Original articles - Universiteit Utrechtdelde102/Lecture14AtmDyn2015.pdf · Original articles:...
Transcript of Original articles - Universiteit Utrechtdelde102/Lecture14AtmDyn2015.pdf · Original articles:...
18/12/15
1
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
18/12/15
2
Baroclinic wave
Baroclinic wave
18/12/15
3
Baroclinic wave
Baroclinic wave
18/12/15
4
Baroclinic wave
Baroclinic wave
18/12/15
5
Warm sector
Warm sector
18/12/15
6
Warm sector
Warm sector
18/12/15
7
occlusion
warm seclusion
18/12/15
8
Upward motion
18/12/15
9
Atmospheric “river”
warm conveyor belt
18/12/15
10
warm conveyor belt
warm conveyor belt
18/12/15
11
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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30
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: