Uniform Distribution Of Heat Sources Research Articles

Onset of Convection in an Inclined Anisotropic Porous Layer with Internal Heat Generation

The onset of convection in an inclined porous layer which is heated internally by a uniform distribution of heat sources is considered. We investigate the combined effects of inclination, anisotropy and internal heat generation on the linear instability of the basic parallel flow. When the Rayleigh number is sufficiently large, instability occurs and a convective motion is set up. It turns out that the preferred motion at convection onset depends quite strongly on the anisotropy ratio, ξ , and the inclination angle. When ξ < 1 the preferred motion is in the form of longitudinal rolls for all inclinations. When ξ > 1 transverse rolls are preferred for small inclinations but, at high inclinations, longitudinal rolls are preferred. At intermediate inclinations the preferred roll orientation varies smoothly between these two extremes.

The effect of anisotropic heat transport in the earth's core on the geodynamo

Intermediate dynamos are axisymmetric, spherical models that evade Cowling's theorem by invoking an α-effect to create the meridional magnetic field from the zonal. Usually the energy source maintaining the motions is a specified thermal wind, but here the dynamo is driven by the buoyancy created by a uniform distribution of heat sources. It has been argued by Braginsky and Meytlis (this journal, vol. 55, 1990) that, in a rapidly rotating, strongly magnetic system such as the Earth's core, heat is transported principally by a small-scale turbulence that is highly anisotropic. They conclude that the diffusion of heat parallel to the rotation axis is then significantly greater than it is in directions away from that axis. A preliminary study of the consequences of this idea is reported here. Solutions are derived numerically using both isotropic and non-isotropic thermal diffusivity tensors, and the results are compared. It is shown that even a small degree of anisotropy can materially alter the character of the dynamo.

The onset of thermal convection in rotating spherical shells

The correct asymptotic theory for the linear onset of instability of a Boussinesq fluid rotating rapidly in a self-gravitating sphere containing a uniform distribution of heat sources was given recently by Jones et al. (2000). Their analysis confirmed the established picture that instability at small Ekman number corrections due to Ekman suction at the outer boundary. All analytic predictions for both stress-free and rigid (no-slip) boundaries compare favourably with our full numerics (always with Prandtl number unity), despite the fact that very small Ekman numbers are needed to reach a true asymptotic regime.

Localised rotating convection induced by topography

The onset of instability of a rapidly rotating, self-gravitating, Boussinesq fluid in a spherically symmetric cavity containing a uniform distribution of heat sources in the small Ekman number limit ( E ⪡ 1) is characterised by the longitudinal propagation of thermal Rossby waves on a short azimuthal φ-length scale O(E 1 3 ) . Here we investigate the onset of instability via a steady geostrophic mode of convection which may occur when the outer spherical boundary is deformed. Attention is restricted to topographic features with simple longitudinal dependence proportional to cos mφ and small height of order ε m . Motion is composed of two parts: the larger is geostrophic and follows the geostrophic contours; the smaller is convective and locked to the topography. Analytic solutions are obtained for the case of rigid boundaries when E 1 2 ⪡ ε 2 ⪡ 1 and E −1 8 ⪡ m ⪡ E −1 3 ; onset of instability is characterised by these modes (with geostrophic motion localised radially on a length scale O(E 1 8 ) ) when ε 2 ⪢ E 2 3 m 3 2 . Solutions are obtained for the case of slippery boundaries in different parameter ranges and, in contrast, these are not localised but fill the sphere. In both cases the critical Rayleigh number grows with decreasing ε, localising the convection of heat in a neighbourhood close to the surface.

Conjugate forced convection in crossflow over a cylinder array with volumetric heating

Numerical simulation of steady forced convection heat transfer in a laminar flow field over an infinite (periodic) and finite in-line array of cylinders is performed. The cylinders are arranged with a pitch to diameter ratio of two, and are heated internally with a uniform distribution of heat sources. The conjugate heat transfer problem is described by two coupled energy equations (one for the fluid, the other for the solid) and the Navier-Stokes equations. This system is solved on curvilinear coordinates via a finite-difference iterative scheme and a domain decomposition procedure. By varying the ratio of fluid-to-solid thermal conductivity (FS) between 0.01 and 100, a parametric study of heat exchange between the solid and fluid (as expressed by the Nusselt number) is reported for Reynolds numbers between 100 and 400 and Prandtl number equal to 0.71. Our results reveal that the internal heat generation case is profoundly different from that of the isothermal cylindrical array if FS is larger than unity. As FS approaches extreme values, e.g. 0.01 (or 100), the temperature field can be approximated by invoking simpler models, e.g. local thermal equilibrium (or concentric) model.

