INFORMATION TO USERS
This manuscript has been reproduced from the microfilm master. UMI films
the text directly from the original or copy submitted. Thus, some thesis and
dissertation copies are in typewriter face, while others may be from any type of
computer printer.
The quality of this reproduction Is dependent upon th e quality of the
copy sulmiitted. Broken or indistinct print, colored or poor quality Mustietions
and photographs, print t)leedthrough, substandard margins, and improper
alignment can adversely affect reproduction.
In the unlikely event that the author did not send UMI a complete manuscript
and there are missing pages, tftese will be noted. Also, if unauthorized
copyright material had to t)e removed, a note will indicate tfte deletion.
Oversize materials (e.g., maps, drawings, charts) are reproduced l>y
sectioning the original, t>eginning at tfie upper left-hand comer and continuing
from left to right in equal sections with small overlaps.
Photographs included in ttie original manuscript have t)een reproduced
xerographically in this copy.
Higher quality 6" x 9" black and vdiite
photographic prints are available for any photographs or illustrations appearing
in this copy for an additional charge Contact UMI directly to order.
Bell & Howell Information and Learning
300 North Zeeb Road, Ann Artx>r, Mi 48106-1346 USA
800-521-0000
M ixing and S econ d ary C irculation in Ju an de Fiica S trait
bv M ic h a e l W illia m O t t B .S c ., U n iv e r s it y of W a te r lo o . 1992 D i s s e r t a t i o n S u b m i t t e d in P a r t i a l F u lf il lm e n t of t h e R e q u i r e m e n t s for t h e D egree o f D O C T O R O F P H I L O S O P H Y in t h e S chool o f E a r t h a n d O c e a n S c ie n c e s W e a c c e p t th i s d is s e r ta t io n as c o n f o r m i n g to t h e re q u ire d s t a n d a r d D r. C h ris G a r r e t t S u p e r v is o r (S ch o o l o f E a r t h a n d O c e a n Sciences) Lueck D e p a r ^ e n t à l W l e m b e r (S c h o o l o f E a r t h a n d O c e a n S cie n c e s ) D r. .-Tndrew W e a v e rD e p a r tr p e f ita l M e m b e r (S ch o o l of E a r t h and O c e a n S cie n c e s )
Dr. P a tr i c k C u m m i n s O u t s i d e M e m b e r ( I n s i t u t e o f O c e a n Sciences) Dr. W a y n e R. G e y e r E x t e r n a l E x a m i n e r ( W o o d s l ^ l e O c e a n o g r a p h i c I n s t i t u t i o n ) © M ich ae l W illia m O t t . 2000 U n iv e rs itv of V ic t o r i a .\11 rig h ts re s e rv e d . T h i s d i s s e r t a t i o n m a y not be r e p r o d u c e d in w h o le o r in p a r t, by p h o to c o p y in g o r o t h e r m e a n s , w ith o u t t h e p e r m i s s i o n o f t h e a u t h o r .
I l
A b str a c t
E s tu a r i e s , t h e re g io n s w h ere runoff of f r e s h w a te r , soil, a n d c o n t a m i n a n t s first e n c o u n t e r t h e o c e a n , a r e also p r i m a r y fishing a n d r e c r e a t i o n a re a s. It is t h e r e f o r e i m p o r t a n t to u n d e r s t a n d t h e d y n a m ic s a s s o c ia te d w i t h m ix in g a n d c u r r e n t s w i t h i n th e s e b o d ie s of w a te r . P o l l u t a n t s a n d fre s h w a te r f r o m riv e r run o ff flow o u t t o s e a in t h e u p p e r layer, w h ile n u tr i e n t - c a r r y i n g o c e a n ic w a t e r r e t u r n s b e n e a t h . W h i l e t h e r e h a v e b e e n m a n y s t u d i e s o f t h e processes involved in t h i s e x c h a n g e flow, t h e d y n a m i c s a n d v ertic a l s t r u c t u r e o f tr a n s v e r s e flows a re m u c h less u n d e r s t o o d , d e s p i t e t h e role th e s e c u r r e n t s p la y in r e d i s t r i b u t i n g w a te r p r o p e r t i e s a n d m o m e n t u m t h r o u g h o u t t h e e s tu a ry .
O n e s u c h e s t u a r y , J u a n d e F uca S t r a i t , is a n id e a l lo c a tio n in w h ic h t o s t u d y e s t u a r i n e e x c h a n g e a n d t h e re s u ltin g c ro s s -c h a n n e l flows in d u c e d by i n t e r n a l f r ic tio n , p r im a r ily b e c a u s e its l e n g t h a n d s m o o th to p o g r a p h y r e d u c e t h e to p o g r a p h i c s t e e r i n g of c u r r e n ts . H is to ric a l c u r r e n t m e t e r d a t a from a n u m b e r o f d e p l o y m e n ts in J u a n d e F uca S tr a i t re v e a l t h a t , w h ile m e a n a lo n g -c h a n n e l c u r r e n t s a r e ro u g h ly c o n s is te n t w ith t h e t h e r m a l w in d e q u a t i o n , cross-ch an n el flows a r e n o t. p a r t i c u l a r l y a t m i d - d e p t h s w here t r a n s v e r s e c u r r e n t s a r e largest.
A m o m e n t u m b a l a n c e usin g histo rical s e a level a n d c u r r e n t m e t e r d a t a s u g g e s ts t h a t th e v e r ti c a l e d d y v isco sity A^. % 0.02 m ^ s ~ ’ a t in te rfa c ia l d e p t h s in M ay. T h e m e a n c i rc u la t io n in J u a n d e F u c a S t r a i t is hig h ly s e a s o n a l in n a t u r e , h o w e v e r, a n d la rg e r values m a y b e m o r e a p p r o p r i a t e in s u m m e r w h e n t h e e s t u a r i n e e x c h a n g e p e a k s d u e to t h e f re s h e t. S t r o n g e r friction is in t u r n a s s o c i a t e d w ith e le v a te d m i x i n g r a te s a n d in c re a s e d t r a n s v e r s e velocities. A n .Acoustic D o p p le r C u r r e n t P rofiler d e p l o y e d in J u a n d e F u c a S t r a i t in t h e s u m m e r of 1996 re so lv e d t h e v e rtic a l s t r u c t u r e o f t h e s e velocities. C o n c u r r e n t C u r r e n t - T e m p e r a t u r e - D e p t h d a t a reveal t h a t n e i t h e r t h e alo n g - nor t h e c ro s s -c h a n n e l c u r r e n t s a r e in g e o s t r o p h i c b a la n c e w ith th e h y d r o g r a p h i c s t r u c t u r e , s u g g e s tin g t h a t
Ill t h e p hysical processes a s s o c ia t e d w ith th e se c u r r e n t s a r e m o r e lo c a lis e d t h a n t h e five k ilo m e tre scales o v e r w h ic h t h e h y d r o g r a p h y was m e a s u r e d .
Z o o p l a n k to n w i t h i n .Juan d e F uca S t r a i t c o m p ri s e a s ig n if ic a n t p a r t of t h e s c a tt e r in g c ro s s -se c tio n u p o n w h ich t h e . \ D C P d e p e n d s . D u r i n g t h e i r d u s k m ig r a tio n in to th e e u p h o t i c z o n e t o feed a n d d a w n d e s c e n t to e s c a p e p r e d a t i o n , t h e y do n ot a c t as passive b a c k s c a t t e r t a r g e t s for t h e .Acoustic D o p p le r C u r r e n t P ro file r. V ertical m i g ra tio n velocities, m e a s u r e d fro m t h e b a c k s c a t t e r i n t e n s i t y r e c o r d , re a c h e d 0.0-3 m s ' , s u g g e s tin g t h a t s ig n ific a n t biases in t h e m e a s u r e d v e r t i c a l v e lo c ity could be i n t ro d u c e d . L ittle effect w as a c t u a l l y seen in t h e v e lo c ity fields, h o w e v e r, e v en t h o u g h t h e cro ss-sectional f r a c t io n o f t h e z o o p l a n k to n was a n o r d e r o f m a g n i t u d e la rg e r t h a n t h e b a c k g ro u n d .
M e a n c u r r e n t s in J u a n d u F u c a S tra it reveal s t r o n g t r a n s v e r s e flows a t m i d d e p t h s . s u g g e s tiv e o f i n t e r f a c i a l E k m a n layers. T h e a l o n g -c h a n n e l e s t u a r i n e e x c h a n g e is significantly e n h a n c e d a t n e a p ti d e , c o n s is te n t w ith w e a k e r m i x i n g u p s t r e a m . T h e c ro ss-ch an n el flows a t in t e r f a c i a l d e p t h s are also s u b s t a n t i a l l y la rg e r d u r i n g n eap ti d e , im p ly in g a f o r tn i g h tly m o d u l a t i o n o f m ix in g r a te s w ith in t h e s t r a i t .
T h e A D C P w as a lso used to m e a s u re t h e R e y n o ld s s tr e s s e s d ire c tly . T h e s e w ere found to b e m o r e t h a n a n o r d e r of m a g n i t u d e la rg e r a t n e a p t i d e t h a n d u r in g s p rin g tid e a n d w ere c o n s i s t e n t w ith changes in t h e m e a n c u r r e n t o v e r t h e s p rin g -n e a p cycle. R ey n o ld s s tre s s e s w ere m a x i m a l a t m i d - d e p t h o n t h e t r a n s i t i o n from e b b to flood, a t w hich t i m e t h e g r a d i e n t R ic h a rd s o n n u m b e r s w ere s m a l l e s t , s u g g e s tin g t h a t c ritic a l layer a b s o r p t i o n o f in t e r n a l waves a re i m p o r t a n t d y n a m ic a l ly .
