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
Baroclinic eddies linear baroclinic instability Charney's model JULE CHARNEY was one of the dominant figures in atmospheric science in the three decades following World War II.Much of the change in meteorology from an art to a science is due to his scientific vision and his thorough commitment to people and programs in this field." -by Norman Phillips Jule Gregory Charney 0年年卡0年9+年年年44。。4年0年004e4 1917-1981 授课教师：张洋
授课教师：张洋 7 Baroclinic eddies - linear baroclinic instability n Charney’s model “ JULE CHARNEY was one of the dominant figures in atmospheric science in the three decades following World War II. Much of the change in meteorology from an art to a science is due to his scientific vision and his thorough commitment to people and programs in this field.” -- by Norman Phillips
Baroclinic eddies Review From linear to nonlinear Linear Reduce the zonal flow Basic flow process temperature gradient; or stablize the lower level Pre-existing flow stratification;enhance the (without zonal variation westerly jet and baroclinic unstable) Small Nonlinear perturbation interactions Perturbations Equilibrated states between grow with time Eddy-mean interactions the adjusted zonal flow (finite amplitude pert.) (Adjust the zonal flow) and baroclinic eddies 授课教师：张洋 8
授课教师：张洋 8 Baroclinic eddies n From linear to nonlinear Basic flow or Pre-existing flow (without zonal variation and baroclinic unstable) Small perturbation Perturbations grow with time (finite amplitude pert.) Eddy-mean interactions (Adjust the zonal flow) Equilibrated states between the adjusted zonal flow and baroclinic eddies Linear process Nonlinear interactions Reduce the zonal flow temperature gradient; stablize the lower level stratification; enhance the westerly jet Review
Baroclinic eddies > Review From linear to nonlinear spinup of Numerical results from a QG model eddies, (Zhang,2009) 400 Basic flow E-P flux or Nonlinear Equilibrium Pre-existing flow adjustment (without zonal variation 20 and baroclinic unstable) 200 x)3d3 Small perturbation 0 100 200 300 400 500 600 day Perturbations Equilibrated states between grow with time Eddy-mean interactions the adjusted zonal flow (finite amplitude pert.) (Adjust the zonal flow) and baroclinic eddies 授课教师：张洋 9
0 100 200 300 400 500 600 40 50 60 70 80 90 100 Time series of KE for SD run MKE ( × 105 J/m2 ) 0 100 200 300 400 500 600 0 5 10 15 20 25 30 EKE ( × 105 J/m2 ) 0 100 200 300 400 500 600 200 400 day Time series of PE for SD run MPE ( × 105 J/m2 ) 0 100 200 300 400 500 600 0 20 40 EPE ( × 105 J/m2 ) Numerical results from a QG model (Zhang, 2009) 授课教师：张洋 9 Baroclinic eddies n From linear to nonlinear Basic flow or Pre-existing flow (without zonal variation and baroclinic unstable) Small perturbation Perturbations grow with time (finite amplitude pert.) Eddy-mean interactions (Adjust the zonal flow) Equilibrated states between the adjusted zonal flow and baroclinic eddies Equilibrium Nonlinear adjustment spinup of eddies E-P flux 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 The energy cycle 授课教师：张洋10
授课教师：张洋 10 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 The energy cycle
The Ferrel Cell -Review< The balance equations [T] b Tropopause a([u*v*]) by +f+E到 a0。_a[0*v] ∂0s. y 20 wp by 00s l ap + ∂(e*v*]) Po R/Cp OY by Cp Boundary layer Ground Subtropics Latitude Subpolar 11
11 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] Observations - Eddy fields 15 \ 授教：洋 ! Heat flux: http://www.adultpdf.com Created by Image To PDF trial version, to remove this mark, please register this software. Transient components: strongest at 40-50 degree, with obvious seasonal variation in N.H.. Stationary components: strongest at mid-latitude in N.H., whose directions are reversed from winter to summer. Zonal mean flow: centered in the tropics, whose directions are reversed from winter to summer. Wednesday, October 30, 2013 ! @✓s @p ⇠ @[✓⇤v⇤] @y 0 Review