Mesoscale Meteorology: Supercell Dynamics 25, 27 April 2017 Overview Supercell thunderstorms are long-lived single-cell thunderstorms, with

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1 Mesoscale Meeorolog: Sercell Dnamics 5, 7 Aril 7 Oerie Sercell hndersorms are long-lied single-cell hndersorms, ih longeiies ranging rom o oer 6 h. In conras o single-cell hndersorms, hich hae no areciable mid-roosheric roaion, sercell hndersorms are characeried b a ersisen mid-roosheric mesocclone, or meso-γ-scale (i.e., cell-scale; O(- m)) area o large erical orici (O( - s - )) collocaed ih he cell s dra oer a signiican erical deh ihin he middle rooshere. Sercell hndersorms occr ihin enironmens o large loer-o-middle roosheric erical ind shear; generall, m s - oer he -6 m laer. As e ill soon demonsrae, he horional orici associaed ih he ericall-sheared lo is he mesocclone erical orici sorce. Large erical ind shear magnides iml ha sercells are aored in enironmens ih long hodograhs. Hodograhs in sercell-soring enironmens oen hae clocise crare (or eering inds ih heigh) ha aors cclonic sercell mesocclones. Conerclocise-cred hodograhs (or bacing inds ih heigh) aors anicclonic sercell mesocclones. Recalling ha bacing inds are associaed ih cold air adecion hrogh hermal ind, since sercell hndersorms reqire large srace-based moisre and insabili, anicclonic mesocclones are less common and oen ond a higher alides (here cold air adecion is less deleerios o srace-based insabili) han heir cclonic conerars. Whereas he moion o single-cell and, o lesser een, mlicell hndersorms has a signiican comonen associaed ih he mean ind oer he loer-o-middle rooshere, sercell moion oen deiaes sbsaniall rom his mean ind ecor. For clocise-cred hodograhs, sorm moion icall lies o he righ o he hodograh; he oosie is re or conerclocise-cred hodograhs. The relaiel large deh ( 6 m) o he erical ind shear laer, ls he magnide o erical ind shear ihin his laer ( m s - ), gies rise o signiican sorm-relaie inds oer mch o he rooshere. These sorm-relaie inds la o imoran roles in sercell mainenance and srcre. Firs, he sorm-relaie inds modlae here reciiaion alls o relaie o he loer-roosheric dra. Recall ha boanc is redced b hdromeeor mass; i his hdromeeor mass is locaed aa rom he dra, hen reciiaion does no signiicanl redce boanc. In general, he middle-o-er roosheric sorm-relaie lo is donind o he loer-roosheric dra. This is resonsible or sercells aearing elongaed donind in radar relecii daa. Second, he sorm-relaie inds modlae ho he gs ron roagaes oard relaie o he loer-roosheric dra. For single-cell and mlicell hndersorms, gs ron roagaion is aa rom he dra. For sercell hndersorms, hoeer, he dra and gs ron remain collocaed, osering qasi-ersisen ard-direced arcel acceleraion. The sorm-relaie lo is generall direced ard o he loer-roosheric dra, oosing he oard roagaion o he gs ron, hereb osering he snergisic collocaion o he gs ron and dra.

