Variational-based data assimilation of initial value problems for ideal magnetohydrodynamic equations subject to solenoidal constraint

Abstract

Recently, the authors have proposed an adjoint-based dat assimilation strategy that optimizes initial conditions from the discrepancy of numerical solutions and observed data in computations of canonical, one-dimensional (1D), MHD problems. However, despite the success of the aforementioned study, all available information associated with the magnetic field data could not be ingested due to issues with the solenoidal property associated with the magnetic field (∇ • B = 0). Therefore, this study proposes an adjoint-based data assimilation procedure that ensures the solenoidal constraint on the magnetic field is satisfied. This new data assimilation schemes involves two novel components: (1) the initial magnetic field is expressed with a divergence-free parametrization; and (2) the ideal MHD equations are augmented with either the Powell source term or a generalized Lagrange multiplier (GLM) formulation. In the case of the former, instead of correcting the initial magnetic field directly, optimal parameters are sought so as to best match the observed data while maintaining the solenoidal property by construction. In the latter, the Powell and GLM approaches are used to remove via transport any non-zero ∇ • B errors generated within the simulation. The adjoint terms corresponding to the Powell and GLM strategies are derived and added directly to the adjoint equations of the original equation system. The proposed data assimilation schemes are assessed for 1D MHD initial value problems, and the ability to correct all components of the magnetic field is demonstrated. In addition, the extension of the data assimilation strategy to 3D simulations is discussed.