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On the mechanics of lake circulation Fofonoff, Nicholas Paul 1951

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ON THE MECHANICS OF LAKE CIRCULATIONS by L c i By NICHOLAS PAUL FOFONOFF A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the Department of PHYSICS We accept this thesis as conforming to the standard required from candidates for the degree of MASTER OF ARTS. Members of the Department of PHYSICS THE UNIVERSITY OF BRITISH-COLUMBIA September, 1951 ABSTRACT Four aspects of the problem of describing the phenomena governing the flow pattern i n a lake are i n v e s t i -gated with the a i d of simple mathematical models. Analogies are pointed out between the two-dimensional models and the corresponding problems i n the e l a s t i o deformation of thin r i g i d plates. A. The dependence of the flow pattern i n a l o n g i t u d i n a l v e r t i c a l cross-section of a lake on a given temperature d i s t r i b u t i o n within the lake i s studied with the a i d of a two-dimensional lake model of semicircular v e r t i c a l oross-section. A stream function s a t i s f y i n g the two-dimensional non-homogeneous Mharmonio equation i s introduced. Solutions are found consisting of a throughflow function having a souroe and sink at the ends of the diameter plus a o i r o u l a -t i o n function whioh i s obtained i n terms of a Green's function and the horizontal density gradient. The t o t a l stream function s a t i s f i e s the conditions of no s l i p p i n g at a s o l i d boundary and zero stress at a free surfaoe. Two oases of simple density structure are considered, and streamlines are drawn for several r a t i o s of c i r c u l a t i o n to throughflow. B. The departure of the lake surfaoe from the horizontal required to maintain a pressure head s u f f i c i e n t to over-oome eddy vi s o o s i t y i s estimated by considering the pressure I d i s t r i b u t i o n i n the case of the throughflow function introduced above* C. The effeot of C o r i o l i s forees on the flow i n a canal • i s investigated with the a i d of a three-dimensional semi-oylindrioal model. I t i s shown that the effeot on the shape of the canal surface i s n e g l i g i b l e . For a s u f f i -c i e n t l y low value of v e r t i c a l density s t r a t i f i o a t i o n the effeot o f G o r i o l i s foroes may show up i n the t i l t i n g of constant density surfaoes. D. The e f f e c t of wind stress on o i r o u l a t i o n i n a v e r t i c a l lake oross-seotion i s investigated mathematically with the aid of a two-dimensional model. The res u l t s are found to agree quite closely with experiments on the deformation of a semicircular plate recently reported i n the l i t e r a t u r e . I I A C O O W L E D G r E M M T I would l i k e to express my sincere appreciation for the guidance and encouragement given me by Professor G. M. Volkoff, who has devoted many hours to d i r e c t i n g my research. I also wish to thank Professors G. L. Pickard and W. M. Cameron of the Institute of Oceanography, who suggested the problem and have generously made the f a c i l i t i e s of the Institute available to me during the summer months, and Dr. C. C. Gotlieb of the Computation Centre, McLennan Laboratory, University of Toronto, who has graciously undertaken an extensive program of computations a r i s i n g out of the work reported i n thi s thesis. I am indebted to the National Research Council who have granted me a bursary and a summer scholarship to carry on t h i s research. I l l TABLE OF CONTENTS Page ABSTRACT I ACKNOWLEDGEMENT . . I l l 1. INTRODUCTION 1 8. FORMULATION OF GENERAL PROBLEM (a) Simplifying Assumptions 3 (b) Eddy Yisoosity 4 (a) Equations of Motion 5 (d) Decomposition of Flow into Throughflow plus C i r c u l a t i o n 8 (e) Analogy to a Problem i n E l a s t i c i t y 9 3. SPECIAL MODELS A" Thermal Circulations (a) The Model Used 11 (b) The Throughflow Function 11 (o) The C i r c u l a t i o n Function i n terms of the Green's Funotion 12 (d) The Flow Pattern for a "Two-Layer Lake" . . 14 (e) The Flow Pattern for a ?Lake? with a Uniform Horizontal Temperature Gradient . . . . . . . . . . 16 B. Pressure D i s t r i b u t i o n for the Throughflow  Funotion . ~. . . . I ii ii 7~~~. ii . . . . . . . 18 C. C o r i o l i s Forces (a) Model Used . E l (b) Longitudinal Flow . . . 23 (c) Transverse E f f e c t s 24 (d) Pressure D i s t r i b u t i o n . . . . . 26 D. Wind-Driven V e r t i o a l C i r o u l a t i o n . . . . . . . 28 4. DISCUSSION OF RESULTS (a) Thermal Circulations 32 (b) Pressure E f f e c t s 34 (o) C o r i o l i s Forces 36 (d) Wind-Driven V e r t i c a l C i r o u l a t i o n . . . . . 38 5. BIBLIOGRAPHY 40 APPENDIX I 41 APPENDIX II 47 ILLUSTRATIONS Plate I following page 12 F i g . 1. Streamlines of the throughflow funotion F i g , 2. Velocity p r o f i l e along v e r t i o a l radius of F i g . 1. Plate II . following page 16 F i g . 3. Streamlines of the thermal o i r c u l a t i o n funotion V • 0.192 for the oase of a "two-layer lake" with a density d i s -continuity along - e « 135°. F i g . 4. Streamlines for (j> « •+ 0.192^ a F i g . 5. Streamlines for s 0f — 0.192 0 t F i g . 6. The dynamic surfaoe of the "lake" . . 20 F i g . 7. Seml-ciroular oanal model . . . . . . 21 Plate III f o i l O H ing page 31 F i g . 8. P r o f i l e s of wind-driven c i r c u l a t i o n functions of Figs. 10 and 11 along radius 0 » 90°. F i g . 9. P r o f i l e s of wind-driven c i r o u l a t i o n functions of Figs. 10 and 11 along r a d i i Q • 45° or 6 * 135°. Plate IV . following page 32 F i g . 10. Streamlines for wind-driven o i r o u l a t i o n (theoretical) F i g . 11. Streamlines for wind-driven o i r o u l a t i o n (experimental) 1. INTRODUCTION L i t t l e work has been done i n the past on the investigation of the mechanics of lake c i r c u l a t i o n s . W. H. Munk and E. R. Anderson^ 5^ have set up a theory of the thermooline and have applied i t to the oase of Sweetwater lake, but the basio flow caused by a potential gradient with the possible thermal currents set up by density differences i n the lake has received l i t t l e attention. Sinoe the v e l o c i t y of the flow i n a lake i s small, dire c t measurements of the velooity d i s t r i b u t i o n with ourrent meters and ourrent drags often f a i l and reoourse must be made to i n d i r e c t methods involving the density struoture and consideration of the forces acting on the f l u i d i n i t s motion. Seasonal changes i n temperature conditions w i l l a f f e c t the flow, for i t i s found that during the summer temperature gradients are considerable. During winter months and early spring the lake i s usually very homogeneous and the flow i s due predominantly to the p o t e n t i a l gradient. C o r i o l i s foroes aoting upon the moving water masses may produce observable e f f e c t s i n the lake flow pattern, although any expected e f f e c t s w i l l be small beoause of the low v e l o o i t i e s involved. In addition to the potential gradient and C o r i o l i s foroes the meohanios of lake flow i s further complicated by - 2 -interactions with the atmosphere i n heat exchange, wind stresses, etc. These e f f e c t s w i l l not be considered e x p l i c i t l y here, with the exception of wind stress, an example of which i s considered for comparison with r e s u l t s obtained from the e l a s t i c deformation of thin r i g i d plates. 