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Internal oscillations of tidal character in certain B.C. inlets Davis, Patrick Austin 1960

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( i ) INTERNAL OSCILLATIONS OP TIDAL CHARACTER IN CERTAIN B.C. INLETS by PATRICK AUSTIN DAVIS B.A., University of B r i t i s h Polumbia, 1960 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF M.A. i n the Department of PHYSICS ¥ 0 accept th i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1960 In presenting t h i s thesis i n p a r t i a l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available f o r reference and study. I further agree that permission for extensive copying of t h i s thesis f o r scholarly purposes may be granted by the Head of my Department or by his representatives. It i s understood that copying or publication of t h i s thesis f o r f i n a n c i a l gain s h a l l not be allowed without my written permission. The University of B r i t i s h Columbia, Vancouver Canada. ( i i ) Abstract A study of series of p r o f i l e s of temperature and current at one station i n Knight Inlet, and of temperature at two simultaneous stations i n Bute, have been made i n conjunction with seasonal p r o f i l e s of temperature, s a l i n i t y , and the associated density. Gn th i s basis certain hypo-t h e t i c a l models have been f i t t e d to the dynamics within the i n l e t s . A c a l c u l a t i o n of the possible modes of progressive i n t e r n a l waves for the two i n l e t s has led to< agreement only i n Bute I n l e t . A q u a l i t a t i v e explanation of the condition i n Knight i n terms of the dynamic c i r c u l a t i o n over the s i l l has led to suggestions as to the mechanism, of exchange i n the deeper waters of the i n l e t , and speculation as to the p o s s i b i l i t y of prediction of states within an i n l e t i n terms of i t s topography, runoff cycle, and communication with the coastal waters. * i i i ) Table of Contests Page<t> Introduction 1 Chapter I 1) D i s t r i b u t i o n of Oceanographic Properties i n 8*C. i n l e t s 3 2) Tides 5 3) The ^ Relation of Tides to the Distribution, of Oceanographic Properties 10 Chapter II 1) The Oceanogrgphic Structure of Bute and Knight Inlets 12 2) S e r i a l D i s t r i b u t i o n of Properties 19 3) Harmonic Analysis 23 4) Theorem of Continuity 28 Chapter III The Equations of Motion 1) Internal Waves 32 ( i ) The solution for a continuous density d i s t r i b u t i o n 32 ( i i ) Approximate forms 41 2) Other Forms of O s c i l l a t o r y Motion 44 ( i ) The solution, for f i n i t e length and variable depth 44 ( i i ) A f i n i t e b a r r i e r at the month of an i n l e t 47 Conclusions 50 Tables 1) Comparison of amplitude calculated from continuity theorem with deflections of isotherms. 2) Wave v e l o c i t i e s and dppths of maximum amplitude. 3) Computed values of sr corresponding to depths for observed amplitudes. 4) Regression c o e f f i c i e n t s (iv) 5) Comparison of computed and observed values of amplitude Kn 5a. 6) Comparison of computed and observed values of current Kn 5a. 7) Comparison of computed and observed values of amplitude Bu 4* 8) Comparison of computed and observed values of amplitude Bu 6. 9) Values for constants of the approximate form for the s t a b i l i t y * 10) Depthi of maximum amplitude and r e l a t i o n of average phase to that of tide f o r anchor stations i n Bute I n l e t for which thereRare harmonic analyses* Figures 1) Location of stations Knight and Bute I n l e t s . 2) P r o f i l e s of temperature Knight and Bute I n l e t s . 3) Series of temperature p r o f i l e s Bute Inlet 1957 to 1958. 4) P r o f i l e s of s a l i n i t y Knight and Bute Inletss 5) P r o f i l e s of density Knight and Bute I n l e t s . 6) Series of s t a b i l i t y plots through Knight I n l e t 1956. 7) T-S diagrams through Knight Inlet 1956. 8) Time Series density at Knight 8-1/2. 9) Time series of plots temperature, temperature gradient, difference from mean temperature, and currents at Knight 5 (July 4 to 6, 1956). 10) Average temperature for time series plots Knight 5, 11) S e r i a l data Knight 5. 12) Brunt frequencies Knight 5, Bute 4. 13) Harmonic analysis of ti d e s at Al e r t Bay (July 4 to 6, 1956). 14) V e r t i c a l o s c i l l a t i o n s and temperature fluctuations Kn 5a. 15) V e r t i c a l o s c i l l a t i o n s Knight 5b ( i ) and ( i i ) . 16) Horizontal currents at Knight 5. (T) Figures (cootAd) 17) V e r t i c a l o s c i l l a t i o n s at Bute 4 and 6 18) O s c i l l a t i o n s exhibited with phase according to period 19) Density destributions and logarithmic p l o t of s t a b i -l i t y Knight 5a 20) Mean densities at Bute 4 and 6 used for integration 21) Logarithmic plot S t a b i l i t y derived from mean density Bute 4 and 6 22) Integration Knight 5a 23) Integration Bute 4 24) Results of regression analysis 25) Synthesis of v e r t i c a l o s c i l l a t i o n s Knight 5 26) Synthesis of horizontal currents Knight 5 27) Synthesis of v e r t i c a l o s c i l l a t i o n s Bute 4 and 6 28) Diagram i l l u s t r a t i n g flow over ba r r i e r i n two-layer system 29) Diagram i l l u s t r a t i n g flow over a barrier of a. continuously s t r a t i f i e d system. (vi) Acknowledgements The research was carried out under the super-v i s i o n of Dr. G.L. Piekard, with the f i n a n c i a l assistance of the Defence Research Board of Canada. F a c i l i t i e s of the ALWAC computer were made available through the mathe-matics department of the University of B r i t i s h Columbia. The author would also l i k e to acknowledge the assistance he has gained through consultation with members of the Inst i t u t e of Oceanography and other departments, i n p a r t i -cular the k i M l y interest the members of the s t a f f of the computing center have always taken i n his work* Introduction' The oceanography of coastal waters, te a much greater extent than that of the open ocean, i s influenced by the astronomical ti d e s , both i n the dynamics of the water body as a whole and i n the r e l a t i v e d i s t r i b u t i o n and mixing of the properties at depths below the surface. This i s p a r t i c u l a r l y true of the B*C* coast, broken as i t i s by many deep i n l e t s of great extent into which the tide must penetrate during every t i d a l cycle* The d i s t r i b u t i o n of properties and the dynamics of the i n l e t s are calculated from surveys which involve ten or twelve hydrographic casts with readings at some dozen depths taken often over a period of a day or mere. I t i s d i f f i c u l t to make a closer g r i d without increasing the p o s s i b i l i t y of change of properties during the time of run, nor to make a swifter survey without losing d e f i n i t i o n through too few stations. Thus arises the need for some sort of model or hypothesis based on some elementary hydrodynamic model to relate the readings both i n time and space* The various models proposed so far have been based on an assumption of steady state and the t i d a l and other r e l a t i v e l y short term periodic motions have either been ignored or averaged out* The d i s t r i b u t i o n of properties shows a change of the phase with depth of the periodic movement of the waters with the ti d e , and a d e f i n i t e s h i f t of phase between stations which i s incompatible with the very short phase lag of the surface t i d e s , indicating some form of internal motion asso-e l a t e d with-, the s t r a t i f i c a t i o n of the deeper water*- S e r i e s of readings of temperature at s e v e r a l anchor s t a t i o n s i n d i c a t e an i n t e r n a l motion 'with .a dbf i & i t e Hiaxiaim of amplitude at same depth below the surface. '<sTsr t h i s t h e s i s calculations' are made'- fsrosi a model o f an' int e r n a l ^aya i a ' a s t r a t i f i e d fluid'making' use ef a method devil sad by,, • Sjaldstad' {1938) and adapted f o r use on a d i g i t a l computer *•'' .'She r e s u l t s of'such an'integration, have boon com-pared at/Several s t a t i o n s with t h e ' d e t a i l e d harmonic a n a l y s i s of v e r t i c a l O s c i l l a t i o n s * At a s t a t i o n some di s t a n c e from the s i l l (a bottom elevation, o c c u r r i n g close to the south of most inlets)''tK©'"calculations f i t " q u i t e ' w e l l * I n the v i c i n i t y of the'••silly oav the other hand* there i s l i t t l e correspondence, A f u r t h e r hypothesis as to- the mcde of f l o w of a s t r a t i f i e d f l u i d over a f i n i t e b a r r i e r s based on model experiments per-* formed by Long (19S4 2 1955} i s suggested as a p o s s i b l e esspla-nation, of 1-the d i f f e r e n t character of the motion between the stations"''la"the- two i n l e t s studied*, In t h i s system, the s i l l height''and'-froude number' ar© c r i t i c a l i a determining tsaethes? the motion-'is s t r i c t l y p e r i o d i c or develops' i n t o a h y d r a u l i c jump f o r a two f l u i d s y s t e c . This goes over t o a system of eddies' f o r "the- case of continuously s t r a t i f i e d flow in. the lee "of such a b a r r i e r * '•'•'Data from c r u i s e s and anchor s t a t i o n s made i n But© and Knight I n l e t s are f i r s t examined q u a l i t a t i v e l y to e s t a b l i s h tho nature of the t i d a l motion. S e r i a l data are then analysed q u a n t i t a t i v e l y through harmonic a n a l y s i s and presented as bar diagrams. C a l c u l a t i o n s frosa the mean, d e n s i t y d i s t r i b u t i o n s a t oao s t a t i o n from each, i n l e t are made to f i n d t he p o s s i b l e modes, o f i n t e r n a l waves.. She unknown parameters are evaluated • from' the o b s e r v e d / o s c i l l a t i o n s and the .synthesis'checked against 'the '<^b&.0yr^-.r^i\iea^ . The p o s s i b l e e f f e c t o f non~ .. uniform depth and hounded end on, thesform o f the motion, axe. examined, and the ,^ two major obstacles'-"'to the a p p l i c a t i o n : ' o f liong^js. .theory for. £ low 'of a s t r a t i f i e d f l u i d over a b a r r i e r are e x p l a i n e d . , An attempt i s made ;i» the c o n c l u s i o n s t o i n d i -cate what f u r t h e r work 'could c o n t r i b u t e t o a ©ore r e a l i s t i c h y p o t h e t i c a l model .for. the i n l e t s , . Chapter I. , . ri: 1). -The • d i s t r i b u t i o n of. Oceaoograph'ie P r o p e r t i e s i n B.C . I n l e t s . • yi:Mk$ d i s t r i b u t i o n o f oceanographic p r o p e r t i e s i n the • •, i n l e t s -governed by two $aajor processes g. the t i d e s and the ' runoffv 'i';dd#^:sed o f th© discharge" 'from- the major r i v e r s and the marginal.';stre'asas.* She c l a s s i f i c a t i o n o f i n l e t s i s p r i m a r i l y by f a c t o r s -which i n f l u e n c e e i t h e r one o f those proeessesss '(Pickard 9-1955) a ) D i v i s i o n - o f i s l e t s fey r u n o f f * I n l a r g e r u n o f f i n l e t s the s t r a t i f i c a t i o n i s most pronounced toward the head of the i n l e t ( a n i n c r e a s e o f 2S p a r t s per thousand i n the f i r s t .haioeline between 5 and 12 meters and airs, i n c r e a s e to 90$ a t 20 meters, from the almost fresh, water a t the s u r f a c e ) . I n the low r u n o f f i n l e t s the s a l i n i t y nowhere decreases a s " a b r u p t l y as th© water© of the other group ( r a n g i n g fro© 7 to-28 p a r t s per thousand i n the s u r f a c e water near th© head).' T h i s marked v a r i a t i o n o f s a l i n i t y in. the s u r f a c e l a y e r s of the i n l e t indicates that at least i n t h i s layer the density w i l l be primarily defined by the s a l i n i t y . b) D i v i s i o n of i n l e t s by location)* Those i n l e t s In connection with the northern coastal waters show i n t h e i r deeper waters a higher s a l i n i t y and lower temperature (temperature of 6*5° C. and a s a l i n i t y of 32*5 parts per thousand) as compared to that i n the i n l e t s south of Knight (which show temperatures of 7.5° C. and s a l i n i t i e s of 30.5 parts per thousand), while Knight remains intermediate between the two. c) D i v i s i o n of i n l e t s by s i l l depth. There are only a few i n l e t s on the B.C. coast which-, l i k e many of the Norwegian ones, have such high s i l l s that the s i l l s prevent the intrusion of heavier water to replace the deeper water of the i n l e t , which on t h i s account becomes stag-nant. However the majority of i n l e t s seem te have free exchange of t h e i r deeper water with that outside. d) Seasonal variations i n the properties* Inlets also divide naturally into those i n l e t s whose runoff i s dominated by d i r e c t discharge from r i v e r s of p r e c i -p i t a t i o n f a l l i n g as r a i n i n their watershed with a maximum im spring and f a l l , and others with runoff composed of the summer melting of p r e c i p i t a t i o n which, f a l l s as snow i n th e i r watershed, with some i n l e t s combining some cha r a c t e r i s t i c s of both. In i n l e t s where the mixing during the spring i s not too great, the e f f e c t of a c c l d winter can be seen u n t i l late into summer as a cold wedge of water, several degrees colder i n . t e m p e r a t u r e t h a n any surrounding w t o r , A • similar d i s t r i -featiera .of' ossygesi has also- fee@B. r«$ o r t o d and has bean .^attributed •' i© i n t r u s i o n o f low oatygsa. weter f r o t a r e g i o n s of high, r e d u c t i o n , ^ . - f a i r l y r e g u l a r esehaago. of a i r l - d a p t h i * a t o r -.-aoacsa-' passies the/.::jtas3iisa i s runoff'* She 'exchange e f deeper water i s ' h o w v e r > .'sioit' irregular:-and &ppooxs t o t a k e .plaoe. by in,t^usioas of denser water*. . ; •©•). S h o r t ' j>03F4oa i r r e g u l a r motion- of. «&tor.-* • ,. •" '.•There i s l i t i l o - i n f o r m a t i o n - abomi such p e r t u r b a t i o n s •to:.-;tho r e g u l a r system as a heavy r a i n f a l l o r very strong- 'Sisd w h i c h may s t r i p away. tho. s u r f a c e water. of an i s l e t ( F i e l d s t a d , -19§2),or t h e -response of the deeper,waters o f an i s l e t - t o a -turbidity' c u r r a n t o r t o tha t i d e * These a r e f a e t o r s : rasioh s a y g i v e a quit© erroneous i d e a o f th© normal s i t u a t i o n s during, a o-iie»=>day";"c: m^i»e' t o suofe' a'region* 2 ) - g j i & a o - , : ; "."