A rather common discretisation with numerical models of the
atmosphere is the terrain following grid, where lines of
grid points follow the mountains. While this method is accurate
with very smooth mountains (mountains which are supported
by many grid points), in cases with realistic mountains
**large errors** are encountered [see 1].
Such errors can be avoided by using lines of grid points which
are horizontal and consequently cut into the mountains.

A test for the cut the cut cells discretisation is the
**advection along a smooth mountain** surface.
For this test it requires some consideration to obtain
reasonable and smooth solutions [2]. For some prediction
schemes **false boundary conditions** are posed, which for fast
waves would be OK, but are not allowed with the advection
equation. Therefore for such false approximations **wrong
solutions** which are very noisy at the mountain surface are
produced. A very smooth surface of a mountain is obtained when the
mountain is a straight line. Fig 1 gives an example of smooth
advection along such a straight line mountain which is
free of the mentioned noise and convergence problems:

Fig 1: left: Cut cells with the straight mountain test and the
sparse **o3o3** method. The initial and predicted fields are
shown for a movement along the straight mountain
(for sparse features of this figure see [1].
Right: the advection of a density field away from the straight mountain.

The advection along a line diagonal to the square area implies a rather regular grid [1]. For any other direction the mountain will cut small and large triangular and pentagonal cells out of a square grid. This leads to difficulties for the numerical treatment [1].

The reader with more serious interest in cut cells will need
some knowledge in the programming language Fortran
and the operating system Unix/Linux. A **login**
to an account on our server will be
necessary.

References:

[1]

[2]Steppeler/Klemp: Advection on cut-cell grids for an idealized mountain of constant slope, in: Mon.Wea.Rev. 145, 1765-1777 (2017)

Contact: send mail to Juergen Steppeler