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In the energy (density) equation, heat and salt molecular diffusion needs likewise to be superseded by the dispersing effect of unresolved turbulent motions and subgrid-scale processes. Using the same horizontal eddy viscosity A for energy as for momentum is generally adequate, because the larger turbulent motions and subgrid processes act to disperse heat and salt as effectively as momentum. In the vertical, however, the practice is usually to distinguish dispersion of energy from that of momentum by introducing a vertical eddy diffusivity E that differs from the vertical eddy viscosity E. This difference stems from the specific turbulent behavior of each state variable and will be further discussed in Section 14. A widely used method to incorporate subgrid-scale processes in the horizontal eddy viscosity is that proposed by Smagorinsky (1963): u x 2 A = x y + v y 2 + 1 2 u v + y x 2, (4. Because the horizontal eddy viscosity is meant to represent physical processes, it ought to obey certain symmetry properties, notably invariance with respect to rotation of the coordinate system in the horizontal plane. We leave it to the reader to verify that the preceding formulation for A does indeed meet this requirement. Accordingly, let us introduce a scale for every variable, as we already did in a limited way in Section 1. By scale, we mean a dimensional constant of dimensions identical to that of the variable and having a numerical value representative of the values of that same variable. Obviously, scale values do vary with every application, and the values listed in Table 4. Even so, the conclusions drawn from the use of these particular values stand in the vast majority of cases. If doubt arises in a specific situation, the following scale analysis can always be redone. It is generally not required to discriminate between the two horizontal directions, and we assign the same length scale L to both coordinates and the same velocity scale U to both velocity components. Geophysical flows are typically confined to domains that are much wider than they are thick, and the aspect ratio H/L is small. The atmospheric layer that determines our weather is only about 10 km thick, yet cyclones and anticyclones spread over thousands of kilometers. Similarly, ocean currents are generally confined to the upper hundred meters of the water column but extend over tens of kilometers or more, up to the width of the ocean basin. L L H We ought to examine three cases: W/H is much less than, on the order of, or much greater than U/L. Indeed, if W/H U/L, the equation reduces in first approximation to w/z = 0, which implies that w is constant in the vertical; because of a bottom somewhere, that flow must be supplied by lateral convergence (see later section 4. In the first case, the leading balance is two-dimensional, u/x + v/y = 0, which implies that convergence in one horizontal direction must be compensated by divergence in the other horizontal direction. The intermediate case, with W/H on the order of U/L, implies a three-way balance, which is also acceptable. In summary, the vertical-velocity scale must be constrained by W and, by virtue of (4. Let us now consider the x­momentum equation in its Boussinesq and turbulence-averaged form (4. H2 the previous remark immediately shows that the fifth term (W) is always much smaller than the sixth (U) and can be safely neglected1. Because of the fundamental importance of the rotation terms in geophysical fluid dynamics, we can anticipate that the pressure-gradient term (the driving force) will scale as the Coriolis terms, i. The fifth term is likely to remain much smaller than some other terms, such as the pressure gradient, but there may be instances when the f term must be retained. Although horizontal and vertical dissipation due to turbulent and subgrid-scale processes is retained in the equation (its last three terms), it cannot dominate the Coriolis force in geophysical flows, which ought to remain among the dominant terms.

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