¾ The occurrence of extra-tropical cyclones is a manifestation of the inherent instability of the zonal ‘westerly’ winds of middle latitudes

Chapter 9

Synoptic-scale instability and cyclogenesis

¾ Look at mean sea level isobaric charts => one notices synoptic-scale vortices or low pressure centres, also called extra-tropical cyclones, depressions, or simply ’lows’.

Extra-tropical cyclones

¾ These vortices play an important role in the dynamics of the atmosphere's general circulation and contribute together with their associated fronts to much of our ‘bad weather’.

¾ The occurrence of extra-tropical cyclones is a manifestation of the inherent instability of the zonal ‘westerly’ winds of middle latitudes.

¾ We begin by considering the energy source for the instability.

¾ Then consider a simple model for cyclogenesis(i.e. cyclone growth).

Extra-tropical cyclones

¾ On average, the tropospheric winds in middle latitudes are westerly and increase in strength with height.

¾ They are also in approximate thermal wind balance with the poleward temperature gradient associated with differential solar heating.

N S

Vρ

Vθ Vθ

Vρ

y

southern hemisphere northern hemisphere

equatorial

region z

. .

. .

south pole north pole

∂u /∂z > 0

The middle latitude 'westerlies'

¾ The atmosphere has an enormous potential energy measured in the usual way:

atmosphere

ρgzdV

∫

¾ Only a small fraction of this is available for conversion to kinetic energy.

¾ The precise amount available is the actual potential energyminusthe potential energy obtained after an adiabatic rearrangement of the density field so that the isentropes (surfaces of constant θ) are horizontal and in stable hydrostatic equilibrium.

Available potential energy

y z

isentropes Vθ

A

B (1)

A B

(2)

Consider the adiabatic interchange of two air parcels A and B in the meridional plane...

ρ ρ ρ= _*+ ₀( )z +ρ( , )y z + ′ρ( , , , )x y z t Let us write:

Either the volume average of ρover the whole flow domain,

or the surface density

The horizontal average of ρ−ρ_*

The zonal average of ρ − ρ_∗ − ρ₀(z)

* 0

[ (z ) ( y, z )]

ρ − ρ + ρ + ρ

A zonal averageis an average in thex-, or eastward-direction

( )⁼_X¹ ∫₀^{X ( )dx}

e.g. the length of a latitude circle

Note that represents the deviation of the density field from hydrostatic equilibrium

Also, by the definition of the averaging operator ρ( , )y z + ′ρ( , , , )x y z t

ρ′ ≡0

In practice, for a Boussinesq fluid

max{ 0(z) , (y,z) , (x, y,z, t)}

ρ >>∗ ρ ρ ρ′

Buoyancy force _{b g} ^0(z)

g g

b b , say

∗

∗

∗ ∗

ρ − ρ − ρ

= − ρ

ρ ρ′

= − − = + ′

ρ ρ

An air parcel displaced a vertical distance ξfrom equilibrium experiences a restoring force b = −N²ξper unit mass.

The work done in producing such a displacement is

2 2 2 2

1 1

2 2

0

bdz N b / N

ξ = ξ =

∫

b = −N²ξ assuming thatNis a constant The change in potential energy due to an adiabatic rearrangement

of the density field from equilibrium is

2

2 2

2 2

atmosphere atmosphere

b 1

dV (b b ' ) dV

2N = 2N +

∫ ∫

A measure of the available potential energy - APE.

When no disturbance (b´ = 0), the APE of a zonal flow is related to the deviation of the local density from the horizontal average at that level.

Zonal flow configuration in the Eady problem (northern hemisphere).

f z

x

warm cold

isentropes

H u

b fU y

= − H

u U

H z

=

Baroclinic instability: the Eady problem

¾ Assumptions:

- Boussinesq liquid.

- N is a constant.

- f is a constant

¾ Basic-state streamfunction => ^Uyz ψ = −H

¾ Basic-state potential vorticity =>

2 2

2h 2 2

q f f f cons tan t

N z

= ∇ ψ + + ∂ ψ = =

∂

The zonal flow satisfies the potential vorticity equation exactly.

The Eady model

We consider small perturbations to the zonal flow: put ψ ψ ψ= + ′ q= + ′q q

(u u ') v ' (q q ') 0

t x y

⎡∂ + + ∂ + ∂ ⎤ + =

⎢∂ ∂ ∂ ⎥

⎣ ⎦

linearize =>

2 2 2 2

2 2 2 2

' ' f '

u 0

t x x y N z

⎡ ⎤

∂ ∂ ∂ ψ ∂ ψ ∂ ψ

⎡ + ⎤ ⎢ + + ⎥=

⎢∂ ∂ ⎥ ∂ ∂ ∂

⎣ ⎦ ⎣ ⎦

Perturbation solution

w = 0 at the ground(z = 0) and at the model tropopause,z = H.

2

g h b N w 0

t

⎡∂ + ⋅∇ ⎤ + =

⎢∂ ⎥

⎣ u ⎦

u u ' v ' (b b ') 0

t x x y

⎡∂ + ∂ + ∂ + ∂ ⎤ + =

⎢∂ ∂ ∂ ∂ ⎥

⎣ ⎦

b fU

y H

∂ = −

∂

b f z

∂ψ′

′ = ∂

' U '

u 0

t x z H x

∂ ∂ ∂ψ ∂ψ

⎡ + ⎤ − =

⎢∂ ∂ ⎥ ∂ ∂

⎣ ⎦ at z = 0, H

z = H

z = 0

Boundary conditions

For maximum simplicity consider 2-D disturbances with ∂/∂y≡ 0.

Assume that an arbitrary disturbance can be expressed as a sum of Fourier modes.

Consider a single Fourier componentwith ψ( , , )x z t = ψ$( )z e^{ik x ct( − )}

k andc are constants and 'the real part' is implied.

The objective is to determinec as a function of wavenumberk, and the corresponding eigenfunction .ψ$( )z

Substitution =>

d dz

N k f

2 2

2 2

2 0

$ $

ψ − ψ =

d dz

N k f

2 2

2 2

2 0

$ $

ψ − ψ=

Put z' = z/H =>

d dz

2 2

4s2 0

$ $

ψ ψ

′ − =

4s²

2 2 2

2 2 2

= N H = f k L_R k

L_R= NH/f is called the Rossby radius of deformation.

d dZ

2 2

4s2 0

$ $

ψ − ψ = A gain in symmetry is obtained if we put

′ = + = +

z Z ¹₂ c ¹₂U UC z = H

z = 0

z' = 1

z' = 0

Z = ¹₂ Z

Z = − ¹₂ Z = 0

nondimensional phase speed of the wave

(C−Z)ψ$ _Z +ψ$ = 0 at Z = ¹₂,− ¹₂ Mathematical niceties

$( ) sinh cosh

ψ Z =A 2sZ+B 2sZ

Solution is:

Boundary conditions give

[ (2s C+¹₂) cosh sinh ]s s A+[coshs−2(C+¹₂)sinh ]s B=0

[ (2s C−¹₂) coshs+sinh ]s A+[coshs−2(C− ¹₂)sinh ]s B=0

A pair of homogeneous algebraic equationsfor Aand B.

Solution exists only ifthe determinant of the coefficients is zero

4s²C² = +1 s²−2s coth 2s

c= ¹₂U±(U/2s) (s−coth ) (s s−tanh )s ^{1 2/}

s s

The expression (s −coth s)(s −tanh s)inside brackets is negative if s < s₀and positive if s > s₀, where s₀= coth s₀.

3.0 2.0 1.0

0.00 0.5 1.0 1.5 2.0

tanh(s)

¾ Fors < s₀, chas the form c= ¹₂U ±ic s_i( ) where i= −1

coth(s)

wave-type disturbances exist and fors < s₀ they propagate with phase speed

′ = ^{± −}

ψ ψ$( )z e ^{kc ti} e^{ik x( 12Ut)} Then

Re(c) = U, ¹₂

= the zonal wind speed at z= ¹₂H This height, at which is called the steering level for the disturbance.

c= u(z)

Such disturbances also grow or decay exponentially with time, the growth rate(or decay rate) being kc_i, or 2sc_i(s)/L_R.

c= ¹₂U±(U/2s) (s−coth ) (s s−tanh )s ^{1 2/}

s s

s

−4s2 ci2

3.0 2.0 1.0

0.00 0.5 1.0 1.5 2.0

tanh(s)

1.0 0.6 0.2

−0.20 0.5 1.0 1.5 2.0

unstable stable

−(0.31U)² coth(s)

The growth rate(or decay rate) is kc_i, or 2sc_i(s)/L_R. For the unstable wave with kc_i> 0, the maximum growth rateoccurs when s = s_m= 0.8.

Our interest is primarily in the amplifying mode.

For s > s₀, both solutions are neutrally stable; i.e., Im(c) = 0.

For the unstable wave with kc_i> 0, the maximum growth rateoccurs when s = s_m= 0.8, and

(kc_i)_max =2s c s_{m i}( _m) / L_R = 0 31. U L/ _R

The half-wavelength of the fastest growing wave is

1

2λ_max = π/ k_m = πL_R /2s_m

this being the distance between the ridge(maximum p'or ψ') and trough(minimum p'or ψ').

The unstable wave mode

Typical atmospheric values aref ~ 10^-4s^-1(45deg. latitude), N ~ 10^-2s^-1,H ~ 10⁴m(10 km) andU ~ 40 ms^-1, giving ,

and

6 3

R

L NH ~ 10 m (10 km)

= f , (kc_i)_max⁻¹ ~ .0 8×10⁵s

These values for and are broadly typical of the observed e-folding timesand horizontal length scales of extra-tropical cyclones in the atmosphere.

1 2

2 106 2000

λ_max ~ × m ( km) (kc_i)_max^{−1 1}_2λmax

(about 1 day) Typical scales

Adding

B=2sCA/ ( tanhs s−1)=iDc s A_i( ) , say.

