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Tham khảo tài liệu 'fundamentals of geophysical fluid dynamics part 6', khoa học tự nhiên, vật lý phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 174 Baroclinic and Jet Dynamics or for continuous height modes 1 iH H Jo dz Gp z Gq z G-.q 5.32 with ỗp q 1 if p q and ỗp q 0 if p q i.e. Ỗ is a discrete delta function . This is a mathematically desirable property for a set of vertical basis functions because it assures that the inverse transformation for 5.30 is well defined as or sN Hh ýn Gm n 5.33 m H dz-f z Gm z . 5.34 m The physical motivation for making this transformation comes from measurements of large-scale atmospheric and oceanic flows that show that most of the energy is associated with only a few of the gravest vertical modes i.e. ones with the smallest m values and correspondingly largest vertical scales . So it is more efficient to analyze the behavior of I m x y t for a few m values than of x y z t at all z values with significant energy. A more theoretical motivation is that the vertical modes can be chosen as explained in the rest of this section so each mode has a independent i.e. decoupled from other modes linear dynamics analogous to a single fluid layer barotropic or shallow-water . In general a full dynamical decoupling between the vertical modes cannot be achieved but it can be done for some important behaviors e.g. the Rossby wave propagation in Sec. 5.2.1. For specificity consider the 2-layer quasigeostrophic equations N 2 to illustrate how the Gm are calculated. The two vertical modes are referred to as barotropic m 0 and baroclinic m 1 . For a N-layer model each mode with m 1 is referred to as the mth baroclinic mode. To achieve the linear-dynamical decoupling between layers it is sufficient to diagonalize the relationship between the potential vorticity and streamfunction. That is determine the 2x2 matrix Gm n such that each modal potential vorticity contribution apart from the planetary vorticity term i.e. qQGm py H n 1Hn qQG n py Gm n 5.1 Layered Hydrostatic Model 175 depends only on its own modal streamfunction field m H n 1 HnlnGm n and not on any other I m0 with m m. This is .
