and a modified gradient Richardson number are then identified for both these instabilities. In the absence of curvature effects, our results are shown to be consistent with previous studies based on normal mode analysis, thus establishing that the local stability approach is well suited to capturing symmetric and double-diffusive instabilities. The paper concludes with a discussion of curvature effects, and the likelihood of monotonic and oscillatory instabilities in typical oceanic settings.
“…The effect of the Schmidt number on the stability of barotropic vortices in a stratified ambient fluid in the absence of rotation have been studied recently by Singh and Mathur 30 with the geometric optics approach developed in [17][18][19][20] .…”
We consider a model of a circular lenticular vortex immersed into a deep and vertically stratified viscous fluid in the presence of gravity and rotation. The vortex is assumed to be baroclinic with a Gaussian profile of angular velocity both in the radial and axial directions. Assuming the base state to be in a cyclogeostrophic balance, we derive linearized equations of motion and seek for their solution in a geometric optics approximation to find amplitude transport equations that yield a comprehensive dispersion relation. Applying algebraic Bilharz criterion to the latter, we establish that stability conditions are reduced to three inequalities that define stability domain in the space of parameters. The main destabilization mechanism is either stationary or oscillatory axisymmetric instability depending on the Schmidt number (Sc), vortex Rossby number and the difference between the radial and axial density gradients as well as the difference between the epicyclic and vertical oscillation frequencies. We discover that the boundaries of the regions of stationary and oscillatory axisymmetric instabilities meet at a codimension-2 point, forming a singularity of the neutral stability curve. We give an exhaustive classification of the geometry of the stability boundary, depending on the values of the Schmidt number. Although we demonstrate that the centrifugally stable (unstable) Gaussian lens can be destabilized (stabilized) by the differential diffusion of mass and momentum and that destabilization can happen even in the limit of vanishing diffusion, we also describe explicitly a set of parameters in which the Gaussian lens is stable for all Sc > 0.
We perform a three-dimensional short-wavelength linear stability analysis of numerically simulated two-dimensional Kelvin–Helmholtz vortices in homogeneous and stratified environments at a fixed Reynolds number of
$Re = 300$
. For the homogeneous case, the elliptic instability at the vortex core dominates at early times, before being taken over by the hyperbolic instability at the vortex edge. For the stratified case of Richardson number
$ Ri = 0.08$
, the early-time instabilities comprise a dominant elliptic instability at the core and a hyperbolic instability influenced strongly by stratification at the vortex edge. At intermediate times, the local approach shows a new branch of (convective) instability that emerges at the vortex core and subsequently moves towards the vortex edge. A few more convective instability bands appear at the vortex core and move away, before coalescing to form the most unstable region inside the vortex periphery at large times. In addition, the stagnation point instability is also recovered outside the periphery of the vortex at intermediate times. The dominant instability characteristics from the local approach are shown to be in good qualitative agreement with the results based on global instability studies for both homogeneous and stratified cases. A systematic study of the dependence of the dominant instability characteristics on
$ Ri $
is then presented. While
$ Ri = 0.1$
is identified as most unstable (with convective instability being dominant), another growth rate maximum at
$ Ri = 0.025$
is not far behind (with the hyperbolic instability influenced by stratification being dominant). Finally, the local stability approach is shown to predict the potential orientation of the flow structures that would result from hyperbolic and convective instabilities, which is found to be consistent with three-dimensional numerical simulations reported previously.
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