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For $k_z=v_z=0$, define the complex vertical vorticity amplitudeTake the curl of the amplitude equations, including ${\dot{k}}_x=2A{k}_y$. Incompressibility cancels the pressure and rotation terms, givingEquivalently, the uniform basic absolute vorticity has no gradient for the perturbation to advect. The general two-dimensional viscous shearing wave solution iswhere $Z_0$ is an arbitrary complex constant and $k_x(t_0)$ is an arbitrary real initial radial wavenumber. The perturbation pressure follows from part (ii), or explicitly $\Pi =(4A{k}_y^2-2\Omega k^2)Z/k^4$.

For $A\ne 0$, shift the time origin to the swing $k_x=0$ and put $T=k_x/k_y=2At$, $T_0=2A{t}_0$ and $\mathrm{Re}=A/(\nu k_y^2)$. ThenThe spatially averaged perturbation kinetic-energy mass density is $E=\rho ⟨|{\mathbf{u}}^{\prime }{|}^2⟩/2=\rho |\mathbf{v}{|}^2/4$, and $|\mathbf{v}{|}^2=|Z{|}^2/k^2$. ConsequentlyThe proportionality constant depends on the initial data; the exponential profile is not a claim of infinite physical growth from the remote past. For $A>0$, leading waves with $T<0$ can gain velocity amplitude as their wavevector magnitude decreases, before viscosity and later winding damp them. This is the Orr mechanism, and it is independent of uniform rotation for these two-dimensional disturbances.

If $A=0$, the wavevector is constant and $\mathbf{v}(t)=\mathbf{v}(t_0)e^{-\nu k^2(t-t_0)}$, so there is no shear amplification. The formulas with the physical-time integral remain valid, although $T=2At$ is no longer a useful coordinate. The high-positive-Reynolds-number case in the next part takes $A>0$.