Ekman-Hartmann Boundary Layers Research Articles

A Mathematical Model for Magnetohydrodynamic Convection Flow in a Rotating Horizontal Channel with Inclined Magnetic Field, Magnetic Induction and Hall Current Effects

Closed-form and asymptotic solutions are derived for the steady, fully-developed hydromagnetic free and forced convection flow in a rotating horizontal parallel-plate channel under the action of an inclined magnetic field and constant pressure gradient along the longitudinal axis of the channel. The magnetic field is strong enough to generate Hall current effects and the magnetic Reynolds number of sufficient magnitude that induced magnetic field effects are also present. Secondary flow is present owing to the Hall current effect. The channel plates are also taken to be electrically-conducting. The conservation equations are formulated in an (x, y, z) coordinate system and non-dimensionalized using appropriate transformations. The resulting non-dimensional coupled ordinary differential equations for primary and secondary velocity components and primary and secondary induced magnetic field components and transformed boundary conditions are shown to be controlled by the dimensionless pressure gradient parameter (px), Hartmann number (M2), Grashof number (G), Hall current parameter (m), rotational parameter (K2), magnetic field inclination (&#920), and the electrical conductance ratios of the upper (&#9201) and lower (&#9202) plates. Solutions are derived using the method of complex variables. Asymptotic solutions are also presented for very high rotation parameter and Hartmann number of order equal to unity, for which Ekman-Hartmann boundary layers are identified at the plates. A parametric study of the evolution of velocity and induced magnetic field distributions is undertaken. It is shown that generally increasing Hall current effect (m) serves to accentuate the secondary (cross) flow but oppose the primary flow. An increase in rotational parameter (K2) is also found to counteract primary flow intensity. An elevation in the Grashof number i.e. free convection parameter (G) is shown to aid the secondary induced magnetic field component (Hz); however there is a decrease in magnitudes of the primary induced magnetic field component (Hx) with increasing Grashof number. Increasing inclination of the applied magnetic field (&#920, is also found to oppose the primary flow (u1) but conversely to strongly assist the secondary flow (w1). Both critical primary (Gcx) and secondary (Gcz)Grashof numbers are shown to be reduced with increasing inclination of the magnetic field (&#920), increasing Hall parameter (m) and rotational parameter (K2). Applications of the study arise in rotating MHD induction power generators and also astrophysical flows.

Compressible Ekman–Hartmann boundary layers

We consider the effect of compressibility on mixed Ekman–Hartmann boundary layers on an infinite plane (z = 0), in the presence of an external magnetic field oblique to the boundary. The aim is to investigate the influence of the magnetic pressure on the fluid density, and hence, via mass conservation, on the mass flow into or out of the boundary layer. We find that if the z-component of vorticity in the main flow, immediately above the boundary layer, is negative, then there is a competition between Ekman suction and the magnetic pressure effect. Indeed, as the magnetic field strength is increased, the magnetic pumping may overcome the Ekman suction produced by anti-cyclonic main flow vortices. Such a mechanism, based on the competition between these effects, may be of importance for understanding the dynamics of the magnetic field in stellar (or planetary) interiors. For the solar tachocline, we find that the analysed magnetic pressure effect is unlikely to play a significant role; however, we give examples of what changes in the assumed scalings would be necessary for the effect to become important.

Stability of Large Amplitude Ekman-Hartmann Boundary Layers in MHD : The Case of Ill-Prepared Data

In this paper, we study an incompressible highly rotating fluid submitted to a high magnetic field between two planes with a Dirichlet boundary condition. We investigate the nonlinear stability of Ekman-Hartmann boundary layers under a spectral assumption for general initial data; this means that the data can be chosen as an arbitrary (but smooth enough) three-dimensional divergence free vector field independent of the small parameter.