I V
Dr. C h r is G a r r e t t
S u p e rv is o r (S c h o q j,« C n d O cean S ciences)
,f Lueck.
entail M e m b e r (S chool o f E a r t h a n d O c e a n S cie n ces)
Dr. A nd rew W e a v e r
D e p a r t m e n t a l M e m b e r (S chool o f E a r t h a n d O c e a n S cie n ces)
Dr. P a t r i c k C u m m i n s
O u ts id e M e m b e r ( I n s t i t u t e o f O c e a n Sciences)
Dr. W a y n e R. G e y e r
T ab le o f C o n te n ts
List of T a b l e s ... viii List of F ig u re s ... ix A c k n o w l e d g m e n t s ...xviii C h a p t e r 1. I n t r o d u c t i o n a n d M o t i v a t i o n ... I C h a p t e r 2. B a c k g r o u n d ... 4 2.1 E s t u a r i e s ... 4 2.1.1 C la s s if ic a tio n S c h e m e s ... 4 2.1 .2 T h e P a r t i a l l y - M i x e d E s t u a r y ... 11 2.1.3 T h e E ffects o f R o t a t i o n ... 12 2.2 T u r b u l e n c e ... 15 2.2.1 R e y n o l d s S t r e s s ... 16 2.2.2 T h e R e y n o l d s N u m b e r ... 19 2 .2 .3 T h e B o t t o m - B o u n d a r y L a y e r ... 20 2.3 S t a b i l i t y in t h e P r e s e n c e of S t r a t i f i c a t i o n ... 22 2.3.1 S h e a r I n s t a b i l i t y ... 23 2 .3 .2 B a r o c l in ie I n s t a b i l i t y ... 25 2.4 T i d e s ... 28 2.4.1 T i d e s in C h a n n e l s ... 29 2 .4 .2 I n t e r n a l T i d e s ... 31 C h a p t e r 3. J u a n d e F u c a S t r a i t ... 36 3.1 B a c k g r o u n d ... 36 3.2 A P r e l i m i n a r y A n a l y s i s ... 44 3.2.1 R e a n a l v s i s of H isto ric al D a t a ... 44VI 3.2.2 M o m e n t u m B a l a n c e ... 51 3.2.3 S t a b i l i t y to I n t e r n a l M i x i n g ... 55 3.2.4 S u m m a r y ... 56 C h a p t e r 4. O b s e r v a tio n s f r o m 1996 ... 57 4.1 H y d r o g r a p h y ... 57
4.1.1 D e n s ity P rofiles in .Juan d e F u c a S t r a i t ... 59
4.1.2 S u r fa c e S a l i n i t y ... 63 4.2 .Acoustic D o p p le r C u r r e n t P r o f i l e r ... 63 4.2.1 A D C P T h e o r y ... 64 4 .2.2 A D C P T e m p e r a t u r e M e a s u r e m e n t s ... 72 4.2.3 A D C P C o m p a s s a n d T ilt A n g le M e a s u r e m e n t s ... 74 4.2.4 C a l c u l a t e d E a r t h V e lo c i tie s ... 76 C h a p t e r 5. T h e V ertic al V e l o c i t y ... SO 5.1 S p a t i a l I n h o m o g e n e i t y ... SO 5.2 H o riz o n ta l C o n t a m i n a t i o n of V ertical V elocities ... S2 5.3 B a c k s c a t t e r I n t e n s i t y a n d T a rg e t S t r e n g t h ... S4 5.3.1 C o r r e c t i o n o f B a c k s c a tte r I n te n s i ty M e a s u r e m e n t s ... S5 5.3.2 T h e D i u r n a l Z o o p la n k to n .V lig r a tio n ... S7 5.4 I n t e r n a l W av e s ... 94 5.5 S u m m a r y ... 96 C h a p t e r 6. T i d e s ... 97 6.1 T i d a l C o n s t i t u e n t s ... 99 6.2 B o t t o m F r i c t i o n ... 101 6.2.1 L o g -L a y e r F i t t i n g ... 102 6.2.2 T h e B o t t o m B o u n d a r y L a y e r ... 106 6.3 I n t e r n a l T i d e s ... 113 6.3.1 T h e .A lo n g -C h an n el M 2 T i d e ... U S
V I I 6.3.2 T h e C r o s s - C h a n n e l A/2 T i d e ... 122 6.3.3 T h e V e r tic a l A /2 T i d e ... 123 C h a p t e r 7. M e a n a n d N o n - T id a l F l u c t u a t i o n s ... 126 7.1 R e s id u a l E s t u a r i n e C i r c u l a t i o n ... 126 '.1.1 C o m p a r i s o n to G e o s tr o p h ic Flow ... 126 '.1.2 V o l u m e t r i c O u t f l o w ... 130 7.1.3 D a ily M e a n s ... 132 7.2 S t a b i l i t y ... 132 .2.1 B a r o c lin ie I n s t a b i l i t y ... 134 .2.2 S h e a r I n s t a b i l i t y ... 135 7.3 R e y n o ld s S t r e s s ... 138 '.3.1 V a r ia t io n w ith A v e ra g in g T i m e ... 139 .3.2 T h e S p r in g - N e a p C y c l e ... 142 .3.3 T h e V e rtic a l E d d y V i s c o s i t y ... 147 .3.4 E ffect on t h e M e a n F l o w ... 149 C h a p t e r 8. C o n c l u s i o n s ... 152 R eferences 158
V I 11
L ist o f T a b le s
3.1 S e a s o n a l V a ria b ility of t h e M e a n E s t u a r i n e F low in J u a n d e F u c a S t r a i t 49
6.1 T i d a l C o n s t i t u e n t s U sed in T i d a l A n a l y s i s ... 98 6.2 O s c i l l a t o r y B o u n d a r y L a y e r M o d e l for t h e A /2 t i d e ... 112 6.3 M o d a l F i t (in m s ~ ' ) t o t h e .A lo n g -C h a n n e l A /2 T i d a l A m p l i t u d e . . 118 6.4 R M S a n d L a rg e st D ifference ( m s~*) t o t h e A l o n g - C h a n n e l M o d a l F i t 119 6.5 A/2 P a r a m e t e r s from 1973 d a t a ( w i t h 95% c o n fid e n ce in te r v a l) . . . . 120 6.6 M o d a l F i t (in m s~*) t o t h e 1973 A /2 T i d e ... 121
I X
L ist o f F ig u res
2.1 L a b o r a t o r y r e s u lt s o f r o t a t i o n a l e x c h a n g e flow ( J o h n s o n a n d O h is e n 1994). T h i c k a rro w s in d i c a t e E k m a n layer flows w h ile t h i n arrow s
in d i c a t e i n t e r i o r flows... 14 2.2 W ed g e o f i n s t a b i l i t y ( s h a d e d region) b e tw e e n s lo p in g is o p y c n a ls (solid
lines) a n d h o r i z o n t a l ( d a s h e d line). D is p la c e m e n t s f ro m o n e re g io n to a n o t h e r re le a s e p o t e n t i a l energy, allow ing for t h e g r o w t h o f b a ro c lin ie
i n s t a b i l i t i e s ... 25
3.1 G e o g r a p h y o f J u a n d e F u c a S tr a i t a n d t h e S t r a i t o f G e o r g ia w ith
h y d r o g r a p h i c , m e te o r o lo g ic a l, sea level g a u g e , a n d c u r r e n t m e t e r sites. 37 3.2 T h e s e a s o n a l c y c le in t h e v o lu m e tric flow of t h e F r a s e r R iv e r, as m e a s u re d a t H o p e . T h e t h i c k solid line is t h e m e a n a n d t h e t h i c k d a s h e d line is t h e m e a n p lu s s t a n d a r d d e v ia tio n o v er t h e y e a r s 1912 to 1996. T h e f r e s h w a t e r i n p u t in to t h e S tr a i t of G e o rg ia is a b o u t 30% la rge r th a n t h e a m o u n t m e a s u r e d a t H o p e... 38 3.3 M id c h a n n e l d e n s i t y (cr^, in kg m~^) alo n g J u a n d e F u c a S t r a i t in J a n u a r y . M ay, a n d J u l y ( C r e a n a n d .Ages 1971)... 41 3.4 .Annual c y c le o f a lo n g -c h a n n e l difference in m e a n t i d a l h e i g h t , a t m o
s p h e ric p r e s s u r e , a n d t o t a l surface p re s s u re . T h e d iffe re n c e in th e m o n t h l y a v e r a g e s e a level h e ig h t b e tw ee n P o r t A n g e le s a n d N e a h Bay c o n ta in s b e t w e e n 8 to 11 y e a rs of d a t a from 1984 t o 1992 for eac h m o n t h : t h e d o t t e d line in d ic a te s th e s t a n d a r d d e v i a t i o n . T h e a t m o s p h e ric p r e s s u r e d iffere n ce uses h o u rly s u rfa c e p r e s s u r e valu es o v e r a tw o y e a r p e r i o d . T h e n e t alo n g -c h a n n e l s u rfa c e p r e s s u r e g r a d i e n t is
