A New Surface Model for Cyclone Anticyclone Asymmetry

Transcription

1 VOL. 59, NO. 16 JOURNAL OF THE ATMOSPHERIC SCIENCES 15 AUGUST 2002 A New Surface Model for Cyclone Anticyclone Aymmetry GREGORY J. HAKIM Univerity of Wahington, Seattle, Wahington CHRIS SNYDER National Center for Atmopheric Reearch, Boulder, Colorado DAVID J. MURAKI Simon Fraer Univerity, Burnaby, Britih Columbia, Canada (Manucript received 19 July 2001, in final form 6 March 2002) ABSTRACT Cyclonic vortice on the tropopaue are characterized by compact tructure and larger preure, wind, and temperature perturbation when compared to broader and weaker anticyclone. Neither the origin of thee vortice nor the reaon for the preferred aymmetrie are completely undertood; quaigeotrophic dynamic, in particular, have cyclone anticyclone ymmetry. In order to explore thee and related problem, a novel mall Roby number approximation i introduced to the primitive equation applied to a imple model of the tropopaue in continuouly tratified fluid. Thi model reolve dynamic that give rie to vortical aymmetrie, while retaining both the conceptual implicity of quaigeotrophic dynamic and the computational economy of two-dimenional flow. The model contain no depth-independent (barotropic) flow, and thu may provide a ueful comparion to two-dimenional flow dominated by thi flow component. Solution for random initial condition (i.e., freely decaying turbulence) exhibit vortical aymmetrie typical of tropopaue obervation, with trong localized cyclone, and weaker diffue anticyclone. Cyclone cluter around a ditinct length cale at a given time, wherea anticyclone do not. Thee reult differ ignificantly from previou tudie of cyclone anticyclone aymmetry in the hallow-water primitive equation and the periodic balance equation. An important ource of aymmetry in the preent olution i divergent flow aociated with frontogenei and the forward cacade of tropopaue potential temperature variance. Thi thermally direct flow change the mean potential temperature of the tropopaue, electively maintain anticyclonic filament relative to cyclonic filament, and appear to promote the merger of anticyclone relative to cyclone. 1. Introduction Obervation of vortical diturbance in the extratropic reveal tructural and population aymmetrie between cyclone and anticyclone from the meocale to the planetary cale. An example of thee aymmetrie i given by meocale undulation of the tropopaue. Tropopaue vortice have typical radii of approximately 500 km, with cyclone characterized by larger preure, wind, and temperature perturbation when compared to anticyclone (Thorpe 1986; Hakim 2000; Muraki and Hakim 2001; Wirth 2001). Moreover, cyclone typically have compact tructure when compared with broader anticyclone. Neither the origin of thee vortice nor the reaon for the preferred tructural aymmetrie are Correponding author addre: Gregory J. Hakim, Dept. of Atmopheric Science, Univerity of Wahington, Box , Seattle, WA hakim@atmo.wahington.edu completely undertood. Here we explore thee problem uing a novel mall Roby number approximation to the primitive equation (PE) applied to a imple model of the tropopaue. Thi model reolve the balanced dynamic that give rie to vortical aymmetrie, uch a vortex tretching of relative vorticity and divergence vorticity feedback aociated with frontogenei. Moreover, the model retain the conceptual implicity of quaigeotrophic dynamic (QG) and the computational economy of two-dimenional dynamic. One hypothei for the origin of tropopaue vortice i derived from numerical olution howing pontaneou vortex emergence from random initial condition in unforced quai-two-dimenional flow (e.g., Mc- William 1984; McWilliam 1990a,b; Bracco et al. 2000). Studie of random initial condition are particularly ueful with regard to cyclone anticyclone aymmetry becaue they allow preferred tructure to be elected by the dynamic. An indirect uggetion of the 2002 American Meteorological Society 2405

