Finite-Amplitude thermal convection within a self-gravitating fluid sphere

Abstract

Finite-difference calculations have been carried out to determine the structure of finite-amplitude thermal convection within a self-gravitating fluid sphere with uniform heat release. For a fixed-surface boundary condition single-cell convection breaks up into double-cell convection at a Rayleigh number of 3 × 10⁴, at a Rayleigh number of 5 × 10⁵ four-cell convection is observed. With a free-surface boundary condition only single cell convection is obtained up to a Rayleigh number of 5 × 10⁶.     

Models of solar differential rotation

Abstract

Models of a differentially rotating compressible convection zone are calculated, considering the inertial forces in the poloidal components of the equations of motion. Two driving mechanisms have been considered: latitude dependent heat transport and anisotropic viscosity. In the former case a meridional circulation is induced initially which in turn generates differential rotation, whereas in the latter case differential rotation is directly driven by the anisotropic viscosity, and the meridional circulation is a secondary effect.

In the case of anisotropic viscosity the choice of boundary conditions has a big influence on the results: depending on whether or not the conditions of vanishing pressure perturbation are imposed at the bottom of the convection zone, one obtains differential rotation with a fast (≥ 10 ms^?1) or a slow (～ 1 ms^?1) circulation. In the latter case the rotation law is mainly a function of radius and the rotation rate increases inwards if the viscosity is larger in radial direction than in the horizontal directions.

The models with latitude dependent heat transport exhibit a strong dependence on the Prandtl number. For values of the Prandtl number less than 0.2 the pole-equator temperature difference and the surface velocity of the meridional circulation are compatible with observations. For sufficiently small values of the Prandtl number the convection zone becomes globally unstable like a layer of fluid for which the critical Rayleigh number is exceeded.     

Distribution of magnetic energy in αΩ-dynamos,I: The method

Abstract

In this paper a method for solving the equation for the mean magnetic energy <BB> of a solar type dynamo with an axisymmetric convection zone geometry is developed and the main features of the method are described. This method is referred to as the finite magnetic energy method since it is based on the idea that the real magnetic field B of the dynamo remains finite only if <BB> remains finite. Ensemble averaging is used, which implies that fields of all spatial scales are included, small-scale as well as large-scale fields. The method yields an energy balance for the mean energy density ε ≡ B ²/8π of the dynamo, from which the relative energy production rates by the different dynamo processes can be inferred. An estimate for the r.m.s. field strength at the surface and at the base of the convection zone can be found by comparing the magnetic energy density and the outgoing flux at the surface with the observed values. We neglect resistive effects and present arguments indicating that this is a fair assumption for the solar convection zone. The model considerations and examples presented indicate that (1) the energy loss at the solar surface is almost instantaneous; (2) the convection in the convection zone takes place in the form of giant cells; (3) the r.m.s. field strength at the base of the solar convection zone is no more than a few hundred gauss; (4) the turbulent diffusion coefficient within the bulk of the convection zone is about 10¹⁴cm²s^?1, which is an order of magnitude larger than usually adopted in solar mean field models.     

A review of: “Problems of Solar and Stellar Oscillations. Iau Colloquium No. 66. Edited by D. O. Gough. D. Reidel Publishing Company, 493 pp. Df1195.00/U.S. $84.50 (ISBN 90-277-1554-8). 1983.”

Abstract

The model equations describing two-dimensional thermohaline convection of a Boussinesq fluid in a rotating horizontal layer are known to support multiple instabilities, depending on the values of certain control parameters (Arneodo et al., 1985). Most of these multiple instabilities have already been studied for double or triple diffusive convection, where behaviours ranging from simple steady to irregular motions have been found. Here we consider the one remaining bifurcation mentioned by Arneodo et al. (1985): the interaction between a steady and an oscillatory convection roll when the linear spectrum for a single wavenumber comprises one zero and one pair of purely imaginary eigenvalues. The method of centre manifolds and normal forms is used to derive evolution equations for the amplitudes of the convection rolls close to bifurcation and the behaviours associated with the equations is discussed.     

Finite amplitude instability thresholds in penetrative convection

Abstract

A nonlinear energy stability analysis is presented for the penetrative convection model of Veronis (1963). For top temperatures between 4°C and 8°C the nonlinear stability boundary obtained is very close to the linear one of Veronis and enables a region of possible sub-critical instabilities to be determined.     

On the onset of convection in rotating spherical shells

Abstract

The linear problem of the onset of convection in rotating spherical shells is analysed numerically in dependence on the Prandtl number. The radius ratio η=r _i/r _o of the inner and outer radii is generally assumed to be 0.4. But other values of η are also considered. The goal of the analysis has been the clarification of the transition between modes drifting in the retrograde azimuthal direction in the low Taylor number regime and modes traveling in the prograde direction at high Taylor numbers. It is shown that for a given value m of the azimuthal wavenumber a single mode describes the onset of convection of fluids of moderate or high Prandtl number. At low Prandtl numbers, however, three different modes for a given m may describe the onset of convection in dependence on the Taylor number. The characteristic properties of the modes are described and the singularities leading to the separation with decreasing Prandtl number are elucidated. Related results for the problem of finite amplitude convection are also reported.     

Travelling wave convection in a rotating layer

Abstract

Small amplitude two-dimensional Boussinesq convection in a plane layer with stress-free boundaries rotating uniformly about the vertical is studied. A horizontally unbounded layer is modelled by periodic boundary conditions. When the centrifugal force is balanced by an appropriate pressure gradient the resulting equations are translation invariant, and overstable convection can take the form of travelling waves. In the Prandtl number regime 0.53 < [sgrave] < 0.68 such solutions are preferred over the more usual standing waves. For [sgrave] < 0.53, travelling waves are stable provided the Taylor number is sufficiently large.     

Physical and numerical experiments on layered convection in a density-Stratified fluid

Abstract

The formation and growth of horizontal layered convection cells in a density stratified solution of salt water subject to an impulsively applied lateral temperature gradient is investigated with physical and numerical experiments. Results indicate that lyers are induced by two mechanisms. One is the successive formation of layers due to the presence of the top and bottom boundaries. The other is the spontaneous occurrence of layers when a suitably defined Rayleigh number exceeds a critical value. It is found that well established layers are homogeneous in temperature and salinity and are separated by sharp gradients in density. Lateral heat transfer is of a periodic nature. Numerical experiments were carried out for finite and infinite geometry cases. For the finite geometry case, convection cells are generated successively inward from the horizontal boundaries. For the infinite geometry case, periodic conditions in the vertical direction are assumed. With continuous input of small perturbations, simultaneous occurrence of the convection cells is obtained at supercritical Rayleigh numbers. Criteria for determining the onset of spontaneous cells numerically are explored.     

Large prandtl number finite-amplitude thermal convection with maxwell viscoelasticity

Abstract

Nonlinear interactions of deviatoric stress components and the velocity field occur in all dynamic flows where convected elasticity is accounted for. By incorporating a linear Maxwellian constitutive relation (Oldroyd ‘B’ type) into a finite-amplitude convection model we quantify the magnitude of some of the effects of these nonlinear interactions. For viscoelastic flows the relevant nondimensional parameter is the ratio of viscoelastic constitutive relaxation time constant, λ₁, to the basic flow process time. The Rayleigh number, Ra, and the nondimensional ratio of λ₁ to thermal conduction time, τ_c, are part of the parameter space investigated. However, shorter basic flow time scales than that for thermal equilibration are of interest since most viscoelastic fluids have relatively small values of λ₁ The ratio of λ₁ to buoyant time [bcirc], or λ₁/[bcirc], is, therefore, a pertinent parameter. Using both lithospheric and aesthenospheric values for λ₁, the ratio appropriate to mantle convection is roughly bounded by O(1)[bcirc]>λ₁/[bcirc]>O(10^?6). Employing these bounds and computing low Rayleigh number time-dependent convective flows in a two-dimensional box, it is demonstrated that viscoelasticity has a negligible influence on quasi-steady heat transport even for λ₁/[bcirc]～O(1) For any time-dependent behavior with time scales as short, or shorter than, the buoyant time, [bcirc], viscoelasticity might be important to the local exchange of mechanical energy. The recoverable strain energy in the descending portion of the lithosphere is comparable to the local viscous dissipation. The magnitude of this recoverable component of shear is proportional to λ₁/[bcirc].     

A theory of convection: Modelling by two buoyant interacting fluids

Abstract

A new model of convection and mixing is presented. The fluid is envisioned as being composed of two buoyant interacting fluids, called thermals and anti-thermals. In the context of the Boussinesq approximation, pairs of governing equations are derived for thermals and anti-thermals. Each pair meets an Invariance Principle as a consequence of the reciprocity in the roles played by thermals and anti-thermals. Each pair is transformed into an average equation for which interaction terms cancel and another very simple equation linking the two fluid properties. An important parameter of the model is the fraction, f, of area occupied by thermals to the total area. A dynamic saturation equilibrium between thermals and antithermals is assumed. This implies a constant values of f throughout the system. The set of equations is written in terms of mean values and root-mean-square fluctuations, in keeping with equations of turbulence theories. The final set consists of four coupled non-linear differential equations. The model neglects dissipation and can be applied to any convective situations where molecular viscosity and diffusivity may be neglected. Applications of the model to mixed-layer deepening and penetrative convection are presented in subsequent papers.     

Intermittent convection

Abstract

A theoretical analysis of pseudo two-dimensional, finite-amplitude, thermal convection is made for an infinite Prandtl number fluid which is subjected to a constant heat flux out of the top boundary and insulated at the bottom. For large Rayleigh numbers the convective flow becomes intermittent and the system is characterized by the following cyclic process: the formation of a thermal boundary layer by diffusion, the instability of this layer when it becomes sufficiently thick, the destruction of the layer by the convective flow, the dying down of the convection, and the reforming of the thermal boundary layer by diffusion. The periodicity and the horizontal wave number of the intermittent convective flow are found to be independent of the depth of the fluid layer but depend on the rate of cooling and the properties of the fluid.     

On the upper bounding approach to thermal convection at moderate rayleigh numbers

Abstract

Two upper bounding problems for thermal convection in a layer of fluid contained between perfectly conducting stress-free boundaries are treated numerically. Since the Euler equations resulting from this variational approach are simpler than the Navier-Stokes equations, they allow numerical calculations to be carried out economically to fairly large values of the Rayleigh number. The upper bounding problem formulated by Howard (1963), which yields a Nusselt number independent of Prandtl number, diverges from the correct behavior as the Rayleigh number increases. In hopes of coming closer to results of previous investigations of the Boussinesq equations of motion, a more restrictive upper bounding problem is formulated. For large Prandtl numbers the momentum equation is linearized and is used as an explicit side constraint on the variational problem, thereby forcing the solutions to more closely resemble the solutions of the Boussinesq equations. Numerical calculations at values of the Rayleigh number up to 1.5 × 10⁵ indicate that the additional constraint decreases the upper bound on the Nusselt number; it appears that this upper bound differs by only a multiplicative factor from that calculated from solutions of the full equations of motion and may be a reasonable approximation for large Rayleigh numbers.     

Instability of flows near the onset of convection in a rotating layer with stress-free horizontal boundaries

We investigate instability of convective flows of simple structure (rolls, standing and travelling waves) in a rotating layer with stress-free horizontal boundaries near the onset of convection. We show that the flows are always unstable to perturbations, which are linear combinations of large-scale modes and short-scale modes, whose wave numbers are close to those of the perturbed flows. Depending on asymptotic relations of small parameters α (the difference between the wave number of perturbed flows and the critical wave number for the onset of convection) and ε (ε² being the overcriticality and the perturbed flow amplitude being O(ε)), either small-angle or Eckhaus instability is prevailing. In the case of small-angle instability for rolls the largest growth rate scales as ε^8/5, in agreement with results of Cox and Matthews (Cox, S.M. and Matthews, P.C., Instability of rotating convection. J. Fluid. Mech., 2000, 403, 153–172) obtained for rolls with k = k _c . For waves, the largest growth rate is of the order ε^4/3. In the case of Eckhaus instability the growth rate is of the order of α².     

Magnetic buoyancy and the Boussinesq approximation

Abstract

The full Boussinesq equations for hydromagnetic convection are derived and shown to include the effects of magnetic buoyancy. Instabilities caused by magnetic buoyancy are analyzed and their roles in double convection are brought out.     

Nonlinear convection in an imposed horizontal magnetic field

Abstract

Nonlinear two-dimensional magnetoconvection, with a Boussinesq fluid driven across the field-lines, is taken as a model for giant-cell convection in the sun and late-type stars. A series of numerical experiments shows the sensitivity of the horizontal scale of convection to the applied field and to the Rayleigh number R. Overstable oscillations occur in cells as broad as they are deep, but increasing R leads to steady motions of much greater wavelength. Purely geometrical effects can cause oscillation: this work implies that strong horizontal field will in general lead to time-dependent convection.     

A benchmark comparison of numerical methods for infinite Prandtl number thermal convection in two-dimensional Cartesian geometry

Abstract

A comparison is made between seven different numerical methods for calculating two-dimensional thermal convection in an infinite Prandtl number fluid. Among the seven methods are finite difference and finite element techniques that have been used to model thermal convection in the Earth's mantle. We evaluate the performance of each method using a suite of four benchmark problems, ranging from steady-state convection to intrinsically time-dependent convection with recurring thermal boundary layer instabilities. These results can be used to determine the accuracy of other computational methods, and to assist in the development of new ones.     

Salt-Finger convection in the surface discharge of heated saline jets

Abstract

Experimental investigations of the surface discharge of two-dimensional heated saline jets into surroundings with stable, constant salt gradients were carried out. The discharge conditions were parameterized with the densimetric Froude number, and the Reynolds number. The evolution of the discharge was monitored by flow visualization methods, and by the measurements of temperature and salinity distributions. For comparison, experiments of the surface discharge of heated water into homogeneous surroundings at the corresponding discharge conditions were also conducted. The results clearly showed that while in the former case, the region away from the vicinity of the discharge manifold was marked by the presence of salt-finger convection, in the latter case this region exhibited stable thermal stratification. Furthermore the occurrence of salt-finger convection considerably retarded the motion of the jet, and increased the penetration depth of temperature and salinity fields.     

An effective scale-dependent dispersivity deduced from a purely convective flow field

Abstract

In the case of straight flow but with hydraulic conductivity varying in a transverse direction, the distribution of hydraulic conductivity has been determined for which the breakthrough curve due to convection only will have the same analytical form as the onedimensional convection/dispersion equation solution at the outlet end of a porous medium. That distribution is found exactly and it is very similar to the lognormal distribution. This result is significant since field evidence indicates that the logarithm of hydraulic conductivity is normally distributed. For the case considered, a simple relation between dispersivity and the coefficient of variation of hydraulic conductivity is found. One can thus determine very simply dispersivity in terms of the parameters of the distribution of hydraulic conductivity. This is particularly useful to estimate dispersivity in various cells of finite difference or finite element models when the distribution of hydraulic conductivity is not stationary, i.e. varies in space.     

Nonlinear internal waves in the upper atmosphere

To investigate the mechanism of mixing in oscillatory doubly diffusive (ODD) convection, we truncate the horizontal modal expansion of the Boussinesq equations to obtain a simplified model of the process. In the astrophysically interesting case with low Prandtl number (traditionally called semiconvection), large-scale shears are generated as in ordinary thermal convection. The interplay between the shear and the oscillatory convection produces intermittent overturning of the fluid with significant mixing. By contrast, in the parameter regime appropriate to sea water, large-scale flows are not generated by the convection. However, if such flows are imposed externally, intermittent overturning with enhanced mixing is observed.