2. FORMULATION OF GENERAL PROBLEM (a) Simplifying Assumptions Assuming that a steady state temperature d i s t r i -bution has been experimentally determined, the density f i e l d may be obtained as a funotion of the pos i t i o n co-ordinates of a point through the dependence of the density on the temperature. (The dependence of density on pressure i s n e g l i g i b l e . ) Heat transfer must be such as to maintain this temperature d i s t r i b u t i o n , i . e . the water gains or loses heat as i t flows along the streamlines at just the ri g h t rate to assume at eaoh point the observed temperature. The transfer of heat between various parts of the f l u i d and heat exchanges with the atmosphere may i n p r i n c i p l e be determined once the temperature d i s t r i b u t i o n and v e l o c i t i e s in the lake are known. Therefore, i t i s not necessary to consider heat transfer i n the following discussion. The observed temperature differences are small so that the f r a c t i o n a l density changes w i l l also be small. From the point of view of the equation of continuity the density w i l l be considered constant, the variations being taken into account only i n oalculating the ef f e c t s of gravity. Mathematically this i s equivalent to saying that the acceleration of gravity varies i n a given way throughout the lake. - 4 -(b) Eddy Yisoosity In desoribing the flow equations involving eddy v i s c o s i t y are used, sinoe large soale flows suoh as those found i n lakes are generally of a turbulent oharaoter. Turbulent flow i s characterized by the presence of numerous eddies superimposed on the mean flow continually transporting f l u i d masses into regions of d i f f e r e n t v e l o c i t y . These eddies are very i r r e g u l a r and complex so that the steady state i n a turbulent flow does not ex i s t i n the same sense as i n laminar flow. Wherever a vel o c i t y gradient occurs i n the mean flow there w i l l be a transport of momentum across the constant momentum surfaces by the turbulent v e l o c i t y com-ponents. P r a n d t l ^ has introduoed a "mixing length", analogous to the mean free path of a gas moleoule, i n which he v i s u a l i z e s f l u i d elements carried by turbulent v e l o c i t i e s a distance " 1 " before l o s i n g their i d e n t i t y by mixing with the surrounding f l u i d . This oonoept has been c r i t i c i s e d by G. I. T a y l o r ^ ) , who considers v o r t l o i t y transfer rather than momentum transfer i n defining the eddy v i s c o s i t y . The two d e f i n i t i o n s lead to the same r e s u l t i f the eddy v i s c o s i t y i s a oonstant, and a discussion of the differences r e s u l t i n g from the two concepts i s given i n Taylor's p a p e r ^ An excellent discussion of the various theories of turbu-lenoe i s given by R o u s e ^ from which the following short - 5 -resume of Prandtl's d e f i n i t i o n of eddy v i s c o s i t y i s adapted. Consider i n the x ty plane a mean motion Vx i * i the x d i r e c t i o n having a v e l o c i t y gradient i n the y d i r e c t i o n . Assuming that a unit f l u i d mass i s ca r r i e d ( 2 h P V x + 1 <* Vx T d v a distance "1" by a 1 * , , 1 turbulent v e l o c i t y ( I P * V x component Yy, the net 0 x momentum transfer to point (2) i s - fYy l 4 Vx . The time average of this quantity gives the mean stress along the constant momentum surfaces. The quantity "jpVy~i" ^ a s d-ini 6 1 1 8*- 0 1 1 8 °f dynamic v i s o o s i t y and i s termed the eddy v i s c o s i t y A. In general i t i s a funotion of position, being affected by the scale and type of turbulence, s t a b i l i t y conditions, magnitude of mean flow, proximity of boundaries, and other faotors. Sinoe turbulenoe i s usually h o r i z o n t a l l y i s o t r o p i c only the horizontal A^ and v e r t i o a l A v eddy c o e f f i c i e n t s are d i f f e r e n t i a t e d . The r a t i o -^v/A^ i s roughly of the order of magnitude of the r a t i o of the v e r t i c a l and horizontal dimensions of the water currents (Rouse p. 186). (o) Equations of Motion In terms of the eddy vi s o o s i t y the hydrodynamic equations for an incompressible f l u i d are: - 6 -In the lake models to he considered the eddy v i s c o s i t y i s assumed oonstant and i s o t r o p i c : A^ =• Ay * A The assumption that A i s is o t r o p i c i s reasonable when the v e r t i c a l and horizontal dimensions are of the same order of magnitude. There i s l e s s physical basis f o r the assumption that A i s independent of po s i t i o n but i t i s necessary to simplify the mathematios. I t has been found by other workers (e.g. Munk^) that assuming A independent of p o s i -t i o n leads to solutions bearing a good deal of resemblance to the observed flow. Taking the external forces to be those of gravity and G o r i o l i s foroes, the hydrodynamio equations are: * V (2) V-V* - ° Consider: — 9 (1) steady state flow ^ Y s o (2) small mean vel o c i t y of flow, i . e . f 5 * V ' can be neglected. The equations reduoe to: (2a) y . * o — ? Introduce a vector potential y such that p^r VKi? and ohoose V - = 0. Taking the c u r l of both sides of equation (2a) a single three dimensional equation for i s obtained *- 13) This equation w i l l be used i n section C below to show that C o r i o l i s foroes are r e l a t i v e l y unimportant i n lake flows. As two dimensional models w i l l play an important role below, equation (3) i s speci a l i z e d to two dimensions i n the x,y plane, with y positive downwards, x positive to the l e f t (the t h i r d z axis i s then directed out of the page). Choose a J^(x,y)tf. This s a t i s f i e s V- m 0, N7- - 0. Then ^ = V x ? - - , where ^ i s the usual two dimensional stream function, i . e . 