2 2406 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 59 turbulence hypothei i contained within Sander (1988) obervational tudy on the origin of tropopaue diturbance: Evidently, the organization and growth of the ytem out of the mall-cale chao of the vorticity field i the mot important proce. Although purely two-dimenional (barotropic vorticity) dynamic may lend upport to a turbulent-cacade hypothei for the origin of vortice, thee dynamic do not reolve cyclone anticyclone aymmetry. The implet three-dimenional repreentation of the tropopaue in continuouly tratified fluid conit of a quai-horizontal interface eparating region of homogeneou potential vorticity (PV) of differing magnitude; mall (large) value of PV are located on the tropopheric (tratopheric) ide of the interface (Rivet et al. 1992; Jucke 1994; Muraki and Hakim 2001). The QG approximation to thi configuration reult in a profound implification that parallel the dynamic of the barotropic vorticity equation: the three dimenional flow i modeled entirely by horizontal advection of potential temperature on the interface (Blumen 1978; Jucke 1994; Held et al. 1995). Following Held et al. (1995), we hall refer to thi approximation a urface quaigeotrophy. An intereting attribute of urface quaigeotrophy i the abence of depth-independent (barotropic) flow; a uch, it provide a ueful comparion to two-dimenional flow dominated by thi flow component. For completene, we alo note that becaue the urface quaigeotrophic (QG) aumption require zero PV gradient away from the interface, Roby wave propagation i confined to the interface, and baroclinic intability i excluded. Although QG dynamic reolve the emergence of vortice out of random initial condition, they fail to produce cyclone anticyclone aymmetry. Here, we allow for cyclone anticyclone aymmetry by extending QG by one order in Roby number ( QG 1 )ain Muraki et al. (1999), which include additional dynamic uch a ageotrophic advection, tretching and tilting of relative vorticity, and gradient-wind effect (centripetal acceleration). Although the model i applied here to pecific problem related to the tropopaue, we emphaize that it applie to a more general cla of problem characterized by balanced dynamic and uniform potential vorticity. Cyclone anticyclone aymmetry and vortex emergence from random initial condition have alo been explored numerically for other dynamical ytem more complete than QG. For example, Cuhman-Roiin and Tang (1990) oberved a trong bia for anticyclone in a generalized geotrophic model. A ubequent tudy by Polvani et al. (1994) uing the hallow-water PE alo found that anticyclone were the preferred vortical tructure. A imilar reult ha been found for three-dimenionally periodic olution of the balance equation (Yavneh et al. 1997). Thee reult tand in contrat to the oberved preference for tropopaue cyclone, and motivate the preent invetigation into the importance of PV dynamic concentrated at an interface a compared to deep ditribution over a layer. A repreentative ample of the main reult to be documented in thi paper i given in Fig. 1. Becaue the olution decay away from the interface, tropopaue dynamic (downward decay) may alo be modeled a urface dynamic (upward decay) with the only difference being that cyclone are aociated with cold (warm) air on the tropopaue (urface). Here and for the remainder of the paper we move from the tropopaue to the urface. A random ditribution of urface potential temperature (Fig. 1a) rapidly evolve into a field of coherent vortice and a region between the vortice filled by incoherent warm and cold filament (Fig. 1b d). By t 1000 (nondimenional time unit), the QG 1 olution exhibit ditinct cyclone anticyclone aymmetrie in both vortex population number and tructure that are abent in the QG olution (cf. Fig. 1c,d). In term of population number, cyclone (warm pot) appear to cluter around a ditinct length cale, wherea anticyclone (cold pot) do not. In term of vortex tructure, cyclone poe a plateau tructure with harply defined edge, wherea anticyclone poe broadly ditributed tructure with poorly defined edge (Fig. 1c). Another triking property of the QG 1 olution i the blue background, which indicate a mean cooling of the urface ha occurred (Fig. 1b,c); the cooling rate i greatet at early time, and then decay lowly with time (Fig. 2a). In the QG 1 potential temperature probability denity function (PDF), urface cooling appear a a hifted peak with a bia toward cold value for mall potential temperature, relative to the QG PDF (Fig. 2b). A we will how, urface cooling i due to divergence vorticity feedback aociated with frontogenei, uch that over time, warm air rie and cold air ink. Thi realitic feedback i alo crucial to the vortex aymmetrie, and i abent in two-dimenional and hallow-water dynamic. The remainder of the paper i organized a follow. Section 2 provide a brief background dicuion of quai-two-dimenional turbulence relevant to the preent work. Section 3 i devoted to defining the new urface model, QG 1, along with a decription of the numerical algorithm. Novel apect of the olution hown in Fig. 1 urface cooling and cyclone anticyclone aymmetrie are documented in ection 4 and 5, repectively. A ummary i given in ection 6, along with everal hypothee to be teted in future reearch. 2. Background on quai-two-dimenional turbulence Thi ection i dedicated to providing a brief background review of certain apect of quai-two-dimenional turbulence a they apply to the preent work. Turbulent flow are dominated by nonlinear (inertia) dynamic, ignificant vorticity, and rapid mixing of paive tracer (e.g., Salmon 1998, ection 4.5). An im-

3 15 AUGUST 2002 HAKIM ET AL FIG. 1. Surface potential temperature evolution for numerical olution of QG 1 [panel (b) and (c)] and QG [panel (d)] freely decaying turbulence. The random initial condition hown in (a) evolve into a field of vortice and filament by t 200 (b). By t 1000, cyclone anticyclone aymmetrie and urface cooling are prominent in the QG 1 olution (c) and are abent in the QG olution (d). All unit are nondimenional, and the Roby number ( ) i 0.1 for the QG 1 olution. portant property of nonlinear dynamic i a propenity to pectrally catter fluid propertie, uch a energy and entrophy, in preferred direction (e.g., to larger or maller cale). At length cale well removed from ignificant forcing and diipation, thi nonlinear cattering may take the form of tatitically teady cacade. In three-dimenional turbulence, for example, energy cacade toward horter length cale on the way to molecular diipation. Two-dimenional turbulence differ from three-dimenional turbulence becaue it conerve entrophy (for weak diipation) and thu upport two cacade: an upcale cacade of energy, and a downcale cacade of entrophy. A kinematic explanation for the downcale cacade i formed around the fact that the area contained within cloed contour of vorticity i conerved, o that vorticity gradient increae a patche of fluid inevitably ditort from axiymmetry. For tatitically teady cacade, the energy pectrum ha a k 3 power law for the downcale entrophy cacade, and a k 5/3 power law for

4 2408 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 59 FIG. 3. Enemble-mean urface potential temperature variance pectra. The QG curve (dahed line) ha been diplaced from the QG 1 curve (olid line) for clarity. Thin vertical line how 1 enemble tandard deviation around the enemble mean, and line with lope of 5/3 and 1 are provided for reference. The pectral peak in the initial condition i denoted by the bold arrow. FIG. 2. (a) Mean urface potential temperature a a function of nondimenional time. Dahed line how 1 enemble tandard deviation about the enemble mean (bold line). (b) Potential temperature PDF at t 1000 for QG 1 (thick line) and QG (thin line). Vertical line how 1 enemble tandard deviation around the enemble mean. the upcale energy cacade, where k i the total wavenumber (e.g., Salmon 1998, ection 4.9). Charney (1971) aert that the two-dimenional inertial-range prediction alo apply to unbounded threedimenional QG dynamic when entrophy i replaced by potential entrophy; that i, PV cacade to mall cale. The effect of rigid horizontal boundarie in the QG ytem ha been tudied by Blumen (1978), Jucke (1994), and Held et al. (1995) for uniform PV above a flat boundary with nonuniform potential temperature ditribution (i.e., the QG ytem). If diipation i weak, boundary potential temperature variance cacade downcale with a k 5/3 pectrum for equilibrium condition. Thi cacade i analogou to the entrophy cacade of two-dimenional turbulence, and i characterized by increaing potential temperature gradient a material line are deformed and elongated. The upcale energy cacade ha a k 1 pectrum in potential temperature variance. Figure 3 how the enemble-mean potential temperature variance pectrum for QG and QG 1 olution at t The pectra do not match the k 1 and k 5/3 power law becaue thee freely decaying (unforced) imulation do not achieve tatitical equilibrium. Furthermore, we note that teeper pectral lope in the forward cacade of turbulent flow are alo aociated with the emergence of highly organized and coherent vortical tructure that dirupt the cacade (e.g., Lumley 1990). In quai-two-dimenional (i.e., two-dimenional QG and QG) freely decaying turbulence, thee vortice emerge pontaneouly from random initial condition, and repreent patche of organized vorticity within a field of diorganized, thin, filament of vorticity (e.g., McWilliam 1984). Vortice of like ign typically merge, producing larger vortice that then undergo an axiymmetrization proce where filament are hed and diipated outide a peritent axiymmetric vortex core. The generation and diipation of filament repreent the forward entrophy cacade to diipation cale, wherea the cale increae of the vortice i a manifetation of the upcale energy cacade. Vortex emergence and evolution propertie for QG turbulence are qualitatively imilar to thoe for twodimenional turbulence, although they have not been tudied a extenively (Held et al. 1995). One important ditinction i that QG filament tend toward intability a they are thinned (Jucke 1995), which i not oberved to occur in two-dimenional turbulence. Held et al. (1995) note that the intability growth rate i et by the magnitude of the filament vorticity, which remain contant for two-dimenional filament and increae without bound for thinning potential temperature filament.