[ (2s C+¹₂) cosh sinh ]s s A+[coshs−2(C+¹₂)sinh ]s B=0 [ (2s C−¹₂) coshs+sinh ]s A+[coshs−2(C−¹₂)sinh ]s B=0 and

Then _{ψ′ =ψ$( )z e}^{±kc ti} _e^{ik x( −12Ut)}

The vertical structureof the disturbance is given by

i ( Z )

ˆ (Z) A[sinh 2sZ iDc (s) cosh 2sZ]i A (Z)e^γ

ψ = + = %

where ^{A Z~( )}=^A[sinh22^sZ+^{D c s2} i^{( ) cosh2 2} 2^{sZ]1 2/} and γ( )Z =arg[sinh2sZ+iDc s_i( ) cosh 2sZ]

The perturbation streamfunctiontakes the form

′ = − +

ψ ( , , , ) ~ γ

( ) cos[ ( ) ( )]

x y Z t A Z e^{kc ti} k x ¹₂Ut Z

Hence the streamfunction (or pressure-) perturbation and other quantities have a phase variation with height.

Note: allthe flow quantitiesare determined in terms of ψ'; e.g.,

[ ¹ ]

x 2 y

2 Z

v sin k(x Ut) (Z) , u 0,

w fH UZ U , and b f ... .

N t x Z x

′ = ψ′ ∝ − − γ ′ = −ψ′ =

⎡ ∂⎛ ∂ ∂ψ′⎞ ∂ψ′⎤

= − ⎢⎣⎜⎝∂ + ∂ ⎟⎠ ∂ − ∂ ⎥⎦ = ψ′ =

To evaluate the expressions for wand σinvolves considerable algebra.

¾ The detailed structure of an unstable two-dimensional Eady waveis shown in the next figure including:

¾ Geostrophic quantities

- the streamfunction(pressure) perturbation - the meridional velocityisotachsv(x, z);

- the buoyancy perturbationb(x, z), proportional to the potential temperature deviationθ´(x,z) = (θ − θ₀(z));

- the vertical component of relative vorticityζ(x, z);

And

¾ Ageostrophic quantities:

Structure of an unstable Eady wave

¾ Ageostrophic quantities:

- the vertical velocityw(x, z);

- the streamfunction of the ageostrophic motionin a vertical plane, denoted here byΦ(x, z) defined byu_a= Φ_z, w =− Φ_x satisfies the two-dimensional continuity equation,

∂u_a/∂x + ∂w/∂z = 0;

-the ageostrophic windu_a(x, z).

Structure of an unstable Eady wave

b > 0 s < 0

p′< 0 p ′> 0

LO

LO z

z

x x HI

HI

¾ The minimum pressure perturbation (the pressure trough axis) and the maximum pressure perturbation (the ridge axis) tilt westwards with height.

¾ This is a characteristic feature of developing cyclonesand anticyclonesin the atmosphere.

¾ The warmest air(b > 0) is rising(w > 0) and the coldest air (b < 0) is subsiding(w < 0), an indication that available potential energy is being reduced.

¾ It is clear also that the warm air moves polewards(v' > 0in NH) and the cold air moves equatorwards(v' < 0in NH) so that the wave effects a poleward heat transport.

¾ Note that cyclonicζcorresponds with negative values in the southern hemisphere.

ψ(x,z) v(x,z)

b(x,z) w(x,z)

LO

LO

LO

LO

x x

x x

z

z

z

z

Φ(x,z) u_ag(x,z)

ζ(x,z)

LO LO

LO

x x

x

z z

z

¾ So far we have assumed that the wave structure is independent of the meridional direction y.

¾ A slightly more realistic calculation vis-á-vis extra-tropical cyclones is to relax this assumption and to investigate wave disturbances confined to a zonal channel with rigid

(frictionless) walls at y = 0and y = Y, say.

¾ Then, ∂/∂y ≠0and u' ≠0, but v' = 0at y = 0and Y.

¾ In this case, the solution procedure is essentially the same as before, but we now take

′ = ⁻

ψ ( , , , )x y z t ψ$( )z e^{ik x ct( )}sin (m y Yπ / )

mis an integer to satisfy the conditionv' = ψ'_x= 0aty = 0, Y.

Three-dimensional waves

4s² =L k²_R( ²+m²π² /Y²)

The next figure shows the pressure patterns corresponding to the total streamfunction

ψ ψ ψ= + ′

The only change to the foregoing analysis is to replace 4s²= L_R² with

at the surface, in the middle troposphere and in the upper troposphere, for the wave with m = l.

Isobar patterns: a) at the surface(z´ = 0) in the middle troposphere(z´= 0.5) and c) in the upper troposphere(z´= 1.0) in the Eady solution for a growing baroclinic wave with m = 1. Shown in d), is the isotach pattern of vertical velocityin the middle troposphere.

ψ_{z= 0} ψ_{z= 1}

ψ_{z= 0.5} w_{z= 0.5}

x x

x x

y y

y y

The End