Development Theory

Chapter 10

Development Theory

» In this section I will discuss:

- further aspects of the structure and dynamics of synoptic-scale disturbances, and

- derive Sutcliffe's development theory, which provides a number of practical forecasting rules.

Development Theory

D

Dt^gu^g f _a + ∧ =k u 0

In quasi-geostrophic theory the ageostrophic wind satisfies the equation:

D

Dt t

g g

g h g

u =

L

NM O

∂

QP

∂ Assume f = constant

Takek∧ =>

u k u

a

g g

f D

= 1 ∧ Dt k 0

u_a

D Dt

gug

NH

In the northern (southern) hemisphere, the ageostrophic wind blows to the left (right) of the acceleration vector

∂

∂w = −∇ ⋅

z ^h u_a

There can be no vertical motionunless there is an ageostrophic component of the wind, assuming that

w = 0 at some height, say at the ground.

Requirement for vertical motion

u k u

a

g g

f

D

= 1 ∧ Dt

0

Consider flow regions, where ^D

D t t

gug ug

≈ ∂

∂

u_{g h}

f p

= − 1 ∧ ∇ ρ k

u_{a h}

f

p

= − ∇

L

NM O

1

QP

0

ρ 2

∂

∂

called the isallobaric wind

» Isopleths of ∂p/∂tare called isallobars.

» Isallobaric charts, are charts on which isopleths ∂p/∂tof are plotted.

» They are useful as forecasting aids and were particularly so before the advent of computer-produced prognostic charts.

The isallobaric wind

» On isallobaric charts, values of ∂p/∂tare normally computed from barometric tendencies for 3hours or 24 hours

» In low latitudes 24hours is more appropriate because diurnal pressure variations due to the atmospheric tide are comparable with, or larger than, typical synoptic changes.

Isallobaric charts

isallobars

falling pressure

rising pressure NH

∂

∂ u_g

t u_a

» Physically, the isallobaric windmay be viewed as a cross- isobaric motionin which the air accelerates or decelerates to take up the geostrophic wind velocity consistent with the new pressure field.

» For example, if the isobars become closer to each other locally, the air must accelerate locally as the geostrophic wind increases.

» To accelerate, work must be done on the airand it must therefore move with a component across the isobars towards low pressure.

» Note that the ageostrophic wind blows towards falling pressure.

» In particular, there is ageostrophic convergencetowards an isallobaric lowand divergencefrom an isallobaric high.

LO

convergence divergence

HI isallobars

NOT isobars

» In regions where flow patternsare approximately stationary, the acceleration experienced by air parcels is represented by the advection term

D Dt

g g

g h g

u ≈u ⋅ ∇ u

» This tends to be the case at upper tropospheric levelswhere wind speeds are generally larger than at lower levels.

u k u

a

g g

f

D

= 1 ∧ Dt

0 u_g⋅ ∇_hu_g

u_a

Thus u_ais perpendicular to the advective acceleration.

Confluence and diffluence

diffluence

diffluence confluence

convergence

divergence

Isobars or geopotential height contours illustrating patterns of confluence and diffluence and associated regions of

convergence and divergence (NH case).

» The foregoing results have application to an understanding of the circulations associated with jet streams.

» These are relatively narrow currents of strong winds which occur in the upper troposphere in association with planetary and synoptic scale wave disturbances.

» The jet stream core, the region of strongest wind, is evident from the distribution of isotachs, which are often

superimposed on upper-level charts, for example, those corresponding with the 500 mband 250 mblevels.

» A typical example is the 500 mbchart shown in the next figure: =>

Jet streams

Regional 500 mbanalysis showing planetary-scale waves. Broken lines are isotachs in knots (2 kn = 1 ms^-1). Note the two “jet-streaks”, one just southwest of Western Australia, the other slightly east of Tasmania.

Mean July wind field over Australia (1956-61). The isotach surfaces of 30, 40and 50 ms^-1westerly wind are drawn and projections on the earth’s surface of their latitude extremes are shown as dotted lines. The values in lines are the heights in

1000’s of feet (300 m) of the isotach surfaces above selected stations.

