Preconditioned Adjoint Data Assimilation for Two-Dimensional Decaying and Forced Turbulence

Abstract

Adjoint-based data assimilation for turbulent Navier-Stokes flows is limited by backward adjoint growth and increasing dominance of small-scale structures, which degrade reconstruction of initial conditions from sparse measurements. We show that the relative weighting of spectral components can be systematically controlled by redefining the inner product under which the adjoint operator is defined. The resulting Fourier-space weighting kernel acts as a preconditioner for the optimization. Specific kernels correspond to fractional integration or diffusion operators on the initial condition. Numerical experiments show that flow-dependent kernel selection substantially improves reconstruction stability and accuracy: exponential kernels suppress high-wavenumber contributions, whereas a fractional integral kernel is particularly effective for forced Kolmogorov flow. Ensemble statistics of adjoint fields reveal scale-dependent backward growth rates, explaining the instability of the standard formulation and how spectral preconditioning attenuates incoherent small-scale amplification.