Starting from a steady-state energy–balance principle for noisy, linearly damped fields, we derive an operator covariance law in Lyapunov–Sylvester form. On the unit disk with self-adjoint boundary conditions, this formulation selects the Zernike polynomials as eigenfunctions. Projecting onto finitely many modes yields a matrix Lyapunov system for the modal covariance. Building on established turbulence models, we rigorously incorporate and leverage physically calibrated forcing in the Zernike basis. We also introduce and prove explicit truncation–error bounds (operator norm for diagonally dominant noise; Hilbert–Schmidt norm in general) with dependence on system parameters and truncation order, specifically tailored for Zernike systems. These bounds enable a priori determination of truncation order for a target accuracy. This provides a pipeline– balance → geometry → finite solve -for steady-state covariance computation in spatially extended dissipative fields. The framework is extended to exponentially correlated (Ornstein–Uhlenbeck) forcing via an augmented-state Sylvester formulation, revealing a simple additive shift in covariance denominators.