  The linear amplification of disturbances is critical in setting up transition
scenarios in viscoelastic channel and Couette flow, and may also play an
important role when such flows are fully turbulent. As such, it is of interest
to assess how this amplification, defined as the steady-state variance
maintained under Gaussian white noise forcing, scales with the main
nondimensional parameters: the Reynolds ($Re$) and Weissenberg ($Wi$) numbers.
This scaling is derived analytically in the two limits of strong and weak
elasticity for when the forcing is streamwise-constant. The latter is the
relevant forcing for capturing the overall behaviour because it was previously
shown to have the dominant contribution to amplification. The final expressions
show that for weak elasticity the scaling retains a form similar to the
well-known O($Re^3$) relationship with an added elastic correction. For strong
elasticity, however, the scaling is O($Wi^3$) with a viscous correction. The
key factor leading to such a mirroring in the scaling is the introduction of
forcing in the polymer stress. The results demonstrate that energy
amplification in a viscoelastic flow can be very sensitive to the model
parameters even at low $Re$. They also suggest that energy amplification can be
significantly increased by forcing the polymer stress, thereby opening up
possibilities such as flow control using systematically designed polymer stress
perturbations.
