Gravity Waves in Shear and Implications for Organized Convection

Transcription

1 SEPTEMBER 2009 S T E C H M A N N A N D M A J D A 2579 Gravity Waves in Shear and Imlications for Organized Convection SAMUEL N. STECHMANN Deartment of Mathematics, and Deartment of Atmosheric and Oceanic Sciences, University of California, Los Angeles, Los Angeles, California ANDREW J. MAJDA Deartment of Mathematics, and Center for Atmoshere Ocean Science, Corant Institte of Mathematical Sciences, New York University, New York, New York (Manscrit received 10 October 2008, in final form 12 Febrary 2009) ABSTRACT It is known that gravity waves in the trooshere, which are often excited by reexisting convection, can favor or sress the formation of new convection. Here it is shown that in the resence of wind shear or barotroic wind, the gravity waves can create a more favorable environment on one side of reexisting convection than the other side. Both the nonlinear and linear analytic models develoed here show that the greatest difference in favorability between the two sides is created by jet shears, and little or no difference in favorability is created by wind rofiles with shear at low levels and no shear in the er trooshere. A nonzero barotroic wind (or, eqivalently, a roagating heat sorce) is shown to also affect the favorability on each side of the reexisting convection. It is shown that these main featres are catred by linear theory, and advection by the backgrond wind is the main hysical mechanism at work. These rocesses shold lay an imortant role in the organization of wave trains of convective systems (i.e., convectively coled waves); if one side of reexisting convection is reeatedly more favorable in a articlar backgrond wind shear, then this shold determine the referred roagation direction of convectively coled waves in this wind shear. In addition, these rocesses are also relevant to individal convective systems: it is shown that a barotroic wind can lead to near-resonant forcing that amlifies the strength of stream gravity waves, which are known to trigger new convective cells within a single convective system. The barotroic wind is also imortant in confining the stream waves to the vicinity of the sorce, which can hel ensre that any new convective cells triggered by the stream waves are able to merge with the convective system. All of these effects are catred in a two-dimensional model that is frther simlified by inclding only the first two vertical baroclinic modes. Nmerical reslts are shown with a nonlinear model, and linear theory reslts are in good agreement with the nonlinear model for most cases. 1. Introdction Corresonding athor address: Samel N. Stechmann, UCLA Mathematics Deartment, 520 Portola Plaza, Los Angeles, CA stechmann@math.cla.ed Troical convection can be organized across a wide range of sace and time scales, from mesoscale convective systems (Hoze 2004) to convectively coled waves on eqatorial synotic scales (Kiladis et al. 2009) to the Madden Jlian oscillation on lanetary/intraseasonal scales (Zhang 2005). These henomena remain challenging both theoretically and ractically; not only are their hysical mechanisms not adeqately nderstood, bt even nmerical simlation of these mltiscale henomena is a challenging endeavor. To better nderstand the hysical mechanisms of organized convection, nmerical simlations are often carried ot sing clod-resolving models (CRMs). Many CRM stdies have docmented the role of gravity waves in organizing convection into wave trains (Oochi 1999; Shige and Satomra 2001; Lac et al. 2002; Li and Moncrieff 2004; Tlich et al. 2007; Tlich and Maes 2008). Convective heating generates a variety of gravity waves that roagate away from the convection, and DOI: /2009JAS Ó 2009 American Meteorological Society

2 2580 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 66 these gravity waves can favor or sress the formation of new convection. New convection is favored when gravity waves romote ascent of air in the free trooshere, which is also accomanied by adiabatic cooling and moistening, and new convection is sressed in the oosite sitation. In addition to gravity waves, density crrents can lay a similar role in triggering new convection (Moncrieff and Li 1999; Tomkins 2001; Fovell 2005). Frthermore, in addition to their role in favoring or sressing new convective systems, gravity waves and density crrents can lay a similar role in favoring or sressing new convective cells within a single convective system (McAnelly et al. 1997; Lane and Reeder 2001; Fovell 2002; Fovell et al. 2006). In addition to (and in some cases receding) these CRM stdies, several theoretical stdies sing simlified models have also been carried ot to better nderstand the role of gravity waves and density crrents in organizing convection (Bretherton and Smolarkiewicz 1989; Nicholls et al. 1991; Pandya et al. 1993; Maes 1993; Li and Moncrieff 2004). These stdies se a simlified set with linearized dynamics and imosed heat sorces to reresent clod heating. The exact linear resonse to the heating can then be fond analytically for any heating rofile, sally chosen to reresent dee convective and stratiform heating. Maes (1993) sed sch a model with two vertical modes to demonstrate how the formation of new convection cold be favored or triggered near reexisting convection by convectively generated gravity waves; sch a mechanism was osited to be imortant for convective organization, and the observations and CRM simlations mentioned in the revios aragrah largely confirm this. For the sake of simlicity, these simlified modeling stdies mentioned above all assmed a motionless, shear-free backgrond wind. The main rose of the resent aer is to investigate the effect of wind shear and barotroic wind on gravity waves by sing the simlified nonlinear model of Stechmann et al. (2008, hereafter SMK08) and a linearized version of this model. The main simlification sed by SMK08, as in the other simlified model stdies mentioned above, is a trncated vertical strctre with only two baroclinic modes; however, nlike revios trncated models, SMK08 inclded nonlinear interactions between the vertical modes, which allows nonlinear interactions between gravity waves and wind shear. The trncated vertical strctre of this model has the imortant advantage that it redces the system dimension, which greatly redces the comtation time needed to solve the eqations nmerically. In addition to reslts with this nonlinear model, an exact linear theory with backgrond shear is develoed here; these reslts are also shown below, and they are in good agreement with the nonlinear nmerical reslts. The imlications of these reslts for organized convection are discssed throghot the aer. The aer is organized as follows: The simlified model of SMK08 is described in section 2. Reslts for varios wind rofiles are resented in section 3. The hysical mechanisms at work are discssed throghot section 3 and in section 4, and some sensitivity stdies inclding higher vertical resoltion are shown in section 5. Finally, conclsions are smmarized in section 6. To streamline the resentation, some analytic details of the model are relegated to the aendices. 2. The simlified model In this section, the simlified model of SMK08 is reviewed. The reresentation of convective heating as an imosed heat sorce is also described; it will allow certain linear theory techniqes to be sed, which are also described below. a. Descrition of the simlified nonlinear model The set considered here is a hydrostatic Bossinesq flid with a backgrond otential temeratre bg (z) that is linear with height. For simlicity, and following revios work mentioned in the introdction, a twodimensional set with zonal coordinate x and vertical coordinate z will be sed here. The flid is bonded above and below by rigid lids where free sli bondary conditions are assmed: Wj z50,h 5 0. The er bondary here is taken to be z 5 H 5 16 km, which is roghly the height of the trooase in the troics, and the height z will be nondimensionalized so that z 5 is the er bondary in nondimensional nits. Standard eqatorial scales are chosen as reference nits so that the horizontal satiotemoral scales are 1500 km and 8 h (Majda 2003), with a reference seed of 50 m s 21 corresonding to the first baroclinic mode gravity wave seed. See Table 1 for the reference scales sed to nondimensionalize the model variables. In this sitation there is an infinite set of horizontally roagating linear gravity waves with hase seeds c j and vertical rofiles sin(jz) for j 5 1, 2, 3,... (Majda 2003). The waves c j are decoled from all waves c k (k 6¼ j) for the case of linear dynamics, bt the waves become coled in the case of nonlinear dynamics, as will be shown below. For the nonlinear model of this aer, only the first two baroclinic modes are ket so that the model variables are exanded as

