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Atmospheric Dynamics: lecture 14 (http://www.staff.science.uu.nl/~delde102/)

Topics Chapter 9: Baroclinic waves and cyclogenesis What is a baroclinic wave? Quasi-geostrophic equations Omega equation

Original articles:

Baroclinic wave

Huib de Swart

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Baroclinic wave

Baroclinic wave

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Baroclinic wave

Baroclinic wave

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Baroclinic wave

Baroclinic wave

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Warm sector

Warm sector

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Warm sector

Warm sector

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occlusion

warm seclusion

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Upward motion

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Atmospheric “river”

warm conveyor belt

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warm conveyor belt

warm conveyor belt

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Quasi-geostrophic theory Quasi-geostrophic approximation Leads to a system of two equations with two unknowns Unknowns: vertical velocity and geopotential. However: neither equation is an explicit equation for the vertical velocity A third equation (the “Omega equation”), the solution of which provides the vertical velocity, is derived. This equation gives physical insight into relation frontogenesis, vertical motion and cyclogenesis

Chapter 9

Quasi-geostrophic approximation

€

d! v dt

= − f ˆ k × ! v −∇Φ

€

∂Φ∂p

= −RTp

€

∂u∂x

+∂v∂y

+∂ω∂p

= 0

€

∂T∂t

+u∂T∂x

+ v∂T∂y

− Spω =Jcp

“Primitive” equations with pressure as vertical coordinate:

Section 1.30:

Section 9.3

€

Sp ≡αcp−∂T∂p

(Box 9.1, 8)

(Box 9.1, 14)

(Box 9.1, 1)

(Box 9.1, 12)

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Quasi-geostrophic approximation

€

d! v dt

= − f ˆ k × ! v −∇Φ

€

∂Φ∂p

= −RTp

€

∂u∂x

+∂v∂y

+∂ω∂p

= 0

€

∂T∂t

+u∂T∂x

+ v∂T∂y

− Spω =Jcp

€

! v g =1f0

ˆ k ×∇Φ

€

f ≅ f0 +dfdyy ≅ f0 +βy

€

d! v dt≅

dg! v g

dt

€

dg

dt≡∂∂t

+! v g ⋅! ∇ =

∂∂t

+ ug∂∂x

+ vg∂∂y

Approximations:

See also GFD

Section 9.3

(Box 9.1, 8)

(Box 9.1, 14)

(Box 9.1, 1)

(Box 9.1, 12) 1

2

3

4

Quasigeostrophic equations

€

d! v dt≅

dg! v g

dt= − f ˆ k × ! v −∇Φ ≅ − f0 +βy( ) ˆ k × ! v g +

! v a( ) + f0 ˆ k × ! v g

Section 9.3

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Quasigeostrophic equations

€

d! v dt≅

dg! v g

dt= − f ˆ k × ! v −∇Φ ≅ − f0 +βy( ) ˆ k × ! v g +

! v a( ) + f0 ˆ k × ! v g

Section 9.3

dg!vgdt

= −βyk̂ × !vg +!va( )− f0k̂ ×

!va ≅ −βyk̂ ×!vg − f0k̂ ×

!va!vg >>

!va

Quasigeostrophic equations

€

d! v dt≅

dg! v g

dt= − f ˆ k × ! v −∇Φ ≅ − f0 +βy( ) ˆ k × ! v g +

! v a( ) + f0 ˆ k × ! v g

dg!vgdt

= −βyk̂ × !vg +!va( )− f0k̂ ×

!va ≅ −βyk̂ ×!vg − f0k̂ ×

!va!vg >>

!va

This is questionable!! (see fig. 1.87 lecture notes)

Quasi-geostrophic approximation is difficult to justify completely from first principals, except under very restricted conditions. The justifications comes from practice: “it works!” (i.e. in hindsight)

Section 9.3

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Jetstreak

€

d! v dt

> 0

€

d! v dt

< 0

€

! v a > 0

€

! v a < 0

If β=0 then

Section 1.32

!va=1f0k̂×dg!vgdt

dgugdt

=1f0va

dg!vgdt

= − f0k̂ ×!va

x-component:

Ageostrophic wind perpendicular to acceleration:

Quasigeostrophic vorticity equation

€

dg! v g

dt= −βy ˆ k × ! v g − f0

ˆ k × ! v a

€

dgugdt

= f0va +βyvg

€

dgvgdt

= − f0ua −βyug

Section 9.4

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Quasigeostrophic vorticity equation

