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The Lecture Notes in Physics Part 7

Mặc dầu đối tượng của vật lý được trải dài trên một khoảng rộng, từ thang vi mô đến thang vĩ mô, song chỉ có một vài lý thuyết vật lý chính, bao quát được hết các hệ thống trong đó. Mỗi thuyết, về cơ bản, đều mô tả đúng trên một phạm vi nhất định. | 116 V. Zeitlin dt u J 0 J-3 -1 0 da u J v 0 4.32 t_3 The eigenvalues of the matrix in the l.h.s. of 4.32 are ịL J 2 and the corresponding left eigenvectors are Ợ J-2 . Hence Riemann invariants are 1 w u 2 J 2 and we have dt w I da w v. 4.33 Expressions of original variables in terms of w are 3 1 16 w w u 2 w w- J w w 2 0 -J . 4.34 In terms of the derivatives of the Riemann invariants r da w we get 9tr fr dar dw r r dw-r-r dav Q a - J 4.35 which may be rewritten using Lagrangian derivatives along the characteristics dt dt ụ- da as dr du. 1 dt w r r wr-r Q a - J 4.36 Wave breaking and shock formation correspond to r rc in finite time. In terms of new variables R eẢr with À 128 log w - w-1 4.35 may be rewritten as dR -e-ẰỊẼ R eẰ Q a - J 4.37 where dw ả w - w- 2 0. The qualitative analysis of these generalized Ricatti equations shows that if initial relative vorticity Q - J da v is sufficiently negative anti-cyclonic rotation does not stop wave breaking which is taking place for any initial conditions. However if the relative vorticity is positive cyclonic case as well as the derivatives of the Riemann invariants at the initial moment there is no breaking. An example of wave breaking due to the geostrophic adjustment of the unbalanced jet is presented in Fig. 4.2. 4 Lagrangian Dynamics of Fronts Vortices and Waves Vjet Vmax 0 L 2L 0 -2L -L 117 Fig. 4.2 Wave breaking and shock formation right panel during adjustment of the unbalanced jet left panel top to bottom consecutive profiles of the free surface with time measured in f-1 units . Length is measured in deformation radius units L Rd f 4.2.1.6 Trapped Waves in 1.5d RSW Pulsating Density Fronts The above-established supra-inertiality of the spectrum of the small perturbations around a balanced 1.5d RSW front means the absence of trapped waves and hence the attainability of the adjusted state by evacuating the excess of energy via inertiagravity wave emission eventually with shock formation . There exist however the .