NUMERICAL INVESTIGATION OF THE DYNAMICS OF SEA OF MARMARA Huseyin Yuce DUDLEY KNOX LIBRARY HAVAL POSTGRADUATE SCHOOL MONTEREY, c- "IMB NAVAL POSTGRADUATE SCHOOL Monterey, California THESIS NUMERICAL INVESTIGATION OF THE DYNAMICS OF SEA OF MARMARA by Huseyin Yuce September 1976 Thesis Advisor: J. B. Wickham Approved for public release; distribution unlimited T174966 SECURITY CLASSIFICATION OF THIS PACE fffhai Dmtm Bntmrmd) REPORT DOCUMENTATION PAGE READ INSTRUCTIONS BEFORE COMPLETING FORM 1. REPORT NUMBER 2. GOVT ACCESSION NO. 1. RECIPIENT'S CATALOG NUMBER 4. TITLE (mnd Subtitle) Numerical Investigation of the Dynamics of Sea of Marmara S. TYPE OF REPORT * PERIOO COVERED Master's Thesis; September 1976 « PERFORMING ORG. REPORT NUMBER 7. AUTHORS •> Huseyin Yuce •. CONTRACT OR GRANT NOMSERO) »- PERFORMING ORGANIZATION NAME AND ADDRESS Naval Postgraduate School Monterey, California 93 940 10. PROGRAM ELEMENT. PROJECT, TASK AREA * WORK UNIT NUMBERS II. CONTROLLING OFFICE NAME AND ADDRESS Naval Postgraduate School Monterey, California 93940 12. REPORT OATE September 1976 13. NUMBER OF PACES 124 14. MONITORING AGENCY NAME * AOORESSfi/ dltterent from Controlling oii'lcm) Naval Postgraduate School Monterey, California 93940 IS. SECURITY CLASS, (ol the report) Unclassified tSa. DECLASSIFICATION/ DOWNGRADING SCHEDULE 16. DISTRIBUTION STATEMENT (ol thlm Report) Approved for public release; distribution unlimited. 17. DISTRIBUTION STATEMENT (of tne momttmct tntorod In Block 30, it dltterent from Report) IS. SUPPLEMENTARY NOTES 19. KEY WORDS (Continue on rewerme aide It neeeemmry mnd Identity or Hook numeer) Sea of Marmara Turkish Straits Regional Circulation Models 20. ABSTRACT (Continue on referee elde II neeeeemry mnd Identity my olmok numeer) Dynamics of the circulation in the Sea of Marmara are investigated with a time dependent, three dimensional numerical model. The empirically inferred hydrologic regimes of the sea and connective straits are discussed. A baroclinic model based on the primitive equations is solved by direct integration of an initial value problem. Do,: (Page 1) 'ST* 1473 EDITION OF I NOV SO IS OBSOLETE S/N 0102-01«-«60t | SECURITY CLASSIFICATION OF THIS PAGE (9hon Dmtm Bntmrmd) 1 futuWTY CLASSIFICATION OF THIS P«G£W«n rtrtm £r»«»r.d The circulation in the sea is driven by surface forces that simulate wind stress and horizontal pressure gradient forces related to internal stratification. The predicted density field, in sigma— t units, is com- pared with data. Detailed three dimensional horizontal velocity patterns and vertical velocity patterns in hori- zontal planes are given. Bottom friction, irregular bottom topography, non— linear terms in the momentum equation and vertical mean part of the horizontal velocity have been omitted. For simplicity density is predicted in place of temperature and salinity. DD Form 1473 . 1 Jan 73 S/N 0102-014-6601 SECURITY CLASSIFICATION OF THIS F»CtrWi»n Dmlm Enfrmd) Numerical Investigation of the Dynamics of Sea of Marmara by Huseyin Yuce Lieutenant, Turkish Navy Turkish Navy Academy, 1970 Submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN OCEANOGRAPHY from the NAVAL POSTGRADUATE SCHOOL September 1976 mis OuniEv KNOY I tor, ABSTRACT Dynamics of the circulation in the Sea of Marmara are investigated with a time dependent, three dimensional numerical model. The empirically inferred hydrologic regimes of the sea and connective straits are discussed. A baroclinic model based on the primitive equations is solved by direct integration of an initial value problem. The circulation in the Sea is driven by surface forces that simulate wind stress and horizontal pressure gradient forces related to internal stratification. The predicted density field, in sigma— t units, is compared with data. Detailed three dimensional horizontal velocity patterns and vertical velocity patterns in horizontal planes are given. Bottom friction, irregular bottom topography, non- linear terms in the momentum equation and vertical mean part of the horizontal velocity have been omitted. For simplicity density is predicted in place of temperature and salinity. TABLE OF CONTENTS I. INTRODUCTION 10 II. PHYSICAL GEOGRAPHY AND HYDROGRAPHY OF THE SEA OF MARMARA 12 III. SELECTION OF THE NUMERICAL MODEL 29 IV. DESCRIPTION OF THE DYNAMIC MODEL 31 A. BASIC EQUATIONS OF THE MODEL 31 B. DESCRIPTION OF THE SOLUTION TECHNIQUES 34 V. DESCRIPTION OF THE NUMERICAL MODEL 42 A. SPACE AND TIME DIFFERENCING TECHNIQUES 42 B. THE FINITE DIFFERENCE EQUATIONS 45 C. SPECIFICATION OF THE BOUNDARY CONDITIONS 55 VI. RESULTS 58 VII. CONCLUSIONS 86 APPENDIX A. COMPUTER PROGRAM DESCRIPTION 89 COMPUTER PROGRAM 99 LIST OF REFERENCES 121 INITIAL DISTRIBUTION LIST 123 LIST OF TABLES Table I. Monthly average fluctuation of sea level differential and sea level differential — — — 18 II. Parameters and constants of the model — 41 III. Layer thicknesses — 47 LIST OF FIGURES Figure 1. Location map of the Sea of Marmara — — 13 2. Schematic diagram of water stratification and current system at two connective straits which has important effect determining hydrological condition of Sea of Marmara — — 16 3. Sigma— t cross section of Strait of Istanbul — — — 20 4. Salinity cross section of Strait of Istanbul — — — 22 5. Temperature cross section of Strait of Istanbul 23 6. Sigma— t cross section of Sea of Marmara for upper 100 meters 26 7. Temperature cross section of Sea of Marmara for upper 100 meters — — 27 8. Salinity cross section of Sea of Marmara for upper 100 meters — — — 28 9. Placement of variables on horizontal grid plane — — — — — ___ 45 10. Vertical structure of the model — ______ 48 11. a. Horizontal location of variables b. Vertical location of variables — — — 51 12. Horizontal velocity vectors for first level, (2.5 meter) 59 13. Horizontal velocity vectors for second level, (7.5 meter) 61 14. Horizontal velocity vectors for fifth level, (40.0 meter) 62 15. Horizontal velocity vectors for ninth level, (700.0 meter) 63 16. Vertical velocity at the base of the first layer (5.0 meter) 65 17. Vertical velocity at the base of the second layer (10.0 meter) 66 18. Vertical velocity at the base of the eighth layer (500 meter) 67 19. Horizontal sigma— t pattern at first level (2.5 meter) 68 20. Horizontal sigma— t pattern at second level, (7.5 meter) 70 21. Horizontal sigma— t pattern at fourth level, (20 meter) 71 22. Meridional cross section for sigma— t at the western part of the basin — 73 23. Meridional cross section for sigma— t at the central part of the basin 74 24. Meridional cross section for sigma— t at the eastern part of the basin — — 75 25. Meridional cross section of the zonal velocity, (u) , at the western part of the basin — — 76 26. Meridional cross section of the zonal velocity, (u) , at the central part of the basin — 77 27. Meridional cross section of the zonal velocity, (u) , at the eastern part of the basin — — — ____________ 78 28. East— west sigma— t cross section for the southern part of the basin —————————— 79 29. East— west sigma— t cross section for the central part of the basin — ___ 80 30. East— west sigma— t cross section for the northern part of the basin _ — ___— 81 31. East— west cross section of the meridional velocity, (v) , at the southern part of the basin 83 32. East— west cross section of the meridional velocity, (v) , at the central part of the basin 84 33. East— west cross section of the meridional velocity, (v) , at the northern part of the basin — — — — — — — — — — 85 34. Descriptive flow diagram of the program — — — 92 ACKNOWLE DGEMEN TS The author wishes to express his appreciation for the guidance and assistance given by Professor J. B. Wickham. He is also indebted to Professor R. L. Haney who generously shared his experience in the field of modelling in numerous invaluable discussions and timely encouragement throughout the entire study. He is grateful to Lieutenant Commander Mustafa Dogusal, Turkish Navy, who made available all data for use in this study. Thanks go to the staff of the W. R. Church Computer Center whose patience and understanding was greatly appre- ciated. Finally, thanks to his wife, Fatos, for her faith, en- couragement and understanding, to his young boys, Kerem, Erhan, for all their efforts which made this study possible, I. INTRODUCTION There is an intense growing interest in recent years in the development of simulation of hydrological conditions in coastal waters and smaller adjacent seas. Advancement in modern technology requires much greater accuracy than can be obtained by the use of classical instrumentation and measure- ment techniques only. Due to the discontinuous nature of such observations these measurements can provide the basis for analysis of the current system, temperature, salinity and other variables on a climatological basis. But climato— logical conditions are not accurate enough in coastal waters for uses such as construction, navigation, tracing of pollu- tants, mining effluent, oil spills, search and rescue opera- tions, agriculture, etc. It is also true that sampling the ocean adequately whether for exploratory or research purpose is a major problem. Great short— term variability of condi- tions in coastal waters makes it even more difficult to make adequate representative measurements. As a result, in addition to the classical methods of analyses of current systems, determination of salinity and temperature distributions etc., new simulation techniques which were applied long before synoptic numerical weather prediction received much more attention (T. Levastu, S. Larson and K. Rabe, 1976) . 10 The objective of this study is to develop a regional circulation model of the Sea of Marmara. The circulation and density distribution, in sigma— t units, in that sea are simulated by a nine- level numerical model with an hori- zontal flat bottom at 1000 m depth. The motion is driven by prescribed wind stress and horizontal pressure gradient forces due to internal stratification. 11 II. PHYSICAL GEOGRAPHY AND HYDROLOGY OF THE SEA OF MARMARA The Sea of Marmara is a small sea located between Asia Minor and Europe and occupies the main part of the Turkish straits system (210 km) . The Strait of Canakkale (Darda- nelles) (60 km) connects the Sea of Marmara to the Mediter- ranean and through the Strait of Istanbul (Bosporus) (30 km) it is connected to the Black Sea at the North. (Figure 1) The structure of the basin which is small in area (about 2 11000 km ) is associated with the Anatolicin fault which lies along its bottom to the North. The trough passing along the Northern deep slope consists of three depressions (max. 1380 m) . These three basins are separated by low connecting sills. A shallow trough lies at the foot of the continental slope which is well pronounced only in the easternmost part of the sea. At the South it becomes shallower having depths of the order 60—80 m. Minimum depths at the Strait of Canakkale and Strait of Istanbul are 57 and 37 m; they have mean widths 4.5 km and 0.7 km and average lengths 60 and 30 km, respectively. The climate of the Sea of Marmara is influenced by the regional atmospheric circulation system of the Eastern Medi- terranean, the Black Sea and Asia Minor Peninsula. During the summer, the Eastern Mediterranean basin is under the ] 2 c3 U fd e U (fl s o id o a) Based on the first detailed survey of Makarov and Merz 1917-18, in Moller (1928), and subsequent studies of Ullyott -* and Ilgaz (1946), Pektas (1956), A. K. Bogdanova (1961, 1965), C. G. Gunnerson and E. Ozturgut (1974) and others, the main -y properties of this straits' system and its vicinities are fairly well understood. One of the main characteristics of these straits is a two— layer structure of the current, which is determined by the differences between the relatively fresh water of the Black Sea and saline water from the Mediterranean which occu- pies the depths of the Sea of Marmara and forms the bottom water of two connective straits (Figure 2). Other external factors such as wind, water level drop along the straits, water balance, topographic influences and the depth and width of the straits also must be considered. The scale of the motion and complexity of these factors make it difficult to determine the current system mathematically. Another main characteristic of these straits is the two- layer stratification of water which is associated with the 15 i : :i o • CD a > v a 3 a < < a: < u_ O < UJ . m ; \ ; ^ < z < r f~}> lb o } S <- < at fr- iz* V3S \ NV393V - < £ 1 33 0 -p P CO TJ >1 >1 m x: ■P a> C .C a) •p u u tn 3 CJ O •H C T3 •H C £ rd p CD c -P 0 CD •H T3 +J (0 C u •H •H *W -P •H C -P 03 rd 4-> P, P -P 0 w 04 p. •H CD • -P a) fd 0 53 •H •H £ 4J P u O ■H w CD T3 C c C 0 9 0 o CM 0 <-* a) 0 fd u £ 0 3 -P ■H cn CP •H +J 0 fa rd H 16 two— layer current system. This stratification is also a consequence of the water mass differences at the two ends of the straits. Detailed observations of temperature and salinity dis- tributions indicate certain features of the diffusion of the Mediterranean water into the Black Sea. This water flows North through the Strait of Istanbul from the Sea of Marmara. It is shown by Bogdanova (1961, 1965) that this underflow of Mediterranean water into the Black Sea is present throughout the year and descends to significant depths in the Black Sea. This water exchange through the straits system fluctu- ates according to seasons, years and periods. This is very important for the hydrologic regimes of the Sea of Marmara and the Black Sea. Water level differences, given with the following expression, are mainly due to excess precipita- tion over evaporation and river runoff in the Black Sea. Ah = ZB - ZM (1.1) Ah : sea level differential between Black Sea and Mediterranean ZR : water level of Black Sea Z., : water level of Mediterranean M Sea level differential (Ah) is always positive and fluctu- ates with time. It can be expressed in two components Ah = Ah + (Ah) » (1.2) 17 where Ah : average sea level differential (Ah) ' : fluctuating part of sea level differential [(Ah)' « f(t)] According to Bogdanova (1965) Ah has a value 42 cm based on the long term observations. The fluctuating part of the sea level differential and total level differential are given in Table I. TABLE I. Monthly average fluctuation sea level differential and sea level differential. 1 Month I II III IV V VI VII VIII IX X XI XII (Ah) ' (cm) 1 0 3 7 11 15 6 0 -5 -7 -7 -4 Ah (cm) 43 42 45 49 53 57 48 42 37 35 35 38 ( Based on the data given by Bogdanova, 1965) The maximum change is -^ (Ah) v = ^r (Ah') v = - 9 cm/month, 3t max 9t max observed during approximately June. This falls approximately in a period when continental runoff begins to decrease. The fluctuating sea level differential increases from March to July. As a consequence during this period the upper current becomes stronger and the lower current becomes weaker. From 18 August to November sea level fluctuation decreases and be- gins to increase again from August to February. The upper current of the Strait of Istanbul becomes weaker and the lower current becomes stronger. According to Bogdanova's (1965) observations the maxi- mum Ah value is observed during June and has a value 57 cm; the minimum Ah value 35 cm is observed during October and November. The average sea level differential between the Black Sea and the Sea of Marmara is about 3 5 cm which is six times Moller's (192 8) estimate (Gunnerson and Ozturgut, 1974) . There is a characteristic stratification with a sharp density transition layer over the whole area. The sharpness is more pronounced at the North at the Strait of Istanbul. A cross section of sigma— t values for the Strait of Istanbul is shown in Figure 3. Fresh warm water of the Black Sea lies in a surface layer 15—20 m thick at Istanbul. This water which is mainly river runoff and excess precipitation overrides the Mediterranean water which is cold and dense. The wedge form of the upper water shows clearly in this strait, but this is not true for the Strait of Canakkale. On the other hand, the wedge form of the lower water is com- paratively stronger and more pronounced in the Strait of Canakkale due to the downward sloping of the sea bed toward the North in the Strait of Istanbul. Average sigma— t values for surface and bottom water are 13.5 and 27.5. (Defant, 1961) 19 p XI c (C +J CO H «4-l 0 -M •H id n +j w Cn O •H XJ w "— * m •H fa -i 1 r °(i/\i) h i °d 3 a * o 20 According to Gunnerson and Ozturgut (1974) , the average salinity of Black Sea waters at a station 9 km from the Northern entrance of the Strait of Istanbul was approximate- ly 17.5 °/oo for an observation period July 1966 to December 1967. Lower values were observed (16—17 /oo) from July 17 to August 23, 1967. This lower surface salinity reflects maximum discharge of rivers into the Black Sea. Higher average surface salinities (17.5 — 19.0 /oo) were observed for the rest of the year and extreme values (as high as 25 /oo) during winter months. These were due to the favor- able conditions for Mediterranean water to flow northward. These conditions include small differences between the levels of the Black Sea and the Mediterranean and southerly or southwesterly winds over the Sea of Marmara and Strait of Istanbul. At the midpoint the average salinity in the lower layer is approximately 38.5 /oo and in the surface layer 17.5 /oo. A salinity cross section is given in Figure 4. Warm and fresh Black Sea water can be identified easily from the temperature cross— section which is given in Figure 5. Below this warm fresh water of the Black Sea cold inter- mediate water of Sea of Marmara can be observed. Mediter- ranean water occupies the deep bottom layer of the Strait of Istanbul. The dynamics of the processes occurring in these two straits is determined by the wind stress over the straits and the drop of the sea levels and densities at the end of the straits. The main driving forces are gradient of water 21 3 XI C (3 ■P en H M-l 0 -P •H td M ■M cn 4-1 0 c 0 • •H , — » -P CO O +J i c -P Eh *-* in CD u 2 en •H Pn (IAI) H 1 d 3 a 8 23 level, the rmo ha line forcing, wind force exerted on the sur- face and coriolis force. According to observations the boundary between the water masses generally does not coincide with the boundary between currents. The boundary between water masses has a greater slope than the boundary between currents (Defant, 1961) . This is probably due to diffusion processes occurring at the current boundary and to the nature of the current (Figure 2) . The average Black Sea inflow and outflow through the Straits of Istanbul according to the analysis of Moller (1928) 3 are 6,100 and 12,600 m /sec, respectively (Gunnerson and Ozturgut, 1974) . Velocity is greatest at the sea surface and decreases rapidly with depth and laterally; it increases from North to South. Under normal conditions it is 40—50 cm/sec at the entrance of the strait and 150 cm/sec at the South end. The lower current is strongest in the central part of the lower water which is at about 16 m from the bottom in the Strait of Istanbul and 45 m in the Strait of Canakkale. These velocities are 100—150 cm/sec and 25 to 10 cm/sec respectively (Defant, 1961) . In general in the Sea of Marmara there is Black Sea water at the surface and Mediterranean water at the bottom. This is the normal state. Except for times when very strong southerly winds bring bottom water to the surface and cause wind mixing these two main water types are both present with a very thin intermediate water layer. The Black Sea water extends to a depth of 15—20 m in the vicinity of the Strait 24 of Istanbul and becomes shallower farther south. There is intermediate water between these two water types which ex- tends at most to about 50—60 m. Below this the basin is filled with Mediterranean water as shown in Figures 6,7 and 8. This water has a temperature 14.9°C, salinity 38.55 /oo and sigma— t 28.75, approximately in June. Surface water has a large annual variability. On the other hand bottom water does not. Due to the described oceanographic conditions bottom water is almost isolated from surface processes and has a large time— scale of variability compared to surface water. Average inflow and outflow rates at the straits give approximately 30 years for "turnover" time or "flushing" time for bottom water and 200 days for surface water. Based on the same rates a typical value for the vertical mean current |u| is 10 cm sec . A characteristic magni- tude for horizontal velocity |u| using the thermal wind relation is 4 cm sec . Due to the large isolated reservoir of Mediterranean water of almost uniform characteristics in the lower part of the basin, the thermohaline circulation is confined to the upper 50—60 m. Z5 T3 u C Q) (d a 1 a -l •H ■P 0 fO co a a) CO C 0) u 1 M-l CD CQ 0 5 ■P >i c (D iH 0 XI a> •H -p -P C! (0 o 0) £ 0) -p 0 0 5-1 •H o -P fe H CO 26 u CO to CU w S CO rfl 0 CO 5-1 *— * O CU • 5-4 CO 3 J-l -P cu A3 4J 5-1 CU CD E & s o a) o ^ rH CU 5-1 3 C7> •H &4 (IAI) H d 3 a 27 p. \ / O O 3 s» in 00 CO f> ft I - 1- o o o GO O o u CD Qu a. 3 u 0 IW rd ,_^ M • rd in ft! S S-l a) n-i •H 0 fa fd en aj c CJ 0 CD •H CO +J o a CD 2 CO fd w w * — CO 0 M • u CO 5-1 >i > |u| and |v| > > |v| (4.13) equation (4.12) takes the form u' (x,y,z,t) = u(x,y,z,t) (4.14) v' (x,y,z,t) = v(x,y,z,t) Thus total u, v component of horizontal velocity can be represented by vertical shear currents u', v'. In the following part primes of u' and v' are dropped. 36 The equations are non— dimensionalized by making the fol- lowing substitutions for new non-dimensional variables (x,y) = (x,y)/L ^ (z) = (z)/h (u,v) = (u,v)/V, where (w) (vO/V^ ~L~ (P,P')= (P,PM PofLVl (t) (t) L Vl (T) = (T) TM (a) (a) (aN"aS) (4.15) V, scale velocity associated with density driven current h scale depth, taken equal to a characteristic thermocline depth L length scale xM wind stress [= MAX(x (x,y) , xy(x,y)] 3 a sigma— t [= (p— 1) x 10 ] a surface sigma— t at northern part of the basin aq surface sigma— t at southern part of the basin 37 A scale velocity associated with the density driven cur- rent is defined by using geostrophic and hydrostatic scaling which were introduced by Bryan and Cox (1968a) . When the geo- strophic relation is differentiated with respect to z and the hydrostatic equation is used, the result is o A scale velocity connected with the density distribution may be defined by V, =2_A£h (4.17, where Ap north— south surface density differences (= PN - PS) A scale velocity for wind driven current is defined by V0 = - " (4.18) 2 P0fhLr where x.. wind stress M L relative characteristic length scale for wind stress [= L/R] and L is a length scale for wind stress and R radius of the earth 38 Substitution of these non-dimensional variables into (4.10), (4.11), (4.4) and (4.5) with some rearrangement, gives these equations in terms of non-dimensional variables of the form 3u 1 3P* EH n2 EV 92u 1 ,, 1Q, 7t " ~ Ito ?x +Ro-Vu+Ro-7-2 + Ro-V (4'19) dZ Sz = - jl 3P' + fs y2v + fx g2v - ^ _i_ t y dz Ro 8y Ro Ro ~7~2 Ro L H s dz r r -Au (4.20) o o o P« = J (a-aQ) dn ~ k f { f (a-a0>dn)dz (4.21) z -Hz, w - J V- V dn (4.22) 8a 3a 9a 3a 1 „2 , K . EV 82a Tt + u^ + v^ + W^=p¥Va+ (!T) ro" ~ 2 V dZ + 6(a) (4.23) where Ro Rossby number which shows the relative importance of local time change term in the equation of motion with respect to that of the coriolis term V, I" IT1 E horizontal Ekman number which shows relative impor- tance of lateral diffusion and coriolis terms AM fir E-. vertical Ekman number, gives the ratio of vertical diffusion of momentum to the coriolis term fh 39 Pe Peclet number, shows importance of advection compared to diffusion of density V L V relative velocity, ratio of barotropic velocity to baroclinic velocity v2 H relative depth, ratio of basin depth to thermocline r depth L hJ The model is governed by these six parameters assuming a value for L is defined. All the parameters of the resultant run and constants of the model are given in table 2. Equations (4.19) — (4.23) are solved for variables u, v, P', w and a. The variables u, v, and a are predicted from (4.19), (4.20) and (4.23). P' and w are calculated diagnosti- cally from equations (4.21) and (4.22), respectively. 40 TABLE 2. PARAMETERS AND CONSTANTS —3 Ro 4.255 x 10 Rossby number —2 E„ 0.532 x 10 Horizontal Ekman number H —2 E 0.851 x 10 Vertical Ekman number Pe 10.0 Peclet number -?— H 50.0 Relative depth V 0.1 Relative velocity r 5 . Ax 10 x 10 cm Zonal grid spacing 5 .... Ay 5 x 10 cm Meridional grid spacing r 10.0 50.0 0.1 10 x 105 cm 5 x 105 cm 8 minute 28.9 day 40° 2 it day 6.37 x 10 cm At 8 minute Time step t 28.9 day Time scale The values Cl = 0.55 and C2 = 0.45 are commonly accepted for these constants which produce very slight damping of 43 inertial oscillations. Equation (5.1) and an analogous equa- tion for u are solved simultaneously for the two unknowns u , v . The time differencing form of the sigma— t equation has the form a (n+1) = a(n-l) + 2At[ADV(n)+ HD + VD(n-l)] + 8c(on+1) (5.3) where ADV advection term HD horizontal diffusion term VD vertical diffusion term The leapfrog scheme is conditionally stable and the time step must satisfy the relation (for two dimensional wave propagation) C — < — (5 4) CAx±^ (5-4) where maximum phase speed of the waves present in the system. External gravity waves are removed and inertial oscilla- tions are rendered neutral by (5.1) and (5.2). Therefore only internal gravity waves are present to determine C. The phase speed of the internal gravity waves is C = /cpH" (5.5) 44 where g' is modified acceleration of gravity and given by the relation g' = g A£ (5.6) yo where g acceleration of gravity Ap a typical value for the vertical density differences p reference density o —2 —3 —3 Assuming g = 1000 cm sec , Ap = 10 x 10 gm cm and —3 ~ —2 p = 1.012 gm cm gives a value for g' = 9.88 cm sec Integrations are carried out with an 8— minute time step for a grid spacing of 5 km, without any stability problem. B. THE FINITE DIFFERENCE EQUATIONS The numerical method is set down in terms of the non- dimensional variables. The finite difference scheme is based on a three— dimensional array of points with indices i,j,k. Time step is indicated by (n) as a superscript. In the nota- tion the letters i,j,k are always integers. Horizontal spacing is such that Ax = 10 km and Ay = 5 km, and uniform in the x and y directions. The horizontal grid pattern is shown in figure 9. Where horizontal velocity com- ponents (u,v) are defined at circled points, vertical veloc- ity (w) and sigma— t (a) are defined at dot points. 45 J:JM H A. fr-j, J:1 1:1 T o «.» Point Point l:IM Figure 9. Placement of variables on horizontal grid plane. The open cir- cles denote the definition points of the horizontal velocity components (u,v) and dots denote definition points of (a,w) . The distance between adja- cent grid point is Ax = 10 km and Ay = 5 km in x and y direction, res- pectively. IM and JM have values 21 and 11 respectively. 46 For an arbitrary interior point, where (u,v) are stored longitude (x) and latitude (y) are defined by x± + \ = ~ [1 + (i-1)] i = 1,2,..., IM-1 yj + \ = ^ [1 + (j-1)] j = 1,2,..., JM-1 (5.7) The vertical pattern of variables is shown in figure 10. The vertical index indicated with k and the vertical velocity w are given along the surface and bottom and along the layer boundaries. The variables (u,v,a) are located at the kth level. Depth of each level is defined as Az, zi - " TT (5.8) m=k Z, = j. m=2 k = Z1 - £ i,j,35 A\ Pijk = Pij,k-1 + f^o'ijk-Js -L\-k k=2'3'KM <5-9» and the vertical average of the departure pressure- is calculated with the relation k=KM 3 .jk ""k P. . = T, P. ., AZ, (5.10) k=l The departure P ■ is calculated from P! .. = P. .. - P. . (5.11) 13k 13k 13 which is the finite difference analog of (4.23). The finite, difference approximation of the continuity equation, in which the vertical velocity between the levels is calculated, has the form Wijk+^ - "ijk-!, + [UX + Vijk AZk (5-12) where U . ., = ^(u . , + u ._,).■, x 13k x j+% x 3— *g lk V . .. = 3g(v i+h + vy i-Vik v ilk v * J (5.13) y ID and ux ij+3gk =: Ax{ui+h u±-h)j+hk vy i+?sjk hy{vj+k Vj-Vi+Jsk 49 (5.14) which are analogs of (-^-) and (•?— ) respectively. Gradients dx dy of horizontal velocity components u and v are defined at half- integer grid points west and south of the u and v storage points shown in figure 11a. Integration from the surface downward, using surface boundary condition w=0, is done by r. .. _,, = y (U + V ) . . i jk+*s La x y lj (5.15) The formula for the finite difference approximation of the equation (4.19) by which u is predicted is n+1 n-1 n P' .._ - P! ... ru — u i 1 r 1+1 j+l 1 j+1 1 2At Ji+3gj+%k ' ' 2Ro L Ax + Pi+1 J "" PiJin . ^H rux i+1 j+^ "" Ux i j+^ Ax Jk Ro L Ay Uy i+^s j + 1 ~ Uy i+^j.n-1 ^v ruz k-1^ "" uz k+^.n-1 Ay Jk Ro L AZ. i+^ j + "2 + 55 [cl v"+1 + C2 v" ^i+a, j+%k (5-16) whe" "yj+l= J 'Ay **>!+* <5'17' Equations (5.16) and (5.17) represent (-A and (yi) and are located half grid distance south and above the velocity (u) storage points, respectively, as shown in figure 11 a— b. 50 > > 3 3 3 b + I CO C 4h o 43 a 0 c •P 0 (0 CD X 4J 43 cd 4J CD -P C P CO G (0 CD CO -H •H -H CD P r 4H T3 T3 C c cd cd rd •H ■H -0 TJ P SH 4-4 0 -H t« tP CD 0, M 43 O 13 rj> o o co T3 •rlH-P 43 •H 4-1 ^ a O U >-i £ CD 43 (d — P rH t3 •H < 43 P CD H - (d rd c rd -* -H •rH > 3 • M CD 4-1 w G rd M CD 4-1 > rd T3 0 4H • 0 4H N to © 0 > c co o-d c CO T3 ■H C 0 £3 C CO P (d -H 0 rd -H rd p •H o * O T3 cd 3 b • iH •• 0 CD u CO P rH C 0 p iH •H <-i • • $3 (flW +JIW D P -H p o c CD r-H C 0 C CD T3 rd T3 -H Q4 0 CO CO u C 0 N -|J CD CD rd 0 "H * rd 0 td CD 3 M P M 0} rH 0 CO • tnp 43 • 4H p rd 0 rd 43 0 CO O rH TS ■H r-\ H CO ^-» CD rH P * rd P > 43 C M > J3 * P CD 0 CD CD 3 M N iH CD •H *-' CD 3 -H O CO *G > tn M M CD td CD 0 •H 0 -H 43 Sh 43 43 h 43 0 P CnP rd 51 The formula for the finite difference approximation of the equation (4.20) in which v is predicted has the form rVn+l _ vp-li j_ .p'i+l 1+1 - P'i+lj 1 2At * ±+hj+kk 2Ro L Ay v ij+1 i^n ~H , x l+l j+k x 13+^ Ay Jk Ro l Ax Vy i+^j+1 Vy i+^jn (n-1) + Ay Jk fv rVz k-% " V: Ro l AZ, Ji+^j+^ JV rvz k-jg z k*1^ Vr Lr y 1 r^,i n+1 , „^ n— 1, + 55 h- t* - *5 [c1u + c2u ]i+%j+^k (5.19) r i+-$ where v . . , = Ii±i^ — litk) J x i+1 Ax vk-i - vk. 'j+k VZ kH* " AZk_^ )i+3gj+3g (5'21) 9v Equations (5.20) and (5.21) are analogues of (-r— ) and (^__) and are stored south and above the velocity storage dZ points, as shown in figure 11a— b. 52 The finite difference approximation of the sigma— t equa- tion has the following form, n+ 1 n— 1 , . -. o ■ . i ■ . i ™" o 1,1 r£_-Z_JL_i = -a (n)+ 1 / x x+ki+k x i-kj+k 1 JEt Jijk ' ljk Pe^ Ax °y i+ki+k °Y i+^j-^n-1 Ay 'k JL _L f ,qz k+k °z k-k. n-1 Kv Ro v( AZk ; i+kj+k + 6c(an+1)ijk (5-22> where Aijk ■ 4S[(ui+%j+% + ui+M-«i) • (aij + "i+lj'k - iu±-hi+h + ui-^j-^' • <°i-lj + ^j'k1 + 4Syt(vi+y+% + vi-%j-%> • (aij-l + aij'k - • <°ij-l + "ij'k1 + 3SzkI(wijW(crijk + 0ijk-l> - (wijk+^» • <°ijk + aijkfl>1 <5-23> 53 and ax i+^jk " Ax (ai+l ai)jk ay ij+^k " Ay" (aj+l aj}ik (5.24) ( 0 In case of unstable stratification, vertical mixing be- comes effectively infinite. At each time step this infinite mixing is included in numerical solution by testing sigma— t profile for unstable lapse rate ( cr, ) . If this condi- tion exists new values of sigma— t for those two layers are set equal to the vertical average sigma— t of the two layers instantaneously. This process is repeated until complete sta- bility is reached for the entire layer. 54 C. SPECIFICATION OF BOUNDATION CONDITIONS As shown in Figure 3—1 cr,w are variables which are de- fined at lateral boundaries; also horizontal gradients of (u,v) are defined at these lateral boundaries. The boundary conditions are satisfied by the following relations which are given for eastern and western boundaries. Similar con- ditions exist for north and south boundaries with the excep- tion of open boundaries. Normal components are set equal to zero by _ 2u. Ux i=1 Ax'i=% 2u> Ux i=IM Ax'i=IM-% (5.26) In the momentum equation zero— slip condition is applied to the valocity tangent to the boundary. On the other hand, when computing the advection term in the density equation free slip condition exists for the velocity tangent to the boundary. This implies the following conditions u. , = u. nl i=l i=l% i=IM IM— \ (5.27) 55 Definition of vertical gradients of (u,v) allows the writing of the vertical stress term in flux form easily. Boundary conditions at the surface and bottom are defined in the following manner: UZ k = 1 = ° (5.28) Zonal wind stress is assumed zero and only a constant meridional wind stress is defined. , V L VZ k - 2 " -if TsY (5-29) On the other hand it is assumed that horizontal velocities vanish at the bottom. The following conditions force veloc- ities (u,v) to reach zero at the bottom 2ui UZ k=KM+?2 ' Az'k=KM (5.30) v = 2V| Z k=KM+% Az'k=KM The boundary conditions of zero mass fluxes across the side walls and bottom are satisfied by imposing the following conditions: e™>i=% = "(<"») 1=3/2] jk (5.31) k=6 <^4> and o has a value 28.5 in sigma— t units. 57 VI. RESULTS Computations are carried out for different values of the parameters with an integration period of 24 hours. Parameters of these runs are given in Table 2. Results of the model are given for the fields of horizontal velocity components (u,v) , vertical velocity (w) and density (sigma— t) in dimensional form. The horizontal velocity field at z = — 2.5 m is given in Figure 12. The horizontal stress exerted on the sea surface by southerly wind causes the surface water to move in a generally North— east direction. This is a consequence of movement of surface water as an Ekman layer, drifting to the right of the wind stress in the Northern Hemisphere. Since the curl of the wind is zero divergence of the Ekman drift is also zero. But horizontal velocity is strongly convergent and divergent near the boundaries. Northeastward drift takes place over the entire basin at this depth. The strongest flow has a magnitude 20 cm/sec in the central part of the basin and diminishes toward the boundaries. Near open boundaries velocities are smaller compared to the velocities at neighbor- ing grid points. The southward flow at these points repre- sents the exchange with the Black Sea water at the north and Mediterranean water at the south. The magnitudes of these currents are 5 cm/sec and 2 cm/sec at grid points just near the entrances of the Straits of Canakkale and the Strait of Istanbul. The applied meridional wind stress has a value of 58 CD > >i +J •H U O H 0) > u CD +J CD e m +J C O N •H M • O CN a ~ CN u •H fa 59 —2 0.3 dyn cm which is not strong enough to move surface water to the north as it does at neighboring grid points. In the second layer z = —7.5 m the horizontal velocity changes direction to the right and becomes weaker. Horizontal velocities at that level are shown in figure 13w; they have a magnitude 5 cm sec . Flow adjacent to the straits is still southerly and has a magnitude 5 cm sec and 3 cm sec at the vicinities of the Straits of Istanbul and Canakkale respectively. The circulation pattern at lower levels differs from the upper level flows both in magnitude and direction. These differences can be observed in the circulation pattern at the z = —40.0 m level (Figure 14). The pattern is not irregu- lar compared to the upper layers. The northeastward surface drift in the first layers is accompanied by a westward and partly southwestward flow in the bottom layers. A northward flow adjacent to the Strait of Canakkale brings Mediterranean water into the basin. A similar northward flow of Mediterranean water does not appear at the northern strait due to the shallow sill depth at that point. Flow in the layers close to the bottom becomes weaker. The influence of the circulation at the straits on the general pattern is small. The circulation pattern at a depth 700.0 m is shown in figure 15. The maximum magnitude is 0.033 cm sec One advantage of the numerical modeling study is that it gives estimates of important oceanographic variables that 60 \ -t L / cm 1 / / CM CN / c-* \ — t \ -1 I 1 I 1 PI i / •V *j tO CM ^M O -t \ / / / ! / / L L \ I / / / / / / I \ •^ ■* •» Ui -«»• •<»■ en << en — i V —1 / / / Ui / Ui / Ui / Ui / I \ -^ / *1 / / US / / / / / \ \ -1 / / / 1 Ui t t 1 i I \ — i L L i / / 1 I V; L \ \ / i / ! / \ I \ ^, — i — ( cm *i pi «0 &i CM CM /• \ — i / i / CM i L to / —I \ \ cm / / CM s rt / CM I / / *s > C O O >i +J •H O 0 <-i CD > n3 U +J <1) C +J 0 CD N 6 •H U ^ 0 • 5B r- cn u •H fa 61 V \ \ X \ X X. T X *V °v \ \ \ X X X ^ \ *»> c^- \ \ X T T X T X *v «>s» \ X r r r r r X \ ■v \ r r r r r r T \ r r r r r r r r T \ — * r r r r r r r r \ o r r r r r r r T \ «*v ^ r r r r r r T \ \. x r r r r r r T \ °v \ r r r r r r t \ ■\ X r r r r r r X \ "\ \ r r r r r r X \ \ X r r r r r r X \ \ X r r r r r T X X \ X r r r r r T X 0) Xi +J •H »W O 4-1 CO U O -P o > >1 •p •H O O rH > — M iH \, X "V, X X "V ->• -* -— ~^ "No \. \> X X \, ~Ss — — X, V \ \ X X \ \, --» -^ \» \ \ \ \ \ \ \ ^o ~^ \, \ \ 1 \ \ X \ ^o "">o \ \ \ i \ \ V \ "So ~^> \ \ 1 1 \ \ X \ Nj N, \ \ X \ \ \ V. \ N. N> \ V. \ \ \ \ \ \ "--« "^ \ V \ \ \ \ \ V N. "No \ \ V V \ X N. --«> -« ~^ \ V V. V. \ \ N. -° y ^» \ \ X \ i 1 J <^- o "V \ \ \ X A 1 4 J I ^> \ \ \ \ \ I \ \ ~Sq ■— \> \ \ \ \ \ \, N» -* --, N» V X \ X V s, --X3 — » -O N> N. X X X \> \> "N> ■^ V 1 i X X \ \ \ \ N> rH 0) > +J c •H u o w u o -p o > >1 +J •H O 0 rH • Q) ^ > u rH -P t* 0 O N • •H O U O 0 I"- S — in U •H P4 63 cannot be obtained by measurements. One such variable is the vertical velocity. The vertical velocity at the base of the first layer is given in Figure 16. The magnitude of the vertical velocity differs markedly between the boundaries and the interior. Rising motion (upwelling) is taking place in the south and southwest and sinking motion (downwelling) is taking place in the north— northeast. Vertical velocity is mainly determined by the Ekman drift current intersecting the boundaries. Southerly wind stress produces upwelling and downwelling at the south and north, respectively. Up- welling extends to the north on the western side and down- welling to the south on the eastern side. The maximum magni- tude of the vertical motion is 358 cm day . Strong upwell- ing and downwelling take place adjacent to the straits. This is consistent with the nature of the horizontal current system in the vicinity of the straits. Vertical velocity at the bottom of the second layer, z = — 7.5 m, is shown in Figure 17. The general pattern is similar to the vertical velocity at the base of the first layer. Strong vertical velocity is present along the lateral walls to the southwest and northeast where it has a magnitude + 300 cm day . With depth the effect of wind on vertical velocity decreases due to the vertical stability. The verti- cal velocity pattern at the base of the eighth level is shown in Figure 18. The sigma— t pattern at the first level, 2.5 m is shown in Figure 19. This pattern at z = — 2.5 m is specified by 64 +J CO u •H «+-) CD x: +j m 0 0 CO rt ja i ^ ■U M -H d) O 4-> 0 0) rH e c • iH in (Q — O -H M -P QJ H >i o rtf > rH VD CD en •H fa 65 T3 C 0 O d) (0 ~ N >i ^ -P d) •H -*-> O CD 0 e rH • o rH rH ra O •H u -P 0) M >1 CD td > H cu Cn -H 66 J2 -P XI Cn •H CD d) £ ■p m 0 Q) 03 fd .Q fd • >i .-^ -P U •H CD O +J 0 c o iH LD ! O rd > ^ 00 0) U •H fa 67 > ■p W u •H «W +J C Q) -P +J (1) u •H fa 68 the surface boundary condition. At this level the surface sigma— t is a function of latitude only and held constant during the integration period. Deviations from the surface pattern begin to appear just below the first level at 7.5m. The sigma— t pattern for the second level is shown in Figure 20. A close relationship is present between this sigma— t pattern and the vertical velocity field shown in Figure 17. At the north and northeast sinking motion brings in low density and water and at the south rising motion brings in high density water to this depth. This pattern is more strongly tied to the surface boundary conditions on sigma— t than at other levels. There is a low density water pool at the north. As a consequence of upwelling a high dense water pool is present at the south extending in an east— west direction. Magnitudes of sigma— t are 16.5 and 18.0 at north and south respectively. The sigma— t pattern at the fourth level z = — 20 m is shown in Figure 21. The effect of vertical motion appears as a low density water pool at the northeast. Below this level z = — 40 (not shown) , sigma— t is uniform and has a value 28.5. This is consistent with the sigma— t pattern given in Figure 6. Further insight into these results may be gained by examin- ing the meridional and east— west cross sections of sigma— t and the (u,v) components of the horizontal velocity. There is also the possibility of comparing the predicted sigma— t pattern to the observations. 69 CD > CD C O O si -p u O 4-1 -P c M 0) +J ■P ■P I e &> •H W U U C 1 id s 0) • CN CO C C >i 0 -P •H -H • TJ U C •H 0 H U r-i to CD CD id S > -Q • in CN CD u 3 Cn •H fe 76 rH U-t i-H 0 +J CQ •H •h fd 'O O Xi •H 0 MHO) -P • CN d) H 3 cr> •H PM 77 t-\ «4-l OJ 0 C O -P M id O a JZ -p c M-4 CD 0 P W c i-H 0 -P M •H •H id T) U J3 •H 0 M H (D Q) a) s: S > -p • CN CO c CD 5-1 r -M -U CO 3 fd 0 H CO • 00 (N CD 5-1 3 & •H fe 79 o c* ■* uj CD x: +j M 0 m c o •H +J O — i a 'n."^,"-j J 111 0) x: ■p M 0 m c 0 •H -M O • (0 c co •H CO w id co .Q 0 U cu O X 4-> -P 1 IW U — ' p * to >i 0) -P f3 • c P 0- H tO r-l CO £} n cu M tr» •H fa 83 <-4 H +J tf M o — ■p * CO >i CD 4-> 5-H • o c -P o- H CO rH CO ro a; rd w > .Q • CM m CD U 3 tn •H fa 84 rt P C O 4-1 •H O •H P e a P Q) 4-1 P O H o c c o •H 0) P £ O 4-> 0) W P 0} OT "» o — n > 0 — * p * W >i X} d) •H • « 85 VII. CONCLUSIONS In order to investigate the dynamics of the Sea of Mar- mara a .circulation model is formulated based on the hydro- static and Boussinesq approximations. Since very little is known about the turbulence characteristics of the ocean or adjacent seas, viscosity and conductivity are replaced by new terms representing the contributions of the smaller scale motions to the exchange of momentum and density. Horizontal eddy transfer of momentum is accomplished using a constant eddy viscosity and diffusivity coefficient. For the exchange of momentum and density in the vertical a constant vertical diffusion coefficient also is used. The vertical mixing of density is enhanced by the use of an instantaneous adjustment mechanism to eliminate unstable lapse rates. A lateral mix- ing coefficient for density is less than that used for momentum. These coefficients should be sufficiently small so as not to obscure lateral transfer of the quantities, and a non zero momentum coefficient is needed to satisfy the zero slip boundary condition. In this study constant values of the coefficients are 7 2—1 iL = .4 x 10 cm sec (density) and A 7 2—1 M = 5 x 10 cm sec (momentum] 86 The vertical diffusion coefficient which is difficult to evaluate either by measurement or in principle, is assigned 2 —1 1.6 and 3.2 cm sec for density and momentum respectively. Other parameters and constants are given in Table II. Further study of thermocline penetration, water exchange, vertical mixing processes and current gradients might provide a better possibility for matching these coefficients. Although many important factors, such as the irregular bottom topography, bottom friction and non— linear terms in the equation of motion have been omitted, the results from the model are generally encouraging. More realistic results may be achieved by incorporating these effects systematically in further studies. _— p. Integration in the model is carried out with a constant southerly wind stress which was chosen arbitrarily. Although the scale of the motion is small compared to a characteristic length scale for meteorological features, wind stress may be significantly variable over the sea due to the geographic location of the sea. Also seasonal variations of the wind may have a considerable effect on the vertical mean part of the current which is neglected during this study. The model does not separately predict temperature and salinity which are sometimes important quantities. Diagnostic calculation of the density with the aid of predicted tempera- ture and salinity fields can be done with a very simple modification of the model. This would require a calculation of the heat and salt fluxes at the surface and meridional 87 salt and heat separately in the model. It would be more difficult, however, to define boundary conditions for these quantities at the open boundaries. It is also difficult to simulate seasonal variations of these quantities at the open boundaries. Once these are defined properly, based on accurate observations, there are no inherent difficulties in simulating the dynamical processes in the sea using the general principles involved in this model. Even though this model depends on the particular hypothesis used for hori— zontal and vertical eddy transport of heat, salt and momentum, it is clear that valuable studies on small scale water bodies can be made. 88 APPENDIX A COMPUTER PROGRAM DESCRIPTION The computer program is written in FORTRAN IV and was used with the IBM 360/67 computer system at the W. R. Church Computer Center, Naval Postgraduate School. The overall program is divided into two basic subpro- grams (1) the main program and associated subroutines (2) an access program which draws and writes the results of the first program. The main program consists of nine subroutines which calculate different terms of the equation of motion and the . density equation, pressure, vertical velocity and changes variables for next time step. FORTRAN IV symbols for the primary program and a brief description of the subroutines are given below: UMI Zonal velocity component at n— 1 time step U Zonal velocity component at n time step UA1 Zonal velocity component at n+1 time step VM1 Meridional velocity component at n— 1 time step U Meridional velocity component at n time step UA1 Meridional velocity component at n+1 time step SGMl Sigma— t at n— 1 time step SGMT Sigma— t at n time step SGMT Sigma— t at n+1 time step 89 ADV Total advection term in density equation ADV Local rate of change of sigma— t ADV Pressure A Surface pressure PBAR Vertical average pressure W Vertical velocity UXG Gradient of zonal velocity in x direction UYG Gradient of zonal velocity in y direction UZG Gradient of zonal velocity in vertical VXG Gradient of meridional velocity in x direction VYG Gradient of meridional velocity in y direction VZG Gradient of meridional velocity in vertical SGX Gradient of sigma— t in x direction SGY Gradient of sigma— t in y direction SGZ Gradient of sigma— t in vertical TX Meridional wind stress TY Zonal wind stress DZ Layer depth DZ Layer BB Depth dependent velocity at the Northern fictitious boundary CA Depth dependent velocity at the Southern fictitious boundary fictitious boundary BO Sigma— t defined outside the domain at North CD Sigma— t defined outside the domain at South DX Horizontal grid spacing in x direction DY Horizontal grid spacing in y direction RO Rossby number 90 EH EV PE1 Wl AK AKV RL HH DT SUBROUTINE SUBROUTINE SUBROUTINE SUBROUTINE SUBROUTINE Horizontal Ekman number Vertical Ekman number P^clet number Relative velocity Vertical eddy diffusivity Vertical eddy viscosity Curl factor Depth of the basin Time step PRES Calculates pressure Calculates vertical velocity VERW ADVEC HDGRD DIFFO SUBROUTINE SIGEQ SUBROUTINE HVGRD SUBROUTINE HORV Calculates advection term in the density equation Calculates gradients of the sigma— t Calculates local time rate of the change sigma— t by subtracting advec- tion term from the diffusion term Calculates new sigma— t and makes convective adjustment for unstable lapse rates. Calculates gradients of the hori- zontal velocities Calculates horizontal velocity components, u,v. A descriptive flow diagram of the program is shown in Figure 34 . 91 ( start ) Layers Depth Calculate Layer Thickness Meridional Wind Stress (TY) Define Zonal Wind Stress (TY) Define Surface Pressure Define Parameters of the Model Figure 34. Descriptive flow diagram of the program, 92 Set Initial Velocities (U,V,W) Define Initial Sigma-T Distribution Define Sigma-T at the Fictitious Grid Points N «-l NMAX -0 U,V,W,SGMT ( STOP Figure 34. Continued Yes SAGLA HDGRD DZ,BO,CD,BB,CA,SGMT,SGX,SGY,SGZ Yes HDGRD ,CA,SGM1,SGX 93 Figure 34. Continued 94 Yes No ADV CALCULATE VERTICAL AVERAGE PRESSURE (PBAR) ADV(I,J,K) = ADV(I,J,K) - PBAR Figure 34. Continued 95 \y Yes HVGRD BB, CA, DZ, TZ, TY, U, V, UXG, UYG, VXG, VYG, UZG, VZG I NO | I sL HVGRD . ..,TY, UM1, VMl, UXG,.. HORV D4T,DZ,ADV, UXG , VXG , UYG , VYG , UZG,VZG,TX, TY, U,V,UA1,VA1 I No X Yes HORV D2T, . . . ,UM1,VM1,UA1,VA1 Figure 34. Continued 96 0 HORV DT, . . . , UM1,VM1,UA1, VA1 CVAR UA1, U, VA1, V, SGP1, SGMT, SGPl, UM1, U, VMl, V, SGM1, SGMT I w CVAR U, UA1, V, VA1, SGMT, SGP1, UM1, U, VMl, V, SGMl, SGMT Figure 34. Continued 97 © Figure 34. 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UJ OOO «» •-• Z-Z.-Z. z o zzzz UJ ooo * i-h •■ >— i 1 II II 1 o OOO •— • -m» 1 II II 1 cc a£ LUUJUUUJ o ^(NJCO >-« w> *■+ M l-KI- z >tino »-» Z l-l-l- 3 CO IZZZ z ■w Z *~ z zzz <. X Z Z.Z.Z H-O -> 1 II II II 1 ooo aik£ CO oaoo X o QQQ 3 3 > > 300 C 200 C 100 C o ooo to 00 600 C 500 C 400 C (XLU 120 LIST OF REFERENCES 1. Bogdanova, A. K. , 1961: The distribution of Mediter- ranean waters in the Black Sea, Okeanologiya , v. 1, no. 6, p. 983—991; English trans., 1963, Deep Sea Research, v. 10, p. 665—672. 2. Bogdanova, A. K., 1965: Seasonal fluctuations in the inflow and distribution of the Mediterranean waters of the Black Sea, Basic features of the geological struc- ture of the hydrologic regime and biology of the Mediterranean Sea, Edited by L. M. Fomin, Academy of Sciences, USSR, Moscow 1965. English trans., 1969 Institute of Modern Languages, Washington, D.C. (AD 686001) . 3. Bryan, K. and M. D. Cox, 1967: A numerical investiga- tion of the oceanic general circulation. Tellus, 19, 1, 54-80. 4. Bryan, K. and M. D. Cox, 1968a: A non— linear model of an ocean driven by wind and differential heating: Part I. Journal of the Atmospheric Sciences, 25, 945— 967.. 5. Bryan, K. and M* D. Cox,, 1968b: A non— linear model of an ocean driven by wind and differential heating: Part II. Journal of the Atmospheric Sciences, 25, 968— 978. 6. Defant, A., 1961: Physical Oceanography, Vol. 1, Pergamon Press, 72 9 pp. 7. Gunnerson, C. G. and E. Ozturgut, 1974: The Bosporus, In E. T. Degens and D. A. Ross, eds., The Black Sea — Geology, Chemistry and Biology. American Association of Petroleum Geologists, (Memoir 20), 1974. 8. Haney, R. L. , 1974: A numerical study of the response of an idealized ocean to large scale surface heat and momentum flux. Journal of Physical Oceanography, 4, 145-167. " 9. Levastu, T. , S. Larson and K. Rabe, 1976: Numerical Models for Synoptic Analysis/Prediction of Currents and Temperature/Salinity Structure in the Oceans. Naval Research Reviews, June 1976. 121 10. Moller, L., 1928: Alfred Merz' Hydrographische Unter— suchungen im Bosporus und Dardanelenn. Verof fentlich— ungen Inst. Meereskunde an der Universitat Berlin/ Nue Folge A, Heft 18 Berlin 1928. 11. Pektas, H. , 1956: The influence of the Mediterranean Water on the Hydrography of the Black Sea: Technical Papers, 4th Meeting Fisheries Center, Istanbul, 1956. 12. Ullyott, P., and O. Ilgaz, 1946: The hydrography of the Bosporus an Introduction: Geo. Rev. , v. 36, no. 1, p. 44-60. 122 INITIAL DISTRIBUTION LIST No. Copies 1. Department of Oceanography, Code 68 3 Naval Postgraduate School Monterey, California 93940 2. Prof. J. B. Wickham 3 Department of Oceanography, Code 68 Naval Postgraduate School Monterey, California 93940 3. Prof. R. L. Haney 1 Department of Meteorology, Code 63Hy Naval Postgraduate School Monterey, California 93940 4. Deniz Kuwetleri Komutanligi 1 Personel Egitim SB. Mudurlugu Ankara, Turkey 5. Dz. Kuwetleri Seyir ve Hidrografi Dairesi Bsk.3 Cubuklu, Istanbul Turkey 6. Istanbul Teknik Universitesi 1 Taskisla, Istanbul Turkey 7. Orta— Dogu Teknik Universitesi 1 Ankara, Turkey 8. Oceanographer of the Navy 1 Hoffman Building No. 2 200 Stovall Street Alexandria, Virginia 22332 9. Office of Naval Research 1 Code 480 Arlington, Virginia 22217 10. Dr. Robert E. Stevenson 1 Scientific Liaison Office, ONR Scripps Institution of Oceanography La Jolla, California 92037 11. Library, Code 3330 1 Naval Oceanographic Office Washington, D. C. 20373 123 12. SIO Library University of California, San Diego P. 0. Box 2367 La Jolla, California 92037 13. Department of Oceanography Library University of Washington Seattle, Washington 93105 14. Department of Oceanography Library Oregon State University Corvallis, Oregon 97331 15. Commanding Officer Fleet Numerical Weather Central Monterey, California 93940 16. Commanding Officer Navy Environmental Prediction Research Facility Monterey, California 93940 17. Department of the Navy Commander Oceanographic System Pacific Box 1390 FPO San Francisco 96610 18. Defense Documentation Center Cameron Station Alexandria, Virginia 22314 19. Library (Code 0142) Naval Postgraduate School Monterey, California 93940 20. Huseyin Yuce Dz. YZB Deniz Harbokulu Egitim Uyesi Heybeliada, Istanbul Turkey 21. Lcdr. J. F. Pfeiffer 77 Adams Place Apt 407 Quincy, Massachusetts 02169 22. Lt. D. Henrickson Marine Sciences Branch Commandant (G— 000) U. S. Coast Guard Washington, D. C. 20591 124 Thesis Y825 Y c.l 2 1 2 uce Numer of the Ma rma ra MN 7T it 166701 ical investigation dynamics of Sea of • ?U838 ; 0 0 2 Thesis 166701 Y825 Yuce c.l Numerical investigation of the dynamics of Sea of Ma rma ra . thesY825 Numerical investigation of the dynamics 3 2768 001 91564 8 DUDLEY KNOX LIBRARY