•*•"• Tr'All" grocesaea that-'- ia&a jfl&ca .at the boundaries o f the ocean 'arav i n f l u e n c e d t o a g r e a t e r o r l e s s o r e x t e n d by o c e a n i c t i d e s * fe lakes and s e a s 9 however s. the r e s p o n s e i s small duo t o t h e l i a i i t a t i o i j t t h e boundaries impose ©n wave l e n g t h * "Is-iBost eases I t a s o s d e s o f the b a s i c twlv© s a d twenty—four' hour o s c i l l a t i o n s s t h e dominant components o f th© oe0as'ic""tide s w i l l occur*' Lofeos' ejad. seas respond s o r e . r e a d i l y to i m p u l s e s o f wind o r o t h e r boundary f o r c e s , as. t h e n a t u r a l p e r i o d " o f 'a water body, which i s det e r m i n e d by t h e rove l e n g t h , ( a f u n c t i o n o f t h e l e n g t h o f l a k e ) and the wave v e l o c i t y "(a f u n c t i o n ' o f the depth.) i s u s u a l l y crack s m e l l e r than the period '&$ the • tide-generating'''forces*'. ,.; > vi,-; •« i ' • S©i©h$s;^ »a>s...i&eae' o s c i l l a t i o n s determined, by.,the -.dimensions of t^;;^t«*'^odgr- i&r'©-. ^e&j^ffj,, .Qtiefu? as ajrospoaoo to. wind? wave,,- or. curfce&t ia-^bays ;and,,gulf s., co.naected, to tho opea oc-eaa.0 ^T/h&s© soicbessGorr.espondv.plosgly'.i^L period, to tbat-ybtdh, 'a bay would haver-it :i^'";^%f;;cop3^1©t©ly(.>i^|og«d<... . • Th©refor@v-it-'lsv&© b© expected thai' the resp0a.se the/'.'-tido^ . . generating forces w i l l be '.'small*;-,1|&:/.#i©v'offiior. ^oasd;-,&h©, BO- . • • c a l l e d c o ^ o s c i i l a t i o a to the ©.esaal^-.t'ide. i s by no^  M^ans.: small* Th© response.- say be, .•increased i f :tSe; length.of ,th©. gjil^:ris- some impropriate- J f r a c t i o n Of the-;^ v4,-.. length*,, th,©. sa^^./cr/it^ria as .-for 'e closed" body of watorf .nmsttfty that the, wave ro.flec;tod • frosi' a tiound&ry ..roinf ore© the .n^^i&coro^ • -';:-" •'"• .^'••'fhel By6,*:',:i-nle/fe)B.i i n .spite of, their_ .considorable loag-&-»»rd'6i-ini^ •.support;>'ilarg© .resonant responses to. the-tldoo -.. In ah;lnlfet'.'.:w;ith an average de^ihv-df 80Oi metor.s th© maximum ;spioeid---fo«'-lb'i^e^lis^l^.^ud^ .'shallow,water •:wave. is,;©G'.sieters per :s^ 'c'pnd'>;o.^ . one"hundred 'and eighteen, kilometers ;p^r;.;hOiUg><. • ior^an>;4jal ;^!^f one^;hun%ed 'and- 'S^ty,\&ilpm^t©rs;.(l0O;..g$ie8) the i iia'tur'ai': period' • %i about .2' hour ej,.,;. very, short j-compared, to . • a' t i d ^ ^ ^ t ^ ' & wave .of IS-liOurs'-'geriod'; with the v e l o c i t y of 60;' 'm'e^ 'fei'.f:. ; ''seoottd' :,h&s;' a w&v©: length -ot- .over, .g^'Oprkilpsotor s e fails'*-©?'. • ti;4©/''^1,1 be./prpppgated. up .ass-, i n l e t so- gu'ickly'that i t r^taia^';''almost the.'s-asjo-! phase,-throughout,- the, ialet,?'.s- length.,, ,. fh'is>'i''^ y^i^QFdl' the .<c«s£. _ Throfrgh'-\th©. 80 .k^^©t©rs^^jp^,a^i^ Alo3bt'';Bay''just outside the .-mouth, of Eaight Inlet, end @l@n.dal'©" •Cove near' its-midpoint t h e r e -is"a change of ; fiir© degrees in-the semi-diurnal .component o f the-tide« Although presumably cbaag£-ag;-tli'©;''sattir.© of t i e f l o o d OT tibhj the s i l l does not s i g n i f i c a n t l y r e t a r d i t , S i m i l a r l y there i s a s h i f t of t e n degrees' in'this' eoapoaeiit 'between F 6 i n t • Atkinson i n the':;Str©'i't''; of Georgia and faddingten Harbour at the head of' Bute, a. d i s - •• tanoo'" of- a^pr'oxiraately '"twice t h a t 'In-'Shigkt. .' '.":"In 'the presence of l a t e r a l boundaries geo s i r ^ p h i e force's'''have';40' other e f f e c t on a /steady f l o w than t o cause th© l e v e l "surfaces to slope ; up from' l e f t to r i g h t of the dire-, c t i o n of f l o w i n t M Northern, gemi'sphera. . Shio i s i n ' a ftnaor' s i m i l a r t o 'that du© t o c e n t r i f u g a l " f o r c e a t a, bend i n th© f l o w , except that, there i s so secondary h e l i c a l f l o w a s s o c i a t e d w i t h ©orlolls'force i n a s t r a i g h t channel, siaee the displacement of the levels"'' i s the ease throughout the length©'of the flow* On the" other hand' a t i d e through any.section w i l l cause-a -' per i o d i c a l l y ' r e v e r s i n g - f l o w which w i l l experience a C o r i o l i a f o r c e p r o p o r t i o n a l to it's v e l o c i t y * thus some considerable secondary;flow i s to be expectedj sweeping the p r o p e r t i e s f i r s t one way' across the i n l e t , then the other, as the t i d e ebbs and flows.' " A t ' i n t e r n a l surfaces t h i s e f f e c t w i l l be.considerably enhanced: i s 7 th© presence' of any s t r a i i f i c a t i o n . These' e f f e c t s have'not been e x t e n s i v e l y studied i n B-.C., i n l e t s though, i t i s ka®«a.'tha,t;''ia the S t r a i t of Georgia the t i d e d e f i n i t e l y f l o o d s along, one" s i d e and ebbs on the other. X^riction can modify the resonant c h a r a c t e r i s t i c s of a t i d a l wave, p r i m a r i l y by i t s a c t i o n on the amplitude.* which r e s u l t s in''only p a r t i a l support of 1 the incoming wave by the r e f l e e t o d one. Thus a node disappears to foe replaced by as area of-ffiisiffiuia . amplitude ? and the.abrupt change of phase at the 'node;-.reduces- to a gradual one.as tho,for® ©f the co-rosci!-* .. latioa.-.approadhoa .that..of &.progressive wave with.an;.exponential decay of paplitudo. . .... -\\ . .'; -.• ••<f.-Vr,;*^ n>.^ 8tu«*ia8 where the .retarding effects arafthas'd ; produced'':byv:i|ie' outflow- of a r i v o r : and .the progressive. ..decrease i n d"epthi£-v:;tha='-tide. has a shortened p e r i o d $ f flood .with,;..-.respect to- tho- shore.;,-. Belative iOi-the river., of course.), tho. period i s the-'-aaiBes as' "the t i d e at i t s : mouth, the., shapo; o-f • tho peak, of'.... -'tho 'flood-is •••steepened as? the higher amplitude parts, of tho .... wavo "overt-site" the early f l o o d * h^os.© wav© v e l o c i t y i s also . decreasing • with' depth. I f the hydraulic head of the following water' is-" s u f f i c i e n t t o increase the. v e l o c i t y above that of. an inertiol:"WavO'in tho some depth of water.* .®. h y d r a u l i c jusap {tidal' box©) may-develop* • Bue to- the- e f f e c t i v e drag; of the bottosr the'-volo'City-of the t i d a l current's i n - i t s v i c i n i t y aro shi f t e d i a phase With respect to. the layers above, and saay i n -fact''"b©"opposite -ia d i r e o t i o a to those at. th© su'$faoe» . •••*•• •• •'-••-l't"-ia not u n l i k e l y • that* socio or.'all of theoo offoots -occur;-at'-'some'-point i n .the passage of a-tide, into- tm, inlet*.; • M'mo.gt"':ail'-'tho i n l e t s -are - separated front- the - coastal waters :.,, •' of-the ooeaa by depths of les s than a - hundred • meters.. l a ; ':lsiigM''aad-';'B\Sst0 l a l a t s t h i s does l i t t l e - to a l t e r the phase or the :amplit:udo.'s though th© amplitude- i s iaeresoed by © foot .or- . two probably.'due to a small c o - o s c i l l a t i o n e f f e c t , Bu-tv sosa© ialotgs aro ;approached- by. entrance paswago-s- of such:-small' , dimeasioairth&t tho flow, aay : a l l . but -cease- to-be iaertia-1. and mav.be. maintained i n s t e a d by the h y d r a u l i c head. between i t s two. ends*;. fhorapsoa. aad Barkey,/(195$5 report, that, ;at-'the eatrane©./channel, t o the/-Hugent, Teliae*'and Seymour-, cotoplejsh . . of • i a l e t s , .the coa@triet.ioa;. i s ouch., that- tho t i d e s r ua o*a,the f l o o d and'-ebbc a t a'...rate.,of. twenty..knots.? ;oao.of :thio.--fei-gfee«,fe.-;--i.'.; t i d a l . ,v#3^i.t'i©s ^ ^asnsaeyo, i a tfe.e, ^ d.rld».; I t i s . iaeoacoivsiblo ..that safch/-&:;-C;paqtric^i3o.a. w i l l h.ave r a o , ' :effect oa ;tae-^ha$© o f the, •^do.st':^.'|%oea Its'i'tw© ends.*;. ;;•;';The-\eurroats': which ecceapany tho t i d e s are-.basically governed"by'rjtho. rospoaso'.sf the waters - i a the for®, of•• 1-oag .< i n e r t i a ! waye"^ . I o t i n . t h e i r .t.ur.a. the. currants;, may iaodify tho foro' of: 'th-a 'tide .through... t h e i r ,;seas a tiv.-ity.,-to gees t r o p h i c ep&\ f r i c t i o a a l , forces*, ,X®%- th© t i d a l ro.sgase of, th©; inlets"'cannot-.be e n t i r e l y ,.i'a-.ertial s, f o r ; the..trevol. tirae< a a l o a l ^ t o d f o r the.". -;;; distf^ c'a";-botwe©a.; A l e r t Boy .and GlendaiOrCov© f o r a wave t r a — ' • : v e i l i n g ; o.a ^:bottom o f 860 meters depth, '.(greater than 'moat -©f the report^a^aepth i n . . t h i s ; section).- i s twice t h a t - reported - in. the tidQ v:tablos*. The• sama.is : tru@->for th® two s t a t i o n s in*tl*©' S t r a i t ' of'Georgia, Thus as. suggested by Dawson (1920) the fl.ood" tid©/'&ots .like, a . p i s t o a on,'th© .aoatSa o f • tho i n l e t ; • f o r - ' " cia-g. .the' t i d e up a t a. r a t e much i a .excess, -of t h a t t o be expected of aa i n e r t i a ' ! ..wave*. This, would iadi-eate that';tho: tid© a^t.ed r a t h e r l i k e one- i n am -estuary 3. and' t h a t the-! p r o f i l e would b© steepened oa the f l o o d ,aad could &©•• longer- toe- t r e a t e d as ; a s i n u s o i d a l " o s c i l l a t i o n f o r which- tho f l o o d and ebb are syaaaetric. - io -3) The Relation of Tides to the D i s t r i b u t i o n of  Qceanographie Properties The simplest theory of the role of tid e and of r i v e r discharge i n the exchange of properties i n and out of an i n l e t i s that of the t i d a l prism. The theory of the t i d a l prism which involves dividing the length of t he i n l e t into segments within which there i s complete v e r t i c a l and horizontal mixing, was o r i g i n a l l y devised for use i n shallow bays and estuaries (Ketchum, 1951)* However observations (Tully, 1949) show that the p r i n c i p a l exchange of the deep B.C. i n l e t s takes place above a depth of eleven meters, a depth possibly subject to further l i m i t a t i o n by s i l l depth* Application of the theory of the t i d a l prism to the layer above this depth, where the i n i t i a l segment i s determined by the r i v e r discharge i n one t i d a l period, and each subsequent volume i s determined by the i n t e r -t i d a l volume of the previous ones, has led to q u a l i t a t i v e agreement for the surface layer. A t y p i c a l i n i t i a l segment length for a coast i n l e t such as Knight i s one mile, each subsequent segment increasing by roughly one-twentieth of the previous one. In the f i f t y mile length of the i n l e t there would then be t h i r t y - f i v e prisms. This i s also the number of t i d a l cycles for which a property discharged at the head would star t being removed from the mouth. A refinement of the tech-nique i n which a mixing length i s determined under the assump-tio n of complete horizontal advection gives a f a i r l y accurate d i s t r i b u t i o n for the surface properties of the i n l e t s (Arons & Stommel, 1951). - i i -The simplest type of exchange between the upper layer and that beneath i t w i l l depend on the r e l a t i v e sheer s t a b i l i t y between the two layers, which i n the f i n a l analysis depends on the density and the transport of the runoff compared to the t i d a l flow inward under t h i s layer* Model experiments (Keulegan, 1949) show that there exists a c r i t i c a l v e l o c i t y between two such layers above which the lower layer w i l l be entrained f r e e l y into the upper. This i s characterized by the breaking into the upper layer of internal waves propogated i n the surface of separation* Above t h i s c r i t i c a l v e l o c i t y an expression for the volume of entrainment i n terms of the c r i -t i c a l v e l o c i t y , and the r e l a t i v e v e l o c i t y of the upper layer derived from a simple model of hydraulic flow, approaches very clos e l y the observed values. A more sophisticated model of estuarine flow which takes account of entrainment through an arbitrary form of eddy d i f f u s i o n , providing that i t i s Fickian, i n spite of not predicting the decrease of depth of the upper layer toward the mouth, does predict the observed c r i t i c a l v e l o c i t y of the flow, and to a good approximation the observed density d i s t r i b u t i o n (Pritchard, 1952). In a single twelve hour period the tide f i l l s and empties an average i n l e t of a volume of water equal to the average runoff i n a month. The passage of t h i s volume through a c o n s t r i c t i o n , such as at the s i l l i n Knight Inl e t , would give an average v e l o c i t y across the whole cross section of t h i r t y centimeters per second. For a regular sinusoidal velo-c i t y t h i s gives a maximum ve l o c i t y of f o r t y - f i v e centimeters - 12 -per second throughout the cross section. Observations at the s i l l (Pickard & Rogers, 1959) suggest that the v e l o c i t y may be neither uniform across the i n l e t nor through i t s depth, but i s concentrated near the center somewhat below the surface, with a maximum v e l o c i t y of over 120 centimeters per second. I f the tide enters and leaves the i n l e t as a prism;, the only apparent mechanism of exchange for water below the surface i s that of d i f f u s i o n . Xet with high v e l o c i t i e s and non-uniform p r o f i l e i t i s quite possible that the flood w i l l be carried by i t s momentum well up the i n l e t i n the form of a modified j e t to mix and return i n some other manner on the ebb. This may i n i t i a t e some considerable exchange with waters outside the i n l e t . Should the waters below the s i l l i n the i n l e t be strongly s t r a t i f i e d , waters that enter i n a j e t w i l l be held by the i r buoyance above th i s layer, and t h i s form of intrusive j e t would be increased i n extent. Should the water of the i n l e t be less dense than the water s p i l l i n g over the s i l l , i t w i l l be replaced by the heavier water and driven out of the i n l e t on the ebb t i d e . Chapter II The Oceanographic Structure of Bute and Knight Inlets In the study of the t i d a l processes of the i n l e t s i t i s important to e s t a b l i s h the reg u l a r i t y of the d i s t r i b u t i o n of properties, i f at a l l possible. When a synoptic survey i s attempted, i t i s hoped that the values of these properties w i l l remain r e l a t i v e l y constant during the period i n which the i n l e t i s sampled., Xet the time taken to observe at stations at f i v e mile intervals through the length of a f i f t y mile i n l e t may be «• 13 ** a matter of one to one and a half days. I f the stations are r e l a t i v e l y evenly spaced i n t e r v a l s , the t i d a l influence may become apparent as a periodic displacement of isopleths as they are plotted against location. On the other hand, i f a series of readings i s taken at one station, the t i d a l effects w i l l be observed as periodic displacements of isopleths i n a time series p l o t . When only a single station i s taken, i t i s v i t a l to the interpretation of these displacements (as horizontal, v e r t i c a l , or turbulent movement of the water) to know the general features of the horizontal d i s t r i b u t i o n of properties. Once the main cha r a c t e r i s t i c s have been established, an attempt can be made to correlate the t i d a l movements with the p r i n c i p a l features of the i n l e t s . Among many i n l e t s on the coast, Bute and Knight (Pig.l) have been studied more extensively than the r e s t . Bute Inl e t , with a length of 41 miles, width of 2 and mean depth of 500 meters, i s the :southernmost of the two. I t possesses a smooth bottom, which slopes i n easy stages from the head to a depth of over 730 meters, erne distance behind a s i l l of 250 meters i n depth. The waters of the i n l e t are connected over t h i s s i l l with the northern waters of the S t r a i t of Georgia. Some connec-t i o n with Johnston S t r a i t i n the north may also possibly influence the deeper waters of the i n l e t , the runoff, which i s a l i t t l e more than average for an i n l e t of Bute's size, originates (as i n most i n l e t s ) from the r i v e r s at the head, the Homathko and the Southgate. To the normal runoff of the peripheral streams along - 14 -i t s length i s added th discharge of the Orford River close to i t s midpoint. Knight i s a l i t t l e longer than Bute, with a length of f i f t y - f i v e miles, average width of 1,6 miles, and average depth of 400 meters. The bottom has the same regular character as Bute with a maximum depth of about 600 meters. Between the two s i l l s , the inner of which i s only 65 meters i n depth, the depth averages some 200 meters and the length i s intersected by Tribune Channel. The outer s i l l forms a narrow con s t r i c t i o n of 60 meters i n depth and between f i v e and ten miles i n length. The e f f e c t of the inner s i l l w i l l be shown to be that of block-ing water outside i t and below i t s l e v e l . On the other hand the deeper water between the s i l l s i s saline enough? to indicate that a possible exchange takes place between i t and Queen Charlotte S t r a i t . I f blocking i s the case for the outer s i l l , exchange could take place only through Tribune Channel, which i n spite of i t s great extent appears to be not much less than 200 meters i n depth. The deep water of Knight Inle t i s midway i n i t s c h a r a c t e r i s t i c s between those of the northern i n l e t s d i r e c t l y connected with the coastal water, and those of the south connected with the S t r a i t of Georgia, whose waters are less saline than those of the coast due to the considerable discharge of the Fraser River. Thus the most s i g n i f i c a n t differences between the two i n l e t s a res(i) the s i l l depth, effecting the degree of exchange of deeper water, and ( i i ) the location, which deter-mines the nature of the deeper water exchanged. •,/ • '- 15 - ' With the exception of Knight Inlet ( i n 1954), both i n l e t s have been v i s i t e d at least once a year since 1951. Prom the re s u l t s of each cruise longitudinal p r o f i l e s of tem-perature, s a l i n i t y , and <7t have been plotted. During 1957 and 1958 some eight cruises were made to Bute i n l e t , repre-senting most phases of the seasonal v a r i a t i o n . Thus although cruises i n Knight Inlet normally were only made i n July, i t i s possible to deduce the seasonal changes nf water structure from Bute Inlet insofar as they are common to> both* To permit t h i s comparison isopleths have been drawn side by side for each i n -l e t and each year. The temperature d i s t r i b u t i o n i n both i n l e t s i s char-acterized during the early part of the year by a d i s t i n c t l y wedge-shaped structure l y i n g at a depth of approximately 100 meters, with i t s t i p to the mouth of the i n l e t and i t s cold core against the head. Xet quantitatively differences do appear between the two, i n p a r t i c u l a r the thickness and the sharpness which defines the wedge's upper and lower surfaces.. However the temperature of the cold core i s the same i n both i n l e t s . For example, the regions bounded by the 9° C. i s o -therm for the two i n l e t s from the diagrams of 1953 and 1956 may be compared. The thickness and length i n Bute are 20 meters and three-quarters of the i n l e t ' s length; i n Knight some 100 meters and three-quarters of the length to the inner s i l l . Above the wedge the values of temperatureeand t h e i r rate of change appear to be mueh the same i n both i n l e t s . Below the wedge the v a r i a t i o n of temperature to the deepest - 16 -point i n Knight i s only 0.5 C.° 9 i n Bute 2 C.°. The gradient of temperature i s much better defined i n Bute, and i s limited to only a few meters just below the wedge, ( F i g . 2) An explanation of the differences i n the temperature structure between the two i n l e t s could possibly be found i n a theory of the formation or decay of thi s wedge. The water* of the wedge core i s colder, i n July, than any surrounding water, either of the surface or the connecting channels from which an. exchange of water could be expected to take place with the i n l e t . On the other hand the formation of the temperature wedge can be accounted for only by a cooling process associated with a boundary, such as the winter cooling of surface water or the intrusion over the s i l l of water cooled i n t h i s way at some other surface. A number of observations indicate that th i s temperature minimum originates at the surface; the large extent of the wedge i n Knight i n spite of i t s shallower s i l l , the formation of a cold layer at the surface i n Bute during; the spring of 1953, and the very close connection of the f o r -mation and character of the wedge with the weather pattern (Tabatai & Pickard, 1957),. The decay of the wedge appears to be from mouth to head, as the series of p r o f i l e s taken from cruises through Bute I n l e t during 1957 and 1958 shows so c l e a r l y (Fig. 3).. Remains of the previous year's wedge are s t i l l present i n March and May of 1958, as signs of the new one appear near the surface. - 11 -The most important implication from the occurence and very slow d i s s i p a t i o n of t h i s structure are the lim i t a t i o n s i t sets on the pattern of c i r c u l a t i o n . The presence" of the wedge for a large part of the year with no source of replenishment indicates that the temperature over r e l a t i v e l y short periods i s very nearly conservative, thus e s s e n t i a l l y dividing, the c i r c u -l a t i o n above from that below i t . .On the other hand the decay of the wedge sets a l i m i t to the Use of temperature as a tracer for water movement, p a r t i c u l a r l y near the t i p where changes seem to take place on every t i d a l cycle. In the upper layer of the i n l e t , energy Considerations indicate that fresh water discharge from runoff should remain at the surface, increasing i n s a l i n i t y by the mixing of more saline water from outside or entrainment from layers below. Isohalines below some 15 meters remain almost horizontal, termi-nating only at the head of the i n l e t (Fig. 4). Therefore the mixing process appears to be limited to the surface 15 meters, l e t a rapid increase of temperature to form the upper surface of the wedge at some twenty-five meters i n Knight and f i f t y meters i n Bute, could be the resu l t of slower adveetion of saline, warmer water from outside the i n l e t , above the wedge and below the layer of most intense mixing. That some sort of exchange of deeper water occurs with the outside i s indicated, since the lower surface of the wedge does not extend to the lbottom of either i n l e t . The difference between the degree of exchange i n the two i n l e t s i s probably the r e s u l t of the marked d i f -ference of size between the s i l l s , . The shallow s i l l i n Knight - 18 -would l i m i t the exchange and thus the nearly homogeneous nature of the deep water would probably be made more nearly so by some iaixing mechanism associated with the ti d e s . The changes i n bottom water seem to be irr e g u l a r i n pattern, and could be caused by an intrusion of more sali n e , warmer, higher oxygen water. In Knight and Bute the s a l i n i t y i n the i n l e t i s nowhere greater than that a few meters below the outer edge of the s i l l , i ndicating an almost t o t a l blocking of water below t h i s l e v e l (Fig. 4)* Exceptional motion of the tides might well be s u f f i c i e n t to produce an intrusion of t h i s blocked water over the s i l l . The low s i l l i n Bute would f a c i l -i t a t e the intrusion of more saline water, which i s also warmer, to mix with and replace the colder, less saline water produced by the mixing down of the lower edge of the wedge, which forms the sharply defined lower edge i n Bute. The T,S diagram i s often used to determine the degree of mixing of water masses (Fig. 7). The data from a series of stations taken i n Knight during July of 1956, when drawn up i n t h i s manner, show, i n the upper stations of the i n l e t , a greater s i m i l a r i t y of water type above and below the wedge than with the wedge i t s e l f * This indicates that more mixing than would normally occur through d i f f u s i o n processes has taken place. The water outside the s i l l shows l i t t l e s i m i l a r i t y to the rest* Station 4 close inside the s i l l shows ch a r a c t e r i s t i c s of well-mixed water below 50 meters. I t i s important to note the defference between stations 5a and 5b taken only a matter of two days apart and at much the same phase of the tide* Two series of temperature and current readings were made at t h i s station.. Yet close to the t i p of the wedge, thi s region shows a considerable change of ch a r a c t e r i s t i c s at a l l l e v e l s , i n spite of the shallow s i l l which might be thought to i n h i b i t motion at any great depth. Closely associated with s a l i n i t y i s density, which i n the B.C. i n l e t s depends on i t to a much greater degree than temperature. This i s s i g n i f i c a n t i n the interpretation of the dynamic motion, which i s only properly represented by deductions made from the form of the isopycnals of <re . Diagrams of isopycnals exhibit a very d e f i n i t e change of l e v e l at most stations, c l o s e l y associated with the phase of the t i d e ( P i g . 5 ) . Nowhere i s t h i s so marked as near the s i l l , where there may occur a drop of as much as 130 meters from one side of the s i l l to the other. 2) S e r i a l D i s t r i b u t i o n of Properties In times of low runoff, the tides at the s i l l may be s u f f i c i e n t to reverse the d i r e c t i o n of the flow i n the upper layer. Indeed i t may be the propagation up i n l e t of the t i d a l e f f e c t s at the s i l l which causes the d i s t r i b u t i o n of properties i n t h i s surface layer to depend on the length of the i n l e t rather than on the distance from the head for i n l e t s of com-parable runoff. The mode of flow of the tide over the s i l l could be of some importance i n determining the intrusion of more saline water into the deeper waters of the i n l e t . The asymmetric e f f e c t of the tide as shown at Knight 3-1/2 i n 1955 might be s u f f i c i e n t to release some of the more saline water - 20 -blocked behind the s i l l ( F ig. 8). Though the p r o f i l e s of i s o -therms at t h i s same location i n the following year give no ind i c a t i o n of t h i s i n the well-mixed properties of the water below 30 meters, near the end of the series there i s some l i t t l e i n d i c a t i o n of an inhomogeneity. Simultaneous readings of s a l i -n i t y would have helped immeasurably i n interpreting t h i s . At stations where there i s a s u f f i c i e n t l y marked v a r i a t i o n of temperature with depth, a comparison of hourly temperatures (which are more speedily sampled than s a l i n i t y ) over several t i d a l cycles may, i f the changes are s t r i c t l y C y c l i c , indicate the dynamic motion of the tide at depths below the sur-face. A basie change of shape from tide to tide could, on the other hand, indicate the extent of the mixing associated with each t i d a l cycle. At Knight 5a (7|- miles behind the inner s i l l ) there i s at low water slack a pronounced minimum, which as the tide floods becomes broader and less pronounced u n t i l at high water slack the water below the surface layer appears almost homogeneous ( F i g . 9 ) . This makes the isotherms i n t h i s period of two or three hours almost impossible te follow. As the tide ebbs, the minimum1 reappears, but net to the same degree. Com-parison of the minima at low water slack, when the water of the wedge would be furthest advanced, indicates a f a i r l y consistant mixing process which takes the form of a broadening, and dimi-nishing of the minimum of temperature. Tet a station at the same location two days l a t e r shows lower temperatures by half a degree. This could only be due to an advance of the wedge, indicating either the d i f f e r e n t extent of the t i d a l excursion - 21 -or an outward advective process at t h i s l e v e l . At a s t a t i o n i n Bute close to the t i p of the wedge a similar proeess i s observed, while some distance along the wedge the minimum re-mains much the same throughout the whole t i d a l cycle, so that the v e r t i c a l displacement i n thi s case would seem to be due en t i r e l y to the displacement of the water l e v e l . In the ease of mixing or some sort of sheer flow i t i s possible that the points of i n f l e c t i o n of the minimum might better define the boundaries of the wedge than the boundary isotherms of the peak of the minimum. An analysis of the derivative of the bathy-thermograms taken at Knight 5a was disappointing ( F i g . 9).. the maxima and minima which correspond to th© i n f l e c t i o n points of the o r i g i n a l traces have l i t t l e r e l a t i o n to one another, and even less te the tides. Characteristic of a l l the p r o f i l e s of isotherms i s the increase of the range i n the v e r t i c a l o s c i l l a t i o n s to a maximum at some depth not infrequently close to the location of the temperature minimum ( F i g . 