3.5 A v erag e, d e - t i d e d along- a n d cross-channel v e lo c itie s in J u a n d e F uca S t r a i t b a s e d o n 1973 d e p l o y m e n ts b e tw ee n P ill a r P o i n t a n d J o r d a n R iv e r ( d e p l o y m e n t I: M a r c h 6 to A pril 16: d e p l o y m e n t 2: A p ril IT to J u n e 14). S t a n d a r d d e v ia tio n s a re in d ic a te d by t h e s h a d e d regions for a lo n g -c h a n n e l c u r r e n t s a n d by th i n lines for c r o s s - c h a n n e l flows. T h e b o t t o m p a n e l also show s 32-day t e m p e r a t u r e a v e ra g e s (d egrees C elsius), a t lo c a tio n s m a r k e d w ith sm all s q u a re s , f r o m r e c o rd s s t a r t i n g
on 15 May, 1973... 48 4.1 1996 C T D a n d A D C P m o o r i n g locations in J u a n d e F u c a S t r a i t a n d
a d e t a ile d s e c ti o n n e a r t h e A D C P m o o rin g s h o w in g t h e b a t h y m e t r y
in m e t r e s ... 58 4.2 M e a n h y d r o g r a p h ic profiles a t C T D s ta t io n s .A1 ( t h i c k s o lid ). .A8
(th ic k d a s h ) , C'4 ( t h i n so lid ), a n d C8 ( th i n d a s h ) : a) t e m p e r a t u r e (°C ). b) s a l i n i t y (p s u ) . c) d e n s ity (cr^. kg m ~ ^). a n d d) T S d ia g r a m , w ith plus sig n s a n d circles in d ic a tin g d e p t h s of 20 a n d 100 m . re s p e c tively. a n d c o n s t a n t sh o w n as d o t t e d lines, w ith v a lu e s g iv e n in kg
m “ ^... 60 1.3 C o n to u r s o f for a) a n alo n g -c h an n el t r a n s e c t , b) a c ro s s -c h a n n e l
t r a n s e c t, a n d c) t h e t i m e series ta k e n a t t h e A D C P s t a t i o n . T h e c o n t o u r in te rv a l is 0.5 kg m w ith t h e n u m b e r s t o t h e rig h t o f each p lo t c o r r e s p o n d in g to t h e solid lines. T h e b o t t o m is i n d i c a t e d by th e th ic k line. P l u s m a r k s in c) in d ic a te in d iv id u a l profiles in t h e ti m e series a n d t h e t i d e ( p o s iti v e to t h e e a s t) a t 80 m d e p t h is sh o w n in d ). 61 4.4 D aily s u rfa c e s a li n ity re c o rd e d a t R ace R ocks for 1996, w ith d e ta ile d
s ectio n d u r i n g t h e c u r r e n t m o o rin g d e p l o y m e n t ... 62 4.5 T h e effect o f n o n -z e ro t i l t a n g le on bin h eight. N o m in a l b in c e n tre s
(p lu s signs) a r e m o v e d (crosses) higher a n d low er for s m a l l e r a n d la rge r d e c lin a tio n s , r e s p e c tiv e ly . N u m b e rs in d ic a te t h e h e ig h t o f t h e bin
X I
4.6 T h e effect o f t h e b in m a p p i n g a l g o r i t h m o n c a l c u l a t e d v e lo c itie s . For t h e t h r e e - d i m e n s i o n a l v e lo c ity s h o w n in a ) , b ) . a n d c). t h e c o r r e s p o n d ing biases a r e s h o w n in d ). e) . a n d f). w i t h d a s h e d lines for t h e RDI n e a re s t n e i g h b o u r b in m a p p i n g a n d solid lin es for l i n e a r i n t e r p o la t io n .
All c u r r e n t s a r e in u n i t s of 10” ^ m s ” ’ ... 70 4.7 T e m p e r a t u r e r e c o r d fro m th e .ADCP. in d e g re e s C e lsiu s; a ) t h e 24
h o u r r u n n i n g m e a n , b) th e re s id u a l (i.e. a c t u a l t e m p e r a t u r e - r u n n in g
m e a n ) ... 72 4.8 C o m p a s s h e a d i n g o f t h e .ADCP. in d e g re e s clo c k w ise fro m t r u e n o rth :
a) t h e 24 h o u r r u n n i n g m e a n , b) t h e r e s i d u a l (i.e. a c t u a l h e a d i n g - r u n n in g m e a n ) ... 74 4.9 T i lt a n g le r e c o r d s fro m t h e A D C P , in d e g r e e s a) p it c h , b) r o ll... 75 4.10 T w e n t y m i n u t e m e a n s o f e a s t, n o r t h , a n d v e r ti c a l v e lo c itie s on .Julian d a y 213, m a x i m u m s p r i n g t i d e ... 77 4.11 T w e n t y - f o u r h o u r r u n n i n g m e a n o f t w o - h o u r a v e r a g e d e a s t , n o r t h , and v e rtic a l c u r r e n t s ... 78 4.12 S q u a r e of b o t t o m c u r r e n t : a) t h e 48 h o u r r u n n i n g m e a n , b) t h e resid ual (i.e. a c t u a l - r u n n i n g m e a n ) 79
5.1 C u m u l a t i v e f r a c t io n for th e v alu e of t h e e r r o r v e lo c ity e to vertical v elo city w r a t i o : a ) c o n t o u rs for e n t i r e r e c o r d , w ith c o n t o u r la b e ls at t h e b o t t o m , b) m e a n c u m u l a t i v e f ra c tio n b e lo w u n i t y a n d 4 for th r e e d a y s a f te r n e a p t i d e (.Julian d a y s 206 to 208. th i c k lines) a n d th r e e d ay s a f te r s p r i n g t i d e ( J u l i a n d a y s 213 t o 215. t h i n lines). T h e th re e - d a y m e a n s a r e i n d i c a t e d by solid lines, w h ile d o t t e d lines d e n o t e th e
X I I
5.2 M in im is a tio n of t h e r o o t - m e a n - s q u a r e v e rtic a l v e lo c ity t o d e t e r m i n e t h e .A.DCP t i lt a n g le b ia s e s : a ) t h e d e p t h v a ria tio n in roll (s o lid lin e) a n d p itc h ( d a s h e d line) a n g l e “c o r re c tio n s " w h ic h m i n i m i s e t h e r m s v e r tic a l velocity, b) t h e r e s u l t i n g r m s w, w ith th i c k line i n d i c a t i n g t h e rm s v e rtic a l v e lo c ity u s in g t h e re c o r d e d t i lt a n g les. T h e d o t t e d lin es in a ) a re th e b e s t lin e a r fit t o t h e c a l c u la te d a n g les o v e r t h e d e p t h
ra n g e s 50 to 60 m a n d 110 t o 120 m ... 82 5.3 B a c k s c a t t e r i n t e n s i t y on .Ju lian d a y 213. m a x i m u m s p r i n g t i d e . T h e
solid line in d ic a te s t h e d e p t h a b o v e w h ich th e u n c o r r e c t e d b a c k s c a t t e r
is less t h a n 60 c o u n t s ) ... 84 5.4 T h e slopes of t o t a l v e r ti c a l v e lo c ity v e rs u s c o r r e c te d b a c k s c a t t e r i n t e n
s ity s c a t t e r p l o t s w ith t i m e : a ) in 20 m i n u t e s e c tio n s o v e r all d e p t h s for all 20 days c o m b in e d , a n d d e p t h a v e r a g e w ith ± 1 s t a n d a r d d e v i a t i o n
fro m b) 50 to 70 m a n d c) 70 t o 90 m ... 88 5.5 S t a n d a r d d e v i a tio n in b a c k s c a t t e r i n t e n s i t y a n d v e r tic a l v e l o c ity fields
o v e r t h e e n t ir e 2 0 -d a y d e p l o y m e n t ... 92 5.6 B a c k s c a t t e r i n t e n s i t y a n d v e r t i c a l v e lo c ity a n o m a l y fields for .July 31.
