Understanding the regeneration stage undergone by surface. cyclones crossing a mid-latitude jet in a two-layer model.

Transcription

1 Generated using version 3 of the official AMS L A TEX template 1 Understanding the regeneration stage undergone by surface cyclones crossing a mid-latitude jet in a two-layer model 3 Gwendal Rivière, CNRM/GAME, Météo-France/CNRS, Toulouse, France 4 Jean-Baptiste Gilet Woods Hole Oceanographic Institution, Woods Hole, Massachusetts, USA 5 and Ludivine Oruba Laboratoire de Météorologie Dynamique/IPSL, ENS/CNRS/UPMC, Paris, France Corresponding author address: Gwendal Rivière, Météo-France, CNRM/GMAP/RECYF, 4 av G Coriolis, 3157 Toulouse Cedex 1, France gwendalriviere@meteofr 1

2 ABSTRACT The present paper provides a rationale for the regeneration stage undergone by surface cyclones when they cross a baroclinic jet from its anticyclonic-shear (warm) side to its cyclonic-shear (cold) side in a two-layer quasi-geostrophic model To do so, the evolution of finite-amplitude synoptic cyclones in various baroclinic zonal flows is analyzed 11 Baroclinic zonal flows with uniform horizontal shears are first considered While the anticyclonic shear allows a much more efficient and sustainable extraction of potential energy than the cyclonic shear, the growth of the lower-layer eddy kinetic energy (EKE) is shown to be highly dependent on the choice of the parameter values An increased vertical shear leads to a more rapid EKE increase in the anticyclonic shear than in the cyclonic shear whereas increasing the vertically-averaged potential-vorticity gradient or the barotropic shear stabilizes more the EKE in the former shear than in the latter Finally, vertical velocities arising from the nonlinear interaction between synoptic cyclones are shown to favor more the EKE growth in the cyclonic shear than in the anticyclonic one The evolution of cyclones initialized on the warm side of a meridionally confined baroclinic 1 jet is then investigated The lower-layer cyclone crosses the jet axis and undergoes two distinct growth stages The first growth stage results from the classical baroclinic interaction and is mainly driven by linear interaction between the cyclones and the jet The second growth stage is mainly a nonlinear process It is triggered by the vertical velocities created by the three-dimensional structure of the cyclonic disturbances when they reach the cyclonic side of the jet 1

3 7 1 Introduction It is well-known that extratropical cyclones develop by extracting energy from the available background potential energy through baroclinic interaction The baroclinicity is the key parameter governing this interaction and is usually quantified by the Eady growth rate which depends on the vertical shear of the background flow (Hoskins and Valdes 199) However, numerous numerical studies have shown that the behavior of extratropical cyclones is modulated by the horizontal shears of the large-scale background flow (James 1987; Davies et al 1991; Thorncroft et al 1993; Wernli et al 1998) as well as by its confluent and diffluent components (Cai and Mak 199; Whitaker and Barcilon 199; Schultz et al 1998) Davies et al (1991) and Thorncroft et al (1993) investigated the nonlinear evolution of unstable normal modes in unsheared, cyclonically-sheared and anticyclonically-sheared baroclinic zonal flows Both the synoptic-scale structure of the disturbances and the fronts were shown to largely differ among the three cases For instance, the cyclonic (anticyclonic) shear favors the formation of the warm (cold) front and the intensification of low (high) pressure systems Wernli et al (1998) found similar results by initializing the models with localized finite-amplitude anomalies Furthermore, the Norwegian cyclone model (Bjerknes and Solberg 19) is found to appear in the cyclonic-shear case while the model of Shapiro and Keyser (199) with a T-bone frontal structure dominates in unsheared cases Despite all this information, it is still unclear what is the more cyclogenetic side of a baroclinic zonal jet Indeed, even though the cyclonic side favors low pressure systems as previously recalled, this side is biased by the low pressure of the background environment and it is not clear if the perturbation low is deeper on the cyclonic side than on the anticyclonic side The purpose of the present study is to numerically investigate this aspect within a quasi-geostrophic framework by modifying the vertical and horizontal shears of the background flow as well as the scale and intensity of the synoptic disturbances in the initial conditions of the model Another motivation of the present study relies on the observations that most Atlantic storms that hit Europe undergo a rapid intensification stage during the crossing of the large-

4 scale jet stream from its warm-air to its cold-air side, that is from its anticyclonic to its cyclonic side (Baehr et al 1999; Wernli et al ; Rivière and Joly 6; Pinto et al 9) Gilet et al (9) have recently shown that such a regeneration of a surface cyclone during the jet-crossing phase can be easily reproduced by performing idealized numerical experiments of a two-layer quasi-geostrophic model on an f-plane or a β-plane The simplest numerical configuration consists in initializing the model with localized finite-amplitude synoptic-scale cyclones on the anticyclonic side of a baroclinic zonal westerly jet The key ingredient that allows a surface cyclone to move across the upper-level jet axis is the vertically-averaged horizontal potential vorticity gradient (hereafter called barotropic PV gradient) If the simulation is performed on an f-plane with a pure baroclinic westerly jet without any barotropic component, the barotropic PV gradient is zero and the surface cyclone is not able to cross 65 the upper-level jet axis On the contrary, if the simulation is done on a β-plane and/or with a westerly jet having a nonzero barotropic component, the barotropic PV gradient is poleward oriented and the displacement across the jet axis may occur The cross-jet speed increases with an intensified poleward-oriented barotropic PV gradient These results were interpreted as a generalization of the so-called β-drift mechanism in a mid-latitude atmospheric context Initially developed from a theoretical perspective by Rossby (1948) and Adem (1956), the β-drift has been extensively studied to explain the behavior of tropical cyclones (eg, Holland 1984; Moustaoui et al ) and oceanic vortices (eg, Morel and McWilliams 1997) and appears now as a key factor to explain the motion of mid-latitude synoptic eddies such as surface cyclones (Gilet et al 9; Oruba et al 1, 13) and upper-tropospheric anomalies (Rivière 8) Note finally that this mechanism was recently confirmed in a much more realistic context by performing numerical sensitivity experiments of the Météo-France global operational forecast model for the particular case of the winter storm Xynthia that occurred at the end of February 1 (Rivière et al 1) Although the mechanism responsible for the motion of a surface cyclone across the upperlevel jet axis was identified in the idealized simulations of Gilet et al (9) and Oruba et al 3