9* = constant defines a streamline, and V% - % • rate of flow per unit width of lake between streamlines B and A. The C o r i o l i s foroes are i n general n e g l i g i b l e , and i n t h e corwpowri/" m ihe vertical pktt? is partioular Aabsent i f the flow i s taken i n the north-south plane. - 8 -Under these conditions, equation (3) reduces to: »v*v*1f = ^? where 7 l » ^ + ^ x (4) The boundary conditions are: (t = tangent, n = normal) (a) r f(x) along lake surfaoe y » 0 . In absence of wind stress f(x) « 0. In the presence of wind stress f (x) i s a given funotion of x. (b) ss 0 along lake bottom 0, i . e . no s l i p p i n g at s o l i d boundary). (o) ¥ • 0 along lake surfaoe. (dj t * ff/b along bottom (IP » rate of flow through aotual lake, b = average width of aotual lake F/b * rate of flow per uni t width) (d) Decomposition of Flow into Throaghflow plus C i r c u l a t i o n The t o t a l stream function i s s p l i t up into two parts ^ = % { 4>, (5) Here i s assumed to be the solution of the homogeneous biharmonio equation obtained by s e t t i n g the right-hand side of equation (4) equal to zero. ^ s a t i s f i e s the boundary conditions a with f(x) a 0 , b, o, and d, replaced by tf>t s 1 along bottom. Thus ^ represents the flow pattern of a lake having no wind stress, no temperature v a r i a t i o n i n the x d i r e c t i o n (although there may be a v a r i a t i o n i n the y d i r e c -t i o n ) , and a unit throughflow per unit time per unit width from r i g h t to l e f t . Consider F / b ^ to be that solution of the non-homogeneous biharmonio equation (4) whioh s a t i s f i e s the boundary conditions (a,b,c) together with ^ a 0 along lake bottom. This corresponds to zero net throughflow and describes the oi r o u l a t i o n i n a "lake" with a given tempera-ture d i s t r i b u t i o n or a given wind stress but with no intake or o u t l e t . In a oase where the temperature d i s t r i b u t i o n i s simple enough to set up a single o i r o u l a t i o n only ( ^ h a s a single positive maximum only), defines a counter-clockwise c i r c u l a t i o n , ^< 0 defines a olookwise c i r c u l a -t i o n . *Vb \ T % _ i s t h e o i r o u l a t i o n rate of flow per unit width of lake. i s chosen to normalize p£t such that = 1. Then: ^ s o i r o u l a t i o n / t h r o u g h f l o w . ( Q) Analogy to a Problem i n E l a s t i o i t y The e l a s t i o deformations o f a thin plate of uniform f l e x u r a l r i g i d i t y damped along a portion of i t s perimeter, hinged (with a given bending moment) along the balance of i t s perimeter, and subject to a variable normal load, are governed by the same equation with the same boundary conditions as the above stream function (Love, A.E. - 10 -H.^ 2 ), page 464). In the oase of the e l a s t i c model analogy discussed above constant <f represents contour l i n e s . Throughflow streamlines are analogous to the contour l i n e s of the de-formation produced i n the absence of normal load when the damped portion of the perimeter i s displaced with respect to the hinged part. C i r o u l a t i o n streamlines are analogous to the oontour l i n e s of the deformation produced by a given normal load d i s t r i b u t i o n or a given bending moment at the hinge when the clamped and hinged parts of the perimeter are at the same l e v e l . 11 -3. SPECIAL MODELS A. Thermal Ci r c u l a t i o n s (a) The Model Used We are interested i n obtaining some idea of the influence of the temperature -inhomogeneity i n a lake on the flow pattern. The mathematical d i f f i c u l t i e s i n t r e a t i n g the actual three-dimensional oase are considerable, so that i n order to obtain a f i r s t o r i e n t a t i o n we consider a two-dimensional model. We further simplify the mathematics by choosing for the p r o f i l e of the "lake bottom" i n the two-dimensional v e r t i c a l cross-seotion the semioirole r s a. We now look for solutions of equation (4) d i s -oussed above subject to the boundary conditions a, b, o, d. As discussed above the solution f a l l s into two separate parts: determination of the throughflow funotion 4>t s a t i s f y i n g the homogeneous biharmonio equation, and the determination of a o i r o u l a t i o n funotion s a t i s f y i n g the non-homogeneous biharmonio equation (4). (b) The Throughflow Junction Using "a" as the unit of length we introduoe ^ = x/a, ) s y/a P a r/a. It may be v e r i f i e d (see Appendix I) that: - 12 -s a t i s f i e s v * " * 7 z ^ / • o and the boundary oonditions: (a) y ~ a 0 along & = © , 7 t (b) \& a 0 along p « l (o) ^ - 0 along o  m °JTC (d) ^ s 0 along p a l 4>, represents the throughflow funotion for a lake of homogeneous horizontal density structure. Streamlines corresponding to <#>t = constant together with the v e l o c i t y p r o f i l e v~x a ~$Jb along x a 0 are given on Plate I, f i g s . (1) and (2). (o) The Ci r o u l a t i o n Function In Terms of the Green's  Funotion To determine <f>^ equation (4) i s rewritten i n terms of coordinates i n which ''a" i s a unit of length. ^ 2 V Z * Z * (8) AF-£ * I with the boundary oonditions: 2 (a) 3 4> 2 e 0 along 6 a Qt7U 3 h 2 (b) ^ 2 s o along p - 1 (o) >^ B 0 along * s 0,7cr (d) «^2 88 0 a l ° n S f " 1 For any given density d i s t r i b u t i o n /g, i s to be ohosen to make 7*2.max - !• This value of then determines the PLATE I - 13 -r e l a t i v e magnitude of the e i r o u l a t i n g flow and the through-flow. The solution of equation (8) for any f g can be expressed as an i n t e g r a l i f the Green's function G(p, e> 0 O) i s known, where p » r / a , q » r , c / a . The Green's funotion s a t i s f i e s : V 2 V 2 G = * ( * - * • ) (9) and the above boundary oonditions. The delta funotion may be replaced by the boundary oondition / " T ( ^ & > ' < * l ? K ^ ] where Oca C->o i s an elementary displacement i n the counterclockwise direotion along a curve surrounding the point (q, © 0 ) , the l i m i t being taken as the curve i s shrunk around that point. The Green's function s a t i s f y i n g the above condi-tions i s : (See Appendix I) where S+_ - P 2 f q 2 - 2pq^o (© ± e© ) •tl * 1 t p 2 q 2 - 2pqCos {G+ ©. ) It may be v e r i f i e d that 0 i n the semicircle. In terms of this Green's function: a 3 b S r 4n - * J qdq J G (/>,*; q e0) D(q, *>) (11) - 14 -where D(g.,3j s evaluated at p » q, © r © 0 (11a) To determine C P , the form of ^2* must be known. (d) The Flow Pattern for a "Two-Layer Lake" (Case Ij The following simple case i s considered: f>3 i s constant i n two parts of the lake with a d i s -continuity along e * &, (two-layer lake). This corres-ponds to a concentrated l i n e load i n the e l a s t i o analogy and i s the most elementary model of the actual s i t u a t i o n . Choose f g « foj» for o s « $ ©. fg s fo 3© (1-f) for <s>,«.*£^ This means that | ^ * 0, >J± = - fa%t <T ( 0 ) . Now, since Ul e Cos6 ^  - 5-^* we obtain Thus from equation (11) ^ becomes ^ 2 a fo ^  f Sln e, / B ( p , © ; q , e j d q <12> AF 1 6 ^ • a g b g»aef Sin^> Ufa.