atmospheric dynamics: lecture 13 - universiteit utrechtdelde102/lecture12atmdyn2014.pdfomega...
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Atmospheric Dynamics: lecture 13 (http://www.staff.science.uu.nl/~delde102/)
Chapter 9 (part 2):
Omega equa2on: interpreta2on
Baroclinic waves and cyclogenesis
Two level model
Rossby waves and baroclinic instability
Interpreta7on in terms of the Q-‐vector
Quasi-geostrophic equations: Section 9.4
€
dg∇2Φ
dt= f0
2 ∂ω∂p
−β∂Φ∂x
€
dg∂Φ∂pdt
= −σω −RJcp p
Quasi-geostrophic vorticity eq.
Quasi-geostrophic thermodynamic eq.
€
σ ≡RpSp
€
Sp ≡αcp−∂T∂p
Quasi-geostrophic equations: Section 9.4
€
dg∇2Φ
dt= f0
2 ∂ω∂p
−β∂Φ∂x
€
dg∂Φ∂pdt
= −σω −RJcp p
Quasi-geostrophic vorticity eq.
Quasi-geostrophic thermodynamic eq.
€
∂ua∂x
+∂va∂y
+∂ω∂p
= 0
€
∂ug∂x
+∂vg∂y
= 0
Continuity equation:
Divergence of geostrophic wind = 0:
€
ug = −1f0∂Φ∂y;vg =
1f0∂Φ∂x
geostrophic wind and geopotential: €
σ ≡RpSp
€
Sp ≡αcp−∂T∂p
€
f ≡ f0 +βy
Omega equation
€
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
Since both T, ug and vg can all be expressed as a function of Φ, we can can calculate the vertical motion (ω) from the distribution of Φ only!!!
Omega equation
€
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
The omega equation is an elliptic partial differential equation
Since both T, ug and vg can all be expressed as a function of Φ, we can can calculate the vertical motion (ω) from the distribution of Φ only!!!
Omega equation
€
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
⎛
⎝ ⎜
⎞
⎠ ⎟ .
Section 9.5
Suppose:
€
ω =W0 sinπppref
⎛
⎝ ⎜
⎞
⎠ ⎟ sin kx( )sin ly( )
upward (downward) motion if Qg-vector is convergent (divergent)
Interpretation of the solution of the omega equation
€
σ∇2ω + f02 ∂
2ω∂p2
= −2Rp ∇ ⋅ Q g
€
−ω ≈ w ≈ − ∇ ⋅
Q gupward (downward) motion if Qg-vector is convergent (divergent)
Omega equation
€
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
In the entrance of a jetstreak we may have:
€
∂T∂x
= 0; ∂T∂y
< 0;
∂vg
∂y< 0;
∂vg
∂y>>
∂vg
∂xArrows: Q-vectors; dashed: isotherms; solid lines: geopotential
y
x
cold
warm
descent
ascent upward motion if Qg-vector is convergent!
Q-vectors in mid-latitude cyclone
Based on a simulation with 36-level model of a the life cycle of an unstable baroclinic wave (details of this model simulation will be given later (see chapter 10 lecture notes)
0 -5
15
1250 1200
1300
46°N
64°N
60°
1000
1400
warm sector
wf bbf
cf
Fig 1.85 (lower panel)
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
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
trough axis
Cyclogenesis and the life-cycle of a mid-latitude cyclone
About 24 hours
March 2 1995
March 3 1995
Conceptual model
about 24-‐4
8 hours
Shapiro and Keyser 1990
Pressure and fronts:
temperature:
Conceptual model
about 24-‐4
8 hours
Shapiro and Keyser 1990
Pressure and fronts:
temperature:
Manifestation of Baroclinic instability of Rossby waves
Analysis of the stability of a zonal current in thermal
wind balance to wave-like perturbations
1.Model formulation 2.Model equations 3.Steady state 4.Linearization around steady state 5.Investigate dynamics of wave like perturbations
Procedure
Two level model
This model is an example of typical numerical weather prediction model of the 1950’s.