€

dg! v g

dt= −βy ˆ k × ! v g − f0

ˆ k × ! v a

€

dgugdt

= f0va +βyvg

€

dgvgdt

= − f0ua −βyug

€

ug = −1f0∂Φ∂y;vg =

1f0∂Φ∂x

Section 9.4

€

∂ug∂x

+∂vg∂y

= 0

Quasigeostrophic vorticity equation

€

dg! v g

dt= −βy ˆ k × ! v g − f0

ˆ k × ! v a

€

dgugdt

= f0va +βyvg

€

dgvgdt

= − f0ua −βyug

€

ug = −1f0∂Φ∂y;vg =

1f0∂Φ∂x

€

ζg =∂vg∂x

−∂ug∂y

=1f0∇2ΦQuasi-geostrophic vorticity:

Section 9.4

€

∂ug∂x

+∂vg∂y

= 0

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Quasigeostrophic vorticity equation

€

dg! v g

dt= −βy ˆ k × ! v g − f0

ˆ k × ! v a

€

dgugdt

= f0va +βyvg

€

dgvgdt

= − f0ua −βyug

€

ug = −1f0∂Φ∂y;vg =

1f0∂Φ∂x

€

dgζgdt

=1f0

dg∇2Φ

dt= − f0

∂ua∂x

+∂va∂y

'

( )

*

+ , −βvgQuasi-geostrophic vorticity eqn:

Section 9.4

€

∂ug∂x

+∂vg∂y

= 0

€

ζg =∂vg∂x

−∂ug∂y

=1f0∇2ΦQuasi-geostrophic vorticity: use this:

Geopotential and omega as unknowns

€

dgζgdt

=1f0

dg∇2Φ

dt= − f0

∂ua∂x

+∂va∂y

'

( )

*

+ , −βvgQuasi-geostrophic vorticity eqn:

Section 9.4

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Geopotential and omega as unknowns

+ continuity equation:

€

dgζgdt

=1f0

dg∇2Φ

dt= − f0

∂ua∂x

+∂va∂y

'

( )

*

+ , −βvgQuasi-geostrophic vorticity eqn:

€

∂u∂x

+∂v∂y

+∂ω∂p

= 0

Section 9.4

Geopotential and omega as unknowns

+ continuity equation:

€

dgζgdt

=1f0

dg∇2Φ

dt= − f0

∂ua∂x

+∂va∂y

'

( )

*

+ , −βvgQuasi-geostrophic vorticity eqn:

€

∂u∂x

+∂v∂y

+∂ω∂p

= 0

€

∂ua∂x

+∂va∂y

+∂ω∂p

= 0becomes: because

€

∂ug∂x

+∂vg∂y

= 0

Section 9.4

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Geopotential and omega as unknowns

+ continuity equation:

€

dgζgdt

=1f0

dg∇2Φ

dt= − f0

∂ua∂x

+∂va∂y

'

( )

*

+ , −βvgQuasi-geostrophic vorticity eqn:

€

∂u∂x

+∂v∂y

+∂ω∂p

= 0

€

∂ua∂x

+∂va∂y

+∂ω∂p

= 0becomes: because

Therefore:

€

dg∇2Φ

dt= f0

2 ∂ω∂p

−β∂Φ∂x

Section 9.4

€

∂ug∂x

+∂vg∂y

= 0

Quasi-geostrophic thermodynamic quation

€

∂Φ∂p

= −RTp

∂T∂t

+u∂T∂x

+ v ∂T∂y

+ω∂T∂p

=αωcp

+Jcp

Section 9.4

€

Sp =αcp−∂T∂p

Eq. 1.195: Eq. 1.194:

∂T∂t

+u∂T∂x

+ v ∂T∂y

− Spω = +Jcp

€

dg∂Φ∂pdt

= −σω −RJcp p

Quasi-geostrophic thermodynamic eq.:

€

σ ≡RpSp

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Two equations and two unknowns

€

dg∇2Φ

dt= f0

2 ∂ω∂p

−β∂Φ∂x

Closed set of equations

€

dg∂Φ∂pdt

= −σω −RJcp p

Quasi-geostrophic vorticity eq.

Quasi-geostrophic thermodynamic eq.

This set of equations was used by the pioneers of numerical weather prediction

Section 9.4

Most important equations

€

dg∇2Φ

dt= f0

2 ∂ω∂p

−β∂Φ∂x

€

dg∂Φ∂pdt

= −σω −RJcp p

Quasi-geostrophic vorticity eq.

