atmospheric dynamics and meteorologyfrontogenesis is forced by the large-scale flow but is also...

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Atmospheric dynamics and meteorologyB. Legras, http://www.lmd.ens.fr/legras

III Frontogenesis

(pre requisite: quasi-geostrophic equation, baroclinic instability in the Eady and Phillips models )

Recommended books:- Holton, An introduction to dynamic meteorology, Academic Press- Houze, Cloud dynamics, Academic Press- Vallis, Atmospheric and oceanic fluid dynamics

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Atmospheric circulation from the viewpoint of a geograph

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EQUATIONS

D t u− f v+∂x ϕ '=0D t v+ f u+∂ yϕ '=0

−b '+∂~z ϕ '=0

Dt b=D tb '+~w N 2

=0∂xu+∂ y v+∂~z

~w=0

with b '= g¯̄θ

θ ' (buoyancy) ,~w=Dt

~z ,

and N 2=g¯̄θd~z θ̄

Since ω=D t p=−pH

Dt z̃ , we have

∂ω∂ p

=∂ pDt p=∂ z̃ w̃−w̃H

Boussinesq approximationconsists here in neglecting

the term in w̃H

FROM HYDROSTATIC EQUATIONSIN PRESSURE COORDINATE

The standard air profile is defined by θ̄and ϕ̄ (geopotential) which are function of p only.ϕ ' is the geopotential deviation: ϕ '=ϕ−ϕ̄ .

A new form of the pressure coordinate is introducedas a pseudo-height by ~z=−H log p/ p0 where H=RT 0/ g .(for an isothermal atmosphere, one would have exactly ~z=z )The hydrostatic equation is now transformed by combining it with the perfect gas law

∂ pϕ=−1ρ=−

RTp

∂ϕ

∂ log p+Rθ( p

p0)κ

=0

∂ϕ '∂ log p

+Rθ '( pp0)

κ

=0

∂~z ϕ '=Rθ ' ( pp0)

κg

RT 0

=gθ '¯̄θ

Notice : ¯̄θ is an isothermal profile , do not confuse with θ̄

b=b̄(~z )+b' (x , y ,~z , t) and N 2=d b̄d~z

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Frontogenesis● Boussinesq equations● Quasi-geostrophic frontogenesis● Semi-geostrophic frontogenesis● Influence of moisture● Large-scale front● Frontogenesis and storms

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Frontogenesis● Boussinesq equations● Quasi-geostrophic frontogenesis● Semi-geostrophic frontogenesis● Influence of moisture● Large-scale front● Frontogenesis and storms

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Quasi geostrophic frontogenesis

Apparent paradox:Q1 is associated to the geostrophic circulationIt destroys the thermal wind balance.It is restored by the non geostrophic wind

Quasi-geostrophic form of the Boussinesq equations

The geostrophic and non geostrophic parts are separatedu=ug+ua and v=v g+va

with f ug=−∂ yϕ and f v g=∂xϕ

HenceDg t ug− f va=0Dg t v g+ f ua=0

D g t b+w N 2=0∂x ua+∂ y va+∂zw=0

with Dg t≡∂t+ug∂x+v g∂ y

THERMAL WIND

f ∂z ug=−∂ y bf ∂z v g=∂ xb

Dg t∂x b=Q1−N 2∂xw

Dg t f ∂ z vg=−Q1− f 2∂zua

where Q1=−∂xu g ∂xb−∂x v g∂ yb

We have 2Q1=N 2 ∂xw− f 2 ∂z ua

Assuming that the geostrophic wind isoriented along y with small variationsin this direction ∂ y≪∂x ,∂z . ug=0 ,∂ y va=0

Let define ua=∂z ψ et w=−∂x ψ ,hence

−2Q1=N 2∂x x

2ψ+ f 2

∂z z2

ψ

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Cold advection

Warm advection

Confluence

Mechanisms of gradient amplification.

