The Pressure Perturbation Equation: Exposed!

Transcription

1 Pressre Perrbain Eqain Page f 6 The Pressre Perrbain Eqain: Esed! The rainal dnamics f sercell srms hae a l d ih he ressre errbains creaed b he air fl. I is his effec ha makes sercells secial. Phase : Deriain Take he diergence f he ecr eqain f min: k d dv ˆ r in cmnen frm: Firs erm: Secnd erm: Third erm:. If e assme ha he erical ind field is linear hen secnd deriaie erms anish. If e assme als ha he ind field is nchanging ih ime I kn a srech gien a sercell in he neighbrhd hen erms inling can als be negleced. Take he remaining deriaies and gr erms as flls: Nice ha e hae sqared erms and sme erms ha are he same i.e. mliles f a erm. N rerie he eqain in erms f he ressre errbain hich is ha e an! A C On he righ-hand side gr A reresens he flid eensin erms; b are he nn-linear erms bisl and c is he erical banc gradien. Ne he reader: This is a rerdcin f nes I k in METR 4433 Messcale Meerlg a he Uniersi f Oklahma. I hae eanded he deriain s ha i migh make a lile mre sense. If find an errr lease ell me! ck

2 Pressre Perrbain Eqain Page f 6 Phase : Mean and Perrbain Fls N le s ariin he inds beeen an enirnmenal mean ind cmnen and he hndersrm-indced errbain inds: We assme he mean fl is hrinal. Tha is all erical min is aribed he hndersrm; mre secificall he erical min indced b he hndersrm is a resl f he ressre errbains ihin he srm. N sbsie he mean and errbain relainshis in he ressre errbain eqain e hae s far deeled. Ne ha in erms A nl he errbains ill be lef in and he mean fl des n ar in he hrinal; in here is n mean fl!. In erms here ill be erms ih mean fl can see hich nes?. Afer his rcess he eqain becmes: Rerie he erms inling mean ind shear b hemseles and n gr all he nn-linear erms geher. This gies Phase 3. Alicain Udraf enhancemen in rain hndersrms and cell diisin This is an elliic diagnsic eqain fr ressre errbain. We can diide he al errbain ressre as flls:

3 dn banc r linear nn linear banc Each ar is aribed cerain erms n he righ hand side f he ressre errbain eqain. We sa earlier ha a srng draf in an enirnmen f significan erical shear rdces a air f cner-raing rices. Cnsider he nn-linear erm nl. I can be erified ha he hree shear erms can be rien in he flling frm: If e assme re rain s here is n diergence nr defrmain and ignre he eensin erms he sqared erms in nn-linear ; i.e. e are lking a he effec f rain nl hen nnlinear ζ Since he lef-hand side cnains a alacian erar ζ herefre h cclnic and anicclnic rain rdce negaie ressre errbain. The l ressre cener is acall reqired hae a PGF ha balances he cenrifgal frce! Negaie ill be larges here rain is he srnges hich is sall a he mid-leels f hndersrms. The magnide f he ressre errbain here can be as high as -4 mb. Earlier figres sh ha becase f iling erical rain is he srnges a he flanks f he draf and he negaie a he mid-leels creaes an draf ressre gradien frce ha rmes ne drafs here a dnamic case fr cell-sliing. N le s cnsider he flid eensin erms.... The righ-hand side ill alas be negaie; herefre he flid eensin erms alas creae a siie ressre errbain. Where is sreching he larges? A he l and high leels f he amshere. Pressre Perrbain Eqain Page 3 f 6

4 S e hae > < > ard erical PGF Therefre he nn-linear ressre errbain de shear and sreching creaes addiinal ard lifing a ressre gradien frce in he ler amshere ha enhances he draf bend ha based n banc alne! Rle f hmb: mb f VPGF er km ~ same frcing as 3 C f banc. Relean Figres: 3. and 3.3 in lesein Vlme. Righ-ming srms Cnsider he linear erm: V s V Fr he sake f simlici assme ha shear is nidirecinal ih heigh fr n: eas Pressre Perrbain Eqain Page 4 f 6

5 On he esern r shear flank f he draf >. n he easern dnshear flank <. As e learned befre he ressre errbain is larges a he midleels. Ths e can eec ne cell grh n he dnshear flank in nidirecinal shear. N a mre cmlicaed scenari. e he hdgrah be clckise cred as flls: We n ms cnsider bh erms. A he l leels rdces A he mid-leels rdces A he er leels rdces Ths here is an ard dnard erical PGF n he righ lef flank f he srm ne cell grh is enhanced he righ raing draf becmes a righ mer. Relean Figres: and 3.5 in lesein Vl. Pressre Perrbain Eqain Page 5 f 6

6 Smmar The nn-linear shear effec rmes ne r cnined cell grh n he flanks f he ld cells. The linear effec f iling biases he cell memen ard he righ lef if he enirnmenal hdgrah is cred in a clckise cnerclckise manner. Unidirecinal shear rmes srms ha sli ih each member f he sli air haing cmnens f min nrmal he shear ecr and sie each her. Ne ban drafs frm ff he ais f he shear becase ard-direced errbain ressre gradiens indce ard accelerains here hs lifing he air is FC. Oing he l-leel cnergence assciaed ih ard-ming air rici increases hrgh he sreching f eising rici and is adeced ard b he draf. Righ ming srms end deel cclnic rain hile lef-ming srms end deel ani-cclnic rain. The cclnic rici rdced in srms ha gr in an enirnmen f clckise rning shear is NOT de he Earh s rain. S is he ressre errbain erm imran? Fr sercells es!!! Pressre Perrbain Eqain Page 6 f 6