5 (13) as explained above, the reason for the regeneration stage during the jet-crossing phase was not deeply investigated The purpose of the present study is to provide a rationale for this growth stage using the same two-layer quasi-geostrophic model as in the latter studies and by making kinetic-energy budgets In section, the model and energy budgets are presented as well as a detailed decomposition of the omega equation into linear and nonlinear terms Section 3 is dedicated to simulations where the background flow is composed of uniform horizontal and vertical shears Sensitivity experiments over a wide range of parameters are made to precisely diagnose the factors favoring the growth rates in anticyclonically-sheared and cyclonically-sheared baroclinic zonal flows In section 4, the background flow is composed of a meridionally confined zonal jet and a detailed energy budget is made to identify the mechanism responsible for the regeneration stage of the lower-layer cyclones when they cross the jet axis and reach its cyclonic side Finally section 5 summarizes the results and provides a discussion 94 Methodology 95 a Model The model is the two-layer quasi-geostrophic model initially developed by Phillips (1951) The two discrete layers are contained between two rigid horizontal plates The model conserves the PV in the lower layer (denoted with subscript l) and upper layer (denoted with subscript u) The prognostic equations can be expressed as: q k t + u k q k =, k [u, l], (1) 1 where q k is the PV in the layer k The PV has the following expression: q u = f + ψ u λ (ψ u ψ l ), () q l = f + ψ l + λ (ψ u ψ l ), (3) 4

6 where denotes the Laplacian operator The streamfunction and geostrophic wind in the layer k are respectively denoted as ψ k and u k = (u k, v k ) λ denotes the Rossby radius of deformation and f = f + βy is the Coriolis parameter As in Gilet et al (9) and Oruba et al (13), the model is spectral, the nonlinear terms are computed on a regular grid A leapfrog temporal scheme is used and the domain is biperiodic The spatial and temporal resolutions are equal to δx = δy = 15 km and δt = 5 s and no diffusion is included in the following numerical experiments Values of f and λ are 1 4 s 1 and 45 km respectively (see Table 1) Depending on the simulations, β can be equal to β a typical mid-latitude value of the y-derivative of f (see Table 1) or to 3β to represent the increased PV gradient in presence of meridionally confined westerly jets 111 b Omega and relative-vorticity equations The omega equation in the two-layer model on a β-plane which determines the vertical velocity as function of the geostrophic variables can be expressed as: ( λ )(λ 1 sω u l ) = λ 1 s Q λ β(v u v l ) = F, (4) 114 where Q = (Q x, Q y ) = λ 1 s ( 1 (u u + u l ) (ψ u ψ l ), 1 ) (u u + u l ) (ψ u ψ l ) x y (5) 115 is the so-called Q-vector (Hoskins et al 1978) and ω u l denotes the vertical velocity in pressure coordinates at the interface between the two layers s h dθ R dp is a stratification ( ) parameter with θ R being the reference potential temperature and h R p R/Cp p p s λ and s are linked by the following relation λ 1 s = f /δp where f is the Coriolis parameter and δp the difference of pressure between the middle of the two layers (Holton 199) 1 Note finally that the sign of F, that is the right-hand side term of Eq(4), determines that of the vertical velocity because F > (F < ) leads to updraft (downdraft) 1 Note that λ has here the dimension of a distance and is the multiplicative inverse of Holton (199) s λ 5

7 In what follows, the total flow is decomposed into a zonal basic state (denoted with an overbar) and a perturbation (denoted with primes) The zonal basic flow depends only on latitude (u u = u u (y)i and u l = u l (y)i) and is solution of Eq(1) Hereafter Q lin and Q nl denote the linear and nonlinear parts of the Q-vector respectively, which can be written as ( 1 Q lin λ 1 s ( u u + u l ) x (v u + v l), 1 ( u u + u l ) y (v u + v l) + 1 (v u v l) ) y (u u + u l ), (6) ( 1 Q nl λ 1 (u u + u l s ) (ψ u ψ x l), 1 (u u + u l ) ) (ψ u ψ y l) (7) F is also decomposed into a linear and a nonlinear part such as F = F lin + F nl According to Eq(4), the linear part of F contains a part without β (F lin λ 1 Qlin ) and another with β (F lin β λ β(v u v l )) and can be expressed as: s F lin F lin + F lin β, = λ [ (u u u l ) (v u + v l) + (β y(u u + u l ))(v u v l) y u l y v u + y u u y v l] (8) The first term of the right-hand side (rhs) of Eq(8) corresponds to the classical baroclinic term that creates updraft (downdraft) to the east (west) of a cyclone embedded in a positive vertical shear (u u u l > ) The second term corresponds to the impact of an effective β on the vertical velocity because it involves the barotropic PV gradient (β y(u u + u l )) The last two terms of the rhs of Eq(8) involving the horizontal shears and their impact on the vertical velocity will be discussed later Finally, the nonlinear part of F can be simply expressed as: F nl = λ 1 s Qnl (9) The linearity of the relation between F and ω u l (see Eq(4)) allows to consider separately the vertical velocity fields induced by the forcings F lin and F nl denoted as ωu l lin and ωnl u l respectively They will be hereafter 6

