.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "api/gallery/plasma/mhd/plot_02_sweet_parker_reconnection.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code. .. rst-class:: sphx-glr-example-title .. _sphx_glr_api_gallery_plasma_mhd_plot_02_sweet_parker_reconnection.py: The Sweet-Parker reconnection-rate bottleneck ================================================== Peter Sweet and Eugene Parker (1957-1958) proposed the first quantitative model of magnetic reconnection: oppositely directed field lines are driven together into a long, thin resistive current sheet of length :math:`L`, where finite conductivity :math:`\eta` finally lets them break and reconnect. The relevant dimensionless control parameter is the Lundquist number :math:`S=Lv_A/\eta`, the ratio of the resistive diffusion time :math:`L^2/\eta` to the Alfven crossing time :math:`L/v_A`. Mass conservation through the sheet's narrow exit throttles the whole process to .. math:: \frac{v_{in}}{v_A} = S^{-1/2}, \qquad \delta = \frac{L}{\sqrt{S}}, a reconnection rate and current-sheet thickness that both fall as :math:`S^{-1/2}` -- for solar-flare conditions (:math:`S\sim10^{12}`), millions of times slower than flares are observed to release their energy, a discrepancy that stood as an open problem until Petschek's 1964 revision (:doc:`plot_04_petschek_reconnection`). :func:`~physicskit.plasma.mhd.lundquist_number` and :func:`~physicskit.plasma.mhd.sweet_parker_rate` / :func:`~physicskit.plasma.mhd.sweet_parker_layer_width` reproduce the model's scaling laws directly. Beyond the steady-state scaling laws, :func:`~physicskit.plasma.instabilities.simulate_reconnection` time-steps an actual X-point reconnecting -- a generic kinematic resistive-induction model closer in spirit to Sweet-Parker's diffusion-throttled picture than to Petschek's localized-shock geometry. .. GENERATED FROM PYTHON SOURCE LINES 36-41 .. code-block:: Python import matplotlib.pyplot as plt import numpy as np import physicskit as pk .. GENERATED FROM PYTHON SOURCE LINES 42-46 Sweep the Lundquist number from laboratory to solar-flare scale -------------------------------------------------------------------- At S ~ 1e12 (solar flare conditions) the predicted inflow speed is millions of times slower than observed flare energy release. .. GENERATED FROM PYTHON SOURCE LINES 46-65 .. code-block:: Python S_vals = np.logspace(4, 14, 50) rate = np.array([pk.plasma.sweet_parker_rate(S) for S in S_vals]) width = np.array([pk.plasma.sweet_parker_layer_width(1e7, S) for S in S_vals]) fig, axes = plt.subplots(1, 2, figsize=(10, 4)) axes[0].loglog(S_vals, rate) axes[0].set_xlabel("Lundquist number S") axes[0].set_ylabel(r"$v_{in}/v_A$") axes[0].set_title(r"Sweet-Parker rate $\propto S^{-1/2}$") axes[1].loglog(S_vals, width) axes[1].set_xlabel("Lundquist number S") axes[1].set_ylabel(r"sheet width $\delta$ (m)") axes[1].set_title(r"Current-sheet thickness $\propto S^{-1/2}$") fig.tight_layout() plt.show() .. image-sg:: /api/gallery/plasma/mhd/images/sphx_glr_plot_02_sweet_parker_reconnection_001.png :alt: Sweet-Parker rate $\propto S^{-1/2}$, Current-sheet thickness $\propto S^{-1/2}$ :srcset: /api/gallery/plasma/mhd/images/sphx_glr_plot_02_sweet_parker_reconnection_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 66-81 An actual X-point reconnecting: the kinematic resistive-induction model ------------------------------------------------------------------------ The scaling laws above describe a *steady-state* reconnection rate; this builds the perturbed Harris current sheet the picture presumes and animates the flux function :math:`\psi(x,y,t)` actually breaking and reconnecting at the X-point under the resistive induction equation .. math:: \partial_t\psi = \eta\nabla^2\psi - \mathbf{v}\cdot\nabla\psi, with a prescribed inflow :math:`\mathbf{v}` and finite resistivity :math:`\eta` -- exactly Faraday's law with an Ohmic (rather than ideal) Ohm's law, restricted to a fixed (kinematic) flow field rather than a self-consistently solved momentum equation. .. GENERATED FROM PYTHON SOURCE LINES 81-87 .. code-block:: Python psi0 = pk.plasma.reconnection_harris_ic(96, 96, Lx=20.0, Ly=20.0, sheet_width=1.0, perturbation_amplitude=0.2) anim = pk.plasma.animate_reconnection(psi0, eta=0.15, v0=0.15, dt=0.05, steps_per_frame=8, n_frames=50, Lx=20.0, Ly=20.0, interval=100) plt.show() .. container:: sphx-glr-animation .. raw:: html .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 3.885 seconds) .. _sphx_glr_download_api_gallery_plasma_mhd_plot_02_sweet_parker_reconnection.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_02_sweet_parker_reconnection.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_02_sweet_parker_reconnection.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_02_sweet_parker_reconnection.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_