chap 4 4 2019 - tcs @ njutcs.nju.edu.cn/yzhang/chap_4_4_2019.pdf · 2019. 11. 13. · 2...

第四章：

中纬度的经向环流系统(IV) - Ferrel cell, baroclinic eddies and the westerly jet

授课教师：张洋2019. 11. 13

!2授课教师：张洋

Baroclinic eddies

! 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;

stabilize the lower level stratification; enhance the

westerly jet

Review

[!] = � @

@y

✓[✓⇤v⇤]

@✓s/@p

◆[v] =

1

f

@

@y([u⇤v⇤])

F ⌘ �[u⇤v⇤] j+ f[v⇤✓⇤]

@✓s/@pk

r · F = 0

f [v]�@([u⇤v⇤])

@y+ [Fx] = 0

[!]@✓s@p

+@([✓⇤v⇤])

@y�✓pop

◆R/cp [Q]

cp= 0


Baroclinic eddies - E-P flux

! In a QG, steady, adiabatic and frictionless flow:

! In a QG, steady flow:

The meridional overturning flow, in addition to the eddy forcing, has to balance the external forcing.

Review


E-P flux, TEM and Residual Circulation - Summary

! E-P flux: F ⌘ �[u⇤v⇤] j+ f[v⇤✓⇤]

@✓s/@pk

! TEM equations:

! Residual mean circulations:

! In a steady, adiabatic and frictionless flow:

@[✓]

@t= � ˜[!]

@✓s@p

+

✓pop

◆R/cp [Q]

cp

@[u]

@t= f ˜[v] +r · F + [Fx] ,

˜[!] = [!] +@

@y

✓[v⇤✓⇤]

@✓s/@p

◆˜[v] = [v]� @

@p

✓[v⇤✓⇤]

@✓s/@p

◆

,

r · F = 0[!] = � @

@y

✓[✓⇤v⇤]

@✓s/@p

◆[v] =

1

f

@

@y([u⇤v⇤])

Review


Baroclinic eddies - TEM

Case 2: Observed circulation

The Ferrel cell in the isentropic coordinate is essentially reflect the

Residual Mean Circulation. (Fig.11.4, Vallis, 2006)

! In isentropic coordinate

D

Dt=

@

@t+ u ·r✓ +

D✓

Dt

@

@✓

=@

@t+ u ·r✓ + ✓

@

@✓

zero for adiabatic flow

= m +[v⇤✓⇤]

@✓s/@p

Review


Baroclinic eddies

! 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

E-P flux, residual mean circulation,

Transformed Eulerian mean equations.

Review

v⇣ = v(@v

@x� @u

@y)

= � @

@yuv +

1

2

@

@x(v2 � u2)q0 =

@2 0

@x2+@2 0

@y2+ f2

o@

@p

✓1

s

@ 0

@p

◆

q =@2

@y2+ �y + f2

o@

@p

✓1

s

@

@p

◆v0q0 = v0⇣ 0 +

fo@✓s/@p

v0@✓0

@p

v@✓

@p=

@

@pv✓ � ✓

@v

@p

(u, v) =

✓�@ @y

,@

@x

◆✓ = �fo

@

@p

=p

R

✓pop

◆R/cp

s = � 1

@✓s@p

v@✓

@p=

@

@pv✓ +

1

2fo

@

@x✓2


Baroclinic eddies - E-P flux: a second view

! E-P flux and the Quasi-geostrophic potential vorticity

F ⌘ �[u⇤v⇤] j+ f[v⇤✓⇤]

@✓s/@pk

From the definition of QG potential vorticity:

⇣ 0 fo@

@p

✓✓0

@✓s/@p

◆

PV flux:

thermal wind relation for

meridional wind

+✓

fo

@✓

@x

Note: ‘ denotes small perturbation

Review

[v⇤q⇤] =@

@y[u⇤v⇤] + fo

@

@p

[v⇤✓⇤]