8 139 The relative-vorticity evolution in each layer (ζ k ) can be written as (Holton 199): ζ u t + u u ζ u + u u ζ u + u u ζ u + βv u = λ 1 sω u l, (1) ζ l t + u l ζ l + u l ζ l + u l ζ l + βv l = λ 1 sω u l, (11) where λ 1 sω u l and λ 1 sω u l are the two stretching terms f ω p appearing in the upper- layer and lower-layer equations of the two-layer quasi-geostrophic model They are propor- tional to the vertical velocity at the interface between the two layers because the lowermost and uppermost frontiers of the model are rigid (Holton 199) Suppose that an upper and a lower disturbance are in quadrature phase with each other and that the upper disturbance is to the west of the lower one Where the lower disturbance reaches its minimum pres- sure (or maximum vorticity), v l = and v u reaches a positive maximum In midlatitudes, since the basic-state vertical shear u u u l is positive as well as the barotropic PV gradient β y(u u +u l ), the first two terms on the rhs of Eq(8) have opposite signs in that region with the first one ( λ (u u u l ) v u) being positive and the second one ( λ (β y(u u +u l ))v u) being negative The effective beta term tends to reduce the ascending motion above the max- imum vorticity of the lower-layer disturbance This reduction in the vertical velocity reduces the positive stretching term in the lower-layer relative-vorticity equation (see Eq(11)) and reduces the relative-vorticity growth of the lower-layer disturbance On the contrary, where the upper disturbance reaches its minimum pressure (or maximum vorticity), v u =, the first two terms on the rhs of Eq(8) have the same sign and are both negative These two terms tend to intensify the relative-vorticity growth of the upper disturbance through the 157 stretching term (see rhs of Eq(1)) Therefore, the effective beta term tends to create an asymmetry between the two layers which is described in the next subsection from an energetic point of view 7

9 16 c Energy budget Following Cai and Mak (199), the time evolution of the perturbation kinetic-energy ( ) (K 1 u g ) and potential energy (P f ψ ) s p in quasi-geostrophic equations can be expressed as D g Dt K = E D (u gp a + f ψ χ ) f p (ω ψ ) + f ω ψ p, (1) D g Dt P = f ) (v ψ ψ ψ ψ + s u p y p p x p ) f ω ψ p, (13) where D g /Dt = / t + (u g + u g) denotes the Lagrangian geostrophic derivative p a and χ represent respectively the perturbation ageostrophic pressure and velocity potential They are both involved in the expression of the perturbation ageostrophic horizontal velocity u a = 1 f k p a + χ where k is the vertical unit vector The scalar product E D is the barotropic energy conversion rate from the basic flow to the perturbation with E and D being defined as follows: E ( 1 (v g u g ), u gv g), (14) D ( u g x v g y, v g x + u g ) (15) y In the two-layer model, the kinetic-energy equation at the two layers can be written as D g Dt K l = E l D l (u lp al + f ψ l χ l) + λ 1 sω u l ψ u + ψ l λ 1 sω ψ u ψ l u l, (16) D g Dt K u = E u D u (u up au + f ψ u χ u) λ 1 sω u l ψ u + ψ l λ 1 sω ψ u ψ l u l, (17) 17 and the potential-energy equation at the interface between the two layers: D g Dt P = λ [(v u + v l)(u u u l ) (u u + u l)(v u v l )] (ψ u ψ l) + λ 1 sω u l(ψ u ψ l) (18) By integrating each equation over the whole domain (operation denoted as < >), advection terms which are included in the Lagrangian geostrophic time derivatives of Eqs (16), (17) and (18) and the horizontal redistribution terms which are the second terms of the rhs of 8

10 176 the same equations disappear Therefore, we obtain and where d dt d dt < K l > =< E l D l > + < CV > + < >, (19) d dt < K u > =< E u D u > < CV > + < >, () d dt < P >=< CB > < >, (1) denotes the time derivative In addition, = λ 1 sω u l(ψ u ψ l), () CV = λ 1 sω u l (ψ u + ψ l ), (3) CB = λ [(v u + v l)(u u u l ) (u u + u l)(v u v l )] (ψ u ψ l) (4) represents the internal conversion term (ie, the energy transfer from eddy potential energy to eddy kinetic energy) Note that appears on the rhs of the kinetic-energy equation of each layer (Eqs(19) and ()) and is equivalent to the fourth term of the rhs of Eq(1) within the two-layer framework logically appears on the rhs of the potential- energy equation at the interface (Eq(1)) CB is the baroclinic conversion term (ie, the baroclinic energy transfer from the basic-state potential energy to eddy potential energy) and is the two-layer formulation of the first term of the rhs of Eq(13) It is only involved in the potential-energy equation at the interface (Eq(1)) CV is the two-layer formulation of the convergence of the vertical ageostrophic fluxes (third term on the rhs of Eq(1)) It corresponds to the exchanges of kinetic energy between the two layers since CV and CV appear respectively in equations (19) and () Note finally that the sum of CV and is simply the product of the stretching term by the perturbation streamfunction f ω p ψ and has the following expression at the lower layer: CV + = λ 1 sω u lψ l (5) 9