2 Sercell hndersorms orm in enironmens ih srace-based CAPE ales rom 5-5 J g -. As sercell dras originae a he srace, sercell hndersorms inherenl are sracebased eares. Sercell hndersorms icall orm in he local lae aernoon and earl eening hors in enironmens o minimal srace-based CIN. Snoic-scale orcing or ascen in sercell enironmens is icall onl large enogh o lead o conecion iniiaion i serosed ih local orcing or ascen. I he snoic-scale ascen is oo ea, conecion ma no iniiae. I he snoic-scale ascen is oo srong, oo mch conecion ma iniiae, and he ineracion beeen hese hndersorms and heir cold ool leads o heir scale groh ino an organied mesoscale comle. This can occr in he cenral Unied Saes as da becomes nigh and he nocrnal lo-leel je deelos, or in he resence o srong liing along an adancing cold ron. Sercell Srcre Sercell hndersorms are characeried b a single dra ha eends hrogh he deh o he rooshere. Collocaed ih his dra oer a sbsanial racion o is deh is a mesocclone. Maimm dra eloci icall eceeds m s -, ih elociies o o 5 m s - ossible in he mos inense sercells in he mos nsable enironmens (gien ha he maimm-heoreical dra eloci is direcl roorional o CAPE). A oering cmlonimbs clod characeries a sercell s isal aearance on he sorm-scale hereas on he dra-scale, he mesocclone resls in signiican clod crare in roimi o he dra. A all clod ma orm donind o he dra as rain-cooled and -hmidiied air is brogh inard oard he dra, ascending and reaching is LCL a a loer alide han he comaraiel less hmid enironmenal air. In radar relecii daa, a sercell dra is characeried b a bonded ea echo region, or a region here he dra is sicienl srong as o ee hdromeeors (i.e., radar arges) eleaed. In he loer rooshere, he ea echo region ma no be bonded on all sides b reciiaion. The ind reciiaion echo bonding he dra is resonsible or he archeal hoo echo signare o sercell hndersorms. In radar eloci daa, mesocclones are characeried b a cole o adjacen inbond and obond elociies. Dal-olariaion daa can be sed o ideni hail (collocaed high Z, lo CC, and negaie o large ZDR) or ornadoes (collocaed srong roaion, lo CC, and negaie or small ZDR) i resen in conjncion ih a gien sercell hndersorm. Sercell hndersorms are characeried b o rimar dra regions, one each in he orard and rearard direcions. The orard-lan dondra (FFD) resls rom donind hdromeeor deosiion b he sorm-relaie ind. Here, hdromeeor eaoraion, meling, and sblimaion resl in laen cooling, he deelomen o negaie boanc, and hs donard-direced arcel acceleraions. The inensi o he negaie boanc, and hs cold ool srengh (here deined as FFD en Tsc Tsc ), is a ncion o he enironmenal moisre roile as i inlences boh enrainmen and eaoraion oenial. The oard sread o he gs ron associaed ih he FFD is o some een resrained b he ind-direced loer roosheric sorm-relaie lo. The cases o he rear-lan dondra (RFD) are less ell-ndersood. One oenial case is he enrainmen o relaiel dr enironmenal air in he middle-o-er rooshere. As in he FFD, his romoes laen cooling, negaie boanc, and donard-direced arcel acceleraions. I he hdromeeor mass becomes oo large ihin a colmn o sor i remaining ssended, is

3 allo ma also resl in a dondra. Finall, a donard-direced erical errbaion ressre gradien orce on he rear lan o he sercell ma also resl in dondra ormaion. Which is dominan liel aries beeen sercells. The resling cold ool srengh aries; laen cooling icall resls in sronger cold ools han hdromeeor loading or ressre gradien orcing. A sercell s loer-roosheric dra is sicienl srong o li boh arm enironmenal air as ell as rain-cooled (and hs sabilied) FFD air. In he laer case, mechanical li is sicienl large as o oercome he negaie boanc generaed b eaoraion, meling, and sblimaion. The near-srace inlo region o a sercell hndersorm is characeried b a dnamic ressre minimm o -3 hpa non as an inlo lo ha sraddles he FFD. This ressre minimm is associaed ih he srong inlo (and hs rearard-direced horional arcel acceleraions) ino he loer-roosheric dra. This connecion can be demonsraed sing a ernolli eqaion, hich assmes sead-sae (or irroaional) lo ha is ricionless and no aeced b he Coriolis orce. Under hese assmions, he olloing eression holds along a srace ( ) sreamline: ρ consan Here, is he ind seed and no he meridional ind comonen. Assming consan densi, i is large, ms be small (and ice ersa). Ths, relaiel as inlo is associaed ih relaiel lo srace ressre. Diereniaing his eqaion along a sreamline (he s direcion), e obain: en nearsorm ρ ( ) en nearsorm For a ind seed increase rom 5- m s - beeen he ambien and near-sorm enironmens, and ρ g m -3, a srace ressre minimm o.88 hpa resls, consisen ih obseraions. To his oin, e hae considered ha can be called classic sercells. Varians on his srcre eis in he orm o high-reciiaion (HP) and lo-reciiaion (LP) sercells, disingished b he aored locaion or reciiaion allo relaie o he dra. For LP sercells, mos reciiaion allo occrs ell donind o he dra, here i icall eaoraes ihin he sbsaraed sb-clod laer beore reaching he grond. The FFD ends o be ea, hereas he RFD ma be absen. For HP sercells, mos reciiaion allo occrs near he dra ihin he hoo echo and on he bacside o he dra, sch ha he dra is oen obscred b rain. LP sercells are aored hen he er-roosheric sorm-relaie ind is relaiel as (> 3 m s - ) and, o some een, in drier ambien enironmens, hereas HP sercells are aored hen he er-roosheric sorm-relaie ind is relaiel slo (< m s - ) and hen man sorms eis close o one anoher. Sercell Mesocclone Deelomen To ndersand sercell mesocclone ormaion, e se he erical orici endenc eqaion. Js as orici is he crl o he eloci, orici endenc is he crl o he eloci endenc manies in he orm o he momenm eqaion. As e are ineresed in roaion in he - lane, e begin b considering onl he comonen o he orici endenc in his lane: 3

