CO N TIN EN TA L SH E LF
RESEA RCH
Vlaams Instituut voor dB Zee
Flanders Merino Institute
PERGAMON
C o n tin e n ta l S h e lf R e s e a r c h 18 (1998) 2 7 9
287
A tw o -co m p a rtm en t m odel for u n derstanding
the sim ulated three-dim ensional circulation
in Prince W illiam Sound, A laska
Eric D eleersnijder3’*, Jia W a n g b, C h risto p h er N .K. M o o e rsb
aIn s titu t d A s tr o n o m ie e t d e G é o p h ysiq u e G. L em a itre, U n iv e rsité , c a th o liq u e de L ou va in ,
2 C h em in d u C y c lo tro n , B - I 3 4 8 L o u v a in -Ia -N e u v e , B elgium
hO cea n P re d ic tio n E x p e r im e n ta l L a b o r a to r y a n d D ivision o f A p p lie d M a r in e P hysics,
R o s e n stie l S c h o o l o f M a r in e a n d A tm o s p h e r ic S cien ce, U n iv e r s ity o f M ia m i, 4 6 0 0 R ic k e n b a c k e r C ausew ay,
M ia m i, F L 3 3 14 9 -1098, U .S .A .
R ec e iv e d 15 J u ly 1997; re c e iv e d in re v is e d fo rm 4 A u g u st 1997; a c c e p te d 14 A u g u st 1997
Abstract
A tw o -c o m p a rtm e n t m o d el o f P rin c e W illia m S o u n d (P W S ), A la sk a , is d ev elo p ed . O n e
c o m p a rtm e n t, c o rre s p o n d in g to th e s o u th e rn P W S , re p re se n ts a d v ec tiv e p h e n o m e n a , w hile th e
o th e r is d o m in a te d by diffusion. T h is sim p le m o d e l is s h o w n to r e p ro d u c e ra th e r well th e
te m p o ra l e v o lu tio n o f th e m ass o f a p a ssiv e tra c e r c o n ta in e d in P W S s im u la te d by a com plex,
th re e -d im e n sio n a l m o d e l u n d e r five ty p e s o f su rfa ce fo rcin g . T h e th re e p a r a m e te rs o f the
b o x -m o d e l h a v e c le a r p h y sic al m e a n in g s, w h ich h e lp s to u n d e rs ta n d th e h y d ro d y n a m ic s o f
P W S . In p a rtic u la r, th e fra c tio n o f th e flow e n te rin g th e n o r th e r n P W S is e stim a te d , as well as
th e tu rn o v e r tim e o f th e tw o re g io n s c o n sid e re d . (Q 1998 E lsev ier S cience L td . A ll rig h ts
re serv e d
1. Introduction
O n e o f th e la rg e st oil spills o n to th e seas o cc u rre d in P rin c e W illiam S o u n d (PW S),
A lask a, on 24 M a rc h 1989, as th e re su lt o f a n a v ig a tio n e rro r o f th e E x x o n V aldez.
Since th e n , c o n sid erab le efforts h a v e been d e v o te d to th e stu d y o f th e eco sy stem o f this
reg io n . In this fra m ew o rk , M o o e rs a n d W a n g (1997)— h ere after referred to as
M W — used a versio n o f th e th re e -d im e n sio n a l P rin c e to n O ce an M o d e l (e.g. Blum b erg a n d M elior, 1987) to sim u la te th e w a te r c irc u la tio n in P W S.
^ C o rre s p o n d in g a u th o r . F a x : -t- 3 2 -1 0 -4 7 4 .7 2 2 ; e -m a il: eric d ifl.a str.u c l.a c .b e .
0 2 7 8 - 4 3 4 3 '9 8 /$ 1 9 .0 0 >0 1998 E l s e \ie r S c ie n c e L td . A ll rig h ts re se rv e d
P U S 0 2 7 8 -4 3 4 3 (9 7 )0 0 0 6 4 -2
280
E. Deleersnijder et a l.1Continental Shelf Research 18 (1998) 279-287
F o r details a b o u t th e to p o g ra p h y o f th e d o m a in o f in te rest, refer to M W . H erein , it
is sufficient to m e n tio n th a t th e la te ra l c o m p u ta tio n a l b o u n d a ry is im p erm eab le,
ex cept for tw o n a rro w passag es in th e so u th e rn p a r t o f P W S , H in c h in b ro o k E n tra n c e
a n d M o n ta g u e S tra it. M o reo v e r, it m u st be u n d e rs c o re d th a t, in g en e ral, w ater
o rig in a tin g from th e G u lf o f A lask a en ters P W S th ro u g h th e fo rm e r a n d leaves
th ro u g h th e latter.