Quasi-geostrophic thermodynamic eq.

Section 9.4

€

∂ua∂x

+∂va∂y

+∂ω∂p

= 0

€

∂ug∂x

+∂vg∂y

= 0

Continuity equation

Divergence of geostrophic wind = 0

9.21

9.22

9.12

9.26

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An equation for omega

Meteorologists are most interested in omega because this variable gives a clear indication of where clouds and precipitation will form. In the following we derive a separate equation for omega, which is called the “omega-equation”

Section 9.5

Frontogenesis as a disturbance to thermal balance

€

∂ug∂p

=Rpf0

∂T∂y;

€

∂vg∂p

= −Rpf0

∂T∂x

Section 9.5

Derive these two equations now

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Frontogenesis as a disturbance to thermal balance

€

dgugdt

= f0va +βyvg

€

∂ug∂p

=Rpf0

∂T∂y;

Using: and assuming β=0, and neglecting ageostrophic motion:

€

∂vg∂p

= −Rpf0

∂T∂x

Section 9.5

Derive this now

9.18

Frontogenesis as a disturbance to thermal balance

€

dgugdt

= f0va +βyvg

€

∂ug∂p

=Rpf0

∂T∂y;

Using: and assuming β=0, and neglecting ageostrophic motion:

€

dgdt

∂ug∂p

#

$ %

&

' ( = −

∂ug∂p

∂ug∂x

−∂vg∂p

∂ug∂y

=Rf0p

−∂ug∂x

∂T∂y

+∂ug∂y

∂T∂x

#

$ %

&

' ( €

∂vg∂p

= −Rpf0

∂T∂x

Section 9.5

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Frontogenesis as a disturbance to thermal balance

€

dgugdt

= f0va +βyvg

€

∂ug∂p

=Rpf0

∂T∂y;

Using: and assuming β=0, and neglecting ageostrophic motion:

€

dgdt

∂ug∂p

#

$ %

&

' ( = −

∂ug∂p

∂ug∂x

−∂vg∂p

∂ug∂y

=Rf0p

−∂ug∂x

∂T∂y

+∂ug∂y

∂T∂x

#

$ %

&

' ( €

∂vg∂p

= −Rpf0

∂T∂x

€

dgdt

∂T∂y#

$ %

&

' ( = −

∂vg∂y

∂T∂y

−∂ug∂y

∂T∂x

* (See also section 1.37, lecture notes)

*

€

(J = 0)

Section 9.5

Frontogenesis as a disturbance to thermal balance

€

dgugdt

= f0va +βyvgUsing: and assuming β=0, and neglecting ageostrophic motion:

€

dgdt

∂ug∂p

#

$ %

&

' ( = −

∂ug∂p

∂ug∂x

−∂vg∂p

∂ug∂y

=Rf0p

−∂ug∂x

∂T∂y

+∂ug∂y

∂T∂x

#

$ %

&

' (

€

dgdt

∂T∂y#

$ %

&

' ( = −

∂vg∂y

∂T∂y

−∂ug∂y

∂T∂x

Subtracting these two equations yields: dgdt

f0pR

∂ug∂p

−∂T∂y

⎛

⎝⎜

⎞

⎠⎟= 2

∂ug∂y

∂T∂x

+∂vg∂y

∂T∂y

⎛

⎝⎜

⎞

⎠⎟= −2

dgdt

∂T∂y⎛

⎝⎜

⎞

⎠⎟ ≡ −2Qg2

€

(J = 0)

€

∂ug∂p

=Rpf0

∂T∂y;

€

∂vg∂p

= −Rpf0

∂T∂x

Section 9.5

€

∂ug∂x

+∂vg∂y

= 0

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Frontogenesis as a disturbance to thermal balance

y-component of the “geostrophic” Q-vector

Q-vector is vector frontogenesis function see section 1.37

Section 9.5

dgdt

f0pR

∂ug∂p

−∂T∂y

⎛

⎝⎜

⎞

⎠⎟= 2

∂ug∂y

∂T∂x

+∂vg∂y

∂T∂y

⎛

⎝⎜

⎞

⎠⎟= −2

dgdt

∂T∂y⎛

⎝⎜

⎞

⎠⎟ ≡ −2Qg2

Frontogenesis as a disturbance to thermal balance

y-component of the “geostrophic” Q-vector`

Disturbance to thermal wind balance

Section 9.5

Q-vector is vector frontogenesis function see section 1.37

dgdt

f0pR

∂ug∂p

−∂T∂y

⎛

⎝⎜

⎞

⎠⎟= 2

∂ug∂y

∂T∂x

+∂vg∂y

∂T∂y

⎛

⎝⎜

⎞

⎠⎟= −2

dgdt

∂T∂y⎛

⎝⎜

⎞

⎠⎟ ≡ −2Qg2

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Frontogenesis as a disturbance to thermal balance

