Atmospheric Dynamics and Meteorology B

Atmospheric dynamics and meteorology B. 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

1 Atmospheric circulation from the viewpoint of a geograph

2 Frontogenesis

● Boussinesq equations

● Quasi-geostrophic frontogenesis

● Semi-geostrophic frontogenesis

● Influence of moisture

● Large-scale front

● Frontogenesis and storms

3 FROM HYDROSTATIC EQUATIONS EQUATIONS IN PRESSURE COORDINATE

Dt u− f v+∂x ϕ' =0 The standard air profile is defined by θ¯ Dt v+ f u+∂ y ϕ '=0 and ϕ¯ (geopotential) which are function of p only. −b ' +∂~z ϕ'=0 ~ 2 ϕ ' is the geopotential deviation: ϕ '=ϕ−ϕ¯ . Dt b=Dt b ' +w N =0 ~ A new form of the pressure coordinate is introduced ∂x u+∂ y v+∂~z w=0 κ p H 0 as a pseudo-height by ~z =[1−( ) ] where H =R θ / g . g p κ 0 with b '= θ' (buoyancy) , 0 θ0 This coordinate is such that ~w=D ~z , κ κ t ~ −dp p g dz p θ0 d z = ( ) H 0= ( ) H 0= dz 2 g p p RT p θ and N = d ~ θ¯ 0 0 θ z 0 (for an adiabatic atmosphere, one would have exactly ~z =z ) ∂ g z The hydrostatic equation is now =g θ ∂~z θ0 p κ p 0 ~ Since ω=Dt p=− ( ) Dt z , we have H 0 p ~ H κ ∂ ω ~ w ~ ~ 0 p =∂ p Dt p=∂~z w− with w=Dt z and H¯ = ( ) ∂ p H¯ 1−κ p0 The Boussinesq approximation consists here in neglecting ~w the term in H 4 Frontogenesis

● Boussinesq equations

● Quasi-geostrophic frontogenesis

● Semi-geostrophic frontogenesis

● Influence of moisture

● Large-scale front

● Frontogenesis and storms

5 Quasi geostrophic frontogenesis

THERMAL WIND Quasi-geostrophic form of the Boussinesq equations f ∂z u g=−∂ y b f ∂ v =∂ b The geostrophic and non geostrophic parts are separated z g x 2 u=u g+ua and v=v g +va D g t ∂x b=Q1−N ∂x w with f u and f v 2 g=−∂ y ϕ g=∂x ϕ Dg t f ∂z v g=−Q1− f ∂z ua Hence D u f v 0 g t g− a= where Q1=−∂x u g ∂x b−∂x v g ∂ y b 2 2 Dg t v g+ f ua=0 We have 2Q1=N ∂x w− f ∂z ua 2 D gt b+w N =0 ∂x ua+∂ y va+∂z w=0 Assuming that the geostrophic wind is with Dg t≡∂t +u g ∂x+v g ∂ y oriented along y with small variations

in this direction ∂ y≪∂x ,∂z . ug=0 ,∂ y va=0

Let define ua=∂z ψ et w=−∂x ψ ,hence Apparent paradox: Q is associated to the geostrophic circulation 2 2 2 2 1 −2 Q1=N ∂x x ψ+ f ∂z z ψ It destroys the thermal wind balance. It6 is restored by the non geostrophic wind Q1 intensifies the temperature gradient by two mechanisms: confluence ( ∂ x ug <0) Confluence Cold advection and gradient shear ( ∂x v g ∂ y b≠0)

x y

Mechanisms of gradient amplification.

2 Dg t ∂x b=Q1−N ∂ x w où Q =−∂ u ∂ b−∂ v ∂ b 1 x g x x g y y x

Q1 frontogenetic Warm advection 7 Q1 frontolytic More generally the temporal evolution of the thermal wind

f ∂z vg =∂x b f ∂ u =−∂ b z g y Let denote Q⃗ =(Q1, Q2) gives 2 Dg t ∂x b=Q1−N ∂x w Generation of the ageostrophic circulation D f v Q f 2 u 2 2 g t ∂z g=− 1− ∂z a 2Q⃗ =N ∇ H w− f ∂z u⃗a in other terms (omega equation) D b Q N 2 w g t ∂y = 2− ∂ y ⃗ 2 2 2 2 2 2 ∇ H .Q=N ∇ H w+ f ∂zz w −Dg t f ∂z ug =−Q2− f ∂z va Convergence of Q⃗ = max of w positive where Divergence of Q⃗ = min of w negative Q1=−∂x ug ∂x b−∂x vg ∂ y b

