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Fundamentals Of Geophysical Fluid Dynamics Part 3

Do chất lỏng là một môi trường liên tục nên các yếu tố chuyển động đều là hàm số liên tục của tọa độ không gian và thời gian. u = u(x,y,z,t); p = p(x,y,z,t); ρ = ρ(x,y,z,t). Trong thủy lực, ta thường xét đến u và p, còn ρ coi như không đổi vì ta coi chất lỏng như không nén được. | 3.2 Vortex Movement 87 Fig. 3.7. Left Co-rotating trajectories for two cyclonic point vortices of unequal circulation strength. Right Trajectories for a cyclonic and anticyclonic pair of point vortices of unequal circulation strength. Vortex 1 has stronger circulation magnitude than vortex 2. X denotes the center of rotation for the trajectories and d1 and d2 are the distances from it to the two vortices. The vortex separation is d di d2. instability in the sense that infinitesimal perturbations will continue to grow to finite displacements . In the limit of vanishing vortex separation the vortex street becomes a vortex sheet representing a flow with a velocity discontinuity across the line i.e. there is infinite horizontal shear and vorticity at the sheet. Thus such a shear flow is unstable at vanishingly small perturbation length scales due to the infinitesimal width of the shear layer . This is an example of barotropic instability Sec. 3.3 that sometimes is called Kelvin-Helmholtz instability. A linear 88 Barotropic and Vortex Dynamics C V 2 d Q - C V - C a bounded domain b unbounded domain Fig. 3.8. a Trajectory of a cyclonic vortex with circulation I C located a distance d from a straight free-slip boundary and b its equivalent image vortex system in an unbounded domain that has zero normal velocity at the location of the virtual boundary. The vortex movement is poleward parallel to the boundary at a speed V C 4 d. normal-mode instability analysis for a vortex sheet is presented in Sec. 3.3.3. Example 5 A Karman vortex street named after Theodore von Karman This is a double vortex street of vortices of equal strengths opposite parities and staggered positions Fig. 3.10 . Each of the vortices moves steadily along its own row with speed U. This configuration can be shown to be stable to small perturbations if cosh b a a 2 with a the along-line vortex separation and b the between-row separation. Such a configuration often arises from flow past an obstacle e.g. a