The effect of eddy-eddy interactions on jet formation and macroturbulent scales

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1 Generted using the officil AMS LATEX templte two-column lyout. FOR AUTHOR USE ONLY, NOT FOR SUBMISSION! 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 The effect of eddy-eddy interctions on jet formtion nd mcroturulent scles REI CHEMKE AND YOHAI KASPI Weizmnn Institute of Science, Deprtment of Erth nd Plnetry Sciences, Rehovot, Isrel ABSTRACT The effect of eddy-eddy interctions on the zonl nd meridionl mcroturulent scles is studied over wide rnge of rottion rtes, using high-resolution idelized GCM simultions with nd without eddyeddy interctions. The eddy-eddy interctions re found to decrese the numer of eddy-driven jets in the tmosphere, without ltering their width. The decrese in the numer of eddy-driven jets occurs due to the nrrowing of the ltitudinl region where zonl jets pper. In oth simultions with nd without eddy-eddy interctions, the width of the jet is found to scle with the Rhines scle through ll ltitudes nd rottion rtes. As the eddy-eddy interctions re mostly importnt polewrd of the ltitude where the Rhines scle is equl to the Rossy deformtion rdius, once they re removed the conversion from roclinic to rotropic eddy inetic energy increses, nd eddy-men flow interctions intrude into these ltitudes nd mintin the jets there. This conversion scle is found to decrese s eddy-eddy interctions re removed, especilly t low rottion rtes. The Rossy deformtion rdius nd this conversion scle do not coincide in oth simultions with nd without eddy-eddy interctions. The energy-contining zonl scle decreses in the simultions without eddy-eddy interctions, mostly polewrd of the ltitude where the Rhines scle is equl to the Rossy deformtion rdius. Moreover, t these ltitudes in the simultions with eddy-eddy interctions, the jets nd the energy-contining zonl scle coincide. Thus, s these interctions re removed, the isotropy res nd the energy-contining zonl scle is smller thn the width of the jet.. Introduction The turulent ehvior of lrge scles in the tmosphere implies on the importnce of eddy-eddy nd eddy-men interctions in controlling the tmospheric energy spectrum (e.g., Chrney 97; Ber 97; Boer nd Shepherd 983; Nstrom nd Gge 985; Shepherd 987). In prticulr, studying their effect on the energy-contining wvenumer is crucil for developing etter understnding of the physicl processes t synoptic scles. As the chrcteristics of two-dimensionl turulence (energy spectrum following 3 slope of forwrd enstrophy cscde; Krichnn 967) occur in the tmosphere (e.g., Chrney 97; Ber 97; Boer nd Shepherd 983; Nstrom nd Gge 985; Shepherd 987), eddy-eddy interctions should hve n importnt role in the lnce. Moreover, Rhines (977) nd Slmon (978) showed tht s energy is converted from the roclinic to rotropic mode t the Rossy deformtion rdius ( scle which liner theory predicts is proportionl to the most unstle wvelength; Edy 949), it cn inverse cscde in the rotropic mode to lrge scles. Such n inverse cscde Corresponding uthor ddress: Rei Cheme, Weizmnn Institute of Science, Deprtment of Erth nd Plnetry Sciences, 34 Herzl St., Rehovot, Isrel 76. E-mil: rei.cheme@weizmnn.c.il ehvior ws lso documented in ocenic oservtions (Koshi nd Kwmur ; Scott nd Wng 5). The inverse cscde will hlt t the Rhines scle due to the opposite dependence of frequency on wvenumer in the turulent nd Rossy wves regimes (Rhines 975; Hollowy nd Hendershott 977; Rhines 979; Dnilov nd Gurrie ; Glperin et l. 6; Kspi nd Flierl 7). Nonetheless, the inverse energy cscde continues up to the zero zonl wvenumer nd formtion of zonl jets occurs (e.g., Rhines 977; Willims 978; Rhines 994), with meridionl wvenumer following the Rhines scle (Rhines 975; Vllis nd Mltrud 993; Pnett 993; Lee 5; O Gormn nd Schneider 8; Cheme nd Kspi 5). Thus, the rtio (scle seprtion) etween the Rhines scle nd the Rossy deformtion rdius, which follows the QG supercriticlity (Held nd Lrichev 996), cn imply on the importnce of inverse energy cscde nd eddy-eddy interctions in the tmosphere (Cheme nd Kspi 5). However, the lc of cler scle seprtion etween the Rhines scle nd the deformtion rdius, nd the fct tht n inverse energy cscde ( spectrl slope of - 5/3 t lrge scles), ssocited with two-dimensionl turulence, is not oserved in the tmosphere (e.g., Ber 97; Boer nd Shepherd 983; Nstrom nd Gge 985), hve rised questions regrding the significnce of eddy- Generted using v4.3. of the AMS L A TEX templte

2 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 eddy interctions (Pnett 993; Schneider nd Wler 6; Frrell nd Ionnou 7; O Gormn nd Schneider 7; Constntinou et l. 4; Mrston ; Srinivsn nd Young ; Bs nd Ionnou 4). It hs een suggested tht in the sence of such scle seprtion (supercriticlity is equl or smller thn one), the eddy-eddy interctions should e negligile in the lnce, implying tht the Rossy deformtion rdius should follow the energy-contining wvenumer (Schneider nd Wler 6; O Gormn nd Schneider 8; Merlis nd Schneider 9) nd the width of the jet (Schneider nd Wler 6). This implies on the mjor role of eddy-men interctions in mintining the jets (Shepherd 987; Hung nd Roinson 998; Frrell nd Ionnou 7), nd in dding energy t lrge scles through sher-induced spectrl trnsfer (e.g., Shepherd 987; Hung nd Roinson 998). Frrell nd Ionnou (7) showed tht while under we forcing the width of the jet follows the most unstle meridionl scle, under strong forcing it follows the Rhines scle; however, not through turulent cscde process, ut sed on the Ryleigh Kuo stility criterion (Kuo 949). Following Stone s roclinic djustment theory (Stone 978), Schneider (4) suggested tht the tmosphere is in mrginlly criticl stte, such tht the eddies tend to eep the supercriticlity round one. This implies tht eddy-eddy interctions my e of less significnce to the min chrcteristics of mcroturulence in the tmosphere. Indeed, O Gormn nd Schneider (7) nd Chi nd Vllis (4) showed using n idelized GCM, tht y removing the eddy-eddy interctions, the spectrl slope of the inetic energy nd the energy-contining wvenumer remin the sme. Severl studies showed tht even y removing the nonliner eddy-eddy interctions, the meridionl structure of the jets remins (Tois et l. ; Constntinou et l. 4; Mrston ; Srinivsn nd Young ; Tois nd Mrston 3; Bs nd Ionnou 4). On the other hnd, other studies hve showed tht the supercriticlity could vry ove one (Zurit-Gotor 8; Jnsen nd Ferrri, 3; Chi nd Vllis 4). Thus, nonliner eddy-eddy interctions ecome importnt (Zurit-Gotor nd Vllis 9; Jnsen nd Ferrri ; Chi nd Vllis 4; Cheme nd Kspi 5), nd the energy-contining wvenumer coincides with the Rhines scle (Jnsen nd Ferrri ; Chi nd Vllis 4; Cheme nd Kspi 5). At lrge supercriticlity the energy-contining wvenumer does not correlte etween simultions with nd without eddy-eddy interctions (Chi nd Vllis 4). Moreover, severl studies (O Gormn nd Schneider 7; Chi nd Vllis 4; Ait-Chll nd Schneider 5) oserved n increse in the numer of jets (the meridionl energycontining wvenumer) s the eddy-eddy interctions were removed. In Cheme nd Kspi (5) we showed the presence of criticl ltitude, where the Rhines scle is equl to the Rossy deformtion rdius. Polewrd