Development of Variational Data Assimilation Methods for the MoSST Geodynamo Code

A range of different approaches to data assimilation for Earth's geodynamo are now being pursued, from sequential schemes based on approximate covariances of various degrees of sophistication, to variational methods for models of varying degrees of physical completeness. While variational methods require development of adjoint (and possible tangent linear) variants on the forward code---a challenging programming task for a fully self-consistent modern dynamo code---this approach may ultimately offer significant advantages. For example, adjoint based variational approaches allow initial, boundary, and forcing terms to be explicitly adjusted to combine data from modern and historical eras into dynamically consistent maps of core state, including flow, buoyancy and magnetic fields. Here we describe development of tangent linear and adjoint codes for the Modular Scalable Self-consistent Three-dimensional (MoSST) geodynamo simulator, and present initial results from simple synthetic data assimilation experiments. Our approach has been to develop the exact linearization and adjoint of the actual discrete functions represented by the computer code. To do this we use a 'divide-and-concur' approach: the code is decomposed as the sequential action of a series of linear and non-linear procedures on specified inputs. Non-linear procedures are first linearized about a pre-computed input background state (derived by running the non-linear forward model), and a tangent linear time-step code is developed. For small perturbations of initial state the linearization appears to remain valid for times comparable to the secular variation time-scale. Adjoints for each linear (or linearized) procedure were then developed and tested separately (for symmetry), and then merged into adjoint procedures of increasing complexity. We have completed development of the adjoint for a serial version of the MoSST code, explore time limits of forward operator linearization, and discuss next steps towards use of the code for variational assimilation, and will use this code in some initial data assimilation tests, adjusting initial conditions to match data (spherical harmonic model coefficients) for a later time.