y-component of the “geostrophic” Q-vector

Disturbance to thermal wind balance

Now: let us include the ageostrophic flow…

Section 9.5

Q-vector is vector frontogenesis function see section 1.37

dgdt

f0pR

∂ug∂p

−∂T∂y

⎛

⎝⎜

⎞

⎠⎟= 2

∂ug∂y

∂T∂x

+∂vg∂y

∂T∂y

⎛

⎝⎜

⎞

⎠⎟= −2

dgdt

∂T∂y⎛

⎝⎜

⎞

⎠⎟ ≡ −2Qg2

Neglecting ageostrophic flow we have (previous slides): Section 9.5

dgdt

f0pR

∂ug∂p

−∂T∂y

⎛

⎝⎜

⎞

⎠⎟= 2

∂ug∂y

∂T∂x

+∂vg∂y

∂T∂y

⎛

⎝⎜

⎞

⎠⎟= −2

dgdt

∂T∂y⎛

⎝⎜

⎞

⎠⎟ ≡ −2Qg2

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Role of ageostrophic flow is to preserve thermal wind balance

Neglecting ageostrophic flow we have (previous slides): Section 9.5

dgdt

f0pR

∂ug∂p

−∂T∂y

⎛

⎝⎜

⎞

⎠⎟= 2

∂ug∂y

∂T∂x

+∂vg∂y

∂T∂y

⎛

⎝⎜

⎞

⎠⎟= −2

dgdt

∂T∂y⎛

⎝⎜

⎞

⎠⎟ ≡ −2Qg2

Role of ageostrophic flow Is to preserve thermal wind balance

Repeating the derivation of the previous slides including ageostrophic flow yields

€

dgdt

f0pR∂ug∂p

−∂T∂y

$

% &

'

( ) = −2Qg2 +

f02pR

∂va∂p

−pσR∂ω∂y

€

dgdt

f0pR∂ug∂p

−∂T∂y

$

% &

'

( ) = 2

∂ug∂y

∂T∂x

+∂vg∂y

∂T∂y

$

% &

'

( ) = −2Qg2 ≡ −2

dgdt

∂T∂y$

% &

'

( )

Neglecting ageostrophic flow we have (previous slides): Section 9.5

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Role of ageostrophic flow Is to preserve thermal wind balance

Repeating the derivation of the previous slides including ageostrophic flow yields

€

dgdt

f0pR∂ug∂p

−∂T∂y

$

% &

'

( ) = −2Qg2 +

f02pR

∂va∂p

−pσR∂ω∂y

€

dgdt

f0pR∂ug∂p

−∂T∂y

$

% &

'

( ) = 2

∂ug∂y

∂T∂x

+∂vg∂y

∂T∂y

$

% &

'

( ) = −2Qg2 ≡ −2

dgdt

∂T∂y$

% &

'

( )

Neglecting ageostrophic flow we have (previous slides): Section 9.5

If there “conservation” thermal wind balance

=0

€

−2Qg2 +f02pR

∂va∂p

−pσR∂ω∂y

= 0

€

−2Qg1 +f02pR

∂ua∂p

−pσR∂ω∂x

= 0

The x-component of thermal wind balance yields:

From previous slide: Section 9.5

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From this we can derive an equation for the vertical motion:

€

−2Qg2 +f02pR

∂va∂p

−pσR∂ω∂y

= 0

€

−2Qg1 +f02pR

∂ua∂p

−pσR∂ω∂x

= 0

The x-component of thermal wind balance yields:

From previous slide:

€

∂∂y

€

∂∂x

+

Section 9.5

Omega-equation

From this we can derive an equation for the vertical motion:

€

−2Qg2 +f02pR

∂va∂p

−pσR∂ω∂y

= 0

€

−2Qg1 +f02pR

∂ua∂p

−pσR∂ω∂x

= 0

The x-component of thermal wind balance yields:

From previous slide:

From the two equations above:

€

σ∇2ω + f02 ∂

2ω∂p2

= −2Rp! ∇ ⋅! Q g

€

∂ua∂x

+∂va∂y

+∂ω∂p

= 0Where we have used and

€

∇2 ≡∂2

∂x2+∂2

∂y2

€

∂∂y

€

∂∂x

+

Section 9.5

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Omega equation:interpretation

€

σ∇2ω + f02 ∂

2ω∂p2

= −2Rp! ∇ ⋅! Q g

€

Qg1 = −∂ug

∂x∂T∂x

+∂vg

∂x∂T∂y

$

% &

'

( ) ; Qg2 = −

∂ug

∂y∂T∂x

+∂vg

∂y∂T∂y

$

% &

'

( ) .