1996 a n d c o m p o s i te a n o m a l y fields o v er t h e 2 0 -d a y d e p l o y m e n t . . . . 93 5.7 B a c k s c a t t e r in t e n s i t y a n d i n t e g r a t e d velocity s u rfa c e s . In a ) t h e m e a
s u r e d v ertical v e lo c ity is u s e d , w h ile in b) th e v e r ti c a l v e l o c ity is c o n t a m i n a t e d w ith h o r iz o n t a l c u r r e n t s by r o t a t i n g it 0.5° d e g r e e s fro m
t h e v e r ti c a l... 95
6.1 T i d a l ellipse p a r a m e t e r s fo r t h e M 2 ( u p p e r p a n e ls ) a n d A 1 (lo w e r p a n e ls ) c o n s t i t u e n t s o f t h e h o r iz o n t a l velocity. D a s h e d lines i n d i c a t e t h e 95% co n fid e n ce in t e r v a ls . T h e in c lin a tio n is t h e a n g l e b e t w e e n t h e s e m i - m a jo r a x is a n d d u e e a s t , w ith p o s itiv e a n g les i m p l y i n g a n t i clockw ise. N e g a ti v e r e l a t i v e p h a s e s i n d i c a te t h e c u r r e n t a t d e p t h la g s
X l l l
6.2 Log-layer fit t o t h e b o t t o m c u r r e n t : a) th e frictio n v e lo c ity v e rs u s t h e referen c e v e lo c ity ( m e a s u r e d 1 1 m a b o v e th e b o t t o m ) , b) t h e r o u g h ness p a r a m e t e r v e rs u s t h e m a g n i t u d e of t h e re fe r e n c e v elocity, c) h is to g r a m o f t h e log-lay er h e i g h t, a n d d) h is to g r a m o f t h e ro u g h n e s s p a r a m e t e r . F i t s for w h ic h e x c e e d s 0.1 m h a v e b e e n o m i t t e d , le a v
ing 1002 lo g -la y e r fits o u t o f t h e 1440 2 0 - m in u te m e a n c u r r e n t profiles. 104 6.3 O s c illa to ry b o u n d a r y la y er fit t o t h e s e m i - m a jo r axis o f t h e M 2 tid a l
ellipse. In a), log-lay er fits t o t h e m e a s u re d M 2 (solid line) y ie ld
u_ = 0.024 m s ^ a n d = 0.003 m (d a sh ed line) w h e n z^ is u n c o n s tr a i n e d a n d u , = 0.026 m s~* w h en z^ is set to 0.006 m ( d o t t e d line). T h e r e s u lt in g m a g n i t u d e a n d p h a s e o f th e S o u lsb y (1983) o s c il la t o r y
b o u n d a r y la y er s o lu ti o n t o t h e ti d e is show n in b) a n d c), re s p e c tiv e ly . 108 6.4 O s c illa to ry b o u n d a r y la y er fit to t h e a n tic lo c k w ise ( t h i c k lines) a n d
clockw ise ( t h i n lines) r o t a r y c o m p o n e n t s of t h e M 2 ti d a l ellipse: a) m a g n i t u d e a n d b) p h a s e . Solid lines a r e d a t a , d a s h e d lines r e p r e s e n t th e b e s t m o d e l fit, a n d d o t t e d lines re p re s e n t t h e fit for z^ s e t t o 0.006 m. T h e s h o r t e r lines in a) r e p r e s e n t t h e log-layer fit t o t h e m a g n i t u d e only, u sed to d e t e r m i n e u , a n d z ^ ... I l l 6.5 V a ria tio n in t h e m e a n d e n s i t y profiles in th e a) c ro s s - c h a n n e l d ir e c tio n :
S ta tio n s C 6 ( t h i n s o lid ), C 7 ( th ic k solid), a n d C 8 ( t h i n d a s h ) , a n d b) a lo n g -c h a n n e l d ir e c tio n : A 4 ( t h i n solid). A5 ( t h i c k s o lid ) , a n d A 6 (th in d a s h ) ... 114 6.6 V ariatio n in t h e v e r ti c a l s t r u c t u r e of th e second a lo n g -c h a n n e l m o d e
at t h e M 2 fre q u e n c y . T h e m o d e s a r e c a lc u la te d by i n t e g r a t i n g t h e T a y lo r-G o ld s te in e q u a t i o n w ith no b a c k g ro u n d flow a t e a c h C T D S t a tion: 0 7 ( th i c k s o lid ). 0 8 (s o lid ), 0 6 ( t h i n solid), A 5 ( t h i c k d a s h ) . A4 (d a s h ), a n d A6 ( t h i n d a s h ) ... 115
X I V
6.7 V a riatio n in v e r ti c a l s t r u c t u r e o f A/2 b a ro clin ie m o d e s a t C T D s t a t i o n 0 7 : a) first a l o n g -c h a n n e l, b) first v ertical, c) s e c o n d a lo n g -c h a n n e l. an d d) sec o n d v e r ti c a l . T h e m o d e s are c a l c u l a t e d by i n t e g r a t i n g t h e T a y lo r- G o ld s te in e q u a t i o n for n o b a c k g ro u n d flow ( t h i c k so lid ), for waves tra v e llin g in t h e p o s itiv e x -d ire c tio n ( e a s t w a r d ) in t h e p re s e n c e of th e m e a n e s t u a r i n e flow ( t h i n solid), a n d for w aves tr a v e llin g in t h e
n e g ativ e x - d ir e c tio n ( t h i n d a s h e d ) ... 116 6.8 T id a l p a r a m e t e r s for t h e A/2 c o n s ti tu e n t of t h e v e r t i c a l v elo city (w ith
95% co n fid e n ce i n t e r v a l s ) ... 123 6.9 M a g n it u d e o f t h e a ) h o r iz o n t a l a n d c o r r e s p o n d in g b) b a r o t r o p i c a n d
c) baro clin ie v e rtic a l m o d e s (fro m th e T a y l o r - G o l d s t e in e q u a t i o n ) a t s ta tio n C7. F or a t i d a l w ave tra v e llin g in t h e p o s iti v e (n e g a tiv e ) d i r e c tion. t h e v e rtic a l v e lo c itie s s h o w n o c c u r a q u a r t e r p e r i o d a f te r (b efo re )
th e h o riz o n ta l v e l o c iti e s ... 124
7.1 M ean re s id u a l (i.e. d e t i d e d ) c u r r e n t s in .Juan d e F u c a S t r a i t o v e r t h e e n tire d e p l o y m e n t. T h e ( x . y ) axes are r o t a t e d 10° clockw ise from
( e a s t , n o r t h ) ... 127 7.2 V ariatio n of m e a s u r e d a ) a lo n g -c h a n n e l a n d b) c ro s s -c h a n n e l c u r r e n t
(th ick ) w ith a x es r o t a t i o n (d e g re e s clockw ise fr o m e a s t - n o r t h ) : 10 (solid). 15 ( d a s h ) , a n d 5 ( d o t- d a s h ) . .Also s h o w n a r e t h e g e o s tro p h ic velocities ( t h i n ) . T h e a l o n g -c h a n n e l g e o s tro p h ic v e lo c ity is b a s e d on t h e s tr a t if ic a tio n a t C T D s t a t i o n s C4 a n d C 8. u s in g t h e c ro s s -s tr a it surface slopes s h o w n , w h ile t h e cross-ch an n el g e o s t r o p h i c v elo city is based on t h e h y d r o g r a p h y a t t h e pairs of C T D s t a t i o n s show n. T h e a lo n g -s tra it s u rfa c e s lo p e u s e d . = 2 x 10 ' . is t h a t for w hich t h e g eo s tro p h ic a n d m e a s u r e d c u r r e n t s m a tc h a t t h e t o p o f t h e E k m a n
X V
7.3 D aily m e a n s of re s id u a ] c u r r e n t in a) a lo n g - c h a n n e l. b ) c r o s s - c h a n n e l, a n d c) v e rtic a l d i r e c t i o n . P lu s signs i n d i c a te z e ro v e lo c ity , a n d th e ± 0 .1 m s * scale is d e n o t e d n e a r th e t o p o f e a c h p lo t . T h e first profile, c e n t r e d a t .Julian d a y 200.5. r e p r e s e n t s t h e m e a n for .Julian
d a y 200, .July I S ... 133 7.4 a) D e n s ity a t C T D s t a t i o n .A.DCP-S on J u l y 17 ( d a y 199. t h i n so lid ,
m e a n o f t h r e e pro files). J u l y 25 (d a y 207. t h i n d a s h , m e a n o f six profiles), A u g u s t 6 ( d a y 219, t h i n d o t, m e a n o f t h r e e pro files), a n d overall m e a n ( th i c k s o lid ) , a n d b) b u o y a n c y f r e q u e n c y o f m e a n profile.
S p rin g ti d e o cc u rs a t J u l i a n d a y s 199 a n d 21 3 ... 134 7.5 Low f re q u e n c y o s c il la t io n s in t h e cro s s -c h a n n e l c u r r e n t . T h e o v e ra ll
m e a n a t e a c h d e p t h is s u b t r a c t e d from t h e 4 8 - h o u r r u n n i n g m e a n of
t h e re s id u a l c u r r e n t ... 135 7.6 V a ria tio n w ith d e p t h a n d o v e r t h e s p rin g -n e a p c y c le o f t h e fr e q u e n c y
(%) w ith w h ich t h e g r a d i e n t R ic h a r d s o n n u m b e r , d e f in e d o v e r 2 m . falls below 0.2-5. In a ) , t h e t h i c k a n d t h i n lines i n d i c a t e t h e p e r io d j u s t a f te r n e a p t i d e ( J u l i a n d a y s 206 to 208) a n d s p r i n g t i d e ( J u l i a n d ays 213 t o 215). r e s p e c tiv e ly . T h e solid lines a r e t h r e e - d a y a v e ra g e s a n d t h e d a s h e d lines i n d i c a t e t h e m e a n ± t h e s t a n d a r d d e v i a t i o n of th e m e a n s for e a c h d a y . In b ), t h e m e a n c u m u l a t i v e f r e q u e n c y {%) for t h e d e p t h s b e t w e e n 80 a n d 90 m o v e r d a y s 206 t o 208 a n d 213 to 215 a r e d e n o t e d b y t h i c k a n d t h i n lines, re s p e c tiv e ly . T h e d a s h e d lines a g a in i n d i c a te t h e m e a n ± t h e s t a n d a r d d e v i a t i o n o v e r t h e t h r e e d ays a n d 6 d e p t h b in s . P a n e l c) plo ts t h e m e a n d a i l y f r e q u e n c y w ith which R i < 1 /4 o v e r t h e d e p t h r a n g e SO t o 90 m (s o lid ) a n d t h e m e a n
X V I
7.7 S h e a r i n s t a b i l i t y o n J u l i a n d ay s a) 207 ( n e a p ti d e ) a n d b) 214 ( s p r i n g tid e ) . S h a d e d reg io n s i n d i c a t e d e p t h s a n d t i m e s for w h ic h t h e c a l c u la te d R i c h a r d s o n n u m b e r fell below 0.25. T h e solid h o r iz o n t a l p rofile is t h e a lo n g -c h a n n e l t i d a l velocity a t SO m d e p t h , w ith t h e ± I m s~^ s c ale p l o t t e d o n t h e r ig h t. T h e v e rtic a l profiles a t 0900. 1200. a n d 2200 in a) a n d a t 0300, 1100. a n d 1500 in b ) a r e t h e h o u r l y a v e r a g e to t a l (i.e. ti d a l a n d r e s id u a l) a lo n g -c h a n n e l c u r r e n t s w ith t h e ± 1 m
s ' scales p l o t t e d a b o v e ... 138 7.8 V a r ia t io n o f a u t o - a n d c ro s s -c o rre la tio n s o f v e lo c ity f lu c t u a ti o n s w ith
a v e r a g in g t i m e , T , o v e r t h e d e p t h ra n g e 50 t o 130 m for J u l i a n d a y
207. j u s t a f t e r n e a p t i d e ... 140 7.9 V a r ia tio n o f a u t o - a n d c ro s s -c o rre la tio n s o f v e lo c ity f lu c t u a t i o n s w ith
a v e r a g in g t i m e . T . o v e r t h e d e p t h ra n g e 50 to 130 m for J u l i a n d a y
214. j u s t a f t e r s p r in g t i d e ... 141 7.10 M a g n it u d e s o f t h e d a i ly m e a n R e y n o ld s s tr e s s a f t e r n e a p t i d e ( J u l i a n
d a y 207) a n d s p r in g t i d e ( J u lia n d a y 214). w ith v p o s iti v e in t h e d ir e c tio n of t h e o r i e n t a t i o n of .A.DCP b e a m 3. 335° fr o m t r u e n o r t h . T h e d a s h e d lines r e p r e s e n t t h e 95% c o n fid e n c e in te rv a ls d e t e r m i n e d fro m a b o o t s t r a p t e c h n i q u e ... 143 7.11 V a r ia tio n o f u ' w ' a n d v ' w ' R e y n o ld s s tre s s t h r o u g h o u t t h e s p r i n g - n e a p
cycle. In e a c h ca s e , t h e solid lines r e p r e s e n t t h e d a ily a v e ra g e s o v e r th e d e p t h r a n g e s s h o w n a n d t h e d a s h e d lines i n d i c a t e t h e s t a n d a r d d e v i a t i o n o v e r t h e d e p t h ran g e. T h e (u , c) a x e s a r e r o t a t e d su ch t h a t
u is p o s itiv e u p - c h a n n e l ... 144 7.12 R e y n o ld s s tr e s s e s a t n e a p ( J u lia n d a y 207) a n d s p r i n g ( J u l i a n d a y
X V I I
7.13 M e a n v e r ti c a l R eyn o ld s s tre s s a n d v e r t i c a l d e riv a tiv e o v er t h r e e d a y s a t n e a p ( J u l i a n d a y 206 t o 208. t h i c k lines) a n d sp rin g t i d e ( J u l i a n d a y 2 13 to 215, th in lines). T h e d a s h e d lines in d ic a te t h e s t a n d a r d d e v i a t i o n o v e r th e t h r e e d a y s ... 146 7.14 M e a n re s id u a l a) a lo n g -c h a n n e l a n d b) cross-ch an n el c u r r e n t w i t h c) a n d d ) a s s o c ia te d sh e a rs . T h e v e rtic a l e d d y viscosities, p a r a m e t e r i s e d as (2 .6 ) a r e p lo t te d in e) a n d f). T h i c k lines a r e t h e th r e e d a y a v e r a g e s a t n e a p t i d e ( J u lia n d a y 206 to 208) a n d t h i n lines d u r in g s p r in g t i d e ( J u l i a n d a y 213 to 2 1 5 )... 148
X V I I I
A c k n o w le d g m e n ts
I w o u ld like t o t h a n k m y s u p e rv is o r D r. C h r i s G a r r e t t for his g u id a n c e a n d g e n e ro u s s u p p o r t , as well as m y e x t e r n a l e x a m i n e r . D r. W a y n e G e y e r . a n d th e m e m b e r s of m y c o m m i t t e e . D r. R o lf L ueck. Dr. .Andrew W e a v e r , a n d D r. P a tr i c k C u m m i n s , for t h e i r c o m m e n t s a n d s u g g estio n s. I w o u ld also like to t h a n k D r. R ic h a r d D e w e y for m a n y h e lp fu l d is c u s s io n s a n d for le n d in g m e his lib ra ry . I a m v e ry g r a te f u l t o m y p a r e n t s . Bill a n d A tie for t h e i r c o n s ta n t e n c o u r a g e m e n t a n d s u p p o r t . M y wife L a u r a has b e e n a g r e a t s o u r c e o f c o m fo r t a n d e n c o u r a g e m e n t for m e a n d h a s been w o n d e r f u l s u p p o r t d u r i n g t h e re s e a rc h a n d w ritin g o f t h i s d i s s e r t a t i o n .
1. In tr o d u c tio n a n d M o tiv a tio n
C h a p te r 1
In tr o d u c tio n an d M o tiv a tio n
C o a s t a l w a t e r s h a v e , sin c e e a r l y tim e s , b e e n very i m p o r t a n t in s h a p in g h u m a n d e v e l o p m e n t a n d lifesty le. N e a r s h o r e u p welling regions a r e a m o n g t h e m o s t b io lo g ically p r o d u c t i v e a r e a s , a c c o u n t i n g for n e a rly -50% o f t h e g lo b a l m a r i n e food c a t c h w ith o n ly 10% of t h e o c e a n i c s u rfa c e area . C o m m e r c ia l s e a tra ffic is in c re a s in g a s t h e w orld e c o n o m ie s b e c o m e f u r t h e r in t e g r a te d . R e c r e a ti o n a l u s e o f c o a s t a l seas is also in te n s ify in g as u r b a n p o p u l a t i o n s c o n tin u e to grow , as d o t h e p r e s s u r e s a s s o c ia te d w ith w a s te d i s p o s a l a n d c o n t a m i n a n t runoff.
E s t u a r i e s , t h e re g io n s w h e re runoff o f f r e s h w a te r, soil, a n d c o n t a m i n a n t s first e n c o u n t e r t h e o c e a n , f r e q u e n t l y e x p e r i e n c e large f lu c t u a ti o n s in e n v i r o n m e n t a l c o n d itio n s o v e r s m a l l t i m e sc a le s. V a r ia t io n s in ti d a l h e i g h ts o f te n c r e a t e la rg e i n t e r t i d a l zones, w h ile ti d a l c u r r e n t s a n d w a v e a c tio n r e s u lt in c o n s t a n t l y c h a n g i n g s a lin itie s , t e m p e r a t u r e s , n u t r i e n t a n d g as c o n c e n tr a ti o n s , a n d s e d i m e n t lo a d s. In a d d i tio n to th e s e f l u c t u a t i o n s , t h e o r g a n is m s w h ich in h a b it th e s e re g io n s , a n d u p o n w h ic h we d e p e n d for m u c h o f o u r food s u p p ly , m u s t in c re a s in g ly c o p e w i t h t h e effects o f p o l l u tio n w h ic h is le a c h e d fro m t h e la n d o r d u m p e d d ir e c tly in to t h e sea. a c c id e n t a lly or o th e rw is e .
.A.long-channel c u r r e n t s , w h ic h u l t i m a t e l y c o n t ro l t h e s a lin ity , n u t r i e n t , a n d c o n t a m i n a n t c o n c e n t r a t i o n s w ith i n c o a s ta l c h a n n e ls v ia e x c h a n g e w ith t h e o p e n o c e a n , h a v e b e e n s t u d i e d t o a fa r g r e a t e r e x t e n t th a n v e rtic a l a n d t r a n s v e r s e flows. V e rtic a l velo cities b r i n g n u t r i e n t s u p w a r d i n t o t h e e u p h o t ic zo n e w h e r e b io lo g ic a l p r o d u c ti o n o c c u rs a n d le a d to e x c h a n g e a n d e n t r a i n m e n t b e tw e e n t h e u p p e r f r e s h e r s u rface la y er a n d low er o c e a n ic w a t e r . C r o s s - c h a n n e l flows t e n d t o slow t h e a l o n g - c h a n n e l e x c h a n g e by t r a n s f e r r i n g m o m e n t u m t o t h e sidew alls. T h e y also e n h a n c e diffusion w ith i n a n e s t u a r y by e x p o s i n g t r a c e r s t o cro s s -c h a n n e l differences in t h e a lo n g - c h a n n e l velocity.
1. In tro d u c tio n a n d M o tiva tio n
T r a n s v e r s e c u r r e n t s c a n b e significant in regions w h e r e fric tio n is i m p o r t a n t , s u c h as in t h e b o t t o m b o u n d a r y a n d a t t h e in te rfa c e b e t w e e n inflows a n d outflow s.
F ric tio n is a s s o c ia t e d w ith t u r b u l e n c e , t h e s m a l l- s c a le h ig h - fre q u e n c y c u r r e n t flu c tu a tio n s by w h ic h e n e r g y t h a t is c o n tin u o u s ly s u p p l i e d t o t h e o ce a n s by g r a v i t a tio n a l a t t r a c t i o n , s o la r ir r a d i a ti o n , a n d w in d s tr e s s is re m o v e d . T u r b u l e n c e e n h a n c e s diffusion by in c r e a s in g t h e e x p o s e d su rfa c e a r e a o f tr a c e r s o v e r w hich m o l e c u la r p r o cesses c a n a c t. C r o s s - c o r re la tio n s in t u r b u l e n t v elo c itie s , o r R ey n o ld s s tre s s e s , c a n d ir e c tly m o d i f y c u r r e n t s a t lo n g e r tim e s c a le s .
b e t t e r u n d e r s t a n d i n g of t u r b u l e n c e a n d o f d y n a m i c s in g en era l is i m p o r t a n t not o n ly for local e s t u a r i e s b u t also for m a n y d iffe re n t ty p e s o f global g e o p h y s ic a l flows o v e r a w id e r a n g e of le n g th a n d t i m e scales. .-\s a sem i-e n c lo s e d b a s in w h ic h is long, s t r a i g h t , a n d has re la tiv e ly s m o o th t o p o g r a p h y . .Juan d e F uca S t r a i t is a n ideal l a b o r a t o r y in w h ich t o e x a m in e s o m e o f t h e s e p h e n o m e n a . In t h e s u m m e r o f 1996. a n o b s e r v a t i o n a l p r o g r a m m e in t h e m i d d le s e c ti o n o f J u a n d e F u c a S t r a i t w as u n d e r t a k e n to s t u d y e x c h a n g e flow, p a r ti c u la r ly t h e fric tio n a n d d y n a m ic s a s s o c i a t e d w ith t h e i n t e r f a c e b e t w e e n t h e inflow a n d o u tflo w layers.
