Strato-hyperbolic instability: a new mechanism of instability in stably stratified vortices

Suzuki, Shota; Hirota, Mitsutomo; Hattori, Yuji

doi:10.1017/jfm.2018.641

Cited by 6 publications

(53 citation statements)

References 43 publications

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“…The largest value is reasonably close to the value obtained by local stability analysis (Suzuki et al. 2018). Since the growth rates of the other modes decrease for large , the strato-hyperbolic instability is dominant when the wavenumber is bounded from below as with large .…”

Section: Stability Of Stratified 2-d Taylor–green Vorticessupporting

confidence: 88%

“…In fact, the resonance condition for the strato-hyperbolic instability obtained by local stability analysis (Suzuki et al. 2018) is where is the period of the fluid particle motion, is the angle between the axis and the wavevector, and and are integers; according to (3.6), when increases, decreases, which implies that increases for a fixed horizontal wavenumber. The growth rate increases with , although viscous damping decreases the growth rate slightly for for .…”

Section: Stability Of Stratified 2-d Taylor–green Vorticesmentioning

confidence: 99%

“…When there are hyperbolic stagnation points, the flow is subject to hyperbolic instability (Friedlander & Vishik 1991; Lifschitz & Hameiri 1991; Sipp & Jacquin 1998; Pralits, Giannetti & Brandt 2013); in this paper, we call this the pure hyperbolic instability in order to differentiate it from the strato-hyperbolic instability introduced below. Our recent work showed that a new type of instability called strato-hyperbolic instability exists in stably stratified vortices having hyperbolic stagnation points (Suzuki, Hirota & Hattori 2018). Since the arrays of vortices in the atmosphere are affected by density stratification and planetary rotation, the role of the strato-hyperbolic instability as well as the pure hyperbolic instability is of much interest in understanding and predicting their motion.…”

Section: Introductionmentioning

confidence: 99%

“…Suzuki et al. (2018) studied the stability of stably stratified vortices. Three base flows which possess hyperbolic stagnation points were considered: the 2-D Taylor–Green vortices, the Stuart vortices and the Lamb–Chaplygin dipole.…”

Section: Introductionmentioning

confidence: 99%

“…In the strato-hyperbolic instability the internal gravity waves rotate the vorticity of the disturbances to change its angle or, in other words, to shift the phase; this phase shift causes the stretching effects near the hyperbolic stagnation points to be maintained when the phase shift satisfies a resonance condition (Suzuki et al. 2018). These three instabilities are short-wave instabilities, whose growth rates converge to finite values in the short-wave and inviscid limits.…”

Section: Introductionmentioning

confidence: 99%

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Modal stability analysis of arrays of stably stratified vortices

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Section: Stability Of Stratified 2-d Taylor–green Vorticessupporting

confidence: 88%

Section: Stability Of Stratified 2-d Taylor–green Vorticesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The pitfalls of investigating rotational flows with the Euler equations

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2021

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Small viscous effects in high-Reynolds-number rotational flows always accumulate over time to have a leading-order effect. Therefore, the high-Reynolds-number limit for the Navier–Stokes equations is singular. It is important to investigate whether a solution of the Euler equations can approximate a real flow at large Reynolds number. These facts are often overlooked and, as a result, the Euler equations are used to simulate laminar rotational flows at large Reynolds number. Based on the Fredholm alternative, an asymptotic perturbation theory is described to establish secularity conditions determined by viscosity for an inviscid solution to approximate a real viscous fluid. Four important classical inviscid solutions are investigated using the theory with the following conclusions. The Stuart cats’ eyes and Mallier–Maslowe vortices are inconsistent with any real fluid at high Reynolds number; whereas Hill's spherical vortex is confirmed to be consistent with a steady state in the spherical core region and the Lamb–Chaplygin dipole is found to be consistent with a quasi-steady state in the circular core region. These solutions have been widely used for analysing the stability of vortex flows and wakes, and their interactions with shock waves or bubbles. Serendipitously, we have revealed an original exact solution of the Navier–Stokes equations which is time dependent, has non-zero nonlinear convective terms and is restricted to a finite domain with the decay rate depending on dipole radius.

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Magneto-gravity-elliptic instability

Salhi

Cambon

2023

J. Fluid Mech.

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Magneto-gravity-elliptic instability is addressed here considering an unbounded strained vortex (with constant vorticity $2\varOmega$ and with ellipticity parameter $\varepsilon$ ) of a perfectly conducting fluid subjected to a uniform axial magnetic field (with Alfvén velocity scaled from the basic magnetic field $B)$ and an axial stratification (with a constant Brunt–Väisälä frequency $N$ ). Such a simple model allows us to formulate the stability problem as a system of equations for disturbances in terms of Lagrangian Fourier (or Kelvin) modes which is universal for wavelengths of the perturbation sufficiently small with respect to the scale of variation of the basic velocity gradients. It can model localised patches of elliptic streamlines which often appear in some astrophysical flows (stars, planets and accretion discs) that are tidally deformed through gravitational interaction with other bodies. In the limit case where the flow streamlines are exactly circular ( $\varepsilon =0),$ there are fast and slow magneto-inertia-gravity waves with frequencies $\omega _{1,2}$ and $\omega _{3,4},$ respectively. Under the effect of finite ellipticity, the resonant cases of these waves, $\omega _i-\omega _j=n\varOmega$ $(i\ne j)$ ( $n$ being an integer), can become destabilising. The maximal growth rate of the subharmonic instability (related to the resonance of order $n=2)$ is determined by extending the asymptotic method by Lebovitz & Zweibel (Astrophys. J., vol. 609, 2004, pp. 301–312). The domains of the $(k_0B/\varOmega, N/\varOmega )$ plane for which this instability operates are identified ( $1/k_0$ being a characteristic length scale). We demonstrate that the $N\rightarrow 0$ limit is, in fact, singular (discontinuous). The axial stable stratification enhances the subharmonic instability related to the resonance between two slow modes because, at large magnetic field strengths, its maximal growth rate is twice that found in the case without stratification.

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Strato-hyperbolic instability: a new mechanism of instability in stably stratified vortices

Cited by 6 publications

References 43 publications

Modal stability analysis of arrays of stably stratified vortices

Modal stability analysis of arrays of stably stratified vortices

The pitfalls of investigating rotational flows with the Euler equations

Magneto-gravity-elliptic instability

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