:orphan:

Random Vector Field Generation
==============================

In 1970, Kraichnan was the first to suggest a randomization method.
For studying the diffusion of single particles in a random incompressible
velocity field, he came up with a randomization method which includes a
projector which ensures the incompressibility of the vector field.


Without loss of generality we assume that the mean velocity :math:`\bar{U}` is oriented
towards the direction of the first basis vector :math:`\mathbf{e}_1`. Our goal is now to
generate random fluctuations with a given covariance model around this mean velocity.
And at the same time, making sure that the velocity field remains incompressible or
in other words, ensure :math:`\nabla \cdot \mathbf U = 0`.
This can be done by using the randomization method we already know, but adding a
projector to every mode being summed:


.. math::

   \mathbf{U}(\mathbf{x}) = \bar{U} \mathbf{e}_1 - \sqrt{\frac{\sigma^{2}}{N}}
   \sum_{i=1}^{N} \mathbf{p}(\mathbf{k}_i) \left[ Z_{1,i}
      \cos\left( \langle \mathbf{k}_{i}, \mathbf{x} \rangle \right)
   + \sin\left( \langle \mathbf{k}_{i}, \mathbf{x} \rangle \right) \right]

with the projector

.. math::

   \mathbf{p}(\mathbf{k}_i) = \mathbf{e}_1 - \frac{\mathbf{k}_i k_1}{k^2} \; .

By calculating :math:`\nabla \cdot \mathbf U = 0`, it can be verified, that
the resulting field is indeed incompressible.


Examples
--------



.. raw:: html

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

.. thumbnail-parent-div-open

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    <div class="sphx-glr-thumbcontainer" tooltip="As a first example we are going to generate a 2d vector field with a Gaussian covariance model on a structured grid:">

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  .. image:: /examples/04_vector_field/images/thumb/sphx_glr_00_2d_vector_field_thumb.png
    :alt:

  :ref:`sphx_glr_examples_04_vector_field_00_2d_vector_field.py`

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      <div class="sphx-glr-thumbnail-title">Generating a Random 2D Vector Field</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="In this example we are going to generate a random 3D vector field with a Gaussian covariance model. The mesh on which we generate the field will be externally defined and it will be generated by PyVista.">

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  .. image:: /examples/04_vector_field/images/thumb/sphx_glr_01_3d_vector_field_thumb.png
    :alt:

  :ref:`sphx_glr_examples_04_vector_field_01_3d_vector_field.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Generating a Random 3D Vector Field</div>
    </div>


.. thumbnail-parent-div-close

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    </div>


.. toctree::
   :hidden:

   /examples/04_vector_field/00_2d_vector_field
   /examples/04_vector_field/01_3d_vector_field



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

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