Adjoint‐Based Marker‐In‐Cell Data Assimilation for Constraining Thermal and Flow Processes From Lagrangian Particle Records

Abstract Geophysical problems often involve Lagrangian particles that follow surrounding flows and record information about the system, such as the pressure and temperature path recorded in metamorphic rocks. These Lagrangian particles can be useful for constraining unknown parameters, such as their sources and the thermal and flow processes of the surrounding fluid. To use information about Lagrangian particles to constrain unknown parameters about the surrounding fluid in an inverse manner, we have developed a 4D‐Var (four‐dimensional variational) data assimilation for thermal convection in a particle‐grid coupled system. Here we consider particles advected in a thermally convecting, highly viscous fluid that mimics geochemical tracers in the Earth's mantle, and estimate time series of thermal and velocity fields only from the particle records, focusing on their high traceability in the laminar flow. We present preliminary 4D‐Var results using a sufficient amount of synthetic particle position and velocity data. The 4D‐Var run achieves a 60‐Myr time reversal of thermal convection at a horizontal wavelength of 6,000 km, without using any temperature data. For complex convection patterns, the cost function tends not to decrease well, likely indicating that the method is successful when the time reversal is much smaller than the mixing time scale, over which information about the initial particle arrangement is lost. Our framework has the potential to constrain thermal, flow, and mixing processes not only in the Earth's mantle but also in any other laminar flow containing Lagrangian particles that record useful information.