LO

A' HI A

B' B

x y

N

Isobaric or geopotential height contours (blue dashed lines), isotachs (red lines) and a typical parcel trajectory (dotted line) in a jet stream core.

A' jet entrance A B' jet exit B

.

SH

z z

y y

Meridional circulation in the entranceand exitregions of a jet stream core.

Arrows denote flow directions in the y - z plane.

With ypointing polewards, the direction of flow along the jet is out of the screen for the northern hemisphere and into the screen for the southern

hemisphere.

cirrus cloud

.

SH

» Typically, the air might accelerate from 20 ms⁻¹to 60 ms⁻¹ in travelling 1500 kmthrough the jet entrance region: =>

f v u u

x v ms

o a g

g

~ ∂ a ~ ~

∂ => ⁺ × ⁻_×

− −

20 60

2 60 20

1500 10 4

3 1

10 10

» Assume that the air 500 kmto either side of the jet is undisturbed and that the velocity maximum occurs 2 km below the tropopause. Then

∂

∂

∂

∂ w

z

v

y^a w ms cm s

= − => ~ 2 × ⁻ = ⁻

500 10 ¹ 4 ¹

» The vertical displacement of a particle is then about 1.5 km, enough to cause condensation in air equatorwards of the axis and to clear any existing cloud on the poleward side.

A calculation

» The cirrus cloud found ahead of the warm front of an extra- tropical cyclone is usually such jet cloud.

» It is formed over the cold front and blown around the upper- level trough to appear ahead of the warm front.

» Note that implicit in the foregoing calculation is the assumption that u(~ u_g) can be computed geostrophically whereas vcannot be.

» This is consistent with scaling analyses which show that the geostrophic approximation holdsin the direction across the jet, but not along it.

» The same is truealso of a front.

Some notes

» Large-scale motions in the atmosphere are in close hydrostatic balance =>

» The pressure at the base of a fixed column of air is proportional to the mass of air in that column; if the total mass decreases, so will the surface pressure, and vice versa.

unit area

p_s= surface pressure

= weight of column

isallobaric convergence

upper-level divergence

Dines compensation

» In a deepening low, the isallobaric wind, andto a lesser extent surface friction, will contribute to low-level convergence.

» If there were no compensating upper-level divergence, the surface pressure would rise as mass accumulated in a column - clearly a contradiction!

» Dinesshowed that low-level convergence is very nearly equal to the divergence at upper levels.

» He pointed out that upper divergence must exceed the low-level convergence when a low deepens.

» Because the integrated divergence is a small residual of much larger, but opposing contributions at different levels, it is not practical to predict surface pressure changes by computing the integrated divergence.

Some notes

p gdz

z

=

z

Note

∂

∂

∂ρ

∂ ρ

p

t g

t dz g

z z

=

z

z

The surface pressure is given by

∂

∂p ρ

t^s = −g

z

In practice, it is of little use for predictionsince observations are not accurate enoughto reliably compute the right-hand-side.

∂ρ

∂ ρ

t + ∇ ⋅( u) =0

for strictgeostrophic motion the surface pressure cannot change!

s

0

p g ( ) dz

t

∂ = − ∞ ∇ ⋅ ρ

∂

∫

strictgeostrophic motion => fρu = ∧ ∇k _hp fconstant=> ∇ ⋅_h (ρu) = 0

» This is consistent with the fact that the geostrophic wind blows parallel with the isobars.

» => any local change in surface pressure is associated entirely with ageostrophic motion.

» => important consequences for the movement of disturbances characterized by their surface pressure distribution, e.g.

cyclones and anticyclones.

» Sutcliffe, an English meteorologist, computed the relative divergence between an upper and lower level in the troposphere using the vorticity equation.

» From this he deduced the distribution of vertical motion.

» A knowledge of the vertical motion at a low level can be used together with the vorticity equation at the surface to study the development of surface pressure systems such as middle latitude cyclones and anticyclones.

» Sutcliffe's theory has proved extremely valuable in the practice of weather forecasting.

Sutcliffe's development theory

» If one can compute the difference between the fields of ageostrophic windat a low leveland at a middle or high tropospheric level, one can deduce the horizontal

divergence, and hence the field of vertical motion which must exist between these levels.

