.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/elliptic_lcs/plot_bickley_elliptic_lcs.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_auto_examples_elliptic_lcs_plot_bickley_elliptic_lcs.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_auto_examples_elliptic_lcs_plot_bickley_elliptic_lcs.py:


Bickley jet Elliptic LCS
========================

Compute the LAVD-based elliptic lcs for the bickley jet.

.. GENERATED FROM PYTHON SOURCE LINES 8-25

.. code-block:: Python


    # Author: ajarvis

    import numpy as np
    from math import copysign, pi
    import matplotlib.pyplot as plt
    from matplotlib.animation import FuncAnimation
    from numbacs.flows import (
        get_predefined_flow,
        get_predefined_callable,
        get_interp_arrays_scalar,
        get_callable_scalar,
    )
    from numbacs.integration import flowmap_n_grid_2D, flowmap_n
    from numbacs.diagnostics import lavd_grid_2D
    from numbacs.extraction import rotcohvrt
    from numbacs.utils import curl_func_tspan, gen_filled_circ, interp_curve, pts_in_poly_mask







.. GENERATED FROM PYTHON SOURCE LINES 26-30

Get flow
--------------
Set the integration span and direction, retrieve the flow, jit-callable for velocity
field, and set up domain.

.. GENERATED FROM PYTHON SOURCE LINES 30-55

.. code-block:: Python


    # set initial time, integration time, integration direction,
    # and integration span (units of days)
    t0 = 0.0
    T = 40.0
    int_direction = copysign(1, T)
    n = 801

    # retrieve function pointer and parameters for bickley jet flow.
    funcptr, params = get_predefined_flow(
        "bickley_jet", int_direction=int_direction, return_domain=False
    )

    # retrieve callable to compute vorticity
    vel_spline = get_predefined_callable("bickley_jet", return_domain=False)

    # set up larger domain to capture all elliptic lcs
    domain = ((-2.0, 6.371 * pi + 2.0), (-3.0, 3.0))
    dx = 0.05
    dy = 0.05
    x = np.arange(domain[0][0], domain[0][1] + dx, dx)
    y = np.arange(domain[1][0], domain[1][1] + dy, dy)
    nx = len(x)
    ny = len(y)








.. GENERATED FROM PYTHON SOURCE LINES 56-59

Integrate
---------
Integrate grid of particles and return positions at n times between [t0,t0+T] (inclusive).

.. GENERATED FROM PYTHON SOURCE LINES 59-61

.. code-block:: Python

    flowmap, tspan = flowmap_n_grid_2D(funcptr, t0, T, x, y, params, n=n)








.. GENERATED FROM PYTHON SOURCE LINES 62-65

Vorticity
---------
Copmute vorticity on the grid and over the times for which the flowmap was returned.

.. GENERATED FROM PYTHON SOURCE LINES 65-70

.. code-block:: Python


    # compute vorticity and create interpolant for it
    vort = curl_func_tspan(vel_spline, tspan, x, y, h=1e-3)
    grid_domain, C_vort = get_interp_arrays_scalar(tspan, x, y, vort)
    vort_spline = get_callable_scalar(grid_domain, C_vort)







.. GENERATED FROM PYTHON SOURCE LINES 71-74

LAVD
----
Compute LAVD from vorticity and particle positions.

.. GENERATED FROM PYTHON SOURCE LINES 74-87

.. code-block:: Python


    # need to pass raveled arrays into lavd_grid_2D
    X, Y = np.meshgrid(x, y, indexing="ij")
    xrav = X.ravel()
    yrav = Y.ravel()

    # since bickley jet is periodic in x-direction, need to pass its period
    # into lavd_grid_2D
    period_x = 6.371 * pi

    # compute lavd
    lavd = lavd_grid_2D(flowmap, tspan, T, vort_spline, xrav, yrav, period_x=period_x)








.. GENERATED FROM PYTHON SOURCE LINES 88-91

LAVD-based elliptic LCS
-----------------------
Compute elliptic LCS from LAVD.

.. GENERATED FROM PYTHON SOURCE LINES 91-96

.. code-block:: Python


    # set parameters and compute lavd-based elliptic lcs
    r = 2.5
    convexity_deficiency = 4e-3
    elcs = rotcohvrt(lavd, x, y, r, convexity_deficiency=convexity_deficiency)







.. GENERATED FROM PYTHON SOURCE LINES 97-100

Plot
----
Plot the elliptic LCS over the LAVD field.

