及图乐 Outline Review 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2
授课教师：张洋 2 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 Review
Baroclinic eddies baroclinic instability -Review Baroclinic Instability-"is an instability that arises in rofating, stratified fluids that are subject to a horizontal temperature gradient". density increasing Energetics: 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3
授课教师：张洋 3 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 Energetics: PE KE n Mathematics: n Linear Baroclinic Instability n Linear baroclinic system n Eady’s model (1949) n Charney’s model (1947) Review
Baroclinic eddies linear baroclinic instability -Review Eady's model (1949) Charney's model (1947) a)The basic zonal flow has uniform vertical shear, U(Z)=AZ,A is a constant The most distinguished difference with Eady's b)The fluid is uniformly stratified, N2 is a constant. model is that beta effect is considered. c)Two rigid lids at the top and bottom, flat horizontal surface,that is w=0 at Z=0 and H. d)The motion is on the f-plane,that is B=0 授课教师：张洋4
d) The motion is on the f -plane, that is = 0 授课教师：张洋 4 Baroclinic eddies - linear baroclinic instability n Eady’s model (1949) n Charney’s model (1947) N2 is a constant. b) The fluid is uniformly stratified, ! = 0 at Z = 0 and H. c) Two rigid lids at the top and bottom, flat horizontal surface, that is The most distinguished difference with Eady’s model is that beta effect is considered. a) The basic zonal flow has uniform vertical shear, Uo(Z) = ⇤Z, ⇤ is a constant Review
Baroclinic eddies linear baroclinic instability Review Small amplitude Variable Basic state Perturbation assumption 小扰动 u(x,t)=U(z)+u(x,t) Linear baroclinic system: u'(x,t)≤U(z) Eady model Linearized PV equation(q=PV): Charney model (品+品 ∂b∂ =0 8x dy Normal mode assumption 02b g- Ps ou 标准波形 0x2+ ∂y2 Ps Oz 82 4二 Obtain the solutions,e.g. 0u2+y+ Ps ov ps∂z instability conditions 标准波形法， 带入方程和边界条件： growth rate most unstable mode (x,t)=Aei(k.x-wt) Find the conditions for non-trivial solutions and Ci >0 授课教师：张洋 5
u(x, t) = U(z) + u0 (x, t) u0 (x, t) ⌧ U(z) ✓ @ @t + U @ @x ◆ q0 + @ @x @q¯ @y = 0 q0 = @2 0 @x2 + @2 0 @y2 + f 2 o ⇢s @ @z ✓ ⇢s N2 @ 0 @z ◆ q¯ = @2 ¯ @y2 + y + f 2 o ⇢s @ @z ✓ ⇢s N2 @ ¯ @z ◆ 授课教师：张洋 5 Baroclinic eddies - linear baroclinic instability Linear baroclinic system: Eady model Charney model Small amplitude assumption ⼩扰动 Normal mode assumption 标准波形 Obtain the solutions, e.g. instability conditions growth rate most unstable mode Variable = Basic state + Perturbation Linearized PV equation (q=PV): 0 (x, t) = Aei(k·x!t) 标准波形法， 带⼊⽅程和边界条件： Find the conditions for non-trivial solutions and Ci >0 Review
Baroclinic eddies linear baroclinic instability Review Conclusions: Necessary condition for baroclinic instability:PV gradient changes sign in the interior or boundaries(Charney-stern theory),according to which the midlatitude atmosphere is baroclinic unstable.Different models.i.e.Eady and Charney models have more rigorous conditions. rowth rate:o=kc0.3 in both Eady and Charney models! Most unstable mode: kmax Ld= ()x BN Eady Charney 授课教师：张洋6
k1 max / Ld = ✓NH fo ◆ 授课教师：张洋 6 Baroclinic eddies - linear baroclinic instability n Conclusions: Necessary condition for baroclinic instability: PV gradient changes sign in the interior or boundaries (Charney-stern theory), according to which the midlatitude atmosphere is baroclinic unstable. Different models. i.e. Eady and Charney models have more rigorous conditions. = kci ⇡ 0.3 ⇤fo N Growth rate: in both Eady and Charney models! k1 max / ⇤ fo N Most unstable mode: Eady Charney Review