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Let H be the mean depth of the fluid, which is invariant with time, and h' be the perturbation height, so that h H  h ' . Furthermore, with a flat surface at the bottom we assume wbottom 0 . We then have wtop dh dt § wu wv · § wu wv · H ¨  ¸  h '¨  ¸ . © wx wy ¹ © wx wy ¹ This equation can be written as dh dt wh wh wh u v wt wx wy § wu wv · § wu wv · H ¨  ¸  h '¨  ¸ . 64) After multiplying by g and defining the geopotential (I ) as I gh , we obtain the continuity equation: § wu wv · wI ' w w  (uI ')  (vI ')  I ¨  ¸ 0 .

0 Evaluate du / dx at x S / 4 using (a) forward differencing, (b) centered differencing, (c) backward differencing, and (d) the second-order finite-difference analog. Analytically, u ( x) represents sin x . Which of the finite-difference formulas represents the best approximation to du / dx x S /4 ? 2. 1, evaluate u ( x) at x 7S /12 , given that the secondorder derivative of u ( x) at x S / 2 is exactly equal to its analytic value. 3. Consider Poisson’s equation ’ 2\ G , where \ \ ( x, y ) and G G ( x, y ) .

The rate of amplification is small if 't is small, and the scheme is weakly unstable. The AdamsBashforth scheme is therefore suitable for short periods of integration with a small time step such as in a squall or flow past a mountain. 5 Implicit Schemes There are two types of implicit schemes, one is the fully implicit scheme discussed below and the other is a semi-implicit scheme discussed in Sec. 4. In the time-differencing schemes discussed so far, the time step should satisfy the CFL condition.

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