11 When the two fields are anticorrelated (strong ascending motions above the lower-layer cyclone), CV + reaches large positive values Our study is hereafter focused on the lower-layer kinetic-energy (EKE) evolution as diagnosed from Eq(19) 195 d Disturbances 196 The perturbation streamfunction ψ k in each layer k has a Gaussian elliptic shape and is defined as ψ k = A exp [ ((x x k) cos φ + (y y k ) sin φ ) ( (x x ] k) sin φ + (y y k ) cos φ ), r (1 e ) r (6) where A is the amplitude, r the characteristic spatial scale, e the excentricity and φ the angle of the major axis with respect to the x-axis The upper and lower-layer disturbances have initially the same amplitude but the lower disturbance is located downstream of the upper one to allow a substantial extraction of potential energy from the basic-state flow 3 Baroclinic zonal flows with uniform horizontal shears To simplify the problem, this section investigates the impact of uniform cyclonic and anticyclonic horizontal background shears onto the growth of lower-layer disturbances in presence of constant vertical background shears To do so, the basic-state vorticity is defined as a constant in each layer over a large area centered in the middle of the domain (two-thirds of the domain) Because the model is biperiodic, the basic-state zonal wind must be periodic and the averaged basic-state vorticity must be zero Therefore, outside the region of constant basic-state vorticity, the basic-state vorticity is defined using cosine profiles in such a way that its spatial average is zero over the whole domain In the present section, all the disturbances are initialized in the center of the domain in the region of constant basic-state vorticity and we have checked that disturbances stay in that region during the whole simulations 1

12 13 14 Within the region of constant basic-state vorticity, the basic-state zonal wind profiles have the following expression: u u = αc x uy + αλc z, (7) u l = αc x l y, (8) 15 where α corresponds to a standard shear value (see Table 1) c x k for k [u, l] and cz correspond to nondimensional parameters determining respectively the horizontal shear in each layer k and the vertical shear Note that αc z = f N u z is an estimate of the baroclinicity (ie the Eady parameter divided by 31) and more precisely the maximum of the baroclinic 19 generation rate CB/(K + P ) (Rivière et al 4; Gilet et al 9) αc x k for k [u, l] is an estimate of the horizontal deformation in each layer k and more precisely the maximum of 1 the barotropic generation rate E k D k /(K + P ) If c z = c x l of the barotropic and baroclinic generation rates are equal = c x u, it means that the maxima 3 In presence of uniform shears, F lin = F lin + Fβ lin depends on the following expressions: F lin = λ [λαc z (v u + v l) α(c x u c x l )y (v u + v l) + αc x l y v u αc x u y v l], (9) F lin β = λ β(v u v l) (3) All the terms in the previous two equations are controlled by the basic-state parameters c z, c x l, cx u and β Note finally that because uniform shears are considered in this section, basic-state relative-vorticity gradients are zero and the effective beta is reduced to β (see second term on the rhs of Eq(8)) β is therefore artificially increased beyond its usual value in midlatitudes to reproduce the stronger effective beta in presence of localized jets 9 a Vertical velocity induced by the various basic-state parameters 3 1) The effect of the basic-state vertical shear and PV gradient 31 Let us first analyze the case without horizontal shears (c x u = c x l = ) with a positive 3 vertical shear c z > Fig1a shows the vertical velocity due to the part of the linear 11

13 term depending on the upper disturbance, that is F lin when v l equals, which can be reduced to F lin (v l = ) = λ 1 αc z v u It shows that there is ascending and descending motion respectively downstream and upstream of a cyclone when the vertical shear c z is positive Because the upper-disturbance is located upstream of the lower one, its velocity induces ascending motions above the lower disturbance that will tend to intensify it This intensification is commonly explained by noting that the perturbation relative vorticity will increase via the stretching term f ω/ p but this can be viewed from a kinetic-energy perspective The lower-layer kinetic-energy evolution (Eq(19)) depends on the sum CV + = λ 1 sω u l ψ l which is mainly positive because the part of λ 1 sω u l related to the upper disturbance is mainly negative above the minimum streamfunction at the lower layer In other words, when there is a dominant westward tilt with height of the perturbation streamfunction isolines with a phase tilt of 9 as in the present case, v u is well anticorrelated with ψ l leading to a strong correlation between λ 1 sωu l lin (v l = ) and ψ l and to large positive values of CV + On the contrary, the vertical velocity due to the part of the linear term depending on the lower disturbance F lin (v u = ) = λ 1 α z v l (Fig1b) is in quadrature phase with ψ l and thus does not modify the lower-layer kinetic energy As already discussed in section b, we expect that the β-term tends to diminish the relative-vorticity growth of the lower-layer perturbation The stabilizing effect of β in the lower layer can be also viewed in terms of kinetic energy because F lin β (Fig1d) is mainly negative in the region where the lower-layer streamfunction is negative The β-term creates descending motions above the lower-layer perturbation that tends to diminish the kinetic- energy growth due to F lin (Fig1c) The Q-vector due to the nonlinear term (arrows in Fig1e) is strong and poleward oriented in the region located between the two cyclonic disturbances because Q nl y 5λ 1 s y (u u + u l )(v u v l ) is positive in that region since (v u v l ) > and y(u u + u l ) < This creates ascending motions to the north and descending motions to the south of the disturbances but λ 1 sω u l due to the nonlinear term and ψ l are in quadrature phase and there is no global 1

