.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/atmosphere/plot_14_vaisala_brunt_buoyancy_frequency.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_14_vaisala_brunt_buoyancy_frequency.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_14_vaisala_brunt_buoyancy_frequency.py:


Väisälä and Brunt's buoyancy frequency (1925)
=============================================

Vilho Väisälä (1925) and David Brunt (1927) showed that a parcel displaced
vertically in stable air oscillates about its level, with angular frequency
``N = sqrt(g / theta * dtheta/dz)``. In the troposphere the period is about
ten minutes. Integrating the parcel equation ``z'' = -N^2 z`` recovers that
period.

.. GENERATED FROM PYTHON SOURCE LINES 11-31



.. image-sg:: /api/gallery/atmosphere/images/sphx_glr_plot_14_vaisala_brunt_buoyancy_frequency_001.png
   :alt: Buoyancy oscillation, period 10.6 min
   :srcset: /api/gallery/atmosphere/images/sphx_glr_plot_14_vaisala_brunt_buoyancy_frequency_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    buoyancy period: 10.6 min






|

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from earthsciencekit.atmosphere import brunt_vaisala_squared
    from earthsciencekit.constants import GRAVITY
    from earthsciencekit.integrators import integrate_ode

    n2 = float(brunt_vaisala_squared(300, 0.003, GRAVITY))
    period = 2 * np.pi / np.sqrt(n2)
    time = np.linspace(0, 3 * period, 601)
    result = integrate_ode(lambda t, y: [y[1], -n2 * y[0]], [100.0, 0.0], time, names=("height", "speed"), units=("m", "m/s"))
    np.testing.assert_allclose(result.values[:, 0], 100 * np.cos(np.sqrt(n2) * time), atol=1e-4)
    print(f"buoyancy period: {period / 60:.1f} min")

    fig, ax = plt.subplots()
    ax.plot(time / 60, result.values[:, 0])
    ax.set(xlabel="time [min]", ylabel="displacement [m]", title=f"Buoyancy oscillation, period {period / 60:.1f} min")

    plt.show()


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

   **Total running time of the script:** (0 minutes 0.112 seconds)


.. _sphx_glr_download_api_gallery_atmosphere_plot_14_vaisala_brunt_buoyancy_frequency.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_14_vaisala_brunt_buoyancy_frequency.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_14_vaisala_brunt_buoyancy_frequency.ipynb <plot_14_vaisala_brunt_buoyancy_frequency.ipynb>`

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

      :download:`Download Python source code: plot_14_vaisala_brunt_buoyancy_frequency.py <plot_14_vaisala_brunt_buoyancy_frequency.py>`

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

      :download:`Download zipped: plot_14_vaisala_brunt_buoyancy_frequency.zip <plot_14_vaisala_brunt_buoyancy_frequency.zip>`


.. only:: html

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

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