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This paper introduces a new operator relevant to input-output analysis offlows in a statistically steady regime far from the steady base flow: the meanresolvent $\mathbf{R}_0$. It is defined as the operator predicting, in thefrequency domain, the mean linear response to forcing of the time-varying baseflow. As such, it provides the statistically optimal linear time-invariantapproximation of the input-output dynamics, which may be useful, for instance,in flow control applications. Theory is developed for the periodic case. Thepoles of the operator are shown to correspond to the Floquet exponents of thesystem, including purely imaginary poles at multiples of the fundamentalfrequency. In general, evaluating mean transfer functions from data requiresaveraging the response to many realizations of the same input. However, in thespecific case of harmonic forcings, we show that the mean transfer functionsmay be identified without averaging: an observation referred to as `dynamiclinearity' in the literature (Dahan et al., 2012). For incompressible flows inthe weakly unsteady limit, i.e. when amplification of perturbations by theunsteady part of the periodic Jacobian is small compared to amplification bythe mean Jacobian, the mean resolvent $\mathbf{R}_0$ is well-approximated bythe well-known resolvent operator about the mean-flow. Although the theorypresented in this paper only extends to quasiperiodic flows, the definition of$\mathbf{R}_0$ remains meaningful for flows with continuous or mixed spectra,including turbulent flows. Numerical evidence supports the close connectionbetween the two resolvent operators in quasiperiodic, chaotic and stochastictwo-dimensional incompressible flows.

Mean resolvent operator of statistically steady flows