Dg t ∂x b=Q1−N 2∂ x w

où Q1=−∂ x ug ∂x b−∂x v g ∂ y b

x

x

x

y

y

y

Q1 frontogeneticQ1 frontolytic

Q1 intensifies the temperature gradient bytwo mechanisms: confluence ( ∂ xug<0)and gradient shear ( ∂x v g∂ y b≠0)

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Let denote Q⃗=(Q1,Q2)

Generation of the ageostrophic circulation2Q⃗=N 2

∇H w− f 2∂z u⃗a

in other terms (omega equation)2 ∇H .Q⃗=N 2 ∇H

2 w+ f 2 ∂zz2 w

Convergence of Q⃗ = max of w positiveDivergence of Q⃗ = min of w negative

More generally the temporal evolutionof the thermal wind

f ∂z v g=∂xbf ∂z u g=−∂ yb

gives Dg t∂x b=Q1−N 2∂xw

Dg t f ∂z vg=−Q1− f 2 ∂z ua

Dg t∂ yb=Q2−N 2∂ yw

−Dg t f ∂z v g=−Q2− f 2 ∂z vawhere

Q1=−∂x ug ∂xb−∂x v g∂ ybQ2=−∂ yug∂xb−∂y v g∂ yb

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Limitation of the quasi-geostrophic approximation : the front is only formed near the ground where w=0 and not inside where the ageostrophic circulation counterbalances Q

1. Any horizontal oscillation of size L is

damped over a depth f L/N in the vertical.

Q

Geostrophic circulation, ageostrophic circulation and total circulation in the transverse plane to the jet in the confluent case.The ageostrophic circulaton would tend to produce a tilted front but cannot do it since it is excluded from the advecting flow upon the quasi-geostrophic approximation.

(entering in the figure)jetCase of frontogenesis by confluence

Q1=−∂x ug ∂x b−∂x v g∂y bQ2=−∂y ug ∂x b−∂ y v g∂ y b

Generation of the ageostrophic circulation2Q⃗=N 2

∇H w− f 2∂z u⃗a

ou in other words (omega equation)2 ∇H .Q⃗=N 2

∇H2 w+ f 2

∂ zz2 w

Convergence of Q⃗ = max of w positiveDivergence of Q⃗ = min of w negative

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Summary (1):- The non divergent quasi-geostrophic flow satisfies the thermal wind balance but the advection by the geostrophic wind tends to destroy this balance.

- The balance is restored by the divergent ageostrophic wind which can be determined from the quasi-geostrophic wind. This slave ageostrophic component is part of the quasi-geostrophic model (with the limitation that its advection is neglected).

Other « free » components of ageostrophic may be superimposed to the « slave » component such as those associated with the gravity waves excluded from the quasi-geostrophic model (defined as a first order approximation in the Rossby number).

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Summary (2):

- The vector Q allows to visualize the slave ageostrophic circulation.-In the case of a rectininear jet, the quantity -Q is proportional to the (mofified) Laplacian of the ageostrophic circulation streamfunction in the transverse plane.

- The horizontal divergence of Q is proportional to the (modified) Laplacian of the vertical component of the ageostrophic velocity: convergence of Q = ascending flow, divergence of Q = subsident flow.

-The geostrophic circulation can reinforce the temperature gradients (frontogenesis) by confluence or shearing of temperature lines. If advection by the ageostrophic flow is neglected, this intensification is limited to the upper and lower boundaries of the domain (surface and tropopause in the Eady model). Taking into account the ageostrophic advection allows to reinforce and tilt the inside part of the front.

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Vecteur QQ and divergence of Q(notice the convergence on the north and east sides of the depression and (warm front) and that on the south-east (cold front)

Geopotential (solid) and temperature (dash) map at 700 hPa

Typical development of a baroclinic perturbation

We see here that the vector Q visualizes the ascending and descending branches of the baroclinic circulation through the temperatue lines.The ascending motion tends to occur at smaller scale than the descending motion, near the developping frontal zones.

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Frontogenesis● Boussinesq equations● Quasi-geostrophic frontogenesis● Semi-geostrophic frontogenesis● Influence of moisture● Large-scale front● Frontogenesis and storms

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Semi-geostrophic frontogenesis

FRONTAL DYNAMICS SCALING (2D version transverse to the jet)

We assume that the length scale of the front is much larger than its transverse scale L y≫Lx and V≫U

We assume that the Lagrangian derivative depends mostly of the crosss front variations

