atmospheric dynamics: lecture 13 - universiteit utrechtdelde102/lecture12atmdyn2014.pdfomega...

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

∂ug∂x

+∂vg∂y

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 = −

∂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 = −

∂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 ≈ − ∇ ⋅

Qg1 = −∂ug

∂x∂T∂x

+∂vg

∂x∂T∂y

⎝ ⎜

⎠ ⎟ ; Qg2 = −

∂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 = −

∂y∂T∂x

+∂vg

∂y∂T∂y

⎝ ⎜

⎠ ⎟ .

Section 9.5

In the entrance of a jetstreak we may have:

∂T∂x

= 0; ∂T∂y

∂y< 0;

∂y>>

∂xArrows: Q-vectors; dashed: isotherms; solid lines: geopotential

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)

1250 1200

warm sector

wf bbf

Fig 1.85 (lower panel)

1250 1200

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

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

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

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

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

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

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

∂∂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

∂∂t

∂ψ∂p⎛

⎝ ⎜

⎠ ⎟ + v g ⋅ ∇ ∂ψ∂p⎛

⎝ ⎜

⎠ ⎟ = −

σf0ω

v ψ ≡ ˆ k × ∇ ψ

Apply to level 2

dg∂Φ∂pdt

= −σω

(ψ=Φ/f0)

∂∂t

∂ψ∂p⎛

⎝ ⎜

⎠ ⎟ 2

+ v g( )2 ⋅ ∇ ∂ψ∂p⎛

⎝ ⎜

⎠ ⎟ 2

= −σf0ω2

Section 9.6

∂∂t

∂ψ∂p⎛

⎝ ⎜

⎠ ⎟ + v g ⋅ ∇ ∂ψ∂p⎛

⎝ ⎜

⎠ ⎟ = −

σf0ω

v ψ ≡ ˆ k × ∇ ψ

Apply to level 2

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

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 ,

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 .

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 .

If δ < 0, cx is imaginary⇒ exponential growth

Baroclinic instability

(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

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 .

special case, β=0

cx =Um ±UTk2 − 2λ2

k2 + 2λ2⎛

⎝ ⎜

⎠ ⎟

Section 9.7

(9.60)

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 .

special case, β=0

k2 + 2λ2⎛

⎝ ⎜

⎠ ⎟

For waves with zonal wave numbers satisfying k2<2λ2, this expression has an imaginary part.

Section 9.7

(9.60)

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 .

special case, β=0

k2 + 2λ2⎛

⎝ ⎜

⎠ ⎟

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:

Lc =2πλ

≈ 3000 km

special case, UT =0:

cx =Um −βk2

cx =Um −β

k2 + 2λ2or

NO THERMAL WIND: BAROTROPIC

Lc =2πλ

≈ 3000 km

special case, UT =0:

cx =Um −βk2

cx =Um −β

k2 + 2λ2or

NO THERMAL WIND: BAROTROPIC

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

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 .

Section 9.7

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

atmospheric dynamics: lecture 13 - universiteit utrechtdelde102/lecture12atmdyn2014.pdfomega...

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