Thermal convection in a spherical shell with a variable radius ratio

Abstract We consider, as a model for the Earth's core, an electrically conducting fluid sphere with a solid concentric inner core. We impose an azimuthal magnetic field B0(r,θ)1ø, azimuthal shear flow U0(r,θ)1 ø and temperature distribution T0 (r), where (r,θ,ø) are spherical polar coordinates, and consider the linear stability of this basic state. We investigate the effect of the growth of an inner core on convection due to a uniform distribution of heat sources, and find that increasing the inner boundary radius is initially beneficial to convection. Further, increasing the magnetic field strength increases the range of this effect. The introduction of differential rotation provides us with results for a sphere corresponding to those in other geometries (Fearn and Proctor, 1983b; Fearn, 1989). For values of the Roberts number, q = K/n, considered here (q = 10−6 and q = 1) convection becomes inhibited when the magnetic Reynolds number Rm (which measures the strength of the shear) is O(q). When Rm>q convection is found to become increasingly localised. For q =10−6 only the temperature perturbation becomes concentrated at some point, but when q =1 all perturbation variables become so. We modify the temperature distribution T0 to model the effect of compositional convection. This is found to be a more efficient mechanism, although the convection patterns produced are very similar to those of the heat source distribution.

Buoyancy Driven Instability Due to Internal Heat Sources in a Horizontal Layer of Binary Fluid Mixture

This paper is concerned with the problem of the onset of buoyancy driven instability in a horizontal layer of binary fluid mixture which is heated internally by a uniform distribution of heat sources. The layer is bounded below by a thermal insulator and above by a rigid wall of constant temperature. Linear stability theory is applied to derive an eigenvalue system of eighth order which is then solved by the power series method. In order to reduce the amount of numerical work, Galerkin's method is also used. For wide ranges of various parameters, the conditions under which instability sets in are determined in detail.

On the propagation and stability of wave motions in rapidly rotating spherical shells: I. The non-magnetic case

Abstract The linear propagation properties of wave motions in a rapidly rotating stratified Boussinesq spherical shell, of outer radius 1 and inner radius η, are studied in the small Prandtl number limit. When η = 0, the various possible motions can be accommodated in two classes: F and D. The class F is closely related to the class of free oscillations of the inviscid unstratified fluid shell while D corresponds to diffusive inertial-gravity waves. Both F and D are subdivided into two infinite sets of modes one of which (E) is symmetric and the other (O) is anti-symmetric about the equatorial plane. The sets E and 0 for class F are further subdivided into two infinite subsets one of which propagates (in phase) eastward and the other westward. Waves of class D propagate eastward. The two classes F and D are decoupled except for one mode which belongs to D on and in the immediate neighbourhood of the axis of rotation and transforms into F away from the axis. This mode provided the already known mode of convection of the linear stability of a fluid sphere containing a uniform distribution of heat sources. When an inner solid core is present (and η is non-zero) all the modes of classes F and D persist outside C, (where C, is the coaxial cylinder whose generators touch the inner core at its equator) but the set E of modes (of both F and D) is suppressed within C c. The stability of the elements of F and D is examined for various forms of the temperature gradient, β, providing the stratification. If the shell is unstably stratified everywhere then every wave can be destabilized if β is large enough. If, however, the shell is composed of an inner region of unstably stratified fluid surrounded by a stably stratified region then only a finite number of modes of F can be made unstable whatever the value of β. The critical mode of convection almost always belongs to the same mode but the location of the preferred cell of convection depends on β. For example, in the case of a differentially heated shell it is situated on C c for all values of η.

