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the Polar Stratospheric Vortex
by
XIAOHONG WANG
M.Sc., U niversity of V ictoria, 1992 B.Sc., .Jilin University, 1988
A D issertatio n S u b m itte d in P a rtia l Fulfillment of the R equirem ents for the Degree of
D O C T O R O F PH IL O SO PH Y in th e D e p a rtm e n t of C om puter Science We accept th is d issertatio n as conform ing
to th e required stan d ard
Dr. ■Iqiîïf'Fyfe, C o-supervisor (School of E a rth and Ocean Science) Dr. D ale Olesky, C o-supervisor [D ep artm en t of C om puter Science)
Dr. F ra ,obei|ts, D e p artm en tal M em ber (D epartm ent of C o m p u ter Science) Dr. AiidVew Weaver, O utside M em ber (School of E a rth an d O cean Science) Dr. C harles M cLandress, E x tern al E xam iner (University of W ashington)
© X IA O H O N G WANG, 1998 U niversity of V ictoria
All rights reserved. This d issertatio n m ay not be reproduced in whole or in p a rt, by photocopying or o th e r m eans, w ith o u t the perm ission of th e a u th o r.
A b s tr a c t
T h e breakdow n of vertically-propagating p lan etary waves in th e strato sp h ere is investigated using a n ultra-high horizontal resolution C o ntour D ynam ics w ith Surgery m odel. In th e m odel, p lan etary waves are forced a t th e tro p o p au se and p ropagate upw ards th ro u g h th e stra to sp h e re and into an absorbing sponge (the first of its kind for such a m odel). In th e context of wave breaking, two aspects of th e system are questioned, namely, 1) w hat is the sensitivity to u p p e r-b o u n d a ry conditions? an d 2) given perfect u p p er-b o u n d ary conditions w hat controls wave breaking?
1) In a Boussinesq environm ent, wave break in g is com pared using: a) a rigid upper- b o u n d ary condition (as in previous work) and b) an absorbing sponge (preventing spurious reflections). In a) b o th local (to th e forcing) and rem ote breaking is evidenced for weak forcing while only local breaking is observed for sufficiently strong forcing. In b) rem ote break in g is absent and local breaking, w hich occurs for sufficiently stro n g forcing, has quite a different ch aracter to th a t seen in a). C om pressibility effects are also investigated.
2) -A. quasi-linear slowly-varying theory is developed which predicts wave breaking if th e zonal m ean flow decelerates by m ore th a n one-half of its initial value (via positive group-velocity/zonal-m ean-flow feedbacks). T his so-called “o n e-h alf’ rule for p lan etary wave breaking is confirm ed th ro u g h fully-nonlinear sim ulations. N um erical sim ulations detail th e precise sequence of events leading u p to and after wave breaking.
supervisor (School of E a rth an d O cean Science)
Dr. Dale Olesky, C o-supervisor (D ep artm en t of C o m p u te r Science)
Dr. Frank R o b erts, D ep artm en tal M em ber (D ep artm en t of C o m p u te r Science)
Dr. A ndrew Weaver, O u tsid e M em ber (School o f E a rth an d O cean Science)
A ck n o w led g em en ts
F irst and forem ost, I would like to express my deep g ratitu d e to D r. John Fyfe, my thesis supervisor, for his rigorous guidance on my doctoral research and invaluable help in this d issertation w riting. His su p p o rt thro u g h o u t my P h .D stu d y is highly appreciated. I am also grateful to Dr. Dale Olesky, my d ep artm en tal supervisor, for all his su p p o rt and encouragem ent in th e p ast few years. T his d issertation is not possible w ith o u t them .
My hearty th a n k s also go to the scientists and staff a t C anadian C en tre for C lim ate Modelling and Analysis. T h ey have provided me w ith su p erb com puting facilities as well as a friendly working environm ent. Dr. R. Saravanan from N ational C entre for A tm ospheric Research kindly provided th e C ontour D ynam ics with Surgery co m p u ter code as well as detailed help in th e earlier stage of this research. I also benefited considerably from discussions w ith Dr. R uping Mo from McGill University on some of th e theoretical aspects of th is dissertation.
Scholarship and other financial su p p o rt from D ep artm en t of C o m p u ter Science, U ni versity of Victoria, an d C lim ate Research Network of th e A tm ospheric Research Service of C an ad a are gratefully acknowledged.
Last b u t not th e least, I give my sincere th an k s to my husband, Ming Li, for his love, patience, u n d erstan d in g and su p p o rt.
A b s tr a c t ii A c k n o w le d g e m e n ts iv D e d ic a t io n v C o n te n ts v i L is t o f F ig u r e s v iii L ist o f T a b le s x iv 1 I n tr o d u c tio n 1 2 T h e m o d e l 9 2.1 G overning e q u a t i o n s ... 9 2.2 Piecew ise-uniform p o ten tial v o r t i c i t y ... 12
2.4 Num erics ... 17
3 U p p e r -b o u n d a r y e ffe c t s on p la n e ta r y w a v e b r e a k in g 23 3.1 .Model s e t u p ...24
3.2 Vertical sponge sensitivity e x p e rim e n ts ...25
3.3 M ain experim ental r e s u l t s ... 26
3.3.1 Rigid u p p er b o u n d a r y ...26
3.3.2 V ertical sponge ...30
3.4 D is c u s s io n ...31
4 A n ew m e c h a n is m for p la n e t a r y w a v e b r e a k in g 48 4.1 T h e o r y ... 51
4.1.1 Q u a s i-lin e a riz a tio n ... 52
4.1.2 D ispersion relatio n and vertical group velocity ...54
4.1.3 Wave activity evolution e c p ia tio n ... 59
4.1.4 Zonal m ean flow e q u a t i o n ...61
4.1.5 T h e one-half rule for p lanetary wave b r e a k in g ... 62
4.2 Nonlinear num erical sim ulations ... 64
4.2.2 Steady c a s e ... 66 4.2.3 U nsteady case ... 66 4.3 S um m ary and d is c u s s io n ...68 5 C o n c lu sio n s 83 B ib lio g r a p h y 86 A W ave a c t iv it y 90 B In te g r a te d P V flu x 91 C Z on al m e a n flow r e la tio n s h ip 94 D S te a d y s t a t e w a v e a c t iv it y 98
1.1 G eopotciitial a t 10 m b for 1993-1997: (a) Dec-.Ian-Feb m ean for th e N o rth ern H em isphere; (b) .June-.Iuly-Aug m ean for the S o u th ern Hemis])here. U nits are m'^s“ ^. D a ta are from U nited K ingdom M eteorological Office (UKM O) analyses (Sw inbank a n d O ’Neill, 1994)... 7 1.2 G eopotential a t 10 m b for F ebruary 22, 1979 (OOZ). U nits are m^s""^. D a ta
are from N ational C entre for E nvironm ental Prediction (N C E P ) reanalyses (K alnay et ah, 199G)... S 2.1 P lan view of th e in itial polar vortex (bold circle) on an /- c a p plane. T he
topography is shown by shaded contours. T he N orth Pole is th e cen tre of the p lo t...21 2.2 A single patch vortex w ith in terio r and exterior vorticity Qi and Qo, re
spectively. C is the b o u n d in g contour an d the area enclosed by C is Tv.