3 SEPTEMBER 2009 S T E C H M A N N A N D M A J D A 2581 TABLE 1. Model arameters and scales. Parameter Derivation Vale Descrition b m 21 s 21 Variation of Coriolis arameter with latitde ref 300 K Reference otential temeratre g 9.8 m s 22 Gravitational acceleration H 16 km Trooase height N 2 (g/ ref ) d bg /dz s 22 Boyancy freqency sqared c NH/ ffiffiffiffiffiffiffi 50 m s 21 Velocity scale L c/b 1500 km Eqatorial length scale T L/c 8 h Eqatorial time scale a HN 2 ref /(g) 15 K Potential temeratre scale H/ 5 km Vertical length scale H/(T) 0.2 m s 21 Vertical velocity scale c m 2 s 22 Pressre scale ffiffiffi ffiffiffi U(x, z, t) (x, t) 2 cos(z) 1 2 (x, t) 2 cos(2z), ffiffiffi Q(x, z, t) 5 1 (x, t) 2 sin(z) 1 2 (x, t)2 ffiffiffi 2 sin(2z), ffiffiffi ffiffiffi P(x, z, t) 5 1 (x, t) 2 cos(z) 1 2 (x, t) 2 cos(2z), and ffiffi ffiffi W(x, z, t) 5 w 1 (x, t) 2 sin(z) 1 w2 (x, t) 2 sin(2z). (1) Note that the ffiffiffi convention ffiffi here is to exand Q on the basis of j 2 sin(jz), not 2 sin(jz). This vertical strctre is illstrated in Fig. 1. The hydrostatic and continity eqations lead to the relationshis j 5 j and w j 5 1 j x j. (2) When the hydrostatic Bossinesq eqations are rojected onto the first two baroclinic modes, inclding the rojections of the nonlinear advection terms, the reslt is (SMK08) 8 1 >< t >: 1 t 8 >< 2 t >: 2 t x x x x 1 4 x 5 3 ffiffiffi x x 5 1 ffiffiffi 2 2 x 2 x 5 0, 2 x ffiffiffi 1 x 1 2 x, 1 x x 1 S 2. 1 x x 1 S 1 ; (3) (In three dimensions, the 2 eqation incldes terms roortional to 1 $ $ 1, which are zero in this two-dimensional case.) Notice that the left-hand side shows the familiar linear dynamics for the baroclinic modes with hase seeds 1 and ½ in the nondimensional nits sed here. The right-hand side shows how the baroclinic modes interact nonlinearly becase of the rojection of the nonlinear advection terms. These nonlinear terms also allow for nonlinear interactions between gravity waves and wind shear. Note that these eqations have a conserved energy, E ( ), (4) and they can be written in matrix form as t 1 A() x 5 S, (5) where 5 ( 1, 1, 2, 2 ) T, S 5 (0, S 1,0,S 2 ), and the matrix A() is written ot in aendix A. Becase these eqations cannot be written as a system of conservation laws, designing a nmerical method for them is a challenge. The nmerical methods sed here were designed and extensively tested by SMK08; the reader is referred there for more details. We also note here that a roagating sorce term S(x 2 st) is eqivalent to a stationary sorce term with a barotroic wind of 0 52s. To see this, start with (5) and assme a sorce term of the form S 5 S(x 2 st). By changing variables into a reference frame moving with the sorce term, with x95x 2 st and t95t, this eqation becomes