M W c o n d u c te d a sensitivity an a ly sis o f th e P W S c irc u la tio n to th e su rface stress. In
th e c o n tro l ru n , n o w in d forcing w as ap p lie d . F o u r a d d itio n a l sim u la tio n s w ere
c a rrie d o u t in w h ich th e w in d w as assu m ed to b e b lo w in g w ith th e sa m e sp eed to w a rd
th e east, th e n o rth , th e w est a n d th e so u th , resp e ctiv ely . F o r ea ch ty p e o f forcing,
a flow in sta tistic a l e q u ilib riu m w as o b ta in ed . T h e la tte r w as th e n u sed to sim u la te the
fate o f a passive tra c e r released a t a c o n s ta n t ra te f o r four d ay s by a lin e so u rc e lo c ate d
in H in c h in b ro o k E n tra n c e . T h is sim u la te d tr a c e r release w ill h elp to u n d e rs ta n d the
circ u latio n in P W S since a passive tra c e r p a rc e l b eh a v es like a w a te r parcel.
In each m o d e l ru n , th e so u rc e released tra c e r in th e u p p e r 40 m a t th e c o n s ta n t ra te
O from tim e t = 0 u n til it w as c u t off, a t tim e t — T , w ith T = 4 day s. T h u s, th e to ta l
a m o u n t o f tra c e r injected in to P W S w as M = <hT. T h e tra c e r c o n te n t o f P W S , ms, w as
sim u lated for 33 d ays by M W (Fig. 1).
T h e te m p o ra l e v o lu tio n o f th e tra c e r m ass p r e s e n t in P W S ex h ib its fo u r d istin ct
p h ases w hich are q u a lita tiv e ly sim ila r in every m o d e l ru n (Fig. 2). F irs t, fro m t = 0
u n til t = T , m s g ro w s a p p ro x im a te ly linearly, s in c e a lm o st n o tra c e r p arcel h a s y et left
P W S th ro u g h M o n ta g u e S tra it. T h e n , as lo n g a s th e o u tg o in g tra c e r flux is negligible,
th e tra c e r m ass re m a in s v irtu a lly c o n s ta n t. T h is reg im e lasts u n til t = r, w h ich is th e
U
l
U
rol
northward
w estw ard-
E 0 .6
ui
Ul
.5c?
o
no wind
eastward "
0 .4
CD
southward.
E
T3
0.2
0.0
0.0
10.0
20.0
30.0
4 0 .0
time (days)
F ig . 1. T h e e v o lu tio n o f th e d im e n s io n le s s tr a c e r m a s s p re s e n t in P W S , m /M , a s s im u la te d b y M W u s in g
th e th re e -d im e n s io n a l P r in c e to n O c e a n M o d e l w ith v a r io u s su rfa c e forcings. T h e d ire c tio n to w a r d w h ic h
th e w in d is b lo w in g is in d ic a te d . (T h e se c u rv e s a r e o b ta in e d b y d e le tin g th e s lig h t o v e rs h o o tin g s o f th e
s im u la te d m a ss, o c c u r r in g f o r u n id e n tifie d n u m e ric a l re a s o n s , a fte r th e s o u rc e w a s c u t off).
E. Deleersnijder et al. j Continental S helf Research 18 (1998) 279-287
281
= linearly increasing mass
mass
m ass ~ constant
rapidly d e crea sin g m ass
slo w ly decreasing mass
IV
T
T
time
F ig . 2. S c h e m a tic illu s tra tio n o f th e fo u r p h a se s , la b e lle d I - I V , o f th e te m p o ra l e v o lu tio n o f th e tr a c e r m a ss
p r e s e n t in P W S .
tim e p erio d th a t a tra c e r p arcel need s to trav e l fro m H in c h in b ro o k E n tra n c e to
M o n ta g u e S tra it follow ing th e m o st d ire c t p a th w a y . T h e n , th e tra c e r m ass d im inishes
m o n o to n ic a lly . D u rin g th e th ird p h ase, th e tra c e r m a ss decreases ap p ro x im ate ly
lin e a rly as tim e progresses. T h e tra n s itio n fro m th e th ird to th e fo u rth p h a s e is
c h a ra c te riz e d by an a b r u p t c h a n g e in th e p ac e a t w h ich th e m a ss dim inishes. F in ally ,
d u rin g th e final p hase, th e ra te o f d ecrease o f th e m a ss is m u c h sm aller.