3.1 T otal wave activ ity density norm alized by rj^ for a range of relaxation rates c ir w ith i'/'= 120k m (i.e., z r = 1 9 .5 ) an d r/o=0.15. W here applicable the vertical sponge b o tto m is indicated by a horizontal dashed line. N ote the nonlinear wave activity scale...34 3.2 As in F igure 3.1 b u t for £7’= 72km (i.e., z r = l l . ~ ) ...35 3.3 P erspective views of the vortex for th e case of a low rigid u p p er boundary:
(a) BCq,45 (7/0=0.4 5); (b) BCq go (7/0=0.CO)... 3C 3.4 T op level (dotted) and b o tto m level (solid) p o ten tial vorticity contours for
th e case of a low rigid u p p e r bo u n d ary : (a) B Cqas (t/o=0.45); (b) B Cq.go
(7/0=0.6 0). T h e b o tto m topography is show n by shaded contours...37 3.5 T otal (top) a n d relative (b o tto m ) wave activ ity density for the case of a low
rigid u p p er boundary: (a) BCqag (7/0=0.45); (b) BCo.eo (7/0=0.6 0 )... 38 3.6 As in Figure 3.3 for th e case of a high rigid u p p er boundary: (a) BTioAS
(7/0=0.4 5); (b) Bno. eo (7/0=0.60)... 39 3.7 As in Figure 3.4 for the case of a high rigid u p p er boundary: (a) B Hqag
(7/0=0.4 5); (b) B y . 0.60 (7/0=0.6 0) ... 40 3.8 .\ s in Figure 3.5 for the case of a high rigid u p p er boundary: (a) B'Hqag
(7/0=0.4 5); (b) B H o m (7/0=0.60)... 41 3.9 As in F igure 3.3 for the case of a vertical sponge ( 0 7 =1.6): (a) fî5o.45
(7/0=0.4 5); (b) B J0.6O (7/0=0.6 0). T h e th in solid contours lie w ithin the
3.10 A.S in Figure 3.4 for th e case of a vertical sponge ( a T = l-0 ): (a) B Sqas
(r/o=0.45); (b) B Sq.go (r/o=O.GO)...43 3.11 .As in F igure 3.5 for th e case of a vertical sponge (a'/-= 1.6): (a) C5o.45
(7/0=0.4 5); (b) i6»So.60 (7/0=0.6 0). T he vertical sponge b o tto m is indicated by th e horizontal dashed line...44
3.12 P erspective views of a compre.ssible vortex. Low rigid u pper boundary cases: (a) C B o.is (r/o=0.15); (b) CBo.i7S (7/o=0.175); (c) CBo.25 (r/o=0.25).
V ertical sponge cases: (d) C.So,i5 (?7o=0.15); (e) CSq.1 7 5 (r/o= 0.175): (f) C(9o.25 (7/0=0.2 5). N ote th a t th e plots do not include the region of the
vertical sponge...45
3.13 U pper-level (d o tted ) a n d bottom -level (solid) p o ten tial vorticity contours: Low rigid u p p e r b o u n d ary cases: (a) C £q.15 (7/0= 0. 1 5 ) ; (b) £ £0 .2 5 (7/0= 0 . 2 5 ) .
V ertical sponge cases: (c) C J o .is ( 7 / o = 0 . 1 5 ) ; (d) £ £0 .2 5 ( 7 / o = 0 . 2 5 )...46
3.14 P erspective views of a com pressible vortex for a very weak forcing am plitude of 7/0=0.0 4. (a) Low rigid lid: £ £0.0 4; (b) High rigid lid: CHo.oi\ (c) Vertical sponge: £5q.o4...47
represent th e relationship betw een .4 and u, (based on quasi-linear slowly- varying theory) th a t m ust exist in a steady s ta te for different forcing am plitudes 7/ (w here r/o < < % < % < T he straig h t line represents the relation betw een .4 a n d ïï th a t exists a t all tim e for a flow th a t evolves from the initial w ind 17°. T h e dashed curve m arks the critical am p litu d e i f above which no steady s ta te s can evolve from th e initial w ind v ° ...70 4.2 P lan view of th e initial p o lar vortex (bold circle) on an /- c a p plane. T he
topography is shaded. T h e centre of the plot is a t th e N orth Pole...71 4.3 Vertical profiles of th e (a) P V ju m p , A Q \ (b) zonal m ean flow a t r = Co, 77o:
(c) vertical wavelength, L \ v and (d) vertical group velocity, Cg. T h e solid curves correspond to the initial sta te while th e dashed curves were obtained diagnostically from a C D /C S sim ulation (r/o = 0.12) evaluated a t t = 100. . 72 4.4 Relation betw een th e initial s ta te as expressed by 7Z^/(7oAQ) and the
critical ratio Ti%/Ti%. T h e dashed horizontal line is th e approxim ate ra tio for sm all 77^/(roAQ). No propagating wave solutions exist beyond 77°/(7oAQ) = 0.5 (shaded area). T he solid dot identifies the initial setup and critical ra tio a t Z* in th e num erical experim ents discussed in Section
4.2 73
4.5 T he initial zonal m ean w ind distribution. T he zero wind line is indicated bv the thick c o n to u r... 74
4.6 Zoiially averaged zonal w ind field calculated from a 3-year ru n w ith the C anadian M iddle A tm osphere Model. C ontour interval is 10 m /s (Beagley ct ah, 1997)... 75 4.7 Wo/ïï° evaluated a t Z* for % = 0 .1 2 ,0.15,0.138, 0.16, 0.17. T h e dash ed line
is the critical ratio as p red icted theoretically... 76 4.8 Particle displacem ent co n to u r for r\o = 0.12 a t (a) t = 40, (b) t = 60,
(c) t = 80 and (d) t = 100. T h e shaded and unshaded contour represent positive and negative value respectively. T he contour interval is 0.5... 77 4.9 Wave activity density A (norm alized by for % = 0.12... 78 4.10 Zonal m ean wind (contours) and its change (shaded) for //o = 0.12 a t t =40,
60, 80 and 100... 79 4.11 Perspective view of th e vortex a t t = 166 for 7/o=0.16... 80 4.12 PV contour a t Z* for 7]o = 0.16 a t (a) t = 160; (b) t = 166; (c) t = 168; (d)
t = 174...81
4.13 PV contour (thick an d solid curve) and the stag n atio n point (solid dot) at Z* for //o = 0.16 at t = 156 [(a) and (c)] and t = 158 [(b) and (d)]. .Arrows in (a) and (b) represent th e velocity field. Shaded contours in (c) and (d) represent the stra in field aro u n d the stag n atio n p o in t...82
integrated zonal m ean flow deceleration, Atldr. D a ta points d enoted by
corresponds to A Q = 3.5, “-t-” to A Q = 4.5 and “o” to A Q = 5.5. The th re e straig h t lines are the linear flts to th e respective d a ta p o in ts... 97
3.1 S um m ary of Boussinesq experim ents perform ed, w here 27- is the dim en sional dom ain height (km ), a y th e relaxation ra te (where applicable) and
7/0 is the topographic am plitude. T he m ain set of experim ents to he dis cussed are in bold-type and the identifying labels are in parentheses. . . . 27
I n tr o d u c tio n
P lan etary waves are th e largest spatial scale waves on E a rth . T hese waves, which are com m only seen in th e N o rth ern H em isphere strato sp h ere, m ostly owe th eir existence to surface forcing associated w ith large-scale topography a n d /o r ocean-continent te m p e ra tu re co n trasts (C harney and Eliassen, 1949; Derom e an d VViin-Nielsen, 1971; G rose and Hoskins, 1979). Indeed, the fact th a t they are relatively weak in th e S o u th ern H em isphere strato sp h ere [see F igure 1.1(b)'] as com pared to th e N o rth ern H em isphere strato sp h ere [see Figure 1.1(a)] is p a rtly a ttrib u ta b le to th e obvious topographic ^isymmetrics existing between the hem ispheres.