4 2582 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 66 FIG. 2. Contor lot of the heat sorce S (x, z) from (7) and (8) sed to reresent dee convective and stratiform heating. Contors are drawn from 50 to 300 K day 21 with a contor interval of 50 K day 21 ; an additional contor at 20 K day 21 is also shown. The minimm heating is 223 K day 21 near x 5 50 km, z 5 4 km, bt no negative contors are shown. This is the standard heat sorce sed nless otherwise secified. The first baroclinic heating S 1 reresents dee convective heating, whereas the second baroclinic heating S 2 reresents stratiform and congests heating. The standard to-heavy heating that will be sed here nless otherwise stated reresents a combination of dee convective and stratiform heating with S 1 (x) 5 a 1 e x2 /2s and S 2 (x) 5 a 2 e x2 /2s 2 2. (8) FIG. 1. Vertical rofiles for the first few modes of (a) velocity and (b) otential temeratre, as described in (1). t9 s x9 1 A() x9 5 S(x9), (6) which is eqivalent to a stationary sorce with a change of 2s to the barotroic wind. Becase of this eqivalence, all of the reslts in this aer cold be interreted in many ways, since, for instance, even the case of zero barotroic wind and a stationary sorce is eqivalent to any case with 0 5 y and s 5 y for any choice of y. b. Reresentation of convective heating Convective heating will be reresented as an imosed heat sorce with comonents from two vertical baroclinic modes, as it was reresented in the revios simlified modeling stdies mentioned in the introdction (Maes 1993): ffiffiffi S(x, z) 5 S 1 (x) 2 sinz 1 S2 (x)2 ffiffiffi 2 sin2z. (7) The standard vales sed here will be a K day 21, 2a K day 21, s km, and s km, nless otherwise secified. These vales are chosen to reresent the mesoscale-filtered convective heating of a sqall line or mesoscale convective system, and they are similar to the vales sed in revios stdies (Nicholls et al. 1991; Maes 1993; Pandya and Drran 1996; Maes 1998); in fact, the ratio of total dee convective to stratiform heating is eqivalent to that sed by Maes (1993), as exlained below. This standard heating S(x, z), given by these standard arameter vales and (7) (8), is shown in Fig. 2. In the time-deendent nmerical simlations shown below, this heating rofile is trned on at the start of the simlation at time t 5 0 and maintained thereafter. A delta fnction aroximation to the zonal heating rofile (8) will be sed for aroximate analytical soltions as described later: ae x2 /2s 2 5 as ffiffiffiffiffiffi 2 (2s 2 ) 1/2 e x2 /2s 2 as ffiffiffiffiffiffi 2 d(x). (9) Notice that the magnitde of the delta fnction, which will be denoted by S j *, is determined by the Gassian rofile s amlitde and standard deviation by

5 SEPTEMBER 2009 S T E C H M A N N A N D M A J D A 2583 ffiffiffiffiffiffi S j (x) a j s j 2 d(x) 5 Sj * d(x) (10) for j 5 1, 2. Notice that s j, the standard deviation of the Gassian rofile, is jst as imortant to the delta fnction magnitde as a j, the amlitde of the Gassian heating. When this is taken into accont, the delta fnction form of the standard heating sed here is eqivalent to that of Maes (1993) becase the larger width of the stratiform heating comensates for its weaker amlitde. c. Exact linear soltions When linearized abot the sheared backgrond state 5 ( 1,0, 2,0) T, the simlified model in (5) becomes t 1 A() x 5 S*d(x), (11) analytical soltions to the jm conditions in (12) will also rovide insight. Comarisons are shown below to demonstrate that the linear aroximation in (11) is accrate in most cases. One way to think of the jm conditions in (12) is to recall the relationshi w j 52 x j /j from the continity eqation in (2). If the jm [ j ] is interreted as a derivative, then [ j ] can be thoght of as being roortional to the vertical velocity at the location of the sorce. The sorce initially excites for waves that sbseqently roagate away from the sorce, and the state [] is the steady balanced state left behind after the adjstment rocess, with the sorce term artially balanced by the vertical velocity [ j ]. For an interretation in terms of the hysical vertical coordinate z instead of vertical modes, see section 4. where a delta fnction sorce term is also inclded. When this linear system is forced by a delta fnction sorce term, for discontinos waves are excited and roagate away from the sorce. These roagating fronts are what Maes (1993) referred to as lses or boyancy bores. The jms associated with these roagating fronts can be fond sing the so-called Rankine Hgoniot jm conditions (Evans 1998; LeVeqe 2002). In addition, there can also be discontinities in the variables at the location of the sorce, x 5 0, and the jms associated with this discontinity can also be fond by sing a form of the Rankine Hgoniot jm conditions (LeVeqe 2002): A[] 5 S*, (12) where A 5 A() is the backgrond advection matrix from (11) and [] is the jm in across the location of the sorce, where 1 and 2 are the vales of jst to the right and left of the sorce, resectively. This is a simle linear system of eqations. Therefore, given the backgrond wind shear, the barotroic wind 0, the sorce roagation seed s, and the magnitde S* of the convective heating, one can se the Rankine Hgoniot jm conditions in (12) to easily determine [], which measres the difference in the variables on the east and west sides of the heat sorce. (Exact soltions to the comlete linear roblem with shear can also be readily fond as shown in aendix A.) In articlar, if there is a jm in 1 and/or 2 across the location of the sorce, this reresents a difference in the thermodynamic state (and therefore a difference in favorability for convection) between the western and eastern sides of the sorce. Determining how this difference in the thermodynamic state deends on, 0, and s is the main rose of the resent aer. In addition to nmerical simlations of the eqations in (3), 3. Reslts The reslts in this section will be of two tyes: (i) nmerical soltions to the nonlinear eqations in (3) and (ii) analytical soltions to the jm conditions in (12). First the simlest case with initially motionless wind is described, after which the effect of wind shear is considered, then barotroic wind, and then both effects combined. a. Simlest case with initially motionless wind As an initial illstration of the gravity wave resonse to diabatic forcing, consider the simlest case of initial conditions with motionless, shear-free winds and a localized heat sorce that is trned on at time t 5 0 and maintained thereafter. The heat sorce sed here is the standard sorce shown in Fig. 2 excet with the dee convective and stratiform heatings widened by factors of 2 and 4, resectively, in order to broaden the roagating fronts for illstration roses. The amlitdes a j of the heatings were redced by factors of 2 and 4 to kee the total heating magnitdes S j * fixed at their standard vales. The reslting gravity wave resonse to this heating is shown in Fig. 3 at time t 5 4 h. The first baroclinic mode resonse can be seen near x km. Its signatre is sbsidence and adiabatic warming throgh the deth of the trooshere, which tend to sress the formation of new convection. This comonent of the resonse roagates away from the sorce at 50 m s 21, whereas the second baroclinic mode resonse roagates at only 25 m s 21 and reaches only x This slower resonse has low-level ascent and adiabatic cooling as its signatre, and these effects tend to favor the formation of new convection (Maes 1993; Tlich and Maes 2008). In a more realistic sitation with water vaor, the

