Variational data assimilation for the initial-value dynamo problem

Abstract

The secular variation of the geomagnetic field as observed at the Earth's surface results from the complex magnetohydrodynamics taking place in the fluid core of the Earth. One way to analyze this system is to use the data in concert with an underlying dynamical model of the system through the technique of variational data assimilation, in much the same way as is employed in meteorology and oceanography. The aim is to discover an optimal initial condition that leads to a trajectory of the system in agreement with observations. Taking the Earth's core to be an electrically conducting fluid sphere in which convection takes place, we develop the continuous adjoint forms of the magnetohydrodynamic equations that govern the dynamical system together with the corresponding numerical algorithms appropriate for a fully spectral method. These adjoint equations enable a computationally fast iterative improvement of the initial condition that determines the system evolution. The initial condition depends on the three dimensional form of quantities such as the magnetic field in the entire sphere. For the magnetic field, conservation of the divergence-free condition for the adjoint magnetic field requires the introduction of an adjoint pressure term satisfying a zero boundary condition. We thus find that solving the forward and adjoint dynamo system requires different numerical algorithms. In this paper, an efficient algorithm for numerically solving this problem is developed and tested for two illustrative problems in a whole sphere: one is a kinematic problem with prescribed velocity field, and the second is associated with the Hall-effect dynamo, exhibiting considerable nonlinearity. The algorithm exhibits reliable numerical accuracy and stability. Using both the analytical and the numerical techniques of this paper, the adjoint dynamo system can be solved directly with the same order of computational complexity as that required to solve the forward problem. These numerical techniques form a foundation for ultimate application to observations of the geomagnetic field over the time scale of centuries.

Variational data assimilation for a forced, inertia-free magnetohydrodynamic dynamo model

“…Using the technique developed in Li et al (2011), we define a positive definite quantity termed the misfit, χ 2 , which measures the deviation between the predictions and the observations…”

“…[∇ 2 ] † is the adjoint operator of ∇ 2 ; we construct the discrete version as the transpose of the appropriate matrix as we did in Li et al (2011). …”

“…The particular approach we adopt is termed 'variational data assimilation' and builds upon our previous work in this area (Li et al 2011).…”

“…until convergence is achieved. A computational scheme is designed around a spectral discretization of the equations, based on spherical harmonics and an expedient radial representation for the full sphere geometry as described in Li et al (2011); this is the topic of Sections 3 and 4 of the paper. In Sections 5 and 6, we describe closed-loop tests, in which the ability to retrieve known I.C.s is quantified under different observational scenarios.…”

Dynamic responses of the Earth’s outer core to assimilation of observed geomagnetic secular variation

“…In addition to the sequential data assimilation systems mentioned above, there are also efforts in developing GDAS systems based on variational data assimilation techniques. For example, Li et al (2011Li et al ( , 2014 have been continuing their effort on a new combined forward and adjoint system towards a full geodynamo model. Encompassed application is the contributions of assimilation results to international geomagnetic reference field (IGRF) ) and efforts to determine field model error statistics (Gillet et al 2013).…”