GME CONVERGENCE TEST

A convergence test of GME Shallow Water Model has been performed. For that the values of the $L_{2}$ normalized error [5,11] for the height and relative vorticity fields at different resolutions for test cases 5 and 6 have been considered. To calculate the $L_{2}$ normalized error [5,11], the (derivative of the NCAR Spectral Transform Shallow Water Model) is used to produce a reference solution considered as a ``true solution''.

CASE 5

The test has been performed at day 15.
The GME runs with the ``spectral mountains'' have been considered.

In the table 4 are the $L_{2}$ normalized values [5,11] for the height and relative vorticity GME fields at day 15.

Table 4: Test case 5: height and relative vorticity $L_{2}$ normalized errors at day 15. $N = 10 * n_{i}^{2}+2$ is the number of grid points

ni	N	height $L_{2}$ normalized errors	r. vorticity $L_{2}$ normalized errors
32	10242	$0.13713 x 10^{-2}$	$0.17342$
64	40962	$0.72106 x 10^{-3}$	$0.67494 x 10^{-1}$
96	92162	$0.67526 x 10^{-3}$	$0.39068 x 10^{-1}$
192	368642	$0.66679 x 10^{-3}$	$0.18496 x 10^{-1}$

CASE 6

The test has been performed at day 10. Not a longer integration has been choosed because the zonal wavenumber 4 Rossby-Haurwitz wave is dynamically unstable (Thuburn and Yong Li, 2000 [10]).

In the table 5 are the $L_{2}$ normalized values [5,11] for the height and relative vorticity GME fields at day 10.

Table 5: Test case 6: height and relative vorticity $L_{2}$ normalized errors at day 10. $N = 10 * n_{i}^{2}+2$ is the number of grid points

ni	N	height $L_{2}$ normalized errors	r. vorticity $L_{2}$ normalized errors
32	10242	$0.90148 x 10^{-2}$	$0.21274$
64	40962	$0.28930 x 10^{-2}$	$0.93839 x 10^{-1}$
96	92162	$0.17858 x 10^{-2}$	$0.73865 x 10^{-1}$
192	368642	$0.96893 x 10^{-3}$	$0.46336 x 10^{-1}$