@✓s/@p

= r · F


! E-P flux and the Quasi-geostrophic potential vorticity

F ⌘ �[u⇤v⇤] j+ f[v⇤✓⇤]

@✓s/@pk

From the definition of QG potential vorticity:

v0q0 = v0⇣ 0 +fo

@✓s/@pv0@✓0

@p

=1

2

@

@x

✓v02 � u02 +

1

✓02

@✓s/@p

◆

+@

@yu0v0

+fo@

@p

v0✓0

@✓s/@p

Zonally averaged PV flux by eddies:


#1

Review

✓@

@t+ U

@

@x

◆q0 +

@ 0

@x

@q

@y= 0

1

2

@

@t[q02] + [v0q0]

@q

@y= 0

A =[q02]

2@q/@y

@A@t

+r · F = 0


! E-P flux and the Eliassen-Palm relation

F ⌘ �[u⇤v⇤] j+ f[v⇤✓⇤]

@✓s/@pk

Linearized PV equation (q=PV):

Multiplying by q’ and zonally average:

Define wave activity density:

Eliassen-Palm relation


#2

Review

✓@

@t+ U

@

@x

◆q0 +

@ 0

@x

@q

@y= 0

Assume U is fixed, and@q

@y= �

✓@

@t+ U

@

@x

◆r2 0 + f2

o@

@p

✓1

s

@ 0

@p

◆�+ �

@ 0

@x= 0

0 = Re ei(kx+ly+mp�!t)

! = Uk � �k

K2

K2 = k2 + l2 +m2f2o /s

cgy =2�kl

K4cgp =

2�kmf2o /s

K4


! E-P flux and the Rossby waves

F ⌘ �[u⇤v⇤] j+ f[v⇤✓⇤]

@✓s/@pk


Exist solutions of the form

Dispersion relation of Rossby waves:

with group velocity

Baroclinic eddies - E-P flux: a second view Review

✓@

@t+ U

@

@x

◆q0 +

@ 0

@x

@q

@y= 0

K2 = k2 + l2 +m2f2o /s

✓ = �Reimfo , q = �ReK2

A =[q02]

2�=

K4

4�| 2|

~F = ~cgA

�[u0v0] =1

2kl| 2| = cgyA

fo[v0✓0]

@✓s/@p=

f2o

2skm| 2| = cgzA


! E-P flux and the Rossby waves

F ⌘ �[u⇤v⇤] j+ f[v⇤✓⇤]

@✓s/@pk


with group velocity

Wave activity density:


#3

cgy =2�kl

K4cgp =

2�kmf2o /s

K4

u = �Reil , v = Reik

Review

@A@t

+r · (A ~cg) = 0


E-P flux, TEM and Residual Circulation - Summary

! E-P flux: F ⌘ �[u⇤v⇤] j+ f[v⇤✓⇤]

@✓s/@pk

! TEM equations:

#1 [v⇤q⇤] =@

@y[u⇤v⇤] + fo

@

@p

[v⇤✓⇤]

@✓s/@p

= r · F

~F = ~cgA#3

! Residual mean circulations:

#2 @A@t

+r · F = 0

! In a steady, adiabatic and frictionless flow:

@[✓]

@t= � ˜[!]

@✓s@p

+

✓pop

◆R/cp [Q]

cp

@[u]

@t= f ˜[v] +r · F + [Fx] ,

˜[!] = [!] +@

@y

✓[v⇤✓⇤]

@✓s/@p

◆˜[v] = [v]� @

@p

✓[v⇤✓⇤]

@✓s/@p

◆

,

r · F = 0[!] = � @

@y

✓[✓⇤v⇤]

@✓s/@p

◆[v] =

1

f

@

@y([u⇤v⇤])


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


~F = ~cgA

@[✓]

@t= � ˜[!]

