Fig. 4.46 Potential vorticity (solid lines) and streamfunction (dashed lines) of the third layer in a multi-layer model. Light (heavy) cross-hatched and stippled area for the source region with intensity weaker (stronger) than 2 x 10-6 m/s (Huang and Bryan, 1987).

the essential boundary conditions. However, there were many difficulties to overcome before we could see light in the darkness. In the following sections of this introduction we will describe some of these historical puzzles.

Ideal-fluid thermocline Basic equations

The so-called ideal-fluid thermocline equations look very simple. They consist of geostro-phy in the horizontal direction, hydrostatic approximation in the vertical direction, the incompressibility, and density conservation equations:

All equations are linear except for the density conservation equation; however, the nonlin-earity associated with the advection term is very strong. Since this equation system was first formulated by Welander in 1959, the challenge had been to formulate suitable boundary value problems for this equation system and to solve them.

Conservation quantities

By cross-differentiating, subtracting Eqns. (4.257) and (4.258), and applying Eqn. (4.260), one obtains the vorticity equation

Taking the z derivative of Eqn. (4.261) yields uz -Vp + u ■ Vpz = 0 (4.263)

Because of the thermal wind relation, the vertical shear of the horizontal velocity is perpendicular to the horizontal density gradient, so the first term on the left-hand side is reduced to

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