.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/atmosphere/plot_13_richardson_numerical_forecast.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_atmosphere_plot_13_richardson_numerical_forecast.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_atmosphere_plot_13_richardson_numerical_forecast.py:


Richardson's numerical forecast (1922)
======================================

In *Weather Prediction by Numerical Process*, Richardson computed by hand a
six-hour forecast for two points over Germany, from the observations of
20 May 1910. He predicted a surface pressure rise of 145 hPa, about a hundred
times too large. The pressure tendency is the column-mean divergence times the
surface pressure, and the divergence at any one level is much larger than the
column mean, because convergence below nearly cancels divergence above. A
column-mean error of 7e-6 per second, the size of the divergence at a single
level, is enough to reproduce his result.

.. GENERATED FROM PYTHON SOURCE LINES 14-37



.. image-sg:: /api/gallery/atmosphere/images/sphx_glr_plot_13_richardson_numerical_forecast_001.png
   :alt: Small divergence errors, huge tendencies
   :srcset: /api/gallery/atmosphere/images/sphx_glr_plot_13_richardson_numerical_forecast_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    six-hour pressure change: 144 hPa






|

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from earthsciencekit.atmosphere import surface_pressure_tendency

    six_hours = 6 * 3600
    surface_pressure = 101325.0
    richardson = float(surface_pressure_tendency(-6.6e-6, surface_pressure)) * six_hours
    assert abs(richardson - 145e2) < 3e2
    divergence = np.logspace(-8, -5, 200)
    change = -surface_pressure_tendency(-divergence, surface_pressure) * six_hours / 100
    assert np.isclose(-surface_pressure_tendency(1e-7, surface_pressure) * six_hours, 218.8, rtol=1e-3)
    print(f"six-hour pressure change: {richardson / 100:.0f} hPa")

    fig, ax = plt.subplots()
    ax.loglog(divergence, -change)
    ax.axhline(145, color="tab:red", ls="--", label="Richardson's 145 hPa")
    ax.axvspan(1e-8, 1e-7, color="tab:green", alpha=0.2, label="realistic column mean")
    ax.set(xlabel="column-mean convergence [1/s]", ylabel="pressure rise in 6 h [hPa]", title="Small divergence errors, huge tendencies")
    ax.legend()

    plt.show()


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

   **Total running time of the script:** (0 minutes 0.172 seconds)


.. _sphx_glr_download_api_gallery_atmosphere_plot_13_richardson_numerical_forecast.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/atmosphere/plot_13_richardson_numerical_forecast.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_13_richardson_numerical_forecast.ipynb <plot_13_richardson_numerical_forecast.ipynb>`

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

      :download:`Download Python source code: plot_13_richardson_numerical_forecast.py <plot_13_richardson_numerical_forecast.py>`

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

      :download:`Download zipped: plot_13_richardson_numerical_forecast.zip <plot_13_richardson_numerical_forecast.zip>`


.. only:: html

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

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