» Then one can locate the regions of falling surface pressure (over rising air) and of rising surface pressure(over subsiding air).

The essence of Sutcliffe's theory

The vertical distribution of horizontal divergence and vertical motion (characterized byω = Dp/Dt) for an extra-tropical cyclone, an anticyclone,

and for example of a more complex disturbance.

conv div

conv ascent

ω div 200 mb 400 mb 600 mb 800 mb 1000 mb

Extratropical cyclone

subsidence conv

div

conv div conv div

subsidence Anticyclone

ω

Complex disturbance

ω

ascent

conv

cyclone z

∂w/∂z > 0

∂w/∂z < 0

anticyclone z

1

2H ¹₂H

H

∂w/∂z < 0

∂w/∂z > 0

» It follows that for an extra-tropical cyclone, ∂²w/∂z²< 0 throughout the depth of the troposphere, and for an anticyclone ∂²w/∂z²> 0.

H

w w

» I will use (x,y,z)rather than (x,y,p)coordinates and adopt a modern approach, using quasi-geostrophic theory.

0

f w ( f )

z t

∂∂ =⎡⎢⎣∂∂ +u^g⋅∇ ζ +⎤⎥⎦

f⁰ z

∂

∂ ⇒ ^{2 2}

0 2 0 0 0

f w f f f ( f )

z t z z z

∂∂ =∂∂ ⎡⎢⎣ ∂ζ∂ ⎤⎥⎦+ ⋅∇⎡⎢⎣∂ζ∂ ⎤⎥⎦+ ∂∂ ⋅∇ ζ +

g g

u u

The plan is to obtain a diagnostic formula for∂²w/∂z² by eliminating the time derivative on the right hand side.

Mathematical derivation

» For an extra-tropical cyclone, ∂²w/∂z²< 0throughout the depth of the troposphere, and for an anticyclone ∂²w/∂z²> 0.

» Now ∂²w/∂z²may be obtained from the vorticity equation:

∂

∂

∂ζ

∂

∂

∂

∂ψ

∂ t f

z t f

h z

0

2 0

L NM O

QP

L

NM O

QP

2 h

b t

= ∇ ∂

∂ using and^{1 p} b z

∂ =

ρ ∂ ψ =p/ρ_∗f₀

( )

2 2

h b N w

= ∇ − ⋅∇ −u_g

using _{g h} b N w² 0 t

⎡∂ + ⋅∇ ⎤ + =

⎢∂ ⎥

⎣ u ⎦

Finally f₀ N^{2 2}_hw ²_h( b) .

t z

∂ ⎡⎢ ∂ζ⎤ = − ∇ −∇⎥ ⋅∇

∂ ⎣ ∂ ⎦ u^g

2

h( b) f0 f0 2 ,

z z

∂ ⎡∂ζ⎤

∇ ⋅∇ = − ∂ ⋅∇ζ + ⋅∇⎢⎣∂ ⎥⎦+ Λ

g

g g

u u u

2Λ = ₀

L

NM O

QP

f E F z F E

z E u

x v

y F v

x u y

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

, , ∂

2 2 2

0 h h

f N w ( b) .

t z

∂ ⎡⎢ ∂ζ⎤ = − ∇ −∇⎥ ⋅∇

∂ ⎣ ∂ ⎦ u^g

2

h( b) f0 f0 2 ,

z z

∂ ⎡∂ζ⎤

∇ ⋅∇ = − ∂ ⋅∇ζ + ⋅∇⎢⎣∂ ⎥⎦+ Λ

g

g g

u u u

The quantity Λis related to the deformation of the flow and can be shown to be small, except in active frontogenetic regions.

Neglecting Λ, substitution

2

2 2 2

0 2 h 0

f w N w f (2 f )

z z

∂ = − ∇ + ∂ ⋅∇ ζ +

∂ ∂

ug

Sutcliffeneglects the adiabatic buoyancy tendency and integrates between the surface and .

N²∇_h²w

1 2H

2

2 2 2

0 2 h 0

f w N w f (2 f )

z z

∂ ∂

= − ∇ + ⋅∇ ζ +

∂ ∂

ug

0

1 2H

z

approximately the level of the 500 mb surface

1 2

1 H2

0 0

H s 0

w w

f f (2 f ) dz

z z z

∂ ∂ ∂

⎡ ⎤ − ⎡ ⎤ = ⋅∇ ζ +

⎢∂ ⎥ ⎢∂ ⎥ ∂

⎣ ⎦ ⎣ ⎦

∫

Small ~level of nondivergence

the surface divergence The surface vorticity tendency

s

s s 0

s 1 H2

s s s 0

( f ) f w

t z

f (2 f ) dz .

z

∂ζ ⎡∂ ⎤

= − ⋅∇ ζ + + ⎢ ⎥

∂ ⎣∂ ⎦

= − ⋅∇ζ − ⋅∇ − ∂ ⋅∇ ζ +

∫ ∂^g u

u u u

1 2H s

s s s f 0 (2 f ) dz .

t z

∂ζ ∂

= − ⋅∇ζ − ⋅∇ − ⋅∇ ζ +

∂ ^{u u} ∫ ∂^ug

Assume a linear vertical shear

∂

∂u u u ζ ζ ζ

u u u

z H

z H

H s

H s s

= −

′ = − = + ′

/

/ , / ,

2

2 2

2

0 12

2 2 4 2

H

s s

H

z

H f dz f

z

u u u

/ ( ζ ζ ) ( ζ ) ζ

∂ζ

∂^s _{s s} ζ _H

t = −(u +2u )′ ⋅ ∇ζ − ′ ⋅ ∇ ′ −u u _/2⋅ ∇f An equation for the surface vorticity tendency ζ^s =ρ1f ∇ p^s

0 2

*

=> The surface pressure tendencycan be diagnosed

» For a wave-like disturbance, V²p_s is proportional to −p_sso that an increase in cyclonic vorticity corresponds with a lowering of the surface pressure.

» There are various ways of interpreting this equation.

» I will discuss the mathematical way.

» The equation has the form D

Dt^∗ζ^s = − ′ ⋅ ∇ ′ − _H ⋅ ∇f ζ

u u _/2

where ^D

Dt t ^s

∗ ≡ ∂ + + ′ ⋅ ∇

∂ (u 2u ) .

Interpretation

D

Dt^∗ζ^s = − ′ ⋅ ∇ ′ − _H ⋅ ∇f ζ

u u _/2 where ^D

Dt t ^s

∗ ≡ ∂ + + ′ ⋅ ∇

∂ (u 2u ) .

» If the thermal vorticity advection−u' ⋅Vζ' and the planetary vorticity tendency−u_H/2⋅Vfare both zero,ζ_swill be

conserved for points moving with velocity u_s+ 2u'.

» => The lines defined by dx/dt =u_s+ 2u'are characteristicsof the equation.

» In the general case, ζ_s changes at the rate −u' ⋅Vζ' −u_H/2⋅ Vf following a characteristic.

The implications of ^D are:

Dt^∗ζ^s = − ′ ⋅ ∇ ′ − _H ⋅ ∇f ζ

u u _/2

(i) At the centre of a surface low, u_s<< 2u' => the low pressure centre will propagate in the direction of the thermal wind, or equivalently the 500 mbwind, with speed proportional to the thermal wind;

This is the thermal steering principle.

It turns out that the constant of proportionality (i.e., 2) is too largeand that a value of unity is more appropriate:

see later =>

(ii) −u_H/2⋅Vf is positive, leading to cyclonic development for an equatorward windand anticyclonic

development for a poleward wind; this term is relatively small.

(iii) Thethermal vorticity advection−u' ⋅Vζ' is the principal contribution to the intensification or decay of systems.

Consider a wave pattern in the thickness isopleths shown in the next figure:

Equator

u'

u'

ζ'_max ζ'_min ζ'_max

div

div

conv

conv

Favorable for: cyclogenesis(−u' ⋅Vζ')_max anticyclogenesis( −u' ⋅Vζ')_min A positive vorticity advection

maximum, or "PVA max"

LO cold

LO cold

HI warm

HI warm

thickness isopleths

» It follows from (i)and (iii)that the propagation and change in intensity of surface depressions can be judged from a thickness chart with a superimposed surface chart.

