On Linear Instability and Stability of the Rayleigh–Taylor Problem in Magnetohydrodynamics

Physica D: Nonlinear Phenomena

2019

Self Cite

We investigate a Parker problem for the three-dimensional compressible isentropic viscous magnetohydrodynamic system with zero resistivity in the presence of a modified gravitational force in a vertical strip domain in which the velocity of the fluid is non-slip on the boundary, and focus on the stabilizing effect of the (equilibrium) magnetic field through the non-slip boundary condition. We show that there is a discriminant Ξ, depending on the known physical parameters, for the stability/instability of the Parker problem. More precisely, if Ξ > 0, then the Parker problem is unstable, i.e., the Parker instability occurs, while if Ξ < 0 and the initial perturbation satisfies some relations, then there exists a global (perturbation) solution which decays algebraically to zero in time, i.e., the Parker instability does not happen. The stability results in this paper reveal the stabilizing effect of the magnetic field through the non-slip boundary condition and the importance of boundary conditions upon the Parker instability, and demonstrate that a sufficiently strong magnetic field can prevent the Parker instability from occurring. In addition, based on the instability results, we further rigorously verify the Parker instability under Schwarzschild's or Tserkovnikov's instability conditions in the sense of Hadamard for a horizontally periodic domain.

Section: Criterion For Stability/instabilitymentioning

confidence: 99%

On the dynamical stability and instability of Parker problem

Physica D: Nonlinear Phenomena

2019

Self Cite

“…(5.34) Recalling the definition of Υ A , we can estimate taht 35) and thus, making use of (4.6), (4.22), (4.49)-(4.51), (4.54) and (5.35), we have…”

Section: Estimates Of Umentioning

confidence: 99%

“…Later, Jiang and Jiang [34,35] further extended the modified variational method to construct unstable solutions to the partial differential equations (PDEs) arising from a linearized RT instability problem. Exploiting the modified variational method of PDE in [35] and an regularity theory of elliptic equations, we obtain the following linear instability result of the TMRT problem. with Λ > 0 being a constant satisfying…”

Section: Linear Instabilitymentioning

confidence: 99%

“…In the last decades, this phenomenon has been extensively investigated from both physical and numerical aspects, see [8,13,59] for examples. It has been also widely investigated how the RT instability evolves under the effects of other physical factors, such as the elasticity [41], the rotation [8], the internal surface tension [14,33,61], the magnetic field [8,35,39] and so on.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Nonlinear stability and instability in the Rayleigh–Taylor problem of stratified compressible MHD fluids

2019

Calc. Var.

Self Cite

We establish criteria of stability and instability for the stratified compressible magnetic Rayleigh-Taylor (RT) problem. More precisely, if under the stability condition Ξ < 1, we show the existence of unique solution with algebraic decay in time for the (compressible) magnetic RT problem with proper initial data in Lagrangian coordinates. The stability result presents that sufficiently large vertical (base) magnetic field can inhibit the development of RT instability. On the other hand, if Ξ > 1, there exists an unstable solution to the magnetic RT problem in the Hadamard sense. This shows that the RT instability still occurs when the strength of base magnetic field is small or the base magnetic field is horizontal with proper large horizontal period cell. Moreover, by analyzing the stability condition in magnetic RT problem for vertical magnetic fields, we can observe that the compressibility destroys the stabilizing effect of magnetic fields in the vertical direction. Fortunately, the instability in vertical direction can be inhibited by the stabilizing effect of pressure, which also plays an important role in the mathematical proof for stability of the magnetic RT problem. In addition, we will extend the results in magnetic RT problem to the (compressible) viscoelastic RT problem, and find that the stabilizing effect of elasticity is stronger than the one of magnetic fields.

“…In the last decades, this phenomenon has been extensively investigated from both physical and numerical aspects, see [3,8,17,19,23,36] for examples. It has been also widely investigated how the RT instability evolves under the effects of other physical factors, internal surface tension [9,38], magnetic fields [3,4,18,21,22,24,25], and so on.…”

Section: Introductionmentioning

confidence: 99%

On the stability of Rayleigh–Taylor problem for stratified rotating viscoelastic fluids

Zhao

2018

Bound Value Probl

We investigate the stability of Rayleigh-Taylor (RT) problem of the stratified incompressible viscoelastic fluids under the rotation and the gravity in a horizontal periodic domain, in which the rotation axis is parallel to the direction of gravity, the two fluids are immiscible, and the heavier fluid lies on the lighter one. We establish a stability condition for the RT problem. Moreover, we prove that, under the stability condition, the RT problem enjoys a unique strong solution, which exponentially decays with respect to time. In addition, we note that the stability condition is independent of rotation angular velocity, and the rotation has no destabilizing effect.