Dt∼ULx

Typically : L y≈1000 km, Lx≈100 km, U≈1 m/s, V≈10 m/sand we take f =10−4 s−1

Hence the ratios between inertial terms and Coriolis areDt u

f v=

U 2

f LxV≈0.01 et

Dt v

f u=

Vf Lx

≈1

Basic approximation: The along front component of the wind is well approximated by itsquasi-geostrophic component but the transverse component is not and we neglect thevariations of the wind along y.Thus we write u=ug+ua where the two components are of the same order, and v=v g.and we take into account the ageostriphic terms in the advection

Dt=∂t+u ∂x+v g ∂ y+w ∂z

The equations areDt v g+ f ua=0 and D tb+N 2w=0

∂xua+∂zw=0

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FRONTAL DYNAMIC EQUATIONS (geostrophic moment equations)

We take u=ug+ua where the two components are of the same order, and v=v g.and we take into account the ageostrophic terms in the advection

D t=∂t+u∂x+v g∂ y+w∂z

The equations are thusDt v g+ f ua=0 et Dt b+N

2w=0∂x ua+∂ zw=0

The time derivatives of the two terms of the thermal gradient are extractedf Dt ∂z v g=−Q1−∂zw ∂x b− f 2

∂zua− f ∂z ua∂x v gD t∂ xb=Q1−(N 2

+∂zb)∂ xw−∂x ua∂ xb

We see new terms (underlines) involving the vertical derivatives of the ageo-strophic circulation which reinforce the temperature gradientAfter combining and rearrangement, we obtain the Sawyer-Eliassen equation:

N s2∂x x

2ψ−2 S2

∂ x z2

ψ+F 2∂ z z

2ψ=−2Q1 ,

with N s2=N 2

+∂zb , F2= f ( f +∂ x v g) and S 2

=∂ xb .This equation is elliptic and can be solved if F 2N s

2−S 4

>0, that isif the potential vorticity is positive (see problem).

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The effect of advection by the ageostrophic equation is● the formation of a tilted front towards the cold region ● le reinforcement of the wind in the cyclone to the expense of the anticyclone● The shrinking of tha area of the warm surface anomaly and the expansion of the area of the cold anomaly● The inverse effect at the top boundary

Frontogenesis is forced by the large-scale flow but is also reinforced by the convergence of the ageostrophic circulation which increases with the reinforcement of the flow. In non dissipative situation, an infinite value of vorticity may appear in a finite time on the front.

Development of a ground frontIt is easier to get a front at the ground because w cannot counterbalance the intensification of temperature gradient.

In gray, location of the front

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SEMI-GEOSTROPHIC EQUATIONS(abridged form)

We define X =x+1fvg , such that Dt X=ug ,

and we change the coordinates ( x , y , z , t ) into (X ,Y= y , Z= z , τ=t )

This leads to ∂x=(1+∂ x v g

f)∂X

∂z=∂Z+∂xb∂X and ∂ y=∂Y+∂ y vgf

∂X

thus (1+∂ xvgf

)(1−∂X v g

f)=1

If Φ=ϕ+12vg

2 , hence ∂X Φ=∂ xϕ= f vg

∂Y Φ=∂ yϕ=− f ug and ∂ZΦ=∂ zϕ=b

The potential vorticity is q=ω⃗ ∇⃗ (b+b̄) with ω⃗=(−∂ z vg ,0, f +∂x vg).It is conserved (D tq=0) and we have

q=∂Z (b+ b̄)( f +∂x vg)= f∂Z 2

2 (Φ̄+Φ)

1−1

f 2 ∂X 22 Φ

or approximately qf≈(N 2+∂Z 2

2 Φ)(1+1

f 2 ∂X 22 Φ)=N 2+∂Z 2

2 Φ+N 2

f 2 ∂X 22 Φ

Under this approximation,

the quantity ∂Z 2

2 Φ+N 2

f 2∂X 2

2 Φ

is conserved by the geostrophic flowin the space (X ,Y , Z ) which is such that

Dt X=ug=−1f

∂Y Φ et D tY≈v g=1f

∂X Φ

In particular, we have

−2Q ' 1=∂X (qf

∂X Ψ)+ f 2 ∂Z 2

2 Ψ

with Q ' 1=−∂X ug∂X b−∂X vg∂Y b

The circulation in the original coordinateis obtained by transformation of the circulationquasi-geostrophic from the coordinates(X ,Y , Z ).

