Adveco
Cycloe A-cycloe Lorez (963) Low dmesoal aracors. Uclear f hey are a good aalogy o he rue clmae sysem, bu hey have some appealg characerscs.
Dscusso Is he al codo balaced? Is here a al adjusme perod? Wha s a rage of predco ( days)? Does hs deped o he al codos? Wha s he average (me mea) emperaure dfferece? How does hs compare o he radave eulbrum emperaure dfferece? Is he poleward hea raspor always posve? Wha s he mea wdspeed? (H: cosder he emporal varace wdspeed) Are your resuls sesve o seleco of free parameers?
Weaher ad clmae I he model how do you chage he weaher? how do you chage he clmae? (how do you DEFINE he clmae?)
Adveco (lear, ad oo-lear) Lear models have relavely smple behavor (cosder a damped harmoc oscllaor) We may fer ha o-leares make models eresg, ad ofe reure umercal mehods The cycloe model has eresg behavor because of he o-lear adveco. (produc VT) Adveco s oe key aspec of hydrodyamc flows
Problems me ad space (classes of d order PDEs) T K T = T c T = ), ( y f y T T = + T c T = Wave euao (hyperbolc) Posso euao (ellpc) Plus suable boudary ad al codos These are he smples caocal forms hey ge asy uckly Dffuso euao (parabolc) Adveco euao (frs order)
Adveco physcal erpreao = u u ud / > 0; / < 0 Adveco descrbes he raspor of a uay o parcel/rego of ar fed place..e., a raslao of he paer (surface), o modfcao followg he moo Toal dervaves for Lagraga vew (followg he moo) Paral dervaves gve Eulara vew (fed space)
Adveco Cosder he coservao of a uay: = (,) [could be CO, poeal emperaure, ozoe, poeal vorcy, waer ] d = d S S s some source d d = + d d = S = u + S e.g., S = 0 -dmesoal adveco euao ( oe way wave euao ) More geerally, = V + S
Fe dfferece schemes Up-sream: Flu flu ou More formal fe dffereces Forward me, ceered space Ceered me, ceered space 0 0 ) / ( ) / ( < > + u u u u u = + u = + + Oe ca of course cosruc may ohers.
Lab eercse How accurae s he fe dfferece predco? Cosruc a -d model of lear adveco (.e., u = cosa) o represe he zoal flow. (.e., perodc aroud oe parcular laude) Develop mercs of error (cosder phase ad amplude!) Do resuls chage wh wave umber, me sep, Coura umber, shape of fuco, me fler? Noe hs assgme wll be useful for he md-erm assgme!
Today s assgme u = = + + u Cosruc a fe dfferece model o es he depedece o he model accuracy o he deals of he scheme. Tes sesvy of he model resuls o:. he Coura umber (each of u,, ad ). sregh of me fler (zero s a good sarg place) 3. dffere shaped al dsrbuos of (sar wh a se wave) Cosder dfferece bewee sably ad accuracy.
Thgs o cosder buldg model Wha ype of fe dfferece scheme? Wha are he boudary codos? Wha oupu do you eed? How ca you measure errors? Wha s ruh? Wha are 6 ways o double check he code!
Tasks Code asks: Defe a array ( space)! Wre a subroue o compue he edecy Wre a subroue o do he me seppg Oupu smulao resuls ( sae ) oly a some fraco of me seps, ad plo wh IDL Oupu error dagoscs, ad plo wh IDL Scece asks: Does he scheme coserve mass? Does he scheme coserve varace? Wha s he crcal Coura umber sably (ry wh/whou a me fler)? Is he forward me dfferece ruly usable (as predced)?
Sably aalyss = u Vo Neuma mehod (he epermeal verso) Cosder kow soluo: = A ep( φ) Check how amplude chages wh me If A coues o grow wh me, he usable If bouded, he sable, bu may o be accurae! Ceraly errors amplude, bu also cosder phase errors. Coura umber C = u / < ~ for sably (Ths s he CFL codo)
Coura-Fredrchs-Levy (CFL) codo Iformao ca o move more ha oe grd bo oe me sep. u C = Tme akes o move across oe bo s v mus be greaer ha he me sep. Thus mamum speed of ay sable moo s lmed by he combao of ad. < Aoher erpreao: + = ( C) + C So advecve raslao s a erpolao space. If C >, he s a erapolao, ad suscepble o epoeally growg errors (amplude grows each me sep)