.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/oceanography/plot_07_sandstrom_helland_hansen_dynamic_method.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_07_sandstrom_helland_hansen_dynamic_method.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_07_sandstrom_helland_hansen_dynamic_method.py:


Sandström and Helland-Hansen's dynamic method (1903)
====================================================

Johan Sandström and Bjørn Helland-Hansen showed how to compute currents from
temperature and salinity alone. Light water stands higher than dense water,
so the sea surface above a warm station rises relative to a cold one until
the pressure at a deep level of no motion is the same under both. The surface
slope then drives a geostrophic current.

.. GENERATED FROM PYTHON SOURCE LINES 11-42



.. image-sg:: /api/gallery/oceanography/images/sphx_glr_plot_07_sandstrom_helland_hansen_dynamic_method_001.png
   :alt: Dynamic method: surface current 0.68 m/s
   :srcset: /api/gallery/oceanography/images/sphx_glr_plot_07_sandstrom_helland_hansen_dynamic_method_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    surface height difference 58 cm, current 0.68 m/s






|

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from earthsciencekit.atmosphere import coriolis_parameter
    from earthsciencekit.constants import GRAVITY
    from earthsciencekit.oceanography import geostrophic_current, hydrostatic_pressure, linear_density

    reference, distance = 1027.0, 100e3
    depth = np.linspace(0, 1000, 2001)
    stations = {"warm": 15 * np.exp(-depth / 300), "cold": 5 * np.exp(-depth / 300)}
    density = {name: linear_density(reference, anomaly, 0, 2e-4, 7.6e-4) for name, anomaly in stations.items()}
    height = {name: np.trapezoid(reference - rho, depth) / reference for name, rho in density.items()}
    bottom = {
        name: hydrostatic_pressure(reference, height[name], GRAVITY) + GRAVITY * np.trapezoid(rho, depth) for name, rho in density.items()
    }
    assert np.isclose(bottom["warm"], bottom["cold"])

    slope = (height["warm"] - height["cold"]) / distance
    speed = geostrophic_current(slope, coriolis_parameter(np.radians(35)), GRAVITY)
    assert 0.3 < speed < 2
    print(f"surface height difference {100 * (height['warm'] - height['cold']):.0f} cm, current {speed:.2f} m/s")

    fig, ax = plt.subplots()
    for name, rho in density.items():
        ax.plot(rho, depth, label=f"{name} station")
    ax.invert_yaxis()
    ax.set(xlabel="density [kg/m^3]", ylabel="depth [m]", title=f"Dynamic method: surface current {speed:.2f} m/s")
    ax.legend()

    plt.show()


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

   **Total running time of the script:** (0 minutes 0.085 seconds)


.. _sphx_glr_download_api_gallery_oceanography_plot_07_sandstrom_helland_hansen_dynamic_method.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_07_sandstrom_helland_hansen_dynamic_method.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_07_sandstrom_helland_hansen_dynamic_method.ipynb <plot_07_sandstrom_helland_hansen_dynamic_method.ipynb>`

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

      :download:`Download Python source code: plot_07_sandstrom_helland_hansen_dynamic_method.py <plot_07_sandstrom_helland_hansen_dynamic_method.py>`

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

      :download:`Download zipped: plot_07_sandstrom_helland_hansen_dynamic_method.zip <plot_07_sandstrom_helland_hansen_dynamic_method.zip>`


.. only:: html

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

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