Q2=−∂y ug ∂x b−∂ y v g ∂ y b

8 Case of frontogenesis by jet vg (entering in the figure) confluence

Q1=−∂x ug ∂x b−∂x v g ∂y b

Q2=−∂y ug ∂x b−∂ y v g ∂ y b Generation of the ageostrophic circulation ⃗ 2 2 Q 2Q=N ∇ H w− f ∂z u⃗a ou in other words (omega equation) 2 2 2 2 2 ∇ H .Q⃗ =N ∇ H w+ f ∂zz w Convergence of Q⃗ = max of w positive Divergence of Q⃗ = min of w negative

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 Limitation of the quasi-geostrophic cannot do it since it is excluded from approximation : the front is only formed near the advecting flow upon the quasi- the ground where w=0 and not inside where geostrophic approximation. the ageostrophic circulation counterbalances Q1. Any horizontal oscillation of size L is 9 damped over a depth f L/N in the vertical. 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).

10 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.

11 Typical development of a baroclinic perturbation

Geopotential (solid) and temperature We see here that the (dash) map at 700 vector Q visualizes the hPa ascending and descending branches of the baroclinic circulation through the temperature lines. The ascending motion tends to occur at smaller scale than the descending Vecteur Q and motion, near the divergence of Q developping frontal zones. (notice the convergence on the north and east sides of the depression and (warm front) and that on the south-east (cold front) 12 Frontogenesis

● Boussinesq equations

● Quasi-geostrophic frontogenesis

● Semi-geostrophic frontogenesis

● Influence of moisture

● Large-scale front

● Frontogenesis and storms

13 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

Ly≫ Lx and V ≫U We assume that the Lagrangian derivative depends mostly of the crosss front variations U Dt∼ Lx

Typically : L y≈1000 km, Lx≈100 km, U ≈1 m/s, V ≈10 m/s and we take f =10−4 s−1 Hence the ratios between inertial terms and Coriolis are 2 Dt u U Dt v V = ≈0.01 et = ≈1 f v f Lx V f u f Lx Basic approximation: The along front component of the wind is well approximated by its quasi-geostrophic component but the transverse component is not and we neglect the ∂u variations of the wind along y , hence =0. ∂ y

Thus we write u=u g+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

Dt =∂t +u ∂x+v g ∂ y+w ∂z The equations are 2 Dt v g + f u a=0 and Dt b+ N w=0

14 ∂ x u a+∂z w=0 FRONTAL DYNAMIC EQUATIONS (geostrophic moment equations)

We take u=u g+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

Dt=∂t +u∂x +v g ∂ y +w∂z The equations are thus 2 Dt v g+ f ua=0 et Dt b+ N w=0

∂x ua +∂ z w=0

The time derivatives of the two terms of the thermal gradient are extracted 2 f Dt ∂z v g =−Q1−∂z w ∂x b− f ∂z ua− f ∂z ua ∂x v g 2 Dt ∂ x b=Q1−(N +∂z b)∂ x w−∂x ua ∂ x b We see new terms (underlines) involving the vertical derivatives of the ageostrophic circulation which reinforce the temperature gradient After combining and rearrangement, we obtain the Sawyer-Eliassen equation:

2 2 2 2 2 2 N s ∂x x ψ−2 S ∂ x z ψ+F ∂ z z ψ=−2Q1 ,

2 2 2 2 with N s=N +∂z b , F = f ( f +∂ x v g ) and S =∂ x b. 2 2 4 This equation is elliptic and can be solved if F N s−S >0, that is if15 the potential vorticity is positive (see problem). The effects of advection by the ageostrophic equation are ● the formation of a tilted front towards the cold region ● the reinforcement of the wind in the cyclone to the expense of the anticyclone (see below) ● the shrinking of the area of the warm surface anomaly and the expansion of the area of the cold anomaly ● the inverse effect at the top In gray, location of the front 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 front It is easier to get a front at the ground because w cannot counterbalance the intensification of temperature gradient.