4 4 ( ) ( ) F ω ω The ossinesq aroimaion as made o obain his erical orici endenc eqaion. The irs righ-hand side erm reresens adecion, he second and hird righ-hand side erms reresen he combined eecs o iling horional orici ino he erical and sreching erical and lanear orici, and he las righ-hand side erm reresens ricion. On sercell scales, he Coriolis orce is icall nimoran and ricional generaion is icall small. Neglecing hese orcings, e obain: η ξ ω here ( ) j i j i η ξ ω. Sbsiing, e obain: The orh and ih righ-hand side erms reresen iling horional orici ino he erical; he sih righ-hand side erm reresens sreching o erical orici. The irs hree erms reresen erical orici adecion in boh horional and erical direcions. Adecion seres o ransor erical orici; sreching amliies re-eising erical orici; and iling can generae ne erical orici rom re-eising horional orici (sch as ih he ericall-sheared lo). To simli or ealaion, e can deelo a linearied orm o his eqaion. Le ( ), ( ), and. Oerbar erms reresen he ambien enironmen; e.g., a horionallhomogeneos ericall-sheared horional ind ih ero erical eloci. Prime erms reresen sorm-scale dearres rom he ambien enironmen. Noe ha gien he deiniions or and, e can sho ha: ( ) ( ) since he oerbar erms ar in he erical direcion onl. I e sbsie hese deiniions ino he erical orici endenc eqaion, e obain: ( ) ( ) ( ) ( ) Neglecing he rodcs o errbaion erms, his simliies o:

5 The irs o righ-hand side erms reresen horional adecion. The second o righ-hand side erms reresen iling o horional orici associaed ih he ericall-sheared lo ino he erical. Seeral erms, noabl he sreching erm, are no resen as he are non-linear orcing erms. We ill reisi he sreching erm shorl. Combining he o adecion erms, he aboe eqaion can be rien as: h I he dra moes ih consan eloci c, he sorm-relaie (dra-relaie) erical orici endenc can be rien as: sr ( c) h Noe ha c onl eners he adecion erm, and hen onl or he horional eloci. Sbracing a consan ale across he enire domain (,, ) does no change he horional or erical gradiens o a gien qani. I one oin iniiall has a ale o and he adjacen oin iniiall has a ale o 5, sch ha heir dierence is 4, sbracing 3 rom each oin does no change heir dierence: he oins no hae ales o - and, he dierence o hich remains 4. As noed beore, adecion canno generae erical orici. I acs onl on, sch ha absen sorm-scale erical orici (as is re in he ambien enironmen), he adecion is ero. Tiling can generae erical orici, hoeer. Consider an enironmen o onl eserl erical ind shear, sch ha he iling erm simliies o he las righ-hand side erm aboe. Ne, consider an isolaed dra ( > ) embedded ihin his enironmen. Norh o he dra, <, sch ha <, generaing anicclonic orici. Soh o he dra, >, sch ha >, generaing cclonic orici. The aboe eamle can be generalied o an erical ind shear, nidirecional or oherise, and o dondras. Since a he grond (no erical eloci hrogh a rigid srace), he orici endenc holds or he loer o middle rooshere. Earlier in he semeser, e demonsraed ha he horional orici ecor oins 9 le o he erical ind shear ecor. Tiling generaes localied erical orici anomalies ha lie along he horional orici ecor, ih boh he horional orici ecor and errbaion erical orici gradien oining rom soh o norh in his eamle. Once erical orici has been generaed b iling, adecion b he sorm-relaie ind becomes imoran. Gien he relaionshi beeen he horional orici ecor and errbaion erical 5

