�����

��

���� ��

.

�� �������

( )zyuu ,=

0),(;,,, =÷÷ø

öççè

涶

¶¶

¶¶

¶¶ yzyu

zyxtL

����

����

irir iccci +=+= ,nnn( ) ( )( )tkxizy nfj -= exp,

( ) ( )( )( ) ( )( )tkxizye

tkxizye

rtkc

rt

i

i

nf

nfj n

-=

-=

exp,exp,

����

�����

������

������

-

®

i

i

kcc ,0!

0=+÷÷ø

öççè

涶

+¶¶

+¶¶ v

yv

xu

tbV

0=¶¶

+¶¶

yv

xu�������

���

Rossby

yVyy 2Ñ=¶¶

=¶¶

-= ,,x

vy

u

02 =¶¶

+Ñ÷øö

çèæ

¶¶

+¶¶

xxu

tjbj

������� ����

! = #$% + '(), %, +)

������������

0=+÷÷ø

öççè

涶

+¶¶

+¶¶ v

yv

xu

tbV

0=¶¶

+¶¶

yv

xu��� ��

�����

������y��

( ) 02

22 =

¶¶÷÷ø

öççè

涶

-+Ñ÷øö

çèæ

¶¶

+¶¶

xyu

xyu

tjbj

( ) ( )yyuu yy == :( ) ( )tyxy ,,jyy +=

�����������

( ) 02

22 =

¶¶÷÷ø

öççè

涶

-+Ñ÷øö

çèæ

¶¶

+¶¶

xyu

xyu

tjbj

01 =¶¶

=x

yy y,

02 =¶¶

=x

yy y,

( ) ( )tkxiey nfj -=

ïî

ïíì

==

=÷øö

çèæ

-¢¢-

--

0

0

21

22

2

f

fbf

yyycuuk

dyd

,

, �����

02

1

22

2

=úû

ùêë

é÷øö

çèæ

-¢¢-

--ò dycuuk

dydy

y

fbff *

022

2

222 2

1

2

1

=-¶¶

--

÷÷

ø

ö

çç

è

æ+ òò dy

cuyu

dykdyd y

y

y

y

fb

ff

( )02

22

2

2

2

1

=+-

¶¶

-

ò dyccu

yu

cy

y iri f

b

[ ]212

2

0 yyydyud

sys

!!,=÷÷ø

öççè

æ-b

��������1: ��� ��sy

���

�����������

( )( ) dycu

ccuyu

dykdyd y

yr

ir

y

yòò -

+-¶¶

-=

÷÷

ø

ö

çç

è

æ+

2

1

2

1

222

2

2

222

fb

ff��

( )dy

ccuyuu

dykdyd y

y ir

y

y

222

2

2

222 2

1

2

1

fb

ffòò +-

÷÷ø

öççè

涶

-=

÷÷

ø

ö

çç

è

æ+

��������2:

���� 2

2

yuu

¶¶

-b

������� ����

. 2

�����������

���� �� ������

�����:� �������

���� :��PV������

��������

p = 0 mb

p = 250 mb

p = 500 mb

p = 750 mb

p = 1000 mb

00 =w

04 =w

2w

1y

3y

��������

To protect the rights of the author(s) and publisher we inform you that this PDF is an uncorrected proof for internal businessuse only by the author(s), editor(s), reviewer(s), Elsevier and typesetter diacriTech. It is not allowed to publish this proofonline or in print. This proof copy is the copyright property of the publisher and is confidential until formal publication.

“Hakim: 11-ch07-213-254-9780123848666” — 2012/7/28 — 12:35 — page 215 — #3

7.2 | Normal Mode Baroclinic Instability: A Two-Layer Model 215

Cold

+ +

Warm(a) (b)

FIGURE 7.1 A schematic picture of cyclogenesis associated with the arrival of an upper-levelpotential vorticity perturbation over a lower-level baroclinic region. (a) Lower-level cyclonic vortic-ity induced by the upper-level potential vorticity anomaly. The circulation induced by the potentialvorticity anomaly is shown by the solid arrows, and potential temperature contours are shown atthe lower boundary. The advection of potential temperature by the induced lower-level circulationleads to a warm anomaly slightly east of the upper-level vorticity anomaly. This in turn will inducea cyclonic circulation as shown by the open arrows in (b). The induced upper-level circulation willreinforce the original upper-level anomaly and can lead to amplification of the disturbance. (AfterHoskins et al., 1985.)

7.2 NORMAL MODE BAROCLINIC INSTABILITY:A TWO-LAYER MODEL

Even for a highly idealized mean-flow profile, the mathematical treatment ofbaroclinic instability in a continuously stratified atmosphere is rather compli-cated. Before considering such a model, we first focus on the simplest modelthat can incorporate baroclinic processes. The atmosphere is represented by twodiscrete layers bounded by surfaces numbered 0, 2, and 4 (generally taken tobe the 0-, 500-, and 1000-hPa surfaces, respectively) as shown in Figure 7.2.The quasi-geostrophic vorticity equation for the midlatitude � plane is appliedat the 250- and 750-hPa levels, designated by 1 and 3 in Figure 7.2, whereas thethermodynamic energy equation is applied at the 500-hPa level, designated by2 in Figure 7.2.

Before writing out the specific equations of the two-layer model, it is conve-nient to define a geostrophic streamfunction, ⌘ 8/f0, for the isobaric form ofthe quasi-geostrophic equations. Then the geostrophic wind and the geostrophicvorticity can be expressed respectively as

V = k ⇥ rh , ⇣g = r2h (7.1)

The quasi-geostrophic vorticity equation and the hydrostatic thermodynamicenergy equation can then be written in terms of and ! as

@

@tr2

h + V ·rh

⇣r2

h ⌘

+ �@

@x= f0

@!

@p(7.2)

To protect the rights of the author(s) and publisher we inform you that this PDF is an uncorrected proof for internal businessuse only by the author(s), editor(s), reviewer(s), Elsevier and typesetter diacriTech. It is not allowed to publish this proofonline or in print. This proof copy is the copyright property of the publisher and is confidential until formal publication.

“Hakim: 11-ch07-213-254-9780123848666” — 2012/7/28 — 12:35 — page 215 — #3

7.2 | Normal Mode Baroclinic Instability: A Two-Layer Model 215

Cold

+ +

Warm(a) (b)

FIGURE 7.1 A schematic picture of cyclogenesis associated with the arrival of an upper-levelpotential vorticity perturbation over a lower-level baroclinic region. (a) Lower-level cyclonic vortic-ity induced by the upper-level potential vorticity anomaly. The circulation induced by the potentialvorticity anomaly is shown by the solid arrows, and potential temperature contours are shown atthe lower boundary. The advection of potential temperature by the induced lower-level circulationleads to a warm anomaly slightly east of the upper-level vorticity anomaly. This in turn will inducea cyclonic circulation as shown by the open arrows in (b). The induced upper-level circulation willreinforce the original upper-level anomaly and can lead to amplification of the disturbance. (AfterHoskins et al., 1985.)

