Observations Summary: Review ■Zonal-mean flow: Ferrel Cell:an indirect cell centered at 40-60 degree,with strong seasonal variation in N.H. Westerly jet:surface westerlies centered at 40-60 degree ■ Eddies:transient eddies are dominant with stationary eddies only obvious in N.H. ■Kinetic energy ■Momentum flux ■Heat flux 授课教师：张洋3
3 Observations \ 授课教师：张洋 n Summary: n Zonal-mean flow: n Ferrel Cell: an indirect cell centered at 40-60 degree, with strong seasonal variation in N.H. n Westerly jet: surface westerlies centered at 40-60 degree n Eddies: transient eddies are dominant with stationary eddies only obvious in N.H. n Kinetic energy n Momentum flux n Heat flux Review
The Ferrel Cell eddy-zonal flow interaction(T)Review The simplified equations: Momentum equation: u= Ot ∂([u*wD+fu+[F] 8y Continuity equation: y+=0 ay *Op Thermodynamic equation: ∂) 00s +0加 ∂(0v* (台 2 R/Cp Q] 8t 8y Cp ()-(,+(品)+()+品 Under the quasi-geostrophic approximation (Ro<1) 授课教师：张洋4
@[v] @y + @[!] @p = 0 Under the quasi-geostrophic approximation (Ro ⌧ 1) @[u] @t = @([u⇤v⇤]) @y + f[v]+[Fx] @[✓] @t + [!] @✓s @p = @([✓⇤v⇤]) @y + ✓po p ◆R/cp [Q] cp 授课教师：张洋 4 The Ferrel Cell eddy-zonal flow interaction (I) ✓ d dt◆ p = ✓ @ @t ◆ p + u ✓ @ @x ◆ p + v ✓ @ @y ◆ p + ! @ @p n Momentum equation: n Continuity equation: n Thermodynamic equation: n The simplified equations: Review
The Ferrel Cell The balance equations: Review Tropopause fu Ofu*v*] ∠0 8y Ferrel Cell: eddy-driven, 70t auoD+f网+F到 by indirect cell a0。_a0*w ∠0 00s Oy wOp a01>0 by 00s a(0*u*]) R/CP IO + Po ∂t ap by Cp Boundary fv ~rusurf>0 layer Ground Subtropics Latitude Subpolar 5
! @✓s @p ⇠ @[✓⇤v⇤] @y 0 fv ⇠ @[u⇤v⇤] @y < 0 5 n The balance equations: The Ferrel Cell @[✓] @t + [!] @✓s @p = @([✓⇤v⇤]) @y + ✓po p ◆R/cp [Q] cp @[u] @t = @([u⇤v⇤]) @y + f[v]+[Fx] Ferrel Cell: eddy-driven, indirect cell Review
及图乐 Outline Observations The Ferrel Cell ■ Baroclinic eddies Review:baroclinic instability and baroclinic eddy life cycle Eddy-mean flow interaction,E-P flux ■ Transformed Eulerian Mean equations Eddy-driven jet Energy cycle 授课教师：张洋6
授课教师：张洋 6 Outline n Observations n The Ferrel Cell n Baroclinic eddies n Review: baroclinic instability and baroclinic eddy life cycle n Eddy-mean flow interaction, E-P flux n Transformed Eulerian Mean equations n Eddy-driven jet n Energy cycle
Baroclinic eddies ■Instability: -baroclinic instability-Review Phenomenon:Given a basic flow with perturbations at the initial moment,if the perturbation grows with time,the basic flow is always taken unstable. Mathematics:P&Aet,a>0 (相对于波动解：P Aeiwt) ■Energy': 能量源→ 扰动动能！ Linear Instability:the instability that arises in a linear system. 授课教师：张洋7
授课教师：张洋 7 Baroclinic eddies - baroclinic instability n Instability: n Phenomenon: Given a basic flow with perturbations at the initial moment, if the perturbation grows with time, the basic flow is always taken unstable. n Linear Instability: the instability that arises in a linear system. n Mathematics: P _ Ae↵t , 9 ↵ > 0 P _ Aei!t (相对于波动解: ) n Energy: 能量源 扰动动能 Review
Baroclinic eddies baroclinic instability Baroclinic Instability-"is an instability that arises in rotating, stratified fluids that are subject to a horizontal temperature gradient". 授课教师：张洋8
授课教师：张洋 8 Baroclinic eddies - baroclinic instability n Baroclinic Instability - “is an instability that arises in rotating, stratified fluids that are subject to a horizontal temperature gradient
Baroclinic eddies baroclinic instability Baroclinic Instability-"is an instability that arises in rotating, stratified fluids that are subject to a horizontal temperature gradient". From A to B:negative buoyant density increasing If A and C are interchanged: low density high temperature PE=∫pgdE density decreasing △PE g(PAZA+PCZC-PCZA-PAZC) = g(zA-ZC)(PA-PC) igh density =g△p△z low temperature △PE = +Ltan g(Lau Ltana 8z warm cold Assume small a andφ APE-9a (1-9 授课教师：张洋 9
P E = g(L@⇢ @y + Ltan ↵ @⇢ @z )Ltan ↵ Assume small ↵ and P E = gL2 @⇢ @y ↵ ✓ 1 ↵ ◆ 授课教师：张洋 9 Baroclinic eddies - baroclinic instability n Baroclinic Instability - “is an instability that arises in rotating, stratified fluids that are subject to a horizontal temperature gradient”. i i i i i i i i ) ) ⇥ A B C high density low density low temperature high temperature density increasing density decreasing Fig. 6.9 A steady basic state giving rise to baroclinic instability. Potential density decreases upwards and equatorwards, and the associated horizontal pressure gradient is balanced by the Coriolis force. Parcel ‘A’ is heavier than ‘C’, and so statically stable, but it is lighter than ‘B’. Hence, if ‘A’ and ‘B’ are interchanged there is a release of potential energy. From Vallis (2006) From Vallis (2006) warm cold From A to B: negative buoyant If A and C are interchanged: y fu b z 0 b g y b z u v w 0 S N X X X X X PE gdz PE g z z z z gz z g z A A B B A B B A A B A B 1 2 y L gL z L y PE g z g L P E = g(⇢AzA + ⇢C zC ⇢C zA ⇢AzC ) = g(zA zC )(⇢A ⇢C ) = g⇢z L