14 kinetic-energy change coming from this term in that particular case In the end, the forcing induced by β can almost nullify the projection of the stretching term λ 1 sω u l onto ψ l created by the linear forcing F lin, that would otherwise lead to kinetic-energy growth in the lower layer 64 ) The effect of the basic-state horizontal shears Without horizontal shears, the Q-vector due to the linear part (Eq(6)), that is Q lin, is eastward oriented along the segment line joining the two centers of the cyclones (see 67 Fig1a) In presence of a cyclonic barotropic shear (c x u = 1, c x l = 1), Q lin will have a 68 strong poleward component in that region between the two disturbances Indeed, Q lin y λ 1 s 1 (v u v l ) (u y u + u l ) = λ 1 s 1 (v u v l )α(cx u + c x l ) is positive and maximum in the region where the dashed and solid black contours are superimposed upon each other (see Figsa-b) Therefore, the convergence of the Q-vector due to the presence of the upper disturbance F lin (v l = ) will not reach its maximum in the region where the meridional velocity v u is maximum, that is on the eastern side of the upper disturbance as in the case without shear (Fig1a), but rather on its northeastern side (Figa) This can be also seen by considering Eq(9) for v l =, that is, F lin (v l = ) = λ 1 αc z v u λ αc x l yv u The second term on the rhs of the previous expression is positive on the northeastern side 77 of the upper disturbance It means that the optimal growth for the lower-layer kinetic energy is not anymore a pure westward tilt with height but rather a southwestward tilt with height Figb shows the vertical velocity forcing due to the lower disturbance Since F lin (v u = ) = λ 1 αc z v l + λ αc x u y v l, the vertical velocities are displaced southward relative to its no-shear counterpart (Fig1b) Figsc-d show the effect of an upper-layer cyclonic shear without shear at the lower layer (c x l = ) Since F lin (v l = ) = λ [λαc z v u αc x uy v u], the upper-layer shear effect due to the upper-layer cyclone is a southward shift of the vertical velocities because of stronger vertical shear to the south which is confirmed by Figc It is interesting to note 13

15 that the upper-layer shear has the opposite effect to the barotropic shear in terms of the vertical velocity forcings induced by the upper-layer disturbance (compare Figsa and c) On the contrary, the vertical velocity forcings induced by the lower-layer disturbance are both displaced southward by increasing the barotropic shear (Figb) and the upper-layer 9 91 shear (Figd) However, these vertical velocities induced by ψ l on EKE because they are still in quadrature with ψ l do not have any net effect Let us summarize the direct effects of the horizontal shears in the linear part of the vertical velocities F lin (Eq(9)) An increase in the upper-layer shear displaces the peak of ascending motions created by the upper cyclone to its southeastern side for the cyclonic shear and to its northeastern side for the anticyclonic shear The converse occurs for an increase in the barotropic shear Another interesting aspect to consider is the nonlinear vertical velocity forcing when the disturbances are horizontally elongated as in Fig3 This effect can be an indirect effect of the horizontal shears As in the no-elongation case (Fig1e), the Q-vector is mainly pole- ward between the two disturbances because Q nl λ 1 s (, 5(v u v l ) y(u u + u nl l )) F (see Eq(9)) is thus mainly positive to the north and negative to the south and the stretching term has a south-north dipolar structure as in the no-elongation case (compare Figs3a and 3b to 1e) However, the forcing term F nl λ [ y (v u v l ) y(u u + u l ) + (v u v l ) y(u u + u l )] 34 is not anymore in quadrature with the lower-layer perturbation streamfunction ψ l as in the 35 no-elongation case Indeed, in the region where ψ l reaches its minimum, horizontal veloci ties (u l, v l ) are close to zero Furthermore, v u being close to its maximum in that region, its y-derivative is close to zero and one can deduce that F nl λ [ y v l yu l + v u yu u] In the case of the cyclonic tilt (Fig3a), v u and u u have opposite signs and the product v u yu u is positive Furthermore, y v l > and yu l < leads to yv l yu l > In the region where ψ l reaches its minimum, F nl is therefore positive This non quadrature structure leads to a pos- itive spatial average of the product between the nonlinear stretching term λ 1 sωu l nl and the lower-layer perturbation streamfuction ψ l for the cyclonic tilt A lower-layer kinetic-energy 14

16 increase (resp decrease) is then induced in the cyclonic (resp anticyclonic) orientation case To conclude, even though the basic-state horizontal shear is not directly involved in the nonlinear stretching term, it may potentially have an impact on this term because it will tend to horizontally elongate the disturbances (cyclonically if the shear is cyclonic as in Fig3a and anticyclonically if the shear is anticyclonic as in Fig3b) We thus expect an increase and a decrease in the lower-layer kinetic-energy due to the nonlinear term for the cyclonic and anticyclonic shear respectively 3 b Time evolution of the control simulations 31 The previous instantaneous diagnoses of the vertical velocities will be useful to analyze 3 the time evolution of the disturbances in the following control simulations The control simulations have the same vertical shear c z = 1 and the same amplitude for the horizontal shears with c x u = 1 and c x l = 5 for the cyclonic-shear case and c x u = 1 and c x l = 5 for the anticyclonic-shear case The lower-layer shear is weaker than the upper-layer one to reproduce the configuration of real mid-latitude jets The perturbation parameters are A = m s 1 and r =9 1 5 m The two columns on the left side of Fig4 correspond to the control simulations with anticyclonic and cyclonic shear without β and the 39 other two on the right side to simulations with β Let us first analyze the anticyclonic case without β (first column) As shown in Gilet et al (9), for the anticyclonic shear, both disturbances are stretched along the southwest-northeast direction and the lower dis- turbance keeps maintaining its position to the east of the upper disturbance (see Fig4m) This three-dimensional structure has some consequences from an energetic point of view as shown by averaging the various kinetic-energy conversion rates over an area S spanning over the lower-layer perturbation streamfunction (hereafter denoted as < > S ) Because of the stretching of the perturbation along the dilatation axes, the barotropic conversion rate <E l D l > S is negative (grey line with stars in Fig5a) and the perturbation is losing energy via barotropic interaction However, because of the maintained westward tilt with height 15

