fronts and cross-frontal circulations - universiteit utrechtdelde102/lecture11atmdyn2011.pdfsolution...

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Atmospheric Dynamics: lecture 11 (http://www.staff.science.uu.nl/~delde102/dynmeteorology.htm)

Fronts and cross-frontal circulations (Chapter 8)

Classical  view  of  fronts  

Frontogene2cally  forced  circula2ons  (sec$on  8.3)  Define  “prototype”  (simplified)  problem  Leads  diagnos8c  equa8on  for  “cross-­‐frontal  circula8on”  (Eliassen-­‐Sawyer  equa8on)  Diagnose  solu8ons  of  this  equa8on  

30 November 2011

9 December 2010 06 UTC Classical weather map

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Correponding satellite image

9 December 2010 08 UTC

Corresponding analysis (850 hPa)

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Potential vorticity at 320 K

Classical conceptual models of fronts

Left panel: a cold front. Right panel: a warm front. (Source: Wikimedia Commons)

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Cross section along 53°N

Cold front

Cross section along 53°N

Cold front

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Cross section along 53°N

Cold front

Cross section along 53°N

Cold front

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Cross section along 53°N

Cold front

Cross section along 48°N

Cold front: cross-frontal circulation

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Vertical motion and fronts

•  Why are fronts associated with clouds and precipitation?

•  Where is precipitation expected, i.e. where is motion upwards?

Vertical motion and fronts

•  Why are fronts associated with clouds and precipitation?

•  Where is precipitation expected, i.e. where is motion upwards?

Sawyer-Eliassen prototype problem of thermal wind adjustment of the atmosphere to frontogenesis

Section 8.3

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Sawyer-Eliassen Prototype problem

geostrophic  flow  

Frontogene$cally  forced  circula$on:  an  illustra$on  

€

v = U (y, z,t)+ Ax,−Ay + va (y, z, t),wa(y, z, t){ }We  assume  that  the  velocity  is  given  by  

geostrophic  deforming  wind  field  

€

v = Ax,−Ay,0{ }

+  

+  

€

v = 0,va(y, z, t),wa(y, z,t){ }

€

v = U (y, z,t),0,0){ }

ageostrophic  wind  

isotherm  

Three  components  of  the  velocity  vector:

Example of deforming wind field

Upper  level  (500  hPa)  weather  map  of  6  August  1996,  00  UTC.  The  temperature  (°C)  and  the  wind  vector  as  measured  by  radiosonde  are  indicated.  The  contours  represent  isopleths  of  500-­‐hPa  height  (labeled  in  dm;  contour  interval  is  2.5  dm). warm  

cold  

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Governing equations

€

dudt

= −θm∂Π∂x

+ fv

€

dvdt

= −θm∂Π∂y

− fu

€

dwdt

= −θm∂Π∂z

+θθm

g

€

dθdt

= 0

Basic  equa2ons  

Section 8.3

Governing equations

€

dudt

= −θm∂Π∂x

+ fv

€

dvdt

= −θm∂Π∂y

− fu

€

dwdt

= −θm∂Π∂z

+θθm

g

€

dθdt

= 0€

θm∂Π∂y

+ fU + fAx = 0

€

θm∂Π∂x

+ fAy = 0

€

θm∂Π∂z

=θθm

g

Steady  state  Basic  equa2ons  

Section 8.3

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Governing equations

€

dudt

= −θm∂Π∂x

+ fv

€

dvdt

= −θm∂Π∂y

− fu

€

dwdt

= −θm∂Π∂z

+θθm

g

€

dθdt

= 0€

θm∂Π∂y

+ fU + fAx = 0

€

θm∂Π∂x

+ fAy = 0

€

θm∂Π∂z

=θθm

g

balance  Basic  equa2ons  

€

f ∂U∂z

= −gθm

∂θ∂y

Thermal wind balance:

This equation must be satisfied at all times!

