Convection in a rotating,laterally heated annulus pattern velocities and amplitude oscillations

Abstract

The velocities of the wave patterns relative to the rotating annulus have been measured with either increasing or decreasing positive radial temperature gradients and different rotation rates, with the fluid in thermal equilibrium and in contact with a rigid lid. The pattern velocities are dependent on initial conditions except in the unique areas of the stability diagram, where the velocities observed with either increasing or decreasing ΔT, overlap. The pattern velocities change discontinuously with each wave number transition, with a particularly large discontinuity at the transition from two to one wave. The frequency of the amplitude oscillations of the waves has been measured also. It has been found that the period of the oscillation of the three wave pattern is inversely proportional to the period of the pattern velocity, which means that in this case the ratio of the frequency of amplitude oscillation and the frequency of the pattern revolution is incommensurate.     

Convection in a rotating,laterally heated annulus transition to lower symmetry

Abstract

The radial temperature differences at which the transition from lower symmetry to the wave regime and the transition from the wave regime to lower symmetry occur have been measured for rotation rates ≦2rad/sec. It was found that the temperature differences at which the transitions occur differ for a fixed rotation rate, depending on whether the radial temperature difference is either increased or decreased with time. There is hysteresis in the transition at lower symmetry.     

Rotating deep annulus convection: The steady high amplitude wave regime—Internal thermal fields

Abstract

In the high-amplitude wave regime, at each point in the axial plane, the temperature has a maximum, a minimum, and a time (or longitudinally) averaged value due to the propagation of the regular waves. The axial-plane profile of each of these three temperature fields was measured for 53 different combinations of ω, Δr _w T and n. A few sample profiles are displayed.

Correlations of several internal thermal parameters (especially Ro _g,t ) with the externally imposed parameters, are given along with the spatial variation of the thermal fields. Ro _g,t is always less than the value which would occur if the symmetrical regime existed, while the variation of Sz _g,t with the imposed parameters is about the same as for the symmetrical regime. The parametric variation of the wave amplitude is the only quantity dependent on n. Based on the parametric variation of the fields, the wave regime diagram can be subdivided into several different regions : a quasi-conductive, a conductive-convective zone, and a region at large rotation rates and temperature differences characterized by Ro _g,t = Ro _g,w ², Sz _o,l = Ro _g,w , and ΔT′ = (z′)^0.6.

The Ro _g, l fields are compared to existing theory and some of the experimental stability diagrams of Fultz et al. (1964) are reconstructed using the internal instead of the externally imposed Rossby number. The transition from three to four waves via a “minor wave” is illustrated.     

Stellar Dynamo Waves: Asymptotic Configurations

We investigate the parameter space of a Parker dynamo with a simple alpha quenching nonlinearity, taking as governing parameters the dynamo number D (D<0) and the ratio of diffusion times in the radial and latitudinal directions in the convective zone. The latter parameter, μ, is connected with the aspect ratio (dimensionless thickness) of the convective zone. We isolate two asymptotic configuration of the dynamo waves excited by the Parker dynamo in the limiting case of strong generation. Apart from the standard case with the solar type dynamo wave travelling from mid-latitudes to the equator, we describe a form of dynamo activity which is basically an anharmonic standing wave. The first situation occurs when μ increases with |D|. With μ fixed and |D| increasing, the second asymptotic configuration occurs. We discuss possibilities of identifying these asymptotic configurations with various types of stellar activity as traced by stellar CaII data.     

Heat transfer and thermal boundary layers in high amplitude annulus waves

Abstract

The heat transfer by a rotating, differentially-heated annulus of fluid is measured throughout the high amplitude wave regime. Only Δr_w T was varied (although v(T₁₅ ).K(T₁₅ ) varied by 46%), and it is found that Nu = C₁(λ)Ra^? away from the symmetry and low amplitude to wave transition curves and this is independent of ω. (λ is the wavelength.) On the wave side of these transition curves a region exists in which Nu (symmetry) λ Nu λ C₁(λ)Ra^?. The local heat transfer rate also varies strongly with wave phase.

Using a selection of measured internal thermal fields in the steady, high amplitude wave regime, the side-wall thermal boundary layer structure is examined. It is found that Nu, = C₂·Gr₂ ^A2; both C ₂ and A ₂ are independent of ω and λ to first order. For the time mean profiles, A ₂ ≈ 0.25; in the high heat transfer portion of the wave A ₂ < ¼ and in the low heat transfer portion of the wave A ₂ > ?. These relations hold over most of the vertical extent of the side walls. The deviations of the boundary layers from the above behavior which occur on the remainder of the walls is illustrated. The average thicknesses of the wall boundary layers ∞ Ra^?¼ except in that phase of the wave in which the wall to mid-gap temperature difference is the largest.     

Nonlinear dynamics of a stellar dynamo in a spherical shell

Abstract

A simple mean-field model of a nonlinear stellar dynamo is considered, in which dynamo action is supposed to occur in a spherical shell, and where the only nonlinearity retained is the influence of the Lorentz forces on the zonal flow field. The equations are simplified by truncating in the radial direction, while full latitudinal dependence is retained. The resulting nonlinear p.d.e.'s in latitude and time are solved numerically, and it is found that while regular dynamo wave type solutions are stable when the dynamo number D is sufficiently close to its critical value, there is a wide variety of stable solutions at larger values of D. Furthermore, two different types of dynamo can coexist at the same parameter values. Implications for fields in late-type stars are discussed.     

An antidynamo theorem for partly symmetric flows

Abstract

It is shown that magnetic fields generated by flows v _r,(r,t)e_r+v_T where v_T is an arbitrary toroidal component (e_r˙v_T≡V≡v_T≡0), cannot be maintained indefinitely against ohmic dissipation. The poloidal field variable max |r ² B _r| is shown to decay strictly monotonically with an undetermined decay rate. A bound on the growth of the toroidal field norm ∥T∥₁ is established solely dependent on the rate of conversion of poloidal to toroidal field, so that when the poloidal field is negligible then ∥T∥₁ decays strictly monotonically. The main application of these results is to models of stellar evolution based on axisymmetric differential rotation and spherically symmetric contraction. This symmetric velocity theorem overlaps with two already known theorems, namely the toroidal velocity theorem where v _r≡0 and the radial velocity theorem where v_T≡0. The new theorem does not entirely include the already established ones, principal differences being in the rates of decay and the field variables for which the decay is proven (see Table 1).     

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.     

Persistence of Planetary Waves in the Lower Ionosphere

The transient planetary waves in the atmosphere and ionosphere seem to occur in the form of bursts of a couple of waves with limited persistence. To study persistence of planetary wave events in the lower ionosphere, data from two radio paths from Central Europe are used, Luxembourg – Panská Ves (f = 6.09 MHz, f _eq = 2.1-2.2 MHz) and Deutschlandfunk – Panská Ves (f = 1539 kHz, f _eq = 650-700 kHz). The absorption along the former radio paths is formed very predominantly at altitudes of about 90-100 km, whereas the latter absorption is formed mostly at altitudes of about 85-90 km. The persistence of planetary wave type oscillations is studied in three period bands centred at 5, 10 and 16 days. Waves with period T near 5 days reveal a typical persistence of wave events around 5 cycles. Waves with T = 10 days are less persistent with a typical persistence of 3-4 cycles. The typical persistence of waves T = 16 days is no more than 3 cycles. In terms of number of cycles, the persistence of oscillations evidently decreases with increasing period. On the other hand, in terms of number of days, the persistence seems rather to increase with increasing period.     

Numerical simulations of mantle convection: Time-dependent,three-dimensional,compressible, spherical shell

Abstract

We describe nonlinear time-dependent numerical simulations of whole mantle convection for a Newtonian, infinite Prandtl number, anelastic fluid in a three-dimensional spherical shell for conditions that approximate the Earth's mantle. Each dependent variable is expanded in a series of 4,096 spherical harmonics to resolve its horizontal structure and in 61 Chebyshev polynomials to resolve its radial structure. A semiimplicit time-integration scheme is used with a spectral transform method. In grid space there are 61 unequally-spaced Chebyshev radial levels, 96 Legendre colatitudinal levels, and 192 Fourier longitudinal levels. For this preliminary study we consider four scenarios, all having the same radially-dependent reference state and no internal heating. They differ by their radially-dependent linear viscous and thermal diffusivities and by the specified temperatures on their isothermal, impermeable, stress-free boundaries. We have found that the structure of convection changes dramatically as the Rayleigh number increases from 10⁵ to 10⁶ to 10⁷. The differences also depend on how the Rayleigh number is increased. That is, increasing the superadiabatic temperature drop, δT, across the mantle produces a greater effect than decreasing the diffusivities. The simulation with a Rayleigh number of 10⁷ is approximately 10,000 times critical, close to estimates of that for the Earth's mantle. However, although the velocity structure for this highest Rayleigh number scenario may be adequately resolved, its thermodynamic structure requires greater horizontal resolution. The velocity and thermodynamic structures of the scenarios at Rayleigh numbers of 10⁵ and 10⁶ appear to be adequately resolved. The 10⁵ Rayleigh number solution has a small number of broad regions of warm upflow embedded in a network of narrow cold downflow regions; whereas, the higher Rayleigh number solutions (with large δT) have a large number of small hot upflow plumes embedded in a broad weak background of downflow. In addition, as would be expected, these higher Rayleigh number solutions have thinner thermal boundary layers and larger convective velocities, temperatures perturbations, and heat fluxes. These differences emphasize the importance of developing even more realistic models at realistic Rayleigh numbers if one wishes to investigate by numerical simulation the type of convection that occurs in the Earth's mantle.     

Kinematic stationary geodynamo models with separated toroidal and poloidal motions

Abstract

This paper is concerned with a three-dimensional spherical model of a stationary dynamo that consists of a convective layer with a simple poloidal flow of the S^2c ₂ kind between a rotating inner body core and solid outer shell. The rotation of the inner core and the outer shell means that there are regions of concentrated shear or differential rotation at the convective layer boundaries. The induction equation for the inside of the convective layer was solved numerically by the Bullard-Gellman method, the eigenvalue of the problem being the magnetic Reynolds number of the poloidal flow (R _M2) and it was assumed that the magnetic Reynolds number of the core (R _M1) and of the shell (R _M3) were prescribed parameters. Hence R _M2 was studied as a function of R _M1 and R _M3, along with the orientation of the rotation axis, the radial dependence of the poloidal velocity and the relative thickness of the layers for the three different situations, (i) the core alone rotating, (ii) the shell alone rotating and (iii) the core and the shell rotating together. In all three cases it was found that, at definite orientations of the rotation axis, there is a good convergence of both the eigenvalues and the eigenfunctions of the problem as the number of spherical harmonics used to represent the problem increases. For R _M1 =R _M3= 10³, corresponding to the westward drift velocity and the parameters of the Earth's core, the critical values of R _M2 are found to be three orders of magnitude lower than R _M1, R _M3 so that the poloidal flow velocity sufficient for maintaining the dynamo process is 10-20 m/yr. With only the core or the shell rotating, the velocity field generally differs little from the axially symmetric case. However, for R _M2 (or R _M3) lying in the range 10² to 10⁵, the self-excitation condition is found to be of the form R _M2˙R ^½ _M1=constant (or R _M2˙R^½ _M3=constant) and the solution does not possess the properties of the Braginsky near-axisymmetric dynamo. We should expect this, in particular, in the Braginsky limit R _M2˙R^?½; _M1=constant.

An analysis of known three-dimensional dynamo models indicates the importance of the absence of mirror symmetry planes for the efficient generation of magnetic fields.     

Heat Waves in the South Moravian Region During the Period 1961-1995

Heat waves (periods of extremely hot summer weather) in the region of south Moravia are in the focus of this study. The introduced definition consists of three requirements imposed on the period that is considered a heat wave: at least three days with T _MAX 30.0°C must be observed; the mean T _MAX over the whole period is at least 30.0°C; and T _MAX must not drop below 25.0°C. To compare the severity of the individual heat waves, various characteristics (duration, number of tropical days, peak temperature, cumulative temperature excess, precipitation amount) are examined. The heat wave index HWI is defined to express the severity of heat waves in the most comprehensive way. An extraordinary heat wave occurred in July and August 1994; it lasted more than a month at several stations, while the duration of a typical heat wave is only 4 - 7 days. The extremely long unbroken period of tropical days, and even of days with T _MAX 32.0°C, represents the most distinct feature of the severe 1994 heat wave. With regard to heat wave characteristics, the summer temperature exceptionality of the early 1990s is indubitable.     

Seasonal climate patterns and their influence on calibration of the Hargreaves-Samani equation

Abstract

Annual patterns in climate parameters were studied to evaluate how these influence the quality of reference evapotranspiration (ETo) estimates obtained from the Hargreaves-Samani (HS) equation, since the method only uses the measured temperature directly. The work evaluates how these patterns can be used to improve the HS ETo estimates. Ten-year moving averages from a set of California Irrigation Management Information System (CIMIS) stations were used to evaluate the relationships between solar radiation (R_s), temperature (T) and ETo. The results indicate that T treads behind solar radiation and its value peaks some 25 days later. Thus, the main irrigation season in the Mediterranean climate (1 May–30 September) can be divided into three phases: increasing R_s and T; decreasing R_s with increasing T; and decreasing R_s and T. Non-univocal annual cycles were observed between R_s and T, ETo and R_s, and ETo and T. These annual patterns result in important seasonal changes in the ratio between the HS and Penman-Monteith (FAO PM) ETo estimates. The changes are particularly important during the irrigation season, where the FAO PM initially calculates greater ETo values than the HS methodology, and from the end of May to early September, where the HS equation overestimates the ETo values (by 17 mm, or 3%). These patterns obtained from 2000–2009 data were used to calibrate and improve HS ETo estimates at new sites for the 2010–2011 period. Calibration based on the proposed seasonal region-wide FAO PM/HS ETo ratios improved both the bias (decreased from 0.40 to 0.36 mm d^-1) and r² (increased from 0.67 to 0.87) of the ETo estimates for the irrigation season. The proposed methodology can be easily applied to other regions, even when the existing weather stations are sparse.

Editor Z.W. Kundzewicz     

Rock temperatures as an indicator of weathering processes affecting rock art

To aid rock art conservation, rock temperatures have been monitored at different depths and at low (30 min) and high (1 min) acquisition rates in a painted rock shelter in the uKhahlamba‐Drakensberg Park (South Africa). Preliminary data for winter (cold and dry) show that in that season cryoclasty is unlikely to occur (rare subzero thermal events and probable reduced moisture availability) and thermal shocks are improbable (highest measured ΔT/Δt < 2 °C min^?1). High amplitude (about 30 °C) rock temperature cycles accompanied by reversals of the thermal gradient have been observed to occur almost daily and hint at the possibility of thermal stress fatigue. Copyright © 2006 John Wiley & Sons, Ltd.     

Magnetic field transforms with low sensitivity to the direction of source magnetization and high centricity

Magnetic data interpretation faces difficulties due to the various shapes of magnetic anomalies and the positions of their extrema with respect to the causative bodies for different directions of the source magnetization. The well‐known transforms — reduction to the pole, pseudogravity field, and analytic signal (total gradient) — help in reducing the problem. Another way to achieve the required effect is the transformation of magnetic data, ΔT or Z, into values of the anomalous magnetic intensity T. In this respect, we have found some transforms based on differential operators such as the gradient of T and its modulus R = |?T|, the Laplacian L = ?²T, the product T ?²T and its square root Q, and the Laplacian ?²(T²) and its square root E, to be useful. They are slightly sensitive to the magnetization orientation and their extrema occur above the sources. For a 2D anomaly of a homogeneous causative body, the proposed transforms do not depend on the inclination of magnetization. In the 3D case, such independence does not exist even for the elementary field of a point dipole. The influence of the magnetization direction is estimated by an integral coefficient of sensitivity. This coefficient takes values of up to 2.0 for ΔT or Z anomalies, while their transforms T, R, E, Q and L have values of less than 0.28, 0.29, 0.24, 0.16 and 0.07, respectively, i.e. on average, 10 times less. The estimation of the centricity is carried out using the relative deviation of the principal extremum of the anomaly or its transforms from the epicentre of the model body at a depth equal to 100 units. For a ΔT anomaly this deviation is up to 67%; for the L transform it is less than 8%; for Q, E, R and T it is less than 10%, 15%, 20% and 25%, respectively. The proposed transforms take only non‐negative values. With respect to their shape, the peripheral magnetic extrema are removed, the anomalous configuration is simplified and the resolution of complicated interference patterns is improved. Their calculation does not require additional data for the direction of magnetization, which is an essential advantage over the reduction‐to‐the‐pole and pseudogravity‐field transforms. A joint analysis of the measured field and its transforms T, E and L offers possibilities for more confident separation of the anomalous effects and direct correlation to their sources. The model tests performed and the 3D field applications to real magnetic data confirm the useful properties of the transforms suggested here.     

It goes both ways: measurements of simultaneous evapotranspiration and fog droplet deposition at a montane cloud forest

Fluxes of latent heat, sensible heat, and water vapor, including turbulent deposition of fog droplets, were measured for two months in autumn 2005 within a subtropical montane cypress forest in Taiwan. The goal of the study was to determine whether significant evapotranspiration can occur during foggy conditions. Water vapor fluxes, Q_W, as determined with the Bowen Ratio method, were compared to those simultaneously measured with the eddy covariance method. The median Bowen Ratio was 1.06, and the median Q_W flux was 5 · 2 × 10^?5 kg m^?2 s^?1. The vertical gradients of temperature and specific humidity over the forest, ΔT and Δq, peaked around noon during days without fog, and were reduced during foggy conditions. For 66% of the data points, ΔT and Δq were negative, corresponding to positive (upward) fluxes of sensible heat Q_H and latent heat Q_E. A Monte Carlo simulation proved that statistically significant evapotranspiration rates, i.e., upward water vapor fluxes, occurred during fog. At the same time, deposition fluxes of fog droplets occurred. Our results show that even during fog events, significant evapotranspiration may occur. Copyright © 2008 John Wiley & Sons, Ltd.     

TheT e-T n difference in the lowerE region and radio-wave absorption

Summary The difference observed between the electron temperature, T _e , and the neutral atmosphere temperature, T _n , is shown to be able to contribute significantly to the observed variability of ionospheric radio-wave absorption (A1 andHF-A3 measurements).     

Estimates of the transition zone temperature in a mechanically mixed upper mantle

Partial melting of mantle peridotite generates a physically and chemically layered oceanic lithosphere that is cycled back into the mantle in subduction zones. Stirring times of the mantle are too long to allow for complete re-homogenization of subducted basalt and harzburgite, given the low chemical diffusivity of the solid mantle. This suggests that the Earth's mantle is a mechanical mixture of basaltic and harzburgitic components. Using a recently developed thermodynamic formulism we determine the phase equilibria and the seismic properties of a mantle comprised of a mechanical mixture of basalt and harzburgite (MM) and a homogeneous mantle (EA) with identical pyrolitic bulk chemistry. We use the theoretical shear velocity profiles as a new thermometer of the mantle below the magma-genetic zone by modeling the difference ΔT_410-660 between traveltimes of shear wave reflections off the 410-km and 660-km with the potential temperature T_P. ΔT_410-660 are measured from waveform stacks. They indicate that, over 1000+ km wave lengths, the temperature varies by about 200 K. Lowest and highest temperatures are resolved for the western Pacific subduction zones and the central Pacific, respectively. This variation is similar for the EA and MM and is in excellent agreement with estimates of transition zone thickness and shear velocity variations. The median value of T_P for the EA is 1720 K. It is about 1625 K for the MM, a value that is in better agreement with the Normal-MORB values of 1610 ± 40 K inferred from olivine-liquid equilibria given that our sampling region encompasses the Western Pacific subduction zones and the oldest parts of the Pacific Plate. We argue therefore that a mechanical mixed mantle, with generally higher velocities and steeper velocities gradients, represents a better physical reference model than a model based on a fully equilibrated assemblage.     

Determining the dilation factor in 4D monitoring of compacting reservoirs by rock-physics models

Hydrocarbon depletion and fluid injection cause compaction and stretching of the reservoir and overburden layers. 4D prestack seismic data can be used to detect these changes because compaction/stretching causes changes in traveltimes and seismic velocities. We show that, by using two different petro‐elastic models at varying effective pressures, a good approximation is to assume that the fractional changes in layer thickness, ΔL/L, and seismic velocity, Δv/v, are related by a linear function of ΔL/L. The slope of this function (the dilation factor, α= (Δv/v)/(ΔL/L) ) is negative and its absolute value generally decreases (shale, low porosity) or increases (sandstone, high porosity) with increasing layer thickness and decreasing effective pressure. The analysis is mainly performed for isotropic deformations. The dilation factor for uniaxial deformations is smaller in absolute value. The dilation factor, which can be calculated from time‐lapse data, can be used to predict reservoir compaction/stretching as a function of depth and surface subsidence.     

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.