On page 12 of Malham's fluid dynamics notes the following flow field is considered: $\boldsymbol u= (u,v) = (kx, -ky)$. It's easy to see in these Cartesian coordinates that this is solenoidal: $\nabla\cdot \boldsymbol u = k-k=0$, and he derives that the stream function is $\psi=kxy$.
Now he moves to polar coordinates of the same flow and denotes $\boldsymbol u=(u_r,u_\theta)=(kr\cos 2\theta, -kr\sin 2\theta)$. I don't see why this parametrisation is acceptable - why the factor of 2? I can kind of derive this if I parametrise the flow as $(u_r,u_\theta)=(kr\cos \alpha\theta, -kr\sin \alpha\theta)$ and then enforce $\nabla\cdot \boldsymbol u=0$ for the polar divergence operator, which yields that $\alpha=2$, but I still don't understand why this is the same flow as the original. Why am I allowed to deviate from $(u_r,u_\theta)=(kr\cos \theta, -kr\sin \theta)$?