T he p a rtic u la r aspect of p lan etary waves which is of in terest to us here is their ability to tra n sp o rt zonal m ean m om entum from the surface vertically, which when deposited aloft induces zonal m ean circulation changes. Given th a t zonal m ean vorticity g radients act as the restoring force for p lan etary waves, changes in th e zonal m ean circulation can in tu rn
so-called “wave-mean” in teractio n s can b e ra th e r ex trem e an d lead to wave breakdow n. W ave-mean interactions leading to wave breaking have been the subject of considerable research in the p ast (A ndrew s and M cIntyre, 1976; Tung an d Lindzen, 1979; D unkcrton, 1981; Fyfe and Held, 1990) an d are the general topic of th is dis.sertation.
O ur prim ary interest here is in wave-mean in teractio n s which lead to a n d /o r p rom ote p lan etary wave breaking. Like surface w ater waves approaching a beach, p lan etary waves are known to break as th ey p ro p ag ate vertically th ro u g h th e m iddle atm osphere. However, unlike surface w ater waves th e m echanism s underlying p lan etary wave breaking are less understo o d . Towards im proving o u r u n d erstan d in g of p lan etary wave breaking we have set th e following objectives for this study:
• To im plem ent an ultra-high horizontal resolution m echanistic m odel of the s tra to sphere which incorporates a n u p p er absorbing sponge. U ltra-high resolution is re- ([uired to o btain accu rate sim ulations of wave breaking, while an u p p er absorbing sponge is required to lim it back reflections of vertically-propagating p lan etary waves. • To develop a new analytical theory for the breakdow n of vertically-propagating p lan etary waves which does not rely on the usual assu m p tio n of “critical levels” in the zonal m ean How. T h e theory will be tested, a n d w here possible extended, througfi fnlly-nonlinear, u ltra-h ig h resolution sim ulations.
W hile the theory we will develop is fairly general in n a tu re we will focus o u r a tte n tio n on one specific application, namely, strato sp h eric sudden w arm ings. S tratospheric sudden
w arm ings arc one of the m ost strik in g exam ples of wave-mean interaction. Norm ally th e w inter strato sp h ere consists of an anticlockw ise vortex centered on a cold polar cap [see F igure 1.1(a)]. T he zonal m ean zonal wind (i.e., east-w est wind) is westerly (i.e., from the west) an d th e zonal m ean te m p e ra tu re decreases towards th e w inter jsole. However, d u r ing som e w inters the polar vortex is g reatly d isru p ted w ithin a few days (see Figure 1.2). D uring these ejjisodes the polar stra to sp h e ric winds decrease rapidly w ith tim e and may even reverse from westerly to easterly. W ith this, the polar vortex breaks up and tem p e ra tu re s rise by tens of degrees. For obvious reasons such events are called strato sp h eric su d d en w arm ings and it is generally accepted th a t the essential dynam ical m echanism re sponsible for them involves th e in teractio n of vertically-propagating p lan etary waves w ith strato sp h eric zonal m ean flows (.Andrews et ah, 1987).
M cIntyre an d Palm er (1983) used m eteorological analyses to show th a t p lan etary wave breaking is characteristic of th e circum stances leading to stratospheric sudden warm ings. T his sem inal observational p a p e r spaw ned a num ber of ultra-high horizontal resolution m odelling studies {.luckes an d M cIntyre, 1987; .luckes, 1989; W augh, 1993; and N orton, 1994) to investigate the dynam ics of pla n e ta ry wave breaking, and specifically the effect of breaking on the erosion of th e p o lar vortex. H orizontal resolution has been an im p o rtan t consideration in modelling stu d ies since observations show th a t when the polar vortex breaks up, small-scales and sh arp -g rad ien ts are rapidly generated over very localized re gions (M cIntyre and Palm er, 1983, 1984; C lough et ah, 1985; D unkcrton and Delisi, 1986; B aldw in and Holton, 1988).
used a “pseudo-spectral” num erical m eth o d which, as with o th er conventional m ethods, suffers from th e co m p u tatio n al disadvantage th a t uniform resolution is required over th e whole dom ain even though, in this physical problem , wave activity is localized near the edge of the polar vortex. Here wo will tak e a very different num erical approach to .IM an d solve the governing equations using th e “C o n to u r D ynam ics w ith Surgery” m ethod (hereafter C D /C S ). The C D /C S m ethod allows one to ad ap t th e resolution to the problem a t h an d and so exploit th e available co m puting resources much m ore fully.
“C ontour D ynam ics” was first proposed by Z abusky et al. (1979) and th en extended by D ritschel (1988) to include a set of “surgical” procedures which effectively allow for infinite horizontal resolution. C D /C S is a L agrangian m eth o d based on the recognition th a t in a conservative system the evolution of a p a tc h of uniform vorticity is fully described by the behaviour of its bounding contour. In practice, th is reduces th e dim ension of the physical problem in question by one, and in so doing greatly accelerates th e com putations. O f course, as the num ber of patches increases, so docs th e num ber of bounding contours and hence also th e co m putational burden. T herefore b est advantage of th e m ethod is realized in physical problem s where a sm all n u m b er of patches (ideally one) i.s ap p ro p riate, such as th e one a t h an d (Legras an d Dritschel, 1993; W augh, 1993).
T h e num erical model to be used here is th e three-dim ensional quasi-geostrophic C D /C S m odel developed by Dritschel and S aravanan (1994, hereafter DS). The m ain m odifica tion we have m ade to th e m odel of DS (to be described in detail in Section 2.3) is the im plem entation of an absorbing sponge a t the top of th e m odel dom ain in order to lim it
1976; K irkw ood an d Derome, 1977). Such absorbing sponges are com m on in num erical m odels [e.g., the C anadian M iddle A tm osphere Model (Beagley et al, 1997)] b u t are very difficult to in co rp o rate in C D /C S models because of th e conservation co n strain ts th a t the teclmicjues o f C D /C S impose. Indeed, th e absorbing sponge described in Section 2.3 is the first of its kind, a n d its design and im plem entation was th e first m ajo r hurdle we needed to overcom e to achieve our objectives. P relim inary results using th e absorbing sponge were published in Fyfe and W ang (1997).
Now for a description of th e key issue a t the h e a rt of this study. T he m ain issue here involves explaining why p lan etary waves are often observed to break a t heights lower th a n expected based on “critical level” theory (B a rn e tt, 1975; Schoeberl, 1978). C ritical level theory p red icts th a t waves will break w hen th eir p hase speed m atches th e speed of th e zonal m ean flow (Dickinson, 1970; W arn and W arn, 1976). For th e special case of a topographically-forced statio n ary wave (i.e., w ith zero phase speed) a critical level occurs w here th e zonal m ean flow is zero. However, as alluded to, sta tio n a ry wave breaking is observed to occur in th e absence of zero zonal m ean flow, leading us to conclude th a t critical levels are n o t a necessary condition for wave breaking. Is th ere an explanation for th e occurrence of statio n ary wave breaking in the absence of zero zonal m ean winds?
Fyfe an d Held (1990, hereafter FH) proposed a m echanism for Ross by wave break ing which pred icts breaking u n d er circum stances where th ere is no critical level (nor the ex p ectatio n of one being generated given linear wave th eo ry estim ates). FH term ed the m ain result of th e ir theory th e “two-fifths” rule for Rossby wave breaking (which will be
discussed in d e ta il in C h a p te r 4). W hile th e tw o-fifths rule is powerful it is lim ited to horizontally-propagating Rossby waves an d as such is n o t directly applicable to vertically- pro p ag atin g p la n e ta ry waves of th e sort discussed here. It is o u r overall aim then to generalize th e th eo ry of FH to allow for an analysis o f vertically-propagating plan etary waves which are relevant to th e strato sp h ere, and in p a rticu lar, to th e problem of s tra to spheric sudden w arm ings. F u rth e r to this we intend to exploit th e power of our modified C D /C S m odel to sim u late the detailed behaviour of wave breaking once initiated. FH used a pseudo-spectral num erical m odel, and as such were un ab le to characterize the breaking once it was in itia te d following the two-fifths rule.
