Asymptotic Analysis for Magnetostrophic Turbulence

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The Earth's magnetic field is vitally important to the existence of life on our planet. It serves to protect the Earth from the charged particles of the solar wind that would otherwise strip away the upper atmosphere. Changes in the magnetic field have significant implications for climatic changes. The source of the Earth's magnetic field is deep inside the Earth where a small solid iron center is surrounded by a huge spherical mass of electrically conducting liquid iron. Convection in this liquid is at the origin of Earth's magnetic field through a dynamo effect: this is the process by which the rotating, convecting, molten iron maintains the geomagnetic field against ohmic decay. The mathematical description of this process is a highly complex system of three-dimensional, nonlinear partial differential equations (PDE) that govern convective magnetohydrodynamics. Computer models have been used to simulate the actual geodynamo; however, no three-dimensional model has yet been run at the spatial resolution required to encompass the broad spectrum of turbulence that surely exists in the Earth's fluid core. The investigator and collaborators study a PDE model for the core where a stochastic force driving the geodynamo is to be interpreted as a source that is continuously regenerating a statistically steady buoyancy distribution throughout the fluid. For scales relevant to the Earth's core, this PDE system has many small parameters. The team exploits this feature by analyzing the asymptotics of the stochastically forced PDE in the limit of vanishingly small parameters. They aim to establish that the PDE system sustains ergodic statistically steady states, thus providing a rigorous foundation for a turbulent geodynamo. A graduate student is involved in the research. The partial differential equations that govern fluid dynamics are notoriously challenging. One aspect of the challenge is the singular behavior of the equations as certain parameters, which are produced by the physics of the problem, vanish. Studying such problems is important both to illuminate the physical processes described by the singular limits and to advance the creation of new mathematical techniques. The investigator and collaborators contribute to this endeavor by studying mathematical models for magnetostrophic turbulence in the Earth's fluid core. The analysis requires a detailed understanding of the delicate interaction between the nonlinearity and the stochastic forcing. The team plans to apply recent developments in a theory of hypoellipticity for stochastic PDE with spatially degenerate forcing to prove the existence of unique ergodic invariant measures for the PDE that model magnetostrophic turbulence.