.. GENERATED FROM PYTHON SOURCE LINES 100-110

.. code-block:: Python


    # sphinx_gallery_thumbnail_number = 1
    fig, ax = plt.subplots(dpi=200)
    ax.contourf(x, y, lavd.T, levels=80)
    ax.set_aspect("equal")
    for rcv, c in elcs:
        ax.plot(rcv[:, 0], rcv[:, 1], lw=1.5)
        ax.scatter(c[0], c[1], 1.5)
    plt.show()




.. image-sg:: /auto_examples/elliptic_lcs/images/sphx_glr_plot_bickley_elliptic_lcs_001.png
   :alt: plot bickley elliptic lcs
   :srcset: /auto_examples/elliptic_lcs/images/sphx_glr_plot_bickley_elliptic_lcs_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 111-115

Advect elliptic LCS
-------------------
Advect an elliptic LCS and nearby filled circle of points to demonstrate
coherence.

.. GENERATED FROM PYTHON SOURCE LINES 115-134

.. code-block:: Python


    # pick an elliptic lcs and create more refined curve by interpolating
    # we pick index 2 because it is the first not on a boundary of the domain,
    # any could be choosen
    c0 = elcs[2][1]
    rcv0 = elcs[2][0]
    rcvi = interp_curve(rcv0, 2500, per=1)

    # create a filled circle which has a center a distance delta away from
    # vortex center
    delta = 5e-1
    circ_delta = gen_filled_circ(1.5, 10000, c=c0 + delta)

    # advect both the elliptic lcs and the nearby filled circle
    frames = 401
    adv_rcv, teval = flowmap_n(funcptr, t0, T, rcvi, params, n=frames)
    adv_circ, _ = flowmap_n(funcptr, t0, T, circ_delta, params, n=frames)









.. GENERATED FROM PYTHON SOURCE LINES 135-138

Animate
-------
Create animation of advected elliptic LCS and nearby filled circle.

.. GENERATED FROM PYTHON SOURCE LINES 138-185

.. code-block:: Python


    # find which points from the filled circle are inside elliptic lcs
    # for plotting purposes
    mask = pts_in_poly_mask(rcv0, circ_delta)

    # create plot
    fig, ax = plt.subplots(dpi=100)
    scatter0 = ax.scatter(adv_rcv[:, 0, 0] % period_x, adv_rcv[:, 0, 1], 1)
    scatter_in = ax.scatter(
        adv_circ[mask, 0, 0] % period_x, adv_circ[mask, 0, 1], 0.5, "purple", zorder=0
    )
    scatter_out = ax.scatter(
        adv_circ[~mask, 0, 0] % period_x, adv_circ[~mask, 0, 1], 0.5, "orange", zorder=0
    )
    ax.set_xlim([domain[0][0] + 2, domain[0][1] - 2])
    ax.set_ylim([domain[1][0], domain[1][1]])
    ax.set_aspect("equal")
    ax.set_title(f"t = {round(teval[0], 1)} days")


    # function for animation
    def update(frame):
        # for each frame, update the data stored on each artist
        x0 = adv_rcv[:, frame, 0] % period_x
        y0 = adv_rcv[:, frame, 1]
        x_in = adv_circ[mask, frame, 0] % period_x
        y_in = adv_circ[mask, frame, 1]
        x_out = adv_circ[~mask, frame, 0] % period_x
        y_out = adv_circ[~mask, frame, 1]
        data0 = np.column_stack((x0, y0))
        data_in = np.column_stack((x_in, y_in))
        data_out = np.column_stack((x_out, y_out))

        # update each scatter plot
        scatter0.set_offsets(data0)
        scatter_in.set_offsets(data_in)
        scatter_out.set_offsets(data_out)

        # update title
        ax.set_title(f"t = {round(teval[frame], 1)} days")

        return (scatter0, scatter_in, scatter_out)


    # create animation
    ani = FuncAnimation(fig=fig, func=update, frames=frames, interval=50)
    plt.show()



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     "


         /* set a timeout to make sure all the above elements are created before
            the object is initialized. */
         setTimeout(function() {
             animdf52d07bab324a32b2cf901ffa65a7ca = new Animation(frames, img_id, slider_id, 50.0,
                                      loop_select_id);
         }, 0);
       })()
     </script>







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

   **Total running time of the script:** (1 minutes 59.318 seconds)


.. _sphx_glr_download_auto_examples_elliptic_lcs_plot_bickley_elliptic_lcs.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

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

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

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

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

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

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


.. only:: html

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

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