T h e p rocesses w hich are a t w o rk d u rin g th e first tw o p h ases a re read ily u n d e rsto o d .
T h a t th e m ass m u s t decrease d u rin g th e s u b se q u e n t p h a se s is o bvious. H ow ever,
ex p la in in g w hy th e ra te of d ec rease o f th e m a ss ch an g es is n o t stra ig h tfo rw a rd . T o do
so, a sim ple, c o m p a rtm e n ta l m o d e l is e sta b lish e d , w h ich will h elp infer m a jo r p ro p e r­
ties o f th e P W S circ u latio n , as w ell as reveal th e in flu en ce o f th e w ind on them .
2. A two-compartment model
A n a tte m p t will b e m a d e to re p re se n t th e e v o lu tio n o f P W S tra c e r c o n te n t by m ean s
o f a n elem e n ta ry m odel. T h e tra c e r m a ss e stim a te d th e re fro m will be d en o ted m so as
to d istin g u ish it from its c o u n te rp a rt, ms, sim u lated by M W u sing a com plex,
th re e -d im e n sio n a l m odel.
T h e tra c e r m ass obeys
=
( 1)
w h e re <//'' is th e tra c e r flux e n te rin g P W S th ro u g h H in c h in b ro o k E n tra n c e while
(j)oul re p re se n ts th e ra te a t w hich tra c e r leaves P W S th ro u g h M o n ta g u e S tra it. T h e flux
E. Deleersnjder et a l.' Continental S helf Research 18 (1998) 279-287
282
(j)m is d u e to th e tra c e r source, w hile th e o u tg o in g flu x d e p e n d s o n th e tra c e r c o n te n t of
th e S o u n d a n d th e h y d ro d y n a m ic processes o c c u rrin g in it, th e m a jo r fea tu re s of
w hich m u st be u n d e rsto o d for a su ita b le p a r a m e te r iz a tio n to be devised.
T h e M W m odelled c u rre n t m a p s clearly suggest t h a t a tr a c e r p arcel m a y follow tw o
d istin ct types o f p a th le ad in g fro m H in c h in b ro o k E n tr a n c e to M o n ta g u e S tra it. T h e
la rg e st c u rre n t speeds o cc u r betw een K n ig h t Is la n d a n d M o n ta g u e Islan d , im p ly in g
th a t th e q u ic k est p a th w a y is a sso c ia te d w ith a d v e c tio n th ro u g h th is region. T h e
second ro u te is th a t o f p arcels g o in g fu rth e r n o r th in to th e S o u n d , w h ere th e
c irc u la tio n is m u ch slow er a n d in tric a te , p a rtly b e c a u se o f n u m e ro u s to p o g ra p h ic
features, su c h as islands, fjords, o r h ea d la n d s. A t th e scale o f th e n o rth e rn P W S , these
ad vective processes m a y a m o u n t to diffusion.
T h is d e sc rip tio n p ro v id es a basis for divid in g P W S in to tw o p a rts , th e so u th e rn
P W S — also te rm ed ‘reg io n 1’ below — d o m in a te d b y a d v e c tio n a n d th e n o rth e rn
P W S — also called ‘reg io n 2’ h e re a fte r — w h ere d iffusive m e ch a n ism s prevail. T h e re ­
fore, th e tra c e r m ass p rese n t in P W S a t a given tim e, m(t), m a y b e re g a rd e d as th e sum
o f m ,(f) a n d m 2(t), w here m f t ) (i = 1, 2) is th e tr a c e r m ass c o n ta in e d in reg io n i. If
(¡)\n a n d (j)0“' d e n o te th e fluxes e n te rin g a n d le av in g reg io n i, respectively, m¡ obeys
2 m¡(£) = (¿j”(f)-,K"(í). ¡ = 1.2
(2)
Since region 1 is assu m ed to be d o m in a te d by a d v e c tio n ,
=
t)
(3)
w h ere th e tim e lag x, as a lre a d y m e n tio n e d in th e p re c e d in g sectio n , is th e p erio d o f
tim e th a t a tra c e r parcel, follo w in g th e q u ic k e st p a th , n eeds to tra n s it fro m H in c h in ­
b ro o k E n tra n c e to M o n ta g u e S tra it.