Boundary layer instability at the top of the Earth's outer core

We investigate the instability of mixed Ekman–Hartmann boundary layers arising in rotating incompressible magnetohydrodynamics flows in a parameter regime relevant to the Earth liquid core. Relying on the small depth of the boundary layer, we perform a local study in a half-space at a given co-latitude θ≠ π/2, and assume a mean dipolar axial magnetic field with internal sources (the geodynamo). Instabilities are driven, for high enough Reynolds number, by the quadratic term in the momentum equation. Nonlinear stability can be proven using energy methods in the neighborhood of the poles (Nonlinearity 12 (2) (1999) 181). Next, following the work of Lilly (J. Atmos. Sci. 23 (1966) 481), we restrict our analysis to the linear growth phase. We describe the dependence of the critical Reynolds number in terms of θ and Elsasser number (measuring the relative strength of Lorentz and Coriolis forces). It turns out that no matter how large the Elsasser number is, there exists a critical band centered on the equator in which instabilities can occur. For geophysically relevant values of parameters, this band could extend as far as 45° away from the equator. This establishes the possibility of boundary layer instabilities near the Earth core-mantle boundary (CMB). We finally present a first attempt of interaction with field maps at the CMB and core flows derived from the secular variation of the field, and discuss the sensitivity of the instability onset not only on the boundary layer Reynolds number, but also on the direction of the flow.

Large mixed Ekman–Hartmann boundary layers in magnetohydrodynamics

In this paper, we study the nonlinear stability of Ekman–Hartmann type boundary layers in a rotating magnetohydrodynamics flow under a sharp spectral assumption. This generalizes the result of Desjardins et al (1999 Nonlinearity 12 181–99) obtained under a smallness assumption on a Reynolds number and the result of Rousset (2003 Arch. Rat. Mech. Anal. in press) about the stability of Ekman layers.

Instability of Ekman–Hartmann boundary layers, with application to the fluid flow near the core–mantle boundary

Our aim is to investigate the instability of mixed Ekman–Hartmann boundary layers arising in rotating incompressible magnetohydrodynamics flows in a parameter regime relevant to the Earth liquid core. We perform a local study in a half space at a given co-latitude θ≠ π/2, and assume a mean dipolar axial magnetic field with internal sources. Instabilities are driven, for high enough Reynolds number, by the quadratic term in the momentum equation. Following the work of Lilly [J. Atmos. Sci. 23 (1966) 481–494], we restrict our analysis to the linearized growth phase. We describe the dependence of the critical Reynolds number in terms of θ and Elsasser number (measuring the relative strength of Lorentz and Coriolis terms). It is found that no matter how large the Elsasser number is, there exists a critical band centered on the equator in which instabilities can occur. For geophysically relevant values of parameters, this band extends over some 45° away from the equator. This study establishes the possibility of boundary layer instabilities near the core–mantle boundary.

Stability of mixed Ekman-Hartmann boundary layers

In this paper we study the nonlinear stability of Ekman-Hartmann-type boundary layers in a rotating magnetohydrodynamics flow in a half-space and between two planes. We prove rigorously that if the Reynolds number defined on boundary-layer characteristics is smaller than a critical value, the boundary layer is nonlinearly stable. It is shown that the normal component of the magnetic field increases the critical Reynolds number for instability.

Strong Potential Magnetic Field Influence on Slightly Differentially Rotating Spherical Shell

An electrically conducting viscous fluid-filled spherical shell is permeated by an axisymmetric strong potential magnetic field with large Elssaser number λ2 ≫ 1. We describe analytically the steady flow driven by a slightly faster rotation of the conducting inner boundary of the shell. The main flow is controlled by Ekman-Hartmann boundary layers with a small thickness δ/λ, where δ2is the Ekman number. Asymptotics based on small λ−1 ≪ 1 reveal the nature of a free shear layer O((δ/λ)1/2) and a super-rotation that allows a part of the fluid to rotate faster than the inner sphere. The free shear is following an imposed field line that is tangent to the inner or outer sphere. Meridional flux is concentrated in the shear and boundary layers. Fluid tends to rotate with the inner sphere and to expel azimuthal magnetic field from an\( \ell \)-region restricted by the free shear in the spherical shell.

