Atmospheric dynamics and meteorology

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

EQUATIONS D t u f v + x ϕ' =0 D t v+ f u+ y ϕ ' =0 b ' + ~z ϕ' =0 Dt b= D t b ' + ~ w N =0 x u + y v+ ~z ~ w=0 g with b ' = θ ' (buoyancy), θ ~ w= D ~ z, FROM HYDROSTATIC EQUATIONS IN PRESSURE COORDINATE The standard air profile is defined by θ and ϕ (geopotential) which are function of p only. ϕ ' is the geopotential deviation: ϕ ' =ϕ ϕ. A new form of the pressure coordinate is introduced as a pseudo-height by ~ z= H log p/ p 0 where H =R T 0 / g. (for an isothermal atmosphere, one would have exactly ~ z= z ) The hydrostatic equation is now transformed t by combining it with the perfect gas law g 1 RT and N = d ~z θ ϕ= = p ρ θ p κ ϕ p +Rθ =0 log p p p 0 Since ω=d t p= Dt z, we have κ H ϕ' p + Rθ ' =0 ω = D p= w w log p p 0 p t z p H κ p g θ' ~z ϕ ' =R θ' =g Boussinesq approximation p0 R T 0 θ consists here in neglecting Notice : θ is an isothermal profile, do not confuse with θ w the term in d b H b= b (~ z )+b' ( x, y, ~ z, t) and N = ~ dz ( ) ( ) ( ) 3

Frontogenesis Boussinesq equations Quasi-geostrophic frontogenesis Semi-geostrophic frontogenesis Influence of moisture Large-scale front Frontogenesis and storms 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 The geostrophic and non geostrophic parts are separated u=u g +u a and v=v g +v a with f u g = y ϕ and f v g = x ϕ Hence D g t u g f v a =0 D g t v g + f u a =0 D g t b+w N =0 x u a + y v a + z w=0 with D g t t +u g x +v g y Apparent paradox: Q1 is associated to the geostrophic circulation It destroys the thermal wind balance. It6 is restored by the non geostrophic wind f z u g = y b f z v g = x b D g t x b=q 1 N x w D g t f z v g = Q 1 f z u a where Q 1= x u g x b x v g y b We have Q 1 = N x w f z u a Assuming that the geostrophic wind is oriented along y with small variations in this direction y x, z. u g =0, y v a =0 Let define u a = z ψ et w= x ψ, hence Q 1= N x x ψ+ f z z ψ

Q1 intensifies the temperature gradient by two mechanisms: confluence ( x u g <0 ) Confluence and gradient shear ( x v g y b 0 ) x Cold advection y x y Mechanisms of gradient amplification. D g t x b=q 1 N x w où Q 1 = x u g x b x v g y b y x 7 Q1 frontogenetic Q1 frontolytic Warm advection

More generally the temporal evolution of the thermal wind f z v g = x b f z u g = y b gives D g t x b=q1 N x w D g t f z v g = Q 1 f z u a D g t y b=q N y w D g t f z v g = Q f z v a where Q 1= x u g x b x v g y b Q = y u g x b y v g y b 8 Let denote Q=(Q 1, Q ) Generation of the ageostrophic circulation N H w f z u a Q= in other terms (omega equation) =N H w + f zz w H.Q = max of w positive Convergence of Q = min of w negative Divergence of Q

Case of frontogenesis by confluence jet (entering in the figure) Q1 = x u g x b x v g y b Q = y u g x b y v g y b 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 quasigeostrophic approximation. 9 Generation of the ageostrophic circulation =N H w f z u a Q ou in other words (omega equation) H. Q=N H w+ f zz w = max of w positive Convergence of Q = min of w negative Divergence of Q 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 Q1. Any horizontal oscillation of size L is 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 (): - 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 (dash) map at 700 hpa Vecteur Q 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) 1 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.

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 (D version transverse to the jet) We assume that the length scale of the front is much larger than its transverse scale L y L x and V U We assume that the Lagrangian derivative depends mostly of the crosss front variations U Dt Lx Typically : L y 1000 km, L x 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 Dt u Dt v U 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 variations of the wind along y. Thus we write u=u g +u a where the two components are of the same order, and v=v g. and we take into account the ageostriphic terms in the advection D t = t +u x +v g y +w z The equations are 14 D t v g + f u a =0 and D t b+ N w=0 x u a + z w=0

FRONTAL DYNAMIC EQUATIONS (geostrophic moment equations) We take u=u g +u a 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 thus Dt v g + f u a =0 et Dt b+ N w=0 x u a + z w=0 The time derivatives of the two terms of the thermal gradient are extracted f Dt z v g = Q 1 z w x b f z u a f z u a x v g D t x b=q1 ( N + z b) x w x u a 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: N s x x ψ S x z ψ+ F z z ψ= Q 1, with N s =N + z b, F = f ( f + x v g ) and S = x b. This equation is elliptic and can be solved if F N s S >0, that is if15the potential vorticity is positive (see problem). 4

In gray, location of the front 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 front It is easier to get a front at the ground because w cannot counterbalance the intensification of temperature gradient. 16

SEMI-GEOSTROPHIC EQUATIONS (abridged form) 1 v, such that Dt X =u g, f g and we change the coordinates ( x, y, z, t ) into ( X, Y = y, Z = z, τ=t ) x v g This leads to x =(1+ ) X f y vg z = Z + x b X and y = Y + X f x vg X v g thus (1+ )(1 )=1 f f 1 If Φ=ϕ + v g, hence X Φ= x ϕ= f v g Y Φ= y ϕ= f u g and Z Φ= z ϕ=b (b+ The potential vorticity is q= ω b ) with We define X =x+ ω =( z v g, 0, f + x v g ). It is conserved ( D t q=0) and we have Z ( Φ+Φ) q= Z (b+ b)( f + x v g )= f 1 1 X Φ f 17 or approximately q 1 N ( N + Z Φ)(1+ X Φ)=N + Z Φ+ X Φ f f f Under this approximation, N the quantity Z Φ+ X Φ f is conserved by the geostrophic flow in the space ( X, Y, Z ) which is such that 1 1 Dt X =u g = Y Φ et D t Y v g = X Φ f f In particular, we have q Q ' 1= X ( X Ψ)+ f Z Ψ f with Q ' 1= X u g X b X v g Y b The circulation in the original coordinate is obtained by transformation of the circulation quasi-geostrophic from the coordinates ( X, Y, Z ).

3D SEMI-GEOSTROPHIC EQUATIONS Let Define X = x+ 1 v f g 1 and Y = y u g, f such that Dt X =u g and D t Y =v g, and we change the coordinates ( x, y, z, t ) into ( X, Y, Z = z, τ=t ) 1 1 If Φ=ϕ + v g + ug, then X Φ= x ϕ= f v g, Y Φ= y ϕ= f u g and Z Φ= z ϕ=b b with Potential vorticity is q= ω ω =( z v g, z u g, f + x v g y u g ). It is conserved ( D t q=0) and we have 1 1 f Φ+ Φ+ Φ=1 q Z f X f Y 18 Let define the geostrophic derivative in the coordinates ( X, Y, Z ): D τ = τ +u g X +v g Y We obtain D τ q+w Z q=0 D τ b+w Z b=0 inside the domain, and D τ b=0 on the upper and lower rigid boundaries. See problem for the derivation of these properties

1 v f z g Maximum tilt where the vertical shear ix maximum Tilt of the surfaces X : z X = Vg(X) X Vg(x) x 1 v shows that, in the real space, the isolines of X, and hence of v g, f x g get closer inside a cyclone ( x v g >0 x X >1 ) and get separated inside an anticyclone x X =1+ 19

Top layer Géopotentiel (solid) Geopotential (trait plein) and temperature et température(tireté) (dash) Linear Eady model Geopotentail and meridional qg Ageostrophic circulation -> development of a cold front but not a warm front Vector Q Ageostrophic streamfunction Meridional wind (solid) et temperature (dash) Meridional wind (solid) et temperature (dash) 0 Bottom layer 0 Geopotential (solid) and temperature(dash) Quasi-geostrophic Surface meridional wind Semi-geostrophic

Frontogenesis: role of the horizontal shear (towards more realism in the eady model) X μ sinh l Z V =Z Z + cosl X sinh l μ=0: flow without horizontal shear; μ=1: full jet, as shown above ( ) m=0 Top row : geopotential (dash) and temperature (solid) at the ground Bottom row: geopotential (dash) and vorticity (solid) at the top 1 m=0.1 m=0.3 Hoskins et West, 1979, JAS m=1.0

Frontogenesis: towards more realism with a better basic flow Davies, Schar, Wernli, 1991, JAS Isobars: dash (int. 3 hpa); Isentrops : solid (int. K) Time interval between two maps: 96 h Development of a baroclinic perturbation under the semi-geostrophic assumption.

Frontogenesis Boussinesq equations Quasi-geostrophic frontogenesis Semi-geostrophic frontogenesis Influence of moisture Large-scale front Frontogenesis and storms 3

Uniform vertical shear Dry air Moist air The heating rate J modifies the tempertuare equation D t b+w N = D τ b+w q= J The Sawyer-Eliassen equation is modified X (q X Ψ )+ f Z Ψ= Q 1 X J and also the potential vorticity equation ζ D t q= Z J f g L For a saturated parcel J = Dr C pt 0 t v 4 Most air with vertical shear decreasing on both sides of the front

Frontogenesis Boussinesq equations Quasi-geostrophic frontogenesis Semi-geostrophic frontogenesis Influence of moisture Large-scale front Frontogenesis and storms 5

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, 000 6

Frontogenèse d'altitude et foliation de tropopause () 7 Tropoapuse fold on a rectilinear jet over 00 degrees of longitude. Baray et al., GRL, 000

Frontogenesis Boussinesq equations Quasi-geostrophic frontogenesis Semi-geostrophic frontogenesis Influence of moisture Large-scale front Frontogenesis and storms 8

Explosive cyclogenesis 9 Frontogenesis et explosive development of a perturbation

30 Trajectories of parcels in the ageostrophic flow