Section 9.5

Omega equation:interpretation

€

σ∇2ω + f02 ∂

2ω∂p2

= −2Rp! ∇ ⋅! Q g

€

−ω ≈ w ≈ −! ∇ ⋅!

Q g

€

Qg1 = −∂ug

∂x∂T∂x

+∂vg

∂x∂T∂y

$

% &

'

( ) ; Qg2 = −

∂ug

∂y∂T∂x

+∂vg

∂y∂T∂y

$

% &

'

( ) .

Since both T, ug and vg can all be expressed as a function of Φ, we can can calculate the vertical motion from the distribution of Φ only!!!

i.e. upward (downward) motion if Qg-vector is convergent(divergent)

Section 9.5

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Next lecture: interpretation of the solution of the omega equation

pe model

Qg-VECTOR, POTENTIAL TEMPERATURE (cyan) and HEIGHT (blue)THICK CONTOURS: HEIGHT:1250.0 m; TEMPERATURE: 0.0 °C;CONTOUR-INTERVAL: HEIGHT: 50.0 m; TEMPERATURE: 5.0 °C

run 2020 864hPa : |Q1|=5*10^-11 K m^-1 s^-1 (min. value:10^-11 K m^-1 s^-1)

60.00 hrs

0 -5

15

1250 1200

1300

46°N

64°N

60°

1000

1400

warm sector

wf bbf

cf

Fig 1.85 (lower panel)

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pe model

Qg-VECTOR, POTENTIAL TEMPERATURE (cyan) and HEIGHT (blue)THICK CONTOURS: HEIGHT:1250.0 m; TEMPERATURE: 0.0 °C;CONTOUR-INTERVAL: HEIGHT: 50.0 m; TEMPERATURE: 5.0 °C

run 2020 864hPa : |Q1|=5*10^-11 K m^-1 s^-1 (min. value:10^-11 K m^-1 s^-1)

60.00 hrs

0 -5

15

1250 1200

1300

46°N

64°N

60°

1000

1400

warm sector

wf bbf

cf Q-vector convergence

Fig 1.62 (lower panel)

pe model

VERTICAL VELOCITY (w) (blue: up; red:down) and WIND VECTORTHICK CONTOURS: / /CONTOUR-INTERVAL: w: 1.0 hPa/hr /

run 2020 8 6 4 h P a 10 m/s

60.00 hrs

warm sector

46°N

64°N

bbf

cf

wf

60°

Q-vector convergence

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Omega equation: example

€

−ω ≈ w ≈ −! ∇ ⋅!

Q gUpward motion if Q-vector is convergent

Analysis of divergence of geostrophic Q-vector at 850 hPa (thick lines, labeled in units of 10-15 K m-2 s-1) and the height of the 850 hPa surface (thin lines labeled in m) on April 4, 2001, 12 UTC.

Trough of a Rossby-wave

Upward motion downward motion

-

+

Satellite image

Meteosat satellite image in VIS-channel, April 4, 2001, 1200 UTC.

Upward motion

downward motion

Section 9.5

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Prepare presentation project 2: 6 January 13:15 Each presentation is 15 minutes and consists of presenting the hypothesis, a description of the data used, the cross-correlation matrix, the eigenvectors and eigenvalues, interpretation of the most important principal component(s) and a conclusion. Project 3: Problem 3.2, hand in answer individually before, or during the next lecture on Friday 8 January 2016 Next lecture (8 January): Baroclinic instability, cyclogenesis and frontogenesis

Next:

Prepare presentation project 2: 6 January 13:15 Each presentation is 15 minutes and consists of presenting the hypothesis, a description of the data used, the cross-correlation matrix, the eigenvectors and eigenvalues, interpretation of the most important principal component(s) and a conclusion. Project 3: Problem 3.2, hand in answer individually before, or during the next lecture on Friday 8 January 2016 Next lecture (8 January): Baroclinic instability, cyclogenesis and frontogenesis Merry Christmas and a happy new year!

Next: