LIS*. NAVAL POSTGRADUATE SCHOOL PEREY, CALIF. 93940 THE UNIVERSITY OF MIAMI ON THE VALIDITY OF THE GEOSTROPHIC APPROXIMATION FOR THE FLORIDA CURRENT BY Edward J. O'Brien, III A THESIS Submitted to the Faculty of the University of Miami in partial fulfillment of the requirements for the degree of Master of Science Coral Gables, Florida June 1967 LIBRARY NAVAL POSTGRADUATE SCHOOL MONTEREY, CALIF. 93940 THE UNIVERSITY OF MIAMI A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science Subject On the Validity of the Geostrophic Approximation for the Florida Current Edward J. O'Brien, III ABSTRACT The object of this thesis is an examination of the validity of geostrophic calculations of the downstream component of the time- averaged Florida Current in the Straits of Florida by comparison of calculated and directly measured current fields. The study is motivated by the assumption made in modern inertial current theory that downstream current speed is in geostrophic balance, plus evidence of recent studies indicating that this region of the Florida Current is primarily inertial in nature. The principal conclusion reached is that geostrophic calculations yield a valid first order approximation to the observed velocity fields, indicating that the assumption made in inertial current theory is valid. In addition, it has been shown that in geostrophic calculations in this region, the density field may be approximated as a parabolic function of temperature only. A general discussion of mass field adjustment to downstream speed changes is offered. in ACKNOWLEDGEMENTS The author is indebted to a multitude of persons for their assistance and patience during the preparation of this thesis. Sandra S. O'Brien alone knows what this enterprise has meant to me. I am deeply grateful to the members of my thesis committee, Dr. E.F. Corcoran, Dr. R.L. Snyder, Dr. L,J. Greenfield, Dr. Saul Broida, and Dr. E.S. Iversen; in particular I wish to express my appreciation to Dr, William S. Richardson, my thesis committee chairman, and all of his associates in the Southeastern Massachusetts Technological Society; to my respected senior, friend and classmate, Cdr. Edward Clausner, USN , MS, and,- to Dr. William J. Schmitz, Jr., without whose patient guidance I could not have completed this work. The financial support for this study was provided by the Office of Naval Research. Edward J. O'Brien III Coral Gables, Florida June, 1967 IV TABLE OF CONTENTS Page LIST OF TABLES vi LIST OF FIGURES .«••*..••••• viii SECTION I. INTRODUCTION ........ 1 II. DATA 5 A. General ........ 5 B . Analysis , 6 C. Errors. 9 III. RESULTS 19 A. Geostrophic Calculations 19 B. Mass Field Adjustment , 37 C. Earlier Work 40 IV. SUMMARY AND CONCLUSIONS 41 LITERATURE CITED 43 APPENDIX 45 v LIST OF TABLES TABLE Page I. SECTION I. 11 A. Cross-stream distance B. Isotherm depth C. Observed velocity along isotherms D. Coriolis parameter II. SECTION II 13 A. Cross-stream distance B. Isotherm depth C. Observed velocity along isotherms D. Coriolis parameter III . SECTION III ... . 15 A, Cross-stream distance B, Isotherm depth C. Observed velocity along isotherms D. Coriolis parameter IV. SECTION IV 17 A. Cross-stream distance B. Isotherm depth C. Observed velocity along isotherms D. Coriolis parameter VI LIST OF TABLES — Continued TABLE Page V. Percent deviation between observed and calculated downstream speeds - Sections I and II 21 VI. Percent deviation between observed and calculated downstream speeds - Sections III and IV......... 23 VI 1 LIST OF FIGURES FIGURE Page 1 . CHART OF SECTION LOCATIONS 3 2. a. GEOSTROPHIC ISOTACHS - Section II 8°C reference isotherm Linear equation of s tate ..,.,. 25 b. GEOSTROPHIC ISOTACHS - Section II 8°C reference isotherm Parabolic equation of state 25 3. a, GEOSTROPHIC ISOTACHS - Section II 8°C reference isotherm VCP equation of state 27 b. GEOSTROPHIC ISOTACHS - Section II 18°C reference isotherm VCP equation of state 27 4. GEOSTROPHIC ISOTACHS - Section II 26°C reference isotherm VCP equation of state 29 5. a. GEOSTROPHIC ISOTACHS - Section I 8°C reference isotherm Parabolic equation of state 31 b. GEOSTROPHIC ISOTACHS - Section I 8°C reference isotherm VCP equation of state 31 viii LIST OF FIGURES— Continued FIGURE Page 6. a. GEOSTROPHIC ISOTACHS - Section III 8°C reference isotherm Parabolic equation of state 33 b. GEOSTROPHIC ISOTACHS - Section III 8°C reference isotherm VCP equation of state 33 7. a. GEOSTROPHIC ISOTACHS - Section IV 8°C reference isotherm Parabolic equation of state 35 b. GEOSTROPHIC ISOTACHS - Section IV 8°C reference isotherm VCP equation of state ,...,.,.,.,., 35 8. Mass field - velocity adjustment Sections I-IV 38 IX I . INTRODUCTION The purpose of this thesis is the examination of the validity of the geostrophic approximation for the Florida Current (within the Straits of Florida) . An experiment was conducted during 1965-1966 having as one objective the critical investigation of such an approxi- mation. Earlier studies of the application of geostrophy in this area have been made in order to calculate the velocity field for the pur- pose of examining various features of the current. Since the development of the free instrument technique (Richardson and Schmitz, 1965), rapid, direct measurement of the current field is possible, and geostrophic calculations are no longer necessary for current field determination. In this thesis, interest is focused on the extent to which the geostrophic approximation describes the downstream component of the flow as postulated in contemporary inertial current theory (Robinson, 1965). Robinson (ibid.) further points out that use of space-temperature (x,y,T) coordinates simplifies the mathematics involved in developing models. The geostrophic equation as examined is unusual in its use of an equation of state with density expressed as a function of temperature only. This thesis constitutes the first quantitative validity deter- mination of geostrophic calculations based on simultaneous hydrographic data for a steady (time-averaged) current. The validity of the geostrophic equation will be determined by comparing calculated downstream velocity fields with observed velocity fields obtained using the free instrument technique. These observed fields will represent a time-averaged Florida Current at four different sections across the Straits of Florida (Figure 1). If the geostrophic approximation is valid for the Straits, it will give meaningful results throughout the area of investigation. Direct measurement of the current shows downstream acceleration (Clausner, 1967). For geostrophy to hold along the length of the current, the mass field structure must adjust as changes occur in the downstream velocity fields, maintaining the balance between pressure gradient and coriolis force. This mass field adjustment will be examined in conjunction with the changing velocity fields. Several previous investigations have invoked geostrophy in the study of the Florida Current (see for example* Wiist , 1924* Parr, 1937). The extensiveness of the free instrument data available from this experiment (four cross-sections over a 225 Km. downstream scale) plus the quality of this data afford a good basis for a critique of earlier investigations. Such a critique will be included as a second objective of this thesis. FIGURE 1: Chart of Section Locations 82° 81° 80* «fj" ■ 1 i 1 t 79° 78° 1 . — - co — ■' ■''.'% ! 3E CO . . \ft pierce MAtANll2CAt>>^ * ' '■' "V4 ' g,, SHOAL i,./~^>..^ v. LITTLE BAHAMA BANK, 27° - \ .- f -27° F LORIDA ''''■':% ' 0^ 26"- \^>5fa- '.•''.- '■fa': ' -26° \ ^%v . /? i n £fc" ..-'': ,, •>--,. V, xX '.■•'•.'••/? MIAMI f^BIMINI \; -../ V. - \ -^ 25°- * v* / |U 1 -25* / .<•■■- " '" / * I'i-^X'i J:.'"' A ..-'■- S O '■I e^ ■ f/.^ q/^.' ■ • ■-< |SW-' ^SOMBRERO ^° Jrf-y £??\: .'•• - VsSSSi^" X'KEY * 1 " ,.,-- s ^WHj - ~-r " \i • f ? "'""* < \ N. ° \^:' GREAT * v^^ CAYSAlH'^" ' *\ v'i WS»W&? BAHAMA 24°- '> \ , ~- - . - 24° o> BANK?' ., ,\ VL %% .' -3* *»•-.. " ' | 1 -23* 82° 81° 80° 79° 78° II. DATA A. General Four sections were selected at locations which provide various combinations of downstream scale and channel geography. It was felt that the geostrophic approximation could be thoroughly tested when applied in such a variety of situations, furthermore , the Florida Straits over the length where sections were sampled is a closed channel, except for the Santaren and Northwest Providence Channel openings. This configuration permits only minimal distortion of the Florida Current by other sources of water. Section I, between Marathon (Vaca Key), Florida and Cay Sal Bank, B.I. has a "V" shaped bottom profile reaching depths in excess of 1000 m. Section II is located about 100 Km. downstream between Fowey Rocks and Cat Cay, B.I. past the point where the axis of the stream has turned toward the north. At this section maximum depth has reduced to just over 800 m. and the bottom of the trough has flattened. Current width at the surface has been reduced from over 100 Km. at Section I to about 85 Km. at this section. Section III is 25 Km. further downstream and has much the same bottom profile as Section II. Current width at the surface, however, has decreased slightly to about 80 Km. Section IV is another 100 Km. downstream, from Fort Pierce, Florida to Matanilla Shoal, northwest of the Little Bahama Bank. The profile here has regained its sharp "V" profile while current width at the surface has diverged to 85-90 Km. , and maximum depth has decreased further to 700 m. Each section consists of twelve (Sections I, IV) or thirteen (Sections II, III) stations. Measurements were taken at these stations using the free instrument technique. This technique yields data in the form of down and cross-stream current fields and vertical tempera- ture profiles (see Tables I-IV) . Each of the 50 stations was sampled over the full surface to bottom range an average of six to twelve times; each sampling consisted of between one and six instrument drops (depending on station depth), plus a surface current measurement. Horizontal station spacing never exceeds 10 Km. , and the spacing is reduced near boundaries and in the cyclonic zone where there is intensified velocity shear. Sampling of each section was conducted over a period of at least three weeks (except for Section IV, where only seventeen days of sampling were conducted due to equipment difficulties). It was hoped, by judicious sampling over this period, to minimize tidal biasing of the data and construct a time-averaged representation of the mean, steady state Florida Current at the selected stations. B. Analysis The free instrument method yields data in a form which can be applied in geostrophic calculations in a space-temperature (x,y,T) reference frame. The x coordinate is horizontal, has its origin at the left shore of the section (looking downstream) and increases cross-stream. The y coordinate is horizontal, parallels the axis of the stream, and is positive in the downstream direction, with its zero line along the section. The T coordinate is vertical, positive upward. Of the raw data obtained from the free instrument measurements, the downstream component of the time-averaged velocity (Vobs) and the temperature profile for each station, depths of selected isotherms were obtained. These have been used to construct smooth isotherm profiles for the four sections. Perusal of the isotherm profiles suggested selection of the 26° isotherm as the upper