scholarly journals THE ADJOINT PROBLEM IN THE PRESENCE OF A DEFORMED SURFACE: THE EXAMPLE OF THE ROSENSWEIG INSTABILITY ON MAGNETIC FLUIDS

2002 ◽  
Vol 16 (08) ◽  
pp. 1155-1170 ◽  
Author(s):  
ADRIAN LANGE
Keyword(s):  
Magnetic Field ◽  
A Priori ◽  
Adjoint Operator ◽  
Horizontal Layer ◽  
Adjoint Problem ◽  
Vertical Magnetic Field ◽  
Amplitude Equations ◽  
Rosensweig Instability ◽  
Pattern Forming ◽  
Almost All

The Rosensweig instability is the phenomenon that above a certain threshold of a vertical magnetic field peaks appear on the free surface of a horizontal layer of magnetic fluid. In contrast to almost all classical hydrodynamical systems, the nonlinearities of the Rosensweig instability are entirely triggered by the properties of a deformed and a priori unknown surface. The resulting problems in defining an adjoint operator for such nonlinearities are illustrated. The implications concerning amplitude equations for pattern forming systems with a deformed surface are discussed.

Numerical study of convection in the horizontal Bridgman configuration under the action of a constant magnetic field. Part 1. Two-dimensional flow

1997 ◽  
Vol 333 ◽  
pp. 23-56 ◽  
Author(s):  
HAMDA BEN HADID ◽  
DANIEL HENRY ◽  
SLIM KADDECHE
Keyword(s):  
Magnetic Field ◽  
Prandtl Number ◽  
Scaling Laws ◽  
Hartmann Number ◽  
Numerical Study ◽  
Flow Region ◽  
Horizontal Layer ◽  
Vertical Magnetic Field ◽  
Two Dimensional Flow ◽  
Thermocapillary Forces

Studies of convection in the horizontal Bridgman configuration were performed to investigate the flow structures and the nature of the convective regimes in a rectangular cavity filled with an electrically conducting liquid metal when it is subjected to a constant vertical magnetic field. Under some assumptions analytical solutions were obtained for the central region and for the turning flow region. The validity of the solutions was checked by comparison with the solutions obtained by direct numerical simulations. The main effects of the magnetic field are first to decrease the strength of the convective flow and then to cause a progressive modification of the flow structure followed by the appearance of Hartmann layers in the vicinity of the rigid walls. When the Hartmann number is large enough, Ha > 10, the decrease in the velocity asymptotically approaches a power-law dependence on Hartmann number. All these features are dependent on the dynamic boundary conditions, e.g. confined cavity or cavity with a free upper surface, and on the type of driving force, e.g. buoyancy and/or thermocapillary forces. From this study we generate scaling laws that govern the influence of applied magnetic fields on convection. Thus, the influence of various flow parameters are isolated, and succinct relationships for the influence of magnetic field on convection are obtained. A linear stability analysis was carried out in the case of an infinite horizontal layer with upper free surface. The results show essentially that the vertical magnetic field stabilizes the flow by increasing the values of the critical Grashof number at which the system becomes unstable and modifies the nature of the instability. In fact, the range of Prandtl number over which transverse oscillatory modes prevail shrinks progressively as the Hartmann number is increased from zero to 5. Therefore, longitudinal oscillatory modes become the preferred modes over a large range of Prandtl number.

Non-linear waves interactions in rotating shallow water magnetohydrodynamics

2020 ◽  
Author(s):  
Dmitry Klimachkov ◽  
Arakel Petrosyan
Keyword(s):  
Magnetic Field ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Plasma Layer ◽  
Vertical Component ◽  
Vertical Magnetic Field ◽  
Amplitude Equations ◽  
Horizontal Magnetic Field ◽  
Non Linear ◽  
Poincare Waves

<p>This study is devoted to the development of the nonlinear theory of the magneto-Poincare waves and magnetostrophic waves in rotating layers of astrophysical and space plasma in the shallow-water approximation. These waves determine the large-scale dynamics of the various astrophysical and space objects such as solar tachocline, as well as  magnetoactive atmospheres of exoplanets trapped by tides of a carrier star, neutron stars atmospheres and the flows in accretion disks of neutron stars. For this purpose we derived magnetohydrodynamic shallow water equations with a rotation and presence of an external vertical magnetic field. The system is obtained from conventional magnetohydrodynamic equations for incompressible inviscid heavy plasma layer with free surface in an external vertical magnetic field. The pressure is assumed to be hydrostatic, and the height of the plasma layer is considered to be much smaller than horizontal scales of the flow. The magnetohydrodynamic equations in the shallow-water approximation play equally important role in the space and astrophysical plasma flows like classical shallow-water equations in the fluid dynamics of a neutral fluid. The magnetohydrodynamic shallow water equations with an external vertical magnetic field are modified by supplementing them with the equation for the vertical component of the magnetic field and divergence-free condition for magnetic field contains its vertical component. Thus the velocity field remains two-dimensional while the magnetic field becomes three-dimensional. It is shown that the presence of a vertical magnetic field significantly changes the dynamics of the wave processes in astrophysical plasma compared to the neutral fluid and plasma layer in a horizontal magnetic field.  We have investigated the interaction of Magneto-Poincare waves and magnetostrophic waves in the magnetohydrodynamic shallow water flows in external vertical magnetic field and in horizontal (toroidal and poloidal) magnetic field. In the absence of the horizontal magnetic field the dynamics of plasma appears to be similar to the neutral fluid dynamics and it is shown that there are four-waves interactions in this case. Using the asymptotic multiscale method we obtained the non-linear amplitude equations for the three interacting Magneto-Poincare waves and magnetostrophic waves. The analysis of the amplitude equations shows that there are two types of instabilities for four different types of three-waves configurations. These instabilities occur in both cases: in the external vertical magnetic field and in the horizontal magnetic field. For all types of instabilities the growth rates are found. In the absence of the vertical magnetic field we obtained the non-linear amplitude equations for the four interacting waves. It is shown that the resulting system of equations has the first integrals that describe the mechanism of energy transfer among interacting waves of small amplitude. This work was supported by the Russian Foundation for Basic Research (project no. 19-02-00016).</p>

Time-periodic Heating of a Rotating Horizontal Fluid Layer in a Vertical Magnetic Field

2005 ◽  
Vol 60 (8-9) ◽  
pp. 583-592 ◽  
Author(s):  
Beer Singh Bhadauria
Keyword(s):  
Magnetic Field ◽  
Fluid Layer ◽  
Vertical Axis ◽  
Horizontal Layer ◽  
Vertical Magnetic Field ◽  
Onset Of Convection ◽  
Electrically Conducting ◽  
Dependent Part ◽  
Horizontal Fluid Layer ◽  
Time Periodic

Thermal instability in a horizontal layer of an electrically conducting fluid heated from below has been investigated under the effects of uniform rotation about a vertical axis and an applied uniform vertical magnetic field. The temperature field between the walls of the fluid layer consists of two parts; a steady part and a time-dependent part, which varies periodically. The effect of modulation of the walls temperature on the onset of convection has been studied using Floquets theory. Stabilizing and destabilizing effects on the onset of convective instability have been found. Some comparisons have been made. - 2000 Mathematics Subject Classification: 76E06, 76R10.

Sign in / Sign up

Export Citation Format

Share Document