A finite-volume solution method for thermal convection and dynamo problems in spherical shells
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Abstract:
We present a novel application of a finite-volume technique to the numerical simulation of thermal convection within a rapidly rotating spherical shell. The performance of the method is extensively tested against a known standard solution at moderate Ekman number. Models at lower Ekman number demonstrate the potential of the method in a parameter range more appropriate to the flow in the molten metallic core of planetary interiors. In addition we present results for the magnetohydrodynamic dynamo problem. In order to avoid the need to solve for the magnetic field in the exterior, we use an approximate magnetic boundary condition. Compared with the geophysically relevant case of insulating boundaries, it is shown that the qualitative structures of the flow and the magnetic field are similar. However, a more quantitative comparison indicates that mean flow velocity and mean magnetic field strength are affected by the boundary conditions by about 20 per cent.Keywords:
Ekman number
Spherical shell
For more than 200 years the origin of Earth's magnetic field was attributed to permanent magnetization. Even today no single argument (e.g., that Earth's deep interior is too hot to sustain permanent magnetization) conclusively rules out the permanent magnetization hypothesis. Nevertheless, when all the evidence is considered, this hypothesis can be safely discarded and replaced with an electric current (dynamo) hypothesis. Surprisingly, this can be done even though there is no adequate dynamo model for Earth. The development of geodynamo models began with the disk dynamo of Larmor in 1919 and expanded to include many classes of models, such as αω, α 2 , α 2 ω, Taylor state, and Model Z dynamos. Because of mathematical difficulties associated with solving the many coupled partial differential equations of dynamo theory, numerous simplifying assumptions are made. The majority of numerical dynamo models assume a three‐dimensional velocity field in an inviscid fluid and use mean field theory to solve for axisymmetric magnetic fields. There is also an increasing number of intermediate and strong field models emerging, in which feedback from the magnetic field to the velocity field is permitted. Nevertheless, these models still require several simplifying assumptions and there are many additional problems. For instance, many core parameters are difficult to estimate; there is debate on whether the top of the core is stably stratified and on the effects such stratification might have; what effects the presence of an inner core have; and whether the coupling across the coremantle boundary significantly affects the geodynamo. Perhaps it is not surprising that dynamo theoreticians, faced with large difficulties in mathematics and many uncertainties in physics, essentially choose to ignore input from fields such as paleomagnetism. However, it is precisely because of such difficulties that paleomagnetism can provide valuable constraints to narrow the range of viable dynamo models. For example, paleomagnetism ultimately should provide constraints on the velocity and magnetic field symmetries of dynamos; determine whether the geodynamo is in the weak, intermediate, or strong field regime; determine if there is a fundamental difference in dynamo processes during superchrons when reversals of the magnetic field essentially cease; and provide valuable information on the growth of the inner core and its possible stabilizing effects on geodynamo processes.
Solar dynamo
Outer core
Inviscid flow
Stratification (seeds)
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Abstract The magnetic field of a rapidly rotating α2 dynamo is computed in a spherical shell as a function of Reynolds and Ekman numbers. It is found that at high enough Reynolds number a Taylor state is reached, after which the solution becomes inviscid. This result confirms the recent work of Hollerbach and Ierley (1991) in a full sphere.
Spherical shell
Inviscid flow
Ekman number
Magnetic Reynolds number
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Solar dynamo
Magnetic Reynolds number
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Mode (computer interface)
Solar dynamo
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