*;*/) AF-£ 16 T c r 15 -where U(P,0; O, ) = /Jdq B(p,©;q , e , ) . (13) Here B i s the function defined by equation (10). To make <f> 2 max - l set: s a 3b f ° % f Sine, u max • c i r c u l a t i o n (14) AF 16 7C throughflow Sinoe U> 0 i n the semicircle, ^ has the same sign as f. Thus the c i r c u l a t i o n i s clockwise for f<^0 and counter-clockwise for f/• 0 i n agreement with the physioal require-ment that dense l i q u i d sinks while l i g h t l i q u i d r i s e s . The evaluation of the i n t e g r a l (13) i s tedious but straight forward and the r e s u l t may be written as: U(p,d; e, ) = F ( p , £ ) - f( p , fy) where <*a O 1L e, (15) and - 16 -The steps i n the integration are given i n Appendix I I . U(p,©; 6,) for &( =135° has been computed on a desk oalculator. U m a x for t h i s case i s found to be 0.192. The streamlines f o r U and f o r the t o t a l streamfunotion f = <f>, -f- ?l are given for - ± 0.192 on Plate I I , f i g s . (3), (4) and (5). Dr. C. C. Grotlieb, of the Computation Centre, MoLennan Laboratory, University of Toronto, has undertaken to compute XT for ©,^ 0(10°) 90°. Unfortunately t h i s material w i l l not be available i n time for inolusion i n th i s thesis and w i l l be published separately. (e) The glow Pattern for a "Lake" with a Uniform  Horizontal Temperature Gradient (Case II) The next oase considered i s a uniform horizontal temperature gradient along the lake, i . e . The v e r t i o a l v a r i a t i o n i s a r b i t r a r y . In the e l a s t i o analogy t h i s corresponds to a uniform load per unit area. This model corresponds somewhat better to the aotual state of a f f a i r s . This gives: D(q, ©o ) = = M 0 C PLATE II Streamlines of the thermal c i r c u l a t i o n function F i g . 3 . Ms 0 . 1 9 2 02. for the case of a "two-layer lake" with a density discontinuity along 9 s 135^ correspond to interv a l s of 0 . 0 2 forV") (Contours F i a 4 Streamlines for ¥> s <P-,+0.1920^. y • (Direction of flow from r i g h t to l e f t , counterclockwise ciroulation) F i g . 5 Streamlines for f e ^ - 0 . 1 9 2 $ % . (Direction of flow from r i g h t to l e f t , clockwise ciroulation) - 17 -From equation (11) $ g becomes A a^bc * Q * 2 B IFY" ^  T , p , e ) ( 1 6 ) where V(p,e>) = J^^J B (P» e? <l» a o ) (16a) As before the r a t i o of oi r o u l a t i o n to throughflow i s given by setting. X = - a ^ s u y - 3 o v m a x (l?) u AF 16 The integration over q i s ca r r i e d out e x p l i c i t l y (See Appendix II) while the integration over &0 i s to be done numerically. v(p,e) = / £ " ( P , * ; e0 (is) where £ C P , © ; © 0 > = hip/*.) - H p / _ ) + p ( ^ ^ F c ^ _ ) - C s ^ / r ^ j j d s ) *P I 3 *p px r J - 18 -and F ( p , $ ) , <f± are defined by equation (15a) above. V ( p , © ) i s also being computed by Dr. C. C. Gotlieb and w i l l be published separately. B. Pressure D i s t r i b u t i o n for the Throughflow Funotion The flow pattern represented by the throughflow function must be maintained by pressure gradients i n the "lake". It i s important to oonsider the pressure to determine the dynamio surfaoe of the lake since the energy necessary to overcome viscous stresses and to maintain the flow i n a lake having a homogeneous horizontal density structure must oome from a loss of g r a v i t a t i o n a l potential of the water, i . e . the surfaoe assumes a slope s u f f i c i e n t to supply the neoessary energy. We oan also v e r i f y the correctness of the assumption made at the outset that terms i n the square of the velooity ( f i e l d acceleration terms) are n e g l i g i b l e . Consider the pressure d i s t r i b u t i o n neoessary to maintain the flow given by the throughflow function ^ = B"/b ^/ where <f>t i s defined by equation (7) above. For a given steady state flow the pressure i s determined from the hydrodynamlc equation: - 19 -Here the left-hand side represents the f i e l d acceleration contribution, while the l a s t term on the right represents the v i s c o s i t y e f f e c t . Consistently with our o r i g i n a l assumption of small mean v e l o c i t y we take the left-hand side equal to zero, and look for the i n t e g r a l of with * - £ £ K > X ^ ^ J and <f>t given by equation (7). This may be integrated e x p l i c i t l y to give: Along the l i n e y » 0 the in t e g r a l i n the above expression vanishes, and the pressure P d i f f e r s from the atmospheric pressure by the second term. This may be represented by the water p i l i n g up above the horizontal surface near the intake and dropping below the horizontal surface near the outlet. The equation of th i s surface i s ; A plot of <js i s shown i n f i g . 6. The dotted portion of the curve i s to be disregarded beoause the i n f i n i t e values of ^ at the two ends are due to the unphysioal feature of our model having o r i f i c e s of zero area leading to i n f i n i t e intake and outlet v e l o c i t i e s . - 20 -To investigate the order of magnitude of the contributions to the pressure of the f i e l d acceleration terms we note that sinoe our unit of length i s "a", the order of magnitude of the ve l o c i t y i s F ^ Q . Thus the f i e l d acceleration terms are of order f F 2 / a 2 b 2, while the viscous term i s of order A F / a 2 ^ , . The r a t i o of the two i s therefore given by * ^ F / A J , . / / / i i i i j i F i g . 6. The dynamic surfaoe of the "lake" - 21 -C. C o r i o l i s Foroes (a) Model Used We are interested i n studying the effects of C o r i o l i s foroes on the flow pattern i n a canal to decide whether or not these foroes play an important role i n determining the v e l o c i t y p r o f i l e i n the oanal. As the two-dimensional model considered i n section 3 A i s not suitable for this investigation we now introduce a three-dimensional model consisting of a semi-oylindrioal "oanal" running i n the North-South d i r e c t i o n . We consider equation ( 3 ) including the C o r i o l i s term A «7Zvz » 2 V x [ -£ 'xr|Vx4\]] - 0 V- ^ = 0 This equation admits a simple approximate solution for the oase of a "oanal" of semi-oiroular transverse oross-section. The coordinate axes are ohosen with Z positive } to the South and \r , O forming a right-handed coordinate system. South « Semi-circular canal model F i g . 7. - 22 -The Laplaoian operator i s defined i n c u r v i l i n e a r coordinates hy: V 2 a V {VV? ) • V z ( V x ^ ) s - 7z[?x+ >) sinoe V - * ^ r 0 . We ohoose ¥ suoh that: V R » 0 H>G = ^ ( r ) ( 2 3 ) ' This s a t i s f i e s V- s 0 . The components of - 7 x ( 7 x f ) are reduced so that: the r oomponent i s zero the 0 oomponent i s j>_ / 1_ S (r f'o)) the Z oomponent i s 1 5 _ i r S 9 ^ i In expanding the C o r i o l i s term v7 i J^iC x $ [ y x ^ j ] —> terms containing derivatives of -XL are neglected sinoe these w i l l involve a faotor where R i s the radius of the earth. To t h i s approximation x ? j y x < f ^ ] J has: f component equal to d_ I Sin© Y0 - CosO YT1 r pe* L J & oomponent equal to - Sin© Y e - Cos© V" rJ Z component equal to - Sin© 3*V 5 r - 23 -Sinoe the v e l o c i t i e s V r, Ye are much smaller than Y & only the z component i s considered i n the f i r s t approximation. Equation (3) under these assumptions reduces to two equations for ^ and ^ 4 . (25) 7 * 7 * ^ s - EAe sin© i v * + >?3 3 r ~5~x where 7*- = 1 a C r 1_ (25a) (b) Longitudinal Flow The l o n g i t u d i n a l flow i n the t. d i r e c t i o n subject to the oonditions that no s l i p p i n g 00curs at the oanal bottom and that the maximum v e l o c i t y i n the oanal i s */0 i s obtained by integrating the €> oomponent of equation (25). The general i n t e g r a l i s K (r) r ^  r 3 f ^ ^ U r + (v/x - c/. )r -f «V r and = 1 2 - ( r^») • B r Z + c A , r - f B r * r 4 Evaluating the constants i n terms o f the boundary oondi-tions above we obtain: / t = (1 - ) (26) - 24 -and Vd = Y«M 2 r - r 3} (26a) 4 1 a a3J In terms of the vector pot e n t i a l f the f l u x through any closed contour C i s by Stokes' theorem F a r where <ts i s an elementary displacement along oontour C. Taking the contour i n the r , © plane, the t o t a l f l u x through the canal i s given by F = f % r<Le for r s a .2 (27) Solving for V V - I F = R F/ i. 2" ° oross-seotional area of oanal (27a) Substituting the values of F and cross-sectional area of Kootenay Lake we obtain values of V 0 of the order of magnitude of 20 f t / h r . (o) Transverse E f f e c t s Equation (25a) i s of the same form as equation (8) solved i n section 3 A. We can obtain a general solution i n terms of the Green's funotion defined i n - 25 -equation. (10). ^ = a ) o qdq J oCL© 0G (p © ; q, ©J D (q, eD) (28) where D(q, © J = > f 3 " 4 * * V. $ sin6>e I f a positive v e r t i o a l density gradient exists i n the lake, the C o r i o l i s foroes w i l l he balanced hy a horizontal density gradient sinoe the transverse o i r o u l a -t i o n induced by the C o r i o l i s foroes w i l l produce a h o r i -zontal density gradient by t i l t i n g the constant density surfaces. In the steady state there i s no transverse oir o u l a t i o n , i . e . ^ t z 0. Since G(p,e ; $,*©) £ ° i n the semicircle this implies D(p,©- ) « 0, i . e . l i s i s x ^ i Integrating t h i s equation we obtain an expression for the density structure. The slope if of the constant density surfaces may be obtained: where X - ~ ° 3 - 2 6 -The constant density surfaces are given by f( i ) + r U = c (29b) ^ * c ~ f ( V •) r * I f the lake i s completely homogeneous i n i t s density structure the C o r i o l i s forces give r i s e to a trans-verse c i r o u l a t i o n whose stream funotion i s : This i n t e g r a l has not been evaluated sinoe C o r i o l i s foroes are very small (see section 4 (c)) and do not contribute s i g n i f i c a n t l y to the flow pattern. (d) Pressure D i s t r i b u t i o n The pressure term i s considered to study the eff e c t of C o r i o l i s foroes on the transverse dynamic surfaoe of the canal i n the oase where the C o r i o l i s foroe i s balanced by a horizontal density gradient. Neglecting the f i e l d acceleration terms the pressure i s given by the hydrodynamio equations: 21 - A oos6> p V 0 (1- + ?9 S i n 0 = - A S i n 0 £ V o (1- £)+HCose (30) - 27 -The z oomponent i s simply the potential drop neoessary to maintain the lo n g i t u d i n a l flow. I t i s inter e s t i n g to oompare the slope of the oanal surfaoe i n the long i t u d i n a l d i r e c t i o n with the slope of the "lake" surfaoe i n the model disoussed i n section 3 B. The expression for the slope at J » 0 i s 16 AF/^t, (30a) while the slope for the oanal i s U V 0 . 16 A F o _ _ _ _ _ _ _ a 2 ^ 7 c a 4 f 9 (30b) Writing the pressure as P = - * A Y , * -f P(r, © ) 4 - P . a 2 we now proceed to f i n d an e x p l i c i t i n t e g r a l for P(r,© ). The pressure term involving C o r i o l i s foroes w i l l be small so that the mathematical s i m p l i f i c a t i o n of considering the v a r i a t i o n i n the term ?_) to l i e i n the acoeleration of gravity w i l l not appreciably a f f e c t the f i n a l r e s u l t but w i l l greatly simplify the expression for P(r,© )• Let a J « , g C S ) " «f*A v o\j«$ Making this s u b s t i t u t i o n for $3 into equation (30) the integration gives: s 1 —» (31) - 28 -at y a 0 P (r,6) 2 $0> Vo<^{\ ~ (31a) This pressure i s produced by a wedge of water p i l e d above y a 0 on the west side and a depression of the surfaoe on the east side of the oanal. The oanal surfaoe under the influence of C o r i o l i s foroes i s given by the equation " 3 1 * F i (3ib) The t o t a l difference i n height of the surfaoe between the west and east sides of the oanal i s f-D. Wind-driven V e r t i o a l C i r o u l a t i o n The lake models considered i n the previous section lend themselves nioely to the consideration of v e r t i o a l o i r o u l a t i o n due to wind stresses.in a lake of homogeneous density d i s t r i b u t i o n . These have been previously neglected but must be considered as a d i s -turbance of the lake flow. The wind stress on the lake surface depends on the wind velooity at the lake surfaoe. I f the mean wind ve l o c i t y v^, were given at eaoh point of the lake surfaoe the stress on the surfaoe may be oaloulated from the equation (Munk % " CD J a i r \ - 2 9 -where Cj, i s the drag c o e f f i c i e n t . In e f f e c t the surfaoe stress as a funotion of x i s given from atmospheric data and i n terms of this funotion the v e r t i o a l o i r o u l a t i o n i n the lake or canal may be deter-mined. A semi-oiroular v e r t i o a l cross-seotion w i l l be oonsidered. This may be either the l o n g i t u d i n a l cross-section of section 3 A or the transverse cross-section of seotion 3 G. The wind stress must be balanoed by a v e l o c i t y shear at the surfaoe y s 0, i . e . T ; ( x ) s - A _ J _ x - - A y j* for y . O ( 3 2 ) We write T s ( x ) = T0t(£") where i s a unit of surfaoe stress, and a dimensionless function. Then *1± - _L2> -- - K t C f c ) f o ^ - o ( 3 2 a The v e r t i o a l c i r c u l a t i o n s are given by the stream-function <fs whioh s a t i s f i e s the homogeneous biharmonio equation V*-?2? a 0. Consider the function <f = - I k y Re£TC«>} where *o = V (33) 2 - 30 -It i s shown i n Appendix I that any funotion of the form *j 4> where £ i s a harmonic function* s a t i s f i e s the biharmonio equation. Choosing T(&>) suoh that: M'S} for > a O (33a) and writing (34) we require that the boundary conditions (a) = -<t(£) f o r \; « 0 i h i (b) *» 0 for p s i s(o) ^ s 0 for ) s O (d) ^ =0 for p s 1 be s a t i s f i e d . must therefore s a t i s f y the boundary oonditions U ) = 0 for J; = 0 < b> 12L - f- {i k R e L r t v a i 3 } f o r p = i 5 ft (o) /t = O for i = O (d) / = 1 Jr Re(Tc-)} for p = 1 Suoh a function may be constructed by the method outlined i n Appendix I. - 31 -The simple oase of uniform wind stress on the lake surfaoe i s considered below mainly to make a quantitative oomparison of t h i s model with the e l a s t i o analogy of a t h i n semicircular plate of uniform f l e x u r a l r i g i d i t y deformed by a constant bending moment applied at the hinged diameter. Deformation contours for this oase were experimentally obtained ( H i d a k a ^ ) using a thi n s t e e l plate clamped along the ciroumferenoe, the diameter r e s t i n g on a knife edge. The bending moment.was applied by extending the st e e l plate past the knife edge i n a series of s t r i p s on which weights of known size were plaoed. The contour l i n e s obtained experimentally are reproduced i n F i g . 11. The biharmonio equation subject to the above boundary oonditions with t( £ ) s C admits an exact solution which i s given below. 7 r L P (35) Contour l i n e s of y> normalized to an arbitrary ^ m a x chosen to agree with the experimental contours are plotted i n F i g . 10, and y (p,$ max i s plotted as a funotion of p for 0 = 90°» and 45°, together with the experimentally obtained deformations for the same angles normalized to unity (broken curves), i n Fi g s . 8 and 9. PLATE I I I P r o f i l e s of wind ^ -driven c i r c u l a t i o n P r o f i l e s of wind-driven c i r c u l a t i o n F i Q. 8 functions of Figs. 10 and 11 along Fig. 9. functions of Figs. 10 and 11 along radius 6 - 90° (normalized to unity r a d i i 6 • 45° or & s 135°. (Same at maximum) normalization as i n F i g . 8) 32 -4. DISCUSSION OF RESULTS (a) Thermal C i r c u l a t i o n s In Figures 1, 5 and 4 of section 3 A we have shown the streamlines for the flow i n a two-dimensional model lake with zero thermal o i r o u l a t i o n , and with a olook-wise and a counterclockwise thermal o i r o u l a t i o n respect-i v e l y . In the clockwise o i r o u l a t i o n oase the flow i s reduoed along the top and increased along the bottom, the streamlines of F i g . 4 a l l being displaoed downwards compared to those of F i g . 1 on the i n l e t side of the lake, and upwards on the outlet side. In the oounterolookwise o i r -oulation oase the flow along the top i s increased, and a closed o i r o u l a t i o n i s set up near the bottom. In both oases the thermal c i r o u l a t i o n has been taken to be 19.2fo of the throughflow. We now discuss the above model i n the l i g h t of observational data taken from a t y p i c a l lake. (In this oase the data i s taken for the northern arm of Kootenay Lake.) The importance of the conveotive o i r o u l a t i o n as oompared to the throughflow i s r e f l e c t e d i n the c o e f f i c i e n t ^ whioh measures the r a t i o of the two flows. We oonsider the "two-layer lake" desoribed i n section 3 A (d) (Case I ) . Here i s given by: 1 a a 2 b fc 9o * sla.0, IT max L A F TS-TC P L A T E I V Streamlines for wind-^driven 1 9 l 0 - c i r c u l a t i o n (theoretical; normalized i n same units as Pig. 11.) Streamlines for wind-driven c i r -ig. 11, culation (experimental contours from e l a s t i c model of r e f . (1); arbitrary normalization) - 33 -Typioal observed values of the quantities are inserted into t h i s equation to get an idea of the order of magnitude of the temperature difference necessary to make thermal c i r c u l a t i o n s important i n determining the flow pattern* I f we take \ * 0.19S • V m a_ for e * 135°, the value used i n Figs. 4 and 5, and solve for f we obtain: f a 16 7E" A j* M . a 3 b sin©, The mean disoharge of the Lardeau River into Kootenay Lake daring the summer months i s 2 x 10 8 cm 3/ S Q 0. The mean width of the lake i s about 2.5 Em. "a" i s taken to be the depth of the lake sinoe the thermal currents w i l l be muoh more sensitive to depth than to the length of the lake. The average depth i s about 400 feet or 1.22 x 104cm. The greatest uncertainty l i e s i n the value of the eddy vis o o s i t y A. The range of values for the horizontal eddy g v i s o o s i t y observed i n ooean currents l i e s from 2 x 10 ** 4 x 10 8 gm/om seo. (Sverdrupl 1 0 *, p. 485). The value i s found to deorease as the size and velocity of the currents decrease. The value of 2 x 10 6 gm/ o m S Q o . was observed for the C a l i f o r n i a ourrent. Kootenay Lake, being muoh narrower with smaller currents than the ocean currents for whioh the above eddy v i s c o s i t i e s were calculated, would be expected to be characterized by a smaller eddy v i s o o s i t y . - 34 -We take A ~ 106 gm/ o m S Q 0 . The corresponding value of f i s f = 2.2 x 10~5 whioh corresponds to a temperature drop from the intake to the outlet of the lake of 0.27°C. at a mean temperature of 10°C Since horizontal temperature variations of this order of magnitude, or even higher, can e a s i l y e x i s t i n the upper layers of the lake during the summer months, our model gives *i 7, 0.2, i . e . that thermal o i r o u l a t i o n i s s u f f i c i e n t l y large to set up flow patterns with olosed oiroulations similar to F i g . 4. This i s consistent with the observed faot that r e l a t i v e l y high v e r t i c a l s t r a t i f i -cation e x i s t s i n the upper layers of the lake during the summer while the lower layers are quite homogeneous, the average temperature not varying appreciably over the seasons. Suoh a s t r a t i f i c a t i o n suggests that during the summer the. warmer r i v e r water flows i n the upper part of the lake while a closed o i r o u l a t i o n takes plaoe i n the bottom homogeneous layer i n a direotion suoh that bottom v e l o c i t i e s are i n the di r e o t i o n of the i n l e t , s imilar to the flow pattern of F i g . 4. (b) Pressure E f f e c t s There i s some question as to the ohoioe of the - 35 -value of A i n the above discussion. Unfortunately there i s a lack of accurate experimental data about aotual conditions i n the lake as well as of the o r e t i c a l knowledge of the v a r i a t i o n of the eddy vi s o o s i t y with d i f f e r e n t flows. However, an estimate of A may be made from the observed i n d i c a t i o n of a sloping lake surfaoe. In seotion 3 C we obtained two equations for the slope of the dynamic lake surfaoe. Equation (30a) ref e r r e d to the two-dimen-sional model of section 3 A, while equation (30b) referred to the three-dimensional canal model of section 3 C. The observed drop of the Kootenay lake surfaoe as found by taking water l e v e l s referred to elevations established by a Geodetic survey from a point near the i n l e t to Queen's Bay, a distance o f about 60 Km., varies between 0 - 5 cms. during the summer months. Using the average value of 2;? cm. for the drop i n the lake surfaoe we obtain a slope of 4.5 x 10~ 7. Using t h i s slope, and the other oonstants for Kootenay Lake as given i n 4 (a), we f i n d A » 2.0 x 10 5 gm/om Seo. for equation (30a). Using a oanal with the same cross-sectional area as the lake we obtain from equation (30b), A = 4.1 x 10 5 gm/ c m S Q 0 # This method of estimation i s subjeot to many oritioisms but does indicate that A l i e s i n the range ohosen for the disoussion i n 4 (a). The r a t i o between the contribution of the f i e l d - 36 -acceleration terms and the eddy v i s c o s i t y term was found i n seotion 3 B to be of the order of magnitude of F / A D» Using the values of F r b and A given i n 4 (a) we obtain for the r a t i o a value of 8 x 10~ 4, so that the f i e l d acceleration terms are about 0.08^ of the v i s o o s i t y terms and may be v a l i d l y neglected i n discussing the mean flow pattern. (o) C o r i o l i s Foroes In the three-dimensional model discussed i n seotion C we found that C o r i o l i s foroes have two main effects on oanal flow: the t i l t i n g of the oanal surfaoe by p i l i n g up of water on the west side and dropping of the surfaoe on the east side, and the t i l t i n g of the constant density surfaoes to counteract the unequal action on the f l u i d of the C o r i o l i s foroes beoause of the v e r t i o a l velooity gradient i n the lake. We oonsider f i r s t l y the t i l t i n g of the oanal surface using v e l o c i t i e s and distances representative of those found i n lakes. I f the e f f e c t s are n e g l i g i b l e i n a canal of the dimensions of the lake then the e f f e c t s would be n e g l i g i b l e i n the lake i t s e l f . The t o t a l difference 11 Jl ^ between the water l e v e l s on the west and east banks of the oanal was found i n seotion 3 C (d) to be given by 8 a V 0a 3" y 3 37 f? = _ * V Q W where If • 2a e width of canal. 