 Solution

For a fieldonly the gradient of the scalar prefactor contributes to its vorticity. At the swimmer,The prescribed rotation law has ${\mathbf{\Omega }}_f={\mathbf{\omega }}^{\ast }/2$. Since $\mathbf{b}\times \mathbf{p}=-\partial \mathbf{p}/\partial \theta$, comparison with $\dot{\mathbf{p}}=(\partial \mathbf{p}/\partial \theta )\dot{\theta }$ givesA pusher has $\mathcal{P}>0$ and rotates toward the stable parallel orientation $\theta =0$; there its image flow attracts it to the free surface. A puller has $\mathcal{P}<0$ and rotates toward a normal orientation $|\theta |=\pi /2$. Depending on which way its polar swimming direction points, it then swims away from the surface or meets it head-on; the dipolar image drift at the exactly normal orientation is toward the surface for this puller sign convention.