Diploma report Implementation and verification of a simple ... - LPAS

4.2 Hydrostatic balance
Hydrostatic equilibrium in vertical direction z can be computed using equation (60). However,
even if the initial pressure is analytical, there will be small numerical imprecisions due
to discretization into finite volumes. Indeed, the pressure is generally not a linear function
of height and small errors arise when approximating average values over each cells. These
imprecisions then lead to the creation of winds at the beginning of the simulation. The maximum
velocity of these winds then increases to stabilise at a constant mean value around
which it oscillates with time.
This behaviour is due to the integration schemes themselves and their associated local
truncation error. In order to obtain good results, numerical filters are used that smooth the
solutions and therefore reduce the wind velocities. Sensitivity tests to filtering using hydrostatic
equilibrium in one dimension were done. The results are presented on semilogarithmic
plots of |w| max versus time, where |w| max is the maximum of z-velocity values over the whole
domain. The plots are shown in Figure 4 and Figure 5. The model is integrated using a
Leapfrog scheme together with centered-in-space differences and the boundary conditions are
a solid wall with background.
The models using a Leapfrog scheme afford an undamped computational mode that leads
to the creation of oscillations and instabilities in the solution. These unwanted effects can
be controlled by applying an Asselin time filter after each step (see section 3.4.2). The value
of the filter coefficient µ must be chosen so that the filter strongly damps the computational
mode while having only little impact on the physical mode. A series of simulations is done for
various values of µ and the results are shown in Figure 4. It is observed that the amplitude
of the oscillations are reduced as µ is increased. However, at the same time, the average
value of |w| max increases a little bit. This can be seen by comparing the plots with time
filter constants of 0.05 and 0.4 in Figure 4. Although the curve for µ = 0.4 is smoother, it
is also higher, meaning that the velocities are bigger. With very large values of µ (e.g. 0.6),
the velocities even continuously increase with time to reach values of 10 −3 ms −1 after 9000 s.
Filter coefficients µ in the range 0.05 - 0.2 will be used in our model.
A second order spatial filter as given by (71) can be used to damp the 2∆x-scale noise
generated by numerical dispersion. Results for different values of the filter coefficient ν are
presented in Figure 5 showing that the velocities can be reduced by increasing the strength of
the filter. However, beside reducing the 2∆x noise, the filtering may also act on longer wave
components, which does contain physical meaningful information and therefore should not
be affected by the filter. In the last plot of Figure 5, the decrease of |w| max with time seems
to indicate that these longer wave components are smoothed by the filter. A very strong filter
may therefore create too much diffusion and lead to the loss of physical information. For
this reason the strength of the filter should be adapted according to the problem being solved.
Sensitivity to source term formulation is also tested and is presented in Figure 6. Three
different techniques for the interpolation of ρ are compared: Midpoint, Simpson rule over
one cell and Simpson rule over two cells. In Midpoint, the average cell value of ρ is used to
compute the source term, leading to a first order approximation. The value of the source term
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