Two-dimensional viscous shearing wave ($\dot{Z}=-\nu k^{2}Z$)

For $k_z=v_z=0$ and $k_y\ne 0$, define $Z=i(k_x{v}_y-k_y{v}_x)$. Uniform rotation does not enter its evolution: $\dot{Z}=-\nu k^{2}Z$, $v_x=i{k}_{y}Z/k^2$ and $v_y=-i{k}_{x}Z/k^2$. With $T=k_x/k_y$ and $\mathrm{Re}=A/(\nu k_y^2)$, the perturbation energy ratio is $E(T)/E(T_0)=[(1+T_0^2)/(1+T^2)]\exp [-(T+T^3/3-T_0-T_0^3/3)/\mathrm{Re}]$. Leading waves can gain energy as shear reduces their wavevector magnitude, before viscosity and subsequent winding damp them.

// Target: hydrodynamic-stability.bigb

For the shearing-wave energy profile, stationary points satisfy $-2\mathrm{Re}T=(1+T^2{)}^2$. At large Reynolds number, the initial minimum is at $T_{-}\sim -(2\mathrm{Re}{)}^{1/3}$ and the subsequent maximum is at $T_{+}\sim -1/(2\mathrm{Re})$. Their energy ratio is $G_{\max }\sim (2\mathrm{Re}{)}^{2/3}{e}^{-2/3}$. The Reynolds number here uses $A$, half the background shear rate; using $2A$ instead changes the coefficient convention.

// Target: astrophysics.bigb