.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/atmosphere/plot_12_richardson_turbulence_criterion.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_12_richardson_turbulence_criterion.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_12_richardson_turbulence_criterion.py:


Richardson's turbulence criterion (1920)
========================================

Lewis Fry Richardson asked when wind shear can stir up a stably stratified
layer. Turbulence must lift heavy air against buoyancy, at a rate set by
``N^2``, using energy drawn from the shear. Their ratio, the Richardson
number, decides: Miles and Howard later proved that a stratified shear flow
can become unstable only where it falls below 1/4. Over a night-time
boundary layer, the critical value marks the height where turbulence dies out.

.. GENERATED FROM PYTHON SOURCE LINES 12-36



.. image-sg:: /api/gallery/atmosphere/images/sphx_glr_plot_12_richardson_turbulence_criterion_001.png
   :alt: A stable night-time layer
   :srcset: /api/gallery/atmosphere/images/sphx_glr_plot_12_richardson_turbulence_criterion_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    turbulence can persist below about 99 m






|

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from earthsciencekit.atmosphere import brunt_vaisala_squared, richardson_number
    from earthsciencekit.constants import GRAVITY

    height = np.linspace(10, 500, 200)
    theta_gradient = 0.01 * (1 - np.exp(-height / 150))
    shear = 8 / 200 * np.exp(-height / 200) + 1e-3
    ri = richardson_number(brunt_vaisala_squared(290, theta_gradient, GRAVITY), shear)
    assert np.all(np.diff(ri) > 0)
    crossing = height[np.argmax(ri > 0.25)]
    assert ri[0] < 0.25 < ri[-1]
    assert np.isclose(richardson_number(-1e-4, 0.01), -1)
    print(f"turbulence can persist below about {crossing:.0f} m")

    fig, ax = plt.subplots()
    ax.semilogx(ri, height)
    ax.axvline(0.25, color="k", ls="--", label="Ri = 1/4")
    ax.set(xlabel="gradient Richardson number", ylabel="height [m]", title="A stable night-time layer")
    ax.legend()

    plt.show()


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

   **Total running time of the script:** (0 minutes 0.108 seconds)


.. _sphx_glr_download_api_gallery_atmosphere_plot_12_richardson_turbulence_criterion.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_12_richardson_turbulence_criterion.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_12_richardson_turbulence_criterion.ipynb <plot_12_richardson_turbulence_criterion.ipynb>`

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

      :download:`Download Python source code: plot_12_richardson_turbulence_criterion.py <plot_12_richardson_turbulence_criterion.py>`

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

      :download:`Download zipped: plot_12_richardson_turbulence_criterion.zip <plot_12_richardson_turbulence_criterion.zip>`


.. only:: html

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

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