If th e diffusive processes ta k in g p la ce in reg io n 2 a re r a th e r stro n g , th e n th e tra c e r
c o n c e n tra tio n th e re in m ay b e a ssu m e d to b e sufficiently h o m o g e n e o u s fo r th e tra c e r
flux leav in g th is reg io n to be p r o p o rtio n a l to th e tra c e r m ass p re se n t in it, i.e.
« “' 0
=
^
(4)
w h ere th e tim e scale 0 is th e tu r n o v e r tim e o f reg io n 2. T h e la tte r is d efin ed to b e th e
av erag e o v er th e n o r th e rn P W S o f th e resid en ce tim e — w hich, a t a given p o in t of
reg io n 2, is th e p e rio d o f tim e th a t a w a te r p arc el, in itially lo c ate d a t th e p o in t
co n sid ered , needs to leave th e n o r th e r n P W S . A d d itio n a l e x p la n a tio n s a n d a p p r o p r i­
a te references on such h y d ro d y n a m ic tim e scales, as w ell as p a ra m e te riz a tio n s sim ilar
to E q. (4), m a y be fo u n d in B o lin a n d R o d h e (1973) a n d T a rtin v ille et al. (1997).
L et a d e n o te th e fra c tio n o f th e tra c e r flux e n te rin g P W S th a t g o es d irectly in to
reg io n (2). H ence,
<ÊÏ(t) = a 0 ta(t)
(5)
T h e tra c e r flux le av in g reg io n 2 m a y be a ssu m ed to jo in th a t ex itin g th e so u th e rn
P W S . H o w ev er, it is rea d ily seen th a t su c h a flux a rra n g e m e n t w o u ld p rev e n t th e P W S
E. Deleersnijder et ál. / Continental Shelf Research 18 (1998) 279-287
283
r
northern PW S
reg io n 2
mv 0
,oul
<P-
southern PW S
reg io n 1
mv r
(1 - a ) <pin ■
>
<P‘
M ontague
Strait
H in ch in b ro o k
E ntrance
F ig . 3. S c h e m a tic illu s tra tio n o f th e tr a c e r flu x e s a c c o u n te d fo r in th e c o m p a r tm e n ta l m o d e l.
tra c e r c o n te n t from gro w in g lin e a rly d u rin g th e first p h a se o f th e tra c e r m ass
e v o lu tio n . In a d d itio n , m co u ld n o t re m a in a lm o st c o n s ta n t fo r a c e rta in p e rio d o f tim e
afte r th e so u rce is c u t off. T o circ u m v e n t th e se p ro b le m s, th e tra c e r p a th w a y in v o lv in g
re g io n 2 m u s t in c lu d e an a p p r o p r ia te tim e lag. F o r th e sak e o f sim plicity, it is d ecid ed
th a t th e flux leav in g region 2 en te rs th e s o u th e rn P W S (Fig. 3), i.e.
0?(t) = d - *)<F(o +
r 2ut(t)
(6)
T h is is a m o d e llin g choice w hich d o es n o t re q u ire in tro d u c in g a tim e lag specific to the
n o r th e r n P W S , since use is m a d e o f th a t asso cia te d w ith re g io n I.