7.2 NORMAL MODE BAROCLINIC INSTABILITY:A TWO-LAYER MODEL

Even for a highly idealized mean-flow profile, the mathematical treatment ofbaroclinic instability in a continuously stratified atmosphere is rather compli-cated. Before considering such a model, we first focus on the simplest modelthat can incorporate baroclinic processes. The atmosphere is represented by twodiscrete layers bounded by surfaces numbered 0, 2, and 4 (generally taken tobe the 0-, 500-, and 1000-hPa surfaces, respectively) as shown in Figure 7.2.The quasi-geostrophic vorticity equation for the midlatitude � plane is appliedat the 250- and 750-hPa levels, designated by 1 and 3 in Figure 7.2, whereas thethermodynamic energy equation is applied at the 500-hPa level, designated by2 in Figure 7.2.

Before writing out the specific equations of the two-layer model, it is conve-nient to define a geostrophic streamfunction, ⌘ 8/f0, for the isobaric form ofthe quasi-geostrophic equations. Then the geostrophic wind and the geostrophicvorticity can be expressed respectively as

V = k ⇥ rh , ⇣g = r2h (7.1)

The quasi-geostrophic vorticity equation and the hydrostatic thermodynamicenergy equation can then be written in terms of and ! as

@

@tr2

h + V ·rh

⇣r2

h ⌘

+ �@

@x= f0

@!

@p(7.2)

To protect the rights of the author(s) and publisher we inform you that this PDF is an uncorrected proof for internal businessuse only by the author(s), editor(s), reviewer(s), Elsevier and typesetter diacriTech. It is not allowed to publish this proofonline or in print. This proof copy is the copyright property of the publisher and is confidential until formal publication.

“Hakim: 11-ch07-213-254-9780123848666” — 2012/7/28 — 12:35 — page 215 — #3

7.2 | Normal Mode Baroclinic Instability: A Two-Layer Model 215

Cold

+ +

Warm(a) (b)

FIGURE 7.1 A schematic picture of cyclogenesis associated with the arrival of an upper-levelpotential vorticity perturbation over a lower-level baroclinic region. (a) Lower-level cyclonic vortic-ity induced by the upper-level potential vorticity anomaly. The circulation induced by the potentialvorticity anomaly is shown by the solid arrows, and potential temperature contours are shown atthe lower boundary. The advection of potential temperature by the induced lower-level circulationleads to a warm anomaly slightly east of the upper-level vorticity anomaly. This in turn will inducea cyclonic circulation as shown by the open arrows in (b). The induced upper-level circulation willreinforce the original upper-level anomaly and can lead to amplification of the disturbance. (AfterHoskins et al., 1985.)

7.2 NORMAL MODE BAROCLINIC INSTABILITY:A TWO-LAYER MODEL

Even for a highly idealized mean-flow profile, the mathematical treatment ofbaroclinic instability in a continuously stratified atmosphere is rather compli-cated. Before considering such a model, we first focus on the simplest modelthat can incorporate baroclinic processes. The atmosphere is represented by twodiscrete layers bounded by surfaces numbered 0, 2, and 4 (generally taken tobe the 0-, 500-, and 1000-hPa surfaces, respectively) as shown in Figure 7.2.The quasi-geostrophic vorticity equation for the midlatitude � plane is appliedat the 250- and 750-hPa levels, designated by 1 and 3 in Figure 7.2, whereas thethermodynamic energy equation is applied at the 500-hPa level, designated by2 in Figure 7.2.

Before writing out the specific equations of the two-layer model, it is conve-nient to define a geostrophic streamfunction, ⌘ 8/f0, for the isobaric form ofthe quasi-geostrophic equations. Then the geostrophic wind and the geostrophicvorticity can be expressed respectively as

V = k ⇥ rh , ⇣g = r2h (7.1)

The quasi-geostrophic vorticity equation and the hydrostatic thermodynamicenergy equation can then be written in terms of and ! as

@

@tr2

h + V ·rh

⇣r2

h ⌘

+ �@

@x= f0

@!

@p(7.2)

To protect the rights of the author(s) and publisher we inform you that this PDF is an uncorrected proof for internal businessuse only by the author(s), editor(s), reviewer(s), Elsevier and typesetter diacriTech. It is not allowed to publish this proofonline or in print. This proof copy is the copyright property of the publisher and is confidential until formal publication.

“Hakim: 11-ch07-213-254-9780123848666” — 2012/7/28 — 12:35 — page 216 — #4

216 CHAPTER | 7 Baroclinic Development

0

1

2

3

4

p = 0

p = 250

p = 500

p = 750

p = 1000

ω0 = 0

ψ1

ω2

ψ3

ω4

(hPa)

FIGURE 7.2 Arrangement of variables in the vertical for the two-level baroclinic model.

@

@t

✓@

@p

◆= �V ·rh

✓@

@p

◆� �

f0! (7.3)

We now apply the vorticity equation (7.2) at the two levels designated as 1and 3, which are at the middle of the two layers. To do this we must estimate thedivergence term @!/@p at these levels using finite difference approximations tothe vertical derivatives:

✓@!

@p

◆

1⇡ ! 2 � ! 0

�p,

✓@!

@p

◆

3⇡ ! 4 � ! 2

�p(7.4)

where �p = 500 hPa is the pressure interval between levels 0 to 2 and 2 to 4,and subscript notation is used to designate the vertical level for each dependentvariable. The resulting vorticity equations are

@

@tr2

h 1 + V1 ·r⇣r2

h 1

⌘+ �

@ 1

@x= f0�p!2 (7.5)

@

@tr2

h 3 + V3 ·rh

⇣r2

h 3

⌘+ �

@ 3

@x= � f0

�p!2 (7.6)

where we have used the fact that ! 0 = 0, and assumed that ! 4 = 0, which isapproximately true for a level-lower boundary surface.

We next write the thermodynamic energy equation (7.3) at level 2. Here wemust evaluate @ /@p using the difference formula

(@ /@p)2 ⇡ ( 3 � 1) /�p

The result is

@

@t( 1 � 3) = �V2 ·rh( 1 � 3) + ��p

f0! 2 (7.7)

��� ���

��������

��� �����1�3���

To protect the rights of the author(s) and publisher we inform you that this PDF is an uncorrected proof for internal businessuse only by the author(s), editor(s), reviewer(s), Elsevier and typesetter diacriTech. It is not allowed to publish this proofonline or in print. This proof copy is the copyright property of the publisher and is confidential until formal publication.

“Hakim: 11-ch07-213-254-9780123848666” — 2012/7/28 — 12:35 — page 216 — #4

216 CHAPTER | 7 Baroclinic Development

0

1

2

3

4

p = 0

p = 250

p = 500

p = 750

p = 1000

ω0 = 0

ψ1

ω2

ψ3

ω4

(hPa)

FIGURE 7.2 Arrangement of variables in the vertical for the two-level baroclinic model.

@

@t

✓@

@p

◆= �V ·rh

✓@

@p

◆� �

f0! (7.3)

We now apply the vorticity equation (7.2) at the two levels designated as 1and 3, which are at the middle of the two layers. To do this we must estimate thedivergence term @!/@p at these levels using finite difference approximations tothe vertical derivatives:

✓@!

@p

◆

1⇡ ! 2 � ! 0

�p,

✓@!

@p

◆

3⇡ ! 4 � ! 2

�p(7.4)

where �p = 500 hPa is the pressure interval between levels 0 to 2 and 2 to 4,and subscript notation is used to designate the vertical level for each dependentvariable. The resulting vorticity equations are

@

@tr2

h 1 + V1 ·r⇣r2

h 1

⌘+ �

@ 1

@x= f0�p!2 (7.5)

@

@tr2

h 3 + V3 ·rh

⇣r2

h 3

⌘+ �

@ 3

@x= � f0

�p!2 (7.6)

where we have used the fact that ! 0 = 0, and assumed that ! 4 = 0, which isapproximately true for a level-lower boundary surface.

We next write the thermodynamic energy equation (7.3) at level 2. Here wemust evaluate @ /@p using the difference formula

(@ /@p)2 ⇡ ( 3 � 1) /�p

The result is

@

@t( 1 � 3) = �V2 ·rh( 1 � 3) + ��p

f0! 2 (7.7)

����������2���

To protect the rights of the author(s) and publisher we inform you that this PDF is an uncorrected proof for internal businessuse only by the author(s), editor(s), reviewer(s), Elsevier and typesetter diacriTech. It is not allowed to publish this proofonline or in print. This proof copy is the copyright property of the publisher and is confidential until formal publication.

“Hakim: 11-ch07-213-254-9780123848666” — 2012/7/28 — 12:35 — page 216 — #4

216 CHAPTER | 7 Baroclinic Development

0

1

2

3

4

p = 0

p = 250

p = 500

p = 750

p = 1000

ω0 = 0

ψ1

ω2

ψ3

ω4

(hPa)

FIGURE 7.2 Arrangement of variables in the vertical for the two-level baroclinic model.

@

@t

✓@

@p

◆= �V ·rh

✓@

@p

◆� �

f0! (7.3)

We now apply the vorticity equation (7.2) at the two levels designated as 1and 3, which are at the middle of the two layers. To do this we must estimate thedivergence term @!/@p at these levels using finite difference approximations tothe vertical derivatives:

✓@!

@p

◆

1⇡ ! 2 � ! 0

�p,

✓@!

@p

◆

3⇡ ! 4 � ! 2

�p(7.4)

where �p = 500 hPa is the pressure interval between levels 0 to 2 and 2 to 4,and subscript notation is used to designate the vertical level for each dependentvariable. The resulting vorticity equations are

@

@tr2

h 1 + V1 ·r⇣r2

h 1

⌘+ �

@ 1

@x= f0�p!2 (7.5)

@

@tr2

h 3 + V3 ·rh

⇣r2

h 3

⌘+ �

@ 3

@x= � f0

�p!2 (7.6)

where we have used the fact that ! 0 = 0, and assumed that ! 4 = 0, which isapproximately true for a level-lower boundary surface.

We next write the thermodynamic energy equation (7.3) at level 2. Here wemust evaluate @ /@p using the difference formula

(@ /@p)2 ⇡ ( 3 � 1) /�p

The result is

@

@t( 1 � 3) = �V2 ·rh( 1 � 3) + ��p

f0! 2 (7.7)

��3����3�����

��

��

To protect the rights of the author(s) and publisher we inform you that this PDF is an uncorrected proof for internal businessuse only by the author(s), editor(s), reviewer(s), Elsevier and typesetter diacriTech. It is not allowed to publish this proofonline or in print. This proof copy is the copyright property of the publisher and is confidential until formal publication.

“Hakim: 11-ch07-213-254-9780123848666” — 2012/7/28 — 12:35 — page 217 — #5

7.2 | Normal Mode Baroclinic Instability: A Two-Layer Model 217

The first term on the right-hand side in (7.7) is the advection of the 250- to750-hPa thickness by the wind at 500 hPa. However, 2, the 500-hPa stream-function, is not a predicted field in this model. Therefore, 2 must be obtainedby linearly interpolating between the 250- and 750-hPa levels:

2 = ( 1 + 3)/2

If this interpolation formula is used, (7.5), (7.6), and (7.7) become a closed setof prediction equations in the variables 1, 3, and !2.

7.2.1 Linear Perturbation Analysis

To keep the analysis as simple as possible, we assume that the streamfunctions 1 and 3 consist of basic state parts that depend linearly on y alone, plusperturbations that depend only on x and t. Thus, we let

1 = �U1y + 01 (x, t)

3 = �U3y + 03 (x, t) (7.8)

!2 = !02 (x, t)

The zonal velocities at levels 1 and 3 are then constants with the values U1and U3, respectively. Hence, the perturbation field has meridional and verticalvelocity components only.

Substituting from (7.8) into (7.5), (7.6), and (7.7) and linearizing yields theperturbation equations

✓@

@t+ U1

@

@x

◆@2 0

1

@x2 + �@ 0

1

@x= f0�p!0

2 (7.9)

✓@

@t+ U3

@

@x

◆@2 0

3

@x2 + �@ 0

3

@x= � f0

�p!0

2 (7.10)

✓@

@t+ Um

@

@x

◆ � 0

1 � 03�� UT

@

@x

� 0

1 + 03�

= ��pf0!0

2 (7.11)

where we have linearly interpolated to express V2 in terms of 1 and 3, andhave defined

Um ⌘ (U1 + U3)/2, UT ⌘ (U1 � U3)/2

Thus, Um and UT are, respectively, the vertically averaged mean zonal wind andthe mean thermal wind.

