.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/atmosphere/plot_15_rossby_planetary_waves.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_15_rossby_planetary_waves.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_15_rossby_planetary_waves.py:


Rossby's planetary waves (1939)
===============================

Carl-Gustaf Rossby explained the slow eastward drift of the large troughs and
ridges in the westerlies. Because the Coriolis parameter grows northward at
the rate ``beta``, a vorticity wave propagates westward relative to the wind
at ``beta / k^2``. Long waves therefore stand still or retrogress, and the
stationary wavelength is ``2 pi sqrt(U / beta)``. These waves are large
scale, with Rossby numbers well below one.

.. GENERATED FROM PYTHON SOURCE LINES 12-37



.. image-sg:: /api/gallery/atmosphere/images/sphx_glr_plot_15_rossby_planetary_waves_001.png
   :alt: Rossby waves in a 15 m/s westerly at 45 °N
   :srcset: /api/gallery/atmosphere/images/sphx_glr_plot_15_rossby_planetary_waves_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    stationary wavelength at 45 °N for 15 m/s: 6048 km






|

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from earthsciencekit.atmosphere import coriolis_parameter, rossby_number, rossby_wave_speed
    from earthsciencekit.constants import EARTH_RADIUS, OMEGA_EARTH

    latitude = np.deg2rad(45)
    beta = 2 * OMEGA_EARTH * np.cos(latitude) / EARTH_RADIUS
    wind = 15.0
    wavelength = np.linspace(1000e3, 10000e3, 300)
    speed = rossby_wave_speed(wind, beta, 2 * np.pi / wavelength)
    stationary = 2 * np.pi * np.sqrt(wind / beta)
    assert abs(float(rossby_wave_speed(wind, beta, 2 * np.pi / stationary))) < 1e-12
    assert rossby_number(wind, 1e6, coriolis_parameter(latitude)) < 0.2
    print(f"stationary wavelength at 45 °N for 15 m/s: {stationary / 1e3:.0f} km")

    fig, ax = plt.subplots()
    ax.plot(wavelength / 1e3, speed)
    ax.axhline(0, color="k", lw=0.5)
    ax.axvline(stationary / 1e3, color="0.5", ls="--", label="stationary")
    ax.set(xlabel="wavelength [km]", ylabel="eastward phase speed [m/s]", title="Rossby waves in a 15 m/s westerly at 45 °N")
    ax.legend()

    plt.show()


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

   **Total running time of the script:** (0 minutes 0.081 seconds)


.. _sphx_glr_download_api_gallery_atmosphere_plot_15_rossby_planetary_waves.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_15_rossby_planetary_waves.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_15_rossby_planetary_waves.ipynb <plot_15_rossby_planetary_waves.ipynb>`

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

      :download:`Download Python source code: plot_15_rossby_planetary_waves.py <plot_15_rossby_planetary_waves.py>`

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

      :download:`Download zipped: plot_15_rossby_planetary_waves.zip <plot_15_rossby_planetary_waves.zip>`


.. only:: html

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

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