6 2584 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 66 FIG. 3. Nmerical soltion of the simlified nonlinear model (3) with motionless initial conditions. Snashots are shown at time t 5 4 h for (a) otential temeratre Q(x, z) and (b) vertical velocity W(x, z). Solid (dashed) contors denote ositive (negative) vales, and the zero contor is not shown. The contor intervals for Q and W are 0.25 K and 10 cm s 21, resectively. sbsidence (ascent) wold be accomanied by drying (moistening), which accentates the dry effects on the favorability for new convection. In short, the effect of the fast wave is to initially sress the formation of new convection, and the effect of the slow wave is to create a nearby neighborhood of the reexisting convection that is favorable for the formation of new convection. These basic hysical mechanisms of the gravity wave resonse have been described in earlier work (Bretherton and Smolarkiewicz 1989; Nicholls et al. 1991; Pandya et al. 1993; Maes 1993), and the role of the slower, shallow second baroclinic mode resonse in triggering convection has been docmented in several nmerical simlations (Lac et al. 2002; Tlich and Maes 2008). This mechanism, whereby the second baroclinic mode reconditions the lower trooshere for convection, has also been inclded in models of convectively coled waves that agree well with observations (Maes 2000; Majda and Shefter 2001; Khoider and Majda 2006). Note that in addition to the imortant effects of sbsidence or ascent at the location of the front on the triggering of new convection, the reslting changes in the thermodynamic state after the assage of the front shold also be imortant becase they set an environment that is favorable or nfavorable for the formation of new convection. In terms of elementary arcel stability theory (Emanel 1994), regions with Q(x, z), 0 (.0) shold be favorable (nfavorable) environments for rising arcels and convection. In addition, the vertical velocity is a less reliable variable in this model becase it mst be comted nmerically as the derivative of horizontal velocity throgh (2). For these reasons, the thermodynamic state Q(x, z) will be sed throghot this aer as a measre of the favorability for the formation of new convection. As demonstrated below, the resence of wind shear or barotroic wind can sbstantially alter the east west symmetry of this ictre, with imortant imlications for the organization of convection. One way in which the east west symmetry can be broken is throgh differences in the strength of the sbsidence (or the low-level ascent) to the east and west of the sorce. Other mechanisms can also case east west asymmetries, as described below. b. Effect of different wind shears The nmerical resonse of the nonlinear model (3) to the standard heat sorce (7) is shown in Fig. 4 for for

7 SEPTEMBER 2009 S T E C H M A N N A N D M A J D A 2585 FIG. 4. Nmerical simlation of waves generated by the localized heat sorce in Fig. 2 sing the simlified nonlinear model (3). (left) Shear rofile sed as the initial conditions. (middle) Vertical rofile of otential temeratre Q(z) to the west (east) of the heat sorce at x km (x km) shown with a dashed (solid) line at time t 5 4 h. (right) First baroclinic mode otential temeratre 1 at time t 5 4h. different initial shear rofiles U(z). All of these cases have zero barotroic wind, and the otential temeratre lots in the middle and right colmns are snashots at time t 5 4h. Figres 4a c show reslts with a shearless initial condition, which can be identified with the contor lots in Fig. 3. Figre 4b shows the otential temeratre Q(z) jst to the west and east of the sorce. Each side of

8 2586 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 66 the sorce has the same Q(z), with Q(z), 0 in the lower trooshere, reresenting favorable conditions for the formation of new convection (cf. Fig. 3). This east west symmetry can also be seen clearly in Fig. 4c, which shows the first baroclinic mode otential temeratre 1 at time t 5 4h. Figres 4d f show reslts with a jet shear initial condition that roghly resembles the shear rofiles of the westerly wind brst hase of the Madden Jlian oscillation (Lin and Johnson 1996; Biello and Majda 2005). In terms of vertical baroclinic modes, this case ses initial conditions of m s 21 and m s 21. For this case there is a large difference in the gravity wave resonse to the west and east of the sorce. As shown in Fig. 4e, Q(z) is more negative to the west of the sorce by roghly 1.5 K throgh most of the trooshere. This asymmetry is also clear in the snashot of 1 in Fig. 4f, which shows that the west (east) of the sorce has become more (less) favorable for the formation of new convection in comarison to the shearless case in Figs. 4a c. This is in agreement with actal observations of the formation of new convection in this tye of shear rofile (W and LeMone 1999). These reslts, when considered in terms of organized convection, rovide redictions for the referred roagation direction of wave trains of convective systems (i.e., convectively coled waves). In a articlar backgrond wind shear, if new convection is reeatedly formed on the more favorable side of reexisting convection, then a wave train of convective systems will take form. For the westerly wind brst like shear in Figs. 4d f, sch a convectively coled wave shold then roagate westward becase the west side of reexisting convection is more favorable. Notice that sqall lines in this wind shear wold be exected to roagate eastward (LeMone et al. 1998; W and LeMone 1999), which is in the oosite direction of the exected roagation of the convectively coled wave. In fact, this arrangement, with the enveloe roagating in the oosite direction of the convective systems within it, is what tends to be seen in observations and simlations (Nakazawa 1988; Grabowski and Moncrieff 2001; Tlich et al. 2007); the reslts shown here sggest that jet shears rovide an ideal environment for this tye of behavior. This referred roagation direction of exlicitly resolved convectively coled waves in different wind shears was also catred in the recent model of Majda and Stechmann (2009), which ses a convective arameterization and does not resolve the sqall lines in detail. Figres 4g i show reslts with a jet shear initial condition that roghly resembles the shear rofiles of the westerly onset hase of the Madden Jlian oscillation (Lin and Johnson 1996; Biello and Majda 2005). In TABLE 2. Nmerical (nm.) and analytical (anal.) redictions of the jm [ 1 ] for different wind shears and different amlitdes of stratiform heating. 1 (m s 21 ) 2 (m s 21 ) 22a 2 (K day 21 ) [ 1 ] (nm.) (K) [ 1 ] (anal.) (K) terms of vertical baroclinic modes, this case ses initial conditions of 1 525ms 21 and m s 21. The reslts in this case are qalitatively similar to the case in Figs. 4d f, excet the east side of the sorce is now more favorable for convection becase an easterly jet is sed for this case. Figres 4j l show reslts with an initial shear that roghly resembles those associated with midlatitde sqall lines (Pandya and Drran 1996; Fovell 2002). In terms of vertical baroclinic modes, this case ses initial conditions of 1 529ms 21 and 2 523ms 21.Figre4k shows that Q(z) is nearly identical to the immediate west and east of the sorce, signifying that each side is nearly eqally favorable for the formation of new convection, althogh Fig. 4l shows that the entire gravity wave resonse is not identical to the west and east of the sorce. In sch a sitation with eqal favorability on each side, one wold not exect wave trains of convective systems to develo becase convection shold form sometimes to the east and sometimes to the west of reexisting convection. This is consistent with the absence of wave trains of convective systems in the midlatitdes where shears like this are common, althogh rotational effects also sress wave trains of sqall lines in midlatitdes (Li and Moncrieff 2004). In smmary, the reslts of Fig. 4 demonstrate that wind shear can (bt does not necessarily) case the east and west of the sorce to be neqally favorable for the formation of new convection. Notice that this asymmetry comes from two arts. In Fig. 4f, for examle, there is greater warming by the 150 m s 21 wave than by the 250 m s 21 wave, and there is warming (cooling) in 1 de to the 125 m s 21 (225 m s 21 ) wave. Both of these