T he plan for th is d issertatio n is as follows. In C h a p te r 2 we describe th e governing ecpiations an d th e C D /C S m odel used to solve them . In C h a p te r 3 we stu d y th e model sensitivity of p lan etary wave breaking to u p p er-b o u n d ary conditions. In C h a p te r 4 we present a new th eo ry for th e breakdow n of vertically-propagating p lan etary waves and provide num erical sim ulations which verify th e theory, a n d as well, show how th e breaking waves behave o u tsid e th e a.ssum ptions of th e theory. In C h a p te r 5 we sum m arize th e main results of this d issertatio n .
D ata arc from U nited K ingdom M eteorological Office (UKM O) analyses (Sw inbank and O ’Neill, 1994).
T h e m o d e l
III this d issertatio n we em ploy a ciuasi-geostrophic potential vorticity (hereafter QG P V ) m odel which except, for one m ajo r m odification (to be described in Section 2.3) is th a t of DS. T his m odel ap p ro x im ately describes ra p id ly -ro tatin g and strongly-stratified flow such as is seen in the ex tra tro p ic al strato sp h ere. In Sections 2.1, 2.2 and 2.3 we intro duce the m odel’s governing equations, PV discretization and in itia l/b o u n d a ry conditions, respectively. In Section 2.4 we describe the num erical approach to solving th e governing equations.
2.1
G o v ern in g e q u a tio n s
(cf. Pedlosky, 1987). T h is equation sta te s th a t QG PV q is conserved following the flow w hen advected by h orizontal velocity u, wliere
, = a n d u = k x V 0 . (2.2)
Po a z \ B az J
Here V is th e horizontal gradient vector, ip the stream fu n ctio n and k th e vertical unit vector. We use a iog-pressurc vertical coordinate z given by 2 = H\n(pao/ v) where H =
RToaIg is th e m ean scale height { R being th e ideal ga.s co n stan t, Too a co n stan t reference
te m p e ra tu re an d g g ravity acceleration), p pressure and poo a c o n stan t reference pressure. T h e reference density Po{z) an d B urger num ber D{z) arc given by
P o ( z ) = P ..e - = /^ ^ and B ( z ) = t V 2 ( z ) / / 2 (2.3) where poo is a co n stan t reference density, f o=2Q th e constant Coriolis p a ra m e te r (fi be ing the a n g u la r ro ta tio n ra te of E a rth ) and No{z) the B runt-V nisala frequency given by
N' ii z) = [ R / H ) { d e j d z ) e ~ ^ ' - I ‘< = { R / H ) { d T o l d z + kT ^ / H ) where 9 ,(z ) = T, {z ) c ^ ' ' l "
and To(z) are th e reference potential te m p e ra tu re an d te m p e ra tu re, respectively {K=R/cp where Cp is specific h eat capacity at co n stan t pressure). In th is d issertatio n we mainly consider a B oussinesq atm osphere, corresponding to taking / / —> oo an d To = Too- In this case N'o = { R / H )kToo/ H = g^/cpToo and
Po — Poo an d 13 = ^ . (2-4)
Cp J oo/o
T his is a reasonable ap proxim ation to th e real atm osphere w hen m otions considered have sm all vertical w avelengths com pared to the density scale height. In this d issertatio n we will m ainly co n cen trate upo n a Boussinesci atm osphere alth o u g h som e asp ects of com pressibility will be discussed in Section 3.4.
Values for th e c o n stan ts used in this dissertatio n are typical ones for the s tra to sphere, i.e., 0 = 7 .2 9 2 x 1 0 " ^ r a d s " '; poo = l-225 kgm "^; poo = 101.3 kPa; Too=210 K; R=2 8 7 .7 k g " 'K " '; fy=9.8 m s" ''; Cp=1012 .J K g " 'K “ '. Derived co n stan ts are therefore set as
/o ~ 1.458 X 10” “' r a d s " ', k % 2 / 7 and H % 6.14 km.
In the poiar-cap /-p la n e considered in th is dissertation, th e centre of the dom ain is at the N orth Pole and is unbo u n d ed horizontally (see Figure 2.1). As will be seen we will be working with th e governing equations in e ith e r C artesian [i.e., (.'r,?y)] or polar cylindrical horizontal coordinates [i.e., (A,r)].
C a r t e s i a n c o o r d i n a t e s
In C artesian coordinates i a n d j arc u n it vectors pointing in th e x and y directions, respectively (see Figure 2.1). In this system th e position and velocity vectors are given by p = z i + yj and u = ?n -f a j, respectively, a n d th e underlying equations take the form,
u = — a nd V = . [2.i)
a y o x
P o l a r c y lin d r ic a l c o o r d i n a t e s
In polar cylindrical coordinates i and j are u n it vectors p ointing in the A (azim uthal) and r (radial) directions, respectively (see F igure 2.1). In w hat follows we use the term s azim uthal interchangeably w ith zonal and m eridional interchangeably w ith radial. In this
system th e position a n d velocity vectors are given by p = Ai + r j an d u = u i + wj, respectively, and the underlying scalar equations take the form
I - ; ! - ! ' » '
. = ...d (2.10)
o r r 0 \
Finally, we nondim ensionalize th e system w ith tim e scale 5 = 47t//o = 1 day; vertical length scale H = BToolg ~ 6.14 km and horizontal length scale L = N a H / f o ~ 902 km. T hus we can define a horizontal velocity scale U = L / S ~ 10.4 m s“ F Unless sta te d , variables o th er th a n those circuniflexed are assum ed nondim ensional.
2.2
P ie c e w ise -u n ifo r m p o te n tia l v o r tic ity
O u r aim is to m odel the evolution of the polar strato sp h eric vortex subject to th e d istu rb in g influence o f upw ard p ro p ag atin g p lan etary waves. To this end we assum e th at PV is sim ply piecew ise-uniform , i.e., q(z) = Q ,(c) inside th e vortex edge C (see Figure 2.2) and q[z) = Q„{z) outside, is the distance from the N orth Pole to a point a t C. T he vortex edge evolves dynam ically u n d er the constraint th a t Qi{~) and Qo{z) is tim e-invariant. Hence A Q = Qi — Qo, which is the P V ju m p across th e vortex edge, is also tim e-invariant. In general one m ight consider m any such vortex edges (or equivalently m any PV ju m p s) b u t in this dis.sertation we will, for com p u tatio n al reasons, consider only one such edge (or ju m p ).
2.3
In itia l a n d b o u n d a ry c o n d itio n s
I n itia l c o n d it io n s
In th is d issertatio n the system is initialized w ith a cylindrical P V colum n centered on th e N o rth Pole (shown in plan view Figure 2.1 by the bold circle centered on the N orth Pole). In m ath em atical term s th is am o u n ts to settin g
r e ( A , 2 , 0 ) = r o ( 2 . 1 1 )
w here Tq is a co n stan t. In all o f the experim ents discussed here we set To = 3. In general we could allow Vo to vary as a function of z b u t for sim plicity we choose not to. We note th a t in C h a p te r 3 we set Qi a n d Qo to co n stan t values independent of height and in C h ap ter 4 we allow Qi and Qo to vary as a function of z.