11),. Comparison with the tide shows that the v e r t i c a l displacement from the mean lags the peak flood and ebb and increases with depth. On the ether hand, observations at Knight 5a show at close agreement with the cur-rents at a l l depths. The l a t e r a l extent of these o s c i l l a t i o n s i s apparent from simultaneous series taken at two stations i n Bute Inlet* The v e r t i c a l o s c i l l a t i o n s d i f f e r i n phase by appro-ximately 180 degrees and th e i r magnitude has been considerably diminished i n the 21 kilometers separating, the stations. In series of bathythermogram casts with half hour - 22 -i n t e r v a l s , there appears, superimposed on the longer period o s c i l l a t i o n s , v e r t i c a l o s c i l l a t i o n s which, because of t h e i r short period of only about two hours, do not appear i n the series taken at hourly i n t e r v a l s . Comparison of t h i s period with that of a free o s c i l l a t i o n i n the given density gradient (Brunt frequency) indicates that t h i s free period i s too short, as i t ranges only from a minute or so near the surface to less than an hour at 300 meters, to explain the observed o s c i l l a t i o n s , ( F i g , 12). Also the actual o s c i l l a t i o n s appear to have much the same period throughout the upper depths of the i n l e t and i n many cases coincide. But they are i r r e g u l a r , as the absence of any term of t h i s period i n the harmonic analysis indicates. A p o s s i b i l i t y i s that these superimposed o s c i l l a t i o n s are the r e -s u l t of seiches. The natural period of Bute I n l e t i s about two hours, and the o s c i l l a t i o n s appear to occur i n bursts of t h i s period, with occasional periods of calm t y p i c a l of seiches* Unfortunately the presence of these short o s c i l l a t i o n s , which often have a f a i r l y considerable amplitude, makes the obser-vation of isopleths over short periods of doubtful value, and the observation of t h e i r horizontal d i s t r i b u t i o n taken along the i n -l e t , ambiguous. Such a series of f i v e bathythermogram casts was taken after an anchor station i n Bute* Short period o s c i l l a t i o n s of s i g n i f i c a n t amplitude Were recorded, revealing a considerable change of depth i n the i s ©therms i n a matter of h a l f an hour's s a i l i n g time. I t i s d i f f i c u l t to judge whether these were d i s -placements of isotherms with time at any given location or the permanent p r o f i l e associated with the water body. A l l the - 23 -evidence of these l a s t two sections seems to favour the f i r s t of these two p o s s i b i l i t i e s . 3) Harmonic Analysis Perhaps the best quantitative approach to the analysis of a series of values i s an attempt to match i t against some known function. The most useful of these are the so-called ortho-gonal functions which can be defined by a d i f f e r e n t i a l r e l a t i o n -ship and which, when integrated over a suitable i n t e r v a l , vanish t when taken as a product with any other solution but i t s e l f . The simplest of these are the sine and cosine functions by which any well-behaved function can be f i t t e d * By the nature of the i r orthogonality properties, the c o e f f i c i e n t s of t h i s so-called Fourier series can be evaluated by the integration of the unknown function over the region of d e f i n i t i o n * I t i s possible to show that such a Fourier series i s the best f i t to any curve, a pro-perty i t has i n common with the results of the Gaussian least squares method which i s more re a d i l y adapted for use with discrete series of readings. In fa c t the two processes can be shown to be en t i r e l y equivalent* A harmonic analysis, as t h i s process i s ca l l e d when the c i r c u l a r functions are used for the regression, w i l l lead to the representation of a series by the f i r s t few terms of a sine series whose periods bear an integer relationship to one ano-ther. The amplitudes when displayed as a spectrum! can be com-pared to others to determine any co r r e l a t i o n between their representations as periodic terms* The series can then be eval-uated and the synthesis compared to the o r i g i n a l data and any _ 24 -aperiodic e ffects isolated*. The synthesis of several analyses of the hourly posi-t i o n of the mean depth of the two 6*7°C. isotherms defining the lower minimum- of temperature at Knight 5a. when plotted against the o r i g i n a l , shows that the basic period i s best f i t t e d by a fundamental harmonic a f a period of 25 hours. The representation by six components shows only a s l i g h t l y better d e f i n i t i o n of the peaks than when represented by four components ( F i g . l l ) * The peaks show the same deviation from the o r i g i n a l as i s charac-t e r i s t i c of t i d a l curves when the impulsive effect of wind i s superimposed on the regular p e r i o d i c i t y of the t i d e s . Thus the difference between the o r i g i n a l and the synthesized curves, when plotted against time shows a regular aperiodic form., which i n the current data has been shown to be related q u a l i t a t i v e l y to the wind action at the surface (Rodgers, 1959).* More exhaustive analysis of the tides, horizontal currents, and v e r t i c a l o s c i l l a t i o n s , as represented by movements of isotherms and horizontal currents, show a rather s i g n i f i c a n t r elationship among t h e i r semi-diurnal components. In a series of discrete readings a method i f i s o l a t i n g the dominant basic frequency i s to run a series of harmonic analyses with a Varying number of terms. The spectrum of the components w i l l , for s i g -n i f i c a n t frequencies, show is o l a t e d peaks against a background of a broad, continuous spectrum c h a r a c t e r i s t i c of noise. The r i s e of the isolated frequencies above the background i s then at measure of i t s significance. Thus i n the analysis of the tides at A l e r t Bay during station Knight 5a, the fourth component, * 25 -which i s close to the twelve hour one, does not increase s i g n i -f i c a n t l y to i t s maximum i n the analysis of 50 terms* but the envelope of the smaller background components decreases to a^  marked extent. (3?ig. 13). The second (diurnal) component i s also very strong. The increase of the t h i r d component for a% smaller number of terms i s assumed to be due to the overlapping of these two larger ones as t h e i r peaks become less well defined* The analysis seems to indicate that the lunar semidiurnal com~ poaent of 25 hours i s the dominant period for the tides during the period of Knight 5a; For t h i s reason and to avoid a rather unproductive excess of labour, the analysis of the remaining; data was r e s t r i c t e d to a basic period of 25 hours. Any very gross change from a 25 hour basic period would appear i n the d i s t r i b u t i o n of the harmonics, and i t appears f a i r l y clear from the q u a l i t a t i v e data that the internal modes of motion are closely related to the tides at least i n period. An analysis of the v e r t i c a l o s c i l l a t i o n s showed, as might be expected, a close relationship to the t i d e , except at Knight 5b ( i i ) , where i n spit© of the marked dominance of the semidiurnal component, the corresponding o s c i l l a t i o n at thuee meters i s p r a c t i c a l l y absent, since i t i s replaced by a very strong twenty-five hour component (Fig, 15). This e f f e c t may possibly be due to strong winds reported during t h i s period. The semidiurnal component appears always to increase with depth to a maximum somewhere i n the v i c i n i t y of the wedge of cold Water, with a continuous change of phase with depth. The other dominant component (the semidiurnal one) seems to vary - 26 -e r r a t i c a l l y with depth i n both phase and amplitude ( F i g . 18). At Knight 5a the semidiurnal component appears to be rather closely related to the temperature var i a t i o n s . Close to the surface the temperature shows a marked d a i l y v a r i a t i o n which decreases with depth, to r i s e sharply i n the region of the lower temperature minimum (Fi g . 14). The v a r i a t i o n at lower depths may be some sort of advective effect connected perhaps with th© change of shape of the minimum;, which tends to have a d a i l y rather than a h a l f - d a i l y period. The behaviour of the semidiurnal component only emphasizes that the v a r i a t i o n of t h i s component must be due to a d i f f e r e n t cause, for the component of temperature v a r i a t i o n bears l i t t l e r e l a t i o n to that of the v e r t i c a l amplitude (Table l ) . Comparison of spectra at two simultaneous stations i n Bute Inlet indicates that the semidiurnal component dominates, and that there i s a change of phase of t h i s component of 170 degrees between the two stations ( Fig. 17). A comparison of the r e l a t i o n to depth of the phase shows a regu l a r i t y only i n the semidiurnal components The currents were observed over a more representative range of depths than was possible for the v e r t i c a l o s c i l l a t i o n s by the method of following distinguishable isotherms, I t i s intere s t i n g to note the mean values for the currents, which indicate a mean flow out from the central region of the i n l e t between one and two hundred meters. This i s compensated by a flow of the same order up i n l e t at a depth of three hundred meters, f i f t y meters from the bottom of Knight I n l e t , and at - 27 -f i f t y meters from the surface i n the r e g i o n between the sur-face mixing area and the temperature wedge. These s e r i e s run only f o r periods of a few days, and thus the means may only be t r a n s i e n t e f f e c t s * v A l l the s t a t i o n s show t h i s tendency t o a greater or l e s s e r degree. However at Knight 5b ( i ) taken two days a f t e r Knight 5a, the f l o w i s out at f i f t y meters and i n at two hundred meters, yet the previous p a t t e r n i s resumed on the f o l l o w i n g day at KnighttSh ( i i ) , as i t i s on the two day s e r i e s taken i n the previous J u l y , Knight 5c ( F i g * 18). At a l l s t a t i o n s the semidiurnal component i s dominant at two hundred meters and a l s o at three hundred except at KMight Sc where the background almost envelops i t . The s i z e ef the com-ponent at Knight 5c i s worth no t i n g as i t i s the more r e l i a b l e , because of c o n t i n u i t y of r e s u l t s . The s e r i e s at Knight Sa and Knight 5b were modified from t h e i r o r i g i n a l form* i n which some readings were mi s s i n g , by adding i n t h e i r place dummy readings c a l c u l a t e d from the two adjacent ones. I t was f e l t t h a t , i n s p i t e of the inaccuracy t h i s would introduce, some i n d i c a t i o n of the o s c i l l a t i o n of the current at these depths would r e s u l t , and the method of a n a l y s i s would not permit empty readings* This a r b i t r a r y method of a s s i g n i n g terms would i n e v i t a b l y l e a d to a p a r t i c u l a r l y l a r g e background, though i t could hardly a l t e r the trends of the main terms. A p o s s i b l e explanation of the very l a r g e values a t three hundred meters i n Knight 5a i s t h a t they are the r e s u l t of some form of i n t r u s i o n . This w i l l be considered l a t e r * The very high background at one hundred meters i n a l l except 5b i s more d i f f i c u l t to i n t e r p r e t , and the readings at f i f t y meters appear to have no d i r e c t r e l a t i o n to the t i d e i n t h e i r general s t r u c t u r e , fhough the semidiurnal component i s l a r g e , the dominant components at Knight 5a, f o r i n s t a n c e , are the 8 , 4 , and 2k hour periods p o s s i b l y a s s o c i a t e d with some form of seiche motion, , 4) Theorem of C o n t i n u i t y Tracer substances are of great use i n the i n t e r p r e t -ation, of dynamics of l a r g e bodies of water* J u s t as dye can be of use i n shallow r i v e r s or bays t o the degree that i t f o l l o w s the movement of the water and i t s r a t e of d i f f u s i o n i s slow i n the time considered, so p r o p e r t i e s which change l i t t l e i n the time scale considered and are w e l l d i s t r i b u t e d i n value through-out a water body can be used to d i v i d e the water by surfaces of constant value; that i s , by i s o p l e t h s , i n t o masses that are e s s e n t i a l l y separated one from the other. For no t r a n s p o r t can take place through such a surface, as t h i s would imply the mixing of the property and therefore i t s e s s e n t i a l l y non-con-s e r v a t i v e nature. Observations of p r o f i l e s of such surfaces over some d i s t a n c e , or a p e r i o d of time, can l e a d t o an idea of the kinematics of th© system* More p a r t i c u l a r l y i f the e q u i -l i b r i u m p o s i t i o n of the surface i s known, then any d e v i a t i o n from i t can be i n t e r p r e t e d q u a n t i t a t i v e l y as a ra©v©a$eat; I n such a case, and i f the motion i s known to be generated by wave motion of given v e l o c i t y and d i r e c t i o n , the associated h o r i -z o n t a l c u r r e n t s ( u ) are defined i n d i r e c t i o n and magnitude at every depth by the v e r t i c a l gradient of the displacement i^,) and the wave v e l o c i t y (cf ) under the assumption t h a t d e n s i t y i s conservative, thus: When a series of readings are taken at a number of isola t e d depths, i t i s frequently impossibly to follow the movement of a given value of a property through i t s v a r i a t i o n with depth. I f the property can be shown to be horizontally homogeneous and to be conservative i n time, the theorem of continuity for that property can again be applied i n the Eul e r i a n form to get: where p i s the difference from the equilibrium value P f o r ggiible i n the period considered* In p a r t i c u l a r the T*S ch a r a c t e r i s t i c of water i n Knight has been shown for July of 1956,, As similar water masses w i l l appear as groups of associated points, there ap-pears to be a d e f i n i t e s i m i l a r i t y of water type for stations higher i n the i n l e t . P r o f i l e s of ch a r a c t e r i s t i c s show that they change only slowly through the seasons. Although v a r i -ations from one station to another occur over the t i d a l c^eUe, i t i s evident, at least away from the region of the s i l l , that they .are i n the nature of displacements, fhe series of pro-f i l e s i n Bute through the years of 1957 and 1958 show the decay of the c h a r a c t e r i s t i c temperature pattern takes place up i n l e t from the s i l l . the given depth j it and r^ty i s the v e r t i c a l displacement from t h i s depth. I t i s of importance to note that both these ^relations are s t r i c t l y applicable only where mixing i s n e g l i -- 30 -An examination i n some d e t a i l of a series of plo t s of temperature against depth over a period of a, day and a h a l f at Knight 5a together with t h e i r gradients, and deviations from the mean, shows that l i t t l e s i m i l a r i t y exists with the deep currents over the same period (Figs. 9 and 10). Study of the p r o f i l e s for that period indicates that the station l i e s at a pos i t i o n close to the most intense decay of the temperature structure, near the t i p . A quantitative analysis of periods shows a remarkably large v a r i a t i o n of phase with depthj t h i s can p a r t l y be explained as negative amplitudes i n regions of negative temperature gradient and the phase can be changed by 180 degrees* When th i s i s done however, the phase at lower depths s t i l l proceeds that of the amplitudes calculated from the movement of the char a c t e r i s t i c p r o f i l e , by almost exactly 90 degrees. I t i s possible then that these temperature v a r i -ations are more closely associated with mixing of the surf&ee tides than to the displacement of the water body. Amplitudes when calculated from the movement of v e r t i c a l plots show a regular change of phase with depth i n th e i r semi-diurnal component and a d e f i n i t e maximum of about one hundred meters. This should correspond to a minimum i n the horizontal current p r o f i l e i f they eaa be simply re l a t e d through a single wave. There i s a suggestion of t h i s , also an i n d i c a t i o n that the si t u a t i o n i s somewhat more complicated. Unlike that i n Knight Inle t simultaneous stations i n Bute Inle t show l i t t l e v a r i a t i o n of the i r v e r t i c a l structure throughout the t i d a l cycle, so that the indicati o n of periodic *** 3 %. ,** motion with a change of phase of 180 degrees and a drop i n amplitude of 60$ between the stations can be accepted with perhaps more credence as the actual r e s u l t of v e r t i c a l d i s -placement of the water mass. Several elteraate hypotheses preseat themselves as an ©xploaatioa of th© inferred movement of the surfaces de-fined by isotheriaals i n these two i n l e t s * l a on assymmetry of movement i a aad out of the i n l e t of the t i d e caused by .geostrophic e f f e c t * the t i d e floods along one side of the i a l e t aad ebbs along the other. Th© respoase of i a t e r a a l surfaces of a s t r a t i f i e d f l u i d to such a motion would be greatly exaggerated*, aad so<-c a i l e d cross i a l e t h e l i c a l cur-rents would develop* A similar e f f e c t would be encountered through c e a t r i f u g a l fore© oa th© sharper beads of the inlet*. Tot t h i s would aot explain the very d e f i n i t e phase lag with depth aor the maximum of amplitude at aa intermediate depth of the v e r t i c a l o s c i l l a t i o a . Another explaaatioa which suf-fe r s from the same defect suggests that th© raoveaeat i a and out with th© t i d a l prism of the temperature structure could well account for th© observed var i a t i o n s , This i s a possi-b i l i t y at the statioas i a Kaightf yet a purely adveeMve movemeat would b© expected to b© th© same ©a th© f l o o d aad ebb* At a position aear th© s i l l i t i s possible that a coabia&tion of the above effects could account f s * the ob-served v a r i a t i o n . For i f th© tide were to flo o d aad ©bb i n a d i f f e r e n t maaner over the s i l l , accompanied by a c e r t a i a amount of mixiag, an e f f e c t which aeed aot mirror the flood oa - 82 -the ebb could occur with a change of phase with depth due to the d i f f e r e n t for© of floo d and ebb, ©n the other hand t h i s could not occur at positions f a r from the s i l l where the temperature structure i s well esta-blished and the mixing therefore minimized. An hypothesis whieh accounts for not only the change of phase with depth, but i t s change from station to station, as well as the maximum of amplitude at an intermediate depth, i n the absence of mixing, i s that of an in t e r n a l i n e r t i a l mode of o s c i l l a t i o n . The aim of the next chapter i s to examine several hypotheses based on t h i s from a more quantitative point of view. Chapter III The Equations of l o t i o n 1) Internal Waves (i ) The solution for a continuous d i s t r i b u t i o n of density. Consideration of the r e l a t i o n of rate of change of momentum to pressure, gravitational p o t e n t i a l , or other int e r n a l forces, gives r i s e to a d i f f e r e n t i a l equation of motion, which defines the response of a water body to these forces i n the presence of d e f i n i t e geometric boundaries. The solution of t h i s equation w i l l usually depend on the three s p a t i a l coordinates and time. When specialized to periodic motion of periodo,, the equations reduce to a solution of p a r t i a l d i f f e r e n t i a l equations i n the spa t i a l coordinates only. Upon the application of t h i s to a canal of constant depth, and motion which proceeds along i t with no secondary ef f e c t s , the equation then w i l l involve only the - 33 -v e r t i c a l coordinate z and define the v e r t i c a l displacement from equilibrium (Fjeldstad, 1933). The d i f f e r e n t i a t i o n of t h i s with respect to time, after interchange of order of d i f f e r e n -tiation, r e s u l t s i n an equation i n v e r t i c a l v e l o c i t y more ea s i l y f i t t e d to the boundary conditions at the top and bottom surface* These require, for internal modes of o s c i l l a t i o n , that the v e r t i -c a l v e l o c i t y be zero* This condition i s not met s t r i c t l y at the upper suffaee, but with respect to the internal motion the d i s -crepancy i s negligable* A solution i s then d i f i n e d , to the order of an a r b i -t r a r y constant m u l t i p l i e r , i n terms of the v e r t i c a l s t a b i l i t y and depth, by the d i f f e r e n t i a l equation and the boundary condi-tions* I f the product of the square of the wave number and the average density i s small at a l l depths with respect to the gra-dient of average density with depth, t h i s term can be neglected i n the equation which then becomes: where w i s the v e r t i c a l v e l o c i t y , g the acceleration of gravity and ^ the inverse of the wave v e l o c i t y , and <f>-~^ " 5 ^ ~ * ' ° V i s the s t a b i l i t y . Should <j> hold a simple functional relationship to depth, an analytic solution of the system of equations can be sought. I f t h i s i s not the case, as i n the B«C* i n l e t s where analytic d i s t r i b u t i o n s have been f i t t e d to the layer only i n the immediate v i c i n i t y of the surface (Cameron, 1951), a numerical method of solution can be applied (Fjeldstad, 1933). This essen-t i a l l y involves d i v i d i n g the i n l e t v e r t i c a l l y into homogeneous lamina© which are then f i t t e d to one another and the boundary conditions by a form of analytic continuation. This process has i n l a t e r years been j u s t i f i e d (Benton, 1956) by showing that the solution i n closed form for a f i n i t e number of laminae goes over i n the l i m i t of large numbers of laminae to the solution for a continuous d i s t r i b u t i o n of density. Through application of t h i s method the associated horizontal currents are determined with the v e r t i c a l ones to the order of an a r b i t r a r y constant mul-t i p l i e r , where i t i s assumed, for the derivation of the equation, that conditions are suitable for the application of the theorem of Continuity. Thus the procedure, i n applying the method to f i n d the possible i n t e r n a l modes i n some sp e c i f i c s i t u a t i o n , i s to deter-mine by t r i a l a value of the parameter A*a which f i t s the boun-dary condition when used for the numerical integration, of the equation* A f i r s t t r i a l parameter i s found by applying the f i r s t step of the W^ KUB* method, which i s equivalent to determining an average value for the numerical function , aad then finding the analytic solution to the system (determiaed by the d i f f e r e a -t i a l equation) i n which t h i s i s a constant* The f i r s t few d i f f e r -ences can be calculated by use of the Taylor series, and continued by alternate use of the d i f f e r e n t i a l equation aad difference formula r e l a t i n g the second difference of the fuactioa w toethe secoad derivative and i t s second difference at various steps of th© c a l c u l a t i o n . The numerical values of <f> must be tabulated for each depth of integration aad chaaged i n accordance with the i n t e r v a l size used* - 35 -The r e s u l t s of the integration give the possible modes of oseil&atioa of the system represented by the numerical func-t i o n <(> i n terms of t h e i r wave v e l o c i t y , and a representative p r o f i l e of v e r t i c a l and horizontal p a r t i c l e v e l o c i t y * There i s th e o r e t i c a l l y an i n f i n i t e sequence of these, but i t i s usual to take the f i r s t few modes only, and to try to f i t the arbit r a r y constant m u l t i p l i e r s of the solutions w to the actual d i s t r i b u -t i o n of amplitude and v e l o c i t y at the given depths for each hourly station, picking out those modes which appear to f i t best. To determine t h i s best f i t , i t i s possible to take advantage of the orthogonal properties of the solutions w and u to determine their c o e f f i c i e n t s when representing an arbitrary function i n terms of them, i n a manner i d e n t i c a l to that of Fourier analysis. This involves integrating over depth, which i s not possible i n th i s case because the Values of the amplitudes are not S u f f i -c i e n t l y well distributed i n depth. An e n t i r e l y equivalent pro-cedure i s to make a regression analysis of these solutions w against the values of the amplitude. Coefficients for each mode used are obtained for each hourly station* The harmonic analysis of these permits the representation of the v e r t i c a l o s c i l l a t i o n s i n terms of the inter n a l modes of motion, under the given hypo-the s i s . An alternative process i s to make a regression of the w's against the product of the harmonic amplitudes and the sine and cosine of the phase angle respectively. Although t h i s i s ce r t a i n l y speedier, i t does not have the advantage of deter-mining the other possible periods besides the one chosen f o r - 36 - .; the analysis. Therefore the former rather tedious process wa£ used,- • \ . \ • •' ' il( . At Knight 5 the s t a b i l i t y could only be determined from a single cast, and i s therefore by no means representative ^ of the average ( F i g , 19), Si m i l a r l y at Bute (Figs. 20, 21), ',[* although f i v e readings were taken during the two day station, ; \ they extended only to a depth of 90 feet* Lower depths were rep* . . ' • v~ resented by an average of six or seven stations taken i n the v i c i -n i t y during that month. The range among these was rather large, and no single deep cast was taken on the exact location of the o r i g i n a l station. To t h i s extent, only q u a l i t a t i v e r e s u l t s can-be hoped for* Fortunately th© values of s t a b i l i t y at lower depths are very small and thus not c r i t i c a l } also i n the determination of the c h a r a c t e r i s t i c s of the wave between the two stations Bute 4 and Bute Bf i t i s rather an advantage than otherwise to have a s t a b i l i t y representative of the surrounding waters. Thus i n spit© of the above reservations, the re s u l t s from Bute 4 may be expected to be more representative than those calculated from the single cast at Knight 5, The maximum values of the numerical function, are some hundred times those i n previous integrations where the value has been given (Fjeldstad, 1933), (Munk, 1941)* On the other hand the depths have been somewhat less and the deeper waters apparently less s t r a t i f i e d . In order to t e s t the hypothesis for th© observed sta-tions, an integration was made for both Knight 5 and But© 4 for the f i r s t four modes (Figs. 22 & 23). A regression of th© accom-- 37 -paaying values of w against the observed values of the v e r t i c a l amplitude i n the particularlease i s somewhat questionable, since the f i t can only be made i n the top one hundred meters, leaving open the p o s s i b i l i t y that a singularity of the determinant of the covariance matrice may occur. Xet probably because the major difference among the four, harmonics occurs i n t h i s upper layer* the d i f f e r e n t modes may very well be s u f f i c i e n t l y well separated by t h i s process, sinee noosingularity does occur* A possible alternative method i s to r e s t r i c t the numerical integration to the upper layer, f i t t i n g i t at the bottom to the boundary condi-tions for a homogeneous bottom layer (Fjeldstad, 1033). Knight 5a At the st a t i o n i n Knight Inlet hourly readings of current were accompanied by bathythermograph casts, which enabled both currents and amplitude® to be observed at t h i s station f o r a period of two days or four semidiurnal cycles* Values of were calculated from a single bottle cast near the end of the station ( F i g . 19). This may be compared to s t a b i l i t i e s calcu-lated for other stations along the length of the i n l e t (Fig* 6). I t i s as well to note that the value of <f> d i f f e r s from that of ^Xio * The former i s ten times greater. Results of numerical integration give the v e l o c i t i e s of propagation of the f i r s t four internal waves as 102, 55*6, 33*2, and 26*0 centimeters per second* (Table 2). Regression analysis of the four solutions against the observed v e r t i c a l o s c i l l a t i o n s (Table 4) shows that for a l l but the t h i r d wave, which i s small for a l l periods, the semidiurnal component dominates. Although the c o e f f i c i e n t for the — 38 -» f i r s t mode i n the diurnal component i s large enough to indicate a possible response of t h i s form, i t i s not s i g n i f i c a n t beside the semidiurnal c o e f f i c i e n t s * The representation of the v e r t i c a l o s c i l l a t i o n L i n terms of the f i r s t four int e r n a l waves of semi-diurnal period a i s ; L a 8.12 sin(at/l^t,+2,55)*^ ^ 5.09 s i n ( a t - k g X 4 0.54)w2 -Y 2.63 s i n C a t - k g X -I. 2»21.)«g. • £ 13.09 sin(at «. k ^ x + 6..i7)w^ where the values w'have been tabulated (Table 3). The c o e f f i c i e n t s of the f i r s t three waves hold much the same r e l a t i o n to one ano-ther as has previously been reported for waves i n Norwegian fi o r d s (Fjeldstad, 1952), but the fourth wave i s remarkably large and dominates the whole motion, i i n c e the f i r s t two waves have essen-t i a l l y opposite phase. The synthesis of t h i s expression when subjected to a harmonic analysis can be compared to the o r i g i n a l data (Table 5). A comparison of the magnitude of the v e r t i c a l o s c i l l a t i o n to that of the observed data shows only q u a l i t a t i v e agreement ( f i g . 25). The values i n the surface layer are of the order of eight meters too large, probably to force the deeper readings down to the cor-rect order of magnitude. The phase follows the observed values more c l o s e l y , but agree mueh better i n shape with the stations Knight 5b ( i ) and ( i i ) i , A synthesis i s made using the derivatives of the w ( i . e . , the u) and the same c o e f f i c i e n t s as for the previous one; t h i s i s compared to the components of the harmonic analysis at the same depths (Table 6). Although these c o e f f i c i e n t s are only eval-uated from values i n the top hundred meters, extrapolation to two and three hundred meters shows close agreement i n phase, although better agreement with the shapes of the v a r i a t i o n with phase exists with Knight 5b ( i ) and 5c ( F i g . 26). On the other hand the magnitude at none of the stations can be considered to be compatible with that expected from t h i s type of o s c i l l a t i o n . Bute 4 and 6 integration with use of the averaged value of the v e r t i c a l s t a b i l i t y ( F i g . 21) leads to values of the wave v e l o c i t y of 106, 70.7, 40.8, 31.2 centimeters per second(Table 2)* I t i s aot surprising i n view of the greater s t a b i l i t y i n the upper layers of the i n l e t that these Wave v e l o c i t i e s should have greater values than the corresponding ones i n Knight. Eegressioa analysis (Table 4) indicates that, although on the whole the semidiurnal component i s dominant, a very large c o e f f i c i e n t for the fourth mod© of the diurnal component .. cannot be dismissed without explanation. The harmonic analysis of the temperature variat i o n s at Knight 5a indicates that there i s a p a r t i c u l a r l y large diurnal component associated with the d a i l y heating aad cooliag of the surface. But the higher modes of o s c i l l a t i o n " have t h e i r greatest values i n the upper layer, and thus w i l l tend to f i t v ariations of t h i s layer i n a regression analysis* Thus the large values of th© fourth component may be attributed to a aon-eonservative v a r i a t i o n of the temperature at the surface, and can be hardly expected to represent the o s c i l l a t o r y motion of a wave. The representation of the v e r t i c a l o s c i l l a t i o n L by four int e r n a l waves of semidiurnal period a i s ; L -s 13.4 si a ( a t kjX 0.94)^ -h 9,2 sin(at - k g x 3*.96)w2 + 2.7 sin(at - k gx +- 1.03)w3 8,2 sin(at -k^x -f- 2«>98)wg where the c o e f f i c i e n t s w are tabulated (Table 3), The synthesis of t h i s (Table 7) compares rather unfavourably ( i n the shape of i t s d i s t r i b u t i o n of magnitude with depth) to the observed d i s t r i -bution, although the phase agrees r e l a t i v e l y well. When the f i r s t mode alone i s f i t t e d to the semidiurnal analysis* a remarkably good f i t i s made (Pig. 27), The wave v e l o c i t i e s are used to compute the phase for the distance of 21 kilometers between the two simultaneous stations i n Bute, and the synthesis i s performed using the corresponding phase changes (Table 8) and the regression c o e f f i c i e n t s calculated for Bute 4. Here the f i t i s no better i n shape and considerably too large i n value for the magnitude* Whea the siagle mode used at Bute 4 i s treated i a t h i s way, aad multiplied by a sealing factor of 0,4, there? i s a much better f i t . This* coupled with the obser-vation that the calculated wave v e l o c i t y (106 centimeters per sec-ond) of the f i r s t mode i s the same as the actual v e l o c i t y of 10'I centimeters per seooad for a 12-§- hour wave to t r a v e l the 21-| k i l o -meters betweea the two stations with a phase change of 170 degrees, seems to coafirm that the o s c i l l a t o r y motioa i a t h i s portioa of the i n l e t i s composed of the f i r s t and possibly the second i n t e r -a a l waves. However* the change of phase betweea the two stations i s somewhat less thaa that to be expected from the the o r e t i c a l wave, aad though t h i s i s withia the possible error of the calcu-1ation, i t may also be accounted for by a decrease i a depth betweea the two statioas, which could have the teadeacy to • 41 -decrease the v e l o c i t y . On the other hand, the average s t a b i l i t y tends to increase, which would tend to compensate for t h i s * I t i s int e r e s t i n g to note that the phase at Bute 4 lags the tid e by a phase angle s l i g h t l y less than the difference of phase between Bute 4 and 6, although the distance i s somewhat greater (34 kilometers as opposed to 21 kilometers)* Bute 6- i s taken as some 22 kilometers from the approximate head of the i n -l e t . Thus a progressive wave generated at the s i l l t ravels with decreasing v e l o c i t y through the length of the i n l e t , This might be due to the dependence of the wave v e l o c i t y on depth and st a -b i l i t y or to the e f f e c t of eddy v i s c o s i t y on the wave number. These questions among others w i l l be discussed for special d i s t r i -butions i n the next subsection, ( i i ) Approximate forms In the previous discussion i t was indicated that the rather large values of the regression c o e f f i c i e n t s for the fourth i n t e r n a l wave were probably due to surface effects not associated with actual internal o s c i l l a t i o n s , and that for t h i s reason they could well be discarded* As the t h i r d component i s i n a l l cases very small, i t seems not improbable that the f i r s t two inter n a l waves could give a good representation of the motion, p a r t i c u l a r l y as the f i r s t one alone i s shown to give an adequate representation^ i n Bute I n l e t . Thus approximations which f i t the f i r s t two waves well could probably be used with some success to represent the actual motion. The comparison of the f i r s t two modes with the r e s u l t for a three layer system (Fjeldstad, 1933) shows that a two layer - 42 -system i n which the lower layer i s homogeneous and the upper layer has a constant gradient might give a good approximation. I f the upper layer i s taken as one hundred meters and the gradient calcu-lated from the average s t a b i l i t y for t h i s layer, a v e l o c i t y of 181 centimeters per second for l u t e and 148 centimeters per second f o r Knight i s found for the f i r s t i n t e r n a l wave. These v e l o c i t i e s are too large. I f the system consists of only one layer, with a s t a -b i l i t y equal to the average over the whole depth, v e l o c i t i e s of 153 and 101 centimeters per second are calculated respectively for the two i n l e t s . Though the value for Bute i s s t i l l too large, that for Knight of 101 centimeters per second i s very close to the one calculated using numerical integration, A better representation for the s t a b i l i t y appears from the logarithmic plots ( F i g . 19 &; 2l) which appear to l i e about a straight l i n e as a rough mean. Thus an analytic representation of the s t a b i l i t y i s A ( x ) where % i s the slope of the mean and A the intercept of the l i n e with the y equal to one abscissa Table 9)• The solution of the wave equation, when t h i s analytic expression i s used for the s t a b i l i t y , i s a Bessel function from which the values of the wave v e l o c i t y can be found by f i t t i n g to the boundary conditions * When th i s i s done for a single layer system, r e s u l t s for the f i r s t two modes at Bute 4 are 266, and 55 centimeters per second, and at Knight 5, 96, and 51 centimeters per second, The values i n Knight seem i n f a i r agreement with those calculated by numerical integration, but the v e l o c i t y of the f i r s t wave i n Bute i s considerably too large, indicating again that t h i s s i t u a t i o n could be better represented by a multi-layer system* - 43 -the modulus of decay of a long wave for a system of two homogeneous layers (Rattray, 1954) i s found to depend on the eddy v i s c o s i t y , difference of density, and the depths of the two layers* the period, and the l a t i t u d e . I f t h i s approximation i s good for Bute I n l e t , the wave v e l o c i t y can also be represented i n terms of the difference of density between two homogeneous layers and t h e i r depths* I f the depth of the top layer i s as-sumed to be the depth of maximum amplitude f o r the f i r s t computed int e r n a l wave at t h i s station (fable 2), the difference of density i n the formula for the modulus of decay can be substituted for i n terms of the wove v e l o c i t y , and the depth of the two layers defined i n t h i s way. When the expression i s evaluated f o r eddy v i s c o s i t y i n terms of the decay of amplitude between stations Bute 4 and 6 for a twelve hour wave i n the l a t i t u d e of 51 degrees north, a value re s u l t s for the eddy v i s c o s i t y some 16 times larger than anticipated by Rattray and twice as large again as values observed i n the i n l e t s (Pickard & T r i t e s , 1957). The approximation which was designed for use on the continental shelf seems u n r e a l i s t i c i n a number of ways when applied to the i n l e t s * I t assumes a constant eddy v i s c o s i t y for a l l depths, whereas the eddy v i s c o s i t y i s shown to have a considerable dependency on depth. But perhaps more s i g n i f i c a n t , i t does not take into account the rather peculiar s t r a t i f i c a t i o n of the i n l e t s , which we have indicated may be rather better f i t t e d by a power function and solutions i n the form of Bessel functions. The importance of the approximate forms of analytic solution i s the two way r e l a t i o n they i n f e r between the oceaho-graphic conditions and the o s c i l l a t o r y motion studied. For as we have i l l u s t r a t e d , the oceanographic parameters may be used i n a numerical analysis to evaluate the modes of the internal waves present, and thus th e i r presence i n a survey may be compen-sated f o r . But i f an approximate form such as Rattray's can be shown to be an adequate representation of the s i t u a t i o n , i t may indeed be possible to in f e r the d i s t r i b u t i o n of the more d i f f i -cult-to-measure parameters from the mode of the o s c i l l a t o r y motion or i t s decay, as has been attempted q u a l i t a t i v e l y i n t h i s sectidn. To t h i s end perhaps the most f r u i t f u l endeavour may be as has already been suggested, the use of a power serie s repre-sentation for the s t a b i l i t y and a solution i n terms of Bessel functions* (See also Benton 1956). 2) Other Forms of O s c i l l a t o r y Motion ( i ) Solution for f i n i t e length and variable depth* In contrast to the theory of a progressive i n t e r n a l wave i n an i n f i n i t e channel of constant depth as treated i n the previous section, a basin of variable depth imposes boundary conditions on a system, with which the solution of the equation of motion as a single progressive wave i s incompatible. Further i t can be shown i n general that the only mode of o s c i l l a t i o n compatible with these conditions for such a basin i s a standing wave (Munk, 1941)* A method has been devised by Munk for finding numeri-c a l l y the modes of a given basin (a proeess used some f i f t e e n years previously for the surface tides (Defant, 1925). The method consists of assuming for a l l the stations a single t r i a l ~ 45 -parameter, which i s used to obtain a value of w and u at the bottom by the same numerical integration technique as used by Fjeldstad, assuming for t he boundary conditions at the surface, w equal to zero and the product of i t s derivative u by some scaling constant equal to one. This i s repeated for other modes. The boundary conditions at the bottom, that i s of no motion normal to the boundary, i s applied to form a set of l i n e a r equations i n these calculated values of w and u summed i n accordance with the various modes* The correct choice of parameter ( i . e . period ef motion) then corresponds to a mini-mum i n the value of the determinant of the c o e f f i c i e n t s i n the corresponding set of equations. In t h i s way free long, period motions were interpreted to be a resonant response of the Gulf of C a l i f o r n i a , Considering the very close period of a l l observed o s c i l l a t i o n s to that of the dominant t i d a l component, i t i s un l i k e l y that they are free standing waves determined only by the boundaries. However the o s c i l l a t i o n s observed at Bute 4 and 6 d i f f e r i n phase by almost 180 degrees. The occurence of the node of an int e r n a l c o o s c i l l a t i n g tide between the two s t a -tions could furnish an alternative explanation to the previous one of progressive waves* In t h i s case the decrease of ampli-tude between the two stations would indicate that Bute 6 was closer to the v i c i n i t y of the node. Taking the predicted velo-c i t y of 106 centimeters per second as the v e l o c i t y of an int e r n a l wave i n the i n l e t , the f i r s t node of a c o o s c i l l a t i o n i a response to the semidiurnal component of the tide would be some 12 k i l o -meters