.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/oceanography/plot_02_laplace_tidal_equations.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_oceanography_plot_02_laplace_tidal_equations.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_oceanography_plot_02_laplace_tidal_equations.py:


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.

.. GENERATED FROM PYTHON SOURCE LINES 11-34



.. image-sg:: /api/gallery/oceanography/images/sphx_glr_plot_02_laplace_tidal_equations_001.png
   :alt: The ocean cannot keep up with the tidal bulge
   :srcset: /api/gallery/oceanography/images/sphx_glr_plot_02_laplace_tidal_equations_001.png
   :class: sphx-glr-single-img





.. code-block:: Python


    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()


.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.141 seconds)


.. _sphx_glr_download_api_gallery_oceanography_plot_02_laplace_tidal_equations.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../lite/lab/index.html?path=api/gallery/oceanography/plot_02_laplace_tidal_equations.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_02_laplace_tidal_equations.ipynb <plot_02_laplace_tidal_equations.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_02_laplace_tidal_equations.py <plot_02_laplace_tidal_equations.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_02_laplace_tidal_equations.zip <plot_02_laplace_tidal_equations.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_