» It is often the case that in most locations, |u´| >> |u_s|, =>

u_500mb≈u´=>

» The thermal wind contribution to the flow at 500 mbmostly dominates that due to the surface wind.

» This is why the thickness isopleths and the isopleths of geopotential at 500 mbhave broadly similar features.

» It means also that the 500 mbisopleths can be used instead of the thickness charts to give an indication as to how

systems will be steered and whether or not they will grow or decay.

Some notes

» Sutcliffe's theory highlights the fact that a surface depression, or trough, and an anticyclone, or ridge, will tend to be displaced in the direction of shear by a process of developmentas distinct from translation: =>

» There is associated divergence and convergence.

» Used with care, it can be a useful guide in weather forecasting, but it suffers the following limitations:

(i) it neglects the effects of adiabatic heating and cooling, represented by the term Ν²V_h²w, and diabatic effects.

These effects may make important contributions to the overall flow evolution.

Limitations:

» The omission ofΝ²V_h²w overestimates the propagation speed by a factor of two.

f₀^{2 2}∂ w/∂z² ≈ −f₀² w H/ ²

f₀^{2 2}∂ w/∂z² +N²∇²_hw ≈ −(f² /H²+N²κ²)w

But for quasi-geostrophic disturbances, f₀² /κ²N H^{2 2} ~1

Both terms are important The omission of Ν²V_h²w

» For a wavelike disturbance with total horizontal wavenumber κ, V_h²w is proportional to −κ²w.

» Since

(ii) the theory is a diagnosticone; it gives an indication of the tendencies at a given instant or, in practical terms, for a few hours or so.

» The thermal field, and hence the thermal vorticity advection

−u' ⋅Vζ' will evolveas the surface flow develops and a more complete solution, such as that provided by numerical integration, must allow this interaction to occur.

» We can understand this interaction from the next figure:

Sutcliffe’s theory is a diagnostic one!

LO

N

cold advection

warm advection mslp

1000 - 500 mb thickness

Some notes

» These considerations show also a further role of the N²w term in the thermodynamic equation which Sutcliffe ignored.

» As we have seen from baroclinic instability theory, the structure of the growing Eady wave is such that, poleward motion is associated with ascent and cooling; equatorwards motion with subsidence and warming.

» In a growing baroclinic wave, the pairs of quantities vand w and vand σare negatively correlated in the southern

hemisphere while wand sare positively correlated.

» Accordingly, the N²wterm in the thermal tendency equation opposes the horizontal temperature advection and hence the rate of increase of −u' ⋅Vζ' .

Other points

» On account of the coupling between the surface and upper- level flow, we would notexpect Sutcliffe's theory to be a substitutefor a good numerical weather prediction model.

(iii) The theory does not work well when |u´|is small; i.e., in the case of "cut-off" lows. These are low pressure systems which develop in or migrate to a region where the upper-level steering winds are relatively light.

(iv) The theory neglectsthe effects of moist processes.

Start from f w

z t f

z f

z f

z f

20 2

2 0 0 0

∂

∂

∂

∂

∂ζ

∂

∂ζ

∂

∂

∂ ζ

=

L

NM O

QP

L

NM O

QP

u u

( )

2 2 2

0 h h

f N w ( b) .

t z

∂ ⎡⎢ ∂ζ⎤ = ∇ −∇⎥ ⋅∇

∂ ⎣ ∂ ⎦ u

2

h( b) f0 f0 2 ,

z z

∂ ⎡∂ζ⎤

∇ ⋅∇ = − ∂ ⋅∇ζ + ⋅∇⎢⎣∂ ⎥⎦+ Λ

u u u

N w f w

z f

z f

h 2 2

0 2

2

2 0 2 2

∇ + ∂ = ⋅ ∇ + +

∂

∂

∂u ζ

( ) Λ

» A diagnostic equationfor the wwhich does not rely on

computing the horizontal divergence from wind observations.

» Called the quasi-geostrophic form of the 'ω-equation' after its counterpart in pressure coordinates.

The omega equation

The End