from the head of the i n l e t , a position 1 1 kilometers further from Bute 4 than i t i s from Bute 6 . The next node would occur some 3 6 kilometers from the head of the i n l e t , a p o s i t i o n approximately f i v e kilometers from Bute 4 , and 1 7 from Bute 6 (Table 9 ) . The v e l o c i t y at. Bute 4 which was used for t h i s calcu-l a t i o n i s probably the greatest v e l o c i t y which such a wave would a t t a i n as the depth decreases from t h i s p osition to the head ( P i g o 5 ) j and the increase i n s t a b i l i t y would probably not be s u f f i c i e n t to compensate for t h i s ; However, any decrease i n v e l o c i t y would s h i f t the position of the node closer to Bute 6 , which would better explain the smaller amplitude at t h i s s t ation. Characteristic of a standing c o o s c i l l a t i o n i i the uniformity of phase between nodes; thus, i f th i s were a direc t response to th* tide s , the phase between any two nodes would either be the same as the tides or 1 3 0 degrees out of phase, or some other constant values, i n contrast to a progressive wave which can be expected to have a continuously varying phase throughout the length of the i n l e t . Thus i n the case of Bute 4 and 6 constant phase Would be expected between the possible location of the nodes at 1 2 , 3 6 , and 6 0 kilometers from the head. Conditions at other times might be expected to vary somewhat i n the pos i t i o n of the nodes and even the phase r e l a t i o n to the tides, due to the dependence of the wave v e l o c i t y on the oceanographic structure, and a possible change i n the driving mechanism. However the mean phase of the semidiurnal component of a harmonic analysis of f i v e stations between the positions of Bute 4 and Bute 6 (Table 1 0 . ) do not show any consistant r e l a t i o n to the ti d e , In f a c t , the general - 47 -gradation of amplitude and phase seems to support the hypothesis of a progressive wave. ¥et, as the stations are spread over a period of two years, t h i s i s by no means conclusive evidence* Only an adequate oceanographic survey, including anchor stations at int e r v a l s throughout the i n l e t or several swift series of bathythermograms over the length of the i n l e t , s u f f i c i e n t l y spacedj would distinguish between the two cases, ( i i ) The e f f e c t of a f i n i t e b a r r i e r at the mouth of an i n l e t . The o s c i l l a t o r y motion of the deeper waters of the i n l e t i s most plausibly associated with the action of the tide over the s i l l , a structure characterizing the mouth of most B.G. i n l e t s . The assumption i s i m p l i c i t l y made i n the numerical integrations and other approximations to the form of the motion, that the depth i s great with respect to the amplitude, and that the approximation of a sinusoidal wave i s j u s t i f i e d , A recent study (Long, 1953, 1954, 1955) shows fro® theory and mode! experiment that sinusoidal waves which r e s u l t from an i n f i n i -tesimal b a r r i e r are not necessarily present for a f i n i t e ampli-tude b a r r i e r . For a two layer system, a hydraulic jump i s l i k e l y to occur i n the surface separating the two f l u i d s , except for a small bar r i e r and small approach v e l o c i t i e s (Fig, 28). In the flow of a continuously s t r a t i f i e d f l u i d large eddies and turbulence are common i n the lee of the b a r r i e r , while j e t s may form upstream i n the case of considerable blocking ahead of the ba r r i e r ( F i g . 29). The method i s to perform the analysis f o r an i n f i n i -tesimal object and then to extend t h i s c a l c u l a t i o n to the f i n i t e «a 48 •* amplitude case by deforming the bottom streamline. The calcu-l a t i o n s are then applied to a ba r r i e r of the desired height and length, but of a shape defined i n t h i s way. The flow i s charac-ter i z e d by the value of the Froude number (which includes the density gradient), the r a t i o of barrier height to t o t a l depth, and for'the two f l u i d ease tthe r a t i o of the i n i t i a l depth of the lower f l u i d to the t o t a l depth. For the case of continuous gradient of density a constant gradient i s used for the Froude number, and also the ha l f length and the i n f i n i t e s i m a l height are s p e c i f i e d , In i t s progress up an i n l e t the tide achieves a rate of progress almost twice that of the theoretical long wave for the same depth. Even when the f i n i t e height of the wave i s taken into account, t h i s i s not compatible with an i n e r t i a l wave, but must progress as some sort of shock wave similar to an hydraulic jump. At the s i l l i t encounters a barr i e r which may block as much as 5/6 of the channel, as i s the case i n i t s approach to Knight I n l e t , or 1/3 to 1/5, i n the case of Bute, I f the main stream moves i n the surface layer, i t i s possible that the s i l l at Bute might appear as an i n f i n i t e s i m a l b a r r i e r . In Knight I n l e t t h i s i s most improbable. I t i s only possible at present to make qu a l i t a t i v e inferences from Long's work, as two conditions hamper i t s application. In the f i r s t place the the o r e t i c a l work makes the assumption of steady state. Secondly, i t i s assumed that the depth i n front of the ba r r i e r i s the same as that behind. The passage of the tide over the s i l l w i l l occur with a continuous increase i n v e l o c i t y which w i l l i n time reach a maximum and f i n a l l y reverse* During t h i s t r a n s i t i o n the flow w i l l probably pass through several of the steady state regimes of motion, and therefore w i l l be ^indefinable i n terms of the c r i t e r i a of Long's theory. Perhaps the best c r i t e r i o n that can be picked i s the value of the Froude number corresponding to< the maximum v e l o c i t y at the apex of the s i l l , as t h i s may give some idea of the regime to whieh '>the motion may develop, low-ever, as thi s i s a transient process, thereeis no knowing Whether the structure of the motions i s the same as the steady state, or i f so, how long i t takes to develop* Because of the o s c i l l a t o r y nature of the tides over the s i l l , the structure which defines the Froude number of the flow w i l l dppend on the flow during the previous h a l f cycle. An asymmetry; of the s i l l w i l l , quite apart from any difference of mixing on either side of th© b a r r i e r , determine d i f f e r e n t forms of motion for the ebb and flood by giving a d i f f e r e n t r a t i o of barri e r height to approach depth for these two phases of th© tide* Thus i t i s quite l i k e l y that the flood and ebb tide behave quite d i f f e r e n t l y at the s i l l . The p r o f i l e s of properties of Knight Inlet show almost complete blocking of the flow below the l e v e l of the s i l l on the flood. Model experiments show that t h i s s i t u a t i o n may r e s u l t i n the r i s i n g of the l e v e l behind the s i l l * and eventual s p i l l i n g over of the f l u i d i f the Froude num-ber of approach i s s u f f i c i e n t l y high (Fig. 28), Thus, abnormally high tides associated with storms or spring t i d e s , may lead to intrusions into the deeper waters of the i n l e t . P r o f i l e s of - 50 -properties indicate a form of hydraulic jump i n tlie lee of the barri e r i n Knight and possibly also i n Bute (Pig* 5^ ) . Experiments with large barriers and a continuous density gradient (Fig. 2S) show that thh presence of such hydraulic jumps i s associated with tumbulence i n large eddies i n the lee of the s i l l . The apparent mean flow i n at the bottom; and out at mid depths reported i n the harmonic analysis of the currents at Knight 5 (Table 5) could possibly be explained by the difference of the mean flow of flood and ebb i n the presence of the s i l l . Model experiments show ( F i g . 29) that for a high barrier a j e t tends to form upstream and above the l e v e l of the b a r r i e r , With stag-nation points above and below i t . The l a t t e r may develop into negative v e l o c i t i e s with respect to the mean flow, A mean deep flow i n on the flood and a flow out at mid dppth on the ebb i s quite compatible with the dynamics of such a system, and may indeed give r i s e to the extraordinary v a r i a t i o n i n the magnitude of the horizontal current fluctuations with depth. Gonelusions I t i s suggested i n standard t r e a t i s e (Sverdrup, 1942) that t i d a l motion i s confined to regions of the i n l e t above the l e -v e l of the s i l l . Studies of the fluctuations i n the horizontal currents and v e r t i c a l o s c i l l a t i o n s of isotherms indicate that motion of t i d a l period takes place at much greater depths. The manner i n which t h i s takes place i s associated with the hydraulic flow over the s i l l . In the presence of a low s i l l the motion appears to be of an i n e r t i a l type, changing phase up i n l e t to lag. the tide by as much as a whole period* This motion can be a t t r i -* 51 — buted to e i t h e r a progressive i n t e r n a l wave or a standing wave. I f the motion were defined by the bottom and l a t e r a l boundaries, the standing wave would be the only one p o s s i b l e . At a p o s i t i o n c l o s e to a s i l l of very great h e i g h t , the motion at depth does not agree with t h a t of an i n e r t i a l o s c i l l a t i o n , but appears t o be defined by some form of h y d r a u l i c jump generated at the s i l l . This i s not incompatible with s t a b i l i t y c o n d i t i o n s as defined by i s o p l e t h s of den s i t y drawn from s t a t i o n s taken i n the same and other years. P i t t i n g of a n a l y t i c s o l u t i o n s of the equations of i n e r -t i a l motion to the i n l e t s show th a t the waters behave toward such a motion i n a manner s i m i l a r to a two l a y e r system, the lower of the two being approximately homogeneous. A power f u n c t i o n approa-ching a homogeneous s t a t e at the bottom i s shown to be a b e t t e r approximation to the a c t u a l d i s t r i b u t i o n of s t a b i l i t y . Motion of S>hh deeper water associated with a progressive i n t e r n a l wave i n such a homogeneous l a y e r should be associated w i t h c u r r e n t s , and subsequent mixing, throughout the length and depth of the l a y e r . Standing waves w i l l on the other hand be associated w i t h p o i n t s of stagnation near the p r i n c i p a l loops. D i s t r i b u t i o n of p r o p e r t i e s i n Bute seems to suggest the f i r s t p o s s i b i l i t y , s i n c e the deeper l a y e r i s w e l l defined from below the bottom of the minimum, and the water appears to show good communication w i t h t h a t outside the s i l l . The motion i n the lee of a steady s t a t e h y d r a u l i c jump or the equivalent to i t i n a continuously s t r a t i f i e d f l u i d i s not s i n u s o i d a l , as the high l o s s of energy tends q u i c k l y to d - 52 -damp the motion with i t s progress horizontally (Long, 1958). Thus such an impulse could not be propagated f a r from i t s source. However due to the periodic nature of the tide over the s i l l * the formation and disappearance of the jump on each cycle of the tide could generate an i n e r t i a l wave that . would t r a v e l with a suitable change of phase throughout the entire length of the i n l e t . Thus i n spite of the intense mixing and turbulence i n the region of the s i l l , which extends at least as far as Knight 5 (seven miles), the c o n d i t i o n ^ further from the s i l l can be expected to be f a i r l y stable. The high s i l l i n Knight blocks, to a much greater extent than the one i n Bute, the entrance of properties from outside, leading to a less well defined, more permanent low temperature wedge, which under the action of the in t e r n a l o s c i l l a t i o n s tends to spread srell into the bottom water. Although the analytic defintion of the motion of a s t r a t i f i e d f l u i d moving p e r i o d i c a l l y over an asymmetric barrier i s well nigh intractable, numerical integration and model studies i n a ti d e basin could show results for a pa r t i c u l a r case which might be of great value i n the interpretation of motion associated with and a r i s i n g from the s i l l . P r o f i l e s of the oceanogrpphic properties i n Knight Inlet suggest that a complete blocking of f l u i d outside and below the l e v e l of the s i l l i s not always the case. The f l u i d appears to s p i l l over on occasions to follow the main flow deep behind the s i l l * Model studies f o r the case of steady state indicate that for a con t i -nuously s t r a t i f i e d f l u i d , such a motion i s accompanied by - 53 -boundary layer separation and turbulence i n the lee of the bar-r i e r . Study of turbulence i n such a region, as well as checking t h i s conjecture, may also reveal what the e f f e c t of a sudden: expansion of aperture would have on the scale of motion i n a s t r a t i f i e d f l u i . d r a consideration of some importance to the study of eddy c o e f f i c i e n t s i n an i n l e t . High t u r b i d i t y might be expected i n association with a bottom swept by such motion, also a s e t t l i n g as sediment i n regions of stagnation. The high degree of mixing i n the region of the s i l l makes i t d i f f i c u l t to determine the motion by following any p a r t i c u l a r c h a r a c t e r i s t i c . Thus a better method i n t h i s region i s a series of current stations close to and over the s i l l of the i n l e t . A single station close to the inside of the s i l l with s u f f i c i e n t l y close depths of readings would be most re-vealing. A study of the v a r i a t i o n of grain size of bottom samples correlated with t u r b i d i t y might also help determine the more common forms of motion. Should i t be possible to determine the motion of the ti d e over the s i l l i n parameters similar to those used by Long for the steady state, and i t s association with the internal o s c i l l a t i o n s of t i d a l period, i t would be possible i n terms of the topography of an i n l e t , i t s runoff, the weather, and the waters with which i t i s i n contact outside the s i l l , to prediet the oceanographie structure of any p a r t i c u l a r i n l e t at any s p e c i f i c time of year. ~ 54 -Observations of the fluctuations of oceanographic properties over several t i d a l cycles indicate t h e i r large ex-tent and the inadequacy of taking i s o l a t e d stations to repre-sent the seasonal properties of any location i n the i n l e t s . For only with continuous surveys through the whole length of an i n l e t , confined i f possible to a single part of the t i d a l cycle, i s i t possible to establish the continuity of readings necessary to smooth out tthe effect of t i d a l v a r i a t i o n s , which are on occasions-greater i n a few hours than the seasonal ones i n a whole year* 55 -Bibliography Arons, A.B. and H. Stommel (1951). A mixing length theory of t i d a l f lushing. Trans. Amer, Geophysical Union, 32(3): 419 - 421.. Benton, G.S. (1956). A general solu t i o n for the c e l e r i t y of long gr a v i t a t i o n a l waves i n a s t r a t i f i e d f l u i d . F l u i d Models i n Geophysics. U.S. Government Pr i n t i n g O f f i c e , Washington 25, B.C. Cameron, W.M. (1951), On the dynamics of i n l e t c i r c u l a t i o n s . Doctoral Dissertation, Scripps I n s t i t u t i o n of Oceano-graphy, University of C a l i f o r n i a , Los Angeles, Cal* Defant, A. (1925). Gezeitenprobleme des Meeres i n landnahe. Probleme der kosmischen physik, VI* Hamburg, pp. 80. Dawson, W.B, (1920), The tides and t i d a l streams with i l l u s t r a t i v e examples from> Canadian waters. Queen*s Pr i n t e r , Ottawa. Fjeldstad, J*E. (1933), Interne wellen, Geofysiske Publikasjoner, 10(6), Fjeldstad, J.E. (1952)* Observations of internal waves. Gravity Waves. Hational Bureau of Standards. Circular 521. Ketchum, B.H, (1951)* The exchange of fresh and s a l t water i n t i d a l estuaries^ J . Mar, Res, 10: 18-38. Keulegan, G.H.