@✓s@p

+

✓pop

◆R/cp [Q]

cp

@[u]

@t= f ˜[v] +r · F + [Fx]

F ⌘ �[u⇤v⇤] j+ f[v⇤✓⇤]

@✓s/@pk

Baroclinic eddy life cycle - An E-P flux view

Initial state:

Jet

temperature

Numerical results from Simmons and Hoskins,

1978, JAS


Baroclinic eddies - baroclinic eddy life cycle

! Eddies’ development

(Thorncroft et al, 1993, Q.J.R.)

Small amplitude perturbations

Finite amplitude perturbations

Wave breaking


~F = ~cgA

F ⌘ �[u⇤v⇤] j+ f[v⇤✓⇤]

@✓s/@pk



1978, JASEddies: generate at lower level, propagate upwards and away from the

eddy source region

ii

ii

ii

ii

Latitude Latitude

(a) (b)

Fig. 12.17 The Eliassen–Palm flux in an idealized primitive equation of the atmo-sphere. (a) The EP flux (arrows) and its divergence (contours, with intervals of2 m s�1/day). The solid contours denote flux divergence, a positive PV flux, andeastward flow acceleration; the dashed contours denote flux convergence and de-celeration. (b) The EP flux (arrows) and the time and zonally averaged zonal wind(contours). See the appendix for details of plotting EP fluxes.

From Vallis (2006)

From Vallis (2006)

Jet

r · F


~F = ~cgA

F ⌘ �[u⇤v⇤] j+ f[v⇤✓⇤]

@✓s/@pk

E-P flux - In the equilibrium state

Wave energies: propagate upwards and away from the center of

the jet

Numerical results from idealized model with pure midlatitude jet

(Vallis, 2006)

ii

ii

ii

ii

Latitude Latitude

(a) (b)


From Vallis (2006)

From Vallis (2006)

r · F


~F = ~cgA

@[u]

@t= f ˜[v] +r · F + [Fx]

F ⌘ �[u⇤v⇤] j+ f[v⇤✓⇤]

@✓s/@pk

E-P flux - The westerly jet


the jet

Divergence

Convergence

Accelerating the lower jet decelerating the upper jet

reduce the vertical shear of U

In the vertical direction:

1

g

Z ps

0dp

@

@t< [u] >= � @

@y< [u⇤v⇤] > �r[usurf]


~F = ~cgA

@[u]

@t= f ˜[v] +r · F + [Fx]

F ⌘ �[u⇤v⇤] j+ f[v⇤✓⇤]

@✓s/@pk

E-P flux - The westerly jet


the jet

Integrate vertically:

< > means vertical average

ii

ii

ii

ii

Latitude Latitude

(a) (b)


From Vallis (2006)

From Vallis (2006)

r · F

@

@t< [u] >= � @

@y< [u⇤v⇤] > �r[usurf]

r[usurf] ⇠ � @

@y< [u⇤v⇤] >


~F = ~cgA

F ⌘ �[u⇤v⇤] j+ f[v⇤✓⇤]

@✓s/@pk


the jet

< > means vertical average

ii

ii

ii

ii

Fig. 12.3 Generation of zonal flow on a �-plane or on a rotating sphere. Stirringin mid-latitudes (by baroclinic eddies) generates Rossby waves that propagate awayfrom the disturbance. Momentum converges in the region of stirring, producingeastward flow there and weaker westward flow on its flanks.

From Vallis (2006)

From Vallis (2006)

Eddy-driven jet: - the momentum budget

In equilibrium:

There MUST be surface westerlies at midlatitudes.

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F ⌘ �[u⇤v⇤] j+ f[v⇤✓⇤]

@✓s/@pk

E-P flux - in the real atmosphere

~F = ~cgA

Vertical component is dominant.

EP divergence in the lower layers; convergence in the upper layers.


~F = ~cgA

@[u]

@t= f ˜[v] +r · F + [Fx]

F ⌘ �[u⇤v⇤] j+ f[v⇤✓⇤]

@✓s/@pk

E-P flux and the eddy-driven jet -summary

• Numerical results and observations: eddies generate in the lower level, propagate upwards and away from the eddy source region.