9 SEPTEMBER 2009 S T E C H M A N N A N D M A J D A 2587 effects involve interactions between the vertical modes and are not seen in the motionless case in Figs. 4a c. A comarison of these nonlinear nmerical reslts with linear theory reslts sing (12) is shown in Table 2. The nmerical and theoretical vales of [ 1 ] are within 10% of each other for each of the cases from Fig. 4, indicating that linear theory rovides a good estimate of the nonlinear reslts. The reason for this is that the wave and sorce amlitdes are weak in a sitable sense. Althogh a jm in 1 of 1 2 K is large in the sense that its imact on the formation of new convection is very imortant, it is small in a nondimensional sense with resect to the reference temeratre a 5 15 K. This resonse is ltimately determined by the strength of the sorce terms; and while the amlitde of the heating is a tyically large vale of O(300 K day 21 ), this heating is actally weak ffiffiffiffiffiffi in terms of the nondimensional form of S* j 5 a j s j 2. This heating magnitde also incldes the effect of the heating width s j, which is small in comarison to the longer far-field length scales of interest here. Also shown in Table 2 are reslts of frther nonlinear simlations with varying shear strength and varying stratiform heating strength (and corresonding reslts sing linear theory). The jm [ 1 ] varies aroximately linearly with the strength of the wind shear for the cases like the westerly jet of Figs. 4d f, with [ 1 ] reaching vales larger than 3 K for the strongest winds considered, and the linear theory vales are again within 10% of the nmerical vales. As the strength of the stratiform heating decreases, with fixed dee convective heating, the jm [ 1 ] decreases and linear theory agrees better with the nonlinear reslts. Many of the reslts from Fig. 4 and Table 2 (and other reslts in this aer) can be seen directly from the exact linear formlas for the jms, which are fond by solving the system of eqations in (12): [ 1 ] denom S* denom S*, 2 [ 2 ] denom S* denom S*, 2 [ 1 ] 5 3 ffiffiffi 2 2 denom S* 1 6 ffiffiffi 2 1 denom S*, 2 and [ 2 ] 5 0, (13) where the denominator (denom) for these fractions is These formlas show that [ 1] 5 0 for the shearless case of , which reresents balance between the vertical velocity [ j ] and the heating S j *. Also notice that [ 2 ] 5 0 for any shear (althogh [ 2 ] 6¼ 0 for nonzero barotroic winds or for shears with higher vertical resoltion, as shown below). c. Effects of barotroic wind and roagating sorce Several interesting effects emerge with a nonzero barotroic wind or a roagating heat sorce [which are eqivalent, as discssed in (6)]. Figre 5 shows reslts of nonlinear simlations with initially and with varying barotroic winds of 0 5 0, 25, 210, and 215 m s 21. (Cases with 0. 0 are mirror images abot x 5 0 for this case with initially.) The left colmn shows Q(z) jst to the west and east of the sorce, and the middle and right colmns show snashots of 1 and 2, resectively, at time t 5 4 h. Notice that the headwind confines the eastward-moving waves to the neighborhood of the sorce, and it hels the westward-moving waves travel farther from the sorce. Ths, the barotroic wind may affect whether the gravity waves are able to roagate far enogh away from a reexisting convective system to excite a new convective system, or whether the gravity waves are held close enogh to an individal convective system to excite new convective cells. Also notice that the barotroic wind creates a large jm in 2. The large vale of [ 2 ] corresonds to a state to the east of the sorce that is very favorable for convection in the lower trooshere, as shown in the left colmn of Fig. 5 at x km by the solid lines. If the easterly barotroic wind is interreted as an eastward-roagating heat sorce, then this reresents a reconditioning of the stream environment, which has been observed and stdied in nmerical simlations of sqall lines (Fovell 2002; Fovell et al. 2006). The large increase in [ 2 ] as the barotroic wind becomes stronger is de to near-resonant forcing of the second baroclinic mode, which has a roagation seed of 25 m s 21. This can be seen in the exact linear theory soltions to (12) for this case of zero baroclinic backgrond winds and varying barotroic wind: [ 1 ] S*, 1 [ 2 ] 5 1 S*, [ 1 ] S*, 1 and [ 2 ] 5 0 S* (14) The jms [ j ] are zero for 0 5 0, bt [ 2 ] becomes large as 0 aroaches 25 m s 21 (which is ½ in nondimensional

10 2588 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 66 FIG. 5. As in Fig. 4, bt with shearless initial conditions with different barotroic wind 0 5 0, 25, 210, and 215 m s 21 from to to bottom. (left) Vertical rofile of otential temeratre Q(z) to the west (east) of the heat sorce at x km (x km) shown with a dashed (solid) line at time t 5 4 h; (middle), (right) 1 and 2, resectively, at time t 5 4h. nits). Notice that this effect is also significant in the 1 resonse, with Fig. 5 showing clear differences in the warming of 1 to the east and west of the sorce. One nonlinear featre in Fig. 5k is the overshooting front that aears in 1 near x km as the barotroic wind becomes stronger. At the same time, the downstream-roagating front in the region km, x, 2500 km is deressed as the barotroic wind becomes stronger. These same trends have also been observed in nmerical simlations of density crrents (Li and