bound, as it stays fairly shallow (never deeper than 130 m.) and rises to the surface only at the extreme left hand edge of the stream. Likewise, the 8° isotherm was chosen as the lower bound because it is present across all of the sections at depths sufficient to include areas of relatively low velocity (less than 20 cm/sec). Intermediate isotherms of 10, 14, 18 and 22 °C were chosen for convenience. Profiles of Vobs have been constructed in the same general way as the isotherms, but from vertical velocity profiles for individual stations. These contours are seen as dashed lines in all figures showing the geostrophically calculated isotachs (Figures 2-7). A detailed derivation of the form of the geos trophic approximation used is given in Appendix A. In brief, one starts with the geostrophic equation for the downstream velocity component, (v) : pQfv = 3P/3x where f is the coriolos parameter (2ftsin) and P is pressure, ft is the angular speed of the earth, and 4> is latitude. Taking the deri- vative of both sides with respect to depth, and applying the hydrostatic approximation: P0f3v/3z = g3p/3x. Transforming to (x,y,T) coordinates and transposing, 3v/3T = g/p0f (3p/3T)(3D/3x) where D is isotherm depth. Isotherm slope 3D/3x is computed for each of the six chosen isotherms at each station. Next the thermal shear 3v/3T is plotted along the selected isotherms, and velocity differences between isotherms determined planime trie ally. Starting from an isotherm along which downstream component of velocity is known from direct measurements (usually the 8°C isotherm is chosen) , the downstream component of the geostrophic velocity field is calculated by numerical integration of 3v/3T values. At this point a density- temperature relationship must be determined. It is desired to construct a relationship of the simplest form capable of producing an acceptable reproduction of the existing velocity field when used in the geostrophic calculations. Three equations of state are compared in the extent to which they achieve this end. One equation of state used is of the form; t K=0* where the coefficients This equation will be called the variable coefficient polynomial equation (herein abbreviated VCP) . The coefficients (AK) have been determined from hydrographic data by fitting the data to a least squares polynomial of order K, where K is the smallest integer such that the RMS deviation of the fit has a maximum of . 1 a t. The hydrographic data used was taken by University of Miami Marine Laboratory personnel at 8 stations in the Fowey Rocks - Gun Cay region, and has been presented in the form of a table of coefficients by Schmitz and Richardson (1966). Interpolation has been made where necessary to account for differences between the locations of the hydrographic and the free instrument stations. Another p-T relationship used is a parabloic equation' p = Po [l+a(T-T0)]f where a = aQ [1+k(T-T0)]. The zero subscript denotes reference quantities (i.e. based on the 8°C isotherm). The constants a0 and k are determined on the basis of mean density characteristics of Florida Current water. A linear equation of state was applied in the calculations at one section. p = p0 [l+y(T-T0)] The constant y was determined as were a0 and k. C. Errors All error estimates discussed in this section are considered upper bounds. It has been shown (Clausner, op. cit.) that errors in observed velocity are up to 5%, and errors in isotherm depth are also up to 5%. Since the error in x is of the order of tens of meters and isotherm slopes are calculated over a minimum distance of 5 Km. , errors in isotherm slope 3D/3x due to errors in x are negligible. Isotherm slope errors are, then twice the error in isotherm depth or 10%. Errors in the 3p/3T calculation depend upon the equation of state. For the VCP equation, a 