T h e tw o -c o m p a rtm e n t m o d e l m e a n t to rep rese n t th e m a jo r fea tu re s o f th e tra c e r
e v o lu tio n in P W S consists o f E qs. (l)- ( 6 ), th e so lu tio n o f w h ich rea d s
o
if m ( 0) = 0. In th e case co n sid ered ab o v e , cf)m = <t> if 0 ^ t < T ( = 4 days), an d <ƒ>"' = 0
o th e rw ise . T h erefo re, if T < r,
(8a)
(8b)
m(f) = 0 ( T + t — r) + a<D0(l - e (t" ')/0),
t < t ^ T + t
(8 c )
an d
m(t) = a ® ö ( e T'° - l ) e (t“ í)/0,
T + x ^ t
(8d)
E. Deleersnijder et al. ¡Continental Shelf Research 18 (1998) 279-287
284
3. Assessment and discussion
A s expected, th e so lu tio n a b o v e ex h ib its fo u r d iffe re n t p h ases. It rem a in s to be seen
w h eth e r o r n o t th e th ree p a ra m e te rs z, 0 a n - I d a m a y be c a lib ra te d in su c h a w ay th a t
th e ev o lu tio n o f th e tra c e r c o n te n t p red ic ted by t h e c o m p a rtm e n ta l m o d el, m, is close
to th a t sim u lated by M W , ms .
T h e dim ensionless d istan c e betw een m a n d m s m a y be defined as
-
1 1 /2
(m _ m s)2 d t
e=
(9)
w here imax = 33 d ays is th e d u r a tio n o f M W ’s th re e -d im e n sio n a l sim u latio n s. O b v i­
ously, th e values o f th e p a ra m e te rs z, 0, a n d a to b e co n sid ered are th o s e th a t m in im ize
s. T h ey are o b ta in ed by m e an s o f a sim ple, ite ra tiv e a lg o rith m . A sim p le in sp e ctio n o f
th e th ree-d im en sio n al m odel resu lts (Fig. 1) su g g e sts selecting 7 d ay s, 50 d ay s a n d 0.7
as th e in itial values o f z, 9 a n d a, respectively. In fa c t, z is th e tim e elap sed a t th e e n d o f
th e second ph ase o f the e v o lu tio n o f th e P W S tr a c e r co n te n t. N ex t, a m a y be estim a ted
from th e slo p e o f th e tra c e r m a ss d u rin g p h a se III — as is illu stra te d b y th e as y m p to tic
ex p a n sio n (10) below . F in ally , k n o w in g z, 6 m ay b e a p p ro a c h e d th ro u g h c a lc u la tio n o f
th e ra tio o f th e tra c e r m ass a t th e b e g in n in g o f th e fo u rth p h ase to th a t a t th e en d o f
th e th ree -d im en sio n al sim u la tio n p erio d .
As m ay be seen in T a b le 1, fo r every ty p e o f su rface forcing, s is r a th e r sm all,
in d ic atin g th a t th ere is a n ex cellen t a g re em e n t b etw e en th e e v o lu tio n o f th e tra c e r
m ass sim u lated by M W a n d th a t d eriv ed fro m th e p rese n t c o m p a rtm e n ta l m odel,
w hich is clearly co n firm ed by th e sim ila rity o f F ig s. 1 an d 4.
T h e p a ra m e te rs o f th e sim p le m o d e l a re such t h a t z — t « 9 d u rin g th e th ird p h ase o f
th e trac er m ass e v o lu tio n , i.e. d u rin g th e tim e in te rv a l z ^ t ^ T + z. As a result,
m ad m its th e follow ing a s y m p to tic ex p a n sio n
m(i) « 0 [ T — (1 - oi)(t - r)],
z ^ t ^ T + z
(10)
w hich is th e rea so n w hy th e tra c e r m ass d ec reases alm o st lin early d u rin g th e th ird
phase, as m ay be seen in F igs. 1 a n d 4.
T a b le 1
T h e v a lu e s o f th e p a r a m e te r s x, 9 a n d a m in im iz in g e a s d e te rm in e d fo r v a r io u s su rfa c e
fo rcin g s. T h e d ire c tio n to w a r d w h ic h th e w in d b lo w s is in d ic a te d . T h e e r r o r b a r s re la te d to
th ese e s tim a te s a rc (A t, AÖ, Act) = (0.1 d a y , 1 d a y , 0.01).
t
(days)
0 (days)
a
£
N o w in d
W e s tw a rd
7.1
48
80
0.68
0.065
7.8
S o u th w a r d
E a s tw a rd
N o r th w a r d
7.4
32
7.1
42
97
7.5
0.78
0.62
0.62
0.78
0 .0 6 6
0.067
0.066
0 .066
E. Deleersnijder ct cd. / Continental Shelf Research 18 (1998) 279-287
285
E
m
m
ca
northward
westward
E
in
in
©
c
o
'm
■no w ind .
eastward
c
cu
E
-a
southward
0.2
0.0
0.0
10.0
20.0
30.0
4 0 .0
time (days)
F ig . 4. T h e e \ o l u t i o n o f th e d im e n s io n le s s tr a c e r m a s s p r e s e n t in P W S , m /M , o b ta in e d fro m th e
c o m p a r tm e n ta l m o d e l w ith v a rio u s su rfa c e fo rcin g s. T h e d ir e c tio n to w a r d s w h ic h th e w in d is b lo w in g is
in d ic a te d .
