How to see this two-arm spiral structure in the vector field $\vec{V}=(x\cos{2rt}+y\sin{2rt})\begin{pmatrix} x \\ y \end{pmatrix}$?

Question

Consider the following 2D vector field on the $xy$-plane $$\vec{V}=(x\cos{2rt}+y\sin{2rt})\begin{pmatrix} x \\ y \end{pmatrix}$$ where $r=\sqrt{x^2+y^2}$. When plotting the vector's angle $\arctan(V_x,V_y)\in[-\pi,\pi]$ by color on the $xy$-plane, it always clearly shows a spiral pattern ($t=10$ in the plot below). How can I prove the appearance of the two spirals?

Answer 1

The spiral boundaries are the points where the scalar factor $x\cos 2rt + y\sin 2rt$ changes sign. That makes the direction of the final vector jump by $\pi$, and so the color changes abruptly.

$x\cos 2rt + y\sin 2rt$ is the dot product product $(x,y)\cdot(\cos 2rt,\sin 2rt)$. This is $0$ when the direction of $(x,y)$ differs by $\pi/2$ from $2rt$, so in polar coordinates the zeroes are at $$ \theta = 2rt \pm \pi/2 $$ You get a double Archimedean spiral whose windings are $\pi/2t$ apart.