SEOM

The problem as posed is inviscid. Although most models will occasionally require finite values of viscosity for numerical reasons, SEOM was successfully run on this non-turbulent problem with zero explicit viscosity. Figure 2 shows the typical behavior of the soliton. Shortly after initiation, the Rossby soliton sheds an eastward-propagating equatorial Kelvin wave as it adjusts from its initial state and begins to propagate westward. During this initial adjustment, the soliton loses approximately 7% of its initial amplitude. Note that the initial state of the model is inexact both because of finite numerical resolution and because the analytical solution is itself approximate.

**Figure 2:** Rossby soliton solutions: Surface displacement for the Rossby soliton problem obtained using SEOM at t=0,8,24,40 time units. The corresponding maximum values are .167, .156, .155, and .153 non-dimensional units.
$\begin{figure} \centerline{ \epsfig{figure=rossby.epsi,width=8.1cm,angle=0} } \end{figure}$

For the ``standard'' run, the average resolution of 0.5 was achieved by using $4 \times 12$ $9 \times 9$ (eighth order) elements. It can also be achieved by using $8 \times 24$ $5 \times 5$ (fourth order) elements. Higher and lower resolutions were also tried, as shown in figure 3 and table 3.

**Figure:** Rossby soliton solutions: Surface displacement at time 40 for the Rossby soliton problem obtained with (a) dx=1.0, $9 \times 9$ (eighth order) elements, (b) dx=1.0, $5 \times 5$ (fourth order) elements, (c) dx=0.5, $9 \times 9$ (eighth order) elements, (d) dx=0.5, $5 \times 5$ (fourth order) elements, (e) dx=0.25, $9 \times 9$ (eighth order) elements.
$\begin{figure} \centerline{ \epsfig{figure=rossby2.epsi,width=8.1cm,angle=0} } \end{figure}$

The solutions show that for the same resolution, the higher-order scheme does a better job of representing the Rossby soliton. Anything less than a resolution of 0.5 will be inadequate to achieve an accurate result. This provides between two and three points in one e-folding scale for the exponential in y. For the under-resolved cases, the animations show that the maximum amplitude oscillates, as does the distance between the two maxima. Initializing with the first-order solution reduces the magnitude of the initial Kelvin wave propogating to the east, at least in the well-resolved cases (run_f.gif), but otherwise produces similar results.

Table 3: SEOM simulations of the Rossby soliton obtained on a Sun UltraSparc 30. The phase speed is computed with a linear fit through the position of the maximum as a function of time. The value of ``b'' is the x-offset of the initial wave computed for this linear fit.

avg. dx	order	rms error	$c/c_{\rm theory}$	b	max/max₀	cpu (sec)	animation
1.0	9	$6.4 \times 10^{-4}$	1.12	0.18	0.66	18	run_a.gif
1.0	5	$1.3 \times 10^{-3}$	0.72	0.29	0.63	16	run_b.gif
0.5	9	$8.3 \times 10^{-5}$	0.99	0.07	0.93	76	run_c.gif
0.5	5	$2.2 \times 10^{-4}$	1.03	0.15	0.80	67	run_d.gif
0.25	9	$3.3 \times 10^{-5}$	1.00	0.12	0.93	310	run_e.gif