17 of the perturbation streamfunction, < > S (grey line with diamonds in Fig5a) is positive (cf section 3a1) and even increases during the simulation The convergence of the vertical ageostrophic fluxes <CV > S is close to zero (grey line with triangles in Fig5a) This can be easily explained by noting that the dominant term of F lin (Eq(9)) is λ 1 αc z (v u + v l ) which is in quadrature phase with ψ u + ψ l This renders the product between λ 1 sω u l and ψ u + ψ l, that is CV (Eq(3)), small on average To conclude on this simulation, since <CV > S is mostly zero and < > S is positive with stronger amplitude than the negative val- ues of <E ld l > S, the sum of the three terms during the whole simulation (solid black lines <K l > S in Fig5a) is positive and close to the kinetic-energy local growth rate (dashed black lines in Fig5a) The difference between the two curves is due to the horizontal redistribution terms (see Eq(16)) which are not exactly zero when the spatial average is locally computed The anticyclonic case with β (third column of Fig4) exhibits similar behavior but the amplitudes of the upper and lower disturbances are significantly reduced compared to the case without β (first column of Fig4) This is consistent with the well-known stabilizing effect of β in baroclinic instability such as on normal modes (Vallis 6) or singular modes (Rivière et al 1) Furthermore, the lower-layer streamfunction decreases more rapidly than the upper-layer one (see Fig 4o) The kinetic-energy budget in Fig 5b shows that <CV > S (grey line with triangles), which redistributes kinetic-energy from lower to upper layer, is the prime driver of this growth difference sum <E ld l > S +<CV > S +< > S <CV > S is negative throughout the simulation is so largely negative that the In the cyclonic-shear case without and with β (second and fourth column of Fig4), as shown in Gilet et al (9), the axes of the two cyclonic disturbances turn cyclonically and there is no elongation along the dilatation axes It leads to smaller negative values for <E l D l > S than in the anticyclonic cases and even slightly positive ones after h (compare the grey curves with stars between Figs5a and c and between Figs5b and d) </> S strongly positive but decreases after a while without and with β (see the grey lines with is 365 diamonds in Figs5c-d) <CV > S is close to zero without β and is strongly negative with β 16

18 but its amplitude decreases with time (see the grey lines with triangles in Figs5c-d) Indeed, as explained before in presence of β, and CV are initially quite large when the lower cyclone is downstream of the upper one but have opposite signs Whereas this configuration is maintained in the anticyclonic shear, it is only transient in the cyclonic shear because the two disturbances turn around each other This largely explains why the absolute amplitudes of both <> S and <CV > S are relatively small and even decrease after 1h in the cyclonic shear Without β, there are not much differences between the EKE growth rates in both shears (compare the black lines in Figs5a and c) However, with β, because CV becomes much more strongly negative in the anticyclonic shear than in the cyclonic shear (compare the grey lines with triangles in Figs5b and d), the EKE growth rate is negative in the former shear and close to zero in the latter one (see the black lines in Figs5b and d) Another large difference between the two shears concerns the eddy potential-energy (EPE) budget (Fig6) Since the westward tilt with height is maintained with time in the an- ticyclonic shear, <CB> S <P > S is maintained with large positive values during the whole simulation (grey curves with squares in Figs6a-b) On the contrary, in the cyclonic-shear case, <CB> S <P > S decreases rapidly with time (grey curves with squares in Fig6c-d) because the westward tilt with height is not maintained This difference explains why the EPE growth rate is much smaller in the cyclonic shear than in the anticyclonic shear This result is valid with and without β even though we notice that the potential-energy growth rates are smaller with β than without β This last result is difficult to interpret because β does not appear in the expression of CB (see Eq(4)) Figure 7 details the budget of the sum of the internal conversion rate and the convergence of the vertical ageostrophic fluxes CV + = λ 1 sω u l ψ l by analyzing the various terms involved in the stretching term λ 1 sω u l During the anticyclonic-shear run, there is an increase with time in the difference between the cases without and with β (compare the black lines in Figs7a and b) for <CV + > S <K l > S On the contrary, during the cyclonic-shear run, there is a slight decrease with time in the difference between the cases without and 17