6 orici gradien ha e js deeloed and he deiniion o he do rodc, he olloing insigh can be obained: For sorm-relaie lo arallel o and along he horional orici ecor, he sormrelaie lo ill adec he cclonic erical orici anomal nderneah he dra. For sorm-relaie lo arallel o and oosing he horional orici ecor, he sormrelaie lo ill adec he anicclonic erical orici anomal nderneah he dra. For sorm-relaie lo erendiclar o he horional orici ecor, he sorm-relaie lo ill adec neiher erical orici anomal nderneah he dra. These are sreamise orici, anisreamise orici, and crossise orici, reseciel. For sorm moion along a sraigh hodograh, he horional orici is enirel crossise. For sorm moion o he hodograh and/or hen he hodograh is cred, sreamise orici is nonero. The greaer he deiaion o sorm moion o he hodograh, he greaer he sreamise orici; sercelllar moion in a direcion oher han ha gien b he mean ind oer he loero-middle rooshere increases he sreamise orici. Here is here he non-linear sreching erm becomes imoran. In is basic orm, i reresens he rodc o he erical orici and he erical ariaion in erical eloci. Consider an dra ih ero erical eloci a he grond and rooase and maimm eloci a mid-leels. This erical disribion or is airl common or dee, mois conecion de o conini and gien a boanc maimm in he middle rooshere (since arcel erical acceleraions are direcl roorional o boanc). For sorm-relaie lo adecing a cclonic erical orici anomal nderneah he dra, sreching amliies he cclonic erical orici hrogho he loer o middle rooshere! From his, i sands o ollo ha cclonicall-roaing sercell mesocclones acqire cclonic erical orici irs b iling horional orici ino he erical lane, adecing i beneah he dra, and hen b amliing i hrogh dra sreching. Noe ha his is no he same rocess ha leads o ornadogenesis, or roaion a he grond, hich is beond he scoe o his class. The reqiremen o iling reqires here o be ambien erical ind shear, ariclarl in he loer o middle rooshere, and he reqiremen o iling reqires ha he sorm-relaie lo be able o adec he iled orici beneah he dra. Single-cell conecion, gien ea ambien erical ind shear, canno generae erical orici b iling. Mlicell conecion, gien he coninal disrion o he loer roosheric dra b he adancing gs ron, canno coninall adec he iled orici beneah he dra o be amliied ia sreching. Onl sercells, ih large ambien erical ind shear and resrained gs ron adancemen, can comlee he rocess. Gien he imorance o sreamise orici o sercell mesocclone deelomen, e desire o obain a qaniaie esimae o sreamise orici ihin he ambien enironmen. Generall, helici is a measre o he degree o hich lid moion is aligned ih he orici o he lid, ms as e demonsraed ih he sorm-relaie adecion erm o he linearied erical orici endenc eqaion. A a single leel, helici is gien b: H ω 6