11 SEPTEMBER 2009 S T E C H M A N N A N D M A J D A 2589 Moncrieff 1996). If there is a resltant change in the vertical velocity at the front, then this cold change the triggering or sressing effect of the front. A comarison of the nonlinear nmerical reslts with linear theory is shown in Table 3. The jms in 2 are in agreement to within 10%, and the jms in 1 show the same trends as 0 varies bt differ qantitatively, robably becase of the nonlinear overshooting front discssed in the revios aragrah. TABLE 3. Nmerical and analytical redictions of the jms [ 1 ] and [ 2 ] across the location of the sorce. Initial conditions are shearless with varying vales of barotroic wind 0. 0 (m s 21 ) [ 1 ] (nm.) (K) [ 1 ] (anal.) (K) [ 2 ] (nm.) (K) [ 2 ] (anal.) (K) d. Effects of both wind shear and barotroic wind The effect of wind shear in combination with a barotroic wind is shown in Fig. 6. The westerly wind brst like shear from Figs. 4d f, with and m s 21, is shown with three vales of the barotroic wind: , 0, and 10 m s 21. These cases can also be interreted in terms of different roagation seeds of the heat sorce. For instance, if a sqall line formed in the wind shear in Fig. 6e and roagated eastward at the jet max seed of roghly 10 m s 21, then the wind U(z)in Fig. 6a wold be that seen in a reference frame moving with the sqall line. This case dislays a combination of the reslts from Figs. 4d f and Figs. 5g i, which showed the searate effects of a jet shear and an easterly barotroic wind, resectively. The stream environment (x. 0) is mch less favorable than the downstream environment (x, 0) for the formation of dee convection (becase [ 1 ] K), bt the stream environment is favorable for low-level convection becase Q(z), 21 K in the lower trooshere (Fig. 6b), which is a henomena stdied by Fovell (2002) and Fovell et al. (2006). Another imortant effect of the barotroic wind in Fig. 6c is that it hels the downstream waves more qickly leave the environment of the reexisting convective system and reach locations where new convective systems cold form. At the same time, it revents the stream waves from leaving the vicinity of the reexisting convection, which cold revent any new convection from forming a new convective system that is distinct from the reexisting system. These effects are likely imortant in the formation of wave trains of convective systems, which tyically show a sccession of distinct convective systems roagating transverse to the enveloe of the wave train (Oochi 1999; Shige and Satomra 2001; Tlich et al. 2007). The case in Figs. 6i l, with m s 21, is ossibly a more realistic aroximation to a westerly wind brst than the case in Fig. 6e becase it has nonzero westerly winds at the lowest levels of the free trooshere (Lin and Johnson 1996; Biello and Majda 2005). The western side of the sorce in this case is mch more favorable for convection than the eastern side (Figs. 6j l), which is in agreement with observations of the actal formation of new convection in similar wind shears (W and LeMone 1999). These nonlinear reslts with the jet shear and varying barotroic wind are comared with linear theory soltions of (12) in Table 4. The reslts agree to within abot 10% excet for the cases with strong easterly barotroic wind, which aear to exhibit nonlinear overshooting fronts (Fig. 6c near x km). Interestingly, the cases with easterly (westerly) barotroic wind are more (less) nonlinear than the case with this jet shear, ossibly de to the difference in the strength of the fast 1 fronts near x and 1700 km in this jet shear in Fig. 6g. Cases with the midlatitde sqall line like shear with different barotroic winds are shown in Fig. 7. The case that most resembles the reference frame moving with the sqall line wold be the left colmn of Fig. 7. In this case, the stream side of the sorce is mch more favorable at low levels (becase of near-resonant forcing) and less favorable at er levels. This tye of thermodynamic environment is consistent with the formation of new low-level convective cells in the stream environment as stdied by Fovell (2002) and Fovell et al. (2006); becase of the redced seed of the streamroagating waves, these effects are confined to the neighborhood of the reexisting convection, which allows the new convective cells to become art of the reexisting convective system. e. Otimal shears Given that different wind shears lead to different vales of [ 1 ], as shown in Fig. 4, two qestions come to mind: Which wind shears give the largest jm [ 1 ]? Which wind shears give [ 1 ] 5 0? To answer these qestions, the jm [ 1 ] was comted sing the exact formla from (13) for the family of two baroclinic mode wind shears satisfying (10 m s 1 ) 2. Reslts are shown in Fig. 8 for three cases of the stratiform heating with the dee convective heating held fixed at its standard vale. The jm [ 1 ] is maximized for a jet shear (Fig. 8b) and is zero for midlatitde sqall line like shears (Fig. 8c). These reslts can also be seen from

12 2590 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 66 FIG. 6. As in Fig. 4, bt sing initial jet shears with , 0, and 10 m s 21 from left to right. (to) (bottom) Wind rofile sed as the initial conditions; vertical rofile of otential temeratre Q(z) to the west (east) of the heat sorce at x km (x km) shown with a dashed (solid) line at time t 5 4h; 1 at time t 5 4 h; and 2 at time t 5 4h. the linear theory formlas in (13). Assming S* j 524S* 2 as for the standard heating rofile sed here, and assming that the 2 j terms in the denominator are negligible, the formla in (13) for [ 1 ] is aroximately [ 1 ] (2 2 1 )6 ffiffiffi 2 S1 *. (15) This will be largest when 1 and 2 have different signs (as in the jet shears in Fig. 4) and it will be smallest in magnitde when 1 and 2 have the same sign (as in the midlatitde sqall line like shear). In the next section, these reslts are discssed in terms of the hysical vertical coordinate z rather than vertical modes.