3-5% error is introduced; the parabolic equation yields errors up to about 10%, and; the linear equation gives 10 typical errors up to about 50%. Conditions are considerably worse at the left edge of the stream than in the middle or right sections, but this area of extreme errors is disregarded in the above estimates. Errors in thermal shear are merely accumulations of the 8p/8T and 3D/9x errors. These values would be 15% for the polynomial equations, 20% for the parabolic equation, and 60% for the linear equation. Final geostrophic velocity is estimated to be in error by about the same amount as the thermal shear, with systematic errors increasing away from the reference isotherm. The validity of the geostrophic approximation is determined by calculating the mean and root mean square deviations of the geostrophic velocity from the observed velocity at intervals along the observed isotachs. The calculations are of the form* N i A = 1/N Z(A.) , Ams = [1/N E(A*)P where A. indicates the derivation at the selected points. The deviations will be expressed as percentages. In relation to any discussion of the Florida Current as a geostrophic current, one should be cognizant that Webster (1962) has shown that horizontal gradients of Reynolds stresses produce non- geos trophic components of velocity which average 10% of the velocity, and may reach a maximum of 25% of the velocity. 11 TABLE I: Section I A. Cross stream distance B. Isotherm depth C. Observed velocity along isotherms D. Coriolis parameter * vO * * * * * ***** O * * * * * eg * * * * * * o * * * * * ***** 12 ******* ******* m c * o-. oo r-. «* * eg m o o o o o o * on o> o\ on a. a * c vo n oon a * h vo co On CO w ►J 9 X o o o o o o * r^ r-- r>. r-» r~* r- * o o o o o o * VO VO vO VO vo vO * O O O O rH Is* * in m m in in m * w rH 3 CI i-h co r>. o m cc * on H r^ O CO O * vo i-i n vo c^ m * i— I i— i eg co m m co on -cr r-. m * o on r-» vo <■ co * co O o> rH CO m vo o o o o * * * * -3- * * * PQ co o r-- co * * * in o m iH * * ■K r-i >H eg c_> on r» co r- * * * ONMON * * * CO on m m o o o o * * * co co co co * * * iH N 00 ON * * * 00 CO * * -K # o O oc -K # * On O CO -K -J: -K O O O * -K # * •K * 4C * r-^ r-^ r- r^ * * iH i— I CM CO * On 00 CO » CO * vO c CO co o. o # rH -H CM CO m 3 3 o vO rH O CO co * 5 VO CTN m CM tN CO * H H CM CO « m st co cm st o ■« m on co o\ oo o -k rH rH CM st C_> CO in CO 00 CM 00 -K cm rH on r-» in co ■* st st m o. in on •$: st cm o r~^ m co -k O. iH CM O. O O. * m co rH r-» in cm * O r^ r-» iH co r^ -jc vO CO iH OC st rH * P- 00 On P- rH CM * LO CO rH CO St r-l * CM CO • vO CM CO • vO w rJ 59 CM CO H • /■— * VO O CM rH CO X • v— <" VO H-l CM » CO o vO m m m m m m m -x CM CM CM CM CM CM -K cm cm o oo st m -k m on cm vo co m # rH rH CM CO 00 o st o l~^ CM -K St CO rH On St rH -X CM CO VO O O O O O O •* CM CM CM CM CM CM -K On O On CO CO U~l * st O. O St O rH -K iH rH CM CO r-» CM 00 CO CO CM -K CM rH On 00 m rH -K CM CO VO m rH m rH m rH m rH IT) rH m rH ■a ■K en CO rH CO o o rH rH CM rH rH rH o r- CM o rH o rH o rH O rH O rH CO rH «3> CM rH o vD CO co m o rH m CO rH CM in CM * co co o> * * * ■K o vD O CO * -K m m m in o. on -K o oo r-. m co -it o on m o o o -k r^. st st * o o o * -a -a * * * * * CM CO vO CM CO vO + B u =fc CO TJ 4-1 •H O U UN CJ M vo CM 00 st O CO CM CM rH rH rH vO CM CO st O CM CM rH rH rH VO CM 00 st O CO CM CM rH rH rH ►J 4 17 TABLE IV: Section IV A. Cross stream distance B. Isotherm depth C. Observed velocity along isotherms D. Coriolis parameter 18 ******* * * * * * * * ******* ******* ******* ******* CM O rH OS 00 IT) CM w ►J 9 X ******* ******* in eg os * * * * rH tH O * * * * in in m m >h * * o o o o o * * m m in m m cm * OS OS OS OS OS Os * m in m in in m * oc oc 00 CO oo oo * in in in m m m * r~ f*. r-» r-^ r» r-~ * m in in in in m * vo vo vo vo vo vo * m m m in m m * m in