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Our interest in this test problem is to investigate spurious dispersion effects, and how they relate to the choice of horizontal resolution and the order of the approximation used in the numerical solution. The model is initialized with the zeroeth order solution and integrated forward in time. The initial wave should propogate nearly unchanged to the west. There are errors due to both the numerical approximation and the neglect of the higher-order terms in the initial conditions. The standards of merit for this problem include the phase speed of the numerical wave and the rms error between the expected solution and the actual solution.

Perturbation solutions to the Rossby soliton problem are available to both zeroeth (Boyd, 1980 [2]) and first order (Boyd, 1985 [3]). The problem as posed uses the zeroeth order solution. Following Boyd, we nondimensionalize with H = 40 cm, L=295 km, T = 1.71 days and U=L/T=1.981 m/s. The asymptotic solution is constructed by adding the lowest and first order solutions:

\begin{eqnarray*}u & = & u^{(0)} + u^{(1)} \\
v & = & v^{(0)} + v^{(1)} \\
\eta & = & \eta^{(0)} + \eta^{(1)} ~~,
\end{eqnarray*}


where u is the zonal velocity, v is the meridional velocity and $\eta$ is the surface height anomaly. The superscripts refer to the order of the asymptotic series.

The zero-order solution is

\begin{eqnarray*}u^{(0)} & = & \phi ({{-9+6 y^2}\over{4}}) e^{-y^2/2} \\
v^{(0...
... \\
\eta^{(0)} & = & \phi ({{3+6 y^2}\over{4}}) e^{-y^2/2} ~~,
\end{eqnarray*}


while the first-order solution is given by

\begin{eqnarray*}u^{(1)} & = & c^{(1)}~\phi~{{9}\over{16}}(3+2y^2)e^{(-y^2/2)}
...
...~{{9}\over{16}}(-5+2y^2) e^{(- y^2/2)}
+ \phi^2 \eta^1 (y) ~~,
\end{eqnarray*}


where

\begin{eqnarray*}c & = & c^{(0)}+c^{(1)},~~ c^{(0)} = -{{1}\over{3}},
~~c^{(1)}...
...partial\phi}{\partial\xi} & = & -2B\mbox{tanh}(B\xi) \; \phi ~~.
\end{eqnarray*}


U1, V1, and $\eta^1$ are given by the infinite Hermite series,

\begin{eqnarray*}\left( \begin{array}{c} U^1 \\ V^1 \\ \eta^1 \end{array} \right...
...in{array}{c} u_n \\ v_n \\ \eta_n \end{array} \right)
H_{n}(y),
\end{eqnarray*}


with the coefficients un, vn and $\eta_n$ listed in Table 2. The Hermite polynomials can be computed with the recurrence formula (Abramowitz and Stegun, 1972 [1]):

\begin{eqnarray*}H_0(y) & = & 1, \\
H_1(y) & = & 2y, \\
H_n(y) & = & 2 y H_{n-1}(y) - 2(n-1) H_{n-2}(y), ~~n\ge 2 ~~.
\end{eqnarray*}


Using the zeroeth order solution as initial conditions, we have computed the evolution of the Rossby soliton for a total of 40 time units. The solution for the lowest symmetric mode wave (n=1) is used. During this interval, the soliton propagates westwards across several of its characteristic widths. (Comparable results are obtained for the more complete, first-order asymptotic solution.)


 
Table 2: Hermite series coefficients.
n un vn $\eta_n$
0 1.789276 0 -3.071430
1 0 0 0
2 0.1164146 0 -0.3508384e-1
3 0 -0.6697824e-1 0
4 -0.3266961e-3 0 -0.1861060e-1
5 0 -0.2266569e-2 0
6 -0.1274022e-2 0 -0.2496364e-3
7 0 0.9228703e-4 0
8 0.4762876e-4 0 0.1639537e-4
9 0 -0.1954691e-5 0
10 -0.1120652e-5 0 -0.4410177e-6
11 0 0.2925271e-7 0
12 0.1996333e-7 0 0.8354759e-11
13 0 -0.3332983e-9 0
14 -0.2891698e-9 0 -0.1254222e-9
15 0 0.2916586e-11 0
16 0.3543594e-11 0 0.1573519e-11
17 0 -0.1824357e-13 0
18 -0.3770130e-13 0 -0.1702300e-13
19 0 0.4920950e-16 0
20 0.3547600e-15 0 0.1621976e-15
21 0 0.6302640e-18 0
22 -0.2994113e-17 0 -0.1382304e-17
23 0 -0.1289167e-19 0
24 0.2291658e-19 0 0.1066277e-19
25 0 0.1471189e-21 0
26 -0.1178252e-21 0 -0.1178252e-21
 


next up previous
Next: Example results Up: Rossby Equatorial Soliton Previous: Problem definition
Kate Hedstrom
2000-01-27