The level of variance maintained in a stochastically forced asymptotic
ally stable linear dynamical system with a non-normal dynamical operat
or cannot be fully characterized by the decay rate of its normal modes
, unlike normal dynamical systems. The nonorthogonality of modes may l
ead to transient growth which supports variance far in excess of that
anticipated from the decay rate given by the eigenvalues of the operat
or. As an example, the variance maintained by stochastic forcing in a
canonical shear Flow is found to increase with a power of the Reynolds
number between 1.5 and 3. This great amplification of variance sugges
ts a fundamentally linear mechanism underlying shear how turbulence.