3 ? 6 ,A s J X s i n f6 where s earth's angular v e l o c i t y , <j^ s l a t i t u d e for Kootenay Lake. <}> * 50°. * * 2 7 T s i n 50° = 5.57 x 10~ 5seo~]- £9 s 980 dynes/ 86400 o m V « • .13 om/seo. (Equation 27a) W * 2.5 x 105om. We obtain h e 2.5 x 10"*3cm. This difference i s completely n e g l i g i b l e so that no t i l t i n g of the lake surfaoe occurs beoause of C o r i o l i s foroes. The second e f f e c t a r i s i n g because of the v e l o c i t y gradient i n the lake i s considered next. We r e c a l l equation (29a). if = r l 34 , where Y- 4* « 3 x 10""8 _ r * _f where a s depth of lake or oanal To get an order of magnitude of -if we a r b i t r a r i l y ohoose the following data, whioh are more representative of summer rather than winter oonditions i n the lake. Suppose that at a depth of 100 feet the lake - 38 -temperature were 7°C and decreasing at the rate of 1° per 100 feet, i . e . £ | i « 4 x 1 0 - 7 / f o o t The slope at the oentre of the lake £ s 0 i s This slope corresponds to a drop of 3 inohes per mile o f the constant density surfaces from the dynamic surfaoe of the lake. This oannot be measured as the lake would have to be completely free from other disturbing effeots before suoh a low slope oould be observed. Sinoe Ve i s e s s e n t i a l l y inversely proportional to the v e r t i o a l s t a b i l i t y faotor h \l the slope might possibly become measurable i n the winter months when the s t r a t i f i o a t i o n i s low. However, no suoh observations are available at present. (d) Wind-driven V e r t i o a l C i r o u l a t i o n Sinoe no wind data has been studied, no attempt i s made here to oaloulate the magnitude of wind-driven c i r c u l a t i o n s . The oase of uniform wind stress i s considered for comparison with a set of oontours experimentally obtained ( H i d a k a ^ ) for the deformation of a damped semi-oiroular s t e e l plate to whioh a constant bending moment was applied at the hinged diameter. The agreement between - 39 -experiment and theory i s quite good. Figures 10 and 11 show the theoretioal and experimental contours with the same arbitrary normalization. In Figures 8 and 9 p r o f i l e s along & » 45° and 6 = 90° are given with the functions normalized to unity at the maximum. I t i s found that the maximum experimental de-formation occurs at 0.25 o f the radius along 6- » 90° while the theoretioal maximum l i e s at 0.3 approximately. Along 9 a 45° the agreement i s not as good, the deformations being larger than t h e o r e t i o a l l y expected. No explanations of the difference between the two functions i s attempted since we have no acourate knowledge of the oonditions under whioh this experiment was performed. - 40 -5. BIBLIOGRAPHY 1. Hidaka, K., and M. Koizumi, 1950: V e r t i o a l o i r o u l a t i o n as infe r r e d from the buckling of e l a s t i c plates. Geophysical Notes, V o l . 3, No. 3, Geo-physical I n s t i t u t e , Tokyo University, Tokyo, Japan. (Included i n Collected Ooeanographioal Papers, V o l . 1, 1949-1950.) 2. Love, A.E.H., 1944: A treatise on the mathematical theory of e l a s t i c i t y . Fourth Ed., Dover, New York. 3. Love, A.E.H., 1929: Biharmonio analysis and i t s applications. Proo. London Math. S o c , Series 2, V o l . 29, pp. 189-242. «4. Munk, W.H., 1950: On the wind-driven ooean o i r o u l a t i o n . Journal of Meteorology, Vol. 7, No. 2, pp. 79-93. 5. Munk, W.H., and E.R. Anderson, 1948: Notes on the theory of the thermooline. Journal of Marine Research, V o l . VII, No. 3, pp. 276-295. 6. Musk h e l i s h v i l i , N.I., 1949: Some fundamental problems i n the mathematical theory of e l a s t i c i t y . (In Russian) Academy of Scienoes, U.S.S.R., Moscow. 7. Prandtl, L., 1925: Bericht Uber Untersuohungen zur ausgebildeten Turbulenz. Z. Angew. Math. Mech., V o l . 5, No. 2, p. 136. B, Rouse, H., 1938: F l u i d meohanios for hydraulic engineers, F i r s t Ed., McGraw-Hill, New York. 9. Taylor, G.I., 1932: The transport of v o r t i c i t y and heat through f l u i d s i n turbulent motion. Proc. Roy. Soo. (London), A, V o l . 135, pp. 685-702. 10. Sverdrup, H.U., M.W. Johnson, and R.H. Fleming, 1942: The oceans, their physios, chemistry and general biology. Prentice-Hall Inc., New York. - 41 -APPENDIX I Biharmonio Analysis Biharmonio functions with prescribed boundary oonditions on the circumference of a c i r c l e may be con-structed by the following general method. I f 0 ( x , y ) , ^ i , ^ 2 a r e harmonic functions ( i . e . VZtf> * 0) then r x^, y ^ , x j ^ y etc. are biharmonio functions s a t i s f y i n g the biharmonio equation V 2 V 2 r ^ = 0. This may be v e r i f i e d by di r e c t s u b s t i t u -t i o n , e.g.: ^ ax* v y ^ T y ^ y Sinoe ^ and therefore S <t> are harmonic functions * y 2 v 2 i i = v s > 7 2 y j £ * 0 > y Sinoe the equation i s l i n e a r the sum or difference of two solutions i s also a solution. Introducing £ s x/a, ) a y/ a, p • *7a we oonsider i n p a r t i c u l a r the two biharmonio functions - 42 -„ ( 2 ) . ^ U ) , .(1) n - l > • n ' r n where » P n sinii0 t«^ 8| « p n Ooa we obtain: = p n V l f cos e s i n ne - s i n e c o s n * } = p n + 1 s i n ( n - l )6 n - i J y,,(2) . pa»-lJ o o s © c o s n © t S i n » s i n n e j s p11"*"1 oos(n-l)© n - i - J A series of suoh biharmonio functions may be written f = ^ { A n ^ n ) + G n * a } " £ [V^ 1 1 1 n a + G n P n + 2 ° o s n d ] Sinoe the set of functions p n s i n nfi , p n cos n * are also solutions of the biharmonio equation we get the general solution of the biharmonio equation t B n [ { A n Pn+"2+ B n P * ] 8 1 * - 9 + [ c a p n* 2+ D n p n ] oos n*] Any biharmonio funotion which i s regular for p < 1 may be expanded i n a series of the above functions and i s deter-mined uniquely by the values of p and 3 & given on the boundary p s i . For the purposes of t h i s thesis we oonsider the function: - 43 -i s a given funotion which may be e i t h e r a harmonic or a biharmonio function s a t i s f y i n g the condition jf ^  = Q __) % s f(j) for £ s 0 , where f££) i s a given function of i . £ J^must s a t i s f y the oonditions 2, f = 0 , « f£) for J . 