5 15 AUGUST 2002 HAKIM ET AL An additional ditinction i that the inverion from potential temperature to treamfunction i more local than the inverion from vorticity to treamfunction, o that vortex interaction decreae more rapidly with ditance for QG dynamic. Changing the ign of the initial condition in twodimenional, QG, and QG turbulence imply change the ign of the olution. Thi property reult in vortice and filament that are ymmetric; there i no dynamical ditinction between cyclone and anticyclone. Thi ymmetry i broken by ytem that retain dynamically active divergent flow or PV inverion more accurate than QG, uch a the hallow-water PE and the balance equation. Specifically, Polvani et al. (1994) find that anticyclone are preferred over cyclone in hallow-water flow a the Froude number (ratio of fluid peed to gravity-wave peed) increae. Yavneh et al. (1997) how that anticyclone alo dominate in three-dimenional periodic f-plane imulation of the balance equation. In particular, anticyclone develop rapidly to larger horizontal cale, a in hallow water, which allow them to interact over deeper layer and vertically align into columnar vortice fater than cyclone. The trong bia for anticyclone in the hallow-water PE and periodic balance equation tand in contrat to the oberved cyclonic bia at the tropopaue. In order to reolve thi dicrepancy, a new model that extend QG i propoed and applied to an idealized repreentation of the tropopaue. The eential novel addition to QG dynamic are aymptotically conitent nextorder rotational and divergent wind. We anticipate that the cacade of potential temperature variance to mall cale will provoke a divergent repone that alo affect the rotational flow. 3. A new urface model: QG 1 We begin with a decription of a new urface model that i a balance approximation of the PE, followed by a decription of the numerical procedure and a dicuion on diipation. The tropopaue i approximated here a an interface eparating region of homogeneou PV of differing value (Rivet et al. 1992; Jucke 1994; Muraki and Hakim 2001). Although in general the interface poition i a function of pace and time, for implicity we hall take the interface to be a rigid urface (equivalent to auming infinite PV on one ide of the interface). Moreover, we invert the tropopaue uch that diturbance decay upward rather than downward from the the rigid interface; thi choice merely revere the ign of warm and cold anomalie. We are intereted in balanced motion upported by the Bouineq, hydrotatic, f-plane, PE in the aumed geometry. Balanced motion are defined here a thoe that atify both balanced dynamic and a balance condition, which render the dynamic free of gravity wave (Warn et al. 1995; Valli 1996). Balanced dynamic may be repreented by the material conervation of Ertel PV q in the interior (z 0), Dq q D q, (1) Dt and potential temperature on the rigid boundary (z 0), D Dt D. (2) The perturbation PV i defined in term of the primitive variable (u,, ) by q ( x uy z) [( u ) u ], (3) x y z z x z y and i a perturbation from a contant tratification reference tate. Subcript denote partial derivative, upercript indicate a urface value (z 0), D q and D are diipation operator, and D u w (4) Dt t x y z i the material derivative for wind vector V (u,, w). The vertical coordinate z i peudoheight (Hokin and Bretherton 1972), and i the Roby number. All variable have been nondimenionalized a in Pedloky (1987, chapter 6), with the exception that w i caled by an additional factor of. For homogeneou q, which we take to be zero, the interior equation (1) i atified exactly. Thi condition reduce the dynamic to (2) with w 0, that i, horizontal advection of urface potential temperature. Depite a reduction to urface advection, the general problem i till three-dimenional, becaue the urface wind (u, ) are determined by a three-dimenional PV inverion. Here we hall aume mall and treat the balance condition aymptotically following the procedure outlined in Muraki et al. (1999). The advecting wind are then given aymptotically by (u, ) (u u, ) O( ). (5) An overview of the principal reult of the inverion follow, with further detail provided in appendix A. The leading-order balance condition yield a tandard QG PV inverion for the leading-order geopotential 0 : q xx yy zz 0; 0 z (z 0). (6) Taking a Fourier tranform and chooing decay in the upward direction, a olution i ˆ (k, l) 0 K z ˆ (k, l) e, (7) K where hat denote pectral variable, k and l are x and