C h a p t e r 2 p r o v id e s t h e th e o re tic a l b a c k g r o u n d for t h e th e sis, a n d in c lu d e s s e c tions o n e s t u a r i e s , t u r b u l e n c e , a n d tides. V arious e s t u a r i n e classification s c h e m e s a n d n o n - d im e n s io n a l p a r a m e t e r s a r e in tr o d u c e d , as a r e a n a l y t i c a l , n u m e ric a l, a n d l a b o r a to r y m o d e ls w h ic h a t t e m p t to e x p la in s o m e o f t h e d y n a m ic s . T h e N a v ie r - S to k e s e q u a t io n s a r e u s e d t o rev ie w t h e c o n c e p ts o f R e y n o l d s s tre s s , n o n - d im e n s io n a l i n s t a b ility p a r a m e t e r s , a n d log-layer d y n a m ic s . T h e la s t s e c tio n c o n s id e rs t h e s t r u c t u r e of b a r o tr o p i c a n d i n t e r n a l tid e s in a ch a n n e l.
C h a p t e r 3 fo cu ses o n J u a n de F u c a S tr a i t specifically, d e t a i l i n g h is to r ic a l w o rk a n d h ig h l ig h ti n g s o m e of t h e scientific q u e s tio n s ra is e d . T h e 1996 o b s e r v a tio n a l p r o g r a m m e is m o t i v a t e d by d e m o n s t r a t i n g t h a t a n a r e a - a v e r a g e d a lo n g -c h a n n e l m o m e n t u m b a l a n c e of t h e u p p e r la y e r c u rr e n t re q u ire s s t r o n g in te rfa c ia l friction a n d im p lie s s tro n g t r a n s v e r s e c u r r e n t s a n d tu r b u le n c e . T h e h y d r o g r a p h i c a n d c u r r e n t m e t e r d a t a co llected t o e x a m i n e t h e d y n a m i c s of J u a n d e F u c a S t r a i t a r e d iscu ssed in C h a p t e r 4.
1. In tro d u ctio n a n d M o tiv a tio n 3
A s h o r t t r e a t m e n t of t h e e r r o r s a s s o c ia t e d w ith t h e m e a s u r e d v elo cities a n d R e y n o ld s s tre s s e s is also in c lu d e d .
S in ce th e v e rtic a l v e lo c ity is s ig n ific a n tly s m a lle r t h a n t h e h o r i z o n t a l c o m p o n e n ts . possible biases a n d e rro rs in t h e m e a s u r e d vertical v elo city m u s t b e e x a m i n e d befo re R eyn o ld s stre s s e s c a n b e c a l c u l a t e d . T h e difficulties in m e a s u r i n g t h e v e rtic a l v e lo c ity a re discussed in C h a p t e r 5. a n d a t t e m p t s are m a d e to v a l id a te t h e s e c u r r e n t s . T h is is d o n e p rim a r ily by u sin g t h e b a c k s c a t t e r intensity, a m e a s u r e o f t h e s c a t t e r ing cross-section u p o n w hich t h e .Acoustic D o p p le r C u r r e n t P ro filer relies t o m e a s u r e velocity. C o n t a m i n a t i o n o f t h e v e r tic a l v e lo c ity by h o riz o n ta l c o m p o n e n t s a n d th e p r o b le m of s p a tia l in h o m o g e n e it y o f t h e flow a r e also c o n s id e re d .
T h e v ertical s t r u c t u r e of t h e tid e s is a n a ly s e d in C h a p t e r 6. p a r t i c u l a r l y for t h e M 2 c o n s ti tu e n t , w h e re a t t e m p t s a r e m a d e t o q u a n tif y t h e s t r e n g t h o f t h e i n t e r nal a lo n g -c h a n n e l, c ro s s -c h a n n e l, a n d v e rtic a l c o m p o n e n ts . B o t t o m b o u n d a r y layer d y n a m i c s a r e also e x p lo re d , in c lu d in g log-layer a n d r o ta r y c u r r e n t a n a l y s e s .
In C h a p t e r 7. n o n - ti d a l flows a r e e x a m i n e d . T h e m e a n flow is c o m p a r e d to h y d r o g r a p h y an d to la b o r a t o r y r e s u lts of a tw o-layer r o t a t i n g flow. S i m p l e m o d e ls a n d c u r r e n t m e te r d a t a su g g e s t t h a t th e a lo n g -c h a n n e l c u r r e n t m a y b e s u b j e c t to b a ro c lin ie insta bility. T h e s p r in g - n e a p cy cle of t h e s h e a r i n s ta b ili ty a n d in t h e o b s erv e d R eynolds stre s s e s a r e e x a m i n e d a n d t h e results a re re la te d to c h a n g e s in th e m e a s u r e d m e a n flow.
2. B a c k g ro u n d
C h a p te r 2
B a ck g ro u n d
2.1
E s tu a r ie s
In m o s t e s t u a r i e s , f re s h w a te r fro m r iv e r r u n o f f m ix e s w ith w a t e r from t h e o c e a n . C a m e r o n a n d P r i t c h a r d (1963) s t a t e t h a t a n e s t u a r y is ' a sem i-e n c lo s e d c o a s ta l b o d y of w a t e r w h ic h h a s a free c o n n e c tio n w ith t h e o p e n s e a a n d w ith in w hich s e a w a t e r is m e a s u r a b l y d i l u t e d w i t h fresh w a te r d e r iv e d f r o m l a n d d ra in a g e ." N e v e r th e le s s , t h e r e exist “n e g a t iv e " e s t u a r i e s , in w hich e v a p o r a t i o n e x c e e d s p r e c i p ita tio n a n d ru n o ff, le ad in g t o s u r f a c e s a li n iti e s la rg e r t h a n th o s e o f t h e o c e a n . T h e d e n s e s u rfa c e w a t e r s w ith in t h e s e b a s in s s i n k a n d flow o u t in to t h e s e a a t d e p t h a n d a r e r e p la c e d w ith o c e a n ic r e t u r n flow a t t h e surface. T h i s th e s i s will, how ever, focus on " p o s i tiv e " e s tu a r ie s , w h e r e o u tf lo w of d il u te d s u rfa c e w a t e r is c o m p e n s a t e d by r e t u r n flow of d en se o c e a n ic w a t e r a t d e p t h .
2.1 .1
C la s s ific a tio n S c h e m e s
C a t e g o r is in g t h e w o rld 's e s tu a r ie s in to d i s t i n c t classes is a difficult, if n o t i m possible. ta s k g iv e n t h e i r w idely vary in g g e o m e t r i e s , b a t h y m e t r i e s . m i x in g r a t e s , a n d c irc u la tio n s . F u r t h e r m o r e , c o n d itio n s w ith i n a n in d i v i d u a l e s t u a r y c a n c h a n g e g r e a t l y w ith lo c a tio n (e.g. r e l a t i v e to t h e h e a d o r m o u t h ) , t i m e o f y e a r (s e a s o n a l c h a n g e s in f re s h w a te r i n p u t ) , a n d t i d a l p h a s e ( m i x in g le v els). N e v e rth e le s s , a n u m b e r o f d if fe r en t c la ss ific a tio n s c h e m e s e x is t, b ased on t o p o g r a p h y , s a lin ity d i s t r i b u t i o n , c i r c u l a t i o n ty p e , o r a c o m b i n a t i o n o f th e se .
a) The T o p o g ra p hy o f E s t u a r i e s
P r i t c h a r d (1952) p ro p o s e s a class ific a tio n s c h e m e based on t o p o g r a p h y w h ic h has t h r e e m a i n c a te g o r ie s : d ro w n e d riv e r valley s, fjo rd s , a n d b a r - b u il t e s t u a r i e s . \
2. B a ckg ro u n d
f o u r t h g r o u p c o m p ri s e s e s t u a r i e s w hich d o n o t fall i n t o t h e g e n e ra l c a te g o rie s , a n d in c lu d e s e s tu a r i e s f o r m e d by la n d slid e s o r m o v e m e n t a lo n g t e c to n ic fa u lt lines.
T h e m o s t c o m m o n t y p e is t h e d r o w n e d r i v e r valley, o th e rw is e k n o w n as a c o a s ta l p la in e s tu a r y . T h e s e w e re fo rm e d w h e n s e a levels rose d u e to t h e m e l t i n g o f g la c ie rs a n d t h u s g e n e r a l l y h a v e d e p t h s less t h a n 30 m . I n itia lly c a rv e d by riv ers, t h e e s t u a r i e s a re o f te n s in u o u s a n d t h e cro s s -se c tio n s t r i a n g u l a r . W i d t h to d e p t h r a tio s a r e t y p i c a ll y la rg e , a n d b o t h t h e w id th a n d d e p t h in c re a s e t o w a r d th e m o u t h . R iv e r flow ( p e r tid a l p e r io d ) is t y p i c a l l y s m a ll c o m p a r e d t o t h e ti d a l p ris m ( t h e differe n ce in v o lu m e of a n e s t u a r y b e t w e e n high a n d low ti d e s ) .
