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The Lecture Notes in Physics Part 2

Vật lý còn được xem là ngành khoa học cơ bản bởi vì các định luật vật lý chi phối tất cả các ngành khoa học tự nhiên khác. Điều này có nghĩa là những ngành khoa học tự nhiên như sinh học, hóa học, địa lý học. chỉ nghiên cứu từng phần cụ thể của tự nhiên và đều phải tuân thủ các định luật vật lý. | 1 Dynamics of Vortices in Rotating and Stratified Fluids 11 Fig. 1.7 Evolution of collapse-induced vortices in a rotating tank from 18 Although vortices with a velocity profile 1.31b were found to be stable Carton and McWilliams 6 have shown that those with velocity profile 1.30b are linearly unstable to m 2 perturbations. It may well be however that the instability is not able to develop when the decay spin-down associated with the Ekman-layer action is sufficiently fast. In the viscous evolution of stable vortex structures two effects play a simultaneous role the spin-down due to the Ekman layer with a timescale Te Ị 1.32 vO 1 2 and the diffusion of vorticity in radial direction which takes place on a timescale L 2 Td - 1.33 v 12 G.J.F. van Heijst with H the fluid depth and L a measure of the core size of the vortex. For typical values V 10-6 m2s-1 1 s-1 L 10-1 m and H 0.2 m one finds Td 104 s Te 2 102 s . 1.34 Apparently in these laboratory conditions the effects of radial diffusion take place on a very long timescale and can hence be neglected. For a more extensive discussion of the viscous evolution of barotropic vortices the reader is referred to 18 and 20 . 1.1.3 The Ekman Layer For steady small-Ro flow 1.8 reduces to 2k X v -Vp EV2v 1.35 with the last term representing viscous effects. Although E is very small this term may become important when large velocity gradients are present somewhere in the flow domain. This is the case for example in the Ekman boundary layer at the tank bottom where d2 E V2 - E r - O 1 . 1.36 d z2 Apparently the non-dimensional layer thickness is ỖE E1 2 and hence in dimensional form L8e LE1 2 1 2 . 1.37 In a typical rotating tank experiment we have V 10-6 m2s-1 water 1 s-1 and L 0.3 m so that E 10-5 and hence LE1 2 10-3 m 1 mm. The Ekman layer is thus very thin. Since the non-dimensional horizontal velocities in the Ekman layer are O 1 the Ekman layer produces a horizontal volume flux of O E1 2 . In the Ekman layer underneath an .