6 2410 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 59 y wavenumber, and K (k 2 l 2 ) 1/2. Leading-order (QG) wind are then determined by an invere Fourier tranform of pectral wind given by (û 0, ˆ 0 ) ( ik ˆ 0, il ˆ 0 ). The QG ytem i defined by (7) and (2), when the advecting wind (u, ) i approximated by (u 0, 0 ) (Held et al. 1995; Jucke 1994). A noteworthy attribute of thi ytem i the dimenional reduction that allow the olution of a three-dimenional flow by a two-dimenional computation. The next-order balance condition yield corrected wind (u 0 u 1, 0 1 ) that, together with (2), define the QG 1 ytem. For the preent dicuion, two important apect of the next-order calculation deerve mention here. Firt, the correction have the remarkable property that, depite the three-dimenional nature of the inverion problem, (u 1, 1 ) are determined entirely from a urface calculation baed on ; thi again reduce the full three-dimenional calculation to a urface problem. Second, although the leading-order wind are trictly nondivergent, the correction poe both divergence and vorticity. The nondivergent correction include thoe due to (approximate) gradientwind balance, while the irrotational correction are econdary circulation and vertical motion that are driven by frontogenei in the leading-order flow. a. Numeric The numerical olution method conit of two tep, inverion and advection. Inverion involve recovering u u 0 u 1 and 0 1 from, a decribed above. Thee approximate wind are then ued to advect in (2). We repreent the diipation operator by eighth-order horizontal hyperdiffuion (e.g., Mc- William 1984): q 8 D D H, (8) 2 where H xx yy. A dicued further in the ubequent ection, thi repreentation for diipation ha little connection to real phyic in the PE, and i ued in computation to prevent the accumulation of energy at the mallet cale. Uing the diipation operator pecified in (8), a Fourier tranform, F, of (2) give the pectral form of the governing equation that i olved numerically: [ ] ˆ F u (k l ) ˆ. (9) t x y In the numerical model, nonlinear term in (9) are quadratically dealiaed to a reolution of 512 by 512 horizontal wavenumber, the Roby number i 0.1, and hyperdiffuion i handled explicitly with Temporal dicretization i by a third-order accurate Adam Bahforth cheme (Durran 1999). An enemble of 25 olution i employed for both QG and QG 1 computation to enure tatitically robut reult. Initial condition are contructed by firt pecifying a ˆ 0 field, and then olving for next-order correction. The ˆ 0 field i pecified with random phae angle and amplitude given by (Polvani et al. 1994): K m/4 1 0, ˆ (k, l), (10) ( K k ) m/2 0 on a domain taken to be 2 k 0 quare. Thi definition for ˆ 0 give a kinetic energy pectrum for urface flow peaked at wavenumber k 0. To maintain the kinetic energy pectrum at next order, the inverion at next order mut have no wind correction. Therefore, the next-order inverion are performed with boundary condition that provide (u 1, 1 ) 0, which force the correction into the field. The initial field i then normalized to have unit total kinetic energy and zero mean potential temperature at z 0. Tet olution for initial condition in which the correction are choen to be zero and the wind are corrected how no qualitative difference from the control olution. Following Polvani et al. (1994), we et k 0 14 and m 25, which give initial condition uch a the one hown in Fig. 1a. b. Role of diipation Some form of diipation i required in the numerical experiment to control the buildup of grid-cale feature aociated with the forward cacade of variance. Our ue of hyperdiffuion to diipate both and q follow common practice in tudie of quai-two-dimenional turbulence. The fact that q remain contant with uch diipation i particularly ueful here [and in QG tudie uch a Held et al. (1995)], ince the calculation become effectively two-dimenional, and thu computationally tractable, only when q i contant. However, for the PE, which QG 1 approximate, mot obviou form of momentum and potential temperature diipation reult in nonzero time tendency of q even for q contant; that i, an initially uniform field of q will develop patial variation a a conequence of the diipation of momentum and potential temperature (e.g., Herring et al. 1994). Thi effect can be ignored at the level of QG theory 1 but enter at next order in. Our olution procedure, which depend on the aumption of contant q, neglect thi diipation contribution to the time tendency of q. The importance of thi neglected PV to the long-term behavior of the flow i an intereting quetion that deerve further invetigation. Conidering the approximation required to reduce the PE to other quai-twodimenional ytem (e.g., the hallow-water equation), we upect that neglecting ome of the diipative effect on PV i a comparatively minor implification. Tet olution for change in the hyperdiffuion coefficient 1 The QG PV i linear, and if heat and momentum are diipated by diffuion (or hyperdiffuion), then QG PV i alo diffued.

7 15 AUGUST 2002 HAKIM ET AL by an order of magnitude howed no qualitative change to our primary concluion. Moreover, the key mechanim that generate aymmetrie rely on advection by the divergent flow component, which operate even in the abence of diipation. 4. Surface cooling Surface cooling 2 i one of the novel feature of QG 1 turbulence hown in Fig. 1 and 2. Thi ection further document urface cooling and provide a imple explanation of why thi phenomenon occur. A ueful tarting point i the equation governing the time tendency of urface-mean potential temperature, which i obtained by area averaging (2) with (8) over the entire urface: t [ H V]. (11) Note that the evolution of ha no explicit dependence on hyperdiffuion, and i conerved by the nondivergent leading-order dynamic. However, the next-order dynamic upport divergence, and net urface cooling occur when and divergence are anticorrelated uch that, on average, warm region contract relative to cold region. Since urface divergence (convergence) i aociated with inking (riing) air jut above the urface, we conclude that urface cooling i related to riing warm air and inking cold air (thermally direct circulation). Converely, urface warming i aociated with thermally indirect circulation, and appear to occur in vortex core (compare the QG 1 and QG PDF for large in Fig. 2b). In our balanced model, thee circulation are due to frontogenei, which i defined a an increae in the magnitude of the horizontal potential temperature gradient following fluid particle. To undertand why thee circulation have a preferred direction, recall from ection 2 that the deformation of potential temperature contour by the leading-order velocity field lead to an increae in potential temperature gradient; equivalently, the turbulence cacade potential temperature variance from large to mall cale. To maintain thermal wind balance, frontogenei require an increae in the vertical hear and therefore the kinetic energy. Since total energy i conerved, thi increae in kinetic energy i balanced by a decreae in potential energy through warm air riing and cold air inking. Although thi proce occur at leading order, the divergent flow i diagnotic and doe not advect; therefore urface cooling cannot occur for QG. At next order, QG 1, divergent flow i dynamically active and may change the area contained within potential temperature 2 Note that the cooling decribed here applie to the area-mean potential temperature, not individual air parcel, and i a reult of adiabatic motion. FIG. 4. Average urface divergence (olid line) and deformation (dahed line) at t 1000 a a function of perturbation urface potential temperature (upper panel). Thee enemble-mean curve repreent average over grid point for within.05 of the central value given on the abcia. The enemble-mean PDF for at t 1000 i hown in the lower panel. Thin olid vertical line how 1 enemble tandard deviation around the enemble mean, and the thin dahed vertical line denote the peak of the PDF; note that thi i a kewed ditribution. contour. Note that thi proce i conitent with the conervation of potential temperature following fluid particle becaue the increae in area aociated with urface cooling i aociated with particle that ink onto the urface. We proceed to tet the hypothei that the main ource of frontogenei, and therefore urface cooling, occur in the filament field. To tet thi hypothei for urface cooling, we conduct an analyi of kinematic quantitie aociated with urface frontogenei: divergence, u x y, and leading-order deformation magnitude [( ) uy x ( ux y) 2 ] 1/2.Aan aid in interpreting Fig. 4, we note that mot vortice are characterized by relatively large value of, while mot filament are found where i relatively mall (cf. Fig. 1); hereafter, we take t 2to repreent the vortex field. The larget contribution to the enemble-mean total deformation magnitude occur in the filament field near the PDF peak of. A divergence dipole flank the ditribution peak, with divergence (convergence) found in relatively cold (warm) air. We conclude that an aymmetry exit in the filament field due to divergent circulation aociated with frontogenei uch that, on average, warm air rie away from the urface and cold air ink toward the urface. Further evidence of thi filament aymmetry i apparent in PDF of and vorticity at t The enemble-mean PDF how a cold bia in the filament field and a warm bia in the vortex field (Fig. 5a). The QG 1 vorticity PDF (Fig. 5b) how that large value of cyclonic (anticyclonic) vorticity occur more (le) frequently than in QG, a one might expect baed on tretching of relative vorticity. Joint PDF of vorticity and how that the large value of cyclonic