Section 8.3

Governing equations

€

dudt

= −θm∂Π∂x

+ fv

Along-­‐front  accelera2on  

€

∂U∂t

+u∂u∂x

+ v∂u∂y

+w∂u∂z

= −θm∂Π∂x

+ fv

Section 8.3

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Governing equations

€

dudt

= −θm∂Π∂x

+ fv

Along-­‐front  accelera2on  

€

∂u∂t

+u∂u∂x

+ v∂u∂y

+w∂u∂z

= −θm∂Π∂x

+ fv

€

∂U∂t

+ A U + Ax( ) + −Ay+ va( )∂U∂y

+wa∂U∂z

= −θm∂Π∂x

+ f −Ay+va( )

Section 8.3

Governing equations

€

dudt

= −θm∂Π∂x

+ fv

Along-­‐front  accelera2on  

€

∂U∂t

+u∂u∂x

+ v∂u∂y

+w∂u∂z

= −θm∂Π∂x

+ fv

€

∂U∂t

+ A U + Ax( ) + −Ay+ va( )∂U∂y

+wa∂U∂z

= −θm∂Π∂x

+ f −Ay+va( )

€

∂U∂t

+ A U + Ax( ) + −Ay+ va( )∂U∂y

+wa∂U∂z

= fvageostrophic  equa$on  

€

θm∂Π∂x

+ fAy = 0

Section 8.3

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Governing equations

€

dudt

= −θm∂Π∂x

+ fv

Along-­‐front  accelera2on  

€

∂U∂t

+u∂u∂x

+ v∂u∂y

+w∂u∂z

= −θm∂Π∂x

+ fv

€

∂U∂t

+ A U + Ax( ) + −Ay+ va( )∂U∂y

+wa∂U∂z

= −θm∂Π∂x

+ f −Ay+va( )

€

∂U∂t

+ A U + Ax( ) + −Ay+ va( )∂U∂y

+wa∂U∂z

= fva

€

∂U∂t

+ va∂U∂y

+wa∂U∂z

= −A U + Ax( ) + Ay∂U∂y

+ fva

Section 8.3

Equation for θ

€

∂θ∂t

+ −Ay+ va( )∂θ∂y

+wa∂θ∂z

= 0€

dθdt

= 0

(1)

Section 8.3

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Equation for θ and continuity equation

€

∂θ∂t

+ −Ay+ va( )∂θ∂y

+wa∂θ∂z

= 0€

dθdt

= 0

€

∂va∂y

+∂wa∂z

= 0

€

va =∂ψ∂z;wa = −

∂ψ∂y

€

ψ : streamfunction

(1)

Equation for θ and continuity equation

€

∂θ∂t

+ −Ay+ va( )∂θ∂y

+wa∂θ∂z

= 0€

dθdt

= 0

€

∂va∂y

+∂wa∂z

= 0

€

va =∂ψ∂z;wa = −

∂ψ∂y

€

∂U∂t

+ va∂U∂y

+wa∂U∂z

= −A U + Ax( ) + Ay∂U∂y

Previous slide:

€

f ∂U∂z

= −gθm

∂θ∂y

€

∂∂t

f ∂U∂z

= −gθm

∂θ∂y

⎛

⎝ ⎜

⎞

⎠ ⎟

€

ψ : streamfunction

Substitute (1) & (2): equation for cross-frontal circulation

Thermal wind balance:

(1)

(2)

