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Rip currents on a circular beach. O'Rourke, John Cameron 1970

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RIP CURRENTS ON A CIRCULAR BEACH by JOHN CAMERON O'ROURKE B.Eng., McGill University, 1966 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE » i n the Department of Physics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December, 1970 o In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Brit ish Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of • Physics The University of Brit ish Columbia Vancouver 8, Canada Date 17 ~J)f>r 7P i i i ABSTRACT A mathematical model i s developed which extends the theory of r i p currents developed by Bowen (1969b) for a straight beach to curved beaches where r a d i i of curvature are large relative to the width of the surf zone. Nine forcing terms are found to cause r i p current systems. The terms are functions of the longshore variation i n wave height and angle of incidence of the incoming waves at the breakers. The model i s applied to the case of a circular beach with conical nearshore bottom topography. A large r i p current component-- i s found to exist which i s inversely proportional to the radius of curvature of the beach. Another significant r i p current component i s found to be proportional to the variation i n the angle of incidence of the waves at the breakers. This component would cause r i p currents on a straight beach where some irregular offshore topography caused some variation i n the incident angle of the incoming waves. Another component r i p current was found which was essentially the same as the one predicted by Bowen (1969b). i v [TABLE OF CONTENTS Page " ABSTRACT i i i TABLE OF CONTENTS i v LIST OF FIGURES v i LIST OF SYMBOLS v i i ACKNOWLEDGEMENTS x LPART I - THEORY »- Chapter 1. INTRODUCTION 1 2. THE MODEL 3 a. The Coastline 3 b. Wave Height and Refraction 3 c. Surf Zone Geometry 6 3. THE EQUATIONS OF MOTION 12 I*. THE FRICTION TERMS 17 a. Inside Surf Zone , 17 b. Outside Surf Zone 20 5. THE FORCING TERMS 23 a. Outside Surf Zone 23 b. Inside Surf Zone 25 c. Radiation Stress Tensor 26 d. Forcing Terms 27 e. Discussion 32 6. SUMMARY OF PART I 36 V Chapter Page PART II - CIRCULAR BAY SOLUTION 7. THE PROBLEM 39 a. The Equations 39 b. The Boundary Conditions 0^ 8. THE SOLUTION • • ^ a. General ^ b. Term g ^ 6 c. Term c 55 d. Term e 58 e. Term a • ^° f. Term f 6 2 g. Term b 63 h. Term i 65 i . Term h 6 ? 9. GENERAL WAVE APPROACH . . . 7° a. General . 70 b. Term g 72 10. SUMMARY OF CONCLUSIONS 7^  LIST OF REFERENCES 8 0 APPENDIX A - DISCUSSION OF SET-DOWN 8 1 APPENDIX B - ORDERING OF FORCING TERMS ^ v i LIST OF FIGURES Figure Page 1. Plan View of Beach k 2. Angle of Refraction k 3. Profile Normal to Beach 7 k. Circular Bay *1 5. Profile of T j f x / O ) 5 3 6. Streamlines, Term g • 53 7. Streamlines, Term c 57 8. Profile of Te (*><>) 5 9 9. Streamlines, Term e 59 10. Streamlines, Term a . . . 61 11. Profile of ^ ( x . O ) 6k 12. Graph of COS 9 - cos 36 6k 13. Streamlines, Terms f and i 6k lk. Streamlines, Term b 66 15. Profile of tyifcO) 66 16. Profile of Mfafi) °9 17. Graph of COS 0 + cos 39 69 18. Streamlines, Term h 6 9 19. Refraction Pattern, General Approach . 71 20. Graph of tfWand tiC(B) for General Approach 73 21. Streamlines, General Approach Term g 73 22. Profile of + 4c a t t h e breaker line 75 23. Profile of Ur at the breaker line due to 75 terms g and c for <J>0» IS9 . v i i LIST OF SYMBOLS A large number of symbols have been used throughout this thesis and they are defined within the text as they are introduced. They are summar-ized below for the convenience of the reader. f\ i constant of proportionality between set-up and wave height (see (2.21)). ' where j s a, b, ..., i : constant coefficients of the forcing terms. C : constant of proportionality for bottom friction (see (3-10)). & : average value of Hk . (J : depth from average water surface ^ to bottomj =f + h E e(e) 3 H Ht U J . K Ic energy density of waves. & . dependent factor of acceleration of gravity, wave height, twice the amplitude, height of waves just before they break, depth from undisturbed water surface to bottom, inertial terms in equations of motion, ratio of surface slope to bottom slope, a constant equal to JL^tK* , (see (5.17)). total horizontal momentum per unit surface area; YY\ : rate of change of d with respect to r, M - ~" ftf^ : slope of ineahcsurface, = inside surf zone. v i i i /77£ : slope of bottom, ml = Jh/Jr. N •• r rl n n « s U,tr •• 6 • n -ratio of set-down at the breakers to surf zone width, amplitude factor of O dependent component of ftl (see (2.1)). wave number of incoming waves. radial co-ordinate of polar co-ordinate system. radial position of breaker line. radial position of shoreline for undisturbed water surface, radial position of shoreline for disturbed water surface, width of surf zone, fsb — Yg — . variable defining distance from average shoreline, s=rs- r ... value of S at K— mean transport velocity, tensor notation. mean transport velocity components in the x and y directions respectively. {Jr^e'' mean transport velocities in the t~ and & directions respectively. cartesian co-ordinate axes defined on the bay (see Figure 1). cartesian co-ordinate axes defined on the incoming wave (see Figure 2). ratio of wave height and depth inside the surf zone, // =yj . (see (2.9)). amplitude factor of & dependent component of d for term a, 6 = mean surface level, f^= O for Undisturbed.surface. amplitude of O varying component of t9 , fi~A Dp for refractive variations. ix amplitude of 0 varying component of ?^ from edge wave effects in term a. angular co-ordinate of polar co-ordinate system (see Figure 1) ratio of wave height to depth just before waves break, tf/>+=2<JL (see (2.9)). longshore scaling distance for refractive variables, longshore scaling distance for edgewave effects in term a. force per unit mass vector due to divergence of radiation stress, tensor (see (3.8) and (3.9)). tensor factor of radiation stress tensor. $+jg i s a function of (p only (see (5.18)). ~ angle of incidence of waves at breaker line (see Figure 2) . transport stream function (see (3*23) and (3*24)). ACKNOWLEDGEMENTS I would like to thank Dr. P.H. Leblond for directing me to this topic and for several helpful discussions concerning this work. I also wish to thank my wife Andrea for patiently typing this thesis. This work was done with the aid of a National Research Council grant. 1 CHAPTER 1 INTRODUCTION On many long sandy ocean beaches signs are posted to warn swimmers of the danger of being swept out by near shore currents to water depths over their heads. These currents flowing away from the beach are commonly known as r i p currents. They are associated with variations of the momentum flux of the incoming surface waves and are found at specific locations along a beach at any one time (see Shepard, Emery and Lafond, 19^ 1 for an introduction to the subject of r i p currents). In this thesis the term " r i p current system" i s used to represent the complete flow pattern of near shore currents, including both seaward flowing currents and longshore currents. This thesis discusses theoretically the effects of bottom topography on r i p current systems along a beach and presents a theoretical solution for a circular bay with a conical bottom. The purpose for studying this subject was to gain some physical insight and understanding of the various effects of bottom topography on r i p current systems. In 19^1 Shepard, Emery and Lafond wrote a mainly descriptive paper on r i p currents, their flow patterns and geological effects on the beach. Various papers have been presented since then attempting physical explanations for the causes of r i p currents. But i t was Bowen's paper (1969b) which for the f i r s t time presented a satisfactory mathematical model of the r i p current process on a straight beach. His model was based on the theory of the radiation stress (Longuet-Higgins and Stewart, 1964) of waves proceeding into shallow water. In this thesis we shall expand upon Bowen's model to include bottom topographical effects and beach curvature. 