7 Or, ericall-inegraed rom he srace (recalling ha sercells are srace-based) o a heigh d aen o reresen he o o he sorm inlo laer (icall -3 m aboe grond leel), d H ωd In he ambien enironmen, sorm-scale errbaions are ero, sch ha and are ncions o onl and. Ths, and helici becomes: ω i j ωh or, in he sorm-relaie sense, H d ω d h H d ( c) d ω here he enironmenal horional orici is nchanged b sbracing he sorm moion c. Recall ha he do rodc beeen o ecors is ero hen he are erendiclar, is a maimm osiie ale hen he are arallel and in he same direcion, and is a maimm negaie ale hen he are arallel and in he oosie direcion. Since sreamise orici is deined b he rojecion o he horional orici ono he sorm-relaie moion ecor, he helici aboe is a ericall-inegraed measre o sreamise orici non as sorm-relaie helici. For an obsered or esimaed c, and since he ambien horional orici is onl a ncion o he ericall-sheared lo, e can calclae he sorm-relaie helici i e no he erical roile o ind ihin he sorm enironmen. On a hodograh, e can esimae he sorm-relaie helici rom he area beeen he cres c o b he srace sorm-relaie moion, he sorm-relaie moion a he o o he inlo laer (e.g., or 3 m), and he hodograh isel. This is erhas he a b hich he saemen aboe regarding ho deian moion rom he mean ind increases he sreamise orici can be bes isalied. For a clocise-cred hodograh, sorm moion rogressiel rher o he righ o he mean ind increases he area c o b he aoremenioned cres, hs increasing he sreamise orici and sorm-relaie helici. Aboe, e emhasied he iling o ambien horional orici associaed ih he ericallshear lo ino he erical. Hoeer, baroclinicall-generaed horional orici associaed ih he FFD ma also be imoran or sercell mesocclone deelomen. Air ihin he FFD is less boan han ha o is soh ihin he ambien enironmen, esablishing a negaie meridional gradien in boanc across he FFD gs ron. Js as e deeloed a erical orici endenc eqaion, so oo can e deelo a endenc eqaion or horional orici across he gs ron: ξ ω i ξ ω i ( F) h 7

8 The righ-hand side orcing erms reresen adecion, sreching and iling o horional orici, iling o lanear orici ino he horional, baroclinic generaion, and ricion, reseciel. Neglecing all b he baroclinic generaion erm or simlici, e obain: ξ For he gien meridional boanc disribion, ξ decreases (or becomes more negaie) ih ime, imling a esard-direced horional orici ecor rom baroclinic generaion along he gs ron s leading edge. This horional ecor reresens norherl lo a he srace, ascen along he gs ron s leading edge, and soherl lo alo, or soherl erical ind shear consisen ih he horional orici ecor. Since sorm-relaie inlo is icall direced oard he dra, hich lies o he es o he FFD s leading edge, he sorm-relaie moion ecor and horional orici ecor associaed ih baroclinic generaion are largel arallel and oined in he same direcion e.g., sreamise! Gien ha a orion o sercell inlo originaes ihin he rain-cooled and -hmiied air along he FFD (since he sercell dra is icall srong enogh o li his air o is LFC desie is non-ero CIN), baroclinic generaion o horional orici can be an imoran conribor o he iling, adecion, and sreching rocesses imoran or sercell mesocclone deelomen. Sercell Proagaion As e ill demonsrae, errbaion erical ressre gradien orces ond ihin sercells are resonsible or heir roagaion characerisics, inclding sliing and deian moion. To do so, e irs need o deelo he aroriae mahemaical relaionshis. We sar ih he momenm eqaion: d Ω g F d ρ Neglecing he erical comonen o he Coriolis orce and a Ω cos ϕ erm in he eansion o he -Ω erm, his erm is eqal o -, sch ha: d d g F ρ I e eand he oal deriaie, neglec ricion, and noe ha g g, e obain: g ρ For a base-sae ha is in hdrosaic balance ih ( ) g ρ ρ, i can be shon ha: 8

9 9 The ransormaion o he horional ressre gradien erms is sraighorard; is no a ncion o or, sch ha arial deriaies in hese direcions are ero. The ransormaion o he erical ressre gradien erm ollos rom he deriaion resened in he reios lecre noes. I e also mae he ossinesq aroimaion, hen e obain: α Ne, e ish o ae he diergence, or, o his eqaion. Doing so, e obain: ( ) ( ) ( ) ( ) α The ossinesq aroimaion reqires ha, sch ha he ime endenc erm is ero. The boanc erm ealaes o. Since α is consan, he errbaion ressre gradien erm ealaes o α, here he gradien oeraor doed ino a gradien resls in he Lalacian. Ths, e obain: ( ) ( ) α Eansion o he irs righ-hand side erm resls in: ( ) Noing ha some erms in he iniial eansion o his erm are ero becase. Eansion o he second righ-hand side erm resls in: ( ) ( ) ( ) β ) ( j i Ths, e obain: β α We ill neglec he o Coriolis erms and β or being small on he mesoscale. I e deine a horionall-homogeneos base-sae ih erical ind shear ha is in hdrosaic balance, sch ha ( ), ( ), and, and sbsie ino he aboe, e obain: α

