» Some of the algebraic details in the Eady solution are complicated - especially:

Chapter 9

Synoptic-scale instability and

cyclogenesis – Part II

» Some of the algebraic details in the Eady solution are complicated - especially:

- the calculation of w and b, and

- the inclusion of a beta effect (∂f/∂y ≠ 0) renders the eigenvalue problem analytically intractable.

A two-layer model

» An even simpler model which does not suffer these

limitations may be formulated at the sacrifice of vertical resolution.

» The procedure is to divide the atmosphere into two layers:

H

w

w

w

u

, v

, ψ

u

, v

, ψ

0 1 2 3 4

H/2

u y v

x t u

x v

y f f w

n

z

n

n

n

n n n

n

= − = L + +

NM O

QP ^{∇ + =} L

NM O

∂ψ QP

∂

∂ψ

∂

∂

∂

∂

∂

∂

∂ ψ ∂

, , (

)

∂

We express [∂w/∂z]

as central differences =>

» Vertical derivatives in the quasi-geostrophic equations are then replaced by central-difference approximations.

» In each layer,

∂

∂

∂

∂ w

z

w w

H

w z

w w

H

L NM O

QP ^{= −} L

NM O

QP ^{= −}

1

0 2

1

2 3

2 4

1 2

,

We impose the boundary conditions w

= 0, w

= 0.

∂

∂

∂

∂

∂

∂ ψ

t u

x v

y f f

H w

+ +

L NM O

QP ^{∇ + = −}

1 1

2 1

0 2

( ) 2

and ^∂

∂

∂

∂

∂

∂ ψ

t u

x v

y f f

H w

+ +

L NM O

QP ^{∇ + = +}

3 3

2 3

0 2

( ) 2

w

satisfies _∂ ^∂ _∂ ^∂ _∂ ^{∂ ∂ψ} _∂ t u

x v

y f

z N w

+ +

L NM O

QP ^{+ =}

2 2

2 2

2

0

Since u

and v

are not carried, we compute them by averaging u

and u

,

∂ψ

/ ∂ z = ( ψ

− ψ

) /

H

∂

∂

∂

∂

∂

∂ ψ ψ

t u u

x v v

y

HN f w

+ +

L NM ^{+ +} O

QP ^{− + =}

1 2

1

2 2 0

1 3 1 3 1 3

2

0

( ) ( ) ( )

The coefficient of w

may be written as 2

4

2

4

0

2 2 0

2

0

2 2

f H

N H f

f H

L

⋅ = = γμ

where γ = 2f

/H and μ = 2/L

∂

∂

∂

∂

∂

∂ ψ ψ

t u u

x v v

y

HN f w

+ +

L NM ^{+ +} O

QP ^{− + =}

1 2

1

2 2 0

1 3 1 3 1 3

2

0

( ) ( ) ( )

L

is the Rossby length

∂

∂

∂

∂

∂

∂ ψ

t u

x v

y f f

H w

+ +

L NM O

QP ^{∇ + = −}

1 1

2 1

0 2

( ) 2

∂

∂

∂

∂

∂

∂ ψ

t u

x v

y f f

H w

+ +

L NM O

QP ^{∇ + = +}

3 3

2 3

0 2

( ) 2

∂

∂

∂

∂

∂

∂ ψ ψ γμ

t u u

x v v

y w

+ +

L NM ^{+ +} O

QP ^{− = −}

1 2

1

1 3

2

2

( ) ( ) ( )

Full set of nonlinear equations

n n

′

ψ = ψ + ψ where

n n

ψ′ << ψ

Let the streamfunction of the basic zonal flow in each layer

n

yU (n

1, 3)

ψ = − =

and consider small perturbations to this

Perturbation method

∂

∂

∂

∂

∂ ψ

∂ β ∂ψ

∂ γ

t U

x x x w

L +

NM O

QP ^{′ + ′ = −}

1

2 1 2

1

2

∂

∂

∂

∂

∂ ψ

∂ β ∂ψ

∂ γ

t U

x x x w

L +

NM O

QP ^{′ + ′ =}

3

2

3 2

3

2

∂

∂

∂

∂ ψ ψ ∂

∂ ψ ψ γμ

t U U

x

U U

x w

+ +

L NM O

QP ^{′ − ′ − − ′ + ′ = −}

1

2

2

1 3

1 3

2

( ) ( ) b g ( )

assuming a perturbation for which ∂/∂y ≡ 0.

