[Python] Generating a discrete polar surface map in cartesian coordinates

I would like to generate a surface plot with discrete arc-shaped cells on a 2D cartesian plane. I am able to get decent results by plotting a 3D surface plot (using plot_surface) and viewing it from the top, but matplotlib can be a bit finicky with 3D, so I'd prefer to do it in 2D. I can also get similar results using pcolormesh on a polar plot, but again, I want a 2D cartesian plane. How can I do this in matplotlib?

MRE:

import numpy as np
import matplotlib.pyplot as plt

r = np.linspace(2, 5, 25)
theta = np.linspace(0, np.pi, 25)
R, Theta = np.meshgrid(r, theta)

X = R*np.cos(Theta)
Y = r*np.sin(Theta)
U = R*np.cos(Theta)*np.exp(R*Theta/500)

fig, ax = plt.subplots(figsize=(8,6), subplot_kw={"projection":"3d"})
surf = ax.plot_surface(X, Y, U, cmap="viridis", rstride=1, cstride=1)
ax.view_init(elev=90, azim=-90)
ax.set_proj_type("ortho")
ax.zaxis.line.set_lw(0.)
ax.set_zticks([])
ax.set_aspect("equalxy")
fig.colorbar(surf, shrink=0.5, aspect=5)
fig.tight_layout()

fig, ax = plt.subplots(figsize=(8,6), subplot_kw={"projection":"polar"})
ax.pcolor(Theta, R, U, shading="nearest")
ax.set_xlim(0, np.pi)
ax.grid(False)
fig.tight_layout()


3D plot version:



2D polar plot version:



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