Thermally driven hydromagnetic convection in a rapidly rotating sphere

Buoyancy-induced convection is investigated in a rapidly rotating, self-gravitating fluid sphere internally heated by a uniform distribution of heat sources and under the influence of an azimuthal magnetic field whose strength is proportional to the distance ϖ * from the rotation axis. Atten­tion is restricted to relatively small magnetic field strengths (as measured by the parameter ∧ ) such that the dominant force balance remains geo-strophic. Convection is then confined to a thin cylindrical annulus, radius ϖ 0 *, about the rotation axis. A linear analysis is used to find the state of marginal stability together with the corresponding minimum critical value of the modified Rayleigh number, R c (a measure of the buoyancy force required to maintain convective motions). Two distinct modes of instability are found to operate: ‘Rossby’ and ‘magnetic’. When no magnetic field is applied ( ∧ = 0), and when the Prandtl number σ ≪ 1, the instability takes the form of a thermally driven Rossby wave propagating eastward but with group velocity westward. Modifications to this mode due to a non-zero magnetic field result in R c being a complicated function R c ( ∧, q, σ ), where q = k/η and σ = v/k with v , k and η denoting the viscous, thermal and magnetic diffusivities. When it has a significant effect [ σ ≤ O (1)], the magnetic field inhibits the Rossby mode and convection moves towards the axis where the field is weaker ( ϖ 0 *→ 0). The presence of the magnetic field also permits new modes of instability which are facilitated by increasing field strength, with R c ∝ ∧ ─1 . These magnetic modes propagate eastwards (westwards) when q < q m ( q > q m ) with q m ≈ 2.8. Of particular interest to the Earth is the limit q ≪ 1 where the marginally stable magnetic mode takes the form of an undamped buoyancy wave in an unstably stratified fluid. No diffusive processes are operating at leading order and the only effect opposing convection is the geostrophic constraint.

Thermal and magnetic instabilities in a rapidly rotating fluid sphere

Abstract A linear analysis is used to study the stability of a rapidly rotating, electrically-conducting, self-gravitating fluid sphere of radius r 0, containing a uniform distribution of heat sources and under the influence of an azimuthal magnetic field whose strength is proportional to the distance from the rotation axis. The Lorentz force is of a magnitude comparable with that of the Coriolis force and so convective motions are fully three-dimensional, filling the entire sphere. We are primarily interested in the limit where the ratio q of the thermal diffusivity κ to the magnetic diffusivity η is much smaller than unity since this is possibly of the greatest geophysical relevance. Thermal convection sets in when the temperature gradient exceeds some critical value as measured by the modified Rayleigh number Rc. The critical temperature gradient is smallest (Rc reaches a minimum) when the magnetic field strength parameter Λ ≃ 1. [Rc and Λ are defined in (2.3).] The instability takes the form of a very...

Axisymmetric convection in a rotating sphere: Part 2. Non-slip surface

Abstract This paper reports numerical experiments on large-scale nonlinear thermal convection in a rotating self-gravitating sphere of Boussinesq fluid containitrg a uniform distribution of heat sources. Solutions are obtained for Taylor numbers in the range 0≦Δ< 105, Rayleigh numbers in the range Rc≦R≲10Rc , and Prandtl numbers in the range 0.1≦P≦10. The present solutions, which are dynamically possible, have a similar form to that which seems to be required by some kinematic models of the earth's hydromagnetic dynamo. The flows with nearly uniform angular momentum and the sustained overstable oscillations, both of which occur when the outer surface is stress-free, are found to be inhibited by a non-slip surface. Most of the solutions qualitatively resemble those predicted by the linear stability equations, with wavenumber increasing with R and A, but other modes, sometimes including a sustained non-linear oscillation, can be produced by different initial conditions. The effect of Prandtl number is small...

Hydromagnetic convective instability of a rotating, self-gravitating fluid sphere containing a uniform distribution of heat sources

The desire to understand better the magneto-hydrodynamics of the Earth's and planetary interiors has recently motivated a number of studies on convective motions in hydromagnetic rotating systems. These studies have, however, been restricted to planar geometry, the convective layer being confined between two horizontal planes in externally applied uniform gravitational and magnetic fields. This paper takes a step further to the geophysical and astrophysical contexts by restoring curvature effects. The linear stability of a uniformly rotating, self-gravitating fluid sphere in the presence of a co-rotating zonal magnetic field is studied when buoyancy is provided by a uniform distribution of heat sources. The analysis is limited to the case where the Chandrasekhar number,Q, and the Taylor number,λ2, are both large. (These are, respectively, dimension-less measures of Lorentz and Coriolis forces relative to the viscous forces.) It is shown that for all values ofλandQthe motions appearing at marginal convection are necessarily time-dependent and associated with a temperature fluctuation which is always symmetric with respect to the equatorial plane. The critical Rayleigh numberRc(λ, Q), which is a dimensionless measure of the temperature contrast necessary for the onset of convection, is found to be qualitatively the same as for the planar modelonlywhenλ≥O(Q), although even in this case certain characteristic curvature effects arise. The motions prevalent at marginal stability, whenO(Q3/2) ≥λ≫Q, occur in the form of a thin cylindrical shell of thicknessO((Q/λ)2/3) and whose distance from the axis of rotation varies between 0.4 and 0.6 spherical radii depending on the value ofq, which is the ratio of the thermal to the magnetic diffusivities. The waves will drift westward or eastward according to whetherq≶ 2.5. (The cause of disagreement in this result with Busse (1975b) is explained in an appendix). Forλ=O(Q)convection occurs in the whole volume of the sphere and the waves drift westward for all values ofq. Whenλ≪Q, not only isRcincorrectly given by that for the plane layer model but also modal degeneracies of convection in the plane layer are removed by the curvature and boundedness of the system. For this range ofλandQconvection again fills the whole sphere but all forms of diffusion are concentrated in multiple boundary layers on the surface of sphere. The waves drift westward. The results are compared with parellel studies, including Braginsky's MAC waves (i. e. Hide's slow magnetohydrodynamic waves) and Busse's recent dynamo model. In particular, it is argued that the last of these may not be representative of planetary magnetism because of a convective growth of field (not considered by Busse) associated with convection patterns occuring in the whole sphere rather than in a cylindrical shell.