10 In he aboe are lid eension, non-linear dnamical, linear dnamical, and boanc orcings. We ish o simli he non-linear dnamical orcing erms. Firs, noe ha he deormaion ensor alied o he errbaion ind can be rien as: i j i j j i ij ε here he diergence is again ero or he ossinesq aroimaion. The hree ses o erms on he hird line are deormaion erms. I e assme ha erical orici (i.e., roaion) dominaes oer deormaion ihin he sercell dra, each o hese erms are assmed o be ero. Ths, Frher, i e assme ha horional orici is ero ihin he sercell dra, hen: ξ and η Comaring hese ses o relaions, e ind ha he hird in he irs and he irs in he second se iml relaionshis o oosing sign. The same is re or he second in he irs and he second in he second se. The onl a or hese ses o eqaliies o be re is i each erm in hem is ero. Sbsiing his ino or ressre errbaion eqaion, e obain: α The remaining non-linear erm can be rerien sing he irs relaion in he irs se aboe and a binomial eansion (ia a cleer algebraic rerie o his orcing erm):

11 Sbsiing his and neglecing he lid eension erms as comaraiel small ihin sercell dras, e obain: α Noing ha, e obain: The righ-hand side erms here reresen non-linear dnamical orcing associaed ih sin, linear dnamical orcing associaed ih he enironmenal erical ind shear, and boanc orcing. In non-sercell enironmens, he boanc erm orcing dominaes. In sercell enironmens, hoeer, he dnamical orcing erms are o comarable magnide o he boanc orcing erm. Here, e are chiel ineresed in he dnamical orcing or erical arcel acceleraions, sch ha e ill neglec he boanc erm. The non-linear orcing erm is resonsible or sliing sercells ha moe o he le and righ o he mean ind. The imac o he linear orcing erm aries deending on hodograh crare: Clocise: aors he righ-sliing sercell; he le-sliing sercell decas. Conerclocise: aors he le-sliing sercell; he righ-sliing sercell decas. Sraigh: neiher he righ- or le-sliing sercell is aored oer he oher. Le s consider he non-linear orcing erm irs. For simlici, consider a eserl enironmenal erical ind shear ih rel onal lo, sch ha he hodograh is sraigh, he erical ind shear ecor is oined o he eas, and he horional orici ecor is oined o he norh. (The insigh belo holds or an sraigh hodograh, no js one ih onl onal lo.) Via iling, he sercell dra resls in he deelomen o cclonic errbaion erical orici o he soh and anicclonic errbaion erical orici o he norh o he dra maimied a mid-leels. Since.5, boh erical orici anomalies are associaed ih <, ih larges ales ond in he middle rooshere. This esablishes ard-direced errbaion ressre gradien orces norh and soh o he main sercell dra ha romoe sercell sliing. Sliing is acceleraed hen a hdromeeor- or eaoraion-indced dondra is ond in roimi o he iniial sercell dra. Sliing icall occrs ihin.5- h o sercell iniiaion and can occr mlile imes dring a sercell s lieccle. Sliing resls in disinc dras ha moe o he le and righ o he mean ind oard he erical orici anomalies. Noe ha his is disinc rom he adecion o he erical orici anomalies beneah he dra ha e said as imoran or sercell mesocclone deelomen; indeed, or he sraigh hodograh assmed in his eamle ih sorm moion along he mean ind ecor, horional orici as iniiall crossise. Hoeer, deian sorm moion osered

12 b sliing resls in horional orici no longer being onl crossise: here is a sreamise comonen or he righ-moer and an anisreamise comonen or he le-moer. The o ne, disinc dras hemseles hen il enironmenal horional orici (hich is sill redominanl crossise in his eamle) ino he erical. The resling erical errbaion ressre gradien orce romoes he sliing o each o hese cells. The le-sli rom he original le-sli and righ-sli rom he original righ-sli romoe conined deian roagaion, hile he righ-sli rom he original le-sli and he le-sli rom he original righ-sli ma inerac ih each oher in eiher consrcie (e.g., merger and scale groh) or desrcie as. Wha abo he case here he enironmenal hodograh is cred and onl sreamise orici is resen? Here, as he original sercell dra ils he horional orici ino he erical, he cclonic erical orici anomal is assmed o near-insananeosl be adeced b he sormrelaie moion nderneah he dra, herein sreching amliies i. This resls in an arddireced errbaion ressre gradien orce ha is collocaed ih he original dra, reinorcing he dra raher han resling in sorm sliing. Conseqenl, he abili or a sercell o sli is direcl roorional o he racion o enironmenal horional orici ha is crossise. No, le s consider he linear orcing erm. For simlici, consider a eserl enironmenal erical ind shear ih rel onal lo, sch ha he hodograh is sraigh and he erical ind shear ecor is oined o he eas. Ths, he linear orcing erm in his eamle simliies o: A he srace,, sch ha horional gradiens o are also ero. In he middle rooshere, >>, sch ha > es and < eas o he dra. Thereore, d decreases ih heigh o he eas o he dra and increases ih heigh o he es o he dra. This esablishes a donard-direced errbaion ressre gradien orce es and an ard-direced errbaion ressre gradien orce eas o he dra. This also esablishes a horional errbaion ressre gradien orce direced oard he eas in he middle rooshere. For sercell roagaion along he mean eserl ind, he errbaion ressre gradien orces do no imac sercell moion, nor romoe he le- or righ-sliing cells ha ma orm de o non-linear orcing. Wha abo i he enironmenal hodograh is cred? Consider he case o easerl srace inds eering o soherl and hen eserl hrogh he loer o middle rooshere. For simlici, e ill also assme ha ind seed remains consan ih heigh. This deines a clocise-cred hodograh ha aes he shae o a semicircle cenered on he origin. In his case, he local erical ind shear ecor is soherl near he srace, eserl in he loer rooshere, and norherl in he middle rooshere. For sorm moion eqal o he mean ind oer his laer (e.g., slighl o he norh), he horional orici is redominanl b no enirel sreamise. The ll linear orcing erm ms be considered in his ealaion; recall ha i aes he orm:

13 Near he srace, > soh and < norh o he dra. In he loer rooshere, > es and < eas o he dra. Finall, in he middle rooshere, < soh and > norh o he dra. The combinaion o he near-srace and middle-roosheric ressre anomalies ses errbaion erical ressre gradiens: ard o he soh and donard o he norh o he dra. Recalling ha his hodograh conains a small amon o crossise orici, sliing is liel. As a resl, he linear dnamic erm romoes righ-sli mainenance, ih addiie ard-direced errbaion ressre gradien orces. The canceling errbaion erical ressre gradien orces romoe he deca o he le-sli. Deian moion o he righ o he mean-ind b he righ-sli decreases he amon o crossise orici, sch ha sliing becomes less liel; hoeer, he linear dnamical orcing sill acs, resling in a single, non-sliing sercell ha roagaes o he righ o he mean ind. Ths, or a clocise-cred hodograh, he linear dnamical orcing erm resls in deian moion o he righ o he mean ind. Wha abo he case o a conerclocise-cred hodograh? For sercell enironmens, sch a hodograh is icall ond in he loer-o-middle rooshere, ih soheserl inds ao he bondar laer ha eer o eserl in he middle rooshere beore bacing o soheserl a higher alides. Ths, he local enironmenal erical ind shear ecor is norherl ao he bondar laer, eserl in he middle rooshere, and soherl a higher alides. From he ll linear orcing erm, < soh and > norh o he dra ao he bondar laer, hereas > soh and < norh o he dra a higher alides. This esablishes errbaion erical ressre gradiens as ell: donard o he soh and ard o he norh o he dra. This can romoe he mainenance o a le-sli and he deca o a righ-sli. Hoeer, as sch hodograhs also icall hae srong clocise crare in he loer rooshere, here is orcing or boh le- and righ-slis o be mainained in his scenario. This oen leads o mess sorm eolions. 3