Added a tutorial for calculating random vector fields

docs/source/tutorial_04_vec_field.rst

@@ -0,0 +1,98 @@
+Tutorial 4: 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.
+
+
+Theoretical Background
+----------------------
+
+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.
+
+
+Generating a Random Vector Field
+--------------------------------
+
+As a first example we are going to generate a vector field with a Gaussian
+covariance model on a structured grid:
+
+.. code-block:: python
+
+   import numpy as np
+   import matplotlib.pyplot as plt
+   from gstools import SRF, Gaussian
+   x = np.arange(100)
+   y = np.arange(100)
+   model = Gaussian(dim=2, var=1, len_scale=10)
+   srf = SRF(model, generator='VectorField')
+   srf((x, y), mesh_type='structured', seed=19841203)
+   srf.plot()
+
+And we get a beautiful streamflow plot:
+
+.. image:: pics/vec_srf_tut_gau.png
+   :width: 600px
+   :align: center
+
+Let us have a look at the influence of the covariance model. Choosing the
+exponential model and keeping all other parameters the same
+
+.. code-block:: python
+
+   from gstools import Exponential
+   
+   model2 = Exponential(dim=2, var=1, len_scale=10)
+   srf.model = model2
+   srf((x, y), mesh_type='structured', seed=19841203)
+   srf.plot()
+
+we get following result
+
+.. image:: pics/vec_srf_tut_exp.png
+   :width: 600px
+   :align: center
+
+and we see, that the wiggles are much "rougher" than the smooth Gaussian ones.
+
+
+Applications
+------------
+
+One great advantage of the Kraichnan method is, that after some initializations,
+one can compute the velocity field at arbitrary points, online, with hardly any
+overhead.
+This means, that for a Lagrangian transport simulation for example, the velocity
+can be evaluated at each particle position very efficiently and without any
+interpolation. These field interpolations are a common problem for Lagrangian
+methods.
+
+
+.. raw:: latex
+
+    \clearpage

docs/source/tutorials.rst

@@ -11,3 +11,4 @@ explore its whole beauty and power.
    tutorial_01_srf.rst
    tutorial_02_cov.rst
    tutorial_03_vario.rst
+   tutorial_04_vec_field.rst