• Accelerating the lower jet, decelerating the upper jet, reduce the vertical shear of U

• Momentum budget indicates that there MUST be surface westerlies in the eddy source latitude.


Outline

! Observations

! The Ferrel Cell




! Eddy-driven jet

! The energy cycle

E = I + �+ LH+K

LH = Lq

K =1

2(u2 + v2 + w2) ⇡ 1

2(u2 + v2)

� = gz


Energy cycles in the baroclinic eddy-mean flow interactions

! Basic forms of energy:

! Kinetic energy (动能):

! Latent energy (相变潜热能):

! Internal energy (内能):

! Gravitational-potential energy (位能):

! Total energy:

I = cvT

K =1

2(u2 + v2 + w2) ⇡ 1

2(u2 + v2)

� = gz

I = cvT





! Internal energy (内能):

! Gravitational-potential energy (位能):

! Total potential energy: Z 1

0⇢(I + �)dz =

1

g

Z ps

0(cvT +RT )dp =

1

g

Z ps

0cpTdp

K =1

2(u2 + v2 + w2) ⇡ 1

2(u2 + v2)





ii

ii

ii

ii

)

)

⇥

�

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 gradientis 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 releaseof potential energy.

From Vallis (2006)

From Vallis (2006)

warm cold

Z 1

0⇢(I + �)dz =

1

g

Z ps

0(cvT +RT )dp =

1

g

Z ps

0cpTdp

! integrating by parts we find for the total potential energy:

ps (z

0

=0) C

g

P Tdp = C

g

P

(1+ " 1

) p0 &'

%'#p"+1

0

ps (z=0)

$ p=p

p

s

! =

,z

0

=0

p"+1d#)*

(*! "

What are the limits on ! ? We now adopt the convention that below the surface, z < zs,

= surface ! , and p = ps = constant, and !" 0 at z = 0. (We implicitly assume that

> 0, i.e. the system is always stably stratified.)

ps

!" p#+1

0 = 0 and

g 1+ C

! P

!

#

$ 0

p!+1d" = TPE( ) p0

However, we note that in a stably stratified atmosphere like ours much of the TPE can

never be released or converted to KE. In particular, consider a typical arrangement of the

isentropic surfaces as shown in the first diagram below.

! < !S

d! dz

Figure by MIT OCW.

!6 > !

5... > !

1. As we see there are horizontal gradients and because of this PE can be

released by the motions in spite of the stable stratification. Consider for example a

displacement like that shown below.

2

From Stone’s class notes

!

Figure by MIT OCW.

In this new state, since !" !z

> 0, there are no displacements which can release PE,

because now for any displacement, w '! ' < 0 . Thus the PE of this state is unavailable.

This led Lorenz (1955) to define the available potential energy as the difference between

the TPE in the actual atmosphere and that in the adjusted state just described. If we call

this P, then for a given system or region,

P = (

CP

) p0

" dxdy %

" 0

d#(p!+1 $ p! !+1 ) g 1+ ! !

A

Since p! is already integrated over the area, we can rewrite the integral as

"+1 "+1A $

% d!(p! # p" ). This is the so-called exact formula. Note that, since 0

!+1! > 0, (p ) > p" , i.e. P is according to our definition always positive.( )!+1

Lorenz derived an approximation for P that is commonly used. Let p = p! + p' , and

similarly for other variables:

4

✓s, Ts

K =1

2(u2 + v2 + w2) ⇡ 1

2(u2 + v2)





! integrating by parts we find for the total potential energy:

ps (z

0

=0) C

g

P Tdp = C

g

P

(1+ " 1

) p0 &'

%'#p"+1

0

ps (z=0)

$ p=p

p

s

! =

,z

0

=0

p"+1d#)*

(*! "

What are the limits on ! ? We now adopt the convention that below the surface, z < zs,

= surface ! , and p = ps = constant, and !" 0 at z = 0. (We implicitly assume that

> 0, i.e. the system is always stably stratified.)

ps

!" p#+1

0 = 0 and

g 1+ C

! P

!

