Stochastic forcing of the linearized Navier–Stokes equations

Abstract

Transient amplification of a particular set of favorably configured forcing functions in the stochastically driven Navier–Stokes equations linearized about a mean shear flow is shown to produce high levels of variance concentrated in a distinct set of response functions. The dominant forcing functions are found as solutions of a Lyapunov equation and the response functions are found as the distinct solutions of a related Lyapunov equation. Neither the forcing nor the response functions can be identified with the normal modes of the linearized dynamical operator. High variance levels are sustained in these systems under stochastic forcing, largely by transfer of energy from the mean flow to the perturbation field, despite the exponential stability of all normal modes of the system. From the perspective of modal analysis the explanation for this amplification of variance can be traced to the non-normality of the linearized dynamical operator. The great amplification of perturbation variance found for Couette and Poiseuille flow implies a mechanism for producing and sustaining high levels of variance in shear flows from relatively small intrinsic or extrinsic forcing disturbances.

Structural Stability of Turbulent Jets

“…The spatial structure of the time-dependent forcing F (t) is given by the structure matrix F so that the ensemble average of the spatial covariance matrix of the stochastic forcing is Q ϭ FF † ( † denotes Hermitian transposition). Perturbation dynamics in turbulent shear flow is dominated by transient growth and the excitation and damping of this linear transient growth by processes including nonlinear wave-wave interactions can be represented by a combination of stochastic driving and eddy damping (Farrell and Ioannou 1993a,b, 1994DelSole 1996DelSole , 1999DelSole , 2001bDelSole andFarrell 1995, 1996). The turbulence theory that results produces an accurate description of the structure and spectra of midlatitude eddies as well as their more subtle velocity covariances and this allows momentum fluxes to be accurately determined Ioannou 1994, 1995;Whitaker and Sardeshmukh 1998;Zhang and Held 1999;DelSole 2001a).…”

Perturbation Structure and Spectra in Turbulent Channel Flow

“…Previously, stochastic analysis was applied to the study of bypass transition by demonstrating very rapid increase in variance (O(R 3 )) with shear based Reynolds number in stochastically excited shear flows which led to a successful prediction of observed transition Reynolds numbers (Farrell and Ioannou, 1994a). A theory based on the dynamics of nonnormal operators has also successfully accounted for maintenance of the large-scale variance in the atmosphere and reproduced the observed synoptic and planetary scale wave spectra and eddy transports (Farrell and Ioannou, 1993c;1994b;1995).…”

Stochastic analysis of subcritical amplification of magnetic energy in a turbulent dynamo

“…We note that the level of second moments maintained in stochastic non-normal dynamical systems associated with linearly stable shear flows has been discussed in [6].…”

“…Traditionally the phenomenon of generation of magnetic fields was analyzed by considering perturbations of the trivial state B = 0 and looking for exponential solutions to the deterministic PDE (1) with appropriate boundary conditions. While this standard stability analysis successfully predicts the dynamo action for the supercritical case, there are situations for which this eigenvalue analysis fails to predict the subcritical onset of instability [6][7][8][9][10]. It was pointed out in [9] that the closure scheme involving only deterministic αβ−parameterization is not completely satisfactory since unresolved fluctuations may produce random terms on the right hand side of the dynamo equation (1).…”