NPS ARCHIVE 1969 STURR, H. CONTINENTAL SHELF WAVES OVER A CONTINENTAL SLOPE by Henry Dixon Sturr DUDLEY KNOX LIBRARY NAVAL POSTGRADUATE SCHOOL MONTEREY, CA 93943-5101 United Stat Naval Postgraduate School THESIS CONTINENTAL SHELF WAVES OVER A CONTINENTAL SLOPE by Henry Dixon Sturr, Jr, October 1969 Thi6 document ka& bten appAjovzd fan. public kq.- leja6t and 6att; ltd ctut/Cibiition >c6 unLunLtzd. i 133303 DUDLEY KNOX LIBRARY NAVAL POSTGRADUATE SCHOOL MONTEREY, CA 93943-5101 Continental Shelf Waves Over a Continental Slope by Henry Dixon Sturr, Jr. Lieutenant Commander, United States Navy B.S., U. S. Naval Academy, 1958 Submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN OCEANOGRAPHY from the NAVAL POSTGRADUATE SCHOOL October 1969 LIBRARY NAVAL POSTGRADUATE SCHOOU MONTEREY, CALIF. 93940 TABLE OF CONTENTS Page I. INTRODUCTION 13 II. ANALYSIS 16 III. RESULTS AND CONCLUSIONS 22 A. REVIEW 22 B. COMPARISON OF MYSAK ' S APPROXIMATE 2 3 SOLUTION WITH AN EXACT SOLUTION FOR MYSAK' S MODEL C. COMPARISON OF THE MYSAK MODEL WITH 2 5 THE TWO-SLOPE MODEL D. COMPARISON OF CONTINENTAL SLOPES 26 E. RECOMMENDED FURTHER STUDY 28 APPENDIX A. NUMERICAL SOLUTION OF ONE-AND TWO- 48 SLOPE MODELS APPENDIX B. NUMERICAL SOLUTION FOR TWO-SLOPE 56 COMPLEX ROOTS LIST OF REFERENCES 61 INITIAL DISTRIBUTION LIST 62 FORM DD 1473 63 LIST OF TABLES Table Page I. ME edgewave wave number cut-off (mW) 24 II. Two-slope edgewave wave number cut-off (mW) 26 LIST OF FIGURES Figures Page 1. Profile of the two-slope model 16 2. Profile of the finite-width model 23 3. Two-slope model 25 4. Two-slope models with different continental 26 slopes 5. Two-slope shelf waves, deep water = 500 29 meters 6. Two-slope shelf waves, deep water = 1000 30 meters 7. Two-slope shelf waves, deep water = 2000 31 meters 8. Two-slope shelf waves, deep water = 3500 32 meters 9. Two-slope shelf waves, deep water = 5000 33 meters 10. Effect of continental slope, deep 34 water = 500 meters 11. Effect of continental slope, deep 35 water = 1000 meters 12. Effect of continental slope, deep 36 water = 2000 meters 13. Effect of continental slope, deep 37 water = 2500 meters Figures Page 14. Effect of continental slope, deep 38 water = 2800 meters 15. Effect of continental slope, deep 39 water = 3000 meters 16. Effect of continental slope, deep 40 water = 3250 meters 17. Effect of continental slope, deep 41 water = 3500 meters 18. Effect of continental slope, deep 42 water = 5000 meters 19. Deep water = 1000 meters. Two-slope 43 edgewaves 20. Deep water = 3000 meters. Two-slope 44 edgewaves 21 Deep water = 5000 meters. Two-slope 45 edgewaves 22. Mode two complex roots, depth = 2800 meters 46 23. Shelf and edgewaves. Depth = 5000 meters, 47 slope = .05 8 LIST OF SYMBOLS a dimensionless variable A constant of integration B constant of integration D depth of deep water e 2.7183 -4 f Coriolis force (0.729 x 10 /sec) Fa (z) Laguerre function of the first kind 2 g gravitational acceleration (9.80 m/sec ) Ga (z) Laguerre function of the second kind h dep th i square root of (-1) k subscript m wave number p dimensionless variable s slope t time u velocity in x direction U portion of u that varies with x v velocity in y direction V portion of v that varies with x W.. width of continental shelf W„ width of continental slope x horizontal coordinate perpendicular to coastline y horizontal coordinate parallel to coastline z dimensionless variable ACKNOWLEDGEMENT The author wishes to thank Assistant Professor Theodore Green III for motivating my interests in this field and for the critical analysis of the numerical solu- tions obtained. The valuable assistance of Professors F. D. Faulkner and L. D. Kovach in understanding the behavior- al patterns of complex functions is also acknowledged at this time. 10 C wave amplitude T7 instantaneous wave height 4 dimensionless variable (J wave angular frequency 05 dimensionless variable (cr/f) 11 I. INTRODUCTION Considerable interest has recently been shown in trap- ped waves travelling along the boundaries of continents. A "waveguide" effect exists over the continental shelf. That is, wave energy is confined (essentially by refraction) to the continental shelf. Two general types of these waves exist: A. Edgewaves which are characterized by wavelengths of hun- dreds of kilometers (km) and periods of hours (almost always less than a pendulum day) . B. Shelf waves, which are generally even longer, have pe- riods greater than one pendulum day, and travel southward along the west coast of an ocean in the northern hemisphere (as do Kelvin waves) . STOKES (1846) showed that such a wave guide effect ex- ists over a uniformly sloping beach or continental shelf, with the amplitude of the gravity waves decaying exponen- tially to seaward. URSELL (1952) showed that Stoke' s edge- waves were the fundamental mode of a family of waves order- ed by the number of modes parallel to the coast. REID(1958) studied long waves on uniformly sloping shelves of infinite width, including the effect of the Coriolis force. Reid showed that the sea surface may react as an "inverse baro- meter" and that atmospheric pressure systems may be a driv- ing force for edgewaves. He found that the Coriolis force could cause the wave period to vary from 46% less than to 13 86% greater than that for the non-rotating case, depending on the direction of travel. A new quasigeos trophic wave is now possible, analogous to a Kelvin wave, having. no small scale counterpart. ROBINSON (1964) initiateda study of the continental shelf wave and studied the data of HAMON (1962, 1963) re- lating tidal and barometric conditions at several stations on the eastern and western coasts of Australia. In this model the continental shelf ends abruptly, at which point the depth becomes infinite. He found that an inverse baro- meter effect was exhibited but that the propagated shelf waves had a celerity double that of his calculations for the western boundary. MOOERS and SMITH (1968) studied the relation of sea level and barometric conditions along the Oregon coast for a period of nearly one year. Their statis- tical results show a barometric factor of -1.2 cm/mb and predominant sea level oscillations of 0.1 and 0.35 cycles per day in the summer. They conclude that a shelf wave of period three days is travelling north. MYSAK and HAMON (1969) found shelf waves off the coast of North Carolina, but found no coupling between the sea surface and air pres- sure in the frequency range 0-0.5 cpd. ADAMS and BUCHWALD (1969) show that an equally suitable driving force for shelf waves is the longshore component of the geos trophic wind. This may account for the exaggerated frequency re- sponse of the sea level on the east coast observed by Hamon. 14 MYSAK (1967, 1968) extended Robinson ' s work, and discus- sed the effect of a continental shelf of finite width on the frequency of Hamon ' s Australian waves. His theoretical so- lutions correspond more closely to the observations, al- though he still cannot account for the extremely low read- ings along the eastern boundary. He attributes the discrep- ancy mainly to the presence of stratified water and currents in the deep water beyond the continental shelf. A signifi- cant discrepancy exists between the dispersion relation and that for waves over an infinitely wide continental shelf. This paper is a study of the effects of a continental slope and finite ocean depth upon the present one-slope models of MYSAK (1968) . A sharp discontinuity in the depth of water beyond the continental shelf is not a common occur- rence in the world ocean. It is interesting to study the two-slope situation where a gently sloping continental shelf (slope, s < 0.002) and steeper continental slope (s w 0.05) form a transition zone between the coast-line and deep water. Three parameters: the slope of the continental shelf, the slope of the continental slope, and the depth of the deep water should have possible effects on shelf waves. These are investigated below. 15 II. ANALYSIS h = s.. x(x ^ W ) TT— ., T l V s2x + W (s - s2) ,2\i K3 (wx * x ^ w2+ Wl) , ,J, ,,r h3= constant I II III = s1VI1 + s2W2 '— ^s -■ \A/| i!**!"*8 W, *^ Fig. 1. Profile of the two-slope model. The model is characterized by a gradually sloping con- tinental shelf of finite width (Region I) adjoining a steep- er continental slope (Region II) which terminates in water of uniform depth (Region III) . A representative slope and width of the continental shelf of .002 and 100 km (Mysak, 1968a) are used below. A representative depth of the deep ocean is 5000m and is the greatest depth of Region III Three slopes will be used for the continental slope: .03, .05 and .08, with .05 used as the standard for comparison with Mysak's model. The shallow water equations are used: Su/at - f v + g3*yax = dv/at + fu + g^/3y = o (1) a/ax(hu) + a/dy(hv) + ac/at = o (2) 16 where (u, v) are the (x, y) velocity components and C is the free surface height. Consider a wave, moving in the y direction, speci- fied by ^ = Uk(x)ei(CTt-m^ vk = Vk(x)ei( (3) Ck = ^(x)ei(CTt-my> where k denotes the region. Using (1) and (3) , u and v are ig *~ „ _ ST? . i(crt-my) ,,. uk = -2~2 (fmT7 - cr ^ )^e (4) -g ,__ - ^T] v i(CTt-my) ,-, vk = -2—2 (omrj - f ^ ) e (5) f -cr J Eliminating u, v from (2) in Region I gives the equa- tion 2_.p2 s, fm ^ hir?i' + siT7i + ( ^T" " a him )T?i = ° (6) After making the substitutions h.. = s,x 2 2 = cr -fz fm pl gs, ' -20 4 1 H (12) Similarly, in Region II, -z /2 T?2 = e {A2Fa2(z2) + B2Ga2(z2)} (13) 18 and u2 = -^:Z2/%2[«.-f,Pa2(22,-2,^|i^] r dGa (z ) -,, + B2L(a-f)Ga2(z2)- 2cr ^ J} (14) where m-p2 z2 = 2mi2 ' a2 = IST ' so = — and p„ = - — — 2 s0 ^2 gs„ (7 In Region III the counterpart of (6) is 2 2 h3r?3" + ( g gf " h3m2 )T73 = ° • <15) Since T] must be bounded for large x, T]3 = A^e where (16) 2 2 -, 1 2 cr ~fz I - \m" - L gh 3 2 and igA? -£(x-W -W ) u^ = " o o (fm+Cf£)e -1 (17) J cr -f There are now equations defining T) and U in each re- gion. The next step is to patch together the solutions for TJ and U at the points x = W_ , and x = (W.+W_) thus eliminat- ing the constants A, , B, . The patching conditions are 19 2 3 CCJ-, = ^J, = ° (surface height continuity) (18) r i2 r i3 LhuJ, = LhuJ2 = 0 (normal flux continuity) (19) where [ :l , t ]. - t ]k . The following abbreviations will be used F. = Fa . (z .) G . s Ga . (z .) 3 3 3 3 D D The subscript, 1, refers to the solution for Region I (continental shelf) where it joins Region II (continental slope) . The subscript, 2, refers to the Region II where it joins Region I. The subscript, 3, refers to the continental slope where it joins Region III, the flat bottom. The func- tions subscripted 3 have the same form as those subscripted 2 with the exception of the variable, z , which is deter- mined by the distance from the origin. Using (18) and (19) between Regions I and II and set- ting A = 1, the constants of integration A and B can be solved for? G„F.. - F, G~ A = -^± ±-±- (20) 2 G F '- F G ' 2 2 2 2 F F ' - F F ' 12 2 1 B2 = ^"T (21) G2F2,_ F2G2" 20 Using (18) and (19) between Regions II and III gives the final equation in terms of m and <7 only: G2Fl'- F1G2' G2F2,_ F2G2 - {(4-m)F3 + 2mF3' F F ' - F F 1 2 2J1 G2F2 " F2G2 - |U-m)G + 2mG '} = 0 (22) Because there is no way to solve (22) analytically, it is necessary to find the roots numerically using the IBM 360/67 computer system at the Naval Postgraduate School. For a fixed cc = cr/f a search routine was used to find the several m's satisfying the equation. The computer work is described in Appendix A. 21 III. RESULTS AND CONCLUSIONS A. REVIEW There are a number of questions to be answered. MYSAK (1968a) in his finite-width model founds 1. Shelf -wave numbers are inversely related to the shelf width, for a fixed frequency. 2. There is a low wave number cut-off for edgewaves which is a function of the shelf width. That is, as the shelf width increases, the smallest possible wave number decreases (the largest possible wavelength increases). 3. The fundamental mode edgewaves of REID (1958) with periods greater than one pendulum day do not exist over a continental shelf of finite width. It should be noted that Mysak's solution is only ap- proximate in that the maximum shelf depth h is assumed much less than the ocean depth, D, leading to an approximation of the equation expressing continuity at the edge of the shelf (Mysak's equation (10)). His results (labeled MA below) depend on this approximation. Exact solutions of Mysak's equation (6) (corresponding to equation (22) in this work) , were also generated so that comparisons could be made among MA, an exact solution for Mysak's model (ME) and results of the two-slope model studied in Section II (TS). 22 B. COMPARISON OF MYSAK'S APPROXIMATE SOLUTION WITH AN EXACT SOLUTION FOR MYSAK'S MODEL Nine cases were studied in order to compare ME and MA. Two of the cases are illustrated in Fig. 2. In all cases, the continental shelf is 100 km wide with a slope of .002, duplicating Mysak's sample calculation. The bottom depth varied from 500m to 5000m. The shelf wave results are shown in Figs. 5-9. T^ > — r/i A fooom \-^—/ookm — »?' ) t } ) i ,, / , /// Y~~-lOC km—m^rrrXf-rrrr- (Vertical exaggeration = 100:1) Fig. 2. Profile of the finite-width model Note that: 1. The approximate solution consistently gives a smaller wave number for a particular cc and mode. It ap- pears to be the limiting condition for the exact solution. 2. Except for the fundamental mode, an error of less than 1% exists between corresponding modes of MA and ME in cases where the depth ratio h/D is smaller than .067. 3. For the fundamental mode there is still a signifi- cant error introduced in m when using MA for large depth ratios and frequencies (>10% for u) > 0.8 when D=5000m) . 23 4. The edgewave results are shown in Figs. 18 - 20. Neither MA nor ME gives a fundamental edgewave mode similar to that of Reid. This can be seen directly from Mysak's equation (10) in the approximate case. Because the argu- ment is positive by definition, a must be negative in order for the Laguerre function to have roots (zeroes) . This in turn requires that either f > a or a » f. 5. A low wave number cut-off does exist for ME edge- waves, which diminishes with an increase in depth and in- creases with an increase of | CO | (Table I) . The cut-offs are always lower than those for MA, and are quite symmetric with respect to direction of travel. Table I. ME edgewave wave number cut-off (mW) Deep-water depth (m) First mode Second mode 1000 0.58 1.32 3000 0.32 0.76 5000 0.26 0.58 As pointed out by Mysak and Reid, the edgewave disper- sion relation is not symmetrical with respect to direction of travel, due to the influence of the Coriolis parameter. 24 C. COMPARISON OF THE MYSAK MODEL WITH THE TWO-SLOPE MODEL /OOOvr, • 7/7? 1 (-*- /ookfr, •56 km /''/ 7/ 4 300O*">n > V > / / (Vertical exaggeration = 100:1) Fig. 3. Two-slope model. The Mysak model was compared with a two-slope model whose continental slope was .05 (Fig. 3) . The results are shown in Figs. 5-9. Note that: 1. Except for the fundamental mode for small oj , there is little similarity between the dispersion relations for the two models, especially for the large deep-water depths (i.e., wide continental slopes). For a deep-water depth of 5000 m, both mode 2 and 3 waves of TS have smaller wave numbers than mode 2 of ME for any given frequency. 2. The fundamental TS shelf wave does not asympto- tically approach CO = 1.0 for large wave numbers as does the corresponding ME wave. In fact, the fundamental wave now behaves like the fundamental edgewave of Reid. 3. A low wave number cutoff is still present for edgewaves, but at a significantly lower wave number (for 25 mode 1, nearly half that of ME) . The cutoffs are no longer symmetric with respect to direction of travel (Table II) . Table II Two-slope edgewave wave number cutoff (mW.. ) Deep-water depth CO < 0 co > 0 m Mode 1 Mode 2 Mode 1 Mode 2 1000 0.42 1.32 0.28 1.28 3000 0.26 0.76 0.16 0.76 5000 0.22 0.58 0.12 0.58 D. COMPARISON OF CONTINENTAL SLOPES 7-ttt Slope =0.03 Slope =0.08 (Vertical exaggeration = 100 si) Fig. 4. Two-slope models with different continental slopes. Three values were used for the value of the continen- tal slope; .03, .05, and .08 (Fig. 4). The results are shown in Figs. 10 - 18, for deep-water depths from 500 m to 5000 m. 26 1. Wave numbers for a particular mode of trapped waves over a fixed deep-water depth decrease with an in- crease of continental slope width. 2. For the fundamental mode over a constant gradient continental slope, wave numbers increase with an increase in slope width (i.e., an increase in deep-water depth). All other modes decrease with an increase in width. 3. A curve is not available for mode 3 for a conti- nental slope of 0.03 and deep-water depth of 5000 m. This is attributed to unknown problems of the computer routine. This problem does not occur elsewhere. 4. Discontinuities appear in the dispersion relations for modes 2 and 3 with a continental slope 0.05 (in the deep-water depth range 2200-3400 m) and with a continental slope 0.08 (for depths greater than 2800 m) , suggesting the presence of complex values of m. A similar phenomenon is not observed for a slope 0.03. Equation (22) was investi- gated for complex values of m and real u) (Appendix B) and complex roots were found for a slope of 0.05 and depth 2800 m (Fig. 22) . Since the surface height is the real part of C (x, y, t) complex values of m imply a spatial growth rate exp Cpm(my) } in the positive y direction. The roots m are complex conjugates, so that one wave grows and one decays at this rate. Then the most unstable wave (i.e., the one with the maximum growth rate) would be ex- pected to dominate the shelf-wave spectrum. 27 5. No complex wave numbers were found for the edge- waves studied. Continental slope width is inversely pro- portional to wave number as in Mysak ' s results. E. RECOMMENDED FURTHER STUDY The next step would be to study further the effect of different continental shelf slopes using the TS model. More study is mandatory in D 4 above, both to? 1. find its limits and 2. determine if this is a mathematical curiosity or a physical reality. Future investigators should seek to avoid approxima- tions to their models. 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GC=9 5K ,T IME.GO= //FORT.SYSIN DD * REAL*8 L,M,H,C0R,SIG,SL0(2) ,W(2 ) ,ANS ,X ( 3 ) , 7 ( 3 ) , P ( 2) , GRAV, DELTA COMMON SLC, M,L,H, COR, SIG,W,X,Z, GRAV, A,KCCL DIMENSION AF(2) ,A(3) , ANSI (3000) ,QM(60 ) , PT(6U ,8), KLUF(8 DATA PT,KLUE/4EC*G. 0,8*0/ READ(5,1C2) KOMAND IF KOMANO EQUALS CCPLANK CARD IN OA^A DECK) THE MYSAK SOLUTION AND APPROXIMATION ARE CCMPUTED. IF KOMAND EQUALS JNY INTEGER ( THROUGH 999) THE 2-SLCPE SOLUTION IS COMPUTED.. IF(KOMANO.NP.O) GOTO 4 CALL MY^K GOTO 5CC A GRAV=S.PDC2 DELTA=C. 2000-08 TW0PI*0ARCQS(-1.Q0>* 2. DO READ(5,52) VI,COP,SLO WRITE( 6,56)WrCOP ,SLO 89 WW=W( 1)+W( 2) KOUNT=C KTPL=1 T Ms fl H=W( 1 )* H.KK.MJ WRITE(7,72)(PT( II, KK) ,11=1, MJ) WRITE (6,92) VJ,(PT(II ,KK) ,II=1,VJ) WRITF( 6,62) CALL PLr*rp(PT(l,l) ,0M,KLUE(1) ,1 ) 00 33 ICK=2,8 !F(KLUF( I SK) .LP. A) GOTO 34 CONTINUE ISK l=fCK-l 00 35 KRIC=2,ISK1 C&LL PLCTP (PTt 1 ,KRIC) ,QM , KLUE ( KRI C ) ,2) CALL PLOTF (PT(1,ISK ) ,QW, KLUE ( ISK ) , 3) WRITE (6,65) REA0(5,PP) W(2) WRITE(6,56> U,CCR,^L0 IF( W( 2).NE.C.D0) STOP GOTP 8 9 •52 FORM^ t( KE1C#3) c3 FORMA T( 5E1C.3) *4 FORMAT ( 1H ,5D2C7) 5* FOPVAT( lHC,12X,lHMf19X,lHL,17X,5HSI0MA,17X,6HANSWER, 215X,1HH,/,5D20. 7) 56 F0PMAT(1H0,5C2G.7) *7 FORMAT* 1H ,3025.12) *8 cqpm^ T(1HC,PX,4HC(1) t16X,4HP(2) , 16 X ,4HM 1 ) , 1 6X , 4HA ( 2 ) , 416X,.LE.G.1D-G2> GOTO 1 ANSWER TOO LAPGF WITHOUT CONVERGENCE? IF (n4BS(DX).GE.l.D65.0R.OABS(nX).LE.1.0-65 > GOTO 1C CONVFRGENCE? IF(DABS(DQ).LE.C.5D-C8. AND.K0UNT.GT.2) GOTO 3 DN=DN+1.00 KOUNT = KOUNT-H GOTH A 1 CALL DLAPG(Y,X,DL,A) GOTO 3 2 WRITE* 6,54>KGUNT 3 RETURN 1C WRITE(6,57) KOUNT,DX,0L GOTO 2 6 WPITF(6,53) GOTO 3 c3 FQRMATfcSH X IS C.L^GLERRE FUNCTION DF THE SECONH KIND DOFS NCT EXI ST) 54 FOPMAT( 1H ,1C0X,I3) *S F0RMAT(1H ,602C7, /f02C.7| 57 F0RMAT(2CH OVF°FLOW APPROACHING. AFTER, 13 « 15W ITER- ATIONS ,DX= ,E 12 . 5 ,9H ANO PL = ,E12.5) END SUBROUTINE DLAPG(Y,X,OL,A) OLAPG SOLVES FOR LAGUERRE FUNCTIONS OF THE SFCCNO KINC WHEN A IS A NEGATIVE INTEGFP.Tn AVOID DIVIDING BY ZERO, THE NTH TFPM IS THF SUV OF N-l MULTIPLICATIONS. 22 21 56 PEAL*S KOUNT= 0N=1.D D0=1.D DX=1.D DL=Y*D DA = A 0Z=1.D 02=0.0 dx=ox* 011 = C DO 1 I 01=1. K0OK=I DD2 K = IF(K.E IF(DAB 01=01* IF(K00 D^ 3 L IF(DAB D1=D1* 011=01 D2=D2 OQ=DQ* IF(0O. 7Z=(D1 KOUNT= IF(KOU DL=DL+ IF(DAB ON=ON+ GHTO 2 RETURN WPITEC G0T0 2 Y,X,D4,Z7,DZ,D1 ,02,PN,0L,Pll,DXf DO 0 c c 0 LOG( X) 0 c x .DC =C,KOUNT 00 ♦ 1 CI Q.I) GOTO 4 S(D1).LE.C.50-1C) GOTH i (OA+DFLOATCK) ) K.GT.KCUMT) GnTC 1 =KOCK,KOUNT S(D1).LE.0.5D-1C) GOTO 1 (D4+DrL0AT(L) ) 1+01 -( 2. 00 /ON) DN*DN GE.C. 1D40) GOTO 56 1 +D1*(DA+DFL0AT(K0UNT) )*02)*DX/DQ KOUNT+1 NT.GT.25) GOTO 56 ZZ S(ZZ) .LE.C.50-C8) GOTO 21 l.DC 2 6,53) ZZ 1 53 51 FQPMATllH ,M5X,E15.8)) 53 FOPMAT(36H G(X) DID NCT CONVERGE. LAST TFRM WAS,E15.8> END SUBROUTINE DRDLAP(A,X, RESULT) DRDLAP SOLVES THE FIRST DERIVATIVE OF THE LAGUEPRE FUNCTION OF THE FIRST KIND. REAL*? YDA,YPD, RESULT, YDB,YDN,X YDA*A*1.DG RESULT=A YDN-l.DO YDP=A 22 YOB=YDN+l.Dj YDD=YDD*YDA*X/ (YDN*YDB) RFSULT=PF^ULT+YDD IHDABM YCD).LT.0.5D-8) GOTO 21 YDA=YDA-H.DO YDN= YDN+l.nO GOTO 2 2 21 RETUPN 23 FORMAT( 1H ,2(5XtE15.Q) ) 54, FORMAT ( 1H ,6020.7) END SUBROUTINE DGDLAP( A , X, RESULT , PLG ,RESIIL ) DGDLAP SOLVES THE FI&ST DERIVATIVE OF THE SECOND KIND OF LAGUERPE FUNCTION. WHEN X=0, THIS FUNCTION DOES NOT EXIST. PEAL*8 X,RESLLtRESULT,XA,PLG,XN,X3 ,X4 ,X2,XX,X?,XC IF(X.EO.C) GCTO 1C XN=2.DG XA=A RESUL=RESULT* OLOG ( X) +PLG/X +1 . CO- { 2 . DO *X A ) X2=1.DG/XA-2.DC XX=XA 21 XA = X^+1.DC IF(DABS(XA).LE.C.5D-M GCTO 22 XX=XX*X/( XN*XN-XN) *X* X2=X2 -2.DC/XN +1.D0/XA XZ=XX*X2 RESUL = PEHL + XZ IF(DABS( X7).LE. 0.50-9) GOTO 10 XN=XN +1.DC IF (DABS( X7) .GE.1.D65) GCTO 23 GO TO 21 22 KISS=1 XN=2.DC XA=A RESUL=PE$ULT*DLOG< X) +■ RLG/X + 1 . 00-( 2. DC*X A ) X0=1.DC XX=1.DC X2=-2.DC 5 XX=XX*X X3=0.DC DO 6 11=0, KISS XA-l.DO KIN=KISS+1 DO 7 JJ=C,II IF(JJ.EO.II) GOTO 9 IF(OABS( X4).LE.G. 50-10) GOTO 6 7 X4=XA*(XA+DFL0AT( JJ) ) 9 IF(KIN.GT.KISS) GOTO 6 DO 8 LL=KIN,KISS 54 IF(PABS( X4).LE.0.50-1C) GOTO 6 8 X4=X4*( XA+OFLOAT(LD) 6 X3=X3+X4 X2=X2-(2.PC/XN) XO=XO*XN'*DFLCAT(KI SS) XZ=( X3+X4*( XA+O FLOAT ( KTSS) ) *X2) *XX/XQ KISS=KISS+1 RFSUL=PESUL+X7 IFCPABSI X7).LE.C.5P-8) GOTO 10 XN=XN+1.D0 IF(KISS.LT.25) WRITE(6,54> XZ 23 N=XN WRITE(6,24) N 10 RFTUPN GOTO 24 FORM^T(ACH OVERFLOW ABOUT TO OCCUR IN OGOLAP AFTeRtI3, 2 7H TERMS.) 51 FOPMATf 1H f4(5X.E15.8) ) 54 FORMA T(1H ,36HGMX) PIP NOT C CNVFRGF. LAST TERM WAS, 1F15.B) ENO //GO.FT06FGG1 PO OCB= ( PECFM = F* ,B LKS I 7 E=l 3 3 ) ,S PACE= ( CYL , ( 15, 1 //GP.SYSIN 00 * 55 APPENDIX B NUMERICAL SOLUTION FOR TWO-SLOPE COMPLEX ROOTS // EXEC FORTCLGP, PARM. FORT= • LI ST, MAP* , REGION. GO= 100K, T IME . GO //FORT.SYSIN DD * COMPLEX* 16 M,L,ANS,X(3),Z(3),P(2),A(3),AE(2),ANSI (1620 REAL*8 H,CnR,SIG, SLO( 2) , W( 2 ) , GRAV, PARI, PAR 2, ANSWER ( 162 COMMON M,L,X,Z,A,H,COR,SIG,SLO,W, GRAV,KOOL DATA ANSI/1620*(0.E0,0.E0)/ THIS PROGRAM SOLVES FOR THE COMPLEX ROOTS OF THE TWO SLOPE MODEL. IN THIS CASE, THE ROOT WAS FOUND FOR OM- EGA=12/60 AT J= 10 AND K= 30. THIS GIVES AN ANSWER OF ( 1.5459r-07,0.1249E-07) FOR THE ROOT. GRAV=9.8D2 READ(5,52) W,COR,SLO WRITE(6,56)W,C0R, SLO 39 WW=W( 1)+W(2) H=W(1)*SL0( 1) +W(2)*SL0(2) WRITE(6,100) PARl=1.541234E-07 SIG=-12.00*C0R/6.00E0 1 DC 2 J=1,2C PAR1 = PAR1 + 0. 5000-10 M=PAR1 *(1.00,0.D0) DO 1 1=1,2 P( I) = ( SIG*SIG-COR*COR) /(GRAV* SLO ( I) )-COR*M/SIG AE(I)=(M-P( I ) )/(2.D0*M) 1 A(I)=AE( I) A(3)=A(2) CALL F1NSIG(ANS,KEY) ANSI ( J)=ANS ANSWER(J)=CnABS(ANSI( J) ) DC 2 K=221,260 PAR2= DFLOAT(K)* C.50CD-10 M=PAR1*(1.CO,O.DO) ♦ PAR2* (O.DO,l.DO) DO 8 1=1,2 P(I)=( SIG*SIG-CQR*COR)/(GRAV*SLO( I) )-CQR*M/SIG AE(I) = (M-P( I ) )/(2.D0*M) 8 AU) = AE( I) A(3)=A(2) CALL FINSIG(ANS,KEY) KCFE=(K-22C)*20 + J PRINTS ANSK21)- ANSK820) ANSI(KOFE)=ANS ANSWER (K0FE)=CDA8S(ANS I (KOFE) ) M=M-(2.DC*PAR2*(0.D0, l.DO) ) DO 9 1=1,2 P( I)=( SIG*SIG-CCR*COR)/(GRAV*SLC( I) )-COR*M/SIG AE( I )=(M-P( I ) )/(2.D0*M) 9 A(I)=AE( I) A(3)=A(2) CALL FINSIG(ANS,KEY) KOFF=(K-180)*20 + J PRINTS ANSM821)- ANSI! 1620) ANSI (KOFF)=ANS ANSWER ( KOFF)=CDABS (AN SI (KOFF) ) 2 CONTINUE DC 4 LPA=1,31 K1=(LPA-1)*20+1 K2=LPA*20 WRITE (6, 101) (ANSI (LP) , ANSWER (LP) ,LP=Ki,K2) 4 CONTINUE 52 FORMAK5E10.3) 56 FORMAT(lH0,5D2C7) 100 FORMATdHl ) 101 FORMAT( 1H0,(9D14.5) ) STCP 56 END SUBROUTINE F INS IG ( ANS ,KEY ) COMPLEX* 16 M,L,X(3),Z(3),A<3) ,F ( 3 ) ,FPRI ( 3 ) ,G(3),GPRI * (3), CANS RFAL*8 H,COR,SIG,SLO(2),W(2) ,GRAV, CRAPS CCMMON M,L,X,Z,A,H,COR,SIG,SLO,W,GRAV,KOOL KOCL=0 CRAPS=0. 1D30 C=(M*M*GRAV*H+COR*CGR-SIG*SIG)/(GRAV*H) L=CDS0RT(C) X( 1)=W( I) X(2)=SLO( 1)*W( l)/SLO( 2) X(3)=X(2)+W(2> DO 2 1=1,3 Z(I)=X(I )*2.DO*M CALL DLAP(F( I ) ,Z( I) ,A< I) 9NN) CALL DLAPGGtF ( I ) ,Z( I ) ,G( I ),A( I ) ) CALL DRDLAP( A(I ) , ZCI) ,FPRI< [J ) CALL DGDLAP(A(I ) ,Z(I) ,FPRI( I ) ,G( I ) ,GPRI< I ) ) IF(CDABS(FU ) ) . GE. CRAPS. OR. CDABS(GU )) .GE .CRAPS. OR CD- ABS(FPRI ( I )) .GE.CRAPS.OR.CDABSCGPRK I ) ).GE. CRAPS) GOTO ? J 2 CONTINUE ANS=(L-M)*{F(3)*( F(1)*GPRI(2)-G(2)*FPRI<1) )*G(3)*(F<2) 2*FPRI ( l)-Fll)*FRRI(2) ) ) +2*M* ( FPRI ( 3 ) * ( F( 1 ) *GPRI( 2 I- 3G(2)*FPRI (1) )+GPRI<3)*(F< I ) *FPR I < 1 )-F ( 1)*FPRI<2) ) ) IF(KEY.EQ.l) WRITE(6,51) ANS,M,L IF(CDABSIANS) .GE. O.IDA) GOTO 4 1 RETURN 4 KOGL=l WRITE(6,61) M GCTO 1 51 FORMATdH ,3D20.7) 61 FCRMAT(3AH0SGLUTIGN DOES NOT EXIST BEYOND M=,E12.4) END SUBROUTINE OLAP( Y , X, A,NN) CLAP SOLVES FOR LAGUEPRE FUNCTIONS OF THE FIRST KIND. GENER- ALLY THE ARGUMENT, A, WOULD BE EXPECTED TO BE NEGAT I VE .WHEN A IS POSITIVE, NN IS SET TO 1. C0MPLEX*16 Y,X,A,YN,YX REAL*8 YY,YZ YN=A YY=1.D0 YK=1.DC YX=1.D0 YZ=1.C0 3 YX=X/YZ*YN/YZ*YX YY=YY+YX YN=YN+1.D0 YZ=YZ*1.D0 IF0/DN) D0=DX*D2 DL=DL+DU DA=DA+l.DO 1/DA APPROACHING INFINITY? IF(CDABStCX) .GE.0.lD65.OR.CDABS(DX ) .L E .0 . 1D-60 ) GOTO 1 ANSWER TOO LARGE WITHOUT CONVERGENCE? IF(CDABS(DA) .LE.0.1D-2) GCTO 1 CCNVERGENCE? 1F(CDABS(DQ) .LE. 0 . 5D- 8. AND. KOLNT. GT .2 ) GOTC 3 DN=DN+1.D0 K0UNT=K0UNT+1 GOTO A 1 CALL CLAPG(Y,X,DL,A) GOTO 3 2 WRITE(6,54)K0UNT 3 RETURN 10 WPITE(6,57) KCUNT,DX,CL GOTO 2 6 WPITE(6,53) GCTO 3 53 FCRMAT(59H X IS 0. LAGUERRE FUNCTION OF THE SECOND KIND H CCES NOT FXIST) 54 FCRMATUH ,100X,I3) 55 F0RMAT(1H , 6D20 . 7 , /, D20 . 7 ) 57 FCRMATI29H OVERFLOW APPROACH ING . AFTER ,I3,15H ITERAT- / I0NS,DX=,E12.5,9H AND DL=,E12.5) END SUBROUTINE DL APG( Y, X, CL , A ) COMPLEX* 16 Y,X,A,DL,DA,ZZ,D1,D11,DX REAL*8 D2,DN,DQ DLAPG SOLVES FOR LAGUERRE FUNCTIONS OF THE SECONC KIND WHEN A IS A NEGATIVE INTEGER. TO AVOID DIVIDING BY ZERO, THE NTH TERM IS THE SUM OF N-l MULTIPLICATIONS. KOUNT=0 DN=1.D0 DQ=1.D0 DX=1.D0 DL=Y*CDLOG 9 IF(KIN.GT.KISS) GOTO 6 DC 8 LL=KIN,KISS IFtCDABS(XA) .LE. 0.50-101 GOTO 6 8 X4=X4*(XA+DFL0AT( LL) ) 6 X3=X3+X4 X2=X2-(2.DC/XN) XC=XQ*XN*DFLOAT(KISS) XZ=(X3+X4*(XA+DFL0AT(KISS) )*X2)*XX/XQ KISS=KISS+1 RESUL=RESUL+XZ IF(CDABS(XZ) .LE. 0.50-8) GOTO 10 XN=XN+1.D0 IF(KISS.LT.25) GOTO 5 WRITE<6,54) XZ 23 N=XN WPITE(6,24) N 10 RETURN 24 F0RMAT(40H OVERFLOW ABOUT TO OCCUR IN DGDLAP AFTER,! 3 + 7H TERMS.) 51 FORMATdH ,4( 5X , E 15. 8 ) ) i 54 FURMATUH ,36HG'lX) DID NOT CONVERGE . LAST TERM WAS, , E15.8) END //GC.FT06F001 DC DC E= ( R ECFM = FA , BLKS I ZE= 1 33) , SP ACE= ( CYL , ( 15, 1 //GO.SYSUDUMP DD SYSOUT=A //GO.SYSIN DD * 0.100D 08 0.520D 07 0.729D-04 0. 2000-02 0.500D-01 60 LIST OF REFERENCES 1. ADAMS, J.K., BUCHWALD V.T. (1969), The Generation of Continental Shelf Waves. Journal of Fluid Mechanics 35 (4), 815-826. 2. HAMON, B.V. (1962), The Spectrum of Mean Sea Level at Sydney, Coff's Harbour, and Lord Howe Island. Journal of Geophysical Research 67 (13), 5147-5155. 3. HAMON,B.V. (1963), Correction to "The Spectrums of Mean Sea Level at Sydney, Coff's Harbour, and Lord Howe Island". Journal of Geophysical Research 68 (15), 4365. 4. LAMB,H. , Hydrodynamic s , Sixth Edition, p. 446. Dover Publications, 1932. 5. MOOERS, C.N.K., SMITH, R.L. (1968), Continental Shelf Waves off Oregon. Journal of Geophysical Research 73 (2), 549-557. 6. MYSAK, L.A. (1967), On the Theory of Continental Shelf Waves. Journal of Marine Research 25 (3), 207-227. 7. MYSAK, L.A. (1968a), Edgewaves on a Gently Sloping Continental Shelf of Finite Width. Journal of Marine Research 26(1), 24-33. 8. MYSAK, L.A. (1968b), Effects of Deep=sea Stratifica- tion and Currents on Edgewaves. Journal of Marine Research 26 (1), 34-42. 9. MYSAK, L.A., HAMON, B.V. (1969), Low-frequency Sea Level Behavior and Continental Shelf Waves off North Carolina. Journal of Geophysical Research 74 (6) , 1397-1405. 10. REID, R.O. (1958), Effect of Coriolis Force on Edge- waves (I): Investigation of the Normal Modes. Journal of Marine Research 16 (2) , 367-368. 11. SLATER, L.J., Confluent Hypergeometric Functions, pp. 1-8, Cambridge University Press, 1960. 12. URSELL, F. (1952), Edgewaves on a Sloping Beach., Proceedings Royal Society, (A) 214, 79-97. 61 INITIAL DISTRIBUTION LIST No. Copies 1. Defense Documentation Center 20 Cameron Station Alexandria, Virginia 22314 2. Library, Code 0212 2 Naval Postgraduate School Monterey, California 93940 3. Oceanographer of the Navy 1 The Madison Building 7 32 North Washington Street Alexandria, Virginia 22314 4. Professor Theodore Green III 3 Department of Meteorology University of Wisconsin Madison, Wisconsin 57306 5. LCDR. H. Dixon Sturr, Jr., USN 3 USS MULIPHEN (LKA-61) Fleet Post Office, New York 09501 6. Professor D. G. Williams, Code 0211 1 Naval Postgraduate School Monterey, California 93940 7. Dept. of Oceanography, Code 58 3 Naval Postgraduate School Monterey, California 93940 62 Unclassified Security Classification DOCUMENT CONTROL DATA -R&D (Security classification ol title, body ol abstract and indexing annotation must be antered when the overall report Is claaailled) originating ACTIVITY (Corporate author) Naval Postgraduate School Monterey, California 93940 2a. REPORT SECURITY CLASSIFICATION Unclassified 2b. GROUP 3. REPORT TITLE Continental Shelf Waves Over a Continental Slope 4. DESCRIPTIVE NOTES (Type of report and.inclusive dates) Master's Thesis; October 1969 5. AUTHOR(S) (First name, middle initial, last name) Henry D. Sturr, Jr, 6. REPORT DATE October 1969 7a. TOTAL NO. OF PAGES 61 7b. NO. OF REFS JUL •a. CONTRACT OR GRANT NO. b. PROJEC T NO. 9a. ORIGINATOR'S REPORT NUMBER'S) 9b. OTHER REPORT NO(S) (Any other numbers that may be aaalgrted this report) 10. DISTRIBUTION STATEMENT This document has been approved for public release and sale; distribution is unlimited. its II. SUPPLEMENTARY NOTES 12- SPONSORING MILITARY ACTIVITY Naval Postgraduate School Monterey, California 93940 13. ABSTRACT A numerical study is made of the effect of a continental slope and shelf of finite width on trapped shelf and edgewaves. A comparison is made between a numerical solution for a continental shelf of finite width and its simplified analytic solution. It is shown that certain modes of these quasigeos trophic waves can undergo exponential growth or decay under special conditions. DD F0"" 1473 1 NOV 65 I "T / *J S/N 0101 -807-681 1 (PAGE 1) 63 Unclassified Security Classification A- 31408 Unclassified Security Classification key wo ROS Trapped Waves Shelf Waves Edgewaves DD ,'°"v\,1473 «»ck> S/N 0101-807-6821 ROLE WT Unclassified 64 Security Classification A-31 409 thesS8579 Continental shelf waves over a continent 3 2768 002 02163 6 DUDLEY KNOX LIBRARY