8 2412 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 59 FIG. 5. Enemble-mean PDF for (a) urface potential temperature, and (b) vorticity and joint PDF of urface potential temperature, and vorticity for (c) QG 1 and (d) QG. In (a) and (b), value for 0 have been folded acro the origin, with poitive value given by thick olid line and negative value given by thick dahed line; the ymmetric QG curve i given by a thin olid line. Thin vertical line how 1 enemble tandard deviation around the enemble mean. Contour in (c) and (d) are given in power of 10, with an innermot contour of 0.1. vorticity occur in the vortex field and that the anticyclonic bia at mall value of vorticity occur in the filament field (Fig. 5c,d). Thee reult demontrate the exitence of a pronounced cold anticyclonic bia in the QG 1 filament field; thi bia ha no counterpart in the ymmetric filament field of two-dimenional, QG, and QG turbulence. The key element that wa previouly miing, but that i included in QG 1, i advection by the divergent flow. Thinning warm filament are accompanied by riing motion and convergence along the filament, which accelerate the thinning of the filament, while the ituation i revered for cold filament where divergence along the filament low the thinning. 3 A a reult of thi aymmetry, warm filament reach mall cale ooner and are diipated more rapidly, o that the oberved filament are predominantly cold. We hypotheize that thi baic aymmetry alo play an important role in determining vortex tructure and population aymmetrie, which are documented in the following ection. 3 We note that becaue the divergent flow i O( ) it cannot overwhelm the O(1) train cold filament do not expand. A imilar reult can be derived from emigeotrophic olution of two-dimenional filament in deformation (Davie and Müller 1988).

9 15 AUGUST 2002 HAKIM ET AL FIG. 6. Vortex cenu data at t 1000 howing (a), (b) enemble-mean vortex amplitude and (c), (d) radiu; note that vortex amplitude i defined relative to. The QG 1 olution are given in (a) and (c), and the QG olution are given in (b) and (d). Thin vertical line how 1 enemble tandard deviation around the enemble mean, which i given by the bold olid line. 5. Cyclone anticyclone aymmetrie Figure 1b and 1c exhibit aymmetrie between cyclonic and anticyclonic vortice in term of both population number and tructure. Cyclone appear to cluter around a ditinct ize at thi time, wherea anticyclone do not, and cyclone tend to have a plateau-like tructure, wherea anticyclone tend to have a broad, prawling, tructure. Thi ection i devoted to quantifying thee qualitative impreion. Vortice are identified by a cenu algorithm that i decribed in appendix B. For each vortex, the algorithm objectively determine vortex amplitude ( extremum), and major- and minor-axi length cale. Hereafter, radiu refer to the vortex major-axi length cale. Vortex population tatitic at t 1000 are dicued firt, followed by an analyi of the tatitic over time, and finally an analyi of vortex tructure. a. Vortex cenu Hitogram of QG 1 enemble-mean vortex amplitude how that cyclone tend to be tronger than anticyclone, with a mean of 3.27 a compared to 1.5 for anticyclone (Fig. 6a). Anticyclone exhibit a ingle peak near 1, wherea cyclone exhibit a bimodal ditribution with a primary peak near 4. Both the anticyclone peak and the econdary cyclone peak near 1 are due to thin-filament intability. We upect that the obervation of fewer cyclone relative to anticyclone for mall cale i due

10 2414 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 59 FIG. 8. Vortex number a a function of time. The thin dahed line repreent QG vortice, the thick olid line repreent QG 1 cyclone, and the thick dahed line repreent QG 1 anticyclone. A olid line with lope 0.55 i provided for reference. FIG. 7. Vortex radiu a a function of amplitude for (a) QG 1 and (b) QG olution at t Each vortex i repreented by a dot, and the olid line are contant-denity contour (0.005, 0.01, 0.02, 0.03) of the joint PDF. to the horter lifepan of warm filament in QG 1 train. The ymmetric QG amplitude hitogram alo exhibit two peak for both cyclone and anticyclone, and both ditribution are dominated by vortice with mall value of (Fig. 6b). The QG 1 anticyclone exhibit no preferred length cale, but rather how an exponential decreae in number away from a peak at mall cale; the tail of the ditribution reache to value more than twice a large a for cyclone (Fig. 6c). Alo, the QG 1 cyclone exhibit two peak, with one near r 3 and another near r 7. The ymmetric QG vortex-radiu plot i harply peaked at mall cale with a minor econdary peak near r 13. The primary peak reflect vortex generation at mall cale due to filamentary intability, wherea the econdary peak reflect the upcale cacade due to vortex merger. Plot of QG 1 vortex radiu a a function of amplitude how that the anticyclonic vortex ditribution peak at mall amplitude and mall radii with a long tail to large radii at modetly larger amplitude (Fig. 7a). In contrat, cyclone tend to cluter around ( 4, r 7) with very few cyclone at larger radii. Fewer cyclone occur near the econd peak at mall amplitude and radiu relative to the anticyclone ditribution. The aymmetric QG 1 vortex amplitude-radiu relationhip i teeper (hallower) for anticyclone (cyclone) when compared to the QG relationhip (Fig. 7b). QG vortice cacade to large cale along an approximately parabolic amplitude radiu relationhip, which mut aymptote to the extrema in the initial condition, becaue i conerved following fluid particle. Thee intantaneou reult ugget that QG 1 anticyclone cacade rapidly to larger cale, wherea QG 1 cyclone appear to cluter around a ditinct cale. Further evidence for thi tendency i given in Fig. 8, which how the total number of cyclone and anticyclone a a function of time. Relative to the QG curve, the number of QG 1 cyclone (anticyclone) decreae le (more) with time. Although filamentary intability alo contribute to variation in the number of vortice, the aymmetrie hown in Fig. 6 8 ugget a difference in how cyclone and anticyclone interact and merge. The ummary ection provide additional analyi of thi vortex merger aymmetry, and a hypotheized frontogenei mechanim to explain it. Figure 8 alo how that the total number of QG vortice approximate a t 0.55 power law. Similar law have been oberved for two-dimenional (McWilliam