13 SEPTEMBER 2009 S T E C H M A N N A N D M A J D A 2591 TABLE 4. As in Table 3, bt initial conditions inclde a jet shear with and m s (m s 21 ) [ 1 ] (nm.) (K) 4. Physical mechanisms In this section the hysical mechanisms of the east west asymmetries are smmarized in terms of vertical baroclinic mode behavior. Then these mechanisms are discssed in terms of the hysical vertical coordinate z rather than vertical modes, and the link with advection by the backgrond wind can be seen more clearly. The reslts in the revios section with two baroclinic modes clearly demonstrate the effects of wind shear and barotroic wind on diabatically forced gravity waves. The effect of wind shear is mainly to create east west asymmetries in 1. This asymmetry comes from two effects: (i) differences in the dee warming of the 50 m s 21 waves to the east and west of the sorce and (ii) differences in the 1 contribtions to the 25 m s 21 waves. The effects of barotroic wind are mainly (i) to increase the low-level cooling on the stream side de to nearresonant forcing, (ii) to create east west asymmetries in the 1 resonse, and (iii) to slow down (seed ) the roagation of stream (downstream) waves. These effects have imortant imlications for the formation of organized convection (both new convective cells and new convective systems), and these effects were highlighted throghot the revios section. Becase these effects are catred well by linear theory, the nderlying mechanisms are resmably de to advection by the backgrond wind, which is described in the following discssion. To see more clearly how the backgrond wind creates east west asymmetries in diabatically forced gravity waves, consider the steady-state linear set of (11) (12) in terms of the hysical vertical coordinate z instead of vertical modes. The horizontal momentm and otential temeratre eqations take the form U(z) dw* dz [ 1 ] (anal.) (K) du 1 W* dz 1 [P] 5 0, and (16) U(z) d[p] dz [ 2 ] (nm.) (K) [ 2 ] (anal.) (K) W* 5 S*(z), (17) after alying the hydrostatic and continity eqations, [Q] 5 d[p]/dz and [U] 52dW*/dz. This set is described in more detail in aendix B. These two eqations can be combined to give a single second-order ordinary differential eqation for [P](z): U 2 d2 [P] dz 2 1 [P] 5 U ds* du S* dz dz. (18) This can be thoght of as the hysical sace analog of the jm conditions in (12) for the vertical modes. In fact, if one assmes vertical strctres with only two baroclinic modes, the right-hand side of (18) is eqivalent to the leading-order terms (ignoring the 2 j terms) of the formlas for [ 1 ] and [ 2 ] in (13). This sggests that the leading-order art of (18) is the simle formla [P] U ds* du S* dz dz, (19) which is a concise descrition of the east west asymmetry of diabatically forced gravity waves de to backgrond wind shear. There is an eqivalent, alternative way to arrive at (19) that draws the connection with advection. If the dominant balance in (17) is W* 5 S*(z), then (19) is simly the horizontal momentm Eq. (16). Ths, US z * and S*U z are simly the terms for advection of diabatically forced horizontal momentm by the backgrond wind U(z). While the simlified Eq. (19) aears to catre the leading-order effects of the soltions with trncated vertical strctres, one mst be carefl abot taking the limit of small U 2 in (18); this is a singlar limit, and the limiting eqations cold ths have qalitatively different soltions than the original eqation. In fact, this aears to be the case in this sitation: the eqation in (18) allows for the ossibility of critical layers when U(z) 5 0 at some height, bt the simlified formla in (19) does not. The absence of critical layer effects also seems to be a roerty of the jm conditions (12) for the vertically trncated system. This is likely de to the model s low vertical resoltion, since critical layer effects tyically occr on small vertical length scales and are associated with high-freqency gravity waves with significant vertical energy roagation (Booker and Bretherton 1967; Lindzen and Tng 1976; Lin 1987; Shige and Satomra 2001). In short, the simlified Eq. (19) and the trncated model (11) (12) effectively filter ot critical layer effects by considering only low-freqency, horizontally roagating

14 2592 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 66 FIG. 7. As in Fig. 6, bt sing initial shear similar to the environment of a midlatitde sqall line with 0 5 (left) 210, (middle) 0, and (right) 10 m s 21. waves. These are sefl aroximations for the roses of the resent aer, since the main focs is on the farfield resonse to diabatic forcing. Critical layer effects cold be stdied in the ftre sing (18) and the associated forced Taylor Goldstein eqation shown in aendix B. Diabatically forced Taylor Goldstein eqations, both nonhydrostatic and hydrostatic, were also recently derived by Majda and Xing (2009). In terms of imlications for organized convection, it was fond by Shige and Satomra (2001) that critical layers lay an imortant role by creating a wave dct, beneath which high-freqency gravity waves can roagate withot losing too mch energy to the stratoshere. This is not inconsistent with the reslts of the resent aer; here the focs is on the far-field effects of low-freqency gravity waves that are forced by the

