南京大学：《大气环流》课程教学资源（课件讲稿）第四章 中纬度的经向环流系统（2/4）

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 de￾creases upwards and equatorwards, and the associated horizontal pressure gradient is balanced by the Coriolis force. Parcel ‘A’ is heavier than ‘C’, and so statically sta￾ble, 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 g
z 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 de￾creases upwards and equatorwards, and the associated horizontal pressure gradient is balanced by the Coriolis force. Parcel ‘A’ is heavier than ‘C’, and so statically sta￾ble, 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 g
z 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 de￾creases upwards and equatorwards, and the associated horizontal pressure gradient is balanced by the Coriolis force. Parcel ‘A’ is heavier than ‘C’, and so statically sta￾ble, 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