A type of eddy‐damped quasinormal Markovian (EDQNM) closure is shown to be potentially nonrealizable in the presence of linear wave phenomena. This statistical closure results from the application of a fluctuation–dissipation (FD) ansatz to the direct‐interaction approximation (DIA); unlike in phenomenological formulations of the EDQNM, both the frequency and the damping rate are renormalized. A violation of realizability can have serious physical consequences, including the prediction of negative or even divergent energies. A new statistical approximation, the realizable Markovian closure (RMC), is proposed as a remedy. An underlying Langevin equation that makes no assumption of white‐noise statistics is exhibited. Even in the wave‐free case the RMC, which is based on a nonstationary version of the FD ansatz, provides a better representation of the true dynamics than does the EDQNM closure. The closure solutions are compared numerically against the exact ensemble dynamics of three interacting waves.
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As in Ref. 35, 5000 realizations were used to reduce the statistical error of the ensemble-averaged results to about 1.4%. The NAG routine D02BBf, based on a Runga-Kutta-Merson method, was employed to perform the numerical integrations of the fundamental equation. A tolerance of and double-precision arithmetic were used to ensure overall numerical accuracy of the ensemble solution. The numerical implementation of the closure solutions is discussed in Part II.
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It can readily be demonstrated that to within a factor symmetric in k, p, and q, Eq. (112) satisfies the steady-state DIA covariance equation. From our numerical results, it appears that the factor is actually unity, but this has not yet been established analytically. (One needs to solve the integral equation for the response function in terms of and ).
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We are not suggesting that these simple expressions for and are consistent with the other EDQNM equations. Our purpose here is to illustrate that the EDQNM form for is not implicitly guaranteed to be positive-semidefinite, in contrast to the inherently positive-semidefinite form of Eq. (7b). A self-consistent that violates positive-semidefiniteness could be determined by inverting the equal-time version of Eq. (18) to obtain in terms of the nonrealizable found in the example of Sec. III B 3.
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