B o u n d a r y c o n d itio n s
T he physical dom ain of the m odel is infinite horizontally b u t bounded vertically by rigid surfaces a t th e b o tto m zg and top zj- of th e dom ain. T he to p surface is Hat while th e b o tto m surface describes a large spatial scale topography of the form /fo'/, where
Ro = U / { f o L ) % 0.08 is the Rosshy num ber and
Here T]o is th e non-dim ensional topographic am p litu d e an d J i a Bessel function. The tapcT-function T ( r ) = 1 for r < 5.0, cos I ^ ^ ^ 7T ) for 5.0 < r < 7.5, (2.13) (2.14) 0 for 7.5 < r, sets the topography to zero a t large r while the sw itch-on function,
[ sin^ for 0 < t < is, S ( t ) =
1 for t, < t,
allows th e topo g rap h y to be sm oothly sw itched-on to its m axim um value a t tim e is- In C h ap ter 3 th e to p o g rap h y is sw itched-on instantaneously [i.e., S { t ) = 1 for t > 0], while in C h ap ter 4, w here we purposely a tte m p t to g en erate a wave “fro n t” , the topography is sw itched-on slowly (w ith t., = 10). A plot of th e topography is show n in Figure 2.1 by the shaded contours. T he m o u n tain is identified w ith a “4-” and the valley w ith a
Since no flow can pass th ro u g h a rigid surface one writes
dtp diji
= - - — // and —
/o u z Z = Z 'J ' 0. (2.15)
Following DS, these troublesom e nonhoniogeneous b o u n d a ry conditions can be replaced by hom ogeneous ones by in tro d u cin g a modified stream fu n ctio n ijt,
N'i i> = Ip + rnin(r, o ) - — rj
Jo
where a is a sm all p a ra m e te r (cr z-y). Taking the z-derivative of i/j yields
(2.1G)
d'4>
â z = 0 and
d4>
F u rth e r to this, V' is replaced in Eq. (2.2) w ith to yield a modified g, i.e.,
- 2 , 7. , ^ d ( Pa dip
a , - V; + - ^ ^ j (2.18)
where J is th e Dirac d e lta function. Like q this modified QG PV satisfies
^ ^ = 0 . (2.19)
D i ^ '
N ote th a t this is the equation to be solved in this dissertation.
A b s o r b in g sp o n g e
We now present th e m ain m odification we have m ade to the m odel of DS. To prevent back reflections of u p w ard-propagating waves from the rigid top b o u n d a ry we have imple m ented an absorbing sponge. W ith o u t such a sponge reflected waves would contam inate th e interior flow and produce unrealistic results (as we shall d e m o n stra te in C h ap ter 3). T he description of th e sponge we will now give is based on th e a ssu m p tio n of a single p atch of PV . T h e generalization to m ultiple patches is straightforw ard.
T he sponge layer used here lies in the range z s < z < z r where z s is th e b o tto m of th e sponge. Over this range of heights th e velocity u is ad ju ste d in such a m anner as to cause upw ard propagating waves to be dam ped, while a t the sam e tim e m aintaining te m p eratu res near a prescribed and fixed profile. In this sense the sponge could be called a “N ew tonian-cooling” sponge. To be m ore specific, th e velocity field at. any in stan t is ad ju sted in a “force-restore” m an n er, w ith forcing velocity u^ a n d restoring velocity u'', i.e.,
where u ’ is th e ad ju sted velocity. In th e sponge, we choose u-^ = p (s ,< )k X u*' and o'" = —o ( z ) ^ j
respectively where u** is th e barotropic (i.e., height independent) com ponent of u. T h e im p o rtan t point here is th a t b o th th e forcing and restoring velocities are by design irro tatio n al an d as such n eith er introduces any external vorticity into the system . T his is an im p o rta n t consideration because of th e fact th a t our num erical approach to solving the underlying equations dem ands vorticity conservation (as we will see in th e next subsection). Indeed, it is this co n strain t w hich until now has prevented th e im plem entation of an absorbing sponge in such a C D /C S m odel.
T h e m agnitude of is d eterm in ed by solving (at each tim e step) the following eciuation [where f-j^{)dTZ' represents an area integral over area TZ enclosed by con to u r C]
o d
+ u ' ) . v , =
(2.2 1)where tp,, a prescribed equilibrium stream fu n ctio n which we take to be th e initial sta te stream function and a{z) is a "N ew tonian d am p in g ” coefficient defined as
i f g
-O f X {e J — g ■'} for < s < z-f,
(2.2 2)
0 for z < Zs,
where a-j- is a co n stan t which sets the m axim um dam ping. In words, Eq. (2.21) eciuates th e advection of r/jot t)y u ’' + to the d am p in g of vertical gradients of any te m p e ra tu re anom alies [i.e., d{t(> — ipe)/dz\ th a t m ay arise in th e sponge layer. In Eq. (2.22) we note th a t a[{zr + z.s’)/2] ~ a n d a ( z r ) ~ «-y.
2.4
N u m er ic s
T h e num erical approach used to solve th e governing equations is th e C D /C S m ethod. T he C D /C S m ethod is a L agrangian m eth o d which is b ased on the assum ption th a t on level surfaces th e P V (which is m aterially conserved) is initially, and for all tim e, piecewise- uniform . O ut of this assum ption comes the trem endous sim plification th a t th e evolution of th e flow th ro u g h o u t th e dom ain is solely determ ined by the movem ent of particles on contours sep aratin g regions of constant PV . We will now describe th e m ethod in detail for th e special case of a single contour on each level surface.
We begin by vertically discretizing the system into N equally thick layers, allowing the m odified stream function tj) to be expanded as
N
•0(X, 2, f) = ^ Xm(2)v5m(x,f) (2.23)
r n = l
where x = [.r,y/]. S u b stitu tin g y „ ,(2)<pm(x, t) into Eq. (2.18) and letting th en leads to a Sturm -Liouville equation for th e eigenfunction Xm,
^ - 7mPo,Y,n = 0 w ith = 0 a t2 = 2/ j ,27’. (2.24)
Because B and Po are positive, the eigenvalues 7/^ are g u aran teed to be real, non-negative
and ordered (cf. Boyce and D iPrim a, 1986). N ext we s u b s titu te Eq. (2.23) into Ecp (2.18) and m ake use of Eq. (2.24) to produce
N
(^V <p,„ Xm — (iltot’ (2.25)
i n — 1
g ratio n (note th a t th e eigenfunctions Xm are o rth o n o rm al) yields
f
Po{z' ) Xn{z) qt ot {y^, z' ,t ) dz . (2.26)iPn is o b tained by inverting th e above equation using th e G reen’s function of the Helm holtz
o p erato r. P u ttin g the back into Eq. (2.23) gives
i>[yi,z,t) =
Po(z')<7tot (x% z', ( ) G ( x , z; x% z ') d z ' (in', (2.27)w ith G reen’s function
N
G{x, z; x% z') = ^ (7mIx' - x |) (2.28)
m=l
w here A'o is th e modified Bessel function. M aking use of Eq. (2.7) we have u = / Po{z') [ { , -% '] G '( x , z; x ', z ') } f/tot (x% z', t)d% ' dz',
J cy VJtz
which after using G reen ’s theorem leads to
u = - ^ P o (r')A Q (z ')
I
G (x , z; x ', z')[f/;j;', * /']j
dz' (2.29) w here A Q [ z ) = Q i { z ) —Q„{z). In this crucial ste p th e velocity d eterm ination reduces from an area integral to a line integral, thereby reducing the dim ension of the problem by one. T he next crucial step follows from K elvin’s circulation theorem which sta te s th a t fluid particles on a m aterial line rem ain on th a t line forever (cf. K undu, 1990). Since the bounding contour C is a m aterial line, th e contour m ovem ent can be tracked by following th e individual particles m aking up the contour, a n d from the contour position the velocity field can be o b tain ed anyw here in th e dom ain using Ecp (2.29).To evaluate Eq. (2.29) we discretize the b o u nding contour C into a finite num ber of nodes connected by cubic spline line segm ents. T he line integral in Eq. (2.29) can be cal culated by ad d in g the line integrals along all the cubic spline lines connecting the nodes on
C. Given th e horizontal location vector of a node x a n d corresponding horizontal velocity
vector u a t level z an d tim e t, we can calculate its subseq u en t location by evaluating
- = u(x,z,().
