.. _sphx_glr_api_gallery_fluids_instabilities:

Instabilities
-------------

Some flows are unstable to *any* perturbation, no matter how small: the
smallest ripple grows exponentially until it reorganizes the whole flow.
Kelvin-Helmholtz instability needs nothing but shear -- two layers sliding
past each other supply their own energy to grow a ripple into a row of
rolled-up vortex cores, the mechanism behind everything from billow clouds
to a flapping flag. Rayleigh-Taylor instability needs nothing but an
unstable density stratification -- heavy fluid sitting on light fluid under
gravity -- and grows a rippled interface into the mushroom-shaped plumes
seen in everything from a lava lamp to a supernova remnant. Both examples
below compare the simulated early-time growth against the exact linear
growth-rate law for that instability, then run well past the point where
that linear theory stops applying, into the fully rolled-up nonlinear state.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

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    <div class="sphx-glr-thumbcontainer" tooltip="Helmholtz (1868) and Kelvin (1871) showed that any interface between two fluid layers sliding past each other is unconditionally unstable: the shear itself supplies the energy to grow a small ripple, with no velocity threshold required. kelvin_helmholtz_ic seeds exactly this thin, rippled shear layer as a vorticity field on a doubly periodic domain, and NavierStokes2D evolves it by integrating the 2D incompressible vorticity-transport equation">

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  .. image:: /api/gallery/fluids/instabilities/images/thumb/sphx_glr_plot_kelvin_helmholtz_thumb.png
    :alt:

  :doc:`/api/gallery/fluids/instabilities/plot_kelvin_helmholtz`

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      <div class="sphx-glr-thumbnail-title">Roll-up of a shear layer into vortex cores</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="A denser fluid layer sits atop a lighter one -- exactly the unstable arrangement -- separated by an interface with a small sinusoidal ripple. The two layers are represented not as a sharp density jump but through the Boussinesq approximation: a dimensionless buoyancy field b=-\delta\rho/\rho_0 (positive where the fluid is locally lighter than the reference density), advected and diffused like vorticity but also exerting a baroclinic torque g\,\partial_x b on it wherever the density gradient is non-vertical:">

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  .. image:: /api/gallery/fluids/instabilities/images/thumb/sphx_glr_plot_rayleigh_taylor_thumb.png
    :alt:

  :doc:`/api/gallery/fluids/instabilities/plot_rayleigh_taylor`

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      <div class="sphx-glr-thumbnail-title">Rayleigh-Taylor plumes from a heavy-over-light interface</div>
    </div>


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    </div>


.. toctree::
   :hidden:

   /api/gallery/fluids/instabilities/plot_kelvin_helmholtz
   /api/gallery/fluids/instabilities/plot_rayleigh_taylor