2 This thesis is in two parts. Part I deals with developing the theoretical model for a general topography and Part II uses this model to solve for rip currents in the particular case of a circular bay with a conical bottom topography. We chose to test our model on a circular bay because i t seemed the next level of complication above a straight beach as modeled by Bowen (1969b). It also provides insight into the effects of beach curvature on rip currents. In section (2c) we derive formulae for the location of the shoreline and the breaker line (the line describing the locus of points at which the waves break), while this is not a very sophisticated development, neverthe-less I could not find similar calculations in any of the literature which I have read. 3 CHAPTER 2 THE MODEL 2a The Coastline In Figure 1 i s shown a sketch of a sample beach and a definition of the co-ordinate systems. It i s assumed that the beach i s smooth. By this i t i s meant that the radius of curvature of the shoreline i s much larger and that the bottom slope i s small. The breaker line i s the locus of points at which the incoming waves f i r s t break on the beach. The shoreline i s the locus of points of the time averaged (over several wave periods) location of the land-sea boundary. The origin of the co-ordinate system i s determined by the average location of the centers of curvature of theobeach within a given region. 2b Wave Height and Refraction When uniform deep water waves proceed toward a non-linear shore, we expect the waves to be refracted by the bottom topography and thereby to have variable wave heights as functions of their longshore location. The subject of refraction i s discussed i n the US Hydrographic Office Publication Number 23k. Analytical and graphical means are available for estimating, with reasonable accuracy, the refraction patterns of waves on beaches of any shape. However, i n this thesis i t i s sufficient to make an educated guess at the refraction pattern of a given beach based on the discussions i n US Hydrographic Office Publication Number 23k. than (at least by a factor of five) the width of the surf zone (rt-ri) The radial distances ^ and w i l l in general be functions of Q Figure 1 Plan View of Beach l wave crest Figure 2 Angle of Refraction L We are interested i n determining the wave height at the breaker line as a function of position on the breaker l i n e . We l e t : /Jt?-- D( 1+J»#.(*)) (2.1) where L) i s the average height along the breaker line, i s the amplitude of the varying component of Hi * and i s the 9 dependent function of unit amplitude describing the variable part of /^^> . In Part II we w i l l consider a circular bay with deep water waves advancing i n the positive y co-ordinate direction and we take to represent a reasonable refraction pattern the relation: /// — < COS Z@ and so: J _ ^ , _ \ HC= D( COsle) (2.2) The functions we choose to represent H,[&) and et*) as discussed i n the preceding and following paragraphs serve merely as examples of possible incoming waves. If one wished to estimate the rip currents on an actual beach one would f i r s t have to measure or otherwise determine the //I (&) and function applicable to that particular beach and wave approach. We are also interested^in the angle of incidence of the wave with the breaker line . To deal with this analytically we define a co-ordinate system as shown in Figure 2. The X axis i s i n the direction,of the phase velocity of the refracted wave at the breaker l i n e . The angle of incidence $ i s the angle between the X axis and the radius vector . The Y axis i s p a r a l l e l to the wave crests. We l e t : where i s the amplitude and G (&) i s the variable function 6 f order unity expressing the 0 dependence of fi . I n our circular beach problem we w i l l l e t : On a large smooth beach we expect the Incoming waves to have had sufficient time to adjust to the bottom topography such that 4>* w i n be at most 15° • Thus we make the approximation: Si* (p = 4> cos <p - / 2c Surf Zone Geometry In this section we discuss the effects of wave height H on the shoreline. To understand the following discussion, i t i s necessary to understand the theory of set-down and set-up as described by Longuet-Higgins and -Stewart (1964). Figure 3 shows a cross section normal to a beach. In general the r co-ordinate w i l l not be normal to the beach everywhere, but for the purposes of this discussion we assume this to be the case and this saves us introducing a new co-ordinate system which would, in general, have components normal and parallel to the beach. The bottom slope ml i s assumed constant and the set-up theory indicates that the surface slope i s also constant for a given bottom slope and wave train (see equation ( A ? ) ) . In the following discussion we shall derive relationships between the surf zone measurements , Yos , fel , IO (see Figure 3) and the height at the breaker line of the incoming waves From Figure 3 we see that: c ML fos — ^Isl - fl the run-up distance to5 i s thus: (2.6) r _ , f ^ y>L \ w (2.7) I OS — ' Figure 3 Profile Normal to Beach 8 Now from (A4) we obtain: 11 li^" fi = - TO*) -where q i s the wave number of the incoming waves. We denote by and HI respectively the wave height at JTS just outside and just inside the surf zone where: Ml+'IA , flf'Kti (2.9) M*/M~ - \/Y where "\ and Y are constants of order unity. Thus from (2.8) and i n shallow water we obtain for the set-down at the breaker line: = 1 2r I MX Again from Figure 3 we have where , then from (2.9) we obtain: = X nL (2,U) or, for the total width of the surf zone, including run-up, we have: Also we have for the undisturbed depth at the breaker l i n e : k - m n. (2-12) Thus using (2.12), (2.11) i n (2.10), we may write the set-down t£0 as: nk = jl 2^Yj$L)ni (2.13) or, for short, ./ / n s where — j . -«a * j/v /V- = i_*L (2.15) and N « / Thus (2.7) becomes: ^S = A^(mt"^ n i (2,16) 9 From Figure 3 we get for the distance from breaker line to the undisturbed shoreline. Ho - r4- = -a/) rsl or Hg_ - / / _ _ /^y./!/ 1 m / ~ —rZf (2.17) Thus from (2.15) and (2.17) we obtains (2.18) which yields: ^ = ^ ^ + j\J must be positive and using the binomial expansion we obtain: Thus /I/ i s a constant. From Figure 3 we see that: y ^ (r-rl) - ?l = /"r(n-rty+y»?(r<-n)'-?i ( 2.i 9) Using (2.17) and (2.14) on (2.19) we obtain: ( >"l 1 ' ^ (2.20) From (2.11) and (2.20) we obtain: f *(*fjg) #4* + m/™)* /?M*+^(r-n) (2.21) 10 Thus, according to our present model, the set-up i s determined by the breaker height and the slope of the bottom (note that KMl, ; see Appendix A, equation (A7)). Also, from (2.1?) and (2.11) we obtain: lassuraing Af rr _/l <^<^ j iFrom (2.16): (2.23) vThus, we see that the distance of run-up on the beach ^os and the position of the breakers JTq^ are dependent on the bottom slope and the breaker height. In order to gain some familiarity with the magnitude and accuracy of these equations, we compare calculated results based on these equations with experimental results taken from Bowen, Inraan and Simmons (1968). These results are shown i n the following table. measured calculated EXP Hi+ cm ht cm K cm cm cm m^  cm cm cm cm cm X 71/3 4.40 4.1f 0.27 75 0.17 1.48 .022 52 19 71 0.30 1.5c 1.06 51/k 6.60 5.0 0.32 85 0.19 2.07 .026 61 19 80 0.50 I.56 1.32 35/7 7.75 5-9 0.39 110 0.18 .032 72 28 100 n.^4 20 1,31 35/15 13.0 9.7 0.37 165 0,43 4.65 .030 120 44 164 0..96 3.6 1.34 where m^  = 0.082 Note that we are able to predict quite accurately. Our predic-tion of ty/na^ i s within 30$ of the measured value and h i s almost 100$ too large. Bowen, Inman and Simmons (1968) explain this discrepancy by noting that near the break point the wave form i s not sinusoidal and therefore, our set-down theory i s not reliable at this point, since we assume a sinusoidal wave form. 12 CHAPTER 3 THE EQUATIONS OF MOTION In the following discussion the cartesian tensor notation w i l l be used. The equations w i l l be kept general - sb\ that they w i l l apply to a general coastline without restriction to a circular bay configuration. However, we shall assume that the radius of curvature of the coastline i s always large; that i s , at least five times greater than the width of the surf zone. In deriving the relevant equations of motion we use the equations of conservation of mass and horizontal momentum as given by Phi l l i p s (1966), and we assume the time averaged motion to take place only i n the horizontal plane. Thesefequations are respectively: and jLg+^(%/^+^)=7: ° # (3.2) where ~fc= -y0jdj£ and fa -=//?(?+J>) % We assume that the short-term time-averaged variables are steady, so we obtain: i . J (3<t 1 - ~ ~7* (3.4) By expanding (3.4), using (3.3) and dividing by ^/Mg+A) we obtain: (3.5) 13 Writing (3.5) i n cartesian co-ordinates with and 2& = ^ where V and are the net velocities i n the x and y directions respectively, we obtain: 7< °LH + ISofji =z - + 7*. ( 3 . 6 ) and where J * J T ^.8) and ( 3 . 9 ) Equations ( 3 . 6 ) and (3*7) were derived for an inviscid f l u i d . For a viscid f l u i d we add the extra f r i c t i o n terms /j** f?y . The f r i c t i o n force should befecomposed of a bottom f r i c t i o n term and an eddy viscosity term. In this paper, to simplify our equations we use only the bottom f r i c t i o n term and assume that the eddy viscosity i s negligible. That this i s not the case for the entire region over which we are seeking a solution w i l l be seen later; nevertheless, we shall s t i l l obtain useful qualitative results using this assumption. The bottom f r i c t i o n force acting on a total column of water of unit area and height d with mean velocity CA ^ we w i l l assume to be a linear function of the mean velocity. Thus the bottom f r i c t i o n force on a column per unit mass of the column i s expressed as: p - cK p <zts-14 where c i s a proportionality constant with units of length over time. In polar co-ordinates we obtain: D _ _ Cl/? / ? , = - g f» (3.10) ~cj~ ' ^ Thus the complete equations of motion i n cartesian co-ordinates are: and iXJj^T^ LrJlT- - * s £ + J Z + R * J# " 7f 7 (3.12) In cartesian tensor notation these are: and i n vector notation (3.13) (3.14) We may eliminate by taking the curl of (3.14) which gives: curl (U = cur/ R + « W 2 Transforming (3.15) into polar co-ordinates gives: JT* Q / J V . +J9 = Jfo - ^ + A - (3.16) 15 where the i n e r t i a l terms are: Tr = tJ^JjS + ^Jj^r-C^ (3.1?) Jr ir j9 r T* = + (3*18) e/r r r and subscripts r* and # indicate components i n the direction of A* and 0 . Equations (3.17) and (3.18) are nonlinear. In his theoretical paper on r i p currents, Bowen (1969b) neglects these nonlinear terms in order to obtain a simple analytic solution. Arthur (1962) shows that the contribution of these nonlinear terms i s to narrow the currents proceeding to deeper water and to widen the currents proceeding into shallow water. Bowen (1969b) solved this nonlinear equation for a linear beach using a computer and the significant difference between his computer solution and his simplified linear analytic solution was this narrowing effect. Therefore, we expect to obtain reasonably good qualitative results by neglecting the nonlinear terms and equation (3.16) becomes: _ curl z R ~ curl* 2" (3.19) or ry& Sr r rje ~T~ (3.20) We c a l l c u r l f c the forcing function. In Part I I , we shall apply equation (3*19) to the case of a circular bay and obtain an analytical solution, however i t i s important to note that (3*19) i s quite general and does not only apply to a circular beach. 16 From (3«3) we get: or using the definition Z/^ = .Af^ w e obtain: T>iv(J2(*) =o (3 .22) which i s the integrated continuity equation. We introduce a transport stream function if (Arthur, 1962) i n polar co-ordinates which Identically satisfies ( 3 . 2 2 ) . We l e t : ^ = ~T T7e (3-23) ^ = l h 7 F ( 3 . 2 . ) where J A as previously defined. 17 CHAPTER 4 THE FRICTION TERMS It i s convenient to speak of two zones: the surf zone ^ A* ^ £ ^ O 4. &c\ and the area outside the surf zone. We shall see i n the following discussions that there are specific differences between these two zones. 4a The Surf Zone Let us examine the individual terms of Ctitl x R = ol£i-JEr *- 4-r • We wish to examine the relative sizes of these terms so we introduce the following ordering scheme. Let the primes denote.non-dimensional variables of order unity. trr= VLwr' r = nt' , i (4.1) where Vj, i s the maximum expected velocity. It mayvbe of the order of 2 to 4 knots for a linear beach (Shepard and Inman 1950)• We introduce two differentials for d representing i t s changes i n the r direction which over the width of the surf zone w i l l be of order /)£ and i t s change i n the & direction which w i l l be mostly due to changes in ^ i n the & direction and are of order (see (2.21) and (2 . 1 ) ) . YjI i s the average width of the surf zone 18 and i s the average s t i l l water depth at the breaker line* From (3.10) we obtain: Jt d jr Jx jr (4.2) Using (4.1) we obtain: Using (3.10) again, we obtain: C».3) slit = - . f ^ f +±<j?d£ and using (4.1) we obtain: (0.5) The terms i n brackets i n (4.3) and (4.5) are of order unity. Thus to compare dE>^/lr\f& to dft&^/t* we take the ratio: s/Et./J£z = - /-ct4 = a ru We assume we have a large bay and so ts&fio / . Thus we can neglect the term relative to the q/P&fif' (4.6) term. 19 (4.8) Again using (3.10) we obtain: sr - C (4.?) and using (4.1) we get: Ss. - - s ^ iy*') Comparing /tj£ with we obtain: ' V o ' r r*MI Tint r„ <M> for a large bay. Hence we can neglect the term relative to O^^P . Therefore i n a large bay and inside the surf zone oft region we have: CUr/z. /T -=r (4.10) Using (3.10) and the transport stream function defined by (3.23) and (3.24) we obtain for (4.10): c*rl * $ = - 2 ^ ) ( l f . u ) where we define: — /?? inside the surf zone. (4.12) 7r 20 Later we shall use (4.11) i n the l e f t hand side of the main equation (3.19) for solving for the current inside the surf zone only. 4b Outside the Surf Zone In this region we cannot in general neglect the same terms of curlft as we did i n the surf zone because we are no longer dealing with a narrow zone at a large radius r from the origin. We-shall use the f u l l expression for curl $ i n terms of the transport stream function *V . We obtain: Jj£ id- + J* -L*Jt 7 (^.13) where JRr-rjs> (4.14) tc/1 21 Equation (4.13) can be simplified. V/e assume that outside the surf zone the depth d of the mean current i s equal to + Ilk . Thus we say that the mean current does not extend below the depth k\j which i s the depth of the undisturbed water at the breaker l i n e . Shepard and Inman (1950) found that r i p currents seaward of the breakers do not extend to the bottom. Although their results do not entirely support our assumption they do point towards that direction. This assumption should be regarded as a crude simplifying one. Thus we get We shall also neglect the term containing 1 rJV JA this we shall compare the -^7 and </t the ordering relationships (4.1): To justify terms using -khi [iLuiJj'+j'jg-:] TT/x. LM J&' je'J (4.15) also -2- <& JJ = -JrVr4/=->VUL 1< (£.<&' (lt-16) 22 The primed terms we assume are of order unity. The ratio - / and so the term containing will be insignificant relative to <J ^/d& term. We therefore neglect the d& term. Thus equation (4.13) becomes: To conclude our discussion of the friction terms, we now have two forms for the expression cutis. R of equation (3.19). These are as follows: inside the surf zone and curl * R = - £ (£!t+.± <J*+J- £!t) <"-20> outside the surf zone. We shall now discuss the radiation stress terms so that we may put equation (3*19) into a workable form and solve for j 23 CHAPTER 5 THE FORCING TERMS In this chapter we shall discuss the C Utf z terms i n equation (3.19) and as defined i n equations (3*8) and (3»9). We c a l l these terms forcing terms because they represent the driving forces, due to the excess momentum flux of the waves, which cause the r i p currents. 5a Outside Surf Zone From equation (3«^ ) we get: For a linear beach on which there i s no variation i n the longshore mass flux ( ic. J$y-0 ) we have from (3.3) <JJJiL =. O . But Mx must be zero at the shore and hence /vx = C? , thus cs(-y? -and (5»1) becomes: ' ' ( 5 * 2 ) since ^ & * ° r a P^ a n e wave traveling i n the positive x direction. From (5»2) we obtain the relation: f - / ^ L <5-3> * tint, a.$f> where /V i s the wave height and ^ is the wave number of the advancing wave (Longuet-Higgins and Stewart (1964)). Bowen (1969b) used this derivation for to show that the gradient of the radiation stress i s balanced by the induced pressure f i e l d and that there are notnet 24 forces outside the?surf zone that might produce circulation patterns. However, (5*3) i s based on the assumption that J^ff- ~0 which i s not i n general true for a r i p current system. The discussion i n Appendix A shows that the i n e r t i a l terms may be neglected i n equation (5.1) for a r i p current system, thereby making (5*2) and (5*3) reasonably accurate where r i p currents are present. Using the y-component of the radiation stress (Longuet-Higgins and Stewart (1964)) where and equation (5.3) we get: which confirms Bowen's finding as stated above providing (5*3) can be used for the r i p current system. For the moment, using the X,iT co-ordinate system shown in Figure 2, equations (5.3) and (5*5) become: dI>2L = e> (5-?> respectively. Note that Sy^ ~ Sxy — O $ (Longuet-Higgins and Stewarti (1964)). Substituting these relations into equations ( 3 « H ) and (3.12) where x and y are replaced by X and I respectively, we obtain: Ux d2<*. + Ky ^ ~ /?x C5,8) 25 A ' (5-9> or i n tensor notation using as a generalized co-ordinate: % ^ ^ ^ (5.10) Taking the curl of (5*10) we obtain: cor/z /? = curlz I (5.1D where _ Z ~ was defined i n polar co-ordinates by (3.1?) and (3.18). (The under a symbol indicates a vector quantity). However, since we neglect the nonlinear terms we obtain: curU R = O (5.12) outside the surf zone where the gradient of the radiation stress balances the pressure gradient. 