Baroclinic eddies baroclinic instability Baroclinic Instability-"is an instability that arises in rotating, stratified fluids that are subject to a horizontal temperature gradient". From A to B:negative buoyant density increasing If A and C are interchanged: low density high temperature PE=∫pgdk density decreasing PE) high density low temperature a is called mixing slope. warm cold If a <o,a loss of potential energy. If a= 20,△E is strongest. 授课教师：张洋10
If ↵ = 1 2 , P E is strongest. ↵ is called mixing slope. 授课教师：张洋 10 Baroclinic eddies - baroclinic instability n Baroclinic Instability - “is an instability that arises in rotating, stratified fluids that are subject to a horizontal temperature gradient”. i i i i i i i i ) ) ⇥ A B C high density low density low temperature high temperature density increasing density decreasing Fig. 6.9 A steady basic state giving rise to baroclinic instability. Potential density decreases upwards and equatorwards, and the associated horizontal pressure gradient is balanced by the Coriolis force. Parcel ‘A’ is heavier than ‘C’, and so statically stable, but it is lighter than ‘B’. Hence, if ‘A’ and ‘B’ are interchanged there is a release of potential energy. From Vallis (2006) From Vallis (2006) warm cold From A to B: negative buoyant If A and C are interchanged: y fu b z 0 b g y b z u v w 0 S N X X X X X PE gdz PE g z z z z gz z g z A A B B A B B A A B A B 1 2 y L gL z L y PE g z g L L P E = gL2 @⇢ @y ↵ ✓ 1 ↵ ◆ If ↵ < , a loss of potential energy.
Baroclinic eddies baroclinic instability Baroclinic Instability-"is an instability that arises in rofating, stratified fluids that are subject to a horizontal temperature gradient. density increasing Energetics: p→k low density density decreasing Mathematics: nigh density low temperature Linear Baroclinic Instability Linear baroclinic system- ■Eady's model(1949) Charney's model (1947) 授课教师：张洋11
授课教师：张洋 11 Baroclinic eddies - baroclinic instability n Baroclinic Instability - “is an instability that arises in rotating, stratified fluids that are subject to a horizontal temperature gradient”. i i i i i i i i ) ) ⇥ A B C high density low density low temperature high temperature density increasing density decreasing Fig. 6.9 A steady basic state giving rise to baroclinic instability. Potential density decreases upwards and equatorwards, and the associated horizontal pressure gradient is balanced by the Coriolis force. Parcel ‘A’ is heavier than ‘C’, and so statically stable, but it is lighter than ‘B’. Hence, if ‘A’ and ‘B’ are interchanged there is a release of potential energy. From Vallis (2006) From Vallis (2006) n Mathematics: n Energetics: n Linear Baroclinic Instability n Linear baroclinic system PE KE n Eady’s model (1949) n Charney’s model (1947)
Baroclinic eddies linear baroclinic instability - Eady's model (1949) Charney's model (1947) JOURNAL OF METEOROLOGY OCTOBER M7 Long Waves and Cyclone Waves THE DYNAMICS OF LONG WAVES IN A BAROCLINIC By E.T.EADY,Imperial College of Science,London WESTERLY CURRENT By J.G.Charney Urhersite ef Cafletria at Lo8 Awode CManucrips recriveeDecerber14) Abstract 时 hown how,by proceslo tn "natural ctiothcompo h the ywe.rhcrTra r ega监h。 你o6o优t原”o The Bo h hepreent papet principles alages of develop of wave-cyones einto account any or all of the te originally omitted.In the present instance long waves.For reasons of space and lability both the argument and the mathe- nent,by comparison cathe which radia ics have been heavily compressed processes (or rather their differential ef 对i tteatment of several of are slow.For a first approximation there points raised will be given in subscquent we cousider the motion as ndiabatic.Also are concerned with the motion of deep la 4 and for a first approximation we neglect 1.Introductien The Equations of Motion effects of internal friction ("turbulence") The large scale weather phenomena in the era wing to the complexity (and non-liearity) skin friction.A rough 121 for V the energy dissipated in frio layer is usually much less tlan the ene s governing atmospheric motion it is supply to the growing disturbance and th of the d rable to simplify these by the omission of eyeloncs.The fint aignificant step toward a solution hose ters which do not nake probably,in most cases,the mjor sor 13.The wne energy os.The present paper wan taken in 1916 by V.Bjerines [8.p.785) b de A this discoery,the syoptic studies of .Bjerkene asd Eady,E.1949.Tellus,1,33-52
授课教师：张洋 12 Baroclinic eddies - linear baroclinic instability n Eady’s model (1949) n Charney’s model (1947) Eady, E. 1949. Tellus, 1, 33-52