du dt

—2rnv cos e

a

uv tan e a

1 dp p0 dx

Scale

U 2/l

foU

foW

UW/a

U 2/a

AP/poL

Gyres

10-8

10-5

10-10

10-14

10-9

10-5

Gulf Stream

10-5

10-4

10-10

10-13

10-7

For large-scale motions in the atmosphere and oceans, the spherical curvature terms associated with factor 1/a are negligible, and the Coriolis force term associated with the vertical velocity is negligible. Therefore, the horizontal momentum equations are reduced to the following forms du

P-plane approximation

Many practical problems studied in dynamical oceanography are associated with meridional scales much smaller than the radius of the Earth. For such problems, the spherical geometry can be approximately represented by a set of local Cartesian coordinates whose horizontal surface is tangential to the local spherical surface. In this new coordinate system, the Coriolis parameter is approximated by its Taylor expansion around the origin of the local Cartesian coordinates f = 2rn sin e ~ 2rn sin e0 + 2rn cos e0 (y — y0) /a x

where P = 2m cos 0o/a is an approximation of the meridional gradient of the planetary vorticity. Using local Cartesian coordinates (Fig. 2.22), we have f = fo + P y (2.132')

Thus, under the P -plane approximation, the corresponding momentum equations are reduced to du 1 dp f v =--JL + Fx (2.133)

dt po y

Note that the Coriolis parameter is a linear function of y; thus, the meridional gradient of f is equal to P. The introduction of the P-plane greatly reduces the complexity of spherical geometry and the associated dynamics, making many simple analytical studies feasible. On the other hand, the P-plane is an approximation, which is valid under the constraint of Eqn. (2.132); thus, for problems with horizontal scales not satisfying Eqn. (2.132), the original equations in spherical coordinates may have to be used.

f-plane approximation For motions of even smaller scale, the change in the Coriolis parameter can be totally neglected; thus, the basic equations (2.133) and (2.134) are further simplified to du 1 p fov =---f+ Fx (2.135)

dt po y

Table 2.8. Comparison of three sets of coordinates

Spherical coordinates j -plane f 2« sin 0

Introducing the j-plane and f -plane provides very simple and useful tools for the study of the atmosphere and oceans without getting into the algebraic complexity of spherical geometry. However, it is important to remember that some essential elements of the problems appear in slightly different forms in these coordinates (see Table. 2.8). Using the local Cartesian coordinates simplifies the equation, but it may also introduce small distortions of the dynamical picture.

For basin-scale circulations, the time-dependent terms are often much smaller than the Coriolis force terms and this can be argued as follows. Let us introduce the Rossby number, which is defined as the ratio of the time-dependent term and the Coriolis force term

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