It contains the bare minimum of physics to produce cyclones by a process called baroclinic instability
Section 9.6
Two level model Unknowns are:
€
Φ1, Φ3 and ω2
(ψ=Φ/f0)
Section 9.6
€
Δp
€
Δp
Two level model
€
dg∇2Φ
dt= f0
2 ∂ω∂p
−β∂Φ∂x
€
dg∂Φ∂pdt
= −σω
Apply vorticity eq. to levels 1 and 3
Apply thermodynamic eq. to level 2
Unknowns are:
€
Φ1, Φ3 and ω2
(ψ=Φ/f0)
Section 9.6
€
Δp
€
Δp
Two level model
€
dg∇2Φ
dt= f0
2 ∂ω∂p
−β∂Φ∂x
€
dg∂Φ∂pdt
= −σω
Apply vorticity eq. to levels 1 and 3
Apply thermodynamic eq. to level 2
Unknowns are:
€
Φ1, Φ3 and ω2
(ψ=Φ/f0)
Section 9.6
€
Δp
€
Δp
Finite differences
€
dg∂Φ∂pdt
= −σω
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ 2
Apply thermodynamic equation to level 2:
(ψ=Φ/f0)
Section 9.6
Finite differences
€
dg∂Φ∂pdt
= −σω
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ 2
€
∂Φ∂p⎡
⎣ ⎢ ⎤
⎦ ⎥ 2≅Φ1 −Φ3p1 − p3
=Φ1 −Φ3Δp
Finite difference approximation of vertical derivative:
Apply thermodynamic equation to level 2:
(ψ=Φ/f0)
Section 9.6
Discretisation of equations
€
dg∇2Φ
dt= f0
2 ∂ω∂p
−β∂Φ∂x
€
∂∇2ψ∂t
+ v g ⋅ ∇ ∇2ψ( ) = f0
∂ω∂p
−β∂ψ∂x
€
v g ≡ ˆ k × ∇ ψ
(ψ=Φ/f0)
Vorticity-equation
Section 9.6
Discretisation of equations
€
dg∇2Φ
dt= f0
2 ∂ω∂p
−β∂Φ∂x
€
∂∇2ψ∂t
+ v g ⋅ ∇ ∇2ψ( ) = f0
∂ω∂p
−β∂ψ∂x
€
v g ≡ ˆ k × ∇ ψ
(ψ=Φ/f0)
€
∂∇2ψ1∂t
+ v 1 ⋅ ∇ ∇2ψ1( ) = f0
∂ω∂p⎡
⎣ ⎢ ⎤
⎦ ⎥ 1−β
∂ψ1∂x
Apply to level 1
Vorticity-equation
Section 9.6
Discretisation of equations
€
dg∇2Φ
dt= f0
2 ∂ω∂p
−β∂Φ∂x
€
∂∇2ψ∂t
+ v g ⋅ ∇ ∇2ψ( ) = f0
∂ω∂p
−β∂ψ∂x
€
v g ≡ ˆ k × ∇ ψ
(ψ=Φ/f0)
€
∂∇2ψ1∂t
+ v 1 ⋅ ∇ ∇2ψ1( ) = f0
∂ω∂p⎡
⎣ ⎢ ⎤
⎦ ⎥ 1−β
∂ψ1∂x
Apply to level 1
€
∂ω∂p⎡
⎣ ⎢ ⎤
⎦ ⎥ 1≅ω2 −ω0p2 − p0
=ω2Δp
Vorticity-equation
Section 9.6
Discretisation of equations
€
∂∂t
∂ψ∂p⎛
⎝ ⎜
⎞
⎠ ⎟ + v g ⋅ ∇ ∂ψ∂p⎛
⎝ ⎜
⎞
⎠ ⎟ = −
σf0ω
€
v ψ ≡ ˆ k × ∇ ψ
Thermodynamic equation
€
dg∂Φ∂pdt
= −σω
(ψ=Φ/f0)
Section 9.6
Discretisation of equations
€
∂∂t
∂ψ∂p⎛
⎝ ⎜
⎞
⎠ ⎟ + v g ⋅ ∇ ∂ψ∂p⎛
⎝ ⎜
⎞
⎠ ⎟ = −
σf0ω
€
v ψ ≡ ˆ k × ∇ ψ
Apply to level 2
Thermodynamic equation
€
dg∂Φ∂pdt
= −σω
(ψ=Φ/f0)
€
∂∂t
∂ψ∂p⎛
⎝ ⎜
⎞
⎠ ⎟ 2
+ v g( )2 ⋅ ∇ ∂ψ∂p⎛
⎝ ⎜
⎞
⎠ ⎟ 2
= −σf0ω2
Section 9.6
Discretisation of equations
€
∂∂t
∂ψ∂p⎛
⎝ ⎜
⎞
⎠ ⎟ + v g ⋅ ∇ ∂ψ∂p⎛
⎝ ⎜
⎞
⎠ ⎟ = −
σf0ω
€
v ψ ≡ ˆ k × ∇ ψ
Apply to level 2
Thermodynamic equation
€
dg∂Φ∂pdt
= −σω
(ψ=Φ/f0)
€
∂ψ∂p⎛
⎝ ⎜
⎞
⎠ ⎟ 2≅ψ1 −ψ3p1 − p3
=ψ1 −ψ3Δp
€
∂∂t
∂ψ∂p⎛
⎝ ⎜
⎞
⎠ ⎟ 2
+ v g( )2 ⋅ ∇ ∂ψ∂p⎛
⎝ ⎜
⎞
⎠ ⎟ 2
= −σf0ω2
€
u2 ≅12u1 +u3( ) = −
12∂∂y
ψ1 +ψ3( ),
Section 9.6
Steady state current + perturbation
€
ψ1 = −U1y +ψ '1 (x,y,t),
€
ψ3 = −U3y +ψ '3 (x,y,t),
€
ω2 =ω '2 (x,y,t),
€
U1 and U3 : geostrophic current
€
U1 and U3
Linear stability analysis of a geostrophic zonal current Section 9.7
Steady state current + perturbation
€
ψ1 = −U1y +ψ '1 (x,y,t),
€
ψ3 = −U3y +ψ '3 (x,y,t),
€
ω2 =ω '2 (x,y,t),
Substitute into equations and linearize around steady state
€
∂∂t
+U1∂∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟ ∇h
2ψ'1+β∂ψ'1∂x
=f0Δp
ω'2 ,€
U1 and U3 : geostrophic current
€
U1 and U3
e.g.:
Linear stability analysis of a geostrophic zonal current Section 9.7
Linearize all three equations around steady state geostrophic current
€
∂∂t
+U1∂∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟ ∇h
2ψ'1+β∂ψ'1∂x
=f0Δp
ω'2 ,
€
∂∂t
+U3∂∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟ ∇h
2ψ'3+β∂ψ'3∂x
=− f0Δp
ω'2 ,
€
∂∂t
+Um∂∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟ ψ'1−ψ'3( )−UT
∂∂x
ψ'1+ψ'3( ) =+σΔpf0
ω'2 ,
With
€
Um ≡U2 ≈U1 +U32
;UT ≡U1 −U32
(9.46)
Section 9.7
€
∂∂t
+Um∂∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟ ∇h
2ψm + β∂ψm
∂x+UT
∂∂x∇h
2ψT = 0,
€
∂∂t
+Um∂∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟ ∇h
2ψT − 2λ2ψT( ) + β
∂ψT
∂x+UT
∂∂x
∇h2ψm + 2λ2ψm( ) = 0,
€
ψm =ψ1'+ψ3 '2
;ψT =ψ1'−ψ3 '2with
We can reduce this set of three equations to
Section 9.7
Substitute wave-like solutions
€
ψm = Aexp i lx + my −ωt( )[ ], ψT = Bexp i lx + my −ωt( )[ ]
Section 9.7
Substitute wave-like solutions
€
ψm = Aexp i lx + my −ωt( )[ ], ψT = Bexp i lx + my −ωt( )[ ]
Leads to a pair of simultaneous linear algebraic equations for the coefficients A and B:
€
cx −Um( )k2 + β[ ]A −UTk2B = 0
€
UT k2 − 2λ2( )A − cx −Um( ) k2 + 2λ2( ) + β[ ]B = 0,
€
k2 ≡ l2 + m2 and cx ≡ω / l.
€
λ2 =f02
σ Δp( )2Inverse of Rossby radius of deformation
Section 9.7
Dispersion relation
€
cx =Um −β k2 + λ2( )k2 k2 + 2λ2( )
± δ1/ 2,
Non-trivial solutions will exist only if the determinant of the coefficients of A and B is zero. Thus the phase speed must satisfy the condition
€
ψm = Aexp i lx + my −ωt( )[ ], ψT = Bexp i lx + my −ωt( )[ ]Remember:
Dispersion relation
€
cx =Um −β k2 + λ2( )k2 k2 + 2λ2( )
± δ1/ 2,
δ ≡ β2λ4
k 4 k 2+2λ 22 - UT
2 2λ 2-k 2
k 2+2λ 2 .
Non-trivial solutions will exist only if the determinant of the coefficients of A and B is zero. Thus the phase speed must satisfy the condition
€
ψm = Aexp i lx + my −ωt( )[ ], ψT = Bexp i lx + my −ωt( )[ ]Remember:
Dispersion relation
€
cx =Um −β k2 + λ2( )k2 k2 + 2λ2( )
± δ1/ 2,
δ ≡ β2λ4
k 4 k 2+2λ 22 - UT
2 2λ 2-k 2
k 2+2λ 2 .
Non-trivial solutions will exist only if the determinant of the coefficients of A and B is zero. Thus the phase speed must satisfy the condition
€
If δ < 0, cx is imaginary⇒ exponential growth
Baroclinic instability
€
ψm = Aexp i lx + my −ωt( )[ ], ψT = Bexp i lx + my −ωt( )[ ]Remember:
(9.58 & 9.59)
Baroclinic instability in a special cases
€
cx =Um −β k2 + λ2( )k2 k2 + 2λ2( )
± δ1/ 2,
δ ≡ β2λ4
k 4 k 2+2λ 22 - UT
2 2λ 2-k 2
k 2+2λ 2 .