The dynamical properties of this system are more clearly expressed if (7.9),(7.10), and (7.11) are combined to eliminate !0

2. We first note that (7.9) and

����� �������������U1� ����������U3

����v���

To protect the rights of the author(s) and publisher we inform you that this PDF is an uncorrected proof for internal businessuse only by the author(s), editor(s), reviewer(s), Elsevier and typesetter diacriTech. It is not allowed to publish this proofonline or in print. This proof copy is the copyright property of the publisher and is confidential until formal publication.

“Hakim: 11-ch07-213-254-9780123848666” — 2012/7/28 — 12:35 — page 217 — #5

7.2 | Normal Mode Baroclinic Instability: A Two-Layer Model 217

The first term on the right-hand side in (7.7) is the advection of the 250- to750-hPa thickness by the wind at 500 hPa. However, 2, the 500-hPa stream-function, is not a predicted field in this model. Therefore, 2 must be obtainedby linearly interpolating between the 250- and 750-hPa levels:

2 = ( 1 + 3)/2

If this interpolation formula is used, (7.5), (7.6), and (7.7) become a closed setof prediction equations in the variables 1, 3, and !2.

7.2.1 Linear Perturbation Analysis

To keep the analysis as simple as possible, we assume that the streamfunctions 1 and 3 consist of basic state parts that depend linearly on y alone, plusperturbations that depend only on x and t. Thus, we let

1 = �U1y + 01 (x, t)

3 = �U3y + 03 (x, t) (7.8)

!2 = !02 (x, t)

The zonal velocities at levels 1 and 3 are then constants with the values U1and U3, respectively. Hence, the perturbation field has meridional and verticalvelocity components only.

Substituting from (7.8) into (7.5), (7.6), and (7.7) and linearizing yields theperturbation equations

✓@

@t+ U1

@

@x

◆@2 0

1

@x2 + �@ 0

1

@x= f0�p!0

2 (7.9)

✓@

@t+ U3

@

@x

◆@2 0

3

@x2 + �@ 0

3

@x= � f0

�p!0

2 (7.10)

✓@

@t+ Um

@

@x

◆ � 0

1 � 03�� UT

@

@x

� 0

1 + 03�

= ��pf0!0

2 (7.11)

where we have linearly interpolated to express V2 in terms of 1 and 3, andhave defined

Um ⌘ (U1 + U3)/2, UT ⌘ (U1 � U3)/2

Thus, Um and UT are, respectively, the vertically averaged mean zonal wind andthe mean thermal wind.

The dynamical properties of this system are more clearly expressed if (7.9),(7.10), and (7.11) are combined to eliminate !0

2. We first note that (7.9) and

�������

To protect the rights of the author(s) and publisher we inform you that this PDF is an uncorrected proof for internal businessuse only by the author(s), editor(s), reviewer(s), Elsevier and typesetter diacriTech. It is not allowed to publish this proofonline or in print. This proof copy is the copyright property of the publisher and is confidential until formal publication.

“Hakim: 11-ch07-213-254-9780123848666” — 2012/7/28 — 12:35 — page 217 — #5

7.2 | Normal Mode Baroclinic Instability: A Two-Layer Model 217

The first term on the right-hand side in (7.7) is the advection of the 250- to750-hPa thickness by the wind at 500 hPa. However, 2, the 500-hPa stream-function, is not a predicted field in this model. Therefore, 2 must be obtainedby linearly interpolating between the 250- and 750-hPa levels:

2 = ( 1 + 3)/2

If this interpolation formula is used, (7.5), (7.6), and (7.7) become a closed setof prediction equations in the variables 1, 3, and !2.

7.2.1 Linear Perturbation Analysis

To keep the analysis as simple as possible, we assume that the streamfunctions 1 and 3 consist of basic state parts that depend linearly on y alone, plusperturbations that depend only on x and t. Thus, we let

1 = �U1y + 01 (x, t)

3 = �U3y + 03 (x, t) (7.8)

!2 = !02 (x, t)

The zonal velocities at levels 1 and 3 are then constants with the values U1and U3, respectively. Hence, the perturbation field has meridional and verticalvelocity components only.

Substituting from (7.8) into (7.5), (7.6), and (7.7) and linearizing yields theperturbation equations

✓@

@t+ U1

@

@x

◆@2 0

1

@x2 + �@ 0

1

@x= f0�p!0

2 (7.9)

✓@

@t+ U3

@

@x

◆@2 0

3

@x2 + �@ 0

3

@x= � f0

�p!0

2 (7.10)

✓@

@t+ Um

@

@x

◆ � 0

1 � 03�� UT

@

@x

� 0

1 + 03�

= ��pf0!0

2 (7.11)

where we have linearly interpolated to express V2 in terms of 1 and 3, andhave defined

Um ⌘ (U1 + U3)/2, UT ⌘ (U1 � U3)/2

Thus, Um and UT are, respectively, the vertically averaged mean zonal wind andthe mean thermal wind.

The dynamical properties of this system are more clearly expressed if (7.9),(7.10), and (7.11) are combined to eliminate !0

2. We first note that (7.9) and

�7.9�-�7.13�������

To protect the rights of the author(s) and publisher we inform you that this PDF is an uncorrected proof for internal businessuse only by the author(s), editor(s), reviewer(s), Elsevier and typesetter diacriTech. It is not allowed to publish this proofonline or in print. This proof copy is the copyright property of the publisher and is confidential until formal publication.

“Hakim: 11-ch07-213-254-9780123848666” — 2012/7/28 — 12:35 — page 218 — #6

218 CHAPTER | 7 Baroclinic Development

(7.10) can be rewritten as@

@t+

�Um + UT

� @@x

�@2 0

1

@x2 + �@ 0

1

@x= f0�p!0

2 (7.12)

@

@t+

�Um � UT

� @@x

�@2 0

3

@x2 + �@ 0

3

@x= � f0

�p!0

2 (7.13)

We now define the barotropic and baroclinic perturbations as

m ⌘� 0

1 + 03�/2; T ⌘

� 0

1 � 03�/2 (7.14)

Adding (7.12) and (7.13) and using the definitions in (7.14) yield@

@t+ Um

@

@x

�@2 m

@x2 + �@ m

@x+ UT

@

@x

✓@2 T

@x2

◆= 0 (7.15)

while subtracting (7.13) from (7.12) and combining with (7.11) to eliminate !02

yield@

@t+ Um

@

@x

� ✓@2 T

@x2 � 2�2 T

◆+ �

@ T

@x

+ UT@

@x

✓@2 m

@x2 + 2�2 m

◆= 0

(7.16)

where �2 ⌘ f 20 /[� (�p)2]. Equations (7.15) and (7.16) govern the evolution

of the barotropic (vertically averaged) and baroclinic (thermal) perturbationvorticities, respectively.