T his equation consists of a system of coupled o rd in ary differential equations which are solved using a fourth-order R u n g e-K u tta scheme to achieve high-order tem poral accuracy.
.\n im p o rta n t feature o f the C D /C S m ethod is th a t it is ad ap tiv e (Zabusky and Over m an, 1983). As a contour deform s under its self-generated velocity field, the num ber of nodes and th eir location are ad ju sted where and w hen necessary to m aintain an adequate resolution everyw here along the contour. For exam ple, ad d itio n al nodes are often inserted and rearranged together w ith existing nodes in regions of increasing curvature. Nodes may also be rem oved from regions where the curvature is weakening. These operations are p er formed (or not) on th e basis of a com puted local density, which is determ ined m ainly by the local cu rv atu re an d the presence of nearby sm all sp atial scale features. A form ula for calculating th e local density a t each nodal point has been o b tained em pirically (Dritschel, 1989).
T he adaptive procedure is appealing b ut lim ited. .\s contours become increasingly complex (as they often will) ad a p ta tio n quickly becom es prohibitively expensive [increas ing as the square of th e num ber of nodes, Dritschel (1988)]. To overcome this lim itation various “surgical” o p eratio n s have been introduced (D ritschel, 1988) which autom atically
remove features sm aller th a n a predefined so-called surgical scale 5, (set to 0.00125 in this d issertatio n ). C ontour surgery consists of two operations: 1) labelling and subsequent removal of corners and 2) sep aratio n a n d reconnection of contours. 1) If th e curvature a t a node exceeds th e node is connected to its neighbours witfi straig h t lines nearly form ing an acu te angle, th e n the node is labelled as a corner. In th is special case the cu rv atu re of th e node an d its two neighbours are set to zero so th a t no additional nodes are added in th e vicinity d u rin g the ad a p ta tio n stage of th e co m p u tatio n . However, when th e angle m ade by the line segm ents connecting th e corner an d its neighbours becomes obtu.se, th e node loses its corner sta tu s an d th e ad ap tiv e p rocedures resum e for this node as for any o th er ordinary node. 2) W hen two p a rts of a con to u r approach each other by a distance less th a n (5.,, th ey are sep arated into two closed contours. O n the other hand, when p a rts of two closed contours approach each o th e r by a distance less th an J ,, they are connected as a single contour. In addition, th e ends of excessively th in contours (som etim es called filam ents) consisting of five or less nodes are p erm an en tly removed.
Surgery allows C D /C S calculations to extend well Ireyond the conventional CD m ethod an d thereby allows for exceedingly com plex vorticity dynam ics to evolve. T he underlying assum ption of surgery is th a t scales sm aller th a n th e predefined scale (5,, are dynam ically insignificant an d will not grow w ith time.
10
forcing
5
0
5
q co n to u r
-10
■10
-5
0
%10
Figure 2.1: P lan view of tlie initial p o lar vortex (bold circle) on a n / - c a p plane. T he topography is shown by shaded contours. T h e N orth Pole is the centre of the plot.
NP
Figure 2.2: A single patch vortex w ith interior a n d exterior vorticity Qi and Qo, respec tively. C is the hounding contour a n d the area enclosed by C is TZ. I'g is th e distance from th e N orth Pole to C. N P denotes th e N orth Pole.
U p p e r -b o u n d a r y e ffe c ts o n
p la n e ta r y w a v e b r e a k in g
In this C h a p te r we describe a series of experim ents designed to assess the effects of np p cr boundaries on p lan etary wave breaking. DS studied th e response of an initially barotropic vortex to topographic forcing of varying am p litu d e in th e presence of a rigid lid. T he m ain result of DS was th a t for eith er Boussiiie.se; or com pressible flow two regimes of wave breaking exist, namely, 1) local breaking near the lower b o u n d ary a t strong forcing or 2) rem ote breaking n ear th e lid a t weak forcing. I) Local wave breaking, which is directly topographically forced, is insensitive to the u jjper-boundary condition and has an ap p aren t shielding effect on upw ard wave propagation. 2) R em ote wave breaking, which involves vertical tra n s p o rt of wave activity, is sensitive to the u p p er boundary condition and is coupled w ith the com pressibility effect w hen com pressibility is invoked. It is the uijper-boundary condition sensitivities th a t we shall investigate by placing our
cooling sponge under th e m odel’s rigid lid. To this end we will m ainly focus on Boussinesq How in ord er to isolate u p p er-b o u n d ary effects when those effects a re im p o rta n t. Towards the end of the C h a p te r we will investigate how the com pressibility effect modifies our conclusions. We again note th a t m ost of the results of this C h a p te r were published in Fyfe an d W ang (1997).
T he plan for this ch a p te r is as follows. Section 3.1 describes th e model setup, Section 3.2 the sponge p a ra m e te r sensitivity ex p erim en ts an d Section 3.3 the wave breaking ex perim ents. In Section 3.4 wo sum m arize th e Boussinesci results and how they are modified by the com pressibility effect.
3.1
M o d e l s e tu p
T he general m odel setu p was described in C h ap ter 2. Here we in tro d u ce som e features of the m odel specific to this C hapter. H ere the b o tto m of the dom ain is fixed a t i y = 12 km (nom inally th e tropopause) while th e top of the dom ain zt ranges from ab o u t 48 to 144 km (depending on th e p a rticu lar experim ent). N um erical solutions are obtained using a layer thickness of ab o u t 2.4 km, which d epending on z r yields 20 to 66 layers. T h e model is initialized w ith a cylindrical P V colum n with radius Vo = 3 a n d w ith Qi = and
Qo = 3.67r. T h e colum n is d istu rb ed topographically w ith an instantaneously sw itched-on
forcing (Figure 2.1). T h e instantaneously sw itched-on forcing enables us to com pare our results w ith those of DS. N ote th a t Q ;, Qo and Co are all independent of height in this C hapter. In th e experim ents w ith th e absorbing sponge the b o tto m of th e sponge layer is at. Zs = 48 km.
3.2
V ertica l sp o n g e s e n s itiv ity e x p e r im e n ts
T he vertical sponge thickness, z t — z s and m axim um relaxation ra te ny- (cpioted in
Section 2.3) have been d eterm in ed by trial an d e rro r using a vertical sponge u n d er a weak forcing condition (i.e., r/o = 0.15). T h e ratio n ale for th e weak forcing, at this stage, is th a t since th ere is no wave breaking (and th e response is nearly linear) th e sim ulations arc relatively easy to in te rp re t an d economical. C onsider F igure 3.1, which for a range of relaxation rates shows th e to ta l wave activity density (hereafter to tal wave activity) when
Z'T — 120 km (giving a b o u t a twelve scale height vertical s])onge). Wave activity is a useful
diagnostic for m onitoring th e mcan-scpiare a m p litu d e of wave d isturbances relative to a circularly sym m etric basic flow (see .\p p c n d ix A for a full definition of wave activity). In th e (Vf = 0.0 (no vertical sponge) sim ulation we see initial upw ard wave p ropagation which after a p p aren t m ultiple b o u n d ary reflections leads to a com plicated and u n stead y vertical stru c tu re . O n the o th er h an d , when 1.6 < a y < 3.2, the b o u n d ary reflections are largely brought u n d er control. .4s seen in Figure 3.2, dro p p in g the to p of the vertical sponge to zy = 72 km (giving aljout a four scale height thickness and a m uch reduced co m p u tatio n al load) does not seriously com prom ise th e absorbing ability of th e vertical s|)onge. It is on th e basis of these experim ents, and o th ers w ith larger topographic am plitudes (see Table 3.1 for a su m m ary of all experim ents conducted), th a t we have selected zy = 72 km and o y = 1.6 as our m ain vertical sponge p aram eters.
