= Solution
In two dimensions at nondimensional streamwise wavenumber one, incompressibility gives $i\widehat u+D\widehat v=0$, so $\widehat u=iD\widehat v$. The spanwise <vorticity> is the $z$ component of the curl:
$$
\widehat\zeta=i\widehat v-D\widehat u=-i(D^2-1)\widehat v.
$$
Consequently
$$
\boxed{\widehat\omega=(D^2-1)\widehat v=i\widehat\zeta.}
$$
The symbol $\widehat\omega$ here denotes a vorticity amplitude, not the temporal frequency used elsewhere. In dimensional variables, $\widehat\zeta^*=-i(D_*^2-(k^*)^2)\widehat v^*/k^*$, with the same sign convention.
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