11 15 AUGUST 2002 HAKIM ET AL FIG. 9.Thet 1000 enemble-mean PDF for magnitude of the horizontal gradient of urface potential temperature, conditioned on 1. Cyclonic QG 1 (poitive) value are given by thick olid line, and anticyclonic QG 1 (negative) value are given by thick dahed line; the ymmetric QG curve i given by a thin olid line. Short vertical line how 1 enemble tandard deviation around the enemble mean. 1990b; Bracco et al. 2000) and three-dimenional-periodic QG turbulence (McWilliam 1990a; McWilliam et al. 1999), but with teeper power law of t 0.72 and t 1.25, repectively. A lower decreae in vortex number for QG dynamic relative to two-dimenional dynamic i conitent with the obervation that two-dimenional vortice have a larger range of influence, which may lead to interaction and merger with other vortice, when compared to QG vortice. The ource of thi larger influence may be attributed to the lower patial decay of the ln(r) two-dimenional Green function, when compared to r 1 for the QG Green function (Held et al. 1995). Becaue the r 1 Green function alo applie to three-dimenional QG dynamic, the ource of the difference may be more cloely related to urface-baed dynamic a compared to deep, barotropic dynamic. Concluion drawn from Fig. 8 mut again be tempered by the caveat that vortex number i changed by filamentary intability, which i largely abent from trictly two-dimenional flow. FIG. 10. Vertical component of relative vorticity at t 1000 for (a) QG 1 and (b) QG olution. The vorticity hown in (b) i leading-order only, wherea in (a) the vorticity i correct to O( ). Here, (a) correpond to the upper left quadrant of Fig. 1c, and (b) correpond to the lower left quadrant of Fig. 1d. b. Vortex tructure A tructural aymmetry apparent in Fig. 1c i the tendency for compact cyclone with harp edge and broad anticyclone with diffue edge. An objective meaure of thi aymmetry i provided by PDF of cyclonic and anticyclonic H conditioned on 1 (Fig. 9). The QG 1 plot how that tronger gradient are found in warm air (cyclone), with gradient in cold air (anticyclone) comparable to the QG reult. A plot of QG 1 vorticity correponding to the upperleft quadrant of Fig. 1c how that, a expected, there i coniderably more mall-cale tructure in the vorticity field when compared to the field (Fig. 10a). The vortex field how larger vorticity value near cyclone compared to anticyclone. Both cyclone and anticyclone exhibit annular-like vorticity tructure in the vortex core, and a tendency toward compenation, with a ring of oppoite-ign vorticity urrounding the core; thee propertie are mot noticeable for QG 1 cyclone. Another intereting property i the tendency for mall anticyclone to become trapped a atellite with nearly contant orbit around cyclone [ee, e.g., (x, y) ( 40,

12 2416 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 59 10) in Fig. 10a]. Such configuration are oberved to lat for hundred of time unit (not hown), and tand in contrat to the two-dimenional tendency for irreverible deformation of vortice of unequal amplitude in cloe proximity 4 (e.g., Guinn and Schubert 1993). The abence of tretching of relative vorticity for QG dynamic i apparent in the QG vorticity plot, which how notably maller magnitude vorticity relative to QG 1 (Fig. 10b; cf. Fig. 5b). The QG vortice are alo compenated, but with le clearly defined ring of vorticity when compared to QG 1 cyclone becaue the edge of QG cyclone are not a harply defined. Moreover, mot QG vortice are nearly axiymmetric, wherea many QG 1 cyclone appear to be elliptical. Lat, we note that compoite average of cyclone and anticyclone were not helpful in quantifying tructural aymmetrie becaue averaging tend to mooth out the harp tructure that vary in location from vortex to vortex. 6. Summary and hypothee We introduce a new model, QG 1, which i devied to tudy the dynamic of continuouly tratified fluid characterized by balanced dynamic, uniform potential vorticity, and a rigid boundary. The model build aymptotically upon urface QG dynamic (e.g., Held et al. 1995) by including next-order correction to the leading-order nondivergent velocity field (Muraki et al. 1999). Thi approach retain conceptually ueful QG concept while alo allowing the olution of three-dimenional dynamic with two-dimenional computational effort. We have applied the QG 1 model to the problem of cyclone anticyclone aymmetry, in order to invetigate the oberved bia for tropopaue cyclone. We ue unbiaed random initial condition, which allow the dynamic to elect aymmetrie; thi i the claic initial-value problem known a freely decaying turbulence. Vortice emerge from the turbulence a in two-dimenional imulation; however, there i a ditinct dynamical aymmetry favoring trong, compact cyclone and weak, broad anticyclone. Cyclone ize cluter around a ditinct length cale at a given time, wherea anticyclone do not, and reach cale much larger than cyclone. The reult ugget that, a urface potential temperature cacade to mall cale, frontogenei produce divergence that i the ource for everal aymmetrie. A Fig. 11 illutrate, a baic aymmetry occur at the level of potential temperature filament, where divergence haten the contraction of warm filament and low the contraction of cold filament. An example of thi aymmetry i given in Fig. 12a and 12b, which how that warm filament are aociated with convergence and cold filament are aociated with divergence. 4 Vortice of oppoite ign and comparable amplitude often merge to form vortex dipole. FIG. 11. Schematic illutration of the divergent-flow repone to (a) cold and (b) warm filament in deformation. Black arrow how the ambient deformation that act to thin the filament (thick gray line) and provoke the divergent repone in the vertical plane normal to the filament (arrow head and arrow tail). The net affect of the divergent motion, given by the gray arrow, i to accelerate the contraction of warm filament relative to cold filament. Thi aymmetry in the divergent flow produce urface cooling a cold air ink and warm air rie in thermally direct circulation; the ene of thee circulation, and of the aymmetry, i determined by the fact that turbulent flow tretch material line and thu increae H (frontogenei). A fundamental reult of QG 1 frontogenetical circulation i a reduced center of ma in the fluid. Thi realitic effect i not captured by the QG and hallow-water equation. We hypotheize that thi divergent-flow aymmetry i the ource of vortex population and tructural aymmetrie. For example, a illutrated in Fig. 12c, when cyclonic vortice approach one another and are cloe to merging, the relatively cold patch of fluid that i pinched between them produce divergence, which oppoe merger. Jut the revere happen for merging anticyclone, which pinch a warm patch of fluid that converge, favoring merger (Fig. 12d). Thi merger aymmetry encourage anticyclone to build to larger cale, wherea cyclone are dicouraged and tend to accumulate around a particular length cale at a given time. A theory that can predict the preferred cyclone cale a a function of time for a given Roby number and initial condition remain an open and intereting problem for future work. We further hypotheize that additional tructural aymmetrie are due to at leat three effect. Firt, urface cooling and conervation of in the vortex core imply that cyclone trengthen relative to anticyclone when their amplitude i meaured againt the mean ( ). Second, vortice peritently weep cold filament near their periphery, which lead to broad anticyclone and compact plateau-like cyclone. Third,