The linearized equations

» The equations form a linear system with constant coefficients and therefore have solutions of the form

ψ ψ

φ φ

′

′

⎡

⎣

⎢ ⎢

⎢ ⎢

⎤

⎦

⎥ ⎥

⎥ ⎥

=

⎡

⎣

⎢ ⎢

⎢ ⎢

⎤

⎦

⎥ ⎥

⎥ ⎥

− 1

3 2

1 3

w w

e ^{ik x ct}

~

( ) , Solution method

constants

» Substitution gives a set of linear homogeneous algebraic

equations:

[ ]

ik c ( − U k

)

+ β φ

+ γ w ~ = 0

[ ]

ik c ( − U

) k

+ β φ

− γ w ~ = 0

[ ] [ ^]

− ik c − U

φ

+ ik c − U

φ

+ γμ

w ~ = 0

These have a non-trivial solution for φ

, φ

and if and only if the determinant of coefficients is zero.

w ~

A quadratic equation for the eigenvalues c:

[ ] [ ]

c k k c k U U k k

U U k k U U k k U U

2 4 2 2 2 2 1

2 1 3 4 2 2

1 3

4 2 2 2

1 3

2 2 2 2

1 3

2

2 2 2

2 0

+ + + − + +

+ + + − + + + − =

μ β μ μ

μ β β μ μ

( ) ( ) ( )

( ) ( ) ( ) ( ) .

Write

1 1

m 2 1 3 T 2 1 3

U = (U + U ), U = (U − U )

c U k

k k

=

− +

+ +

β μ

μ δ

( )

( )

/

2 2

2 2 2

1 2

2

δ β μ

μ

μ

= μ

+ − −

+

2 4

4 2 2 2

2

2 2

2 2

2

2

2

k k U k

T k

( )

( )

( )

Eigenvalue equation

2 2

2 2 1/ 2

2 T 2 2

3 1

2 2

2

2 2 1/ 2

3 1

T 2 2

(k )

U k k

(c U )k k 2

(k )

(c U )k

U k k

k 2 β + μ

+ + δ

− + β

φ = − = − + μ

β + μ

φ − + β − − δ

+ μ

[ ]

ik c ( − U k

)

+ β φ

+ γ w ~ = 0

[ ]

ik c ( − U

) k

+ β φ

− γ w ~ = 0

Put

2 2 2

1 / 2 2 2

2 2 2 T 2 2

T

2 k

q U p p q

k (k 2 )U k 2

⎡ ⎤

βμ μ −

= + μ δ = = − ⎢ ⎣ + μ ⎥ ⎦

φ

1 φ

= 1 + +

− −

⎡

⎣⎢

⎤

⎦⎥

q p

q p

1. No vertical shear, U

= 0, i.e., U

= U

.

δ βμ

μ

1 2 2

2 2 2

2

/

( )

= ± k k +

Then c U ß

m

k

= −

or c U ß

m

k

= −

+

2 2

2 μ

Some special cases

No vertical shear, U

= 0, i.e., U

= U

.

w ~ = 0 φ

= φ

,

ψ'

and ψ'

are exactly in phase =>

the ridges and troughs are in phase.

This solution corresponds with a barotropic Rossby wave as the dispersion relation suggests.

w ~ = 0

c U ß

m

k

= −

there is no interchange of fluid

between the two layers.

No vertical shear, U

= 0, i.e., U

= U

.

c U ß

m

k

= −

+

2 2

2 μ

» The waves in the upper and lower layers are exactly out of phase, i.e., ψ'

= ψ '

e

.

» Thus at meridians (x-values) where the perturbation velocities are poleward in the upper layer, they are equatorward in the lower layer and vice versa.

» This mode is called a baroclinic, or internal, Rossby wave.

» The presence of the free mode of this type is a weakness of the two-layer model; see Holton, p. 220.

» The mode does not correspond with any free oscillation of the atmosphere, but such wave modes do exist in the oceans.

φ

= − φ

2. No beta effect, β = 0 , finite shear U

≠ 0.

δ μ

μ

1 2

2 2

2 2

2

2

/

/

= −

+

⎡

⎣ ⎢ ⎤

⎦ ⎥ U k

T

k

imaginary if k

< 2μ

Let δ

= ic

e ^{ik x ct ( − )} = e ^{ik x U t ( −}

⁾ e ^{kc t}

The wave grows or decays exponentially with time,

according to the sign of c

, and propagates zonally

with phase speed U

.

In , q = 0

No beta effect, β = 0 , finite shear U

≠ 0.

When k

< 2μ

, p

= − (2μ

− k

)/(2μ

+ k

) = −p

, say.

φ

1 φ

= 1 + +

− −

⎡

⎣⎢

⎤

⎦⎥

q p q p

φ

φ

2 02

1

1

1

= + 1

− = +

+ =

ip ip

ip

p e

( )

where θ = tan

p

.

Note that | p

| < 1 and if p

> 0, 0 < θ < . ^π ₄

No beta effect, β = 0 , finite shear U

≠ 0.