I t is clea r th a t th e tu rn o v e r tim e o f th e s o u th e rn P W S , as m o d e lle d ab o v e, is
e q u a l to t
2 . T h a t, in all cases, th e tu rn o v e r tim e o f reg io n 1 is m u c h sm aller
th a n th a t o f reg io n 2 enables su g g e stin g th e fo llo w in g scen ario : d u rin g th e th ird
p h ase, th e o u tg o in g flux a t M o n ta g u e S tra it is esse n tia lly d u e to th e tra c e r p arcels
th a t fo llo w ed th e q u ic k est ro u te, i.e. th o se th a t d id n o t e n te r reg io n 2; m o st o f th e
tra c e r p arc els w hich p e n e tra te d in to th e n o r th e rn P W S leave P W S d u rin g th e final
p h ase.
A s m a y b e expected, th e fractio n o f th e tra c e r p arcels e n te rin g th e n o rth e n P W S , a,
is la rg e st, o r sm allest, w hen th e w in d is n o rth w a rd , o r s o u th w a rd , respectively. W h en
th e w in d is n o r th w a r d o r w estw a rd — w hich ind u ces a n e t n o rth w a rd E k m a n tr a n s ­
p o r t — , it is co n c eiv ab le th a t tra c e r p arc els leave th e n o r th e n P W S less fre q u en tly to
e n te r th e s o u th e rn reg io n , so th a t it is in su c h circ u m sta n c e s th a t th e tu rn o v e r tim e of
reg io n 2, 0, is la rg e st (T able 1). C onv ersely , 0 is sm allest w h e n th e w in d is s o u th w a rd o r
ea stw a rd .
F o r re a so n s w h ich are still u n clea r, th e advectiv e tim e la g x d ep en d s w eak ly o n the
su rfa ce fo rc in g co n sid ered (T able 1). H o w ev er, it m ay b e th a t x is d o m in a te d b y th e
specified c o n s ta n t th ro u g h flo w a n d n o t m u c h in flu en ced by th e E k m a n flow w hich
v arie s fro m case-to -case, a h y p o th e sis co n siste n t w ith M W ’s m o d elled c u rre n t
fields.
I t c a n n o t b e claim ed th a t th e sim p le m o d e l ab o v e is m o re rea listic th a n th e th reed im e n sio n a l P rin c e to n O c e a n M o d e l a p p lie d b y M W to P W S . T h a t th e c o m p a rtm e n ­
ta l m o d e l p ro v id e s a t a lm o st no co st a n e stim a te o f the tr a c e r tr a n s p o r t th ro u g h P W S
is m a rg in a lly in te restin g , since m o s t th ree -d im en sio n al m o d els m a y n o w be ru n
286
E. Deleersnijder et al. / Continental S helf Research 18 (1998) 279-287
ro u tin e ly in d o m a in s like P W S . T h e m o st a p p e a lin g fe a tu re o f th e tw o -c o m p a rtm e n t
m o d e l is th a t it involves o n ly th ree p a ra m e te rs , w h ich a ll h av e a clear ph y sical
m e an in g , h elp in g to u n d e rs ta n d th e h y d ro d y n a m ic s of P W S — since th e tra c e r a n d
w a te r parcels b eh av e sim ilarly. So, it h a s been p o s s ib le to e s tim a te , fo r every ty p e o f
su rface forcing considered, th e fra ctio n o f th e w a te r flux cro ssin g H in c h in b ro o k
E n tra n c e w hich first goes n o rth w a rd , in ste a d o f flo w in g d ire c tly to w a rd M o n ta g u e
S tra it. In a d d itio n , th e tu rn o v e r tim es o f th e s o u th e r n a n d th e n o rth e rn reg io n s h av e
b een ev alu ated . F in ally , it h a s been su ggested th a t th e h y d ro d y n a m ic pro cesses ta k in g
p lace in th e la tte r reg io n a m o u n t to diffusion, w h ile th e fo rm e r is d o m in a te d by m ere
ad vective m echanism s.