Magnetohydrodynamic Ekman and Stewartson layers in a rotating spherical shell

I investigate numerically the flow of an electrically conducting fluid in a differentially rotating spherical shell, in the presence of an imposed magnetic field. For a very weak field the flow is seen to consist of an Ekman layer on the inner and outer spherical boundaries, and a Stewartson layer on the cylinder circumscribing the inner sphere and parallel to the axis of rotation, in agreement with the classical non-magnetic analysis. As the field strength is increased, the non-magnetic Ekman layers merge smoothly into magnetic Ekman-Hartmann layers, and the Stewartson layer is suppressed. In the fully magnetic régime the interior flow consists essentially of a solid-body rotation, with the precise rate determined by a torque balance between the inner and outer Ekman-Hartmann boundary layers.

Ekman-Hartmann Boundary Layers and the Length of Day Variations

In the first issue of Journal of Fluid Mechanics Ian Proudman published a paper on the dynamical properties of a fluid between two concentric rotating spheres (Proudman, 1956). The angular velocities of the spheres were assumed only slightly different and the Reynolds number of the flow was large. It was found under non-slip boundary conditions that the cylindrical surface that touches the inner sphere and parallel to the axis of rotation is a singular surface in which velocity gradients are very large. Outside the cylinder the fluid rotates as a rigid body with the same angular velocity as the outer sphere. Inside the cylinder the fluid rotates with an angular velocity intermediate to the angular velocities of the spheres and there is also a meridional circulation. Later Stewartson (1966) presented a detailed investigation of structure of the shear layer near the cylindrical surface. One of the present authors (Ruzmaikin, 1989) pointed out a possible geophysical importance of these solutions. The liquid part of the Earth core occupying a shell between the inner solid core and the rock mantle can be considered as the fluid between two rotating spheres.

Nonlinear hydrodynamic and hydromagnetic spin-up driven by Ekman-Hartmann boundary layers

Finite amplitude, impulsively started spin-up and spin-down is analysed for axially symmetric flow of a viscous, incompressible, electrically conducting fluid confined between infinite, flat, parallel, insulating boundaries. A uniform axial magnetic field is present in the initial state, but is subsequently distorted by fluid motions. The method of matched asymptotic expansions reduces the problem to a first-order, ordinary, nonlinear, integro-differential equation for the transient development of the interior angular velocity on the time scale of spin- up, as driven by quasi-steady nonlinear Ekman-Hartmann boundary layers. This two-parameter equation is solved analytically in certain limits and numeric-ally in general. The solutions show that nonlinear non-magnetic spin-up and spin-down take longer than for linearized flow, spin-down occurring more rapidly in the early stages but requiring more time for completion than spin-up. A magnetic field promotes both spin-up and spin-down, but a weak field is relatively ineffective for spin-down yet very effective for spin-up. A strong magnetic field dominates nonlinear processes and gives identical spin-up and spin-down times, which coincide with that found from linear hydromagnetic theory.

On the spin-up of an electrically conducting fluid Part 2. Hydromagnetic spin-up between infinite flat insulating plates

The linear spin-up of a homogeneous electrically conducting fluid confined between infinite flat insulating plates is analyzed for the case in which a uniform magnetic field is applied normal to the boundaries. As in part 1 (Benton & Loper 1969), complete hydromagnetic interaction is embraced even within linearized equations. Approximate inversion of the exact Laplace transform solution reveals the presence of several flow structures: two thin Ekman–Hartmann boundary layers (one on each plate), which are quasi-steady on the time scale of spin-up, two thicker continuously growing magnetic diffusion regions, and an essentially inviscid, current-free core, which may or may not be present on the spin-up time scale, depending upon the growth rate of the magnetic diffusion regions. When a current-free core exists, it is found to spin-up at the same rate as the fluid within magnetic diffusion regions, although different physical mechanisms are at play. As a result, a single hydromagnetic spin-up time is derived, independently of the thickness of magnetic diffusion regions; this time is shorter than in the non-magnetic problem.