The science
A soaring bird in a banked turn at angle phi flies at V = V0 / sqrt(cos phi), sinks at s = s0 / cos(phi)^1.5 and circles with radius
r = V^2 / (g tan phi) = V0^2 / (g sin phi)
The updraft is modelled as w(r) = w0 (1 - (r/R)^2) inside the thermal and zero outside. The net climb is the updraft averaged over the flight circle minus the sink; a tight turn stays in the strong core but pays more sink, which gives an optimum bank angle.
Honest simplifications: the numbers are illustrative glider-theory values for a large soaring seabird, not measurements of a particular frigatebird; the compensating downdraft, wind drift of the thermal, turbulence and the chasing or stealing behaviour of the 3D scene are not modelled here. Checks: integrated altitude over one lap, the turn equation, the optimum found two ways and, when the circle fits in the core, the analytic mean updraft.