ψ φ

ψ φ

θ

1 3

2

3 3

′ − +

′ −

=

=

e e

e e

kc t ik x U t i

kc t ik x U t

i m

i m

( )

( )

, ,

» If c

> 0, the upper wave is 2θ radians in advance of the lower wave => again the trough and ridge positions are displaced westwards with height, as in the growing Eady wave.

» The threshold for instability occurs when k

= 2μ

, or k = 2.82/L

, waves of large wavenumber (shorter wavelength) being stable.

» This should be compared with the Eady stability criterion

which requires that s

< 1.2 or k < 2.4/L

.

» It is evident that the growth rate of a disturbance is related to the degree of westward displacement of the trough with

height.

» This accords with synoptic experience and provides

forecasters with a rule for judging whether or not a lower pressure centre will intensify during a forecast period.

» This rule is based on a comparison of the positions of the upper-level trough (which may even have one or two closed isopleths) and the surface low.

» The rule will be investigated further in the next chapter.

Forecasting rule

3. The general case, β ≠ 0 , U

≠ 0.

Algebraically more complicated.

c U k

k k

=

− +

+ +

β μ

μ δ

( )

( )

/

2 2

2 2 2

1 2

2

2 4 2 2

2

4 2 2 2 T 2 2

(2 k ) k (k 2 ) U (k 2 )

β μ μ −

δ = −

+ μ + μ

Neutral stability <=> δ = 0

β μ μ

μ μ

μ

β μ

2 4

4 2 2 2

2

2 2

2 2

4 4

2

4 2

1 2

2

2 2

2 1 1

4

k k U k

k k

U

T

T

( )

( )

( ) ,

.

/

+ = −

+

= ± − ⎡

⎣ ⎢ ⎤

⎦ ⎥

As before

5.0

1.0 4.0

3.0

2.0

1.0

0.0

0.0 0.5

2 μ

β

U

k

2 μ

δ = 0 δ > 0

Stable

δ < 0

Unstable

» The relative simplicity of the two-layer model makes it especially suitable for studying the energy conversions associated with baroclinic waves.

» The following discussion closely parallels that of Holton,

§9.3.2.

The energetics of baroclinic waves

∂

∂

∂

∂

∂ ψ

∂ β ∂ψ

∂ γ

t U

x x x w

L +

NM O

QP ^{′ + ′ = −}

1

2 1 2

1

2

∂

∂

∂

∂

∂ ψ

∂ β ∂ψ

∂ γ

t U

x x x w

L +

NM O

QP ^{′ + ′ =}

3

2 3 2

3

2

∂

∂

∂

∂ ψ ψ ∂

∂ ψ ψ γμ

t U U

x

U U

x w

+ +

L NM O

QP ^{′ − ′ − − ′ + ′ = −}

1

2

2

1 3

1 3 2

( ) ( ) b g ( )

Multiply by −ψ'

Multiply by −ψ'

Multiply by ψ '

− ψ '

omit primes, and take zonal averages denoted by

< ^{( )} > = _λ ¹ z

^{( ) dx}

λ is the perturbation wavelength.

The energy equations

d

dt < 1 v >= < w >

2

2

2 1

γ ψ

d

dt < 1 v >= − < w >

2

2

2 3

γ ψ

d

dt U

x w

< − >=

< − + >

−

< − >

1

2

2

1 3 1 3

2

2 1 3

( ) ( ) ( )

( ) .

ψ ψ ψ ψ ∂

∂ ψ ψ

γμ ψ ψ

2 2

1 1 1

1 2 1 2 1

1 1 2

1

2 1 2

1 1 2 1 1

t x x t x x t

0 1 (v ) .

x t x 2 t

U 1 U v 0.

x 2 x

x

⎡ ⎤

∂ ∂ ψ ∂ ⎡∂ψ ⎤ ∂ ⎡ ∂ ∂ψ⎡ ⎤⎤

− < ψ ∂ ⎢⎣ ∂ ⎥⎦>= − < ψ ∂ ⎢⎣ ∂ ⎥⎦ >= − < ∂ ⎢⎣ψ ∂ ⎢⎣ ∂ ⎥⎦⎥⎦ > +

∂ψ ∂ ∂ψ⎡ ⎤ ∂

< ∂ ∂ ⎢⎣ ∂ ⎥⎦ >= + < ∂ >

∂ ⎡∂ψ ⎤ ∂

− < ψ ∂ ⎢⎣ ∂ ⎥⎦ >= < ∂ >=

Note that

Likewise

Define K ′ =

H < v

+ v

>

the perturbation kinetic energy averaged over a wavelength per unit meridional direction.

d

dt < 1 v >= < w >

2

2

2 1

γ ψ d

dt < 1 v >= − < w >

2

2

2 3

γ ψ

+

2 1 3 2 2

dK 1 1

H w ( ) H w b

dt 2 2

′ = γ < ψ − ψ >= < >

an equation for the rate-of-change of the

average perturbation kinetic energy.

σ ₂ = f ₀ a ∂ψ ∂ / z f b ₂ = f ψ ₁ − ψ ₃ g / ¹ ₂ H = γ ψ b ₁ − ψ ₃ g

In the continuous model, available potential energy is defined as

We approximate the contribution to this from the perturbation by defining

2

2 2

1 2

1 3

2 2

b

1 1

P H H ( )

2 N 4

< >

′ = × = μ < ψ − ψ >

2 2

b 1

v 2 N dV

∫

Recall that

to be the average perturbation available potential energy per

unit meridional direction.

You may wonder why the operation is replaced here z dV by rather than H. ¹ ₂ H

» It turns out to be necessary to do this for energy consistency.

» Since b is defined only at one level (i.e., level 2), the system knows only about the available potential energy between levels 3 and 1.

» With this definition for P', the model is formally equivalent to the two-layer model assuming immiscible fluids with a free fluid interface as studied by Pedlosky (1979; see §7.16).

» Holton does not point out this subtlety in defining P´.

gives

T 2 2 2 2

2

dP 2f H

U b v w b

dt N 2

′ = < > − < >

d

dt U

x w

< − >=

< − + >

−

< − >

1

2

2

1 3 1 3

2

2 1 3

( ) ( ) ( )

( ) .

ψ ψ ψ ψ ∂

∂ ψ ψ

γμ ψ ψ

Then

0

T 2 2 2 2

2

2f

dP H

U b v w b

dt N 2

′ = < > − < >

» This term correlates upward motion with positive buoyancy and downward motion with negative buoyancy.

» It represents a conversion of perturbation available potential energy into perturbation kinetic energy.

» It is the only source of K' and is the sink term of P'.

2 1 3 2 2

dK 1 1

H w ( ) H w b

dt 2 2

′ = γ < ψ − ψ >= < >

Interpretation

0

T 2 2 2 2

2

dP 2f H

U b v w b

dt N 2

′ = < > − < >

» This term correlates poleward motion with positive buoyancy between levels 1 and 3, and equatorward motion with

negative buoyancy.

» It is proportional to the vertical shear of the basic flow U

, or, equivalently, to the basic meridional temperature gradient.

» It represents the conversion of mean available potential

energy of the basic flow into perturbation available potential energy and is a source term in the above equation.

» Clearly, this term must exceed the second term in the

equation if the disturbance is to grow.

» Introduce the notation C(A, B) to denote a rate-of-conversion of energy form A to energy form B.

» Then C(A, B) = −C(B, A), and

» In particular

o

T 2 2

2

C(P, P ) 2f U b v

′ = N < >

2 2

C(P , K ) 1 H w b

′ ′ = 2 < >

Then

2 1 3 2 2

dK 1 1

H w ( ) H w b

dt 2 2

′ = γ < ψ − ψ >= < >

0

T 2 2 2 2

2

dP 2f H

U b v w b

dt N 2

′ = < > − < >

Energy conversions in a block diagram

dK

dt ′ = C P K ( ′ ′ , ) ^dP

dt ′ = C P P ( , ′ − ) C P K ( ′ , ′ )

and

P

′ P

′ K

C P P ( , ′ ) C P K ( ′ ′ , )

Adding the above equations gives

d

dt ( K ′ + ′ = P ) C P P ( , ′ )

The rate-of-change of total perturbation energy

K' + P', is just . C P P ( , ′ )

» Since the available potential energy of the basic flow per unit volume is finite, exponential growth of a perturbation cannot continue indefinitely.

» The foregoing theories assume that the perturbation remains sufficiently small so that changes in the mean flow due to the presence of the wave can be ignored.

» When the wave amplitude grows to a significant amplitude, its interaction with the mean flow cannot be ignored and the depletion of the mean flow available potential energy is

reflected in a reduced growth rate.

Large amplitude waves

» To study such finite amplitude effects necessarily requires a nonlinear analysis in which mean flow changes are determined as part of the solution.

» Such analyses are algebraically complicated and beyond the scope of these lecture.

Nonlinear Theory

» Differential solar heating between the equatorial and polar regions helps to maintain the available potential energy

associated with the middle latitude westerly winds.

» The baroclinic instability of the westerlies leads to the growth of extra-tropical cyclones.

» These cyclones transport heat polewards and upwards, reducing the mean meridional temperature gradient. (i.e., depleting available potential energy) and increasing the vertical stability.

The role of baroclinic waves in the atmosphere's

general circulation

» Extra-tropical cyclones act together with planetary waves to reduce the meridional temperature contrasts which would occur if the

earth's atmosphere were in radiative equilibrium.

» Therefore, both types of waves are important components of the

atmosphere's "air-conditioning" system.

The End