of this ltitude, the Rhines scle is lrger thn the Rossy deformtion rdius nd the QG supercriticlity is lrger thn one. Hence, herefter we refer to this ltitude s the supercriticlity ltitude. At these polewrd ltitudes, the length scle of the energy-contining zonl wvenumer coincides with the width of the jet nd the Rhines scle nd the eddy-eddy interctions ply n importnt role in trnsferring energy oth upscle nd down scle. Thus, the rotropic eddy inetic energy spectrum follows 5/3 slope from the Rossy deformtion rdius up to the Rhines scle. Using stellite ltimetry, Scott nd Wng (5) lso showed tht n inverse energy cscde y eddy-eddy interctions occurs polewrd of this ltitude. Equtorwrd of the supercriticlity ltitude, the eddy-men interctions ecome importnt nd the energy-contining zonl scle ws found to e lrger thn the jets nd Rhines scles. At these ltitudes the rotropic eddy inetic energy spectrum follows 3 slope from the Rhines scle down to smller scles. Theiss (4) nd Eden (7) lso showed tht the flow is isotropic (nisotropic) polewrd (equtorwrd) of this ltitude. The width of the jet ws found to coincide with the Rhines scle through ll ltitudes, however with etter greement polewrd of the supercriticlity ltitude. Under Erth-lie simultions the supercriticlity ltitude is plced polewrd of the roclinic zone, producing only the 3 slope of enstrophy cscde with no hint of inverse cscde, s oserved in Erth s tmosphere (Nstrom nd Gge 985). In this study, we exmine the importnce of the eddyeddy interctions in controlling the meridionl nd zonl mcroturulent scles. Due to the ltitudinl dependence of oth scles (Cheme nd Kspi 5), we study their effect y compring simultions with nd without eddyeddy interctions (e.g., O Gormn nd Schneider 7; Chi nd Vllis 4), oth polewrd nd equtorwrd of the supercriticlity ltitude. The eddy-eddy interctions re found to decrese the numer of eddy-driven jets in the tmosphere, without ltering their width, y nrrowing the region where these jets pper. Consistently, oth the conversion from roclinic to rotropic eddy inetic energy nd eddy-men interctions decrese, especilly t high ltitudes, where the eddy-eddy interctions re found to e importnt. While the eddy-eddy interctions hve minor effect on the Rhines nd width of the jets, they tend to increse oth the energy contining scle (so it would equl the width of the jet), nd the energy conversion scle. Section descries the idelized GCM nd presents riefly the formultion of the qusi-liner simultions. Sections 3 nd 4 discuss the effect of eddy-eddy interctions on the jet nd mcroturulent scles, respectively, y compring results from simultions with nd without

3 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 3 σ σ Ltitude FIG.. Snpshot of the men zonl wind (ms ) s function of sigm nd ltitude for the 8Ω e simultions with eddy-eddy interctions () nd without eddy-eddy interctions (). eddy-eddy interctions. The results re further discussed in section 5.. Model We use n idelized quplnet moist glol circultion model (GCM), sed on the GFDL flexile modeling system (FMS). This is sphericl coordinte primitive eqution model of n idel-gs tmosphere similr to Frierson et l. (6) nd O Gormn nd Schneider (8). The lower lyer of the model is n ocen sl which hs no dynmics ut only exchnges energy with the lowest tmospheric lyer. Shortwve rdition is imposed s perpetul equinox nd long-wve rditive trnsfer is represented y two-strem gry rdition scheme (Goody 964; Held 98). In order to exmine the effect of the nonliner eddyeddy interctions on mcroturulent scles, we remove these interctions from the momentum nd temperture equtions (O Gormn nd Schneider 7). We refer to these simultions s the qusi-liner (QL) simultions nd to the simultions with eddy-eddy interctions s the full simultions. The men temperture nd momentum equtions re similr in the QL nd full simultions, s they oth contin ll the eddy-men flow interctions, including the contriutions of the men eddy fluxes (i.e., v u y, where u nd v re the zonl nd meridionl velocities, nd the r nd prime denote zonl men nd devition from this men, respectively). However, the eddy equtions differ in the full nd QL simultions s the purely eddy interctions (i.e., ) re removed in the QL ( ) simul- v u y tions. In this wy, the interctions involving three different wvenumers re removed in the QL simultions, nd the spectrum is uilt only through eddy-men interctions. We crry out set of experiments where we systemticlly increse the plnetry rottion rte up to 8 times Erth s rottion rte (Ω e ). Incresing the rottion rte enles studying multiple jet plnets t ll ltitudes (due to the polewrd migrting jets, Cheme nd Kspi 5), nd thus to ccumulte etter sttistics on the meridionl nd zonl contining scles. At high rottion rtes the eddy length scle is reltively smll compred to the plnet size (e.g., Schneider nd Wler 6; Kspi nd Showmn 5), which provides wide seprtion mong the importnt mcroturulence scles (e.g., the Rhines scle, Rossy deformtion rdius, scle of rotropic to roclinic energy conversion, etc.). Ech simultion hs 3 verticl sigm lyers t T7 horizontl resolution (.7.7 ). Unless otherwise stted, the results represent the time verge of the lst 5 dys of dy simultions. 3. The width of the jet As discussed in section, previous studies (O Gormn nd Schneider 7; Chi nd Vllis 4) showed tht in Erth-lie simultions when the eddy-eddy interctions were removed the jets were compressed towrds the equtor nd n dditionl jet emerged t higher ltitudes. Figure shows snpshot of the meridionl-height cross section of the men zonl wind in the full nd QL 8Ω e simultions. The men zonl wind intensifies in the QL simultion (Tois nd Mrston 3), prticulrly t high ltitudes, s in Mrston (). In order to see how roust this phenomenon is, we clculte the numer of eddy driven jets through series of simultions with different rottion rtes (Fig. ). As the rottion rte increses, the eddy length scle decreses, which increses the numer of jets in the tmosphere (Willims nd Hollowy 98; Nvrr nd Boccletti ; Kspi nd Schneider ; Cheme nd Kspi 5), since their meridionl scle correltes with the Rhines scle (Rhines 975; Vllis nd Mltrud 993; Pnett 993; Lee 5; O Gormn nd Schneider 8; Cheme nd Kspi 5). The numer of the eddy driven jets in the QL simultions (red dots in Fig. ) is lrger y 58% thn in the full simultions (lue dots in Fig. ). This cn lso e seen y oserving the men zonl wind in the full nd QL 8Ω e simultions (Fig. ). O Gormn nd Schneider (7) suggested tht the meridionl wvenumer of the men flow is smller in the full simultion, due to the tendency of eddy-eddy interctions to inverse cscde energy ner the zero zonl We define n eddy-driven jet to e where its mximum verticl nd zonl verged velocity is lrger thn 5% of the mximum verticl nd zonl verged velocity of the strongest jet. This voids low vlues of zonl wind for the clcultion of the numer of eddy-driven jets. Ting lower or higher percentges does not ffect the result. Defining n eddy-driven jet s locl mximum of the zonl wind tht resides in the roclinic zone (e.g., the ltitudinl extent of the roclinic zone is defined where 3% of the mximum vlue of the eddy het flux is oserved ner the surfce, Schneider nd Wler 6), produces the sme results.