(1949)* I n t e r f a c i a l i n s t a b i l i t y and mixing i n a s t r a t i f i e d flow. J . Tes. Natl . Bur. Stand, 43, PR 2040 j 487 - 500, Long, R.R. (1953), Some aspects of the flow of s t r a t i f i e d f l u i d s . I. (A the o r e t i c a l investigation). T e l l u s , 5(1)s 42-58. (1954) , Some aspects of the flow of s t r a t i f i e d f l u i d s * I I . (Experiments with a two-fluid system). Tellus, 6(2) : 87 - 115. (1955) , Some aspects of the flow of s t r a t i f i e d f l u i d s * III . (Continuous density gradients). Tel l u s , 7(3) : 341-357. Munk, W.H. (1941), Internal waves i n the Gulf of C a l i f o r n i a . Jour. Mar, Res, 4 : 81 - 91. Pickard, G.L. (1955), B r i t i s h Columbia Inlets* Trans. Amer. Geo-physical Union 36(5) s 897 - 901. (1956). Physical features of B r i t i s h Columbia i n l e t s * Trans. Roy, Soc* Canada, 50, Ser. 3: 47-58. ~ 56' -Pickard, G.L. aad R.W. T r i t e s (1957). Fresh water transport determination from the heat budget with applications to B r i t i s h Columbia i n l e t s . Jour. Fish.Res, Bd, Canada, 14(4) s 605 - 616. Pickard, G.L. arid K. Rogers (1959). Current measurements i n Knight I n l e t , B r i t i s h Columbia. J.Fish.Res.Bd.Canada, 16 (5) j 635 - 678. Pritchard, D.W. (1952), Estuarine hydrogrpphy. Advances i n Geophysics. Vol I, pp. 243-280, Academic Press Inc., New York, N.X. Rattray, M. (1954). Propagation and di s s i p a t i o n of long i n t e r n a l waves. Tech, Rep. 27. University of Washington Depart-ment of Oceanography, Seattle, Washington. Sverdrup, H.U., H.W, Johnson, R.H. Fleming (1942). The Oceans, th e i r physics, chemistry and general biology. Prentice H a l l , Inc., Englewood C l i f f s , N.J. Tabata, S. and Pickard, G.L, (1957), The physical oceanography of Bute I n l e t , B r i t i s h Columbia. J . Fish. Res, Bd. Canada, 14(4) s 487 - 520. Thompson, T.G. and K.T. Barkey (1938), Observations on f j o r d -waters. Trans. Amer, Geophys* Union, 19 : 254 - 260. T r i t e s , R.W. (1955). A study of theooceanographic structure i n B r i t i s h Columbia i n l e t s and some of the determining fa c t o r s . Doctoral Dissertation, University of B r i t i s h Columbia, Vancouver* T u l l y , J.P. (1949), Oceanography and prediction of pulp m i l l p o l l u t i o n i n Alberni I n l e t . Fish, Res. Bd. Canada, B u l l . No. 88, pp. 169. Table 1 Comparison of amplitudes calculated from continuity theorem with deflections of isotherms. Depth Amplitude from Deflections (meters) temperature analysis isotherms magnitude phase magnitude (meters) (radians) (meters) 9,1 1,2 (4,6) 1.2 13,7 4.6 (4.5) 3.4 18.3 7.3 (4.2) 30.4 48.8 (3.9) 9.1 45.7 30,4 (0.7) 13 i 4 76.2 38.1 (2.3) 91,5 38 fl (1.7) 16.5 106 *7F 35,6 (2.1) 122.0 25.9 (4.3) 19,8 152.4 30,4 (4.9) Table 2 phase (radians) (4.2) (3.7) (2.5) (2,1) (1.2) (0,8) Wave v e l o c i t i e s and depths of maximum amplitude. Station Bute 6 (observed) Bute 4 (observed5 (calculated) Knight 5a (observed) (calculated) Mode I II III IV Max I Max II Wave Velocity (cm/sec) 101 101 106 71 41 31 I II III IV 102 56 33 26 Depth of Maximum Amplitude (meters) 75 80 120 140 280 320 70 130 65 100 ISO 140 Table 3 (See Table of Contents) Knight 5a Depth *1 W2 W 3 W4 (meters) 4*2 0.405 -0.420 0,846 -0*120 5 .8 0.569 —0.662 0*857 -0.103 10.1 0.953 -0.908 0*360 0.012 14.9 1.324 -0*971 -0.440 0.S35 32.9 2.058 0.269 -0.736 -0.238 38.1 2.181 0,648 -0.623 -0,345 48.6. 2.427 1.344 -0.1S6 *-0.359 51*7 2.473 1.501 -0.020 -0.299 98.0 2.383 2.088 -1.334 0.948 125.9 2.163 1.991 1*512 1.230 148.4 1.960 1*834 1.471 1.254 u l u 2 n 3 u4 50 -1.462 *-5,107 -5*291 -2.043 100 0.711 0.108 -1.160 -1.600 200 0.954': .849 0.553 0.377 300 0.993 .977 0.929 0,398 lu t e 4 Depth (metersf W2 *3 *4 8.4 1.271 -1.473 -0.131 0.166 10.4 1*465 -1.414 -0.419 0.258 13.9 1.747 -1.244 -0.739 0.325 21.2 2*298 -0.748 -1*464 -0.318 45.2 3.574 1.068 -1.505 -0.286 80.6 4*188 2.157 -0.885 -0.576 67.5 4.408 2.569 -0.413 -0*619 76.6i 4.651 3.077 -0*002 -0*602 80.1 4.734 3.253 C0*204 -0.585 83.8 4.841 3*414 0.418 -0.528 99.1 5.017 3.940 1.208 -0*245 Table 4 Regression Coefficients (I) Mode Diurnal (2) Semidiurnal amp phase amp phase (radians) (radians) 8.33 Hour amp phase 6.25 Hour amp phase (radians) (radians) Unight 5a w" Bute 4 16.9 8.1 9.1 3.4 (0.18) (3.82) (0.14) (3.42) (2.55) (0.54) 13.0 (4*15) 44.0 13.4 (1.52) 30.1 *4 5.2 . (2.76) 8.8 35.0 (2.12) 26*8 28.3 16.7 8.3 (2.21 42.9 (6.17 (0.94) *96) 4.7. (2.80) 6*3 (5.53) 2.4 (0.33) 7.2 (2.65) 3.0 (3.37) 5.7 (5.96) 10*9 (4.88) 7.6 (2.83) (3 1.03) 2*99) 8.5 9*9 1.4 22 • 2 ;o.55j ,3.68J [4.82 3.57, 1) The c o e f f i c i e n t s should only be compared with those of the same: station as the scaling of the solution ws i s di f f e r e n t for the two stations, when f i t t e d to the c o e f f i c i e n t s the readings are i n feet. 2) This i s the lunar semidiurnal component and has a period of 25 hours. Table 5 Comparison of computed and observed values of amplitude at Knight 5a. Depth (meters) 4.2 5.7 10.0 14*6 32.5 37,7 48,0 51.1 96.8 124.3 146.6 Observed Amplitude Phase (meters) (radians) Calculated J Amplitude Phase / (meters) (radians)!> 1.2 1.6 3.4 4,6 e.2 i l . o 13.5 14.0 16.6 19*9 20.8 :4.1s; ,3.78' ;3*70 3.29 2.50J 2*47) 2.11) (2,13) "jl.19) 0.80) ,0*49) 6*5 8.3 10*5 11.7 15.5 17.0 19.9 20*4 19,1 18,1 16.6 |2*84 2*84 2.90) 2*93) ;2.57) 2*49) ,2.32) (2.28) ,1.77) ,1.52) ,1.51) Tide 1.3 (3*25) Table: # Synthesis of currents Knight 6 Depth (M) 5 0 I O C ) 2 0 0 0 3 0 0 Observed Amp Phase (cm/sec) (rad) 2 . 3 6 . 0 1 0 * 5 5.8 ( 3 * 3 0 ) ( 2 * 7 6 ) ( 1 . 8 8 ) ( 1 . 7 3 ) Table- 7 Synthesis Bute 4 Depth jjep (M) 8 , 4 1 0 . 4 1 3 . 9 2 1 . 2 4 5 . 2 6 0 . 6 6 7 * 5 7 6 . 6 8 0 * 2 8 3 . 8 Tide Observed Amp Phase (rad) p (Ml 1 . 1 1*4 2 . 0 3 . 1 6 * 6 6 . 7 6 . 9 7 . 2 7 . 5 7 . 7 ( 1 . 5 6 ) ( 1 . 7 1 ) ( 1 . 5 9 ( 1 . 3 6 . ( 0 . 9 9 ( 0 . 9 3 ] ( 0 * 9 1 f ( 0 , 8 4 ; ( 0 * 7 4 (Q . 83 J 2 . 2 ( 2 , 8 3 ) Table 8 Synthesis Bute 6 Depth i»e  Am 7 . 3 9 . 4 1 8 . 2 3 1 . 6 4 7 . 9 5 7 . 8 6 3 . 4 7 2 . 9 7 7 1 1 8 0 . 7 8 9 . 4 1 0 9 . 5 Tide Observed Phase (rad) ( 4 . 4 3 ) ( 4 * 4 2 ] ( 4 * 1 9 ( 4 * 4 2 ( 3 . 7 6 ( 4 . 2 1 ( 4 . 2 8 ( 3 . 8 7 ] ( 3 . 9 2 ( 4 . 0 5 ( 3 . 8 3 ] ( 3 . 6 9 ) 2 , 2 ( 2 . 8 3 ) 1 . 0 1 . 2 1 . 5 1 . 9 1 . 9 2 . 6 2 . 9 3 . 8 3 . 3 3 .0» 3 . 3 2 . 8 Predicted Amp Phase (cm/sec) (rad) 2 0 * 4 8 . 7 8 . 0 8 * 0 Calculated Amp Phase (M) (rad) ( 0 . 9 1 ) 6) 4 . 8 5 . 0 5 . 4 5 . 0 7 , 7 5 , 6 5 . 9 0 . 6 5 . 6 5 . 6 ( 0 . 8 6 , ; ( 0 , 8 7 ; 1 0 , 8 8 ( 0 . 8 3 ] ( 0 . 8 3 ) ( 0 . 8 4 ) ( 0 . 8 6 ) ( 0 . 8 6 ) ( 0 * 8 7 ) Amp (M) 3 * 6 4 . 0 4 . 8 5 * 5 8 . 1 Predicted Phase (rad) 5 . 4 6 ) 5 . 4 3 ) 5 . 3 9 ) 8 * 1 8 . 7 9 * 0 9 * 5 1 0 * 9 Mean (cm/sec) - 1 . 9 1 . 8 0 * 6 - 0 . 8 Table 9 Values for Constants of the approximate form f o r the s t a b i l i t y (from figures 19 & 21) - N Stabilxty *» Ax • Station Depth A N Depth of surface (meters) homogeneous layer (meters) Bute 4 700 10,000 2.47 4 Bute 6 500 85,000 1*89 7 Knight 5a 350 2,000 1.89 4 • -£able 10 Depth of maximum amplitude and r e l a t i o n of average phase to that of the tide for anchor stations which have been given har-monic analysis. Series Date Distance Maximum amplitude Phase from s i l l (Km) from (Km) head Magnitude • (M); depth (M) to tide (degrees Bute 4 3/7/53 34 43 7 . j f 90 (283) Series 2 8/6/52 37 40 5.0 88 (129) Series 3 13/8/52 39 38 5.8 84 ($07) Series 1 31/5/52 46 31 2.7 80 (182) Bute 6 3/6/53 56 21 3.9 72 (72) S T A T I O N L O C A T I O N S ANNUAL TEMPERATURE STRUCTURE C O Fig. 2o KNIGHT J U N E 1 9 5 1 ' BUTE ANNUAL TEMPERATURE STRUCTURE (CONT.M 'C) F i g . 2b KNIGH1 Jl'LV I9S6 BUTE JUNE 30,I9»6 FEBRUARY 1958 MARCH 1958 MAY 1958 JUNE 1988 S E P T E M B E R 1957 ANNUAL DENSITY STRUCTURE (a.) FJO.S t KNIGHT G R A D I E N T O F D E N S I T Y ( ~ T ) STATION | 2 3 4 5 6 7 6 t 10 1 0 6 0 1 0 0 -k. o> ••— a* E 2 0 0 -X »-a. U J Q 3 0 0 -- 5 - 1 I / 0 0 0 5< , 1 0 0 1 , 1 30 1000 I Ab 0 20 • i Be / • ove 60m low 6 0 m 1 7 [ { T,S DIAGRAM, KNIGHT INLET K N I G H T - 5 o TIME SERIES Of PLOTS K N h G H T - 5 a F i 9 , , ° A V E R A G E T E M P E R A T U R E 7 8 50 CO a> +- 100 F I RST 24 HOURS X a. LU a 1501 Fl RST 36 HOURS 2 0 0 250J-TIME SERIES KNIGHT—5d T I D E ( A L E R T B A Y . OBSERVED) K N I G H T - 5 Q Period (minutes) B U T E - 4 I I I I L. 1 1 1 ' 1 1 T I D E S . H A R M O N I C A N A L Y S I S P e r i o d 4 -I H o u r Fig.13 A l e r t Bay ( S P E C T R A 1 6 J u l y , 19 5 6 R e a d i n g s 4 8 L S T A . K N I G H T - 5 a T e m p e r a t u r e O s c i l l c r t i o r 25 R e a d i n g s (I h r. apart) - M ( )" °C 4 . 6 m (9..°C) 9 . I (7.6) 13 .7 (7.,) 111., 3 0 . 5 ( 6 . 6 ) L 45.7 (6.6) L 6 1 0 76.2 (6.6)1 SLA (6.5) IQ6.7 (6.4) I Z\.9 (6. 3)L Iff 2. 4 16T5)L. 18 2.9 fe .7) c 289 I (6.7j L J L 1 1 1 1 1 . 1 1 j 1  1 1 1 I 1 1 1 1 [ M HiO.O) c [6 91 *™ S T A , K N I GH T-5a Fig.l4 V e r t i c a l 0 t c i l l a t ions 50 Readings (I hr. apart) -L 7 8 (72) L 32 5 16.7) 37 6 (MIN.) L 48 0 (MIN.) L 51.1 (6.7) L 96.8 12 4.3 (MIN.) 146.5 m (6 .7* C) L _ _ L J L J L I 1 l F i g . 15 V E R T I C A L O S C I L L A T I O N S K n i g h t - 5 b ( i ) 28 READINGS ( I HR. A P A R T ") K n i gh t - 5 b (i i) ( lO.O) ( 8 . 0 ) 1 17.5) 18 .0 ( 7 . 0 ) , 3 4 . 3 L ( 6 . 5 ) 7 9 . 2 ' ( M I N . ) 115.0 ( 6 . 5 ) , 136.7 1 1 1 1 1 1 1 . 1 J L ( 1 0 . 0 ) 9 . 6 10 ( 8 . 0 ) 16.4 ( 7 . 5 ) 2 0 . 2 * ~ 0 m e t e r s ( 7 . 0 ) 3 4 . 9 ( ) - o c e n t i g r a d e l 6 5 ) - M e t e r s 8 1 0 (MIN.) _LLCL7_ ( 6 . 5 ) 139 .8 J I J L. T I DE u TIOE u V E R T I C A L O S C I L L A T I O N S B U T E - 4 B U T E - 6 I O O R e a d i n g s (1/2 hr . a p a r t ) - 0 m e t e r s (MIN!) 8 0 . 2 l M B3.R1 (7.8) 99-' { ) - o c - M T I D E in nl' ' ' • . . 1 -a -^l I 1 1 . . 1 IO 4. 1 1 1 . 1 1 (lO.O) 1 V l 3 9 l 1 1 1 . 1 1 '94) 21.21 1 1 , 1 , 1 M 1 45.2l 1 1 i 1 . 1 (8.3) 1 6 0 Gl 1 1 1 . 1 (7 8) | 6 7.51 1 1 , 1 , ! Hi 1 1 1 I (8.9) 47 9 M 7 2 8 (MIN; 77.1 8 9 . 3 (9.4) J 0 9 . 5 L PA I 80 .7L L F i g . 16 3 0 -m e t e r s (9.4 9.4 1 1 1 1 1 (I0 0)l 1 IB ? l 1 1 • 1 1 (94) 31 B 1 , , , I 1 (B .3) 5 7.8 1 1 1 (7.8) 63 . 4 1 1 1 J_l J L ( — U PSTREAM) _e KNIGHT 5a Depth L 3 0 CM./SEC. 50 m. I i_ C URRENTS 5b(i) - 9 6 3 -0 CM../SEC - L 100 m. 200 m. L - i L 1 I 300 m. I I i ( + DOWNSTREAM) 5b(H) 9 6 3 o c my SEC. J _ i 5c r 6 - 3 0 CM. /SEC. I I J i L J_L HARMONIC ANALYSIS (DEPENDENCE ON DEPTH) Fla. 18 sa Ampl i tude T . z s KNIGHT-5 P h a s e it i I I. Il T « 8 I I I 1 II J I .1 . » _ _ o _ _ _ —o- o S C A L E S ; i . » 1 . 1 -1 0 l 28 1 60 .1 • 0 — •e . A . o o o 00 0 o o » oo 0 o T « 6 » » «• i -L o o oo ° B . ° »b i T . 2 S • 1 1 1 • 1 — — ' a a T » I 2 . S . 1 1 o o o > • • 1 • « ° o * • T « 6 . 1 . 1 1 1 S B o o o » • » ' ' T . 2 8 , . l l 1 e o o o o o T • I2.S ... 1 e e o . . . • 1 - 3 2 • _ _ .» o o T « 6 1 . 1 1 1 T(«if « ratur« B t I I 1 . . . . I I . . * U • T«lt.» 0 0 0 O 0 O 1 0 15 - i — I I *- 2 0 I ' 1 r-100 B U T E - 4 L_ 2 0 0 E LU Q 3 0 0 L_ 4 0 0 D E N S I T Y K ) 5 0 0 6 0 0 25 5 T 10 L _ 2 0 0 Q . U J Q -i 1 1 r 100 3 0 0 L_ 4 0 0 \— 5 0 0 BUTE 6 0 0 i' A M P L I T U D E R E G R E S S I O N A N A L Y S I S B U T E - 4 w, w, T * 2 5 T • 12.5 T « 8 I T » 6 I I — .5 ' — 0 T « 2 5 T - 1 2 . 5 T « 8 L T = 2 5 w s w, T-I2.5L 1 1 . 1 1 1 1 T « 8 T « 6 K N I G H T - 5a T « 2 5 J L T = I 2 . 5 T » 8 T " 6 , | I I T « 6 P H A S E F i g . 2 4 r— 3 6 0 ' - 1 8 0 ' 1 — 0 ° i K N I G H T - 5 S Y N T H E S I S FOR I N T E R N A L W A V E S A M P L I T U D E ( m e t e r s ) T E M P E R A T U R E ( ° C ) P H A S E (degrees) D E P T H ( m e t e r s ) l 100 2 0 0 3001 V E L O C I T Y ( c m . / s e c . ) — r . 2 - T . 3 4 5 6 X o \ \ \ \ C U R R E N T S - K N I G H T - 5 S Y N T H E S I S \ \ X \ \ \ o - - o 5 a ( o b s e r v e d ) • • 5 d ( c a l c u l a t e d A . — A 5 b(i) ( o b s . ) \ D — o 5 b( i i ) ( o b s . ) \ x — x 5 c ( o b s . ) \ \ / / \ / I \ l \ I \ \ • \ / / / / / / y 1 V P H A S E ( d e g r e e s ) I 8 0 1 3 6 0 ro Fig . 2 8 F L O W O F S T R A T I F I E D F L U I D TWO FLUID SYSTEM A B S O L U T E SUBCRITICAL A B S O L U T E SUPER-F L O W C R I T I C A L F L O W ( LOW FROUDE NUMBER) (HIGH FROUDE NUMBER) M m r n m r , S M A L L BARRIER S I N E WAVE L A R G E BARRIER H Y D R A U L I C JUMP C O M P L E T E BLOCKING S P I L L I N G OF F L U I D O V E R B A R R I E R ( A F T E R LONG 1954) FIG. 2 9 FLOW OF S T R A T I F I E D FLUID C O N T I N U O U S D E N S I T Y G R A D I E N T S U P E R C R I T I C A L F L O W p :> —L~ S T A G N A T I O N ( 2 7 T < . F T R E V E R S A L OF V E L O C I T Y E D D I E S A N D A N D F O R M A T I O N O F TURBULENCE O V E R T U R N 277- 1 IT B L O C K I N G B L O C K I N G W I T H FORMATION O F J E T B O U N D A R Y L A Y E R S E P A R A T I O N T S V - < F < 4 l F ^ •*= F < = 2 7T '^rrTIflJlMmirt^ ^ ^ ^ ^ ^ ( A F T E R L O N G ! 9 5 5 ) 

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