"""
Laplace's tidal equations (1776)
================================

Pierre-Simon Laplace replaced Newton's static bulge with the dynamics of a thin
ocean on a rotating Earth. Long waves in that ocean travel at ``sqrt(g h)``.
To keep pace with the Moon's bulge at the equator the ocean would have to be
about 22 km deep; the real ocean, about 4 km deep, cannot follow it, so real
tides are a forced dynamic response, not an equilibrium.
"""

import matplotlib.pyplot as plt
import numpy as np

from earthsciencekit.constants import EARTH_RADIUS, GRAVITY, OMEGA_EARTH
from earthsciencekit.oceanography import shallow_wave_speed

surface_speed = OMEGA_EARTH * EARTH_RADIUS
critical_depth = surface_speed**2 / GRAVITY
assert np.isclose(shallow_wave_speed(critical_depth, GRAVITY), surface_speed)
assert 21e3 < critical_depth < 23e3
assert shallow_wave_speed(4000, GRAVITY) < 0.5 * surface_speed

depth = np.logspace(1, np.log10(40e3), 300)
fig, ax = plt.subplots()
ax.semilogx(depth / 1e3, shallow_wave_speed(depth, GRAVITY), label="long-wave speed sqrt(g h)")
ax.axhline(surface_speed, color="k", ls="--", label="equatorial surface speed")
ax.axvline(4, color="tab:blue", alpha=0.4, lw=6, label="mean ocean depth")
ax.axvline(critical_depth / 1e3, color="tab:red", ls=":", label=f"resonant depth {critical_depth / 1e3:.0f} km")
ax.set(xlabel="ocean depth [km]", ylabel="speed [m/s]", title="The ocean cannot keep up with the tidal bulge")
ax.legend()

plt.show()