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The science
A gliding bird loses height relative to the air at the sink rate s = n V (CD0 + k CL^2) / CL, where the load factor in a banked turn is n = 1/cos(phi) and the speed is V = sqrt(2 n W / (rho S CL)). The turn radius is R = V^2 / (g tan phi). A thermal is modelled as a Gaussian updraft w(r) = w0 exp(-(r/Rt)^2), so the net climb rate on a circle of radius R is the average updraft around the circle minus the sink rate. Steeper banks give a smaller circle (stronger lift near the core) but a higher load factor and faster sink, so there is an optimum bank angle.
For a circle of radius R whose centre is offset d from the thermal core the mean updraft is w0 exp(-(R^2 + d^2)/Rt^2) I0(2 R d / Rt^2) (I0 is the modified Bessel function), which is why centering matters. The centering reflex compares the updraft at the two wing tips and banks harder toward the stronger side.
Simplifications: quasi-steady turn aerodynamics with a parabolic drag polar (CD0 = 0.035, aspect ratio 7, efficiency 0.9, wing area scaled with mass^(2/3)), air density 1.1 kg/m^3, a fixed circular Gaussian thermal in the air-mass frame (wind drift is not drawn), no turbulence, no sink outside the thermal. The 3D lab on this site is a flocking visualisation without these aerodynamics, so the numbers here are not copied from it.