5b Inside Surf Zone Bowen (1969) assumed that inside the surf zone the height of the broken wave i s directly proportional to the mean water depth d. // = / J ^ .13) Also i n the surf zone the water i s shallow and hence from (5«4): 26 Using these relationships Bowen (1969b) showed that the longshore component of the radiation stress i s not i n equilibrium with the pressure component there are current generating forces inside the surf zone. 5c Radiation Stress Tensor As the waves approach the bay from deep water, they are refracted and line (Figure 2 ) . I f our beach i s large, as we have assumed, then the waves should be almost plane. We shall assume this to be the case at the breaker line and inside the surf zone. Refering to Figure 2 where the X axis i s the axis of advance of these plane waves, we have the radiation stress for the X and Y axis given by Longuet-Higgins and Stewart (1964) as: longshore. Thus we have Ctirl * f$ ^- O inside the surf zone and inclined at an angle to the normal of the beach at the breaker sitK = zjr , s-. yy — a. •** 5 ; x y y* where E i s the energy density of the waves (5,15) or using (5-13) for H inside the surf zone we obtain: (5.16) (5.17) , a constant. 27 Transforming -S^y^ from the X, Y co-ordinates to the f^G -ordinates by a rotation of co-ordinates through the angle § co. obtain: 3 ~«2 sv*/* *A> sin 2& 5 ' * (5.18) or (5.19) inside the surf zone. T U / S i s the tensor shown inside the square brackets in (5.18). Here the indices o< and each take on the values 1 and 2. We assume that J and $ are known variables for a given beach with a particular wave system. They can be measured, calculated or assumed, (see Breakers and Surf, HO Pub. 234). 5d Forcing Terms (Inside Surf Zone) Since we shall be using polar co-ordinates we use the identity: curl z I ~* rjt role r (5.20) where from (3.8) and (3«9) we have: (5.21) i n cartesian tensor notation. Prager (1961, p. 37) gives the polar co-ordinate components of the vector . Using this relation on (5*21) we obtain: 28 (5.22) and (5.23) Now we must determine - ^ o £ ^ i n polar co-ordinates i n -terms of independent variables which can be somehow measured directly from the surf zone or calculated. These variables are <t> and Ml* Thus we are now prepared to calculate cut/ a: 0" i n terms of known variables. From (5«23) we obtain: Thus substituting (5.24) into (5.22) and (5.23) wo obtain: J t * j t rje rile r t (5-25) %*-M$xJd- 4 ^ - 2 k L $ * Jr F7& rj* r (5.26) 29 Notice that we have not neglected the *J<IJQ/& terms here as we did in discussing the friction terms because they provide significant forcing terms to the overall forcing function Ctftl 2 T . From (5.25) and (5.26) we obtain: ^ -U ih^dd- AJS% - (5.27) and (5.28) and 2 * = -2kin. &-3Lk$* M (5.29) Notice that equation (5*30) is a general equation regardless of the shape of the bay or its bottom contours. The terms have been numbered from 1 to 9 and grouped into three categories, I, II and III. Category I includes a l l the terms which would be non-zero i f the waves advanced normal to the beach, that is <P = & . Category II includes a l l the terms which would be non-zero for a wave system in which 4^  will not be zero everywhere and the mean water depth d is not a function of & Thatjis.j these terms represent forcing functions due only to the non^zero angle. 31 of incidence $ of the wave system. Category III i s the l e f t over terms which are non-zero when Categories I and II terms are both non-zero. We can calculate the mean water depth d i n terms of the breaker wave height as discussed i n section 2c. We obtain from equation (2.21) the following: (5.31) where $ i s a constant; and are the mean slopes of the water surface and beach bottom respectively. Thus we obtain from (5.31): ~~fP K n (5.32) and ot& </& y & '5.33) Also, inside the surf zone: (5.3*0 32 and i f we assume a circular bay, we have: = O (5.35) Using (5.35),'(5.34) and (5.18) i n (5.30) we obtain: / , ® © 1 © ® 7T I ® © © (5*36) where ~ J^. , a constant. 5e Discussion of Forcing Terms for a Circular Bay Equations (4.19) and (4.20) are both linear i n V Thus we may take each term of CLfr/z T* and solve for i t s corresponding value of and add the solutions to obtain the to t a l solution for ^ . We can regard each term i n (5»30) or (5.36) as a forcing function for i t s own r i p current system. I t i s of interest to 33 examine each of these terms and interpret physically what causes them to exist. We f i r s t describe the relative magnitudes of the terms. We do this by ordering each of the terms and comparing them. The exact calculations for (5.36) are given i n Appendix B. The only difference between (5»36) and (5.30) i s that we assumed /ty ^  }71(G) for (5.36)which gives us a circular bay. For a non-circular shoreline (a straight beach i s considered circular with JT~* «0 ) the effect of may be large; however, i n this thesis we consider only the next level of complication above a straight beach and that i s a circular beach. Thus we order the terms for a circular bay. For a circular bay we«assume the independent variables (ft and ftb to change significantly over the angle 5?- = "fir/fX and we use this i n our ordering scheme i n Appendix B. The results of this ordering process are given below. The terms are arranged i n descending order of magnitude. Beside each pair of terms i s the multiplication factor relating the two terms. For L example: * j _ if means /-/w " f = -ty X tern* k> CL+ e (5.37) 34 We sha l l discuss each term i n descending order of magnitude. Term d, the largest term, exists i f dp ^ O and the bottom i s nonlinear i n the r direction. We expect that on most sandy beaches the bottom w i l l be almost linear and hence term d w i l l be small on such a beach. We shall neglect this term i n our circular beach problem by assuming the bottom to be linear. Term g then w i l l be the largest term for a beach with a linear bottom. We see that term - g vanishes for a straight coastline For a curved coastline, term g exists only i f $ & and there exists a bottom slope. Thus we may conclude that term g r i p currents on a curved beach are produced by the angle of incidence ^ of the waves onto the beach. And these currents w i l l be the ones most easily observed since they w i l l be the largest. Term c depends not only on the existence of (f) but that $ i s a variable function of @ . Thus on a straight coastline where i t i s possible to have $ non-zero but constant i n the longshore direc-tion term c would be zero and thereby not produce a r i p current. In our problem 4= 4(e) and term c i s significant. Along a straight coastline there w i l l s t i l l l i k e l y be some variation i n the bottom topography such as the La Jol l a Canyon off the coast of Southern California. This w i l l make (j? vary i n the longshore direction, thereby creating the forcing term c. Term c w i l l be of order of magnitude of term a or greater i f d * i s of the order of 3 degrees or more where £o i s the scaling value of and A $ i s the expected change i n over the longshore scaling distance 0~o 35 Thus we see that term c is very sensitive to changes in <jp Term e depends on the existence of /fy (&) being a nonlinear function of O (ie. JXi/jS^ ^  & ). This will usually be the case where the bottom topography is irregular. Term a is the forcing term Bowen (1969b) found to produce rip currents along a straight beach. It depends on the longshore variation of . We have discussed this under surf zone geome-try and i f the r direction is normal to the beach (as i t is for a straight or circular beach) then the bottom topography effects will not make yyl a function of ^ . A t this point we just let be a function of & and say i t is due to edge wave, effects. This should be the subject of a future study; however, we shall assume i t to be true for now. Quantitatively we let: (5.38) where £ is of order fa/fa =? 0*01 . We see that for a curved beach term g will likely dominate the scene but as the beach becomes more of a straight one, terms c, e and a play a more dominant role. If the beach were perfectly straight without irregular bottom topography, then term a would be the only term available to produce rip currents. This is the case described by Bowen (1969b). The remaining terms f, b, i and h are of less significance; however, i t will be interesting to note in our circular bay solution the character of the flows these terms produce. 