Variational necessary and sufficient stability conditions for inviscid shear flow

Hirota, Mitsutomo; Morrison, Philip; Hattori, Yuji

doi:10.1098/rspa.2014.0322

Cited by 9 publications

(8 citation statements)

References 46 publications

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“…Based on this method, we have derived new stability criteria (Criteria 1,2, and 3) as extensions of the Rayleigh-Fjørtoft criterion to when N 2 B (x c ) = 0. By making use of the abundant knowledge about Rayleigh's equation [16], more detailed stability criteria tailored for these applications will be discussed elsewhere.…”

Section: Discussionmentioning

confidence: 99%

“…2. Conversely, for the case (v), J R has a minimum that is below 1/4, but U > 0 is large enough to satisfy (16) with D w ⊂ [b, L].…”

Section: Sufficient Conditions For Stabilitymentioning

confidence: 99%

“…are found to be stable by Criterion 2. For the case (iv), the sign of U violates the Rayleigh-Fjørtoft criterion and U has a maximum in D w , but the stratification J R contributes to stabilization so that the inequality (16) with D w = [b, L] holds, as shown in Fig. 2.…”

mentioning

confidence: 96%

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Stability boundaries and sufficient stability conditions for stably stratified, monotonic shear flows

Hirota

Morrison

2016

Physics Letters A

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Linear stability of inviscid, parallel, and stably stratified shear flow is studied under the assumption of smooth strictly monotonic profiles of shear flow and density, so that the local Richardson number is positive everywhere. The marginally unstable modes are systematically found by solving a one-parameter family of regular Sturm-Liouville problems, which can determine the stability boundaries more efficiently than solving the Taylor-Goldstein equation directly.By arguing for the non-existence of a marginally unstable mode, we derive new sufficient conditions for stability, which generalize the Rayleigh-Fjørtoft criterion for unstratified shear flows.

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Section: Discussionmentioning

confidence: 99%

“…2. Conversely, for the case (v), J R has a minimum that is below 1/4, but U > 0 is large enough to satisfy (16) with D w ⊂ [b, L].…”

Section: Sufficient Conditions For Stabilitymentioning

confidence: 99%

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Stability boundaries and sufficient stability conditions for stably stratified, monotonic shear flows

Hirota

Morrison

2016

Physics Letters A

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“…as given in Timofeev (2000). Equation ( 21) is a form of Rayleigh's equation (see e.g Chandrasekhar 1961), a well known previously obtained expression in hydrodynamics concerning inviscid shear flows (Rayleigh 1879; Hirota et al 2014). The corresponding expression for the total pressure perturbationP T inside a magnetic slab with a spatially varying background plasma flow is given by:…”

Section: Coronal Slab In Presence Of Inhomogeneous Background Flowmentioning

confidence: 95%

I. The effect of symmetric and spatially varying equilibria and flow on MHD wave modes: slab geometry

Skirvin

Fedun

Verth

2021

Monthly Notices of the Royal Astronomical Society

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Realistic theoretical models of magnetohydrodynamic wave propagation in the different solar magnetic configurations are required to explain observational results, allowing magnetoseismology to be conducted and provide more accurate information about local plasma properties. The numerical approach described in this paper allows a dispersion diagram to be obtained for any arbitrary symmetric magnetic slab model of solar atmospheric features. This proposed technique implements the shooting method to match necessary boundary conditions on continuity of displacement and total pressure of the waveguide. The algorithm also implements fundamental physical knowledge of the sausage and kink modes such that both can be investigated. The dispersion diagrams for a number of analytic cases that model magnetohydrodynamic waves in a magnetic slab were successfully reproduced. This work is then extended by considering density structuring modelled as a series of Gaussian profiles and a sinc(x) function. A further case study investigates properties of MHD wave modes in a coronal slab with a non-uniform background plasma flow, for which the governing equations are derived. It is found that the dispersive properties of slow body modes are more greatly altered than those of fast modes when any equilibrium inhomogeneity is increased, including background flow. The spatial structure of the eigenfunctions is also modified, introducing extra nodes and points of inflexion that may be of interest to observers.

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“…The study of the stability of shear flows goes back to Rayleigh [21], who introduced the stability criterion named after hime. More recently improvements of such criteria were given in [16] and [20]. Our methods are not as general and somewhat specific to the particular shear flows under consideration.…”

Section: Introductionmentioning

confidence: 99%

Stability Theory of the 3-Dimensional Euler Equations

Dullin¹,

Worthington²

2019

SIAM J. Appl. Math.

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The Euler equations on a three-dimensional periodic domain have a family of shear flow steady states. We show that the linearised system around these steady states decomposes into subsystems equivalent to the linearisation of shear flows in a two-dimensional periodic domain. To do so, we derive a formulation of the dynamics of the vorticity Fourier modes on a periodic domain and linearise around the shear flows. The linearised system has a decomposition analogous to the two-dimensional problem, which can be significantly simplified. By appealing to previous results it is shown that some subset of the shear flows are spectrally stable, and another subset are spectrally unstable. For a dense set of parameter values the linearised operator has a nilpotent part, leading to linear instability. This is connected to the nonnormality of the linearised dynamics and the transition to turbulence. Finally we show that all shear flows in the family considered (even the linearly stable ones) are parametrically unstable.

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Variational necessary and sufficient stability conditions for inviscid shear flow

Cited by 9 publications

References 46 publications

Stability boundaries and sufficient stability conditions for stably stratified, monotonic shear flows

Stability boundaries and sufficient stability conditions for stably stratified, monotonic shear flows

I. The effect of symmetric and spatially varying equilibria and flow on MHD wave modes: slab geometry

Stability Theory of the 3-Dimensional Euler Equations

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