Axisymmetric convection in a rotating sphere. Part 1. Stress-free surface

This paper examines large-scale nonlinear thermal convection in a rotating selfgravitating sphere of Boussinesq fluid containing a uniform distribution of heat sources. Conservative finite-difference forms of the equations of axisymmetric laminar motion are marched forward in time. The surface is assumed to be stress free and at constant temperature. Numerical solutions are obtained for Taylor numbers in the range 0 ≤ Λ ≤ 104 and Rayleigh numbers with \[ R_c \leqslant R\lesssim 10R_c. \] For high Prandtl number (P &gt; 5) the solutions are steady and most of them resemble the solutions of the linear stability equations, though other steady solutions are also found. For P [lsim ] 1, the steady solutions have horizontal wavenumber l = 1 and nearly uniform angular momentum per unit mass, rather than nearly uniform angular velocity. This rotation law seems to be independent of many details of the model and may hold in the convective core of a rotating star.

Maximum Internal Temperatures in One-Dimensional Geometries with Nonuniform Heat Generation and Equal Surface Temperatures

The maximum temperature in an infinite slab, an infinite cylindrical shell, or a spherical shell, whose two bounding surfaces are kept at the same constant temperature, depends upon the distribution of heat sources in the region. We find precise upper and lower bounds for the error involved in replacing this maximum temperature by the maximum temperature for a uniform distribution of heat sources with the same average power density.

Convection in an internally heated layer

A numerical study has been made of steady laminar convection in an infinite horizontal layer of fluid, bounded above by a rigid plate held at constant temperature, and below by a thermal insulator. The fluid is heated by a uniform distribution of heat sources. Recent methods for the solution of the Navier-Stokes equations by finite differences are discussed, and the results of integrations in two and three space dimensions and time are presented in an attempt to determine the planform and other flow properties as functions of the Rayleigh and Prandtl numbers.These results are compared with a preliminary study by Roberts (1967) using a mean-field equation approach and with experimental observations by Tritton &amp; Zarraga (1967) and Hooper (private communication).

On the thermal instability of a rotating-fluid sphere containing heat sources

The theory of marginal convection in a uniformly rotating, self-gravitating, fluid sphere, of uniform density and containing a uniform distribution of heat sources, is developed to embrace modes of convection which are asymmetric with respect to the axis of rotation. It is shown that these modes are the most unstable, except for the smallest Taylor numbers, T (a measure of the rotation rate); i.e. for any T and o) (Prandtl number), the lowest Rayleigh number (a measure of the temperature gradients in the sphere) is associated with an asymmetric motion. This is demonstrated both by an expansion method suitable for small T, and by asymptotic theory for T oo. For large T, the eigenmode most easily excited is small in amplitude everywhere except near a cylindrical surface, of radius about half that of the sphere, and coaxial with the diameter parallel to the angular velocity vector.

Finite amplitude convection in a self-gravitating fluid sphere containing heat sources

AbstractThe non-linear equations governing the convective flow of heat, beyond marginal stability, in a fluid sphere containing a uniform distribution of heat sources, and cooled at its surface, are expressed in the Boussinesq approximation as a perturbation of the steady-state conduction solution. A steady-state axisymmetric poloidal flow is assumed, and the equations for an approximate solution, involving the first spherical harmonic only, are obtained using a special case of the general evolution criterion of Glansdorff and Prigogine. The solution is found numerically using an iterative procedure, the results showing good agreement with those obtained by Stuart's ‘shape assumption’ for values of the Rayleigh number just above the critical value for the onset of marginal stability.