36 CHAPTER 6 SUMMARY OF PART I We are now prepared to use our mathematical model on a given topographical beach. F i r s t , however, i t w i l l be useful to summarize our mathematical model. , In deriving i t , we made the following assumptions: a. A l l averaged motion i s in-the horizontal plane. b. Wave reflection i s neglected because of the small bottom slope. c. Eddy viscosity i s neglected. d. Bottom f r i c t i o n i s linear. e. At the breaker line the waves are assumed plane. f. We assume a large bay such that g. The energy density E of the waves i s equal to inside and outside the surf zone. h. The breaker height H i s directly proportional to the mean water depth d inside the surf zone. i . The rip currents do not extend below J - hi outside the surf zone. j . The non-linear i n e r t i a l terms are neglected. Apart from these assumptions our model using equations (3.19). (4.19). (4.20), (5.12) and (5.30) i s for a general beach with arbitrary bottom topography. For a circular beach we assumed only that was not a function of @ . Thus for a circular beach our model consists of equations: (3.19)» (4.19), (4.20), (5.12) and (5.36). For a circular beach we expect term g to produce the most significant r i p currents. As the beach straightens out we see that terms a and c w i l l become dominant. 37 Term a is likely generated by edge wave interaction effects (Bowen 1969b). The approximated form of CUtlz R (equations (4.19) and (4.20)), does not contain any derivatives with respect to £ . This implies that the & dependence of will exactly correspond with the & dependence of each of the forcing terms. PART II CIRCULAR BAY SOLUTION CHAPTER 7 THE PROBLEM 39 (7.1) 7a The Equations We shall solve for using the mathematical model derived i n Part I for a circular hay. The equations to be solved are as follows: Inside the surf zone we have from equations (3.19). (^ .19) and (5«36) where ^ -The forcing terms i n (7.1) are arranged i n descending order of magnitude (see ( 5 . 3 7 ) ) . Outside the surf zone we have from ( 4 . 2 0 ) and (5.12): (a) In polar co-ordinates: (b) i n cartesian co-ordinates: We must solve these equations according to the boundary conditions discussed in the following section. 40 7b The Boundary Conditions As discussed i n Section l c , the width of the surf zone fsh w i l l change s i g i f i c a n t l y due to wave height changes at the breaker l i n e . For example, i f the wave height varies by 50$ over an interval, the surf zone width w i l l vary by the same percentage. Thus i f we were attempting to obtain accurate quantitative results we would have to make the boundaries of the shore and breaker line as accurate as possible. This would involve tedious matching problems at the boundary involving Fourier series. The result would give us stream lines which wiggle with the shoreline, but would not change the overall r i p current patterns which we are attempting to determine. Therefore we w i l l take the space average values of the radial components of the shoreline and breaker l i n e , denoted by and respectively, to represent the boundaries of the surf zone. We assume that the deep water waves have a phase velocity parallel to the y axis of Figure 4. Since water does not flow across the shoreline Ys = ts , y  ^ t the shoreline must be a stream line and we a r b i t r a r i l y l e t have the value of zero at the shoreline. We divide the bay into three regions, A, B and C as shown i n Figure 4. In regions A and B we use polar co-ordinates and i t i s convenient to use cartesian co-ordinates i n region C. I t i s not necessary to assume deep water waves with phase velocity parallel to the y axis; however^ since we must assume some sort of wave approach, this one seems the most convenient as i t w i l l lead to symmetric r i p current distributions. Later we solve for a case where the wave approach i s not par a l l e l to the y-axis. 41 Figure 4 Circular Bay 42 At the shoreline must be zero. Therefore at IT- K t 0 . At the shoreline (Je w i l l be zero due to the bottom f r i c t i o n - (see (3.10) )•. V Therefore at . At the breaker l i n e Of must be continuous because of mass conservation, hence i s continuous at )T— f£ O £ . Also *P i t s e l f must be continuous at f s tk since we cannot have which would imply 01 oo Also we know IT* , i n physical r e a l i t y , must be continuous, otherwise we would have an i n f i n i t e shear which i s not physically possible. However, this continuity i s based on there being eddy viscosity present which we have not included i n our model. Thus we w i l l not i n s i s t that Ol or <W/Jr be continuous at the breaker l i n e or any other boundary; however, i t turns out that i t w i l l be continuous on the breaker line using only bottom f r i c t i o n i n our model. The boundary conditions are summarized as follows: ty-o at r=n}oie£?r (7^) b.. Jt^o at r = £ > o^e^Tr- ( 7 # 5 ) at , oielw (7.6) at r--n , oleitr (7-7) where subscripts A and B denote values i n Regions A and B respectively. h = Va at r'rt , o£ei?r (v.8) I f we can allow to be continuous everywhere, then we should do so; however, the physical cause of this being so was not b u i l t into our model (the cause being eddy viscosity). 44 CHAPTER 8 THE SOLUTION 8a General There are none forcing terms in equation (?.l). Since (?.l) is linear in then we may solve for a for each term of the forcing function and add a l l the resulting 9^  terms and obtain the total 7^ due to the total forcing functions of (7.1). In Section 2b we discussed the effects of the topography on wave height //i and angle of incidence 4^ at the breaker line. We approximate these effects by assuming the following distributions for //4* and & cos & (8.2) The solutions for outside the surf zone are given in the following: Region C: Here we use cartesian co-ordinates as shown in Figure 4. We have: cut/ x. R = <)fiy _ — O where ^ = - f> = -CJf ) ^ ^ and the transport stream function / is defined as i»5 Thus we obtain: CUtl2 R ~ " — d ^ — — <—^ =0 <U </*"- di Jf* This i s Laplace's Equation and i t s solution i s : % = ( f i e * * / ? * + Bi*t»/0x)(k>e + ) and we assume V i s f i n i t e at $ ~ ~ thus Ki~ & and we obtain: =• (AC"S* + ^ ' " ^ * ) « (8.3) Region B: From (4.20) and (5.12) we obtain: HI + J- at + -i £i = o '8-« Jt^ r J>~ •"• d^ Solving by separation of variables we l e t : y - R(r) <8>lo) (8.5) ii / if and (8.4) becomes: ~f- I. B -h -L ® =z O R h If (B> R ~R If ^ where ^U. i s a constant. Thus we obtain: 46 (8.6) (8.7) Thus f/^zCo^ ^^S/^X^^^^) (8.8) Since ^ must be f i n i t e at IT* = 0 , we must have i(*f = ® thus we obtain: = (/9X coS/u& + Bx s>»sue) rA (8.9) where A 3 i s absorbed i n A and 3. We must match this solution with the various term solutions we obtain inside the surf zone. We assume a linear bottom so term d = 0. We now solve for for each term of equation (7.1) i n descending order of magnitude, (see Section 5© for physical interpretation of each term). 8b Term g From equation (7.1) we obtain: or and which i s Euler's equation and the solution i s : # = for* +forM (8.10) 47 where we l e t \T on the right hand side (RHS) be . Since we assume the bay i s large the error resulting from this approximation should be small. Now but we are assuming the shore to be circular as was discussed i n Section 7b, therefore we have an approximation: r3 ^ rs ( 8 . U ) . and a — "'{.'S I (8.12) where 5* —— Trans forming the variable from A to JD we obtain: Ht - A ±1 = 2 km3 s " sin24> (8.13) where B* = q km2 is and <S (6>) i s the (9 dependence factor of (fy as shown i n (8.2). We l e t Sin 2 & 2 (p = 2 <fc e(6) The complfijnentary solution i s determined from the homogeneous equation: illt — -A. IJt = O (8.14) Js* s 7s and i s found to be tffoj= (C-hCx s ) Wl (8.15) and the particular solution i s : f= (C, s3 + Ms. s') efe) 48 Thus Y/i ~ '~7j~ ~ J (8.16) We now apply the boundary conditions: (a) y= O at or 5 = & Thus Ci ~ O and I f o r r ' e(*) [ C* * % + B£ s"] (8.17) (b) M= O at 5 = ^ Thus (8.17) gives which is identically true at 5^  -~ & (c) = O at 3 which is identically true using (8.17) or e(£>)[cx si* * Bp si*] (see (8.9)) Now i f ^ were not sinusoidal we would use a Fourier series expansion. To demonstrate this we let: . n where Fn(9) = /)h C*S/?& -r-%, Since (8.13) is linear in *f we say: (P. = where (8.18) 49 We also l e t : = jE. ^8n inhere Y a , - ( ft'n cosne + B'„ sir, »») r" 5 G. fr)r" At the boundary S — we have: — ^Hn which gives us: • j Q y v) Ti*v v and d $9jf — 7^fl which gives us: Therefore from these two expressions for Gn (&) we obtain values for CZyi , a n d our boundary problem i s solved. When we discuss term f we use this method of solution for n = 1 and 3« However, i n this problem we l e t COS & a n ( j hence: where si =r n - Fi • (8.19) (e) Ujb(rl) = ife (rl) or C O S or > 3 , = - 5 C i 5^ - &j 0 6 (8.20) Using (8.20) and (8.19) we obtain: (8.21) for large bays r?«k. I 50 and or /2 ri (8.22) (8.23) Therefore we obtain: ^ r cos B (8.24) (8.25) Note that by not including eddy viscosity and s t i l l saying i s continuous at , we are forcing a value on £z and r*i for which the model was not spe c i f i c a l l y designed. I f we did not have the boundary condition . then Cr i we could chop off before i t reached i t s maximum value. I f eddy viscosity were included, then the equation (7.1) would be of higher order and would enable us to state with certainty the continuity of However, since we do get a highly satisfactory solution without eddy viscosity, then perhaps i t i s negligible relative to the bottom f r i c t i o n . This seems reasonable although i t should receive more attention i n a future study. (f) % = % at y Thus we obtain from (8.3) and (8 .25): 51 since ^tj = o) i s an odd function. Using Fourier series expansion theory we obtain: (8.27) where s jjjr Thus * (8.28) J7 From (8.28) we obtain: Vjtc * ft (8.29) From (8.25) (8,30) Thus we cannot match (8.29) and (8.30) at j / = ^ because (8.29) w i l l not in; general be identically zero. Therefore we have a discontinuity i n fJlr at — tP . I n real l i f e this i s impossible; however, since we have not included eddy viscosity i n our model we obtain this discontinuity. In summary then, we have for term g of equation (7 .1): / z rl (8.31) 52 At/the boundary between regions B and C we used a Fourier series method to match the solutions. This makes region C periodic i n x with period *2 J* . T o eliminate this periodicity, we could have used a Fourier integral method by letting: which identically satisfies Laplace*s equation and we have: which satisfies the boundary condition at ^ = O for any function . This same method can be used at the boundary between regions A and B where we would l e t : % = f FM * (X f j n *»d F(») = -jL- J VCrt, 9) <s where we l e t ty(fit #) =0 for & > and <£> < ° We have no need to use this method i n this paper for the boundary at because we l e t <P(£>) and HL (9) be simple trigonometric functions. We find the location of ^ 9 maximum as follows: Ujr * 1 a n si) therefore the maximum occurs at: /2n si, or J / ? » K For a large bay ^/ti / and so S/na* = Si For & = O we plot ^ VS f* as shown in Figure 5« From (8.32) and (8 .3D we get: ^ tu= w ) * - ^ r (8-33) . f(r,o) 53 Figure 6 Streamlines, Term g 54 The e n t i r e f i e l d of I f i s sketched according to equations (8.31) i n Figure 6. We see that there i s only one r i p current, a t Q = • The streamlines represent the average paths f l u i d p a r t i c l e s would follow providing the streamline pattern does not change s i g n i f i c a n t l y over the time i t would take f o r the p a r t i c l e s to complete a round t r i p along the streamlines. The r i p current flows out past the mouth of the bay (where y = o) and then curves around and enters the surf zone with a strong v e l o c i t y as indicated by the high density of the streamlines. TKe (+) and (-) symbols i n Figure 6 show where the transport stream function i f i s p o s i t i v e or negative respec-t i v e l y . The e f f e c t of the non-linear terms, which we neglected (Chapter 3 ) , w i l l be to strengthen the r i p current a t because the current there i s flowing i n t o deeper water, and to decrease the current flowing i n t o the s u r f zone a t & — O and because the flow there i s i n t o shallow water (Arthur 1962). A sample magnitude of i s computed as follows: UrU*,t)~ d N & i %tj '* jm SvT ( n } where J ~ M St> m Q = 32 ftftec Si = /t>° f * li - J K rtJ c-Bowen (1969b) estimates c = 0.2 cm./sec. = 0.0067 f t . / s e c . which gives: Let Then /sec 55 From Figure 6 we see that the maximum velocity occurs in the surf zone at 0 — Ot "77" . To calculate this velocity we do the following: tSff occurs at S = Si } @ — ^ Then ana tgr*^*) = ^  -d- -tnen ^ f f ^ = ^ ^ The effects of the nonlinear terms wil l be to lessen OQ maximum and increase Ur(f^^j -2" ^ 8c Term c The equation for inside the surf zone for term c is from (7*1): dUt -h ilL = % km dz sin *4 i£ ( 8 . 3 ^ ) Using the same procedure as for term g we obtain: Js* * js « JB where g = & /C M <Yo, s = rs -r The solution i s obtained i n the same manner as for term g and i s 1/ 56 (8.36) or using £ 3 — COS @ w e S6'*'2 Subjecting (8.37) to the boundary conditions (BCs) with ^ y^c given by (8.3) and (8.9) we obtain: *"%[%-'^gL)-s!]s;*2* (8.3a, % •= SJL. l £ ^ S'm X& (8.39) s „ a t = s l O -1>~. * % (rl) - # «>.*) Again there i s a velocity discontinuity at ^ — & as was the case for term g . This indicates that eddy viscosity i s significant at this location i n the model. A sketch of the r i p current system for term c i s shown i n Figure The dotted lines are estimated flow lines which are expected to exist i f we include eddy viscosity i n our model. We obtain r i p currents where i s maximum or minimum for a sinusoidal distribution. (8.40) (8.41) 57 Figure 7 Streamlines, Term c 58 8d Term e From (7.1) we obtain: sr d Jr crx which we alter as we did for terms g and c and obtain: £t-±l± = Bes3JjL w h e r e SU = k»Uk 5 =* n - r Solving we obtain: 2 or with — COS & we obtain: Subjecting (8.46) to the BCs we obtain: f* = Be r cos 9 '5- n and Ay where - n 3r 0 (8.43) (8.44) (8.45) (8.46) (8.47) (8.48) Jx 59 rtx.O) Figure 8 Profile of e^(x,0) Figure 9 Streamlines, Term e 60 From 7^9 and ^ff we obtain the graph of ty(as shown i n Figure 8. From (8.47) we obtain: A sketch of the r i p current system for term e i s shown i n Figure 9. Again we have a discontinuity at (j O because of the absence of eddy viscosity i n our model. 8e Term a From (7.1) we obtain: * t . £± + 3kcL coslb <J * <8-5o> or with S ~ — t we obtain: where = 2A a n d //,(p)zr~C6S Here we l e t where 6 i s of order fab • Tnis i s similar to the function Bowen(l969b) used, although i t cannot be derived from the theory discussed i n our section of surf zone geometry. One may circumvent this d i f f i c u l t y by saying Q / a r i s e s from an edge wave effect. This i s speculative, but i t gives us a f i n i t e value for term a. The solution to (8.51) i s : 61 Figure 10 Streamlines, Term a 62 (8.55) A sketch of ^ i s shown in Figure 10. Just as Bowen (1969b) found, we see that the r i p current occurs where the waves are the lowest. (at e = o, w) . I t i s important to realize that we assumed term a to have a d9 dependence the same as . I f term a i s influenced by phenom-enon other than just refraction, then i t may very l i k e l y have a & dependence different than that of fJi+ . I f term a varies significantly over a longshore distance of the order of the surf zone width, then we must reconsider our assumptions i n obtaining equation (4.10). In particular, we would not be able to neglect the term i n carl*. R 8f Term f From .("7-1) we obtain: Alt + 2 J B J 1 - - M ^ ^ n i i f i i f Altering this as done for previous terms we obtain: where Bf = ^ M (8.56) (8.57) We obtain the solution: (8.58) 63 Letting rr Cos 0 , we get: e(2^)l= C€>s& - cosZe ( 8 < 5 9 ) Since (8.56) i s linear in we may treat each of the terms of as separate forcing functions and add their solutions to obtain ^ We obtain: (jVf 2 ^.S^fcOS & ~ c c s l ^ ) ( 8 < 6 o ) (8.61) and - S /«. ' (8,62) where jS* ,/i=/,2,\•••• and from (8.61) and (8.60) we obtain the sketch of C'**^) shown in Figure 11. A sketch of C05 & — COS 3&is shown i n Figure 12. We use these sketches to sketch tflfi. as shown i n Figure 13. 8g Term b From (7.1) we obtain: £J> ^ J ^ f ^ 2$ Jl£ ^ d 7? crx We alter (8.63) to the form: 3, ^z//(£) (8.64) where /-J(* = DC^P M(&)) and we have discussed i n Section 2b the form of //, (#) = — C OS 2 & TUO) Figure 12 Graph of COS 0 - Cos39 Figure 13 Streamlines, Terms f & i Hence we obtain the solution: ? . . 65 (8.65) -fc, +CiS* +BI s^ftsinZe (8.66) Using the boundary conditions we obtain: 67) <PB = -I l?l s£rXs/»Z9- (8.68) and ~ ° (8.69) is sketched in Figure 14. Notice that the rip current occurs where the wave height is greatest. This is contrary to the result Bowen (1969b) got. However, we obtain Bowen's result for our term a which is similar to his forcing term. Term b is a result of the variation of wave height and the curvature of the beach. It goes to zero value as the beach becomes straight fe * *°) . 8h Term i From (7.1) we obtain: + Z>2<M = Sk/fScoUbjM Jg£ (8. 7 0) We alter this to the form: , » . . ,» * \ where D£ - 3£# J99' <$>„ fo D_ Thus we obtain: . . . Figure 14 Streamlines, Term b Figure 15 Profile of <fi (x, o) { 67 or with C = C>OS & and M (fls ~ Cos^& we get: % = -(c, f ^ jJf tos'X*"*-'0*3*) (8'72) In a similar manner to the f terms we allow each of the trigonome-t r i c terms to represent a forcing function and solve for each. We then add the solutions and obtain: _ u v % ^ & ( s l S ~ j ^ S )(cosO-<*>s30) (8.73) = S / ^ J v V / ^ ^ " (8.75) and where = JLZ^ _ n rs U From (8.73) and (8.74) we obtain the sketch of tyfa/d) as shown i n Figure 15. The sketch of ^ for term i i s similar to the sketch for term f i n Figure 13. 8i Term h From (7.1) we obtain: , which upon rearrangement yields: arrangement yxexas: \* // / \ ^ _ * l<£ - B k s e(#) eLAfl where Bi = Z M * P*/* * (8.76) (8.77) We obtain the solution: = (cl+cls\M.su)e(4^) (8-78) 68 Lor substituting for e(tJ) and //,(&) : Similarly to term i we obtain: and (8.79) (8.80) n and C fc = S fl» 2'f»>3»3C e**^ (8.81) n where From (8.79) and (8.80) we obtain the sketch of fy(~*Cf0) shown i n Figure 16. A sketch of COS & +• COS ~$Q i s shown i n Figure 17. A sketch of i s shown i n Figure 18. 44<><.o) 69 Figure 16 Profile of %(X.,0) Figure 18 Streamlines, Term h 70 CHAPTER 9 GENERAL WAVE APPROACH 9a General In the previous solutions we assumed the deep water wave approach to be i n the direction of the radius vector with . Let us now examine a more general wave approach i n the direction of the radius vector at & ~ &o. . This,does not introduce any new d i f f i c u l t y . We merely determine the functions Hf(9) and $(0) for the new wave approach and solve equations (7.1), (7.2) and (7.3) subject to the same boundary conditions. Figure 19 shows a sketch of a sample refraction pattern for waves approaching with the angle Bo . The sample bottom topography i s shown by contour lines (the dotted l i n e s ) . The solid lines with direction arrows represent the direction of the wave phase speed and are called orthogonals (see H 0 pub 234). Where the orthogonals diverge we w i l l have smaller wave heights. From this sketch we assign the following functions to represent the wave characteristics /VI and 4> : where yU = } &0 ^ and 4> = <Po efe) (9.2) &<&) = cos = cos##> <§ I f 0O ^ we would have: and efp) = czos (/ro°-& ) ( /2O0-0o ' 71 Figure 19 Refraction Pattern, General Approach 72 Figure 20 i s a sketch of these functions. We assume trigonometric functions, however we need not have and for an actual situation we would not expect these functions to be purely trigonometric. I f the functions are not trigonometric, then we w i l l have to use Fourier series to solve the boundary conditions. This i s unnecessarily complicated for our present discussion (see Section 8b). Using (9.1) and (9.2) in (7.1), (7.2) and (7*3), we arrive at solutions for a l l the forcing terms. Term g i s discussed i n the following section. 9b Term g (9.3) = - B? s? r^cos^e (9.4) and where (9.5) Note that our solution here i s the same as the former case where (p0 ~ 2?%. except that here — instead of unity as for the previous example. This i s due to the function 4) (6) we chose. However, the result i s meaningful providing our i s f a i r l y r e a l i s t i c . A sketch of the transport stream lines i s shown i n Figure 21. The remaining r i p current systems for each of the other forcing terms can be calculated i n a manner similar to that done in CHAPTER 8. Term g, as already calculated, w i l l be the dominant r i p current and should be readily observable on a real curved beach. 73 Figure 21 Streamlines, General Approach, Term g 74 CHAPTER 10 SUMMARY OF CONCLUSIONS In this thesis we derived a mathematical model for r i p current systems along beaches which takes into account the effects of bottom topography. These effects are: the slope and curvature of the bottom inside the surf zone; the curvature of the shoreline; and the variations i n wave height Ml and angle of incidence sr of the waves at the breaker line due to wave refraction outside the surf zone. In this model nine forcing terms caused nine component r i p currents whose sum gives the t o t a l predicted r i p current system. One of these terms (term a) i s equivalent to the forcing function Bowen (1969b) used i n his straight beach problem. The model was applied to the special case of a circular bay and the r i p current patterns which occured were discussed and their streamlines sketched. When one interprets these sketches, i t should be remembered that the i n e r t i a l terms which were neglected i n the equations of motion w i l l cause the streamlines to come together where the velocity i s towards deeper water and to separate where the velocity flows into shallow water. This effect tends to strengthen the outward flowing r i p currents. I t was found that as long as the variations i n and t^l were caused by the circular bay topography only, the dominant flow pattern in the circular bay would be that due to term g (Figure 6). Ternu c was the next lower term on the magnitude scale (See (5«37)). Figure 22 shows a sketch of the sum of the transport stream functions and tyc along Figure 22 P r o f i l e of ^ 3 + % a t the breaker l i n e $ 0 = 1 5° F i g u r e 23 P r o f i l e o f Ur a t the breaker l i n e due t o terras g and c f o r tJ>o~/5"* . the breaker lin e for ^ — /-~ . The differences between the curves for % and % are seen to be small. The remaining stream functions due to terms e, a, f, b, i and h w i l l be insignificant since they are at least an order of magnitude smaller than term c (see (5«37)). Thus we see that the sketch of tyj i n Figure 6 represents a f i r s t order approximation of the total flow pattern that one would expect to see in a circular bay as we have described i t . A sketch of the sum of the velocities normal to the breaker line at the breaker line due to terms g and c is shown in Figure 23. We see that the tendency i s to establish a uniform seaward velocity along the breaker line. However, one must be careful when adding these velocities because the effect of the non-linear i n e r t i a l terms, which we have neglected (Chapter 3), w i l l be to weaken the current due to term c between and to strengthen the currents due to term g i n the same region thereby tending to cause a maximum combined seaward current at which is not apparent from Figure 23. The currents due to the remaining forcing terms are at least an order of magnitude smaller than those due to terms g and c and can be neglected i n the f i r s t order current approximation. When discussing this model one must r e c a l l that the equation we used to represent cu/"/sz R inside the surf zone (4.10) assumes that the distance &Z of significant longshore variations i s much larger than the width of the surf zone. This enables us to neglect the term i n cUt/jc £ . This assumption is valid for refractive effects in large circular bays with conical bottom topography; however, this may 77 not be a valid assumption i f there were irregular topographic features present to give small fluctuations to the refractive variables ( and (fy ). Edge wave disturbances may also give significant fluctuations over longshore distances Oei which are comparable to the surf zone width. In our circular bay problem we l i m i t ourselves to the case where Q£ 0- • thereby making (4.10) valid. We find that forcing term g, which i s proportional to w i l l be the most significant forcing term providing that the radius of curvature /« of the beach and the longshore distance &o over which 4> and HI change significantly are of the same order of magnitude (See (B8)). This i s the case i n our circular bay problem. Because of the sin $ dependence of term g we obtain along the^breaker lin e to be largest where <P i s largest. This necessitates the longshore current being largest where <P i s largest. When the radius of curvature of the beach f° i s much larger than the longshore distance we have the beach tending towards a straight beach solution and terms c and a will.become more significant than term g (See (B8) and ( B 7 ) ) . In this case the refraction effects would be caused by irregular-i t i e s i n the l o c a l bottom topography such as the La Jol l a Canyon near the Scripps Institution of Oceanography i n California (see Shepard and Inman, 1950). Itt.is interesting to note that term a, which i s the forcing function Bowen (1969b) used for a straight beach, i s smaller than term c providing 4> exists and (fa/lfi)*'^- where 4>o i s the scale of (fy and ^ ^ s the change in (fy over the longshore 78 distance C?Z , and ££. i s the significant longshore distance for changes i n d i n term a (See (Bl)). This suggests that term c, as well as term a, may be responsible for the r i p currents along the straight beach opposite La Jol l a Canyon near San Diego, California due to any small refraction caused by the offshore topography. Term c i s i n fact very sensitive to any small amounts of refraction and must be considered as a potential cause for r i p currents even on a so-called straight beach where some small degree of refraction w i l l almost always be present. The following paragraphs state some recommendations for future studies. I t i s suggested that this linear analytical model of a circular beach could be extended to more r e a l i s t i c a l l y shaped beaches by f i t t i n g the shoreline with circular arcs, concave or convex, and solving for each circular arc region i n a similar manner to our circular beach solution and matching each arc solution to i t s adjacent arc solution. It would be interesting to use a numerical computer method to solve this circular bay problem including nonlinear i n e r t i a l terms and eddy viscosity as did Bowen (1969b) for his straight beach model. A computer program could also be made, using our general mathemati-cal model, to predict r i p current systems for general beaches with irregular bottom topography. A part of this program could predict refraction patterns for given wave approaches and wave heights and determine the location of the breaker line and compute It should be worth studying the effects of edge waves on the mean , thus giving us more insight i n discussing term a. Experimental work should be done to verify our mathematical model and to examine the effects of a nonlinear bottom slope as predicted by term d. Experiments on straight beaches could be conducted to examine the relative sizes of currents produced by forcing terms c and a. 80 LIST OF REFERENCES Arthur, R.S. (1962). A note on the dynamics of r i p currents. J . Geophys. Res., 67(7), 2777-2779. Bowen, A.J. (1969a). The generation of longshore currents on a plane beach. J. Mar. Res. 27, 206-215. Bowen, A.J. (1969b). Rip currents, theoretical investigations, labora-tory and f i e l d observations. J. Geophys. Res., 74(23), 5467-5490. Bowen, A.J., Inman, D.L. and Simmons, V.P. (1968). Wave set-down and set-up. J. Geophys. Res. 73(8), 2569-2577. Breakers and surf, principles i n forecasting. H.O. Pub. No. 234, 5^  PP» Washington, D.C. Johnson, J.W., O'Brien, M.P. and Issacs, J.D. (1948). Graphical construc-tion of wave refraction diagrams. H.O. Pub. No. 605, 45 pp, Washington, D.C. Longuet-Higgins, M.S. and Stewart, R.W. (1964). Radiation stress in water waves, a physical discussion with applications. Deep-Sea Res., 11(4)| 147-151. P h i l l i p s , O.M. (1966). The dynamics of the upper ocean. Cambridge, Cambridge University Press, 277 pp. Prager, W. (1961). Introduction to mechanics of continua. Ginn and Company, 230 pp. Shepard, F.P., Emery, K.0. and Lafond, E.C. (1941). Rip currents: a process of geological importance. J. Geol. 49(4), 337-369. Shepard, F.P. and Inman, D.L. (1950). Nearshore water circulation related to bottom topography and wave refraction. Trans. Amer. Geophys. Union, 31(2)1, 196-212. 81 APPENDIX A DISCUSSION OF SET DOWN The momentum equation (3«4) gives us: I f we have a straight beach and the waves are advancing normal to the beach, we have: 1U i — o (A2) as discussed i n (5*1) and we obtain: from which Longuet-Higgins and Stewart (1964) got JS''4_ s?<fd AjL (A3) j - -J-JLgL .  m as described i n (5*3) of this paper for the region outside the surf zone for = O . Inside the surf zone we assume: // YJ (A5> and S - * , - * * ^ > ( A 6 ) as did Bowen (1969b). That i s we assume that the radiation stress i s the same function of wave energy inside the surf zone as outside. Using (A5), (A6) and (A3) we obtain: . °Ll = - KsS. =m> = / c r t ( A 7 ) where . _/ 82 and Wlf is the absolute value of the surface slope. Jtf^ is the absolute value of the bottom slope. A problem arises when we have a rip current system where we know . Does (A3) s t i l l apply and give us (A4) and (A7) which Bowen (1969b) used in his rip current analysis? If (A3)is s t i l l applicable for our rip current system, we must have (A8) Now J-(7}lf%)=%,<L%>+X4'Ha ™ and using (3*3) we obtain: ^ ^^c/fZ/, +£l4X>) ( A l l ) by the definition of M# in ( 3 . 2 ) . Near the surf zone in a large bay, we expect that velocity changes in the onshore direction will be much larger than in the longshore direction. Thus we neglect the term of (All) and obtain: J-(#,/%) s /o^Zt.ilL (A12) Using (A6) and (A5) we obtain: J s * J J k . = ^ r / i > 9 y y ^ d ( A 1 3 ) Thus by forming a ratio we obtain from (A12) and (A13): (A10) Rip current measurements were made by Shepard and Inman (1950) and to estimate the magnitude of (A14) we use some of their sample results: Let Y = 1 Jl - 10 f t . 2c/ = 2 to. = 3.4 ft;/sec-. . a. ^ = 3 2 ft;/sec. ^ Thus (A14) becomes with J<///x - > « ^ ^ TTT ; or in general equals /X in units of feet and seconds. Hence i t is expected that by an order of magnitude and therefore (A4) and (A7) may be used with reasonable accuracy when rip currents exist. APPENDIX B ORDERING OF FORCING TERMS We sha l l discuss the relative magnitudes of the forcing terms given in equation (5.36). From (2.21) and (2.1) we see that $l)/o i s the amplitude of the & varying component of ^ and also of d i f we l e t M be independent of £P as i t is for a circular bay (see (5«33))« We introduce the-ordering scheme» — /9j)/t> for refractive variables. y / = fa. for term a only. <f> = <t>0<p'(e) jTa) •= A(y ff (P ^ general, and for * a circular bay. /^L -T>~ ~ i for the. refraction variables d & — (j and q> i n a circular bay. v% /o — rfcT({h" * ° r refraction variables on a ^ general shoreline. •for a circular bay. 5 long only. (Z-i /~sr~— sT~ jCrr f°r the longshore variable of /Vi? = <J^ - ^ <^ 4erm . only" J -U d'(r,<s>) (BD 85 where the primed symbols are variables of order unity. The differentials atanc* ( J d a r e introduced to allow for the different rates of change of or i n the r and & component directions as we did i n (4.1). We introduce two longshore scales 6*9 and C?oL to allow us to differentiate between the longshore variation due to refraction and any possible longshore variation i n term a due to edge waves. We also l e t fit. f° r "term a. This i s because may not be caused by refraction but by edge waves and therefore may not equal Terms d and g Jerrys _y - - ~ ~ 7 — Using the ordering scheme we obtain: f Terms = - -JL £L H frJX ) M/ni no (B2) (B3) (B4) therefore Terms A- z_-JL 1L(r (B5) J * *sl ^  t / / x I f we have a large bay, and i f A/S' i s of order unity we see that term d i s much larger than term g. However, i n a beach with a linear bottom and term d i s thus zero. However, this ratio does show us that i f the bottom were not linear then term d could be of great influence. Terms g and a . where we l e t 17? — hi/rA If £ r/fX then terms ^ -f ) Now i f we say kk^ then *" ° CL ' 86 (B6) (B7) In the absence of short wave length edge wave modulation of the incoming swell, we can say that the longshore length scale w i l l be determined by the beach curvature and then we can say roughly that fo — and so term g = 50 term a. As the bay gets large — ^ 0°) then term a w i l l be more significant than term g, providing edge waves are generated and l^cu i s of f i n i t e value. Terms g and c , For a circular bay we have A ^ If/ IX and &o — 7f/x and so we obtain: Terms c and e 6 kl C O * ^ 4 ' r7- je-87 and for a circular bay ($L = £ • - J*" w e g e t : » / / / / / \ (BIO) Thus, i f the primed terms are of order unity and we expect them to be so, we see that 7Z,m c £.2M%rMe < B U > Terms c and a where <fcr - f the longshore arc length. We say <P and d change by d $ and over the distances d~o and respectively, then: 0^4^ ~ d & To1 ~~ ~az o>ar' f and JlsL = - jgs=-Thus ^ _ 3(0j(K )<t<t><k (W/WT )/(iy/S/J^') (B12) I f 4(f/= 2=J?y'/2^ ^ and — then ~7er*s ~~ ~ 'if hi/*?a. thus 7er»* C- *??femQ. for our circular bay problem. Also, Tf/VM £ = w h e n ^<^> <Po)%~ 3 assuming = and = CTZ . Terms a and e Terms ± = = V*L)f*S& ^/Jfjr ] I f we assume the primed terms to be of order unity and assume a circular bay where 6§= tyo - and = A we get: 88 ^ fsi (B13) and If ^ ^ JlA and C" ^ ^ /£> and = / then we obtain: O.b Term e ^ Term a. ^ 1.2 l&rrn e _Thus terra a i s of the order of magnitude of term e i f i s of order unity. However, i n our problem of the circular bay, we assumed to be a constant and so Bowen ( 1 9 6 9 b ) admitted a & dependence to /?? and thereby obtained a r i p current system; however, according to our present theory of set-up, we do not obtain a variable /?? 1. Perhaps i f one were to consider the effects of edge waves and not just bottom topography effects, one would get ffl to be a function of & . I f this was found to be the case, then i t i s also possible that 0cL , the longshore scaling parameter for term a, would not equal , the longshore scaling factor for the refraction variables /V^and 4> . I f <%/<K>l then term a would be more significant by a factor of This factor i s also relevant for comparing terms c and a (see (B12)). Terms e and f £. I y \ t. We expect the primed term to be of order unity, so for a circular bay: -rr e ^ _ I 2z U (B15) 89 Terms f and b Tervs ^Jllf^  We expect the primed terms to be of order unity and ^ / f i ^  ^OO thus for a circular bay: ^ W^-^S-JLJU 3f _y ( a i ) Terms i and b je _ - 3 IL A$fJb \ or 7e~P/*fj ~ 2. f° r a circular bay. (B17) Terms i and h Ud cos i t 4 _t£ We expect the primed terms to be of order unity and A $ (fro for a circular bay, therefore: . • - (B18) Jems — = _ 4 2~ 90 Thus we obtain a magnitude structure as follows: Term Circular Bay Factor General Factor 3d<p n \ 2.r./r,i, c •• at I i hi ni & e .1 Oo I 3 n 4<p (B19) I 


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