13 15 AUGUST 2002 HAKIM ET AL FIG. 12. Surface potential temperature (color) and divergence (black line) for example of (a) warm filament, (b) cold filament, (c) merging cyclone, and (d) merging anticyclone. Solid line how divergence, and dahed line how convergence. thee tructural aymmetrie may perit when perturbed, becaue they imply that vortex Roby wave (e.g., Guinn and Schubert 1993; Montgomery and Kallenbach 1997) are trapped at the edge of cyclone, which give cyclone an elliptical appearance a thee wave propagate around the vortex. In contrat, broad anticyclone upport radial propagation and, ultimately, thee wave break at the edge of the vortex, further contributing to broad tructure. Future work i needed to addre thee hypothee, and alo to link thi work with earlier tudie howing anticyclone dominance in flow trongly influenced by barotropic dynamic; QG 1 dynamic repreent a baroclinic two-dimenional limit in that they contain no barotropic (depth independent) velocity component. For example, Polvani et al. (1994) find that anticyclone are preferred over cyclone in hallow-water flow a the Froude number (ratio of fluid peed to gravity-wave

14 2418 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 59 peed) increae. They attribute thi aymmetry to the maller (larger) Roby deformation radiu for cyclone (anticyclone), which allow anticyclone to interact/ merge more readily than cyclone. Arai and Yamagata (1994) how that iolated elliptical cyclone tend to eject more filamentary material during axiymmetrization when compared to anticyclone, and elliptical cyclone are alo more prone to plitting. The anticyclone bia in hallow-water flow i trengthened on the plane due to a diperion nonlinearity balance that i not attainable to cyclone (e.g., William 1996). Yavneh et al. (1997) how that anticyclone alo dominate in threedimenional periodic f plane imulation of the balance equation. In particular, anticyclone develop rapidly to larger horizontal cale (a in hallow water), which allow them to interact over deeper layer and collect into columnar vortice fater than cyclone. A natural extenion of the preent work that link with tudie of barotropically dominated flow involve the introduction of a econd horizontal boundary; thi addition allow barotropic flow to develop naturally, and promote columnar organization on the larget cale. Becaue anticyclone build to larger cale fater than cyclone, we anticipate that anticyclone will be firt to engage a partner on the oppoing boundary, producing a barotropic tructure; the vortex horizontal length cale for which thi proce begin i expected to depend on boundary eparation. In the limit of infinite boundary eparation, QG 1 dynamic are recovered. For finite boundary eparation larger cale will tend toward barotropy; however, there will alway be cale hort enough uch that the oppoing boundary i never felt and QG 1 dynamic prevail. Exploration of thee and imilar problem hould promote deeper undertanding of the variou regime of cyclone anticyclone aymmetry, and the role of barotropic motion in continuouly tratified fluid. Acknowledgment. GJH acknowledge upport through NSF Grant ATM DJM acknowledge upport through NSF DMR , DOE DE-FG02-88ER25053, the Alfred P. Sloan Foundation, and the NCAR Viiting Scientit program through the MMM Diviion. We thank Martin Jucke and an anonymou reviewer for comment that helped clarify portion of the manucript. APPENDIX A Next-Order PV Inverion Becaue the general QG 1 model i thoroughly dicued in Muraki et al. (1999), only the key reult and implification relevant to the QG 1 PV inverion are ummarized here. The QG 1 approach replace primitive variable (u,, ) with potential (, F, G), where G, u F, x z y z G F, z x y and the continuity equation implie that (A1) w Fx G y. (A2) Next-order correction in Roby number are incorporated by expanding the potential a the perturbation erie , F F, G G. (A3) For balanced tate, thee potential are obtained by the olution of three-dimenional Poion equation, ubject to appropriate boundary condition: , z ; F 2J( z, x), 1 F 0; G 2J( z, y), 1 G 0; z, z ; (A4) where uniform (zero) PV ha been aumed, and 2 xx yy zz. The firt equation in (A4) ha olution (7), which define wind for QG dynamic. Although the correction potential ( 1, F 1, G 1 ) are determined by Poion inverion of three-dimenional inhomogeneou term, we demontrate the following urpriing reduction to Laplace (urface) inverion. Uing the fact that the inhomogeneou term for ( 1, F 1, G 1 ) involve only 0, which atifie a Laplace problem, we may pecify the following exact particular olution: F F, G G, y z x z z z. (A5) 2 The homogeneou term (F 1, G 1, 1 ) atify a Laplace problem, with boundary condition that allow (F 1, G 1, 1 ) to atify (A4): F 0, F ; y z G 0, G ; x z 0,. (A6) z z zz Next-order urface wind may then be evaluated from F ( ) F { K F [ ]}, z y z z y z G ( ) F { K F [ ]}, z x z z x z ik x xz z F F[ z zz], K il y yz z F F[ z zz], (A7) K where F 1 i the invere Fourier tranform. Note that

15 15 AUGUST 2002 HAKIM ET AL the (k, l) (0, 0) contribution to x, and y i O( 2 ) and i neglected. APPENDIX B Vortex Cenu Algorithm Vortice are defined in term of grid point, ince they are clearly ditinguihed from the background turbulence in that field. The vortex cenu algorithm proceed through the following tep. 1) At each grid point (i, j), determine whether (i, j) min, where min i a lower bound that i ueful in rejecting local maxima aociated with filament. 2) Determine whether grid point (i, j) repreent a local maximum by earching outward along radial arm. The radial arm are eparated azimuthally by 45, o no interpolation i required. If any grid point (m, n) along any radial arm atifie (m, n) (i, j), point (i, j) i rejected. The earch along each arm top uccefully when (m, n) crit. An additional filament filter i derived from the radial arm by computing the ditance ( diameter ) along four pair of arm, where the pair are elected a ( x, y) reflection through the vortex center; in the cae of a circular vortex, thee value are equal to the diameter. If any of the four diameter value fall below the threhold d min, then the vortex i rejected. 3) The vortex centroid ( x, y) i determined by firt e- tablihing a local mak that define the vortex a crit. The mak i etablihed by piraling outward from (i, j) and etting all point where (m, n) crit to zero; if the point adjacent to (m, n) and cloer to (i, j) i zero, (m, n) i et to zero. The vortex centroid i then calculated by x(m, n) (m, n) m,n x, (m, n) m,n y(m, n) (m, n) m,n y. (B1) (m, n) m,n Merging vortice are filtered by requiring that {[x x(i, j)] 2 [ y y(i, j)] 2 } 1/2 r minor /4, where r minor i the vortex minor-axi length cale a defined ubequently. 4) The vortex major- and minor-axi length cale are determined a moment about the centroid. A 2 N deviation matrix X i contructed for the N grid point defining the vortex. Entrie in the firt and econd row of X are given by (m, n) 1/2 [x(i, j) x] and (m, n) 1/2 [y(m, n) y], repectively. The 2 2 covariance matrix i defined a [ ] 1 S (m, n) XX T. (B2) m,n The vortex major- and minor-axi length cale are defined in term of the caled eigenvalue of S. For example, the major axi i given in term of the leading eigenvalue, 1,a(2 1 ) 1/2. The parameter ued in thi tudy are: min 0.5, max[( (i, j)e 2 crit, 0.4)], and d min 2.0. Thee value were determined empirically, and appear to apply over a wide range of numerical reolution. Moreover, thi tuning of the algorithm i conervative in that we noticed it would occaionally omit a mall, weak vortex, but it would never include a filament. REFERENCES Arai, M., and T. Yamagata, 1994: Aymmetric evolution of eddie in rotating hallow water. Chao, 4, Blumen, W., 1978: Uniform potential vorticity flow: Part 1. Theory of wave interaction and two-dimenional turbulence. J. Atmo. Sci., 35, Bracco, A., J. C. McWilliam, G. Murante, A. Provenzale, and J. B. Wei, 2000: Reviiting freely decaying two-dimenional turbulence at millennial reolution. Phy. Fluid, 12, Charney, J. G., 1971: Geotrophic turbulence. J. Atmo. Sci., 28, Cuhman-Roiin, B., and B. Tang, 1990: Geotrophic turbulence and emergence of eddie beyond the radiu of deformation. J. Phy. Oceanogr., 20, Davie, H. C., and J. C. Müller, 1988: Detailed decription of deformation-induced emi-geotrophic frontogenei. Quart. J. Roy. Meteor. Soc., 114, Durran, D. R., 1999: Numerical Method for Wave Equation in Geophyical Fluid Dynamic. Springer-Verlag, 465 pp. Guinn, T. A., and W. H. Schubert, 1993: Hurricane piral band. J. Atmo. Sci., 50, Hakim, G. J., 2000: Climatology of coherent tructure on the extratropical tropopaue. Mon. Wea. Rev., 128, Held, I. M., R. T. Pierrehumbert, S. T. Garner, and K. L. Swanon, 1995: Surface quai-geotrophic dynamic. J. Fluid Mech., 282, Herring, J. R., R. M. Kerr, and R. Rotunno, 1994: Ertel potential vorticity in untratified turbulence. J. Atmo. Sci., 51, Hokin, B. J., and F. P. Bretherton, 1972: Atmopheric frontogenei model: Mathematical formulation and olution. J. Atmo. Sci., 29, Jucke, M., 1994: Quaigeotrophic dynamic of the tropopaue. J. Atmo. Sci., 51, , 1995: Intability of urface and upper-tropopheric hear line. J. Atmo. Sci., 52, Lumley, J. L., 1990: Whither Turbulence? Turbulence at the Croroad. Springer-Verlag, 535 pp. McWilliam, J. C., 1984: The emergence of iolated coherent vortice in turbulent flow. J. Fluid Mech., 146, , 1990a: The vortice of geotrophic turbulence. J. Fluid Mech., 219, , 1990b: The vortice of two-dimenional turbulence. J. Fluid Mech., 219, , J. B. Wei, and I. Yavneh, 1999: The vortice of homogeneou geotrophic turbulence. J. Fluid Mech., 401, Montgomery, M. T., and R. J. Kallenbach, 1997: A theory for vortex Roby-wave and it application to piral band and intenity change in hurricane. Quart. J. Roy. Meteor. Soc., 123,