m m m m * o o o o o o * in m in m m m * in m m m m oo * o- -* -a- » «4-: OS OS vO CM o m * 00 rH m - PQ rH H CM CM CO CJ> iH rH O VO O r^ O CO O o * OS r^ CM co o. O * rH 00 rH rH in rH rH CO CM rH * CO O tH r^ in CO * • vO o m CM O o * * CO r^ CM CO o * * rH m CO rH rH rH rH CM * * OS vO 0 10 20 30 40 50 60 70 80 90 100 110 120 27 FIGURE 3. a, GEOS TROPHIC ISOTACHS - Section II 8°C reference isotherm VCP equation of state b. GEOSTROPHIC ISOTACHS - Section II 18°C reference isotherm VCP equation of state SECTION II ISOTACHS (v-cm/%) KM CROSS -STREAM * 0 10 20 30 40 50 60 70 80 90 100 110 120 SECTION II ISOTACHS (v-cm/s) KM CROSS - STREAM » 0 10 20 30 40 50 60 70 80 90 100 110 120 I I I 29 FIGURE 4. GEOSTROPHIC ISOTACHS - Section II 26°C reference isotherm VCP equation of state SECTION! ISOTACHS (v-cm/s) KM CROSS -STREAM — 0 10 20 30 40 50 60 70 80 90 100 110 120 OH irc* -J L 31 FIGURE 5, a, GEOSTROPHIC ISOTACHS - Section I 8°C reference isotherm Parabolic equation of state b. GEOSTROPHIC ISOTACHS - Section I 8°C reference isotherm VCP equation of state SECTION I ISOTACHS (v-cm/s) KM CROSS -STREAM 0 10 20 30 40 50 60 70 80 90 100 110 120 100 200 c/) 300 QC UJ I- 400 UJ 500 X 600 I- UJ 700 Q 800 900 1000- 1100- SECTION I ISOTACHS (v-cm/s) KM CROSS - STREAM » 0 10 20 30 40 50 60 70 80 90 100 110 120 33 FIGURE 6, a. GEOSTROPHIC ISOTACHS - Section III 8°C reference isotherm Parabolic equation of state b. GEOSTROPHIC ISOTACHS - Section III 8°C reference isotherm VCP equation of state SECTION IE ISOTACHS (v-cm/s) KM CROSS- STREAM — 10 20 30 40 50 60 70 80 90 100 110 120 section m ISOTACHS (v-cm/s) KM CROSS- STREAM 0 10 20 30 40 50 60 70 80 90 100 110 120 0-f ^ — . ' . ~ '■ 35 FIGURE 7, a, GEOSTROPHIC ISOTACHS - Section IV 8°C reference isotherm Parabolic equation of state b. GEOSTROPHIC ISOTACHS - Section IV 8°C reference isotherm VCP equation of state SECTION H ISOTACHS (v-cm/s) KM CROSS -STREAM > 0 10 20 30 40 50 60 70 80 90 100 110 120 U" ^^^c- ^80 /^ / i\ 100- ^^^^^^y J '/, /// 200- \\\^>-~J00^^ / // / CO 300- ce UJ H400- 500- x600- h- UJ700- Q \\ \ ^^-jooy' /// \\v S\ - - - " /' ll 1 W x\ /' /' / \\ \^ Jl0^'' // 800- 900- 1000- noo- SECTION 12 ISOTACHS (v-cm/s) KM CROSS -STREAM » 0 10 20 30 40 50 60 70 80 90 100 110 120 37 VCP equation for the bulk of the current. B. Mass Field Adjustments Mass field adjustment in response to changing downstream velocity components is displayed as a plot of average isotherm slope and average downstream speed along the isotherms at each of the sections (Figure 8). Average slope of each of the six selected isotherms is plotted, as is the average velocity along each isotherm. The average isotherm slope was determined graphically from isotherm profiles for the sections. Average velocity was computed as a number average of the observed downstream speeds along each isotherm. From Section I to Section II speeds increase from the upper waters (26°C region) down through the 10°C region. Isotherm slope, though less at the 26°C and 22°C lines, increases markedly below the 22°C level indicating, for reasons of continuity, deepening of the isotachs as the isotherms rise. From Sections II to III little difference appears in the plots, except for the increased slope of the 10°C isotherm at Section III, which reflects moderate speed increase in the mid layers (22°C - 10°C). Between Sections III and IV the velocity profile changes quite differently. Speeds are lower at Section IV down to the vicinity of the 14°C isotherm, and below this, Section IV speeds are higher than those of Section III. Down to just above the 14°C isotherm, Section IV isotherm slopes are greater. Below this point, the Section III isotherm slopes are greater, especially the 10°C isotherm. This reflects the first marked increase in speeds along the lower isotherms. 38 FIGURE 8. Mass field - velocity adjustment Sections I- IV SECT ION I SECTION n AD/Ax(m/km) — *- 0 2 4 v (cm/sec)— o— 0 40 80 120 160 T(«C) I 26 22 18 14 10 8 AD/AX(m/km) -*- 0 2 4 v (cm/sec) — o— 0 40 80 120 160 section m AD/Ax (m/km) — *- 0 2 4 6 8 v (cm/sec) — °— 0 40 80 120 160 SECTIONS AD/A x (m/km) -x- 0 2 4 6 8 v (cm/sec) — o — 0 40 80 120 160 26 22 18 14 10 8 40 C. Earlier Work The geos trophic calculations of WUst (op. cit.) and the agreement of his results with the direct measurements of Pillsbury (1890) have done much to convince oceanographers of the value of the geostrophic approximation. It is remarkable that, using temperature measurements taken in 1878-81, some salinity measurements taken in 1914 and a T-S correlation based on North Atlantic waters (to cover zones where no salinity values were available) WUst was able to construct a density field which yielded such good results when compared with the Pillsbury measurements of 1885-1886, His results agree (in the vicinity of our Section II) with those obtained using the free instrument method to roughly 20%. It is interesting that the isotachs resulting from his calculations are skewed to the right (as compared with the observed isotachs) much as are those calculated in this thesis. In his analysis of five hydrographic stations between Miami and Bimini, B.I., Parr (op. cit.) showed the variations in the density field with time. In particular he described the cross-stream oscillations of a high salinity core. Broida (1966) has shown how this variation in the mass field can result in a distorted picture of the current field as calculated geostrophically . One example of such a distortion is the appearance of a bi- or multi-axial surface current profile cross-stream, a phenomenon which is not evident using the more nearly synoptic free instrument technique. 41 IV, SUMMARY AND CONCLUSIONS During the summer months of 1965 and 1966, four sections across the Florida Straits (over a 225 Km. downstream distance) were sampled using the free instrument technique, over time and space scales designed to yield a picture of a time-averaged, steady state Florida Current, From this series of measurements profiles were constructed for each section of the downstream component of velocity and of depth of selected isotherms to provide a basis for evaluation of geostrophic velocity fields. Based on hydrographic data obtained in 1962-63 in the Fowey Rocks - Gun Cay area, three equations of state were developed in which density was expressed as a function of temperature only. A fifth order polynomial equation was used with coefficients a function of cross- stream distance; a simple parabolic equation was applied, and; a linear equation was employed at one section. Each of these equations was used in the geostrophic equation for downstream velocity in a space- temperature coordinate system, to determine the validity of this form of the geostrophic equation. Three different levels were tested as reference for the calculations. These were the 8°C, 18°C and 26 °C isotherms, along which velocity was accurately known from the direct measurements. Velocity fields from the geostrophic calculations have been compared with those obtained by direct measurement. Lastly, an attempt has been made to show the adjustment of the 42 mass field (specifically the temperature field) to changes in the downstream component of the velocity field as the current is accelerated downstream. Results of the experiment discussed in this thesis point to the conclusion that, under certain conditions, the geostrophic approxi- mation yields a model of the downstream component of the velocity field which is a first order approximation to the directly observed field. For most of the current, geostrophically calculated velocities agree within experimental error with directly measured velocities. Necessary conditions for such a conclusion include the use of an accurately known reference level and use of an equation of state in which density is expressed in terms of temperature by a second (or higher) order equation of state. Constants for the equation must be determined on the basis of local hydrographic conditions. These results confirm the validity of using the geostrophic approximation in theoretical models of inertial flow. Furthermore, it has been shown that there is a direct relation- ship between the mass distribution (specifically the temperature) and the downstream component of the velocity field. A3 LITERATURE CITED 44 Broida, Saul 1966. Interpretation of geos trophy in the Straits of Florida. Doctoral dissertation. Univ. of Miami. Clausner, Edward 1967. Characteristic features of the Florida Current. Masters thesis. (in preparation). Univ. of Miami. Finlen, James R. 1966. Transport investigations in the Northwest Providence Channel. Masters thesis. Univ. of Miami. Parr, A. E. 1937. Report on hydrographic observations at a series of anchor stations across the Straits of Florida. Bull, Bingham Oceanogr, Coll., 6 (3): 1-62. Pillsbury, J, E. 1890. The Gulfs tream - A description of the methods employed in the investigation and the results of the research, 459-620, Report of the Superintendent of the U. S. Coast and Geodetic Survey. Richardson, W. S. and W. J. Schmitz, Jr. 1965. A technique for the direct measurement of transport with application to the Straits of Florida. Jour. Mar. Res., .23. (2): 172-185. Robinson, A. R, 1965. A three dimensional model of inertial currents in a variable-density coean. Jour. Fluid Mech. 21_ (2)' 211-223. Schmitz, W. J. Jr. and W. S. Richardson 1966. A preliminary report on operation Strait Jacket. Univ. of Miami, Mar. Lab. Data Report. 66-1. 222 pp. Webster, Ferris 1962. Departures from geos trophy in the Gulf Stream. Deep Sea Res., 9. (2): 117-119. Wttst, G. 1924. Florida and Antilles Current System. W. von Dunser, trans., Veroff Inst. Meereskunde-naturwiss. , Heft 29. 70 pp. 45 APPENDIX A: DERIVING THE PARTICULAR FORM OF THE GEOSTROPHIC EOUATION 46 Robinson (ibid,) has stated that the downstream component of an inertial flow is in geostrophic balance. In (x,y,z) coordinates' pQfv = 3P/3x where the use of a constant p is an approximation resulting in less than 1% error. Taking the derivative with respect to depth of the above equation and applying the hydrostatic approximation; p0f(3v/3z) = -g(3p/3x). One may transform to (x,y,T) coordinates by: (3v/3z)x>y = (3v/3T)x>yOT/3z)X)y and; (3p/3x)v = -(3p/3T)(3D/3x)v T(3T/3z)v which yields : 3v/3T = g/fPo(3p/3T)(3D/3x) where D is the depth of an isotherm T. It now remains to take the temperature derivative of the three equations of state, which is straightforward and gives the results shown below. Linear equation; 3p/3T = p0y Parabolic equation; 3p/3T = p0a0[l+2K(T-T0)] VCP equation; 3p/3T = Z [KArT^"1^] VITA LT. Edward J. O'Brien III, USN, was born in Pittsburgh, Pennsylvania, on November 25, 1939. His parents are Edward J. O'Brien Jr. and Barbara B. O'Brien. He received his elementary education at St. Mary School, Baltimore, Maryland, and his secondary education at Loyola High School, Towson, Maryland. In July, 1957, he entered the U.S. Naval Academy, Annapolis, Maryland. Upon graduation in June, 1961, with a B.S., he was commissioned an Ensign in the U.S. Navy. Subsequent Naval service included three years in the Destroyer Force of the Pacific Fleet followed by a year as Instructor of Naval Science at College of the Holy Cross, Worcester, Massachusetts. He was admitted to the Graduate School of the University of Miami in September, 1965. He was granted the degree of Master of Science in June, 1967. Permanent address: 7103 Wardman Road Baltimore, Maryland 21212 / 9 -*"