"""
Ekman's dead water (1904)
=========================

On Nansen's Fram expedition the ship slowed to a crawl in fjords where a thin
layer of meltwater lay over salty sea water. Vagn Walfrid Ekman showed that
the ship was making waves on the hidden density jump. Those internal waves
travel at about ``sqrt(g' h)`` with the reduced gravity ``g' = g d_rho / rho``,
a few tens of centimetres per second, so a slow ship pours its energy into
them.
"""

import matplotlib.pyplot as plt
import numpy as np

from earthsciencekit.constants import GRAVITY
from earthsciencekit.oceanography import internal_wave_speed, reduced_gravity, shallow_wave_speed

upper = np.linspace(0.5, 10, 200)
fig, ax = plt.subplots()
for contrast in [5, 10, 20]:
    gprime = reduced_gravity(contrast, 1025, GRAVITY)
    deep_limit = internal_wave_speed(gprime, upper, 1e6)
    np.testing.assert_allclose(deep_limit / shallow_wave_speed(upper, GRAVITY), np.sqrt(contrast / 1025), rtol=1e-5)
    ax.plot(upper, internal_wave_speed(gprime, upper, 100), label=f"density jump {contrast} kg/m^3")
ax.axhspan(0.5, 0.8, color="tab:red", alpha=0.15, label="Fram in dead water")
ax.set(xlabel="fresh-layer thickness [m]", ylabel="internal wave speed [m/s]", title="Waves on a hidden density jump")
ax.legend()

plt.show()