Preconditioned Adjoint Data Assimilation for Two-Dimensional Decaying Isotropic Turbulence

TL;DR

The paper tackles the instability of adjoint-based data assimilation in turbulent flows caused by backward Lyapunov growth that amplifies high-wavenumber content. It introduces a scale-aware strategy by redefining the inner product with a Fourier-space kernel $\mathcal{G}$, equivalently transforming the control variable to $\boldsymbol{s}_0=\mathcal{G}^{-1/2}\boldsymbol{u}_0$, and implementing spectral preconditioners. Two kernel families are evaluated: algebraic $\hat{G}_{p}(k)=k^{-2\alpha}$ and exponential $\hat{G}_{e}(k)=\exp(-\nu\beta k^{2})$, within a $2$D decaying HIT test. Results show that the exponential preconditioner with an optimal $\beta$ (around $0.8$) significantly improves initial-condition reconstruction (up to about $35\%$ reduction in error) and provides robust smoothing of gradient noise, outperforming the algebraic variant. Ensemble analyses reveal a spectral catastrophe in standard adjoint sensitivity, which the preconditioned framework attenuates by suppressing small-scale incoherent growth and preserving large-scale, physically meaningful gradients.

Abstract

Adjoint-based data assimilation for turbulent Navier-Stokes flows is fundamentally limited by the behavior of the adjoint dynamics: in backward time, adjoint fields exhibit exponential growth and become increasingly dominated by small-scale structures, severely degrading reconstruction of the initial condition from sparse measurements. We demonstrate that the relative weighting of spectral components in the adjoint formulation can be systematically controlled by redefining the inner product under which the adjoint operator is defined. The inverse problem is formulated as a constrained minimization in which a cost functional measures the mismatch between model predictions and observations, and the adjoint equations provide the gradient with respect to the initial velocity field. Redefining the forward-adjoint duality through a Fourier-space weighting kernel preconditions the optimization and is mathematically equivalent to changing the representation of the control variable or, alternatively, introducing a smoothing operation on the governing dynamics. Specific kernel choices correspond to fractional integration or diffusion operators applied to the initial condition. Among these, exponential kernels provide effective regularization by suppressing high-wavenumber contributions while preserving large-scale coherence, leading to improved reconstruction across scales. A statistical analysis of an ensemble of adjoint fields from different turbulent realizations reveals scale-dependent backward growth rates, explaining the instability of the standard formulation and clarifying the mechanism by which the proposed preconditioning attenuates incoherent small-scale amplification.

Figures (15)

• Figure 1: A toy problem demonstration of the change of inner product and the change of control variables on the impact of optimization. Subscripts $l$ and $h$ mark low- and high-sensitivity parts in the initial state $\boldsymbol{u}_0$.
• Figure 2: Data assimilation framework with preconditioner $\mathcal{G}$.
• Figure 3: Illustration of the underlying grid resolution in a two-dimensional periodic domain of size $x=y=2\pi$. (a): a representative patch of the full $512\times512$ grid. (b): the observed grid obtained by downsampling with a grid gap of $\Delta I = 8$, corresponding to an effective $64\times64$ resolution.
• Figure 4: Snapshots of true fields and reconstructed fields from t=0 to $t=T$.
• Figure 5: Energy spectra $E(k)$ under different choice of control variables $\boldsymbol{s}_0$. Each subplot corresponds to a specific control strategy or regularization method. For each color group, the curves represent the temporal evolution of the energy spectrum, with lighter shades indicating later times ($t =0\to T$).