4 4 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 Numer of jets Jet region width Rottion rte [Ω e ] FIG... The numer of eddy-driven jets in oth hemispheres s function of rottion rte, for simultions with (lue) nd without (red) eddy-eddy interctions.. The ltitudinl width of the jet region for simultions with (lue) nd without (red) eddy-eddy interctions. The two lc lines in pnel follow C, Ω.8 where C =.58C. The width of the jet region is defined s the distnce etween the most equtorwrd nd polewrd jets. Only simultions where the eddy driven jets re clerly seprted re ten into ccount for the nlysis of pnel (Ω > Ω e ). wvenumer to smller meridionl wvenumers. The increse in the numer of eddy-driven jets in the QL simultions (Fig. ) could e explined y nrrowing of the jet s width s the eddy-eddy interctions re removed. However, similr to Constntinou et l. (4) nd Srinivsn nd Young (), when the eddy-eddy interctions re removed, the width of the jet remins similr to the width of the jet in the full simultions (95% of the jets hve either increse or decrese their width in less thn %), through ll ltitudes nd rottion rtes (Fig. 3). The width of the jet is defined s the meridionl distnce etween two consecutive minim vlues of the men zonl wind. The increse in the numer of eddy driven jets in these simultions (Fig. ), is due to n increse in the ltitudinl width of the region where these jets pper. We refer to this region s the jet region. By compring the men zonl wind for the QL nd full 8Ω e simultions (Fig. ), the jet region ecomes wider (stronger eddy-driven jets pper t higher ltitudes) s the eddy-eddy interctions re removed. This increse in the width of the jet region is found to occur through ll rottion rtes (Fig. ). The ltitudinl jet region increses in the QL simultions (red dots in Fig. ) y 7% compred to the full simultions (lue dots in Fig. ). Hence, most of the chnge in the meridionl structure of the zonl wind in the QL simultions comes from the expnsion of the jet region (Fig. ) nd not from chnge in the jets width (Fig. 3). The increse in the QL simultions of the jet region ( 7%) is lrger thn the increse in the numer of eddy-driven jets ( 58%), since the jets tht pper t higher ltitudes re wider (Fig. ) due to the et term in the Rhines scle, s discussed in previous studies (Hung nd Roinson 998; Kidston nd Vllis ; Cheme nd Kspi 5,). The width of the jet region displys non-monotonic dependence on rottion rte, in oth full nd QL simultions. For the lower rottion rtes, the jet region increses with rottion rte (Fig. ) minly due to the decrese in the ltitudinl extent of the Hdley cell (Held nd Hou 98; Wler nd Schneider 6). On the other hnd, t higher rottion rtes, the jet region slowly decreses with rottion rte, which might e due to the weening of the polewrd eddy het flux with rottion rte (Hunt 979; Kspi nd Showmn 5), which limits the edge of the roclinic zone. The width of the jet in the QL simultions is found to scle with the Rhines scle, nd not with the Rossy deformtion rdius (Fig. 3) through ll ltitudes nd rottion rtes, s ws shown to occur in the full simultions s well (Cheme nd Kspi 5). Following O Gormn nd Schneider (8), the Rhines scle is defined s, ( ) (EKE) L β = π, () β where EKE is the verticlly verged eddy inetic energy. The eddies re defined s perturtion from the zonl men, nd β is the meridionl derivtive of the Coriolis prmeter, f. The Rossy deformtion rdius is defined s, L D = π NH, () f where H is the tropopuse height clculted s the height where the sttic stility reches threshold vlue of.5 s, nd N = g ϑ ϑ z is the verticlly verged sttic stility elow the tropopuse height (similr to Frierson et l. 6), where g is grvity nd ϑ is the potentil temperture. The Rossy deformtion rdius is similr to the Rossy deformtion rdius clculted when pplying the WKBJ pproximtion on the eigenvlue prolem for the verticl structure of the QG stremfunction (Gill 98; Chelton et l. 998). The constnt of proportionlity (π) is empiriclly chosen to est seprte the rotropic inverse cscde of EKE nd forwrd enstrophy cscde (Cheme nd Kspi 5). Rhines scles clculted using the rotropic EKE (e.g., Hidvogel nd Held 98; Schneider nd Wler 6; Jnsen nd Ferrri ; Chi nd Vllis 4), the energy cscde rte (Held nd Lrichev 996) or using the Ryleigh Kuo stility criterion (Frrell nd Ionnou 7),

5 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 5 The correltion of the width of the jet nd the Rhines scle, ltitude y ltitude through ll rottion rtes, in the QL simultions, implies tht the eddy-eddy interctions do not ply n importnt role in setting the width of the jet. This is similr to the correltion of the jet nd the Rhines scles, equtorwrd of the supercriticlity ltitude in the full simultions, where eddy-eddy interctions were found to e we. Nevertheless, in the full simultions the width of the jet coincides etter with the Rhines scle polewrd of the suwere found to e less consistent with the width of the jet nd with the hlting scle of the inverse energy cscde (Cheme nd Kspi 5) thn the Rhines scle in Eq.. The constnt of proportionlity (π) in Eq. is empiriclly chosen to est fit with the jet nd the inverse energy cscde scles. J s J s LβJ s, LDJ s c LeL e Ltitude FIG. 3. Scle properties s function of ltitude for ll rottion rtes. In ll pnels ech dot represents rtio t single ltitude nd rottion rte.. The rtio etween the width of the jet in the QL (denoted with n steris) nd full simultion.. The rtio etween the Rhines scle nd the width of the jet (green), nd etween the Rossy deformtion rdius nd the width of the jet (ornge) for simultions without eddyeddy interctions. c. The rtio etween the length scle of the energycontining zonl wvenumer, clculted from the zonl spectrum of the rotropic eddy meridionl velocity (Eq. 7), in the full nd QL (denoted with n steris) simultions. In pnels nd c the lue (red) dots represents ltitudes polewrd (equtorwrd) of the ltitude where the Rhines scle is equl to the Rossy deformtion rdius. percriticlity ltitude, where eddy-eddy interctions were found to e importnt (Cheme nd Kspi 5). In the QL simultions, on the other hnd, the width of the jet coincides with the Rhines scle similrly t ll ltitudes (Fig. 3). This implies tht the eddy-eddy interctions do hve some effect on the width of the jet, mostly polewrd of the supercriticlity ltitude. This effect turns out to e we s the rtio etween the width of the jets in the full nd QL simultions ehves similrly polewrd (lue dots in Fig. 3) nd equtorwrd (red dots in Fig. 3) of the supercriticlity ltitude. The presence of eddy-driven jets in the QL simultions t ltitudes where these jets were sent in the full simultions (Fig. ), implies tht the production of the zonl jets through locl processes in spectrl spce of nonliner eddy-eddy interctions (e.g., Rhines 977; Willims 978; Rhines 994) is less importnt in such tmospheric simultions (e.g, Frrell nd Ionnou (7); Constntinou et l. (4); Mrston (); Srinivsn nd Young ()). The mintennce of jets y eddy-men flow interctions (e.g., Shepherd 987; Pnett 993; Hung nd Roinson 998; Roinson 6; Frrell nd Ionnou 7), is further discussed in the next section. Even though eddy-eddy interctions re not prerequisite for the formtion of zonl jets (Figs. nd ), t strongly nonliner flow regimes (i.e., polewrd of the supercriticlity ltitude) they do shpe, lthough wely, the meridionl scle of the jets nd hve lrge ffect on the width of the jet region (Fig. ). 4. Mcroturulent scles The effect of eddy-eddy interctions on the meridionl structure of the flow could e further understood y compring the different components in the udget of the zonl rotropic EKE (following the nlysis of Lrichev nd Held 995, ut here s function of ltitude, s in Cheme nd Kspi 5). Even though the meridionl structure of the men flow nd eddies re ffected y importnt meridionl mcroturulent scles (Simmons 974; Vllis nd Mltrud 993; Hung nd Roinson 998; Hung et l. ; Brry et l. ), here we study the effect of eddy-eddy interctions on the meridionl structure of the jets using the zonl spectrum; this is minly due to two resons: first, this llows studying the effect of eddy-eddy interctions s function of ltitude (oth polewrd nd equtorwrd of the supercriticlity ltitude). Second, the eddy-eddy interctions lso ffect other zonl mcroturulent scles (e.g., the roclinic conversion nd energy-contining scles) s further discussed elow. The zonl spectrl rotropic EKE is computed using onedimensionl Fourier nlysis for ech ltitude (Sltzmn 957) s follows, EKE = [u] + [v], (3)

6 6 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 where ngle rcets denote time men, squred rcets denote verticl verge, nd prime denotes the devition from zonl men 3. The suscript denotes the zonl spectrl components with zonl wvenumer. Two components of the udget of the rotropic EKE re presented in Fig. 4: the conversion of roclinic EKE to rotropic EKE, { ( CT = Re [u] [u+ u +] [ u + v + tnθ ( [v] [u+ v +] + [ u + u + tnθ nd the eddy-men interctions, ] ) ] ) }, (4) { ( EM = Re [u] ([u] [u] ) + ( [u] [u] ) ( [u][v] tnθ ) ( [u] [v]tnθ ) ) [v] ( ([u] [v] ) + ( [u] [v] ) + ( [u][u] tnθ ) ) }, (5) where u denotes the three dimensionl velocity vector, θ is ltitude, the steris denotes complex conjugte, the overr denotes zonl men, is Erth s rdius nd the plus sign superscript denotes devition from verticl verge. The ove components re presented only for the 8Ω e simultion, s it cptures the properties oth ove nd elow the supercriticlity ltitude (Fig. 4). A similr nlysis ws conducted in previous studies (e.g., Lmert 984; Koshy nd Hmilton ; Jnsen nd Ferrri ; Chi nd Vllis 4) only for the EKE using two-dimensionl spectr. The lc line in ech pnel of Fig. 4 shows the supercriticlity ltitude from the full simultion, where the Rhines scle (Eq. ) is equl to the Rossy deformtion rdius (Eq. ). The white nd gry lines show the Rossy nd the conversion from roclinic to rotropic EKE zonl wvenumers, respectively. The conversion etween wvenumers () nd length scles (L) is computed t ech ltitude s follows, L = πcos(θ), (6) 3 Studying the rotropic energy provides n understnding on the trnsfers of EKE to lrger scles y oth eddy-eddy nd eddy-men interctions (Rhines 977; Slmon 978). Thus, in order to study only the rotropic components, it is necessry to first te verticl verge, nd only then compute the devition from zonl men. Ltitude Ltitude c 5 5 Zonl wvenumer 5 5 Zonl wvenumer FIG. 4. Components of the rotropic EKE eqution ( 5 m s 3, Eqs. 4 nd 5) in the 8Ω e run s function of ltitude nd zonl wvenumer. The left (right) column presents simultion with (without) eddy-eddy interctions of conversion of roclinic EKE (top), nd eddy-men interctions (ottom). Ech component is multiplied y the wvenumer nd smoothed with -point running men. The lc, white, green, lue nd gry lines correspond to the supercriticlity ltitude (where the Rhines scle is equl to the Rossy deformtion rdius) from the full simultion, the conversion wvenumer of roclinic to rotropic EKE, the Rhines scle, the mximum roclinic growth wvenumer nd the Rossy deformtion wvenumer (see text for definitions), respectively. nd the conversion wvenumer of roclinic to rotropic EKE is clculted s the centroid of Eq. 4. This definition provides conversion wvenumers tht re closest to the mximum of Eq. 4. The green lines show the scle of mximum roclinic growth (similr to Jnsen nd Ferrri ), which is the wvenumer corresponding to the fstest growth rte computed using liner norml-mode instility nlysis (Smith 7). This is done y solving the linerized qusigeostrophic potentil vorticity eqution using the meridionl potentil vorticity grdient, men zonl wind nd density from the idelized GCM simultions. The eigenvlues re computed t ech ltitude nd height, nd only the mximum growth rte t ech zonl wvenumer nd ltitude is ten into ccount. As in Smith (7), in order to consider only the most unstle wvenumers tht re energeticlly importnt, we normlized the mximum growth rte of ll levels y the eddy ville potentil energy (EAPE). The EAPE is estimted using the vrince of the temperture field (e.g., Lorenz 955; Sltzmn 957; Boer 975; Schneider nd Wler 6). In this wy the most unstle wvenumer is found to correspond to the first locl mximum growth rte (i.e., the longest locl most unstle wve) 4. 4 Applying shortwve cutoff t ech ltitude y filtering out the unstle wvenumers where the conversion from roclinic to rotropic EKE is less thn 85% of the mximum conversion, produces the sme results. d 5

7 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 7 L β, L D, L g 7 x L β, L D,L g x 6 FIG. 5. The Rhines scle (lue), the Rossy deformtion rdius (red) nd the conversion scle of roclinic to rotropic EKE (ornge) in the QL simultions (denoted with n steris) compred to the full simultions, for ll rottion rtes. For compring the scles in the QL nd full simultions, the meridionl verge of ech scle is ten over the jet region clculted from the full simultions (defined s the distnce etween the most equtorwrd nd polewrd jet). It hs een suggested tht the presence of eddy-eddy interctions reduces the conversion of potentil to inetic energy, which cn explin the increse in the inetic energy in the QL simultions (e.g., O Gormn nd Schneider 7; Chi nd Vllis 4). In our QL simultions the conversion from roclinic to rotropic EKE indeed strengthens, compred to the full simultions, mostly t intermedite scles (Berloff nd Kmenovich 3,), nd weens t lrge nd smll scles (Fig. 4 nd 4). This is consistent with the increse of rotropic EKE in the QL simultions t intermedite scles, nd decrese t lrge nd smll scles (Cheme nd Kspi 5). Due to this opposite ehvior of the EKE t different scles, nd since the width of the jet in the QL nd full simultions is similr (Fig. 3), nd follows the Rhines scle (Fig. 3), the men EKE in the full nd QL simultions should lso e similr. Indeed, the Rhines scle (computed using the EKE, Eq. ) is found to e similr for ech rottion rte in the full nd QL simultions (Fig. 5, lue dots). The increse in roclinic conversion to the rotropic mode in the QL simultions is consistent with the tendency of the eddy-eddy interctions to me the strtifiction less uniform. A more uniform strtifiction, s occurs in the tmosphere compred to the ocen, enles stronger roclinic conversions (Fu 98; Smith nd Vllis, ). The conversion from roclinic to rotropic EKE not only strengthens, ut lso shifts to lrger zonl wvenumers in the QL simultion, nd reches higher ltitudes (Fig. 4). The eddy-men interctions lso strengthen nd intrude into higher ltitudes in the QL simultions (replcing the strong eddy-eddy interctions polewrd of the supercriticlity ltitude, Cheme nd Kspi 5, which only spred the input of roclinic EKE loclly in spectrl spce), nd mintin the jets there (Figs. nd 4c nd 4d). Not only the trnsfer of rotropic EKE to the men flow increses in the QL simultion, ut lso the ddition of rotropic EKE to lrge scles increses (Figs. 4c nd 4d). While in the QL simultions the ddition of rotropic EKE y the eddy-men interctions t lrge scles coincides with the Rhines scle (Fig. 4d), in the full simultions, on the other hnd, the ddition of rotropic EKE y eddy-eddy interctions coincide with the Rhines scle. As in the full simultions, lso in the QL simultions the Rossy deformtion rdius does not coincide with the conversion scle of roclinic to rotropic EKE, s expected from the theory of Slmon (978) (white nd gry lines in Figs. 4 nd 4 ). This cn e more roustly seen in Fig. 5 (where ll scles decrese monotoniclly with rottion rte), y compring the deformtion rdius (red) nd conversion scle (ornge) t ech rottion rte simultion (e.g., the right most deformtion scle with the right most conversion scle). At low rottion rtes (lrger scles in Fig. 5) oth the conversion scle nd the Rossy deformtion rdius re lrger in the full simultions. At high rottion rtes, on the other hnd, while the conversion scle etter coincides in the QL nd full simultions, the Rossy deformtion rdius is lrger in the QL simultions (Fig. 5). The conversion scle decreses with rottion rte, s the inverse cscde of the rotropic EKE is eing rrested t smller Rhines scles (Chi nd Vllis 4). Note, tht LeJ s Ltitude 4 6 Ltitude FIG. 6. The rtio of the length scle of the energy-contining zonl wvenumer, clculted from the zonl spectrum of the rotropic meridionl velocity (Eq. 7), nd the jet spce s function of ltitude for ll rottion rtes presented in Fig. for simultions with eddy-eddy interctions () nd without eddy-eddy interctions (). The lue (red) dots represents ltitudes polewrd (equtorwrd) of the ltitude where the Rhines scle is equl to the Rossy deformtion rdius.

8 8 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 Totl wvenumer Zonl wvenumer Zonl wvenumer FIG. 7. The D spectrum, computed using sphericl hrmonics s sis functions (Boer nd Shepherd 983), of the rotropic EKE ( m s ) s function of zonl nd totl wvenumers in the 8Ω e simultion with eddy-eddy interctions () nd without eddy-eddy interctions (). the conversion from roclinic to rotropic contins the conversions from ll roclinic modes nd not only from the first roclinic mode s occurs in the clssic two-lyers model (e.g., Slmon 978). This could e one reson for the pprent discrepncy etween the deformtion rdius nd the conversion scle. Even though the Rossy deformtion rdius does not coincide with the conversion scle, the scle of mximum roclinic growth does show generl ltitudinl ehvior s the conversion scle, consistent with Jnsen nd Ferrri, through ll ltitudes (Figs. 4 nd 4). In the QL simultion the conversion scle nd the scle of mximum roclinic growth coincide t ll ltitudes, s oth scles re derived from models tht neglect the contriution of nonliner interctions (Fig. 4). In the full simultion, on the other hnd, the scle of mximum roclinic growth is smller thn the conversion scle polewrd of the supercriticlity ltitude (Fig. 4). This is consistent with the tendency of the eddy-eddy interctions to shift the conversion scle to lrger scles. However, equtorwrd of the supercriticlity ltitude, where eddy-eddy interctions re weer, the scle of mximum roclinic growth nd the conversion scle pproximtely coincide. Using crtesin coordinte models, severl studies showed tht the Rossy deformtion rdius does coincide with the conversion scle (Slmon 978; Lrichev nd Held 995; Jnsen nd Ferrri, 5). Thus, one possile explntion for the difference etween the conversion scle nd the Rossy deformtion rdius, in our simultions, is the effect of the sphericity of the plnet. Similr to the meridionl direction, the energycontining zonl wvenumer lso increses in the QL simultions (Fig. 3c). As in O Gormn nd Schneider (8) nd Cheme nd Kspi (5), the energycontining zonl wvenumer is computed using the squred inverse centroid of the zonl spectrum of the rotropic eddy meridionl velocity s follows 5, e = [v] [v]. (7) The length scle of the energy-contining zonl wvenumer is not only lrger in the full simultions, (consistent with Chi nd Vllis 4 QL simultions under strong supercriticlity), ut lso its rtio with the length scle of the energy-contining zonl wvenumer in the QL simultions increses with ltitude (Fig. 3c, lue dots). Equtorwrd of the supercriticlity ltitude the length scle of the energy-contining zonl wvenumer is found to e lrger in the full simultions y fctor of.3 (Fig. 3c, red dots). At these ltitudes, the width of the jet is smller thn the length scles of the energycontining zonl wvenumers in oth full nd QL simultions (Fig. 6, red dots). The effect of eddy-eddy interctions on the rtio etween the zonl energy-contining nd meridionl width of the jets, cn e explined y the tendency of the eddy-eddy interctions to spred the rotropic energy long lines of constnt totl wvenumers (Fig. 7, turulence isotropiztion) s discussed in Shepherd (987) nd Hung nd Roinson (998). In the QL simultion, the lc of eddy-eddy interctions reduces the isotropiztion of the flow, nd the rotropic EKE is no longer spred long lines of constnt totl wvenumers (Fig. 7). Thus, once the eddy-eddy interctions re removed, nd the isotropy res, the meridionl width of the jet does no longer coincide with the zonl energy-contining scle polewrd of the supercriticlity ltitude (Fig. 6, lue dots). Furthermore, the energy contining wvenumer is isolted in the spectrl spce (Fig. 7), once these interctions re removed. 5. Conclusions In this study we show tht eddy-eddy interctions hve lrge effect on oth the meridionl nd zonl structure of jets in series of idelized GCM simultions where we systemticlly compre simultions with (full simultions) nd without eddy-eddy interctions (qusi-liner simultions, QL), t different rottion rtes. The different rottion rtes llow preforming the nlysis over continuous rnge of ltitudes nd width of the jets. Our min conclusions re s follows: The eddy-eddy interctions re found to decrese the numer nd intensity of eddy-driven jets in the tmosphere (Figs. nd ), y limiting the ltitudinl extent of the region where these jets pper (Fig. ). 5 This definition produces n energy-contining zonl wvenumer closest to the pe of the zonl spectrum of the rotropic eddy meridionl velocity, nd its length scle (Eq. 6) est coincides with the jets scle (Fig. 6, lue dots).

9 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 9 This implies tht not only tht eddy-eddy interctions re not prerequisite for the formtion of zonl jets in the tmosphere, ut lso on their tendency to inhiit jet formtion. As eddy-eddy interctions re found to e importnt mostly polewrd of the ltitude where the Rhines scle is equl to the Rossy deformtion rdius (the supercriticlity ltitude, Cheme nd Kspi 5), once they re removed the conversion of roclinic to rotropic EKE increses t these ltitudes (Figs. 4 nd 4). The increse in the conversion term is consistent with n increse of the rotropic eddymen flow interctions t these ltitudes (Figs. 4c nd 4d). These interctions replce the eddy-eddy interctions (which efficiently spred the input of roclinic EKE loclly in spectrl spce) t these ltitudes nd mintin the jets (trnsfer energy to the zero zonl wvenumer) t high ltitudes, nd thus, widen the jet region (Fig. ). In the full Erth-lie simultion, the supercriticlity ltitude ws found to occur polewrd of the roclinic zone. As result, eddy-eddy interctions were found to ply negligile role in the lnce, nd to hve minor effect on the spectrl slope of the rotropic EKE (Cheme nd Kspi 5). Nonetheless, in the QL Erth-lie simultion, n dditionl jet emerges t high ltitudes (Fig. ) s in O Gormn nd Schneider (7) nd Chi nd Vllis (4). This implies tht eddy-eddy interctions do hve some effect on the zonl men flow, even under Erth s prmeters. The eddy-eddy interctions re found to hve minor effect on the meridionl width of the jets through ll ltitudes nd rottion rtes (Fig. 3). As in the full simultions (Cheme nd Kspi 5), lso in the QL simultions the width of the jet coincides with the Rhine scle through ll ltitudes nd rottion rtes (Fig. 3). Thus, the Rhines scle lso remins similr in oth full nd QL simultions. The conversion scle of roclinic to rotropic EKE does not coincide with the Rossy deformtion rdius, for ll rottion rtes, oth in the full nd QL simultions (Figs. 4, 4 nd 5). The effect of the eddy-eddy interctions on these scles depends differently on rottion rte. At low rottion rtes the eddy-eddy interctions increse oth the Rossy deformtion rdius nd the conversion scle. On the other hnd, t high rottion rtes they decrese the Rossy deformtion rdius nd hve minor effect on the conversion scle (lthough still lrger in the simultions with eddy-eddy interctions, Fig 5). The conversion scle, however, does coincide ltitude-y-ltitude in the QL simultions with the scle of mximum roclinic growth (Fig. 4). In the full simultions, the scle of mximum roclinic growth is smller thn the conversion scle, polewrd of the supercriticlity ltitude where eddy-eddy interctions ply mjor role in the lnce (Fig. 4). However, equtorwrd of the supercriticlity ltitude, oth of the scles pproximtely coincide. The energy-contining zonl scle lso decreses for ll ltitudes nd rottion rtes, s the eddy-eddy interctions re removed (Fig. 3c). Equtorwrd of the supercriticlity ltitude, in oth full nd QL simultions, the energy-contining zonl scle is lrger thn the width of the jet (Fig. 6). On the other hnd, polewrd of the supercriticlity ltitude (where eddyeddy interctions were found to e importnt), in the simultions with eddy-eddy interctions, the length scle of the energy-contining zonl wvenumer coincides with the width of the jet (Fig. 6). Thus, s the width of the jet coincides in the full nd QL simultions, t these ltitudes once the eddy-eddy interctions re removed, the width of the jet is lrger then the energy-contining scle (Fig. 6). This is consistent with the tendency of eddy-eddy interctions to spred the rotropic energy long lines of constnt totl wvenumer (Shepherd 987,), nd hence to me the flow more isotropic (Fig. 7). Acnowledgments. We thn Jnni Yuvl nd Mlte Jnsen for very fruitful discussions during the preprtion of this mnuscript. This reserch hs een supported y n EU-FP7 Mrie Curie Creer Integrtion Grnt (CIG- 34), the Isreli Science Foundtion (grnts 3/ nd 859/), nd the Isreli Ministry of Science. References Ait-Chll, F., nd T. Schneider, 5: Why eddy momentum fluxes re concentrted in the upper troposphere. J. Atmos. Sci., 7, Ber, F., 97: An lternte scle representtion of tmospheric energy spectr. J. Atmos. Sci., 9 (4), Bs, N. A., nd P. J. Ionnou, 4: A theory for the emergence of coherent structures in et-plne turulence. J. Fluid Mech., 75, Brry, L., G. C. Crig, nd J. Thuurn, : Polewrd het trnsport y the tmospheric het engine. Nture, 45 (6873), Berloff, P., nd I. Kmenovich, 3: On spectrl nlysis of mesoscle eddies. Prt I: Liner nlysis. J. Phys. Ocenogr., 43, Berloff, P., nd I. Kmenovich, 3: On spectrl nlysis of mesoscle eddies. Prt II: Nonliner nlysis. J. Phys. Ocenogr., 43,

10 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 Boer, G. J., 975: Zonl nd eddy forms of the ville potentil energy equtions in pressure coordintes. Tellus, 7 (5), Boer, G. J., nd T. G. Shepherd, 983: Lrge-scle two-dimensionl turulence in the tmosphere. Journl of Atmospheric Sciences, 4 (), Chi, J., nd G. K. Vllis, 4: The role of criticlity on the horizontl nd verticl scles of extrtropicl eddies in dry GCM. J. Atmos. Sci., 7 (7), Chrney, J. G., 97: Geostrophic turulence. J. Atmos. Sci., 8 (6), Chelton, D. B., R. A. Deszoee, M. G. Schlx, E. Nggr, nd N. Siwertz, 998: Geogrphicl vriility of the first roclinic Rossy rdius of deformtion. J. Phys. Ocenogr., 8 (3), Cheme, R., nd Y. Kspi, 5: Polewrd migrtion of eddy driven jets. Journl of Advnces in Modeling Erth Systems, in press. Cheme, R., nd Y. Kspi, 5: The ltitudinl dependence of tmospheric jet scles nd mcroturulent energy cscdes. J. Atmos. Sci., in press. Constntinou, N. C., B. F. Frrell, nd P. J. Ionnou, 4: Emergence nd equilirtion of jets in et-plne turulence: pplictions of stochstic structurl stility theory. J. Atmos. Sci., 7 (5), Dnilov, S. D., nd D. Gurrie, : Qusi-two-dimensionl turulence. Uspehi Fizichesih Nu, 7 (9), Edy, E. T., 949: Long wves nd cyclone wves. Tellus, (3), Eden, C., 7: Eddy length scles in the North Atlntic Ocen. J. Geophys. Res.,. Frrell, B. F., nd P. J. Ionnou, 7: Structure nd spcing of jets in rotropic turulence. J. Atmos. Sci., 64 (), Frierson, D. M. W., I. M. Held, nd P. Zurit-Gotor, 6: A gryrdition quplnet moist GCM. Prt I: Sttic stility nd eddy scle. J. Atmos. Sci., 63 (), Fu, L. L., 98: Nonliner energy nd enstrophy trnsfers in relisticlly strtified ocen. Dyn. Atmos. Ocens, 4 (4), Glperin, B., S. Suorinsy, P. Red, Y. Ymzi, nd R. Wordsworth, 6: Anisotropic turulence nd zonl jets in rotting flows with et effect. Nonliner Processes in Geophysics, 3 (), Gill, A. E., 98: Atmosphere-ocen dynmics. Vol.3. Acdemic Press. Goody, R. M., 964: Atmospheric rdition. pp. 46. Clrendon Press. Hidvogel, D. B., nd I. M. Held, 98: Homogeneous qusigeostrophic turulence driven y uniform temperture grdient. J. Atmos. Sci., 37 (), Held, I. M., 98: On the height of the tropopuse nd the sttic stility of the troposphere. J. Atmos. Sci., 39 (), Held, I. M., nd A. Y. Hou, 98: Nonliner xilly symmetric circultions in nerly inviscid tmosphere. J. Atmos. Sci., 37, Held, I. M., nd V. D. Lrichev, 996: A scling theory for horizontlly homogeneous, rocliniclly unstle flow on et plne. J. Atmos. Sci., 53 (7), Hollowy, G., nd M. C. Hendershott, 977: Stochstic closure for nonliner Rossy wves. J. Fluid Mech., 8 (4), Hung, H., B. Glperin, nd S. Suorinsy, : Genertion of men flows nd jets on et plne nd over topogrphy. Phys. of Fluids, 3, 5 4. Hung, H. P., nd W. A. Roinson, 998: Two-dimentionl turulence nd persistent jets in glol rotropic model. J. Atmos. Sci., 55 (4), Hunt, B. G., 979: The influence of the Erth s rottion rte on the generl circultion of the tmosphere. J. Atmos. Sci., 36 (7), Jnsen, M., nd R. Ferrri, : Mcroturulent equilirtion in thermlly forced primitive eqution system. J. Atmos. Sci., 69 (), Jnsen, M., nd R. Ferrri, 3: Equilirtion of n tmosphere y ditic eddy fluxes. J. Atmos. Sci., 7 (9), Jnsen, M., nd R. Ferrri, 5: Dignosing the Verticl Structure of the Eddy Diffusivity in Rel nd Idelized Atmospheres. Q. J. R. Meteorol. Soc., 4 (687), Kspi, Y., nd G. R. Flierl, 7: Formtion of jets y roclinic instility on gs plnet tmospheres. J. Atmos. Sci., 64 (9), Kspi, Y., nd T. Schneider, : Downstrem self-destruction of storm trcs. J. Atmos. Sci., 68, Kspi, Y., nd A. P. Showmn, 5: Three dimensionl tmospheric dynmics of terrestil exoplnets over wide rnge of oritl nd tmospheric prmeters. Astrophys. J., 84, 6. Kidston, J., nd G. K. Vllis, : Reltionship etween eddy-driven jet ltitude nd width. Geophys. Res. Lett., 37 (). Koshi, F., nd H. Kwmur, : Sesonl vrition nd instility nture of the North Pcific Sutropicl Countercurrent nd the Hwiin Lee Countercurrent. J. Geophys. Res., 7, 6. Koshy, J. N., nd K. Hmilton, : The horizontl inetic energy spectrum nd spectrl udget simulted y high-resolution troposphere-strtosphere-mesosphere GCM. J. Atmos. Sci., 58 (4), Krichnn, R. H., 967: Inertil rnges in two-dimensionl turulence. Phys. of Fluids,, Kuo, H. L., 949: Dynmic instility of two-dimensionl nondivergent flow in rotropic tmosphere. J. Meteor., 6 (), 5. Lmert, S. J., 984: A glol ville potentil energy-inetic energy udget in terms of the two-dimensionl wvenumer for the FGGE yer. Atmosphere-Ocen, (3), Lrichev, V. D., nd I. M. Held, 995: Eddy mplitudes nd fluxes in homogeneous model of fully developed roclinic instility. J. Phys. Ocenogr., 5 (), Lee, S., 5: Broclinic multiple zonl jets on the sphere. J. Atmos. Sci., 6 (7), Lorenz, E. N., 955: Aville potentil energy nd the mintennce of the generl circultion. Tellus, 7 (), Mrston, J. B., : Plnetry tmospheres s non-equilirium condensed mtter. Ann. Rev. Condens. Mtter Phys., 3, 85 3.

11 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 Merlis, T. M., nd T. Schneider, 9: Scles of liner roclinic instility nd mcroturulence in dry tmospheres. J. Atmos. Sci., 66 (6), Nstrom, G. D., nd K. A. Gge, 985: A climtology of tmospheric wvenumer spectr of wind nd temperture oserved y commercil ircrft. J. Atmos. Sci., 4 (9), Nvrr, A., nd G. Boccletti, : Numericl generl circultion experiments of sensitivity to erth rottion rte. Clim. Dyn., 9 (5-6), O Gormn, P. A., nd T. Schneider, 7: Recovery of tmospheric flow sttistics in generl circultion model without nonliner eddyeddy interctions. Geophys. Res. Lett., 34 (), L 8. O Gormn, P. A., nd T. Schneider, 8: The hydrologicl cycle over wide rnge of climtes simulted with n idelized GCM. J. Climte, (5), O Gormn, P. A., nd T. Schneider, 8: Wether-lyer dynmics of roclinic eddies nd multiple jets in n idelized generl circultion model. J. Atmos. Sci., 65 (), Pnett, R. L., 993: Zonl jets in wide rocliniclly unstle regions: Persistence nd scle selection. J. Atmos. Sci., 5 (4), Rhines, P. B., 975: Wves nd turulence on et plne. J. Fluid Mech., 69 (3), Rhines, P. B., 977: The dynmics of unstedy currents. The se, 6, Rhines, P. B., 979: Geostrophic turulence. Ann. Rev. Fluid Mech.,, Rhines, P. B., 994: Jets. Chos, 4 (), Roinson, W. A., 6: On the self mintennce of midltitude jets. J. Atmos. Sci., 63 (8), 9. Slmon, R., 978: Two-lyer qusi-geostrophic turulence in simple specil cse. Geophys. Astrophys. Fluid Dyn., (), 5 5. Sltzmn, B., 957: Equtions governing the energetics of the lrger scles of tmospheric turulence in the domin of wve numer. J. of Meteorology, 4 (6), Smith, K. S., 7: The geogrphy of liner roclinic instility in Erth s ocens. J. Mr. Res., 65 (5), Smith, K. S., nd G. K. Vllis, : The scles nd equilirtion of midocen eddies: Freely evolving flow. J. Phys. Ocenogr., 3 (), Smith, K. S., nd G. K. Vllis, : The scles nd equilirtion of midocen eddies: Forced-dissiptive flow. J. Phys. Ocenogr., 3 (6), Srinivsn, K., nd W. R. Young, : Zonostrophic instility. J. Atmos. Sci., 65 (9), Stone, P. H., 978: Broclinic djustment. J. Atmos. Sci., 35 (4), Theiss, J., 4: Equtorwrd energy cscde, criticl ltitude, nd the predominnce of cyclonic vortices in geostrophic turulence. J. Phys. Ocenogr., 34 (7), Tois, S. M., K. Dgon, nd J. B. Mrston, : Astrophysicl fluid dynmics vi direct sttisticl simultion. Astrophys. J., 77 (), 7. Tois, S. M., nd J. B. Mrston, 3: Direct sttisticl simultion of out-of-equilirium jets. Phys. Rev. Let., (), 4 5. Vllis, G. K., nd M. E. Mltrud, 993: Genertion of men flows nd jets on et plne nd over topogrphy. J. Phys. Ocenogr., 3 (7), Wler, C. C., nd T. Schneider, 6: Eddy influences on Hdley circultions: Simultions with n idelized GCM. J. Atmos. Sci., 63, Willims, G. P., 978: Plnetry circultions:. rotropic representtion of the Jovin nd terrestril turulence. J. Atmos. Sci., 35 (8), Willims, G. P., nd J. L. Hollowy, 98: The rnge nd unity of plnetry circultions. Nture, 97 (5864), Zurit-Gotor, P., 8: The sensitivity of the isentropic slope in primitive eqution dry model. J. Atmos. Sci., 65 (), Zurit-Gotor, P., nd G. K. Vllis, 9: Equilirtion of roclinic turulence in primitive equtions nd qusigeostrophic models. J. Atmos. Sci., 66 (4), Schneider, T., 4: The tropopuse nd the therml strtifiction in the extrtropics of dry tmosphere. J. Atmos. Sci., 6 (), Schneider, T., nd C. C. Wler, 6: Self-orgniztion of tmospheric mcroturulence into criticl sttes of we nonliner eddy eddy interctions. J. Atmos. Sci., 63 (6), Scott, R. B., nd F. Wng, 5: Direct evidence of n ocenic inverse inetic energy cscde from stellite ltimetry. J. Phys. Ocenogr., 35 (9), Shepherd, T. G., 987: Rossy wves nd two-dimensionl turulence in lrge-scle zonl jet. J. Fluid Mech., 83, Shepherd, T. G., 987: A spectrl view of nonliner fluxes nd sttionry-trnsient interction in the tmosphere. J. Atmos. Sci., 44 (8), Simmons, A. J., 974: The meridionl scle of roclinic wves. J. Atmos. Sci., 3 (6),