19 with β (compare the black lines in Fig7c-d) This can be explained in large part by the β-term F lin β (Eq(3)) For the anticyclonic shear, in Fig7b) becomes more and more negative with time <CV + > S due to F lin β (grey solid line This could be explained by the stretching of the cyclones which are more and more meridionally tilted (see the third column of Fig4) increasing the ratio between the perturbation meridional and zonal velocities The ratio <CV + > S velocities and K l could be thus more negative because of a greater ratio between meridional For the cyclonic shear, <CV + > S due to F lin β (grey solid line in Fig7d) becomes less and less negative with time because the two disturbances are turning around each other without being significantly stretched This makes the correlation between v u and ψ l small and the value of <CV + > S due to F lin β becomes less and less negative during the simulation The linear part F lin for the cyclonic shear becomes less positive in the case with β (grey dashed lines in Figs7c and d) while it does not change so much for the anticyclonic shear (grey dashed lines in Figs7a and b) The nonlinear part of <CV + > S due to F nl (dash-dotted lines in Fig7) participates in the difference between the anticyclonic-shear and cyclonic-shear cases It is negative and positive in the anticyclonic-shear and cyclonic-shear case respectively as expected from Fig3 The presence of β amplifies this nonlinear effect (compare Figs7a to b and Figs7c to d) 41 which can be interpreted as follows In the anticyclonic-shear case, cyclones are slightly less meridionally stretched and more southwest-northeast oriented with β than without β This leads to a more negative value of CV + due to F nl with β In the cyclonic-shear case, cyclones less rapidly rotate cyclonically with β and therefore stay a longer time with a northwest-southeast elongation The effect is to increase the positive value of CV + due to F nl This might explain part of the amplifying difference between the cyclonic and anticyclonic shear cases in presence of β To conclude, the stabilizing effect of β is less efficient in the cyclonic-shear case than in the anticyclonic-shear case as the lower-layer kinetic-energy growth rate is greater in the former case than in the latter in presence of β while the two growth rates have equivalent 18

20 4 amplitudes when β is zero This is explained in large part by the direct effect of β in 41 4 the vertical velocity forcing (F lin ) as well as by the indirect effect in the nonlinear vertical β velocity forcing (F nl ) which are both involved in the sum CV + 43 c Sensitivity to various parameters The previous control simulations were obtained for specific values of the parameters The purpose of the present section is to change the value of each parameter to identify its role in the difference between cyclonic- and anticyclonic-shear cases Fig8 shows the sensitivity of the EKE amplification between t=h and t=4h, <K l > S(t=4h), to various parameters Spa- <K l > S(t=h) tial averages are still computed over a square S whose center is the lower-layer perturbation streamfunction minimum When spatial averages are made over the whole domain amplifica- tion rates are greater than when they are made over the area S because energy redistribution terms (second term on the rhs of Eq(5)) tend to disperse energy outside the region where 43 finite-amplitude disturbances are Note that this horizontal dispersion of energy is even stronger when β is present (not shown) because of the so-called Rossby radiation of energy in presence of PV gradient (Oruba et al 13) Despite these distinct features between the global and local averages, the results below are valid for both averages and only local ones are shown in Fig8 Generally speaking, the presence of β reduces EKE amplification rates (compare the dashed and solid lines in Fig8) The stabilizing effect of β is more efficient in the anticyclonicshear case than in the cyclonic-shear case and confirms the results found before Let us now look at the sensitivity to individual parameters Since the difference between the two horizontal shears is a nonlinear phenomenon, it is natural to change the value of the initial amplitude of the disturbances Figure 8a shows the sensitivity to the perturbation amplitude A In the case without β (dashed lines), changing the perturbation amplitude does not modify the result In presence of β (solid lines), the EKE amplification is systematically larger for the cyclonic-shear case than the anticyclonic-shear case and the difference between 19

21 the two cases slightly increases with the perturbation amplitude The last result was expected because the two cases converge toward each other for very small disturbances Figure 8b shows the sensitivity to the horizontal scales of the disturbances r, keeping constant the distance between them The two simulations without β are generally similar (dashed lines) With β, EKE amplification rates are stronger for smaller length scales This result is consistent with the previous sensitivity test to A Indeed, the eddy vorticity simply scaling as A /r, an increase in A would have the same effect on vorticity amplitude as a decrease in r For all the length scales in presence of β, the growth rates are greater in the cyclonic-shear case than in the anticyclonic-shear case and the difference between the two cases slightly increases by increasing the length scale This last result cannot be simply explained in terms of vorticity amplitude because an increase in r (which decreases the vorticity amplitude) leads to larger differences between the cyclonic and anticyclonic-shear cases similarly to an increase in A (which increases the vorticity amplitude) The sensitivity to the basic-state vertical shear c z is shown in Fig8c As expected, all growth rates become larger by increasing c z because becomes larger In the case without β (dashed lines), the EKE amplification rate becomes larger for the anticyclonic shear than for the cyclonic shear for c z 15 With β (solid lines), the same thing occurs for c z This is consistent with the fact that the westward tilt with height is maintained with time in the anticyclonic shear while it is not in the cyclonic shear Indeed, following Eqs(4) and (9), the term involving the vertical shear in CV + can be expressed as CV + = λ 1 αc z ψ l ( λ ) 1 (v u + v l ) which is close to λ 1 αc z ψ l v u where ψ l is minimum For the anticyclonic shear, since the westward tilt with height is well maintained, v u and ψ l are strongly anticorrelated and an increased c z leads to a rapid increase in λ 1 αc z ψ l v u and thus in CV + On the contrary, for the cyclonic shear, because v u and ψ l are not well anticorrelated, CV + λ 1 αc z ψ l v u increases less rapidly with c z The same reasoning holds when looking at the vertical velocity forcing term involving β in Eq(8) whose effects in CV + can be written as λ βψ l ( λ ) 1 (v u v l ) which is also highly dependent

