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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$on 8.3) Define “prototype” (simplified) problem Leads diagnos8c equa8on for “cross-‐frontal circula8on” (Eliassen-‐Sawyer equa8on) Diagnose solu8ons of this equa8on
30 November 2011
9 December 2010 06 UTC Classical weather map
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Correponding satellite image
9 December 2010 08 UTC
Corresponding analysis (850 hPa)
3
Potential vorticity at 320 K
Classical conceptual models of fronts
Left panel: a cold front. Right panel: a warm front. (Source: Wikimedia Commons)
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7
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Cross section along 53°N
Cold front
Cross section along 53°N
Cold front
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Cross section along 53°N
Cold front
Cross section along 53°N
Cold front
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Cross section along 53°N
Cold front
Cross section along 48°N
Cold front: cross-frontal circulation
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Vertical motion and fronts
• Why are fronts associated with clouds and precipitation?
• Where is precipitation expected, i.e. where is motion upwards?
Vertical motion and fronts
• Why are fronts associated with clouds and precipitation?
• Where is precipitation expected, i.e. where is motion upwards?
Sawyer-Eliassen prototype problem of thermal wind adjustment of the atmosphere to frontogenesis
Section 8.3
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Sawyer-Eliassen Prototype problem
geostrophic flow
Frontogene$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
14
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
15
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
16
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
17
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
19
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
20
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
21
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
22
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
23
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?
Retake on Wednesday 21 December 09:30-12:30, room 165 BBL
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