I wonder how difficult it is to compute geodesics on Dini's Surface, a twisted pseudosphere?

Here is one parametrization, from Alfred Gray's Modern Differential Geometry of Curves and Surfaces, p.495: \begin{eqnarray} x(u,v) &=& a \cos (u) \sin (v)\\ y(u,v) &=& a \sin (u) \sin (v)\\ z(u,v) &=& a \left[\cos (v)+\log \left(\tan \frac{v}{2}\right)\right]+b u \end{eqnarray} Dini's surface has constant curvature of $\frac{-1}{a^2+b^2}$.

And here is an image, for $a=1,\; b=\frac{1}{12}$, with $u \in [0,8\pi]$ (Added: now with Will Sawin's $v=\pi/2$ geodesic shown green):
       
What I especially wonder is if there is a geodesic that spirals down through every turn, which would be kinda cool. :-)
(Added: See Will Sawin's answer.)