F jo rd s , c r e a t e d w h e n g la c ie rs s u b s t a n t i a l l y d e e p e n e d e x i s t in g riv er valleys, of te n h a v e shallo w sills a t t h e m o u t h w h ere g la c ie rs d e p o s i t e d m o r a i n e . T h e s e e s t u a r i e s a r e u p to SCO m d e e p , a r e g e n e r a lly s t r a i g h t a n d lo n g ( u p to 100 k m ) , a n d h a v e a s m a l l w id th to d e p t h r a t i o . R iv e r i n p u t p e r ti d a l c y c l e is u s u a lly large c o m p a r e d to t h e ti d a l p ris m sin c e t i d a l ra n g e s a r e o fte n r e s t r i c t e d , b u t a r e v ery s m a ll c o m p a r e d to t h e w a te r v o lu m e w i t h i n t h e fjord.
In a b a r - b u i l t e s t u a r y , s e d i m e n t is d e p o s i t e d a t t h e m o u t h , f o rm in g a b a r a cro ss t h e e s tu a r y . S e d i m e n t a t i o n r a te s a r e large a n d t h e e s t u a r i e s a r e g e n e ra lly o n ly a few m e t r e s d eep . O c c a s io n a lly , b a rs a r e also fo r m e d w i t h i n t h e e s t u a r y w hich m a y c r e a t e la g o o n s, s t r e t c h e s o f s a l t w a t e r c u t off fro m t h e m a i n e s tu a r y . R iv e r i n p u t is large, a n d v aries c o n s id e r a b ly t h r o u g h o u t t h e y e a r. D u r i n g p e r io d s of u n u s u a lly la rg e riv er flow, t h e b a r m a y b e t e m p o r a r i l y m o v e d o r d e s t r o y e d .
b) T h e S a l i n i t y S t r u c t u r e W i t h i n E s t u a r i e s
T h e h y d r o g r a p h y o f e s t u a r i e s ranges fro m v e r t i c a l l y h o m o g e n e o u s t h r o u g h c o n ti n u o u s l y s tr a t if ie d to e s s e n ti a lly tw o-layer. . \ s m e n t i o n e d , t h e s t r a t if ic a tio n w ith in an in d i v id u a l e s t u a r y c a n v ary o v er s e a s o n a l a n d t i d a l cycles, a n d e v e n a lo n g its le n g th . P r i t c h a r d (I95Ô) a n d C a m e r o n a n d P r i t c h a r d (1963) d is tin g u is h four m a in t y p e s o f e s tu a r i e s b a s e d o n t h e o b s e rv e d s a li n ity s t r u c t u r e : h o m o g e n e o u s ( w e ll-m ix e d ), p a r t i a l l y - m i x e d ( p a r t i a l l y - s t r a t i f i e d ) , fjords, a n d s a l t w ed g e , w h e re t h e l a t t e r tw o a r e s o m e t i m e s c o llectiv ely t e r m e d h ig h ly - s tr a tifie d .
2. B a c k g ro u n d ____________________________________________________________________6
F or e s tu a r ie s o f s m a ll d e p t h , t u r b u l e n c e a s s o c ia te d w ith b o t t o m fric tio n a c t i n g o n t h e tid e s m a y b e sufficient t o v e r t i c a l l y m i x th e e n t i r e w a t e r c o l u m n . T h e s e v e r t i c a lly h o m o g e n e o u s e s tu a r i e s c a n b e e i t h e r la te r a lly h o m o g e n e o u s , w h ic h o c c u rs w h en t h e w i d t h is s m a ll e n o u g h t h a t fric tio n l a t e r a l l y m ixes t h e e s tu a r y , o r in h o m o g e n e o u s . in w h ic h case t h e flow m a y b e h o r iz o n t a ll y s e p a r a t e d . In t h e f o r m e r case, flow is o u t w a r d a t all d e p t h s , a n d t h e o u t w a r d a d v e c t i o n of salt by t h e m e a n flow is b a l a n c e d by t h e in w a r d t u r b u l e n t diffusion o f s a lt d u e to eddies. In la t e r a l l y in h o m o g e n e o u s e s t u a r i e s , h o r iz o n ta l c i rc u la t io n t e n d s t o t r a n s p o r t salt u p - e s t u a r y a lo n g t h e left side of t h e c h a n n e l (in t h e N o r t h e r n h e m is p h e r e ) d u e to r o ta t io n .
T h e p a r ti a ll y - m ix e d e s t u a r y is c h a r a c te r i s e d by l i m it e d v e r ti c a l s tr a t if ic a tio n : t u r b u l e n t m ix in g , w hile in te n s e , is n o t sufficient to c o m p le te ly h o m o g e n is e t h e w a t e r c o l u m n . S a lt a n d w a te r a r e e n t r a i n e d in to t h e u p p e r layer, in c re a s in g b o t h s u rfa c e s a l i n i t y a n d w a te r t r a n s p o r t . T h e r e s u l t i n g d e n s it y s t r u c t u r e d riv e s a r e t u r n flow in t h e low er layer to b a la n c e t h e s a lt loss in t h e u p p e r layer.
T h e h ig h ly -s tr a tifie d e s t u a r y is s u b d i v i d e d in to tw o ty p e s : t h e s a lt - w e d g e a n d t h e fjo rd . In s a lt-w e d g e flow, r iv e r flows a r e c o n s id e ra b ly la rg e r t h a n ti d a l flows ( D y e r 197.3). a n d t h e s ea w a te r i n t r u d e s u p s t r e a m as a w edge below t h e fresh layer. T h e p o s iti o n o f th is sa lt w edge d e p e n d s on t h e s t r e n g t h of t h e riv e r flow, a n d o s c illates h o r iz o n t a ll y w ith t h e tid e . S a lty w a t e r will b e e n t r a in e d in to t h e u p p e r layer, i n c re a s ing t h e d is c h a rg e r a t e as t h e m o u t h is r e a c h e d a n d re q u irin g a l a n d w a r d flow in t h e s a lt w e d g e to b a la n c e th e s a lt loss. T h e s a li n ity inside t h e s a lt w ed g e d o e s not c h a n g e in t h e a lo n g -c h a n n e l d ir e c tio n sin ce t h e r e is no e n t r a i n m e n t in to t h e low er layer. T h e fjord e s t u a r y is s im i la r to t h e s a lt - w e d g e flow, ex c e p t t h a t t h e low er la y er is o fte n m u c h d e e p e r , a n d a sill u s u a lly e x i s t s a t t h e m o u t h of t h e e s tu a r y . O c c a s io n a lly , t h e sill is sh a llo w e n o u g h to c u t off t h e r e t u r n flow, a n d t h e lower la y e r in t h e e s t u a r y s t a g n a t e s . R en ew al s o m e t im e s o c c u r s o n ly an n u a lly , w h e n t h e riv e r flow is a t its p e a k . .At th e s e tim e s , e n t r a i n m e n t is large, a n d t h e d e n s it y d iffe re n c e b e tw e e n t h e low er la yers o f t h e e s t u a r y a n d t h e o c e a n is g r e a t e s t.
2. B a ckg ro u n d
c) E s t u a r i n e P a r a m e t e r s
M a n y a t t e m p t s h a v e b e e n m a d e to d ev e lo p a c la s s ific a tio n s c h e m e b a s e d o n ly on e x t e r n a l p a r a m e t e r s s u c h as th e tid a l c u r r e n t s t r e n g t h , t h e f re s h w a te r i n p u t , a n d d i m e n s i o n s o f t h e e s t u a r y . T h e goal, essentially, is to i d e n t i f y t h e physical p r o c e s s e s le a d in g to th e o b s e r v e d s a l i n i t y s t r u c t u r e .
P e r h a p s t h e s i m p l e s t s c h e m e is t h a t p ro p o s e d b y S i m m o n s (I9Ô5) a n d b a s e d on t h e r a t i o of th e r iv e r i n p u t p e r ti d a l cycle to t h e t i d a l p r is m . T h e fo r m e r is Q j T . w ith Q / t h e f re s h w a te r v o l u m e t r i c i n p u t r a te a n d T t h e t i d a l p e r io d . T h e ti d a l p r i s m for a n e s t u a r y of w i d t h I F a n d d e p t h H a t t h e m o u t h is = W H u j - T . w h e r e Uj- is t h e m e a n tid a l flow. F o r S i = Q j / { W H u - p ) of o r d e r u n ity , a r r e s te d ( s a lt - w e d g e ) flow is u s u a lly fo u n d , a s i t u a t i o n in w hich a t h i n l a y e r o f fresh w a te r o v e r la y s a s t a t i o n a r y layer of o c e a n ic w a te r , w ith little to no e n t r a i n m e n t o r m ix in g b e t w e e n t h e tw o layers. R a t io s o f 10 ^ a n d I0~^ c o rre s p o n d t o p a r ti a ll y - s t r a ti fi e d a n d well-
mi.xed e s tu a r i e s , re s p e c tiv e ly . H ow ever, in o m i t t i n g g r a v i t y a n d t h e d e n s ity d iffe re n c e b e tw e e n fresh a n d o c e a n ic w a te r , t h e s ta b ilis in g in flu e n c e o f b u o y a n c y h as n o t b e e n p r o p e r ly co n sid e re d .