#

$ 0

p!+1d" = TPE( ) p0

However, we note that in a stably stratified atmosphere like ours much of the TPE can

never be released or converted to KE. In particular, consider a typical arrangement of the

isentropic surfaces as shown in the first diagram below.

! < !S

d! dz

Figure by MIT OCW.

!6 > !

5... > !

1. As we see there are horizontal gradients and because of this PE can be

released by the motions in spite of the stable stratification. Consider for example a

displacement like that shown below.

2

－

State A State B

= Available potential energy

Z 1

0⇢(I + �)dz =

1

g

Z ps

0(cvT +RT )dp =

1

g

Z ps

0cpTdp

K =1

2(u2 + v2 + w2) ⇡ 1

2(u2 + v2)

� = � R

cpp

✓psp

◆ Rcp

✓@✓s@p

◆�1

= (�d/Ts) (�d � �s)�1

Z 1

0⇢(I + �)dz =

1

g

Z ps

0(cvT +RT )dp =

1

g

Z ps

0cpTdp





State A State B = Available potential energy

! Available potential energy (有效位能):

－

From the “approximate” expression of Lorenz (1955)

P =1

2

Z ps

0

Ts

�d � �s

✓T � Ts

Ts

◆2

dp =cp2g

Z ps

0� (T � Ts)

2 dp

K =1

2(u2 + v2 + w2) ⇡ 1

2(u2 + v2)

P =1

2

Z ps

0

Ts

�d � �s

✓T � Ts

Ts

◆2

dp =cp2g

Z ps

0� (T � Ts)

2 dp






! Tendency equations:

@

@t

ZPdm = R

Z!T

pdm+

Z�(T � Ts)(Q�Qs)dm

@

@t

ZKdm = �R

Z!T

pdm+

Z(uFx + vFy)dm

Q - diabatic heating





! Tendency equations:

@

@t

ZPdm = R

Z!T

pdm+


@

@t

ZKdm = �R

Z!T

pdm+

Z(uFx + vFy)dm

! Zonal mean and eddy components:

KE =1

2

⇣[u⇤2] + [v⇤2]

⌘KM =

1

2

�[u]2 + [v]2

�

PM =cp2� ([T ]� Ts)

2PE =

cp2�[T ⇤2]

Q - diabatic heating

@

@t

ZPMdm = R

Z[!][T ]

pdm+ cp

Z�[v⇤T ⇤]

@[T ]

@ydm+

Z�([T ]� Ts)([Q]�Qs)dm

@

@t

ZPEdm = R

Z[!⇤T ⇤]

pdm� cp

Z�[v⇤T ⇤]

@[T ]

@ydm+

Z�[T ⇤Q⇤]dm

@

@t

ZPdm = R

Z!T

pdm+


@

@t

ZKEdm = �R

Z[!⇤T ⇤]

pdm�

Z[u⇤v⇤]

@[u]

@ydm+

Z([u⇤F ⇤

x + v⇤F ⇤y ])dm

@

@t

ZKMdm = �R

Z[!][T ]

pdm+

Z[u⇤v⇤]

@[u]

@ydm+

Z([u][Fx] + [v][Fy])dm

@

@t

ZKdm = �R

Z!T

pdm+

Z(uFx + vFy)dm



! Equations under the Quasi-geostrophic assumption:

@

@t

ZPMdm = R

Z[!][T ]

pdm+ cp

Z�[v⇤T ⇤]

@[T ]

@ydm+

Z�([T ]� Ts)([Q]�Qs)dm

@

@t

ZPEdm = R

Z[!⇤T ⇤]

pdm� cp

Z�[v⇤T ⇤]

@[T ]

@ydm+

Z�[T ⇤Q⇤]dm

@

@t

ZPdm = R

Z!T

pdm+


@

@t

ZKEdm = �R

Z[!⇤T ⇤]

pdm�

Z[u⇤v⇤]

@[u]

@ydm+

Z([u⇤F ⇤

x + v⇤F ⇤y ])dm

@

@t

ZKMdm = �R

Z[!][T ]

pdm+

Z[u⇤v⇤]

@[u]

@ydm+

Z([u][Fx] + [v][Fy])dm

@

@t

ZKdm = �R

Z!T

pdm+

Z(uFx + vFy)dm

< PE,KE >



! Equations under the Quasi-geostrophic assumption:

< PM,KM >

< PM,KM >

< PE,KE >

< KE,KM >

< KE,KM >

< PE, PM >

< PE, PM >

D(KE)

D(KM)

G(PE)

G(PM)

G(PE) D(KE)

D(KM )G(PM )

R[!⇤T ⇤]

p

R[!][T ]

p

cp�[v⇤T ⇤]

@[T ]

@y

@[u]

@y[u⇤v⇤]



PE KE

@PM

@t

@PE

@t

@KM

@t

@KE

@t

< PM,KM >

< KE,KM >

< PE,KE >

< PE, PM >

! Energy cycles in eddy life cycle:


~F = ~cgA

F ⌘ �[u⇤v⇤] j+ f[v⇤✓⇤]

@✓s/@pk



1978, JASEddies: generate at lower level, propagate upwards and away from the

eddy source region



! Westerly jet and energy cycle:

R[!⇤T ⇤]

p

cp�[v⇤T ⇤]

@[T ]

@y

@[u]

@y[u⇤v⇤]

@PM

@t

@PE

@t

@KM

@t

@KE

@t

< KE,KM >

< PE,KE >

< PE, PM >



! Westerly jet and energy cycle:

R[!⇤T ⇤]

p

cp�[v⇤T ⇤]

@[T ]

@y

@[u]

@y[u⇤v⇤]

@PM

@t

@PE

@t

@KM

@t

@KE

@t

< KE,KM >

< PE,KE >

< PE, PM >


1978, JAS

ii

ii

ii

ii

4

3

2

1

0

-3

-2

-1

Wm

-2

0 2 64 12 148 10

50

40

30

-1

0 3 6 9 12 15 (days)

C(AZ AE)

C(AE KE)

C(AZ KZ)

C(KZ KE)

time

u (m

s

)En

ergy

con

vers

ions

Fig. 9.9 Top: energy conversion and dissipation processes in a numerical simulationof an idealized atmospheric baroclinic lifecycle, simulated with a GCM. Bottom: evo-lution of the maximum zonal-mean velocity. AZ and AE are zonal and eddy availablepotential energies, and KZ and KE are the corresponding kinetic energies. Initiallybaroclinic processes dominate, with conversions from zonal to eddy kinetic energyand then eddy kinetic to eddy available potential energy, followed by the barotropicconversion of eddy kinetic to zonal kinetic energy. The latter process is reflected inthe increase of the maximum zonal-mean velocity at about day 10.6

From Vallis (2006)

From Vallis (2006)

Strengthen of westerly jet

< PE, PM >

< PE,KE >

< KE,KM >

< PM,KM >

Eddy momentum flux grows, which extracts kinetic energy from the

eddies to the zonal mean flow, then the growth of

the eddy energy ceases.

G(PE) D(KE)

D(KM )G(PM )

R[!⇤T ⇤]

p

R[!][T ]

p

cp�[v⇤T ⇤]

@[T ]

@y

@[u]

@y[u⇤v⇤]



! Energy cycles in equilibrium:

@PM

@t

@PE

@t

@KM

@t

@KE

@t

< PM,KM >

< KE,KM >

< PE,KE >

< PE, PM >

PE KE

Lorenz energy cycle


PE KE

Energy cycles in Hadley Cell

D(KM )G(PM )

R[!][T ]

p

@PM

@t

@KM

@t

< PM,KM >

If assume no eddies.

G(PE) D(KE)

D(KM )G(PM )



! Energy cycles in real atmosphere:

@PM

@t

@PE

@t

@KM

@t

@KE

@t

< PM,KM >

< KE,KM >

< PE,KE >

< PE, PM > GLOBAL ANNUAL

energy: 105Jm�2

33.3

11.1 7.3

4.5conversion: Wm�2

0.15

2.0

0.33

1.7

0.18

1.27

1.12

0.7


U.S. All NYTU.S.WORLD U.S. N.Y. / REGION BUSINESS TECHNOLOGY SCIENCE HEALTH SPORTS OPINION ARTS STYLE TRAVEL JOBS REAL ESTATE AUTOS

POLITICS WASHINGTON EDUCATION

M.I.T. News Office

Edward N. Lorenz

Edward N. Lorenz, a Meteorologist and a Father ofChaos Theory, Dies at 90By KENNETH CHANGPublished: April 17, 2008

Edward N. Lorenz, a meteorologist who tried to predict the weather

with computers but instead gave rise to the modern field of chaos

theory, died Wednesday at his home in Cambridge, Mass. He was 90.

The cause was cancer, said his

daughter Cheryl Lorenz.

In discovering “deterministic chaos,”

Dr. Lorenz established a principle that

“profoundly influenced a wide range

of basic sciences and brought about

one of the most dramatic changes in mankind’s view of

nature since Sir Isaac Newton,” said a committee that

awarded him the 1991 Kyoto Prize for basic sciences.

Dr. Lorenz is best known for the notion of the “butterfly

effect,” the idea that a small disturbance like the flapping

of a butterfly’s wings can induce enormous consequences.

As recounted in the book “Chaos” by James Gleick, Dr. Lorenz’s accidental discovery of

chaos came in the winter of 1961. Dr. Lorenz was running simulations of weather using a

simple computer model. One day, he wanted to repeat one of the simulations for a longer

time, but instead of repeating the whole simulation, he started the second run in the

middle, typing in numbers from the first run for the initial conditions.

The computer program was the same, so the weather patterns of the second run should

have exactly followed those of the first. Instead, the two weather trajectories quickly

diverged on completely separate paths.

At first, he thought the computer was malfunctioning. Then he realized that he had not

entered the initial conditions exactly. The computer stored numbers to an accuracy of six

decimal places, like 0.506127, while, to save space, the printout of results shortened the

numbers to three decimal places, 0.506. When typing in the new conditions, Dr. Lorenz

had entered the rounded-off numbers, and even this small discrepancy, of less than 0.1

percent, completely changed the end result.

Even though his model was vastly simplified, Dr. Lorenz realized that this meant perfect

weather prediction was a fantasy.

A perfect forecast would require not only a perfect model, but also perfect knowledge of

wind, temperature, humidity and other conditions everywhere around the world at one

moment of time. Even a small discrepancy could lead to completely different weather.

Dr. Lorenz published his findings in 1963. “The paper he wrote in 1963 is a masterpiece of

clarity of exposition about why weather is unpredictable,” said J. Doyne Farmer, a

professor at the Santa Fe Institute in New Mexico.

The following year, Dr. Lorenz published another paper that described how a small

twiddling of parameters in a model could produce vastly different behavior, transforming

regular, periodic events into a seemingly random chaotic pattern.

At a meeting of the American Association for the Advancement of Science in 1972, he gave

a talk with a title that captured the essence of his ideas: “Predictability: Does the Flap of a

Butterfly’s Wings in Brazil Set Off a Tornado in Texas?”

Dr. Lorenz was not the first to stumble onto chaos. At the end of the 19th century, the

mathematician Henri Poincaré showed that the gravitational dance of as few as three

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Summary & Discussion

! Observations

! The Ferrel Cell




! Eddy-driven jet

! The energy cycle

1. The role of moisture;2. Quantify (parameterize) the relation between eddies and mean flow; 3. Zonal variations.

chap 4 4 2019 - tcs @ njutcs.nju.edu.cn/yzhang/chap_4_4_2019.pdf · 2019. 11. 13. · 2...

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