Nonlinear evolution of a weakly unstable 
wave in a free shear flow with a weak parallel magnetic field

Shukhman, I. G.

doi:10.1017/s0022112098001748

Cited by 3 publications

(1 citation statement)

References 21 publications

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“…The strong amplitudes in the critical layers suggest that the effects of nonlinearity and diffusion may also be important. Shukhman (1998 a , b ) has constructed weakly nonlinear theories for the evolution of magnetic critical layers, which we may adopt to extend the study in the present paper.…”

Section: Conclusion and Remarksmentioning

confidence: 92%

Critical-layer instability of shallow-water magnetohydrodynamic shear flows

2022

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In this paper, the instability of shallow-water shear flow with a sheared parallel magnetic field is studied. Waves propagating in such magnetic shear flows encounter critical levels where the phase velocity relative to the basic flow, $c-U(y)$ , matches the Alfvén wave velocities $\pm B(y)/\sqrt {\mu \rho }$ , based on the local magnetic field $B(y)$ , the magnetic permeability $\mu$ , and the mass density of the fluid $\rho$ . It is shown that when the two critical levels are close to each other, the critical layer can generate an instability. The instability problem is solved, combining asymptotic solutions at large wavenumbers and numerical solutions, and the mechanism of instability explained using the conservation of momentum. For the shallow-water magnetohydrodynamic system, the paper gives the general form of the local differential equation governing such coalescing critical layers for any generic field and flow profiles, and determines precisely how the magnetic field modifies the purely hydrodynamic stability criterion based on the potential vorticity gradient in the critical layer. The curvature of the magnetic field profile, or equivalently the electric current gradient $J' = - B''/\mu$ in the critical layer, is found to play a complementary role in the instability.

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Section: Conclusion and Remarksmentioning

confidence: 92%

Critical-layer instability of shallow-water magnetohydrodynamic shear flows

2022

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A weakly nonlinear theory of the spatial evolution of disturbances in a shear flow with a parallel magnetic field

Shukhman

1998

Physics of Fluids

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A study is made of the spatial downstream evolution of a weakly unstable disturbance excited by an external source of a frequency ω close to the frequency of a marginally stable (neutral) mode, in a mixing layer of conducting, nearly inviscid fluid with a uniform parallel magnetic field. The nonlinear dynamics of such a disturbance is governed largely by two critical layers, i.e., by two narrow regions on the flow profile vx=u(y) near critical levels y=yj (j=±1), on which the resonance condition u(y±1)=c±cA is satisfied (c being the wave’s phase velocity corresponding to the neutral mode, and cA is the Alfven velocity). A nonlinear integrodifferential evolution equation is derived for a complex amplitude of the disturbance, and its solutions are analyzed (numerically and analytically) with different relationships between problem parameters. It is shown that the nonlinearity can play both a stabilizing and destabilizing role depending on the magnitude of the magnetic field and on the degree of supercriticality of the wave. With not too large a magnetic field (cA<cA*), the future of a disturbance depends only on the relationship between the reciprocal of the Reynolds number ν and the wave’s linear spatial growth rate γL (provided that the magnetic Prandtl number is of order unity). With a sufficiently small growth rate (γL≲ν1/3), the critical layer regime throughout the evolution remains a viscous one, and the wave amplitude, upon reaching a very smooth maximum, goes very slowly to zero. At larger values of γL(γL≳ν1/3), when the nonlinearity threshold is attained, a peculiar kind of self-maintaining unsteady regime of critical layers is established, and the amplitude grows explosively, |A|∝(x0−x)−2 (and ultimately reaches values where weakly nonlinear theory becomes invalid). With a sufficiently large magnetic field (cA>cA*, but cA<12Δu, where Δu is the velocity difference), the nonlinearity leads to an “explosion” with any relationship between γL and ν1/3.

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On the predominance of oblique disturbances in the supersonic shear flow instability of the geomagnetic tail boundary

Мишин

2003

Nonlin. Processes Geophys.

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Abstract.A study is made of the influence of the longitudinal magnetic field and density inhomogeneity on the supersonic shear flow instability at the magnetospheric tail boundary. It is shown that the most unstable are slow oblique (3D) disturbances, with a phase velocity approaching at a sufficiently large angle (with respect to the flow direction) the magnetosonic velocity. Their growth rate and spectral width are much larger than those of the usually considered longitudinal (2D) supersonic disturbances. The magnetic field reduces the compressibility effect and, unlike the subsonic case, has a noticeable destabilizing effect on the excitation of oblique disturbances.

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Nonlinear evolution of a weakly unstable wave in a free shear flow with a weak parallel magnetic field

Cited by 3 publications

References 21 publications

Critical-layer instability of shallow-water magnetohydrodynamic shear flows

Critical-layer instability of shallow-water magnetohydrodynamic shear flows

A weakly nonlinear theory of the spatial evolution of disturbances in a shear flow with a parallel magnetic field

On the predominance of oblique disturbances in the supersonic shear flow instability of the geomagnetic tail boundary

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