0 V' = oonstant, _____ = 0 for p = 1 *P The constants A n, B n are obtained by Fourier analysis and the r e s u l t i n g series i n a l l oases oonsidered below may be summed by substituting w for p s i n n © where « s pe'* » ^ + imaginary part of the r e s u l t -ing complex function of <-a gives us the required function. For a more detailed disoussion of biharmonio analysis see Love, A.E.H.^ or M u s k h e l i s h v i l i , N . I . ^ . (a) The Throughflow Funotion Choose a 2/^ arctan [" 2 p sin6 1. L i - P* J This harmonic function i s well known from potential flow theory and gives the streamlines of non-viscous flow due to a souroe and sink at the ends of the diameter of a unit o i r o l e . Choosing the c i r c u l a r streamline p • 1 as a s o l i d boundary we require that - 44 -= 2/^ arc tan j 2 p s i n 6 1 +. Z ( A^ p*1*2*. B n pnJ s i n n 6 L i - P 2 J n s a t i s f y the oonditions ^ » 1, « 0 at p s 1 i . e . that no s l i p p i n g occur at the s o l i d boundary. The r e s u l t i n g series i s : ZL. H ( p 2 * + 1 - p 2 * * 3 ] sin(2n4-l)© Although t h i s series i s not convergent f o r P/• at p s 1 P (1-p ) p s i n 6 \ to whioh this series l - 2 p 2 oosZ** p 4 i converges for p < l s a t i s f i e s a l l the oonditions at p = 1. (b) The Green's Funotion This function i s well known from e l a s t i c theory and describes the d e f l e c t i o n of a clamped c i r c u l a r plate due to equal and opposite point loads at conjugate points (Love, A . E . H . ^ ) . It may be constructed by the above method by choosing « C £ s £ In s_ - s 2 In s 2 } where s_» s«- a r e the distances from the conjugate points |, {&- 0. ); ^  ,0+e»)respectively. This s a t i s f i e s the biharmonio equation and the neoessary boundary oondi-tions along the diameter. The Green's function i s then: G - C [B£ i n s£ - s 2 i n s 2 } ->I|A_ p n*K B_ p n j s i n n 0 - 45 -The c o e f f i c i e n t s A n, B, are determined by Fourier analysis and the r e s u l t i n g series i s m^ s i n m 0 o s i n m © •h[2 p 3 q 3 - 4 p 3q - 4 p q 3 ] s i n © c s i n * j which converges to: C [ In t 2 . - i n t _ \ where t 2 s l + p 2 q 8 - 2 p t oos (9 ± ©•) The normalization i s accomplished by the method outlined i n section 3 A (o). (o) Wind Stress Funotion Choosing the biharmonio funotion P 2 p « - 1 K p s i n 9 s a t i s f y i n g the oonditions we require that the t o t a l function f s - l K p 2 sin 2© -t £ $ A n pn<"2«- B n p nj s i n n © 2 * s a t i s f y ^ r 0, j>J_ s 0 along p s i . > P The r e s u l t i n g series i s : - __ 21 f f j ; 1 1 _2n+l r 1 2n+3l X n * °l Lin^T 2n-TTj » jg-J* J < ^ s i n (2n-»-l)EJ - 46 -which converges to: sin© f 2p 2 sin2© tan" 1^ 2P S J^«f ] - (1-P 2) 2r sjg_2©ln[ l»2p oos»-l-p2~| 2p2 - L Z. Ll-2p oose + p * J - oos 2 ^ t a n _ 1 f 2p sin© 1 | [ l i z p — J J j - 47 -APPEMIZ II Evaluation of Thermal Cirou.lati.on Functions  Case I. The evaluation of V"(p,0; 9t) as defined by equation (13). X T « / d q B ( p , * ; q,©,) where B(p t d ; q ^ J i s defined by equation (10), We oonsider l T ( p , 0 ; 6,) where Of i s replaced by £ 0 . Let ^£.a © ±fl>0 , s 2 r p 2+q 2-2 pq oos^ , t 2« 1+p2 q 2-2pq oos^ B(p , 0 ; q^^©) m a y t e written: B t p ^ j q ^ c V , ) = B l (P» 4iJt) " B l (P» ^ . ^ ) where: B x (p, q,* ) = s 2 In s 2/ t2 Then: where: S ( P t 0 ) = / ^ ( p , q, f* ) d q o Making the transformation z . s ^ -^poos^, and wr i t i n g p z - ( l ^ p 2 ) oos^ s w, we obtain: B_(p,q.*) - {z 2^p 2sin 2^/ln|_z 2+i> 2sin 2^f - l n C w V e i n V j ) * ( p . 0 - / 1 b 2 f P 2 i s i n 2 ^ { l n L z 2 * . p 2 s i n 2 f ] - l n [ w 2 * s l n 2 * j } dz -poos^> - 48 Integrating by parts and decomposing the r e s u l t i n g integrand into p a r t i a l fractions whioh may be integrated d i r e o t l y , we obtain the i n d e f i n i t e i n t e g r a l : | z 3 , + p S sinV*]fln£z 24-p 2 sinV] - ln|> 2«-sin 2<£] j -f. (1-p 2) oosf z 2+ 2 fo-P2>2 o o s 2 * - U-p4)sin2<}>} z 3p 3p^"c ^(lj_ p l) 2{(l-p 2 ) o o s V ~ ad$p2) s in 2 ^ lnLAsin 29tJ 3p 3 3 4 __ s i n < 3 /sin 0/ L /sin<*>/. 4- 4 p s i n ^  arotanf z 1 L j s i n * | J _ 2 sin 2<fr(3(l-p 2) 2 oosV - (l"3p 4) sin») 1 arotanf" w "1 3p 3 |sin£| Osin^J Evaluating t h i s expression between the l i m i t s z =-p cos# and z a 1-p oos ^  we obtain: F(P»*) = p_/oos 3f> -f- 3 sinVoos*} In p 2 3 1 , 1-p 2 i -2(l*p 2) 4 P e o s * + 2(2-p 2) c o s 2 ^ ] 3-p 2 1 .2. „.(l-p 2) 2QQS<?>f(l-p 2) o o s 2 ^ -3(l^p 2)sin 2fli}ln[l4-p 2-2p cosj_) 3 p-4- 4p 3 sin 4 ft arotanf fslngj "j 3 )sin £>| Lp-oos^J - 2 sin2^3(l-p 2)^oos 2 -(l-3p 4 ) s i n 2 / j_l arotanf (sing/1 3p 3 (sinftj t'l-oos^J P - 49 -Making use of multiple angle relations the above equation may be simplified, to the following: 6" Etp,^) = P f 3 cos^ - oos 3ft} In p 2 f (1-P2> { -3p2-tP oos / + (2-p2) oos 2 ft} 3p< 2,2 f (1-P2) j (2<-p2) cos 3/ - 3p 2 oosft} In [l#-p2-2p o o s + J-3 j |3 sinft - sin 3ft/j arotanf j s i n ^ t l 3 Lp-oos^J _ 1 f(2-3p 2) sin 30*3p 2(2p 2-l)sinftj sln ft arotan ~3p2~* |sin?| f Is in ft I 1 I _L -cosft I Case II. The evaluation of $ (p,©; © 0 ) a s cLsf—iscL by equation (18). iMp.e; - / X qaq B(p,©;q,<90) o As before, we write this as a sum of two integrals $"(?,©;©.) *f B-jJp.q,^ )qdq - / Bitp,q, ftv )qdq o -/o . • - . Making the transformation defined in Case I above we obtain - 50 -/l •* l-pCOS«5 /L-pOOSft Bl(p»<l»^ )<iag. = J B;_(z,^)zdz +-p oosft/ B;_(z,ft)dz o -poos <^  -^poos^ * hl ( p . ^ ) f P oosft F(p,ft) where F(p,ft) is defined in Case I above. Integrating the expression for h;_ by parts and decomposing the resultant integrand into partial fraotions whioh may be integrated directly, we obtain the following indefinite integral: / Bx (z,ft)zdz « £ z V p 2 s l n 2 f z 2 j [In[z 2+p 2sin 2ft] -ln[w2-*-sin2«ftj} (1-P8) oosft z 3 ^  f ( l - p 2 ) 2 o o s V - ( l - p 4 ) s i n V } z 2 6 p ( [ J 4p' ^ (l^p 2)oos<* ( (l-p 2) 2cos 2ft + (2p 4-3)sin 2 f t j z " 2p^ <• 4__. 4^ , r 2 2_, 2 + P sin <P In [_ z -M? sin ftJ 4 f 1 |"[(l-p 8) 8oos^f (l-p 4Jslay 2-4((l-p 2) 2(2-p 4) oosV sin2ftj -p 8sin 4*} ln(w2-<-sin2*) ^ 2 J ( l - p 2 ) 2 s i n 2 ^ cos^[(14p 2)sin 2g-(l-p 2)oos 2<l>J p4"( * j 1 arotan j w j i 1 |sinft| L IsinftlJJ Evaluating this expression between the limits z • -p cosft * 51 -and z = 1-p cos 56 we obtain: ^ ( P . f t ) s - p_, i n p 2 4 .(!_£_)[(lj.p2)f (9-.4p2-9p4) cos^ -(l-p 2)oos 2ft-4(l-p 2)oos^ 2p ( 2p • 3p a p p 2 f ( 1 - P 2 ) 2 | [(l-p 2)cos 2Wl+p 2)sin 2«J 2 - 4(2-p4)oos2ftsinVj* 4 p K { m f i + p 2 - 2 P costf} 4_ 2(l-p 2) 2sln 2*oosft 5 (H-p2)sin2ft - (l-p 2)oos 2^} * x ( 1 arotanf fsin ft< "] I |sinft| I 1 -oosft I Introducing h r h i *- 'p4 In p2+- ( l ~ P 2 ) (1+P2) A A . ^ p p 4 m p 24 - ( i z r 2 ^ M j L " 2 ' 4 4*p< and expressing the other terms i n multiple angle form, we obtain a much simpler expression for h(p,* ): h(P.*) = - (lr_l)((5-9p 2) oosft - (l - p 2 ) c o s 2^- (1-p 2)oos 3f] 2p t 3 2p - pk (1-P 2) (fp 4-2p 2oos 2ft-*-oos 4 ft] In fl+p 2-2p oosftj r ~ 1 2~ 2p - T s i n 4tf> - 2 D 2 s i n 2 1^ . \aiai\ [l-oos<|, [ k ft p ] sinft>arctan J* lsin</>| 1 J F i n a l l y $(p,*;*<>) = h(p,£) - h(p,<^)i-p{aos^F(p, ftj-oosj4fF(p, ftj j 

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