Variance maintained by stochastic forcing of non-normal dynamical systems associated with linearly stable shear flows

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Abstract
The level of variance maintained in a stochastically forced asymptotically stable linear dynamical system with a non-normal dynamical operator cannot be fully characterized by the decay rate of its normal modes, in contrast to the case of normal dynamical systems. The nonorthogonality of modes in a non-normal system may lead to transient growth which supports variance far in excess of that anticipated from the decay rate given by the eigenvalues of the operator. As an example, the variance maintained by stochastic forcing in a canonical shear flow is examined in this work and found to increase with a power of the Reynolds number between 1.5 and 3. This great amplification of variance suggests a fundamentally linear mechanism underlying shear flow turbulence.

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