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Eliassen-Sawyer equation

€

F2 ∂2ψ

∂z2− 2S2 ∂

2ψ∂y∂z

+ N 2 ∂2ψ

∂z2=−2Agθm

∂θ∂y

= −2AS2

equation for cross-frontal circulation Section 8.3

Eliassen-Sawyer equation

€

F2 ∂2ψ

∂z2− 2S2 ∂

2ψ∂y∂z

+ N 2 ∂2ψ

∂z2=−2Agθm

∂θ∂y

= −2AS2

€

F2 = − f∂Mg

∂y;N 2 =

gθm

∂θ∂z;S2 =

gθm

∂θ∂y

F: Inertial frequency; S: baroclinic frequency; N:Brunt Väisälä frequency

€

Mg ≡ u − fy =U + Ax − fy

equation for cross-frontal circulation Section 8.3

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Eliassen-Sawyer equation

€

F2 ∂2ψ

∂z2− 2S2 ∂

2ψ∂y∂z

+ N 2 ∂2ψ

∂z2=−2Agθm

∂θ∂y

= −2AS2

Elliptic equation if

€

q = F2N 2 − S4 > 0

€

F2 = − f∂Mg

∂y;N 2 =

gθm

∂θ∂z;S2 =

gθm

∂θ∂y

F: Inertial frequency; S: baroclinic frequency; N:Brunt Väisälä frequency

€

Mg ≡ u − fy =U + Ax − fy

Solution can be obtained by numerical method (successive relaxation)

Boundary condition:

€

ψ = 0

equation for cross-frontal circulation Section 8.3

Eliassen-Sawyer equation

€

F2 ∂2ψ

∂z2− 2S2 ∂

2ψ∂y∂z

+ N 2 ∂2ψ

∂z2=−2Agθm

∂θ∂y

= −2AS2

equation for cross-frontal circulation

€

FF ≡ −2AS2 = 2∂vg∂y

S2 = 2∂vg∂y

gθm

∂θ∂y

= 2 gθm

∂vg∂y

∂θ∂y

= 2 gθm

Qg2

frontogenetical function, FF:

geostrophic Q-vector

Section 8.3

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Prescribed jet

€

U = U 0 exp −y − y0

Y⎛ ⎝ ⎜

⎞ ⎠ ⎟

2⎧ ⎨ ⎩

⎫ ⎬ ⎭

exp −z − z0

Z⎛ ⎝ ⎜

⎞ ⎠ ⎟

2⎧ ⎨ ⎩

⎫ ⎬ ⎭

if z ≤ z0

€

U =U 0 exp −y− y0

Y⎛ ⎝ ⎜

⎞ ⎠ ⎟

2⎧ ⎨ ⎩

⎫ ⎬ ⎭

if z > z0

€

f ∂U∂z

= −gθm

∂θ∂y

€

z0 =10 km

€

y0 = 0; Y = 500 km; Z = 5 km

tropopause

Section 8.3

Solution of the Eliassen-Sawyer equation

Vertical motion and forcing

wa labels in cm/s The frontogenetic forcing function, -2AS2, is shown in black (labels in units of 10-11 s-3)

Forcing is prescribed by prescribing A

Section 8.3

Fig 8.10

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Solution of the Eliassen-Sawyer equation

Motion perpendicular to front and parallel to front

labels in m/s U and va

Section 8.3

Forcing is prescribed by prescribing A

Fig 8.11

Action at a distance

The solution of the Sawyer-Eliassen equation at y=0. The frontogenetic forcing function, FF=-2AS2, is shown in red, and the ageostrophic horizontal velocity, va, is shown in blue. Values of other parameters are given in previous slides.

Section 8.3

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Action at a distance

The solution of the Sawyer-Eliassen equation at z=7 km. The frontogenetic forcing function, FF=-2AS2, is shown in red, and the ageostrophic vertical velocity, wa, is shown in blue. Values of other parameters are given in the previous slides.

front

warm cold

Section 8.3

Some properties of the solution

•  If frontogenesis warm air rises and cold air sinks (direct circulation)

•  Upward velocity is one to two orders of magnitude smaller than horizontal velocity implying very slanted motion leading to the formation layered clouds

•  Action at a distance: cross-frontal circulation penetrates into region where no forcing occurs

•  Cross-frontal circulation is frontolytic if warm air rises!

Section 8.3

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Solution of the Eliassen-Sawyer equation: qualitatively in accord with this case

Aug.6  1996,  00  UTC  

up  

down  

warm

cold  

Section 8.3

“Sinterklaas-storm”?

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warm sector

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“stau”

“stau”

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“stau”

What next? Topics of the presentations?

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