As in Chapter 5, we assume that wave-like solutions exist of the form

m = Aeik(x�ct), T = Beik(x�ct) (7.17)

Substituting these assumed solutions into (7.15) and (7.16) and dividing throughby the common exponential factor, we obtain a pair of simultaneous linearalgebraic equations for the coefficients A, B:

ikh�

c � Um�k2 + �

iA � ik3UTB = 0 (7.18)

ikh�

c � Um� ⇣

k2 + 2�2⌘

+ �i

B � ikUT

⇣k2 � 2�2

⌘A = 0 (7.19)

Because this set is homogeneous, nontrivial solutions will exist only if the deter-minant of the coefficients of A and B is zero. Thus, the phase speed c must satisfythe condition

����(c � Um) k2 + � �k2UT�UT

�k2 � 2�2� (c � Um)

�k2 + 2�2� + �

���� = 0

To protect the rights of the author(s) and publisher we inform you that this PDF is an uncorrected proof for internal businessuse only by the author(s), editor(s), reviewer(s), Elsevier and typesetter diacriTech. It is not allowed to publish this proofonline or in print. This proof copy is the copyright property of the publisher and is confidential until formal publication.

“Hakim: 11-ch07-213-254-9780123848666” — 2012/7/28 — 12:35 — page 218 — #6

218 CHAPTER | 7 Baroclinic Development

(7.10) can be rewritten as@

@t+

�Um + UT

� @@x

�@2 0

1

@x2 + �@ 0

1

@x= f0�p!0

2 (7.12)

@

@t+

�Um � UT

� @@x

�@2 0

3

@x2 + �@ 0

3

@x= � f0

�p!0

2 (7.13)

We now define the barotropic and baroclinic perturbations as

m ⌘� 0

1 + 03�/2; T ⌘

� 0

1 � 03�/2 (7.14)

Adding (7.12) and (7.13) and using the definitions in (7.14) yield@

@t+ Um

@

@x

�@2 m

@x2 + �@ m

@x+ UT

@

@x

✓@2 T

@x2

◆= 0 (7.15)

while subtracting (7.13) from (7.12) and combining with (7.11) to eliminate !02

yield@

@t+ Um

@

@x

� ✓@2 T

@x2 � 2�2 T

◆+ �

@ T

@x

+ UT@

@x

✓@2 m

@x2 + 2�2 m

◆= 0

(7.16)

where �2 ⌘ f 20 /[� (�p)2]. Equations (7.15) and (7.16) govern the evolution

of the barotropic (vertically averaged) and baroclinic (thermal) perturbationvorticities, respectively.

As in Chapter 5, we assume that wave-like solutions exist of the form

m = Aeik(x�ct), T = Beik(x�ct) (7.17)

Substituting these assumed solutions into (7.15) and (7.16) and dividing throughby the common exponential factor, we obtain a pair of simultaneous linearalgebraic equations for the coefficients A, B:

ikh�

c � Um�k2 + �

iA � ik3UTB = 0 (7.18)

ikh�

c � Um� ⇣

k2 + 2�2⌘

+ �i

B � ikUT

⇣k2 � 2�2

⌘A = 0 (7.19)

Because this set is homogeneous, nontrivial solutions will exist only if the deter-minant of the coefficients of A and B is zero. Thus, the phase speed c must satisfythe condition

����(c � Um) k2 + � �k2UT�UT

�k2 � 2�2� (c � Um)

�k2 + 2�2� + �

���� = 0

To protect the rights of the author(s) and publisher we inform you that this PDF is an uncorrected proof for internal businessuse only by the author(s), editor(s), reviewer(s), Elsevier and typesetter diacriTech. It is not allowed to publish this proofonline or in print. This proof copy is the copyright property of the publisher and is confidential until formal publication.

“Hakim: 11-ch07-213-254-9780123848666” — 2012/7/28 — 12:35 — page 218 — #6

218 CHAPTER | 7 Baroclinic Development

(7.10) can be rewritten as@

@t+

�Um + UT

� @@x

�@2 0

1

@x2 + �@ 0

1

@x= f0�p!0

2 (7.12)

@

@t+

�Um � UT

� @@x

�@2 0

3

@x2 + �@ 0

3

@x= � f0

�p!0

2 (7.13)

We now define the barotropic and baroclinic perturbations as

m ⌘� 0

1 + 03�/2; T ⌘

� 0

1 � 03�/2 (7.14)

Adding (7.12) and (7.13) and using the definitions in (7.14) yield@

@t+ Um

@

@x

�@2 m

@x2 + �@ m

@x+ UT

@

@x

✓@2 T

@x2

◆= 0 (7.15)

while subtracting (7.13) from (7.12) and combining with (7.11) to eliminate !02

yield@

@t+ Um

@

@x

� ✓@2 T

@x2 � 2�2 T

◆+ �

@ T

@x

+ UT@

@x

✓@2 m

@x2 + 2�2 m

◆= 0

(7.16)

where �2 ⌘ f 20 /[� (�p)2]. Equations (7.15) and (7.16) govern the evolution

of the barotropic (vertically averaged) and baroclinic (thermal) perturbationvorticities, respectively.

As in Chapter 5, we assume that wave-like solutions exist of the form

m = Aeik(x�ct), T = Beik(x�ct) (7.17)

Substituting these assumed solutions into (7.15) and (7.16) and dividing throughby the common exponential factor, we obtain a pair of simultaneous linearalgebraic equations for the coefficients A, B:

ikh�

c � Um�k2 + �

iA � ik3UTB = 0 (7.18)

ikh�

c � Um� ⇣

k2 + 2�2⌘

+ �i

B � ikUT

⇣k2 � 2�2

⌘A = 0 (7.19)

Because this set is homogeneous, nontrivial solutions will exist only if the deter-minant of the coefficients of A and B is zero. Thus, the phase speed c must satisfythe condition

����(c � Um) k2 + � �k2UT�UT

�k2 � 2�2� (c � Um)

�k2 + 2�2� + �

���� = 0

�� � � �

������

�����

To protect the rights of the author(s) and publisher we inform you that this PDF is an uncorrected proof for internal businessuse only by the author(s), editor(s), reviewer(s), Elsevier and typesetter diacriTech. It is not allowed to publish this proofonline or in print. This proof copy is the copyright property of the publisher and is confidential until formal publication.

“Hakim: 11-ch07-213-254-9780123848666” — 2012/7/28 — 12:35 — page 218 — #6

218 CHAPTER | 7 Baroclinic Development

(7.10) can be rewritten as@

@t+

�Um + UT

� @@x

�@2 0

1

@x2 + �@ 0

1

@x= f0�p!0

2 (7.12)

@

@t+

�Um � UT

� @@x

�@2 0

3

@x2 + �@ 0

3

@x= � f0

�p!0

2 (7.13)

We now define the barotropic and baroclinic perturbations as

m ⌘� 0

1 + 03�/2; T ⌘

� 0

1 � 03�/2 (7.14)

Adding (7.12) and (7.13) and using the definitions in (7.14) yield@

@t+ Um

@

@x

�@2 m

@x2 + �@ m

@x+ UT

@

@x

✓@2 T

@x2

◆= 0 (7.15)

while subtracting (7.13) from (7.12) and combining with (7.11) to eliminate !02

yield@

@t+ Um

@

@x

� ✓@2 T

@x2 � 2�2 T

◆+ �

@ T

@x

+ UT@

@x

✓@2 m

@x2 + 2�2 m

◆= 0

(7.16)

where �2 ⌘ f 20 /[� (�p)2]. Equations (7.15) and (7.16) govern the evolution

of the barotropic (vertically averaged) and baroclinic (thermal) perturbationvorticities, respectively.

As in Chapter 5, we assume that wave-like solutions exist of the form

m = Aeik(x�ct), T = Beik(x�ct) (7.17)

Substituting these assumed solutions into (7.15) and (7.16) and dividing throughby the common exponential factor, we obtain a pair of simultaneous linearalgebraic equations for the coefficients A, B:

ikh�

c � Um�k2 + �

iA � ik3UTB = 0 (7.18)

ikh�

c � Um� ⇣

k2 + 2�2⌘

+ �i

B � ikUT

⇣k2 � 2�2

⌘A = 0 (7.19)

Because this set is homogeneous, nontrivial solutions will exist only if the deter-minant of the coefficients of A and B is zero. Thus, the phase speed c must satisfythe condition

����(c � Um) k2 + � �k2UT�UT

�k2 � 2�2� (c � Um)

�k2 + 2�2� + �

���� = 0

To protect the rights of the author(s) and publisher we inform you that this PDF is an uncorrected proof for internal businessuse only by the author(s), editor(s), reviewer(s), Elsevier and typesetter diacriTech. It is not allowed to publish this proofonline or in print. This proof copy is the copyright property of the publisher and is confidential until formal publication.

“Hakim: 11-ch07-213-254-9780123848666” — 2012/7/28 — 12:35 — page 218 — #6

218 CHAPTER | 7 Baroclinic Development

(7.10) can be rewritten as@

@t+

�Um + UT

� @@x

�@2 0

1

@x2 + �@ 0

1

@x= f0�p!0

2 (7.12)

@

@t+

�Um � UT

� @@x

�@2 0

3

@x2 + �@ 0

3

@x= � f0

�p!0

2 (7.13)

We now define the barotropic and baroclinic perturbations as

m ⌘� 0

1 + 03�/2; T ⌘

� 0

1 � 03�/2 (7.14)

Adding (7.12) and (7.13) and using the definitions in (7.14) yield@

@t+ Um

@

@x

�@2 m

@x2 + �@ m

@x+ UT

@

@x

✓@2 T

@x2

◆= 0 (7.15)

while subtracting (7.13) from (7.12) and combining with (7.11) to eliminate !02

yield@

@t+ Um

@

@x

� ✓@2 T

@x2 � 2�2 T

◆+ �

@ T

@x

+ UT@

@x

✓@2 m

@x2 + 2�2 m

◆= 0

(7.16)

where �2 ⌘ f 20 /[� (�p)2]. Equations (7.15) and (7.16) govern the evolution

of the barotropic (vertically averaged) and baroclinic (thermal) perturbationvorticities, respectively.

As in Chapter 5, we assume that wave-like solutions exist of the form

m = Aeik(x�ct), T = Beik(x�ct) (7.17)

Substituting these assumed solutions into (7.15) and (7.16) and dividing throughby the common exponential factor, we obtain a pair of simultaneous linearalgebraic equations for the coefficients A, B:

ikh�

c � Um�k2 + �

iA � ik3UTB = 0 (7.18)

ikh�

c � Um� ⇣

k2 + 2�2⌘

+ �i

B � ikUT

⇣k2 � 2�2

⌘A = 0 (7.19)

Because this set is homogeneous, nontrivial solutions will exist only if the deter-minant of the coefficients of A and B is zero. Thus, the phase speed c must satisfythe condition

����(c � Um) k2 + � �k2UT�UT

�k2 � 2�2� (c � Um)

�k2 + 2�2� + �

���� = 0

�����

To protect the rights of the author(s) and publisher we inform you that this PDF is an uncorrected proof for internal businessuse only by the author(s), editor(s), reviewer(s), Elsevier and typesetter diacriTech. It is not allowed to publish this proofonline or in print. This proof copy is the copyright property of the publisher and is confidential until formal publication.

“Hakim: 11-ch07-213-254-9780123848666” — 2012/7/28 — 12:35 — page 218 — #6

218 CHAPTER | 7 Baroclinic Development

(7.10) can be rewritten as@

@t+

�Um + UT

� @@x

�@2 0

1

@x2 + �@ 0

1

@x= f0�p!0

2 (7.12)

@

@t+

�Um � UT

� @@x

�@2 0

3

@x2 + �@ 0

3

@x= � f0

�p!0

2 (7.13)

We now define the barotropic and baroclinic perturbations as

m ⌘� 0

1 + 03�/2; T ⌘

� 0

1 � 03�/2 (7.14)

Adding (7.12) and (7.13) and using the definitions in (7.14) yield@

@t+ Um

@

@x

�@2 m

@x2 + �@ m

@x+ UT

@

@x

✓@2 T

@x2

◆= 0 (7.15)

while subtracting (7.13) from (7.12) and combining with (7.11) to eliminate !02

yield@

@t+ Um

@

@x

� ✓@2 T

@x2 � 2�2 T

◆+ �

@ T

@x

+ UT@

@x

✓@2 m

@x2 + 2�2 m

◆= 0

(7.16)

where �2 ⌘ f 20 /[� (�p)2]. Equations (7.15) and (7.16) govern the evolution

of the barotropic (vertically averaged) and baroclinic (thermal) perturbationvorticities, respectively.

As in Chapter 5, we assume that wave-like solutions exist of the form

m = Aeik(x�ct), T = Beik(x�ct) (7.17)

Substituting these assumed solutions into (7.15) and (7.16) and dividing throughby the common exponential factor, we obtain a pair of simultaneous linearalgebraic equations for the coefficients A, B:

ikh�

c � Um�k2 + �

iA � ik3UTB = 0 (7.18)

ikh�

c � Um� ⇣

k2 + 2�2⌘

+ �i

B � ikUT

⇣k2 � 2�2

⌘A = 0 (7.19)

Because this set is homogeneous, nontrivial solutions will exist only if the deter-minant of the coefficients of A and B is zero. Thus, the phase speed c must satisfythe condition

����(c � Um) k2 + � �k2UT�UT

�k2 � 2�2� (c � Um)

�k2 + 2�2� + �

���� = 0

To protect the rights of the author(s) and publisher we inform you that this PDF is an uncorrected proof for internal businessuse only by the author(s), editor(s), reviewer(s), Elsevier and typesetter diacriTech. It is not allowed to publish this proofonline or in print. This proof copy is the copyright property of the publisher and is confidential until formal publication.

“Hakim: 11-ch07-213-254-9780123848666” — 2012/7/28 — 12:35 — page 218 — #6

218 CHAPTER | 7 Baroclinic Development

(7.10) can be rewritten as@

@t+

�Um + UT

� @@x

�@2 0

1

@x2 + �@ 0

1

@x= f0�p!0

2 (7.12)

@

@t+

�Um � UT

� @@x

�@2 0

3

@x2 + �@ 0

3

@x= � f0

�p!0

2 (7.13)

We now define the barotropic and baroclinic perturbations as

m ⌘� 0

1 + 03�/2; T ⌘

� 0

1 � 03�/2 (7.14)

Adding (7.12) and (7.13) and using the definitions in (7.14) yield@

@t+ Um

@

@x

�@2 m

@x2 + �@ m

@x+ UT

@

@x

✓@2 T

@x2

◆= 0 (7.15)

while subtracting (7.13) from (7.12) and combining with (7.11) to eliminate !02

yield@

@t+ Um

@

@x

� ✓@2 T

@x2 � 2�2 T

◆+ �

@ T

@x

+ UT@

@x

✓@2 m

@x2 + 2�2 m

◆= 0

(7.16)

where �2 ⌘ f 20 /[� (�p)2]. Equations (7.15) and (7.16) govern the evolution

of the barotropic (vertically averaged) and baroclinic (thermal) perturbationvorticities, respectively.

As in Chapter 5, we assume that wave-like solutions exist of the form

m = Aeik(x�ct), T = Beik(x�ct) (7.17)

Substituting these assumed solutions into (7.15) and (7.16) and dividing throughby the common exponential factor, we obtain a pair of simultaneous linearalgebraic equations for the coefficients A, B:

ikh�

c � Um�k2 + �

iA � ik3UTB = 0 (7.18)

ikh�

c � Um� ⇣

k2 + 2�2⌘

+ �i

B � ikUT

⇣k2 � 2�2

⌘A = 0 (7.19)

Because this set is homogeneous, nontrivial solutions will exist only if the deter-minant of the coefficients of A and B is zero. Thus, the phase speed c must satisfythe condition

����(c � Um) k2 + � �k2UT�UT

�k2 � 2�2� (c � Um)

�k2 + 2�2� + �

���� = 0

To protect the rights of the author(s) and publisher we inform you that this PDF is an uncorrected proof for internal businessuse only by the author(s), editor(s), reviewer(s), Elsevier and typesetter diacriTech. It is not allowed to publish this proofonline or in print. This proof copy is the copyright property of the publisher and is confidential until formal publication.

“Hakim: 11-ch07-213-254-9780123848666” — 2012/7/28 — 12:35 — page 219 — #7

7.2 | Normal Mode Baroclinic Instability: A Two-Layer Model 219

which gives a quadratic dispersion equation in c:⇣

c � Um

⌘2k2

⇣k2 + 2�2

⌘+ 2

⇣c � Um

⌘�⇣

k2 + �2⌘

+h�2 + U2

Tk2⇣

2�2 � k2⌘i

= 0(7.20)

which is analogous to the linear wave dispersion equations developed inChapter 5. The dispersion relationship in (7.20) yields for the phase speed

c = Um ���k2 + �2�

k2�k2 + 2�2

� ± �1/2 (7.21)

where

� ⌘ �2�4

k4�k2 + 2�2

�2 �U2

T

�2�2 � k2�

�k2 + 2�2

�

We have now shown that (7.17) is a solution for the system (7.15) and (7.16)only if the phase speed satisfies (7.21). Although (7.21) appears to be rathercomplicated, it is immediately apparent that if �< 0, the phase speed will havean imaginary part and the perturbations will amplify exponentially. Before dis-cussing the general physical conditions required for exponential growth, it isuseful to consider two special cases.

As the first special case we let UT = 0 so that the basic state thermal windvanishes and the mean flow is barotropic. The phase speeds in this case are

c1 = Um � �k�2 (7.22)

and

c2 = Um � �⇣

k2 + 2�2⌘�1

(7.23)

These are real quantities that correspond to the free (normal mode) oscillationsfor the two-level model with a barotropic basic state current. The phase speedc1 is simply the dispersion relationship for a barotropic Rossby wave with no ydependence (see Section 5.7). Substituting from (7.22) for c in (7.18) and (7.19),we see that in this case B = 0 so that the perturbation is barotropic in structure.The expression of (7.23), however, may be interpreted as the phase speed for aninternal baroclinic Rossby wave. Note that c2 is a dispersion relationship anal-ogous to the Rossby wave speed for a homogeneous ocean with a free surface,which was given in Problem 7.16. However, in the two-level model, the factor2�2 appears in the denominator in place of the f 2

0 /gH for the oceanic case.In each of these cases there is vertical motion associated with the Rossby

wave so that static stability modifies the wave speed. It is left as a problem forthe reader to show that if c2 is substituted into (7.18) and (7.19), the resultingfields of 1 and 3 are 180� out of phase so that the perturbation is baroclinic,

To protect the rights of the author(s) and publisher we inform you that this PDF is an uncorrected proof for internal businessuse only by the author(s), editor(s), reviewer(s), Elsevier and typesetter diacriTech. It is not allowed to publish this proofonline or in print. This proof copy is the copyright property of the publisher and is confidential until formal publication.

“Hakim: 11-ch07-213-254-9780123848666” — 2012/7/28 — 12:35 — page 219 — #7

7.2 | Normal Mode Baroclinic Instability: A Two-Layer Model 219

which gives a quadratic dispersion equation in c:⇣

c � Um

⌘2k2

⇣k2 + 2�2

⌘+ 2

⇣c � Um

⌘�⇣

k2 + �2⌘

+h�2 + U2

Tk2⇣

2�2 � k2⌘i

= 0(7.20)

which is analogous to the linear wave dispersion equations developed inChapter 5. The dispersion relationship in (7.20) yields for the phase speed

c = Um ���k2 + �2�

k2�k2 + 2�2

� ± �1/2 (7.21)

where

� ⌘ �2�4

k4�k2 + 2�2

�2 �U2

T

�2�2 � k2�

�k2 + 2�2

�

We have now shown that (7.17) is a solution for the system (7.15) and (7.16)only if the phase speed satisfies (7.21). Although (7.21) appears to be rathercomplicated, it is immediately apparent that if �< 0, the phase speed will havean imaginary part and the perturbations will amplify exponentially. Before dis-cussing the general physical conditions required for exponential growth, it isuseful to consider two special cases.

As the first special case we let UT = 0 so that the basic state thermal windvanishes and the mean flow is barotropic. The phase speeds in this case are

c1 = Um � �k�2 (7.22)

and

c2 = Um � �⇣

k2 + 2�2⌘�1

(7.23)

These are real quantities that correspond to the free (normal mode) oscillationsfor the two-level model with a barotropic basic state current. The phase speedc1 is simply the dispersion relationship for a barotropic Rossby wave with no ydependence (see Section 5.7). Substituting from (7.22) for c in (7.18) and (7.19),we see that in this case B = 0 so that the perturbation is barotropic in structure.The expression of (7.23), however, may be interpreted as the phase speed for aninternal baroclinic Rossby wave. Note that c2 is a dispersion relationship anal-ogous to the Rossby wave speed for a homogeneous ocean with a free surface,which was given in Problem 7.16. However, in the two-level model, the factor2�2 appears in the denominator in place of the f 2

0 /gH for the oceanic case.In each of these cases there is vertical motion associated with the Rossby

wave so that static stability modifies the wave speed. It is left as a problem forthe reader to show that if c2 is substituted into (7.18) and (7.19), the resultingfields of 1 and 3 are 180� out of phase so that the perturbation is baroclinic,

������������ ��

���������

!����� ��������

0,0,

221 ====vyyCBA

9

0

1 5

2220

221

220

22,

,2

2,

yl

bs

vyy

lb

s

+D-=-=

+D-=-=

kpfi

Bkp

fiCBA

8 9

0 5 1

50 0 5

1 0,0 ¹= b�TU

a)21 k

Uc mb

-= 222 2lb+

-=k

Uc mb)

2�21

22

22

22

÷÷ø

öççè

æ+-

±=ll

kkUUc Tm

l2=ck

ckk ! ����

ckk ! �����;�������

21

22

22

22

÷÷ø

öççè

æ+-

=kkkUkc Ti l

l

max

,

ii kckck

=

-= 1220 l

,0,0 ¹= TU�b

��������

T

i

Ukcl2

l2k

�����

21

22

22

22

÷÷ø

öççè

æ+-

=kkkUkc Ti l

l

���������

( )

KmL

Kmk

L

hPapsf

Nsm

c

cc

4700

3000250010

1022

1

40

3321

~

.~

,,/~

,/~

p

s

=

=D

´-

-

-------������

-----����

�� 00 ¹¹ TU,b ( )( )

( )( )2224

444242

21

222

22

242

lllbd

dllb

+

--º

±++

-=

kkkkU

kkkUc

T

m ,

l2=ck,ckk ! ��

,ckk ! ���

( ),

4 21

442

2

kkUT

-l

bl!

To protect the rights of the author(s) and publisher we inform you that this PDF is an uncorrected proof for internal businessuse only by the author(s), editor(s), reviewer(s), Elsevier and typesetter diacriTech. It is not allowed to publish this proofonline or in print. This proof copy is the copyright property of the publisher and is confidential until formal publication.

“Hakim: 11-ch07-213-254-9780123848666” — 2012/7/28 — 12:35 — page 221 — #9

7.2 | Normal Mode Baroclinic Instability: A Two-Layer Model 221

relative strength of the accompanying vertical circulation must increase as thewavelength of the disturbance decreases. Because static stability tends to resistvertical displacements, the shortest wavelengths will thus be stabilized.

It is also of interest that with � = 0 the critical wavelength for instabilitydoes not depend on the magnitude of the basic state thermal wind UT . Thegrowth rate, however, does depend on UT . According to (7.17) the timedependence of the disturbance solution has the form exp(–ikct). Thus, theexponential growth rate is ↵ = kci, where ci designates the imaginary part ofthe phase speed. In the present case

↵ = kUT

✓2�2 � k2

2�2 + k2

◆1/2

(7.25)

so that the growth rate increases linearly with the mean thermal wind.Returning to the general case where all terms are retained in (7.21), the sta-

bility criterion is understood most easily by computing the neutral curve, whichconnects all values of UT and k for which � = 0 so that the flow is marginallystable. From (7.21), the condition � = 0 implies that

�2�4

k4�2�2 + k2

� = U2T

⇣2�2 � k2

⌘(7.26)

This complicated relationship between UT and k can best be displayed bysolving (7.26) for k4/2�4, yielding

k4.⇣

2�4⌘

= 1 ±h1 � �2

.⇣4�4U2

T

⌘i1/2

In Figure 7.3 the nondimensional quantity k2/2�2, which is proportionalto the square of the zonal wave number, is plotted against the nondimensionalparameter 2�2UT/�, which is proportional to the thermal wind. As indicatedin Figure 7.3, the neutral curve separates the unstable region of the UT , kplane from the stable region. It is clear that the inclusion of the � effect servesto stabilize the flow, since now unstable roots exist only for |UT | > �/(2�2).In addition, the minimum value of UT required for unstable growth dependsstrongly on k. Thus, the � effect strongly stabilizes the long wave end of thewave spectrum (k ! 0). Again, the flow is always stable for waves shorter thanthe critical wavelength Lc =

p2⇡/�.

This long-wave stabilization associated with the � effect is caused by therapid westward propagation of long waves (i.e., Rossby wave propagation),which occurs only when the � effect is included in the model. It can be shownthat baroclinically unstable waves always propagate at a speed that lies betweenmaximum and minimum mean zonal wind speeds. Thus, for the two-level modelin the usual midlatitude case where U1 > U3 > 0, the real part of the phase speedsatisfies the inequality U3 < cr < U1 for unstable waves. In a continuous atmo-sphere, this would imply that there must be a level where U = cr. Such a

�����

Unstable

Stable

22lb /TU

2

2

2lk