Dispersion relation:
Section 9.7
Baroclinic instability in a special cases
€
cx =Um −β k2 + λ2( )k2 k2 + 2λ2( )
± δ1/ 2,
δ ≡ β2λ4
k 4 k 2+2λ 22 - UT
2 2λ 2-k 2
k 2+2λ 2 .
Dispersion relation:
special case, β=0
€
cx =Um ±UTk2 − 2λ2
k2 + 2λ2⎛
⎝ ⎜
⎞
⎠ ⎟
1/2
Section 9.7
(9.60)
Baroclinic instability in a special cases
€
cx =Um −β k2 + λ2( )k2 k2 + 2λ2( )
± δ1/ 2,
δ ≡ β2λ4
k 4 k 2+2λ 22 - UT
2 2λ 2-k 2
k 2+2λ 2 .
Dispersion relation:
special case, β=0
€
cx =Um ±UTk2 − 2λ2
k2 + 2λ2⎛
⎝ ⎜
⎞
⎠ ⎟
1/2
For waves with zonal wave numbers satisfying k2<2λ2, this expression has an imaginary part.
Section 9.7
(9.60)
Baroclinic instability in a special cases
€
cx =Um −β k2 + λ2( )k2 k2 + 2λ2( )
± δ1/ 2,
δ ≡ β2λ4
k 4 k 2+2λ 22 - UT
2 2λ 2-k 2
k 2+2λ 2 .
Dispersion relation:
special case, β=0
€
cx =Um ±UTk2 − 2λ2
k2 + 2λ2⎛
⎝ ⎜
⎞
⎠ ⎟
1/2
For waves with zonal wave numbers satisfying k2<2λ2, this expression has an imaginary part. Therefore, all waves with wavelengths greater than a critical wavelength will amplify.
Section 9.7
(9.60)
Critical wavelength:
€
Lc =2πλ
≈ 3000 km
special case, β=0, continued:
Critical wavelength:
€
Lc =2πλ
≈ 3000 km
special case, UT =0:
€
cx =Um −βk2
€
cx =Um −β
k2 + 2λ2or
NO THERMAL WIND: BAROTROPIC
special case, β=0, continued:
Critical wavelength:
€
Lc =2πλ
≈ 3000 km
special case, UT =0:
€
cx =Um −βk2
€
cx =Um −β
k2 + 2λ2or
NO THERMAL WIND: BAROTROPIC
special case, β=0, continued:
This is the dispersion relation for barotropic stable Rossby waves
Hovemöller diagram The 500 hPa geopotential (given in dm) as a function of time and longitude. The values are average values of geopotentials between 35°N and 60°N. Ridges are shown by horizontal hatching; troughs are shown by vertical hatching. The slanted straight lines indicate a succession of maximum develop-ment of troughs and ridges (Figure 7.13).
group velocity
phase velocity
Dispersive waves!! group velocity>phase velocity
x
time
November 1945
Fig. 9.13
Section 9.7
Neutral stability curve
Neutral stability curve, δ=0, for the two-level baroclinic model
€
cx =Um −β k2 + λ2( )k2 k2 + 2λ2( )
± δ1/ 2,
δ ≡ β2λ4
k 4 k 2+2λ 22 - UT
2 2λ 2-k 2
k 2+2λ 2 .
Dispersion relation:
Section 9.7
δ=0
Instability: optimum wavelength
Short waves are stabilized by static stability
Long waves are stabilized by beta-effect
Section 9.7
€
k =2πLx
€
Lx = 3.8×106 m
Assuming m=0:
Instability: manifestation in a simplified numerical model
Geopotential, Temperature, Q-vector at 864 hPa
http://www.staff.science.uu.nl/~delde102/BaroclinicLifeCycle.htm
Geopotential field and Vertical motion (taken from Holton, 2004)
Given the following expression for the geopotential field:
where Φ0 is a function of p alone, c is a constant speed, l a zonal wave number, and p0=1000hPa. (a) Use the quasi-geostrophic vorticity equation to obtain the horizontal divergence field consistent with this Φ-field. (Assume df/dy=0) (b) Assuming that ω(p0)=0, obtain an expression for ω(x, y, p, t) by integrating the continuity equation with respect to pressure. (c) Sketch the geopotential fields at 750 hPa and at 250 hPa. Indicate regions of positive and negative vertical velocity at 500 hPa.
€
Φ =Φ0 p( ) + cf0 −y cos πp / p0( ) +1[ ]( ) + l−1 sin l x − ct( )[ ]
Problem 9.3 Homework
Comment: you will see how vertical motion is related to the geopotential wave (phase, phase speed and wavelength)
Next week:
• Interpretation of baroclinic instability, using the omega equation and Q-vectors
• Solutions of the omega equation • Numerical simulation of idealized
unstable baroclinic wave