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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. 相似文献
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We consider the effect of including both dipole and quadrupole parities in the previous mean-field model of Hollerbach and Jones (1995), which considered dipole parity only. Allowing for both parities, we find that the onset of dynamo action occurs at 0 6, in the form of a purely quadrupolar dynamo wave. A symmetry-breaking bifurcation then occurs at 0 11, beyond which the solutions are of mixed parity. The quadrupolar component still oscillates about a zero time-average, but the dipolar component about a non-zero average. For even greater 0
we obtain an unconnected upper-branch solution. In sharp contrast to the HJ95 pure-parity upper branch, however, this mixed-parity upper branch is steady-state rather than periodic. Although it does not appear to be possible to connect these two upper branches by any simple sequence of bifurcations, we nevertheless suggest how aspects of the mixed-parity branch may help in understanding features of the previous pure-parity branch. 相似文献
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Gary A. Glatzmaier 《地球物理与天体物理流体动力学》2013,107(2):223-264
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. 相似文献
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The validity of the anelastic approximation has recently been questioned in the regime of rapidly-rotating compressible convection in low Prandtl number fluids (Calkins, Julien and Marti, Proc. R. Soc. A, 2015, vol. 471, 20140689). Given the broad usage and the high computational efficiency of sound-proof approaches in this astrophysically relevant regime, this paper clarifies the conditions for a safe application. The potential of the alternative pseudo-incompressible approximation is investigated, which in contrast to the anelastic approximation is shown to never break down for predicting the point of marginal stability. Its accuracy, however, decreases close to the parameters corresponding to the failure of the anelastic approach, which is shown to occur when the sound-crossing time of the domain exceeds a rotation time scale, i.e. for rotational Mach numbers greater than one. Concerning the supercritical case, which is naturally characterised by smaller rotational Mach numbers, we find that the anelastic approximation does not show unphysical behaviour. Growth rates computed with the linearised anelastic equations converge toward the corresponding fully compressible values as the Rayleigh number increases. Likewise, our fully nonlinear turbulent simulations, produced with our fully compressible and anelastic models and carried out in a highly supercritical, rotating, compressible, low Prandtl number regime show good agreement. However, this nonlinear test example is for only a moderately low convective Rossby number of 0.14. 相似文献
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C.A. Jones P. Boronski A.S. Brun G.A. Glatzmaier T. Gastine M.S. Miesch J. Wicht 《Icarus》2011,216(1):120-135
Benchmark solutions for fully nonlinear anelastic compressible convection and dynamo action in a rotating spherical shell are proposed. Three benchmarks are specified. The first is a purely hydrodynamic case, which is steady in a uniformly drifting frame. The second is a self-excited saturated dynamo solution, also steady in a drifting frame. The third is again a self-excited dynamo but is unsteady in time, and it has a higher Rayleigh number than the steady dynamo benchmark. Four independent codes have been tested against these benchmarks, and very satisfactory agreement has been found. This provides an accurate reference standard against which new anelastic codes can be tested. 相似文献
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Over the past 10 years, geodynamo simulations have grown rapidly in sophistication. However, it is still necessary to make certain approximations in order to maintain numerical stability. In addition, models are forced to make assumptions about poorly known parameters for the Earth's core. Different magnetic Prandtl numbers have been used and different assumptions about the presence of radiogenic heating have been made. This study examines some of the consequences of different approximations and assumptions using the Glatzmaier–Roberts geodynamo model. Here, we show that the choice of magnetic Prandtl number has a greater influence on the character of the magnetic field produced than the addition of a plausible amount of radiogenic heating. In particular, we find that prescribing a magnetic Prandtl number of unity with Ekman number limited by current computing resources, results in magnetic fields with significantly smaller intensities and variabilities compared with the much more Earth-like results obtained from simulations with large magnetic Prandtl numbers. A magnetic Prandtl number of unity, with both the viscous and magnetic diffusivities set to the Earth's magnetic diffusivity, requires a rotation rate much smaller than that of the Earth for currently reachable Ekman numbers. This results in a reduced dominance of the Coriolis forces relative to the buoyancy forces, and therefore, a reduction in the magnetic field intensity and the variability compared to the large Prandtl number cases. 相似文献
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Abstract The geodynamo simulation of Glatzmaier and Roberts (1996, Physica D97, 81) is driven by the cooling of the model Earth, which releases latent heat and light components of core fluid at the freezing surface of the inner core as it advances outwards. At some time in the past, the inner core was only a quarter of its present size and at some time in the future it will be twice its present size. The geodynamo operating during those epochs are studied here, the three models (past, present and future) being tied together in an evolutionary sense. The time taken for the models to evolve from past to future depends on the cooling rate, which is controlled by the dynamics of the mantle and is not studied here. All three models generate external fields of comparable strength and all three appear to be close to Taylor states. Unexpectedly, the future model showed considerable variability in time, while the past model does not. Deviations from axisymmetry in the external field increase with inner core radius and the relative predominance of the centered dipole over other multipole components declines. 相似文献