C iv il e n g in e e rs h a v e long u sed th e " e s tu a ry n u m b e r ' . E^ = P p F ~ / { T Q for a ti d a l c h a n n e l, w h e re E^ = U p j y / g H is th e e x t e r n a l F r o u d c n u m b e r , " to c o r r e l a t e m o d e l e x p e r i m e n t s a n d field d a t a " ( T u r n e r 1973). W i t h t h e a b o v e defin itio n s for t h e S i m m o n s p a r a m e t e r a n d t h e t i d a l p ris m . E^ = F ^ / S i = I F n ^ / ( < / Q y ). V alues o f 0.03 to 0.3 d e fin e th e t r a n s i t i o n b e tw e e n s tra tifie d a n d w e ll-m ix e d e s tu a rie s , w ith la rg e r
E^ i m p l y i n g g r e a t e r v e r ti c a l h o m o g e n e ity ( T u r n e r 1973).
re la te d p a r a m e t e r , b a s e d on th e " p ip e R i c h a r d s o n n u m b e r ‘d ( E llis o n a n d T u r n e r 1960) is th e " 'e stu a rin e R ic h a rd s o n n u m b e r '' R i ^ . T h i s is th e r a t i o o f t h e b u o y a n c y i n p u t ( d u e to a riv er) p e r u n it w id th { A p g Q j / W . w h e re A p is t h e d e n s i t y diffe re n c e b e tw e e n fresh a n d o c e a n w a te r) to t h e m i x i n g p o w e r of t h e tid e s (u'^). U sin g t h e m e a n d e n s it y p^ t o n o n -d im e n s io n a lis e .
2. B a c k g ro u n d S
E s t u a r i n e o b s e r v a ti o n s s u g g e s t t h a t “t r a n s i t i o n fro m a w e ll - m ix e d t o a s tr o n g ly s t r a t ified e s t u a r y o c c u rs in t h e r a n g e 0.08 < Ri^ < 0.8" ( F i s c h e r et al. 1979).
T h e e s t u a r y n u m b e r a n d e s t u a r i n e R ic h a r d s o n n u m b e r a r e closely r e la te d :
Ri^ = A p / . W i t h t h e d e n s i t y r a t i o ù^p/p^ % 0.02 in m o s t e s tu a r i e s ( T u r n e r
1973). t h e t r a n s i t i o n reg io n s for t h e tw o p a r a m e t e r s a r e . n o t s u rp ris in g ly , a l m o s t id e n tic a l. O n t h e o t h e r h a n d , t h e r a t i o b etw ee n t h e e s t u a r i n e R ic h a rd s o n n u m b e r a n d t h e p a r a m e t e r p r o p o s e d by S i m m o n s (I9ÔÔ) is R i ^ / S i = g ' R / = A F r ~ ^ .
w h e r e F r is t h e i n t e r n a l F r o u d e n u m b e r based on t h e t i d a l s p e e d a n d th e i n t e r n a l w ave s p e e d for a tw o -la y e r flow w ith f re s h w a te r o f d e p t h H j ' 2 a b o v e o c ea n ic w a t e r o f e q u a l d e p t h .
In h is a n a ly s is o f e s t u a r i n e a d j u s t m e n t to c h a n g e s in r i v e r flow a n d tid a l m i x in g M a c C r e a d y (1999) c o n s id e rs t h e t h r e e f u n d a m e n t a l v e l o c ity scale s in a n e s tu a r y : t h e riv e r v e lo c ity TT = Q j / { W H ) . t h e ro o t m e a n s q u a r e ( r m s ) ti d a l v elocity u ^ . a n d t h e m a x i m u m i n t e r n a l w ave s p e e d = y j g ' H / A. ]\xsX. as H a n s e n a n d R a t t r a y (1966) h a d d o n e e a rlie r. F ro m t h e s e t h r e e velocities, tw o n o n - d i m e n s i o n a l p a r a m e t e r s c a n b e d e fin e d . W h e r e a s H a n s e n a n d R a t t r a y (1966) chose t h e " d e n s i m e t r i c F ro u d e n u m b e r "
= Ti(C^. a n d t h e flow r a t i o P = Tl/uj.. M a c C r e a d y f o u n d it m o r e co n v e n ie n t to
u se a n d F = u - f / C ^ . as t h e s e y ie ld in d e p e n d e n t m e a s u r e s of t h e river flow a n d ti d a l a m p l i t u d e , re s p e c tiv e ly . O f c o u rs e , th e s e choices a r e a r b i t r a r y a n d th e r e s u lt in g p a r a m e t e r s a r e cle a rly r e la te d : F = F ^ / P .
It is also not s u r p r i s i n g t h a t th e s e p a r a m e t e r p a irs a r e r e l a t e d to th e e a r l ie r set
( R i ^ . F r ' ) . w ith = R i ^ F r ^ / A a n d F = F r . T h is b egs t h e q u e s t i o n o f th e n u m b e r of s e p a r a t e n o n - d im e n s io n a l p a r a m e t e r s n e e d e d to a d e q u a t e l y d e s c r ib e th e r a n g e of e s t u a r y ty p e s . T h e w o rk of H a n s e n a n d R a t t r a y ( 1966) a n d M a c C r e a d y ( 1999) su g g e s t t h a t t h e r e a r e tw o. unless t h e y a p p e a r in o n ly o n e c o m b i n a t i o n in th e g o v e rn in g e q u a t i o n s . In th is case, w h ic h m a y o c c u r only for s im p lifie d fo rm s o f t h e e q u a t i o n s , o n ly o n e p a r a m e t e r m a y b e n e e d e d .
2. B a c k g ro u n d
Ig n o r in g t h e effects of w in d s tre s s , t h e c h a r a c t e r i s t i c e x t e r n a l p a r a m e t e r s for a n e s t u a r y a r e Q ^ . W . H , L. u ^ . g . a n d A p . w h e r e L is t h e le n g th o f t h e e s t u a r y . N o t ing t h a t it is th e f r e s h w a te r i n p u t p e r u n i t w id th w h ich defin e s t h e local b u o y a n c y a n o m a l y , t h a t A/5 n e e d s to b e d iv i d e d by a b a c k g ro u n d d e n s i t y as it is t h e o n ly p a r a m e t e r in v o lv in g m a s s , a n d t h a t t h e le n g th of t h e c h a n n e l s h o u ld n o t affect t h e t i d a l c u r r e n t s or f r e s h w a te r i n p u t , n o n -d im e n s io n a l p a r a m e t e r s c la ss ify in g e s t u a r i e s s h o u ld b e b a s e d on Q ^ j W . H . u-j-.g. a n d A p / p ^ . F u r th e r m o r e , g r a v i t y a c t s u p o n t h e d e n s i t y difference, so t h e l a t t e r tw o t e r m s s h o u ld be c o m b i n e d as g' = g 2 ^ p j p ^ . t h e e s t u a r y n u m b e r n o t w i t h s t a n d i n g . T h e fo u r re m a in in g p a r a m e t e r s inv o lv e o n ly tw o d i m e n s i o n s (le n g th a n d t i m e ) , i m p ly in g tw o n o n - d im e n s io n a l n u m b e r s . N e v e rth e le s s , ■'the id e a o f being a b le to p r e d i c t e s t u a r i n e s t r u c t u r e from a few s im p l e e x t e r n a l p a r a m e t e r s r e m a in s e lu siv e" ( M a c C r e a d y 1999).
R a t h e r t h a n u s in g e x t e r n a l p a r a m e t e r s , H an sen a n d R a t t r a y (1966) p ro p o s e d a c la s s ific a tio n s c h e m e w h ic h u tilis e s p a r a m e t e r s involving t h e m e a s u r e d h y d r o g r a p h y a n d c u r r e n t s t r u c t u r e w ith i n a n e s tu a r y . T w o d im e n s io n le ss p a r a m e t e r s , b a s e d o n ly on s a l i n i t y a n d velocity, classify e s t u a r i e s a lo n g a c o n t in u u m r a t h e r t h a n in to d i s t i n c t classes. T h e tw o -d im e n s io n a l p a r a m e t e r s p a c e is d iv id ed in to c h a r a c t e r i s t i c regions, s im i la r to t h o s e p ro p o s e d by P r i t c h a r d ( 1955). based o n t h e r a ti o o f t h e t i d a l diffusion s a lt flux to t h e t o t a l u p - e s t u a r y s a lt flux. T h e s tr a t if ic a tio n p a r a m e t e r is S S / S ^ . w h e re 6 S is t h e s u rfa c e to b o t t o m d iffere n ce in s a lin ity a n d is t h e m e a n cross- s e c ti o n a l salinity. T h e c i r c u la t io n p a r a m e t e r u ^ / u ^ . w ith t h e n e t s u r f a c e c u r r e n t a v e r a g e d o v e r a tid a l cycle, a n d t h e m e a n cro s s -se c tio n a l v e lo c ity (i.e. IT. t h e riv e r i n p u t d iv i d e d by t h e c ro s s - s e c tio n a l a r e a ) , is a m e a s u re o f t h e a m o u n t o f e n t r a i n m e n t in t o t h e u p p e r layer. For p a r t i a l l y - m i x e d e s t u a r i e s of r e c t a n g u l a r c ro s s - s e c tio n , t h e c i r c u l a t i o n a n d s tr a t if ic a tio n p a r a m e t e r s c a n b e r e la te d to t h e e x t e r n a l p a r a m e t e r s