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See problem for the derivation of these properties

3D SEMI-GEOSTROPHIC EQUATIONS

Let Define X= x+1fv g

and Y= y−1fug ,

such that Dt X =u g and D tY=v g ,

and we change the coordinates ( x , y , z , t ) into (X ,Y , Z= z , τ=t )

If Φ=ϕ+12vg

2 +12ug

2 ,

then ∂X Φ=∂x ϕ= f v g ,

∂Y Φ=∂ y ϕ=− f ug and ∂Z Φ=∂zϕ=b

Potential vorticity is q=ω⃗ ∇⃗ b withω⃗=(−∂ z vg ,∂z ug , f +∂ x vg−∂ yug).It is conserved (D tq=0) and we have

1

f 2∂X 2

2 Φ+1

f 2∂Y 2

2 Φ+fq

∂Z 2

2 Φ=1

Let define the geostrophic derivativein the coordinates (X ,Y , Z ):

D̂τ=∂ τ+ug ∂X+v g∂Y

We obtainD̂τq+w ∂Z q=0

D̂τb+w ∂Z b=0

inside the domain,and D̂τb=0on the upper and lowerrigid boundaries.

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X

x

Tilt of the surfaces X :∂z X =1f

∂ z vg

Maximum tilt where the vertical shear ix maximum

∂ x X =1+1f

∂ x vg shows that, in the real space, the isolines of X , and hence of vg ,

get closer inside a cyclone ( ∂x vg>0→∂x X >1 ) and get separated inside an anticyclone

Vg(X)

Vg(x)

20 Quasi-geostrophic Semi-geostrophic

Linear Eady model-> development of a cold front but not a warm front

Toplayer

Bottom layer

Ageostrophic streamfunction

Géopotentiel (trait plein) et température(tireté)

Geopotential (solid) and temperature(dash)

Meridional wind (solid) et temperature (dash)

Géopotentiel (trait plein) et température(tireté)Geopotential (solid) and temperature (dash)

Surface meridional wind

0

Ageostrophic circulation

Vector Q

Geopotentail and meridional qg

Meridional wind (solid) et temperature (dash)

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V =Z−μ

2 (Z +sinh l Zsinh l

cos l X )μ=0: flow without horizontal shear; μ=1: full jet, as shown above

Frontogenesis: role of the horizontal shear(towards more realism in the eady model)

Top row : geopotential (dash) and temperature (solid) at the groundBottom row:geopotential (dash) and vorticity (solid) at the top

m=0 m=0.1 m=1.0m=0.3

Hoskins et West, 1979, JAS

X

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Development of a baroclinic perturbation under the semi-geostrophic assumption.

Isobars: dash (int. 3 hPa); Isentrops : solid (int. 2K)Time interval between two maps: 96 h

Frontogenesis: towards more realism with a better basic flow

Davies, Schar, Wernli, 1991, JAS

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Frontogenesis● Boussinesq equations● Quasi-geostrophic frontogenesis● Semi-geostrophic frontogenesis● Influence of moisture● Large-scale front● Frontogenesis and storms

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Dry air Moist air

Most air with vertical shear decreasing on both sides of the front

Uniform vertical shear

The heating rate J modifiesthe tempertuare equation

D t b+w N 2=D̂ τb+W q=J

The Sawyer-Eliassen equation is modified∂X (q∂X Ψ)+ f 2

∂Z 2

2Ψ=−2Q1−∂ X J

and also the potential vorticity equationD t q=

ζf

∂Z J

For a saturated parcel J=−g LC pT 0

Dt r v

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Frontogenesis● Boussinesq equations● Quasi-geostrophic frontogenesis● Semi-geostrophic frontogenesis● Influence of moisture● Large-scale front● Frontogenesis and storms

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Upper level frontogenesis at tropopause. Generation of a tropopause flod and injection of strtatospheric air in the troposphere. Baray et al., GRL, 2000

Upper level frontogeneis and tropopause foliation

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Tropoapuse fold on a rectilinear jet over 200 degrees of longitude.Baray et al., GRL, 2000

Frontogenèse d'altitude et foliation de tropopause (2)

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Frontogenesis● Boussinesq equations● Quasi-geostrophic frontogenesis● Semi-geostrophic frontogenesis● Influence of moisture● Large-scale front● Frontogenesis and storms

29 Frontogenesis et explosive development of a perturbation

Explosive cyclogenesis

30 Trajectories of parcels in the ageostrophic flow