3 .3
M ain e x p e r im e n ta l r esu lts
C onsider T able 3.1 which sum m arizes th e various experim ents conducted (over 45 in to ta l). T h e m ain set o f exp erim en ts to be discussed (in bold type in Table 3.1) build upon th e Boussinesci rigid u p p er-b o u n d ary experim ents of DS, w here tjo = 0.45 or 7/0 = 0.60. T h e p lan here is to co n trast wave breaking seen w ith and w ith o u t a vertical sponge.
3 .3 .1 R i g i d u p p e r b o u n d a r y
Here we consider two sets of experim ents: rigid u p p e r b o u n d ary a t 1) 27- = 48 km (low
rigid u p p er b o u n d ary ) and 2) i y = 72 km (high rigid u p p e r b o u n d ary ). In subseciuent subsections we in tro d u ce a vertical sponge in th e region 48 km < z < 72 km, an d so in a sense the present experim ents can be considered as lim iting vertical sponge cases w ith 1) corresponding to a-y —» 0 0 and 2) corresponding to «7— > 0.
L o w rig id u p p e r b o u n d a r y
C onsider Figure 3.3(a) and (b) which show perspective views of a weakly forced (here a fter B £o.4 5, BC sta n d in g for "Boussinesci low rigid u p p er b o u n d ary ") an d strongly forced
(hereafter BCo.eo) vortex, respectively. T hese are O S ’s .54 an d B 6 experim ents respec tively, using th eir nom enclature. To 1 = 5 th e B C 0.45 and BCq go vortices arc very sim ilar w ith the lower (upper) contours m oving southw ard in th e positive y direction (x direction). .A.t < = 10 (not show n) lower-level breaking has ju s t begun in the B £ o.4 5 sim ulation b u t is ciuitc advanced in th e BCq.go sim ulation. By t — 15 upper-level contour deform ations d om inate the B £ q.45 sim ulation while lower-level breaking dom inates th e 5£o.oo sim
u-Table 3.1: Sum m ary of Boussinesq experim ents perform ed, where i r is the dim ensional dom ain height (km ), a r the relaxation rate (w here applicable) and % is the topographic am plitude. T he m ain set of experim ents to be discussed are in bold-type and th e identi fying labels are in parentheses.
U p p e r b o u n d a r y
z r (krn)
a r
Tjo = 0.15 Tjo = 0.45 Tjo = 0.60
R ig id L id 48 0.0 0 .0 ( B Co a s) 0.0 (BCo.eo) 60 0.0 0.0 0.0 72 0.0 0 .0 [B-Ho a s) 0 .0 [ISKo do) 96 0.0 - -120 0.0 - -144 0.0 - -V e r tic a l S p o n g e 60 0.8, 1.2, 1.6 0.8, 1.2, 1.6 0.8, 1.2, 1.6 72 0.4, 0.8, 1.2, 1.6 2.0, 2.4, 2.8, 3.2 0.8, 1.2 1 .6 ( B 5o.45), 2.8 0.8, 1.2 1 .6 (g Jo .o o ), 2.8 06 1.6 - -120 0.4, 0.8, 1.2, 1.6 2.0, 2.4, 2.8, 3.2 -
-latioii. A t t = 15, th e upper-level contours in the B Cq as sim u latio n are stretched way- o u t [see th e plan view of th e to p contour (dotted) in F igure 3.4(a)] while the lower-level contours are filarnented [see th e plan view of the b o tto m contour (solid) in Figure 3.4(a)]. At t = 15 th e upper-level contours in the BCo.eo sim ulation arc relatively quiescent [see the plan view of the top co n to u r (dotted) in Figure 3.4(b)] while th e lower-level contours are broken and dramatically- deform ed [see the plan view of th e b o tto m contour (solid) in Figure 3.4(b)]. Here, th e stro n g lower-level breaking evidently shields th e upper-level contours from the d ra m a tic deform ations seen in the corresponding BCq.45 contours (as suggested by DS). A physical reason is th a t th e low level breaking homogenizes the PV, which in effect removes th e resto rin g force needed for th e vertical p ro p ag atio n of planctary waves. O ur vertical sponge ex perim ents to come will show which asp ects of this picture are artifacts of the rigid u p p e r boundary- and which are not.
.Another way to look a t th ese sim ulations is through height-tim e cross sections of total an d relative wave activity (F igure 3.5). Relative wave a ctiv ity is th e com ponent of the to ta l wave activity which is clue to m otions relative to th e cen tro id of th e vortex, i.e., m otions th a t leads to the sh a p e changes of the vortex. In e ith e r th e B C q a s or BCo eo
cases we see an accum ulation of th e to ta l wave activity a t th e top o f the dom ain up to ( % 5. It is during this tim e th a t the vortex obtains its stro n g tilt w ith height as the upper-level contours shift off th e Pole in the direction o f low topographic heights (see Figure 2.1). .At f. % 10 th e to ta l wave activity again builds, mostly- at upper levels in the B £ q. 4 5 sim ulation an d a t lower levels in the BCq.bo sim ulation (fu rth er evidence of the shielding effect alluded to in the last paragraph). T h e relative wave activity plots
indicate th a t these behaviours are due to contour shape changes as m uch as they are to changes in th e position of th e centroid o f th e vortex. We also note th e im pression of waves reflecting downwards from th e top a t t « 5 (the m axim a of to tal wave activity a t th e top around th a t tim e), reaching th e b o tto m a t t % 1 0 (the m axim a o f to ta l wave activity at the b o tto m around th a t tim e) an d th en in the B Cq a^ sim ulation reflecting upw ards to
cause th e subsequent upper-level havoc and in th e BCo.eo case s e ttin g off the dramatic- lower-level breaking (and p e rh a p s suppressing fu rth er wave p ro p ag atio n as suggested by DS). F u rth e r diagnostics would be required to confirm these hypothesized chains of events.
H ig h r ig id u p p e r b o u n d a r y
How does the p ictu re change w hen th e rigid u p p er b o undary is lifted to iy- = 72 km? C onsider Figures 3.6(a) an d (b) w hich arc th e perspective view s of a weakly forced (hereafter BT-Lo.ao, where H sta n d s for high rigid u p p er boundary) an d strongly forced (hereafter BHo.eo) vortex, respectively, w hen i f = 72 km . N othing app reciab ly different from th e low u p p er b o u n d ary case h ap p en s to the vortex before f % 3, w hether weakly or strongly forced. A fter f % 3 th e BHo.eo vortex evolves much as th e BCo.eo one, in the sense th a t the breaking is confined to th e lower half of the dom ain a n d is very complex, [i.e., involving vortex displacem ent, reshaping, filam entation and secondary developm ent (com pare Figure 3.7(b) and F igure 3.4(b) solid curves)]. O n the o th e r hand, after < w 3 the B'Ho. 4 5 vortex evolves m uch differently th an th e ^ £0 .4 5 case insofar iis the form er’s
upper-level contours undergo relatively little shape change [com pare Figure 3.7(a) and Figure 3.4(a), dashed curves]. We note in the BH0.45 case subseq u en t to f % 13 (not
show n), two distin ct regions of breaking develop, one near a kink in th e vortex at z % 7.5 a n d an o th e r near th e b o tto m . A n early h int of this behaviour is seen in the b o tto m left panel o f Figure 3.8 which shows relative wave activity m axim a a t z % 7.5 and z % z« b eginning around t = 10.
