How to create a contour plot on the surface of a sphere in Python?

Question

I have plotted a sphere in Python to be used as a graphical representation of some stochastic variables u_1, u_2 and u_3. The plot can be found here.

This is achieved using the following Python code:

import numpy as np
import matplotlib.pyplot as plt

fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.set_xlabel(r'$u_1$'), ax.set_xlim([-4, 4]), ax.set_xticks([-4, 0, 4])
ax.set_ylabel(r'$u_2$'), ax.set_ylim([-4, 4]), ax.set_yticks([-4, 0, 4])
ax.set_zlabel(r'$u_3$'), ax.set_zlim([-4, 4]), ax.set_zticks([-4, 0, 4])

# Plot sphere using spherical coordinates
beta = 4.498
theta, phi = np.mgrid[0:np.pi:50j, 0:2*np.pi:50j]
xcoord = beta * np.sin(theta) * np.cos(phi)
ycoord = beta * np.sin(theta) * np.sin(phi)
zcoord = beta * np.cos(theta)
ax.plot_surface(xcoord, ycoord, zcoord, color='yellow')

I now want to plot the following function over the surface of the sphere:

v_3 = np.sqrt(-2 * m0(u_1, u_2, u_3) * np.log(2 * np.pi / T_tilde * np.sqrt(m0(u_1, u_2, u_3) / m2(u_1, u_2, u_3)) * (1 - phi(u_3)))

Here m0(u_1, u_2, u_3), m2(u_1, u_2, u_3) and T_tilde denote variables calculated earlier in the code and phi denotes the standard normal operator.

The result should be something like this.

Any suggestions on how this can be achieved?