22 on the correlation between v u and ψ l However, cz and β have opposite effects The term involving c z increases EKE while the one involving β decreases it c z increase leads to a more rapid EKE increase in the anticyclonic shear than in the cyclonic shear while β increase leads to a more rapid EKE decrease in the former shear than in the latter 477 Fig8d shows the sensitivity to the barotropic shear, that is when c x l and c x u are changed but maintained equal to each other While the EKE amplification does not change much in the cyclonic shear, a strong barotropic shear practically eradicates any lower-layer cyclo- genesis in the anticyclonic shear, especially in the case with β (black solid line) This can be explained in large part by the barotropic conversion rate For large horizontal shears, the lower-layer cyclone embedded in the anticyclonic shear is strongly stretched leading to negative kinetic-energy growth rates because of negative barotropic conversion rates In the cyclonic shear, the lower-layer cyclone is not that stretched and its axis is not maintained along the dilatation axes: the barotropic conversion rate is thus relatively small CV + (not shown) is also responsible for the stabilizing effect of the barotropic anticyclonic shear When this shear is increased, the nonlinear part of the vertical velocity leads to more and more negative tendencies which significantly reduces CV + Differences in the nonlinear term are even larger in presence of β (not shown) It is consistent with Fig7 showing larger magnitudes of the nonlinear terms in presence of β (compare the dash-dotted lines in Figs7a and b) Fig8e shows the sensitivity to the upper-layer shear c x u for c x l = Since the lower- layer shear is zero, there is no barotropic sink and EKE amplification rates are greater than in the case with barotropic shear The anticyclonic shear exhibits even stronger EKE amplification rates with increasing the upper-layer shear This is mainly due to the linear part of the vertical velocity (not shown) As shown in Figc, the ascending motions due to the presence of the upper-layer cyclone is to the southeast for the cyclonic shear and to the northeast for the anticyclonic shear Since the lower-layer cyclone is maintained to the northeast of the upper-layer cyclone in the anticyclonic shear (see the first and third 1

23 columns in Fig4), there is a good correlation between the linear vertical velocity triggered by the upper-layer cyclone and the lower-layer streamfunction Another effect that renders the previous correlation strong is due to the horizontal stretching of the disturbances that increases the ratio between the eddy meridional velocity and the eddy kinetic energy Since F lin (v l = ) = λ [λαc z v u αc x uy v u], the associated linear stretching term λ 1 sω lin u l depends strongly on v u Therefore, the ratio CV + depends on the ratio v u K l which is greater for meridionally stretched cyclones To conclude, the reinforcement of the upper- layer anticyclonic shear facilitates the EKE increase via the linear vertical velocity forcings Since Figs8d and e show that the increased barotropic and upper-layer shears respectively decrease and increase the EKE amplification rates in the anticyclonic-shear case, we expect a strong decrease in EKE amplification rates when the lower-layer shear is increased This is precisely what Fig8f shows (see the black lines) The cyclonic-shear cases are less sensitive to the lower-layer shear (grey lines) To conclude, increasing the basic-state vertical shear destabilizes more the anticyclonic- shear cases while increasing β or the lower-layer horizontal shear stabilizes them more than the cyclonic-shear ones The sensitivity to perturbation parameters, such as to the spatial scale or the amplitude, shows less drastic differences between the two shears Meridionally confined zonal jet In the present section, the jet-crossing phase undergone by a surface cyclone is investigated To do so, a unique simulation is performed which is similar to the one described by Gilet et al (9) The initial conditions are the following: the basic-state flow consists of a meridionally confined zonal jet having a Gaussian horizontal profile with maximum wind speeds at the lower and upper layer equal respectively to 5 m s 1 and 5 m s 1 (see the basic-state wind profile in Fig1 of Gilet et al 9) The perturbation flow is composed of the same two zonally-aligned disturbances as in previous sections but localized to the south

24 of the jet at a distance of 875 km The distance between the two disturbances is 1 km and other perturbation parameters are A = 1 7 m s 1 and r = 1 km As shown by Gilet et al (9), the two disturbances move northward through a generalized β-drift mechanism in presence of a positive barotropic PV gradient The lower-layer cyclone crosses the jet axis around t=36h Besides, during the evolution between t=h and t=6h, there exist two peaks in the EKE growth rate (see the dashed black line in Fig9a), one before the jet-crossing phase around t=1h and another at about t=6h terms <E l D l > S +<CV > S +< > S The average of the sum of the three (solid black line in Fig9a) follows the same evolution as the EKE growth rate with the former being slightly greater than the latter because of horizontal energy outward flux at the boundary of the considered area Note in particular that there exist also two peaks in <E l D l > S +<CV > S +< > S conversion rate <E l D l > S around t=1h and t=6h The barotropic (grey line with stars in Fig9a) is mainly negative and its fluctua- tions play a minor role, albeit not negligible, in the evolution of <E l D l > S +<CV > S +< the two disturbances get closer to the jet axis, because the vertical shear increases, < > S (grey line with diamonds in Fig9a) is more and more positive and <CV > S > S As (grey line with triangles in Fig9a) is more and more negative because of a stronger barotropic PV gradient close to the jet axis consistent with section 3 Note this evolution is also consistent with the uniform anticyclonic-shear case of Fig5b On the contrary, once the disturbance is located on the cyclonic side, the amplitudes of < > S and <CV > S cyclonic-shear case of Fig5d This is the sum of <CV + > S decrease similarly to the uniform that explains the two growth stages, one on the anticyclonic side and the other on the cyclonic side, as shown by the sim- ilarity between <CV + > S (solid black line in Fig9b) and <E ld l +CV + > S (solid black line in Fig9a) However, the two peaks are related to different terms and mechanisms During the first growth stage, the peak of <CV + > S curve in Fig9b) while during the second growth stage, the peak of to the nonlinear term F nl (dash-dotted curve in Fig9b) is mainly due to the linear term F lin (dashed <CV + > S is mainly due The snapshot of Fig1 at t=1h details the spatial structure of the various vertical 3