3.3.2
V e rtic a l spon ge
Lot us now consider placing a vertical sponge in th e region 48 km < z < 72 km. C om paring F igure 3.9 and Figure 3.6 th ere a p p ears little difference betw een the BT-L (high rigid u p p er b o u n d ary ) and th e B S (vertical sponge) cases up to ( % 5 (i.e., w ith or w ith o u t a vertical sponge th e vortex is tilted w ith height and there is som e m inor lower- level breaking w hen strongly forced). Beyond t % 5 the evolution is cjuite different. For exam ple, the B5o.45 vortex [Figure 3.9(a) a n d F igure 3.10(a)] rem ains in a m ore or less ste a d y sta te (except for som e relatively m inor lower-level breaking) to t % 15 (and in fact to th e end of the sim ulation a t < = 25, not show n). O n the o ther hand, the corresponding
B Ho a^ vortex [Figure 3.6(a) an d Figure 3.7(a)] shows advanced breaking a n d /o r contour
d isto rtio n th ro u g h o u t the lower tw o-thirds o f the dom ain during th e same tim e. The BJo.GO [Figure 3.9(b) and Figure 3.10(b)] a n d BTïo.eo [Figure 3.6(b) and Figure 3.7(1))] sim ulations arc also cpiite different, .\lth o u g h bo th sim ulations exhibit significant lower- level breaking, th e ch aracter o f th e breaking is quite different. Specifically, the B Sq.oo
vortex rem ains m ore or less in ta c t despite w inding and then expidsion of a th in and seem ingly passive filament. T h e BT^o.ao vortex, on the o ther hand, shows a m uch less recognizable m ain vortex an d th e developm ent of a secondary vortex. For com pleteness we
show th e B S wave activity plots in Figure 3 .I I , which w hen com pared to th e corresponding
B H plots in F igure 3.8, fu rth er p u n ctu ates o u r p o in t th a t the evolution of a barotropic
vortex can be highly sensitive to the u p p e r-b o u n d a ry condition.
3.4
D isc u s sio n
T h e stu d y in th is C h ap ter was prim arily m otivated by DS who used a three-dim ensional QG C D /C S num erical model to stu d y th e response of a barotropic vortex to topographic forcing of varying am plitude. O u r specific question, in th e context of DS, is w hat is the role of th e rigid u p p er-b o u n d ary condition in wave breaking? We have addressed this (luestion by placing a vertical sponge under th e rigid u p p er boundary. O u r m ain results are:
• Given a vertical sponge a n d a forcing a m p litu d e g reater th an a certain critical value, breaking is as in the corresponding rigid u p p e r b o u n d ary case insofar as it is confined to th e lower half of the vortex. However, im p o rta n t differences exist. For exam ple, w ith a vertical sponge the vortex rem ains fairly tru e to its initial circular shape b u t is w rapped by a long narrow hlam ent a t lower levels. W ith o u t a vertical sponge the vortex is so deform ed a t lower levels as to be nearly unrecognizable.
• Given a vertical sponge an d a forcing a m p litu d e less th a n the critical value no break ing exists. T his contrasts w ith the rigid u p p er b o u n d ary case which shows significant deform ations near the u p p er boundary. E vidently th e upper-level deform ations arc an artifact of the rigid u p p er-b o u n d ary condition.
O ur conclusions so far are based on th e assu m p tio n of a co n sta n t density profile. Now we consider an isotherm al (i.e., To — 210K) atm o sp h ere in which the density po has an exponential profile given ;xs Eq. (2.3). T h e degree to which our Boussinesq results hold will d epend upo n th e point a t which wave am p litu d es becom e large enough, due to the density effect, to break. If th a t point is situ a te d well above the m odel dom ain then our Boussinesq resu lts should hold to first appro x im atio n . If, on the o th er hand, th a t point lies w ithin th e m odel dom ain our conclusions will have to be modified. For exam ple. F igure 3.12(a)-(c) show a series of experim ents (CZZo.is, CC,o.i7S and CTq.2 5 where C stan d s for “com pressible” ) w ith an exponential d en sity profile and a rigid lid located ju st afjove th e top-m ost contour (as considered in DS). To ju d g e w h eth er or not the upper-level breaking is due to th e density effect (i.e., wave am p litu d es increasing exponentially w ith height) o r th e rigid-lid effect (discussed earlier) we re p e a t th is set o f sim ulations b u t w ith our absorbing sponge replacing D S’s rigid lid [see F igure 3.12(d)-(f)]. Given the close correspondence betw een th e rigid-lid and absorbing sponge sim ulations we conclude th a t for this range of forcing am plitudes the density effect dom inates th e upper-level response, w ith u p p er-b o u n d ary condition playing only a m inor role (consider also Figure 3.13 for the corresponding p lan views). F u rth er in this vein, consider Figure 3.14 where, given a much reduced top o g rap h ic am p litu d e (i.e., rjo = 0.04), we co n trast low rigid-lid C E0.0 4, high rigid-lid C'Hq,04 an d absorbing sponge C.$o.04 sim ulations. For tin s topographic forcing
it ap p ears th a t th e density effect only becomes significant w ithin th e region of sponge as shown in Figure 3.14(b). T he upper-level breaking seen in th e low rigid-lid experim ent is therefore in te rp re te d as a local am plification due to b o u n d a ry reflections (as in our small
am p litu d e Boussincsq sim ulations).
In this C h a p te r we have stu d ie d u p p er-b o u n d ary effects on p lan etary wave breaking. To elim inate spurious reflections from th e to p boundary, we have developed and tested a N ew tonian cooling sponge layer. Using the C D /C S m odel w ith this sponge layer, we now investigate th e dynam ics leading to p lan etary wave breaking.
« ^ = 0.0 15 10 5 0 5 10 15 2 0 25 15 10 5 0 5 10 15 20 25 15 10 5 0 5 10 15 20 25 15 10 5 0 5 10 15 2 0 25 a y = 2.0 15 10 5 0 5 10 15 20 25 = 2.4 15 10 5 0 5 10 15 20 25 15 10 5 0 5 10 15 20 25 3.2 15 10 5 0 5 10 15 20 25
Figure 3.1: T otal wave activity d en sity norm alized by % for a range of relaxation rates
a r w ith z/ =:120kni (i.e., z/-=19.5) a n d Oo=0.15. W here api.dicable th e vertical sponge
a,- = 0.4 a,. = 0.0 aj- = 0.8 0 5 10 15 2 0 25 aj-= 2.0 0 5 10 15 2 0 25 0 5 10 15 20 25 15 10 5 0 5 10 15 20 25 15 10 5 0 5 10 15 20 25 0 7-= 2.8 15 10 5 0 5 10 15 20 25 15 10 5 0 5 10 15 20 25
(a)
N
IX
F igure 3.3: Perspective views of th e vortex for th e case of a low rigid u p p er boundary: (a) B£o,45 (7/0=0.4 5); (b) BCo.eo (7/0=0.6 0).
(a)
4 5{t — 15)
b o ttom
top
-10
10
-5
0
5
10
Figure 3.4: Top level (d o tted ) and b o tto m level (.solid) p o ten tial vorticity contours for the case of a low rigid u p p er b oundary: (a) B Cqas (% =0.45); (b) -STo.eo (?/o=0.60). T he b o tto m topography is shown by shaded contours.