16 SEMI-GEOSTROPHIC EQUATIONS or approximately (abridged form) 2 q 2 2 1 2 2 2 N 2 ≈(N +∂Z 2 Φ)(1+ 2 ∂X 2 Φ)=N +∂Z 2 Φ+ 2 ∂X 2 Φ 1 f f f We define X =x+ v , such that D X =u , f g t g Under this approximation, 2 and we change the coordinates ( x , y , z ,t) into 2 N 2 the quantity ∂ 2 Φ+ ∂ 2 Φ ( X ,Y =y , Z =z , τ=t) Z f 2 X ∂x v g is conserved by the geostrophic flow This leads to ∂x=(1+ )∂X f in the space ( X ,Y , Z ) which is such that ∂x b ∂ y v g 1 1 ∂z=∂Z + 2 ∂X and ∂y=∂Y + ∂X D X =u =− ∂ Φ et D Y ≈v = ∂ Φ f f t g f Y t g f X ∂x v g ∂ X v g In particular, we have thus (1+ )(1− )=1 f f q 2 2 −2Q ' 1=∂X ( ∂X Ψ)+ f ∂ 2 Ψ 1 2 f Z If Φ=ϕ+ v , hence ∂ Φ=∂ ϕ= f v 2 g X x g

∂Y Φ=∂ y ϕ=− f ug and ∂Z Φ=∂z ϕ=b with Q '1=−∂ X ug ∂ X b−∂X v g ∂Y b ⃗ The potential vorticity is q=ω⃗ ∇ (b+b¯) with and U a=∂Z Ψ , W =∂ X Ψ

ω⃗=(−∂z v g ,0, f +∂x v g ). the ageostrophic circulation in the transformed

It is conserved (Dt q=0) and we have frame. 2 ∂Z 2 (Φ¯ +Φ) The circulation in the original coordinate q=∂Z (b+b¯)( f +∂x v g )= f 1 2 is obtained by transformation of the circulation 1− ∂ 2 Φ f 2 X quasi-geostrophic from the coordinates ( X ,Y , Z ). 17 1 Tilt of the surfaces X :∂ X = ∂ v z f z g Maximum tilt where the vertical shear ix maximum

Vg(X) X

Vg(x) x 1 ∂ X =1+ ∂ v shows that, in the real space, the isolines of X , and hence of v , x f x g g

19 get closer inside a cyclone ( ∂x v g >0→∂x X >1) and get separated inside an anticyclone Top layer Geopotential and meridional qg Linear Eady GéopotentielGeopotential (solid)(trait plein)and temperature et température(tireté) (dash) model Ageostrophic circulation -> development of a cold front but not a

warm front Vector Q

Ageostrophic streamfunction

0 0

Meridional wind (solid) et temperature (dash) Meridional wind (solid) et temperature (dash)

0 Bottom layer

Geopotential (solid) and temperature(dash) Surface meridional wind 20 Quasi-geostrophic Semi-geostrophic Frontogenesis: role of the horizontal shear (towards more realism in the eady model) μ sinh l Z X V =Z− Z + cosl X 2 ( sinh l ) Hoskins et West, 1979, JAS μ=0: flow without horizontal shear; μ=1: full jet, as shown above

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

Top row : geopotential (dash) and temperature (solid) at the ground Bottom row: geopotential (dash) and vorticity (solid) at the top 21 Frontogenesis: towards more realism with a better basic flow

Davies, Schar, Wernli, 1991, JAS

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

Cold front Warm front

Development of a baroclinic perturbation under the semi-geostrophic assumption. 22 Frontogenesis

● Boussinesq equations

● Quasi-geostrophic frontogenesis

● Semi-geostrophic frontogenesis

● Influence of moisture

● Large-scale front

● Frontogenesis and storms

25 Upper level frontogeneis and tropopause foliation

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

26 Frontogenèse d'altitude et foliation de tropopause (2)

Tropopause fold on a rectilinear jet over 200 degrees of longitude. Baray et al., GRL, 2000 27 Frontogenesis

● Boussinesq equations

● Quasi-geostrophic frontogenesis

● Semi-geostrophic frontogenesis

● Influence of moisture

● Large-scale front

● Frontogenesis and storms

28 Explosive cyclogenesis

29 Frontogenesis et explosive development of a perturbation Trajectories of parcels in the ageostrophic flow 30