Transitions to chaotic thermal convection in a rapidly rotating spherical fluid shell

Abstract

Numerical simulations of thermal convection in a rapidly rotating spherical fluid shell heated from below and within have been carried out with a nonlinear, three-dimensional, time-dependent pseudospectral code. The investigated phenomena include the sequence of transitions to chaos and the differential mean zonal rotation. At the fixed Taylor number T _a =10⁶ and Prandtl number Pr=1 and with increasing Rayleigh number R, convection undergoes a series of bifurcations from onset of steadily propagating motions SP at R=R _c = 13050, to a periodic state P, and thence to a quasi-periodic state QP and a non-periodic or chaotic state NP. Examples of SP, P, QP, and NP solutions are obtained at R = 1.3R _c , R = 1.7 R _c , R = 2R _c , and R = 5 R _c , respectively. In the SP state, convection rolls propagate at a constant longitudinal phase velocity that is slower than that obtained from the linear calculation at the onset of instability. The P state, characterized by a single frequency and its harmonics, has a two-layer cellular structure in radius. Convection rolls near the upper and lower surfaces of the spherical shell both propagate in a prograde sense with respect to the rotation of the reference frame. The outer convection rolls propagate faster than those near the inner shell. The physical mechanism responsible for the time-periodic oscillations is the differential shear of the convection cells due to the mean zonal flow. Meridional transport of zonal momentum by the convection cells in turn supports the mean zonal differential rotation. In the QP state, the longitudinal wave number m of the convection pattern oscillates among m = 3,4,5, and 6; the convection pattern near the outer shell has larger m than that near the inner shell. Radial motions are very weak in the polar regions. The convection pattern also shifts in m for the NP state at R = 5R _c , whose power spectrum is characterized by broadened peaks and broadband background noise. The convection pattern near the outer shell propagates prograde, while the pattern near the inner shell propagates retrograde with respect to the basic rotation. Convection cells exist in polar regions. There is a large variation in the vigor of individual convection cells. An example of a more vigorously convecting chaotic state is obtained at R = 50R _c . At this Rayleigh number some of the convection rolls have axes perpendicular to the axis of the basic rotation, indicating a partial relaxation of the rotational constraint. There are strong convective motions in the polar regions. The longitudinally averaged mean zonal flow has an equatorial superrotation and a high latitude subrotation for all cases except R = 50R _c , at this highest Rayleigh number, the mean zonal flow pattern is completely reversed, opposite to the solar differential rotation pattern.     

Turbulent transport of magnetic fields. V. Distribution of magnetic energy in a simple α2-dynamo

Abstract

In this paper we analyse the stationary mean energy density tensor T_ij = B_iB_j for the x ²-sphere. This model is one of the simplest possible turbulent dynamos, originally due to Krause and Steenbeck (1967): a conducting sphere of radius R with homogeneous, isotropic and stationary turbulent convection, no differential rotation and negligible resistivity. The stationary solution of the (linear) equation for T_ij is found analytically. Only T_rr , T _θθ and T _φφ are unequal to zero, and we present their dependence on the radial distance r.

The stationary solution depends on two coefficients describing the turbulent state: the diffusion coefficient β≈?u²?τ_c/3 and the vorticity coefficient γ ≈ ?|?×u|²?τ_c/3 where u(r, t) is the turbulent velocity and _c its correlation time. But the solution is independent of the dynamo coefficient α≈??u·?×u?τ_c/3 although α does occur in the equation for T_ij . This result confirms earlier conclusions that helicity is not required for magnetic field generation. In the stationary state, magnetic energy is generated by the vorticity and transported to the boundary, where it escapes at the same rate. The solution presented contains one free parameter that is connected with the distribution of B over spatial scales at the boundary, about which T_ij gives no information. We regard this investigation as a first step towards the analysis of more complicated, solar-type dynamos.     

Penetrative convection in the upper ocean due to surface cooling

Abstract

A new non-linear model of mixing and convection based on a modelling of two buoyant interacting fluids is applied to penetrative convection in the upper ocean due to surface cooling. In view of simple algebra, the model is one-dimensional. Dissipation is included, but no mean shear is present. A non-similar analytical solution is found in the case of a well-mixed layer bounded below by a sharp thermocline treated as a boundary layer. This solution is valid if the Richardson number, R _i , defined as the ratio of the total mixed-layer buoyancy to a characteristic rms vertical velocity, is much greater than unity. The model predicts a deepening rate proportional to R _i ^?3/4. The thermocline remains of constant thickness, and the ratio thermocline thickness to mixed-layer depth decreases as R _i ^?3/4 as the mixed layer deepens. If the surface flux is constant, the mixed-layer depth increases with time as t ^½. The vertical structure throughout the mixed layer and thermocline is given by the analytical solution, and vertical profiles of mean temperature and vertical fluxes are plotted. Computed profiles and available laboratory data agree remarkably well. Moreover, the accuracy of the simple analytical results presented here is comparable to that of sophisticated turbulence numerical models.     

Numerical simulations of thermal convection in a rapidly rotating spherical shell cooled inhomogeneously from above

Abstract

Numerical simulations of thermal convection in a rapidly rotating spherical fluid shell with and without inhomogeneous temperature anomalies on the top boundary have been carried out using a three-dimensional, time-dependent, spectral-transform code. The spherical shell of Boussinesq fluid has inner and outer radii the same as those of the Earth's liquid outer core. The Taylor number is 10⁷, the Prandtl number is 1, and the Rayleigh number R is 5R_c (R_c is the critical value of R for the onset of convection when the top boundary is isothermal and R is based on the spherically averaged temperature difference across the shell). The shell is heated from below and cooled from above; there is no internal heating. The lower boundary of the shell is isothermal and both boundaries are rigid and impermeable. Three cases are considered. In one, the upper boundary is isothermal while in the others, temperature anomalies with (l,m) = (3,2) and (6,4) are imposed on the top boundary. The spherically averaged temperature difference across the shell is the same in all three cases. The amplitudes of the imposed temperature anomalies are equal to one-half of the spherically averaged temperature difference across the shell. Convective structures are strongly controlled by both rotation and the imposed temperature anomalies suggesting that thermal inhomogeneities imposed by the mantle on the core have a significant influence on the motions inside the core. The imposed temperature anomaly locks the thermal perturbation structure in the outer part of the spherical shell onto the upper boundary and significantly modifies the velocity structure in the same region. However, the radial velocity structure in the outer part of the shell is different from the temperature perturbation structure. The influence of the imposed temperature anomaly decreases with depth in the shell. Thermal structure and velocity structure are similar and convective rolls are more columnar in the inner part of the shell where the effects of rotation are most dominant.     

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.     

Penetrative convection in the planetary mantle

Abstract

The hydrodynamic equations for thermal convection in a plane layer of viscous, heat conducting fluid are scaled using the normalization of Ostrach (1965) in which the magnitude of the non-dimensional group τ = gαd/c_p determines the importance of compression work and viscous dissipation in the energy balance of the flow. A linear asymptotic theory valid in the limit τ → ∞ is constructed for the Bénard problem and this is shown to be analogous to Couette flow between contra-rotating cylinders. For sufficiently large τ the flow becomes penetrative. This fact is illustrated for homogeneous fluids by the numerical integration of a set of coupled 1st order differential equations, both for the Bénard and internally heated configurations. The effect of viscosity and thermal conductivity in-homogeneity on the depth of penetration of the main cell in the circulation pattern are assessed and it is concluded that such interactions may be sufficient to effectively limit the depth extent of mantle convection. Finally a discussion of the effect of phase transitions is given following the technique of Busse and Schubert (1971).     

Non-linear marginal convection in a rotating magnetic system

Abstract

An inviscid, electrically conducting fluid is contained between two rigid horizontal planes and bounded laterally by two vertical walls. The fluid is permeated by a strong uniform horizontal magnetic field aligned with the side wall boundaries and the entire system rotates rapidly about a vertical axis. The ratio of the magnitudes of the Lorentz and Coriolis forces is characterized by the Elsasser number, A, and the ratio of the thermal and magnetic diffusivities, q. By heating the fluid from below and cooling from above the system becomes unstable to small perturbations when the adverse density gradient as measured by the Rayleigh number, R, is sufficiently large.

With the viscosity ignored the geostrophic velocity, U, which is aligned with the applied magnetic field, is independent of the coordinate parallel to the rotation axis but is an arbitrary function of the horizontal cross-stream coordinate. At the onset of instability the value of U taken ensures that Taylor's condition is met. Specifically the Lorentz force, which results from marginal convection must not cause any acceleration of the geostrophic flow. It is found that the critical Rayleigh number characterising the onset of instability is generally close to the corresponding value for the usual linear problem, in which Taylor's condition is ignored and U is chosen to vanish. Significant differences can occur when q is small owing to a complicated flow structure. There is a central interior region in which the local magnetic Reynolds number, R_m , based on U is small of order q and on exterior region in which R_m is of order unity.     

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.     

Nonlinear dynamics of boussinesq convection in a deep rotating spherical shell-i

Abstract

Convection in a rotating spherical shell has wide application for understanding the dynamics of the atmospheres and interiors of many celestial bodies. In this paper we review linear results for convection in a shell of finite depth at substantial but not asymptotically large Taylor numbers, present nonlinear multimode calculations for similar conditions, and discuss the model and results in the context of the problem of solar convection and differential rotation. Detailed nonlinear calculations are presented for Taylor number T = 10⁵, Prandtl number P = 1, and Rayleigh number R between 1 |MX 10⁴ and 4 |MX 10⁴ (which is between about 4 and 16 times critical) for a shell of depth 20% of the outer radius. Sixteen longitudinal wave numbers are usually included (all even wave numbers m between 0 and 30) the amplitudes of which are computed on a staggered grid in the meridian plane.

The kinetic energy spectrum shows a peak in the wave number range m = 12–18 at R = 10⁴, which straddles the critical wave number m = 14 predicted by linear theory. These are modes which peak near the equator. The spectrum shows a second strong peak at m = 0, which represents the differential rotation driven by the peak convective modes. As R is increased, the amplitude of low wave numbers increases relative to high wave numbers as convection fills in in high and middle latitudes, and as the longitudinal scale of equatorial convection grows. By R = 3 |MX 10⁴, m = 8 is the peak convective mode. There is a clear minimum in the total kinetic energy at middle latitudes relative to low and high, well into the nonlinear regime, representing the continued dominance of equatorial and polar modes found in the linear case. The kinetic energy spectrum for m > 0 is maintained primarily by buoyancy work in each mode, but with substantial nonlinear transfer of kinetic energy from the peak modes to both lower and higher wave numbers.

For R = 1 to 2 |MX 10⁴, the differential rotation takes the form of an equatorial acceleration, with angular velocity generally decreasing with latitude away from the equator (as on the sun) and decreasing inwards. By R = 4 |MX 10⁴, this equatorial profile has completely reversed, with angular velocity increasing with depth and latitude. Also, a polar vortex which has positive rotation relative to the reference frame (no evidence of which has been seen on the sun) builds up as soon as polar modes become important. Meridional circulation is quite weak relative to differential rotation at R = 10⁴, but grows relative to it as R is increased. This circulation takes the farm of a single cell of large latitudinal extent in equatorial regions, with upward flow near the equator, together with a series of narrower cells in high latitudes. It is maintained primarily by axisymmetric buoyancy forces. The differential rotation is maintained at all R primarily by Reynolds stresses, rather than meridional circulation. Angular momentum transport toward the equator for R = 1–2 |MX 10⁴ maintains the equatorial acceleration while radially inward transport maintains the opposite profile at R = 4 |MX 10⁴.

The total heat flux out the top of the convective shell always shows two peaks for the range of R studied, one at the equator and the other near the poles (no significant variation with latitude is seen on the sun), while heat flux in at the bottom shows only a polar peak at large R. The meridional circulation and convective cells transport heat toward the equator to maintain this difference.

The helicity of the convection plus the differential rotation produced by it suggest the system may be capable of driving a field reversing dynamo, but the toroidal field may migrate with lime in each cycle toward the poles and equator, rather than just toward the equator as apparently occurs on the sun.

We finally outline additions to the physics of the model to make it more realistic for solar application.     

Estimating time parameters of overland flow on impervious surfaces by the particle tracking method

Abstract

Travel time and time of concentration T_c are important time parameters in hydrological designs. Although T_c is the time for the runoff to travel to the outlet from the most remote part of the catchment, most researchers have used an indirect method such as hydrograph analysis to estimate T_c. A quasi two-dimensional diffusion wave model with particle tracking for overland flow was developed to determine the travel time, and validated for runoff discharges, velocities, and depths. Travel times for 85%, 95% and 100% of particles arrival at the outlet of impervious surfaces (i.e. T_t85, T_t95, and T_t100) were determined for 530 model runs. The correlations between these travel times and T_c estimated from hydrograph analysis showed a significant agreement between T_c and T_t85. All the travel times showed nonlinear relationships with the input variables (plot length, slope, roughness coefficient, and effective rainfall intensity) but showed linear relationships with each other.

Editor D. Koutsoyiannis; Associate editor S. Grimaldi     

Generation of magnetic fields by convection in a rotating sphere,I

The magnetohydrodynamic dynamo problem is solved for an electrically conducting spherical fluid shell with spherically symmetric distributions of gravity and heat sources. The dynamics of motions generated by thermal buoyancy are dominated by the effects of rotation of the fluid shell. Dynamos are found for low and intermediate values of the Taylor number, T ? 10⁵, if the scale of the nonaxisymmetric component of the velocity field is sufficiently small. The generation of magnetic fields of quadrupolar symmetry is preferred at Rayleigh numbers close to the critical value R_c for onset of convection. As the Rayleigh number increases, the generation of dipolar magnetic fields becomes preferred.     

Calculation of magnetic field turbulent diffusivities and α-effect

Abstract

First the exact numerical solutions of DIA system of equations describing the transportation of magnetic field in an infinite medium are presented. It is assumed that the turbulence is stationary, homogeneous, isotropic and incompressible. The spectra of turbulence of δ-type and Kolmogorov's type were used. The steady state values of magnetic field diffusivity D_T and the α-effect coefficient α _T were calculated for various values of space-scale and lifetimes of these spectra and the spectra of helicity. Also investigated is the dependence of D_T and α _T on the degree of helicity. The corrections to the α _T -coefficient due to the contribution of four-order velocity correlators are given. The results are compared with those due to the self-consistent technique.     

Thermal convection in a twisted horizontal magnetic field

The onset of convection in a layer of an electrically conducting fluid heated from below is considered in the case when the layer is permeated by a horizontal magnetic field of strength B ₀ the orientation of which varies sinusoidally with height. The critical value of the Rayleigh number for the onset of convection is derived as a function of the Chandrasekhar number Q. With increasing Q the height of the convection rolls decreases, while their horizontal wavelength slowly increases. Potential applications to the penumbral filaments of sunspots are briefly discussed.     

Thermal regime of the Earth's core

The outer core is assumed to consist of iron and sulfur, with a small amount of potassium that generates heat by radioactive decay of sim||pre|40 K. Two cases are considered, corresponding respectively to a high rate of heat production (Q = 2 · 10¹² cal./sec, about 0.1% K), and to a low rate (Q = 2 · 10¹¹ cal./sec). The temperature at a depth of 2800 km in the mantle is taken to be 3300°K (Wang, 1972). The temperature T_c at the core-mantle boundary depends on whether or not a density gradient in the lowermost layer D″ of the mantle prevents convection in that layer. In the first case, and for high Q, T_c = 4500–5000°K. In the second case, or for low Q, T_c ≈ 3500°K.The heat-conduction equation is used to calculate the temperature T_i at the inner-core boundary in the absence of convection. For high Q, T_i ? T_c ≈ 1600°K; for low Q, T_i ? T_c ≈ 160°K. Corresponding temperature gradients at r = r_c and r = r_i are listed in Table I.The adiabatic gradient at the top of the core is calculated by the method of Stewart (1970). It strongly depends on the parameters (ρ₀, c₀, γ₀, etc.) that characterize core material at low pressure. Stewart has drawn graphs that allow the selection of sets of parameters that are consistent with seismic velocities and a given density distribution in the core. Some acceptable sets of parameters are listed in Table II. Many sets yield temperatures T_c in the range 3500–5000°K; some give an adiabatic gradient steeper than the conductive gradient and are compatible with convection; others do not. Since properties of FeS melts remain unknown, there is at present no way of selecting any set in preference to another.Properties of the FeS system at low pressure suggest the possible appearance of immiscibility at high temperature in liquids of low sulfur content; accordingly, the inner-core boundary is thought to represent equilibrium between a solid (FeNi) inner core and a liquid layer containing only a small amount of sulfur; layer F in turn is in equilibrium with another liquid (forming layer E) containing more sulfur, and slightly less dense, than F. The temperature T_i at the inner-core boundary is about 6000–6500°K for high Q and T_c ≈ 4500–5000°K. It is consistent with Alder's (1966) and Leppaluoto's (1972) estimates of the melting point of iron at 3.3 Mbar, but not with that of Higgins and Kennedy (1971).     

Modelling Rupture Dynamics of a Planar Fault in 3-D Half Space by Boundary Integral Equation Method: An Overview

In this article, we first reviewed the method of boundary integral equation (BIEM) for modelling rupture dynamics of a planar fault embedded in a 3-D elastic half space developed recently (ZHANG and CHEN, 2005a,b). By incorporating the half-space Green's function, we successfully extended the BIEM, which is a powerful tool to study earthquake rupture dynamics on complicated fault systems but limited to full-space model to date, to half-space model. In order to effectively compute the singular integrals in the kernels of the fundamental boundary integral equation, we proposed a regularization procedure consisting of the generalized Apsel-Luco correction and the Karami-Derakhshan algorithm to remove all the singularities, and developed an adaptive integration scheme to efficiently deal with those nonsingular while slowly convergent integrals. The new BIEM provides a powerful tool for investigating the physics of earthquake dynamics. We then applied the new BIEM to investigate the influences of geometrical and physical parameters, such as the dip angle (δ) and depth (h) of the fault, radius of the nucleation region (R_asp), slip-weakening distance (D_c), and stress inside (T_i) and outside (T_e) the nucleation region, on the dynamic rupture processes on the fault embedded in a 3-D half space, and found that (1) overall pattern of the rupture depends on whether the fault runs up to the free surface or not, especially for strike-slip, (2) although final slip distribution is influenced by the dip angle of the fault, the dip angle plays a less important role in the major feature of the rupture progress, (3) different value of h, δ, R_asp, T_e, T_i and D_c may influence the balance of energy and thus the acceleration time of the rupture, but the final rupture speed is not controlled by these parameters.     

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.     

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 R_c. The critical temperature gradient is smallest (R_c reaches a minimum) when the magnetic field strength parameter Λ ? 1. [R_c and Λ are defined in (2.3).] The instability takes the form of a very slow wave with frequency of order κ/r ² ₀ and its direction of propagation changes from eastward to westward as Λ increases through Λ _c ? 4.

When the fluid is sufficiently stably stratified and when Λ > Λ_m ? 22 a new mode of instability sets in. It is magnetically driven but requires some stratification before the energy stored in the magnetic field can be released. The instability takes the form of an eastward propagating wave with azimuthal wavenumber m = 1.     

A model of thermal convection in the major planets with a strongly density-stratified atmospheric layer

Abstract

As an extension of a model by Busse (1983a), a two-layer model of thermal convection in the self-gravitating rotating spherical fluid is considered. The upper layer with arbitrary vertical distributions of density and potential temperature representing the atmospheric layer of major planets is imposed on the spherical Boussinesq fluid. The Prandtl number P and the ratio of the mass of the upper layer to that of the lower layer are used as small expansion parameters. The modification of the critical Rayleigh number by imposing the upper layer are clearly separated into two parts, proportional to (1) the mass of the upper layer and to (2) an integral representing a measure of convective instability of the upper layer. Some implications for atmospheric dynamics of the major planets are also presented.