 Solution

In two dimensions at nondimensional streamwise wavenumber one, incompressibility gives $i\hat{u}+D\hat{v}=0$, so $\hat{u}=iD\hat{v}$. The spanwise vorticity is the $z$ component of the curl:ConsequentlyThe symbol $\hat{\omega }$ here denotes a vorticity amplitude, not the temporal frequency used elsewhere. In dimensional variables, ${\hat{\zeta }}^{\ast }=-i(D_{\ast }^2-(k^{\ast }{)}^2){\hat{v}}^{\ast }/k^{\ast }$, with the same sign convention.