15 SEPTEMBER 2009 S T E C H M A N N A N D M A J D A 2593 FIG. 8. Linear theory calclations of [ 1 ] for the family of wind shears with and (10 m s 1 ) 2 ; (a) [ 1 ] as a fnction of 1 ; (b) wind shear U(z), which maximizes [ 1 ]; (c) wind shear U(z) for which [ 1 ] 5 0. Plots are shown for three different vales of the stratiform heating amlitde: 2a (dashed dotted), 250 (dashed), and 275 K day 21 (solid). mesoscale filtered heating of a convective system rather than on the high-freqency waves forced by flctations of convective cells. Both high- and low-freqency gravity waves are imortant for triggering or favoring the formation of new convection, with high- (low) freqency waves robably most imortant for triggering (favoring). Lowfreqency gravity waves, nlike high-freqency waves, can roagate long distances horizontally withot losing too mch energy to the stratoshere (Shige and Satomra 2001). Several stdies have shown that the trooase itself is enogh to tra low-freqency waves in the trooshere (Maes 1993, 1998; Xe 2002; Li and Moncrieff 2004), and the se of an er rigid lid in this aer is meant to be an aroximation of this effect, bt it cold also be a stand-in for a wave dct. In smmary, both high- and low-freqency waves are imortant for convective organization, bt the main focs of the resent aer is on low-freqency waves, in which case critical layers are not an essential effect. Many other assmtions were made in the models here that shold be ket in mind when interreting the reslts, sch as hydrostatic balance, two-dimensional flow, and the simle imosed form of the convective heating. These assmtions have been commonly sed in the earlier stdies mentioned in the introdction as well; the reader is referred to these references for frther discssion. The effects of higher vertical modes are also not stdied in detail here (bt some reslts are shown in the next section). Other stdies have shown that the third baroclinic mode may lay an imortant role in convective organization (Lane and Reeder 2001; Tlich and Maes 2008). This might be exlained by convective forcing of the third baroclinic mode de to convection with lower clod-to heights. It is also ossible that the effect of the higher baroclinic modes, in the context of wave trains of convective systems, is limited by their slow roagation seeds, which confine them to the close neighborhood of the sorce. In any case, the simlified model sed here is not meant to catre every detail of this roblem; it is meant to isolate a few imortant effects and demonstrate the basic effects of wind shear and barotroic wind in a simlified setting. Indeed, one imortant strength of this model is its simlicity, which allows clear, inexensive comtation of a wide range of ossibilities as well as analytical nderstanding. This simlified model and the reslts shown here shold hel gide frther stdies that inclde additional hysical effects. 5. Sensitivity stdies with higher vertical resoltion In this section reslts are shown sing the linear theory from (12) extended to inclde the effects of the third and forth baroclinic modes. a. Sensitivity to nmber of vertical modes Earlier reslts in this aer emhasize that wind shear and/or nonlinearities lead to interactions between vertical modes. One might then exect that higher vertical modes beyond the first two modes wold be excited even if the sorce term incldes contribtions from only the first two modes. For instance, the 25 and 50 m s 21 waves might inclde contribtions from higher vertical modes, or wind shear cold case the 12 and 17 m s 21 waves to be excited in the resence of wind shear even if the sorce term incldes contribtions from only the first two modes. To check that the reslts of section 3 do not change mch when the gravity wave resonse is allowed to inclde more vertical modes, several cases are reeated here with the same wind shears and sorce terms with only two vertical modes, bt with a wave resonse that can inclde contribtions from the third and forth vertical modes. This is done by comting the jms [ j ] sing the linear theory (12) with an advection matrix A() that is and for the cases inclding

16 2594 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 66 TABLE 5. Linear theory calclations with two, three, and for baroclinic modes for the case of a jet shear with and m s 21. All vales are in K. Nm. of modes [ 1 ] [ 2 ] [ 3 ] [ 4 ] TABLE 6. As in Table 5, bt for the case of a midlatitde sqall line like shear with and ms 21. Nmber of modes [ 1 ] [ 2 ] [ 3 ] [ 4 ] resonses in modes to the third and forth, resectively. These larger advection matrices are given in aendix A. Table 5 shows a comarison of linear theory reslts with two, three, and for baroclinic modes for the westerly wind brst like wind shear with and m s 21. Althogh the jms [ j ] change deending on the nmber of modes, the changes are not drastic. Similar reslts for the wind shear that resembles conditions for midlatitde sqall lines (with and 1 523ms 21 ) are shown in Table 6. Again, althogh the jms [ j ] change deending on the nmber of modes, the changes are not drastic. These reslts sggest that the simle case with two modes rovides a reasonable aroximation to [Q](z) when the wind shear and sorce term are comosed of two modes. b. Sensitivity to jet height U to this oint, the wind shear and sorce term inclded contribtions from only the first and second vertical modes. However, the wind shear rofiles associated with sqall lines tyically have lower troosheric shear that cannot be adeqately reresented by the first two baroclinic modes (Pandya and Drran 1996; Fovell 2002; Majda and Xing 2009). In this section, backgrond wind shears will be sed with contribtions from the third and forth baroclinic modes, so wind shear rofiles with jets in the lower trooshere can be sed. The heat sorce, however, will still be the standard case from Fig. 2 with contribtions from only two vertical modes. As in the revios sbsection, the linear theory reslts (12) will se the advection matrix A() with for vertical modes, which means that the gravity wave resonse incldes contribtions from the third and forth vertical modes. The to row of Fig. 9 illstrates three families of backgrond shears. These families are the rojections onto for baroclinic modes of wind rofiles with constant shear of 2.5 m s 21 km 21 in the lower trooshere to heights of 8, 6, or 4 km. The mid- and er-troosheric winds above this height also have constant shear, and reslts are comted for a range of er-troosheric shears that incldes jet shears, midlatitde sqall line like shears, and constant shears. These shear rofiles are similar to those sed by Li and Moncrieff (2001). Whereas the to row of Fig. 9 shows U(z) inclding a barotroic comonent, the calclations of [ j ]sezero barotroic wind. The middle row of Fig. 9 shows the jms j[ j ] for these different wind shears. [The factor j is inclded in j[ j ] to give each mode s vertical strctre fnction an amlitde of 1; see the exansion in (1).] The circles reresent the vales of j[ j ] corresonding to the shear rofiles in the to row of Fig. 9. For each shear family, [ 1 ]. 0 for the jet shear, [ 1 ] 0 for the midlatitde sqall line like shear, and [ 1 ], 0 for the constant shear. These reslts are consistent with the two-mode reslts with similar shears in Fig. 4. The jms 2[ 2 ] and 4[ 4 ] are relatively small for all of these shears, and they do not vary mch for the different shears considered here. The jm 3[ 3 ] also changes little excet for the different jet rofiles; it is negative for the midtroosheric jet in Fig. 9b and ositive for the lowertroosheric jet in Fig. 9h. Althogh there are some changes as the jet height changes, the jms [Q](z) are generally determined by [ 1 ], as shown in the middle row of Fig. 9. No matter what the jet height is, it is aroximately tre that [Q](z). 0 for the jet rofiles, [Q](z) 0 for the midlatitde sqall line like rofiles, and [Q](z), 0 for the linear rofiles, which is the trend shown in [ 1 ]. 6. Conclsions The effects of wind shear and barotroic wind on gravity waves were stdied with a focs on the imlications for organized convection. It was shown that becase of advection by a backgrond wind shear or barotroic wind, eastward- and westward-roagating gravity waves can have different roerties, and this asymmetry can create differences in the environments to the east and west of a heat sorce. Therefore, becase of wind shear or barotroic wind, one side of reexisting convection can be more favorable for the formation of new convection than the other side. These effects were discssed in the context of two settings for convective organization: the formation of new convective systems near a