T o design th e tw o -c o m p a rtm e n t m o d el, it h a s n o t b e e n n ecessary to precisely
d elin ea te th e b o u n d a ry betw een th e tw o reg io n s c o n s id e re d . In fact, th e o nly clea r-cu t
difference betw een th e tw o b oxes is th a t th e y a r e d o m in a te d by h y d ro d y n a m ic
processes o f a different n a tu re , w hich d oes n o t p re v e n t th em from o v erla p p in g
g eographically. In o th e r w o rd s, it is h y d ro d y n a m ic s r a th e r th a n to p o g ra p h y th a t
allow s d istin g u ish in g th e tw o c o m p a rtm e n ts. T h e re fo re , it w o u ld b e difficult to
v alid ate th e sim ple m odel by d irec t c o m p a riso n w ith fluxes c o m p u te d , o r m e asu red in
situ, th ro u g h a given sectio n o f P W S . T h is c o n s id e ra tio n is w h y a n in v erse a p p ro a c h
h a s been — a n d sh o u ld c o n tin u e to be — p referre d .
In spite o f th e excellent a g re e m e n t betw een th e e v o lu tio n o f th e tra c e r m ass
sim u lated by M W ’s co m plex th re e -d im e n sio n a l m o d e l a n d th a t p red ic ted by th e
p resen t sim ple m odel, th e la tte r m ay n o t be d e e m e d to be fully v alid ated . In fact, it
w o u ld be necessary to c o m p a re th e p re d ic tio n s o f th e c o m p a rtm e n ta l m o d e l to th o se
o f th e th ree -d im en sio n al m odel o v er p e rio d s o f tim e la rg e r th a n th e la rg e st tu rn o v e r
tim e, i.e. a t le ast 100 days. O n th e o th e r h a n d , fo r ev ery ty p e o f flow, o nly o n e tra c e r
release sc en a rio h as been co n sid ered so far, w h ic h is n o t sufficient. T h erefo re, for
a given flow in P W S , it sh o u ld b e verified th a t th e sa m e set o f p a ra m e te rs en ab les th e
tw o -b o x m odel to re p re se n t th e e v o lu tio n o f th e tra c e r m a ss in a sa tisfac to ry w ay
th r o u g h o u t a series o f tra c e r releases. F o r in sta n c e , a c o n s ta n t so u rce lo c ate d a t
H in c h in b ro o k E n tra n c e o f m a g n itu d e $ m a y be c o n s id e re d , in w hich case th e tra c e r
m ass sh o u ld te n d to ( t + aö) <1>. A n o th e r in te re s tin g ex am p le co n sists o f a so u rce
releasing a n a m o u n t M o f tra c e r in a n a r b itr a r ily s h o rt p e rio d o f tim e, w hich
c o rre sp o n d s to th e case d e ta ile d a b o v e in th e lim it T -> 0. In th is case, a c c o rd in g to th e
sim p le m odel, th e tra c e r m a ss sh o u ld re m a in e q u a l to M u n til t = t ; th e n m sh o u ld
u n d erg o a d isc o n tin u ity , a n d finally d ec rease as « ex p [(x — t)/0 ] for % < t.
Acknowledgements
E D is a R esearch A sso ciate w ith th e N a tio n a l F u n d for Scientific R esearch of
B elgium . JW a n d C N K M a p p re c ia te th e fin a n c ia l s u p p o rt o f th e S o u n d E cosystem
A ssessm ent (SEA) P ro g ra m o f th e E x x o n V ald ez O il Spill (E V O S ) T ru s te e s C o u n cil
th ro u g h P rin c e W illiam S o u n d S cience C e n te r, A lask a. E D is in d e b ted to B enoît
T a rtin v ille a n d M a ria n S co tt for th e c o m m e n ts th e y p ro v id ed o n v a rio u s asp ects of
th is w ork.
E. Deleersnijder cl al. ¡Continental S helf Research 18 (1998) 279-287
287
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