I'm interested in the properties of direction vector fields that are defined by a scalar phase field $\theta(\vec{r})$ as follows:
$$ \vec{v}(\vec{r}) = \{ \cos \theta(\vec{r}), \sin \theta(\vec{r}) \} $$
That is, the scalar field defines the angle of the vector at each point in space. Writing this as a differential equation
$$ \frac{d\vec{r}(t)}{dt} = \vec{v}(\vec{r}(t)) $$
One notices that there are a number of features of these vector fields that are interesting. As an example, consider the following phase field:
and its corresponding streamlines:
There are curves that locally appear to be convergent contours for all streamlines in their vicinity, and they alternate in direction. These curves spiral out from local extrema of the phase function and are roughly parallel when the phase function is nearly linear.
I'm wondering what is known about these curves, what they're called, and any existing literature about this topic.