Tidal Propagation in the Gulf of Carpentaria Are Developed
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tolrol¿Õ i TIDAL PROPAGATION IN THE GULF OF CARPENTARIA by Michele Marie Rienecker B.Sc. (ttons. ), University of Queensland Thesis submitted for the degree of Doctor of PhilosoPhy in the University of Adelaide Department of Applied Mathematics December L978 tli" 1-qT0 [^t.ro{r.i 11ì 0,q J TABLE OF CONTENTS SUMMARY (i) SIGNED STATEMENT ( ii) ACKNOIIILEDGEMENTS (íii) CHAPTER 1 INTRODUCTION I CHAPTER 2 THE PROBLEM: A RECTÆ{GULAR RESONATOR ON A SEMI-INFINITE CHANNEL 2.L The Tidal Equations 5 2.2 Frequency Response Analysis 6 2.3 Reformulation of the Equations for an Analytic Model 8 CHAPTER 3 THE ANA],YTIC SOLUTION 3.1 The Method of Solution and the Boundary Conditions 10 3.2 Solution for the Channel Region 11 3.3 Solution for the ResonaÈor Region 13 3.4 Solution folthe Junction Region 15 3.5 The Remaining Matchíng Conditions and the Galerkin Technique 20 CHAPTER 4 AN EXTENSION TO THE MODEL: AN ADJOINING CHANNEL 4.1 The Equatíons 35 4,2 The Solutions 37 4.3 Determination of ô 46 CHAPTER 5 TI4IO NUMERICAL MODELS 47 5.1 A Linear Fínite-oifference Numeríca1 Model 4B 5,2 The EVP Method 52 5.3 Stability, Consistency and Convergence 55 5.4 The Friction Parameter 59 5.5 A Non-Linear Model 6L 5.6 Consistency, Convergence and Stability 64 CHAPTER 6 APPLICATION TO TIIE GI]LF OF CARPENTARIA 6.1 The Gulf of CarPentaría 68 6.2 The AnalYtic Model of ChaPter 3 72 6.3 The AnalYtic Model of ChaPter 4 89 6.4 The Linear Numerical Model 95 6.5 The Non-linear Numerical Model 105 6.6 The Programs 110 6.7 The Response of the Gulf to Tidal Forcing 110 116 CHAPTER 7 CONCLUSION 1t_8 APPENDIX 1 The Representation of Bottom Friction L27 APPENDIX 2 \te Galerkin and Collocation Methods APPENDIX 3 Evaluation of the Integral Form for 6z(x,y) 130 APPENDIX 4 The Classes of Elements for the Non-linear Model and Their Associated Finite-Difference Equations 135 138 BIBLIOGRAPHY (i) SUMMARY This thesis considers tidal propagation in a rectangular resonator-channel system, with specific reference to the Gulf of Carpentaria, situated to Èhe North of Australia. The linearízed form of the trdo-dimensional depth-averaged equations of continuity and momenEum conservation is used. An analytic solution is found by dividing the area into regions of constant deprh. In this manner, a solution is found for Èhe case of a síngle connecting channel and then for the exEended case of two connecting channels, associated with either neglecÈing or including the effect of tidal flux from Torres Strait into the Gulf. Results from the analytic model are used to provide tidal inputs for two numerical models, both of which use explicir finite- difference approximations. The first numerical model is linear and is developed to account for realistic boundary and bathynetry variations, with the emphasis on obtaining a model with small- compuÈer time and memory requirements. Since, in shallower coastal areas, the non- linear Lerms in Èhe continuity and momentum equations become more important, the second numerical model developed includes these terms to determíne their effect on the resonator as a whole. The two models show favourable agreement, thus verifying the usefulness of the linear mode 1 . ( ii) SIGNED STATEMENT I hereby declare that this thesis contains no material which has been accepted for the award of any other degree or diploma in any University and, to the best of my knowledge, it contains no material previously published by any other person, except where due reference ís made in the text of the thesie. M.M. RIENECKER. (iii) ACKNOI,üLEDGEMENTS I would like to thank my st¡pervisor, Dr. B.J. Noye, for his advice and guidance in completíng this thesis. Many thanks are also due Èo Dr. Michael Teubner for his help and encouragement throughout, Èo Mrs. Angela McKay for her excellent, accurate typing, to Mr, G. de Vries for preparing the diagrams and to Mr. Phil Leppard for his advice on computing. The work associated wíth this thesis was carried Òut from February 1975 to November 1978, during which time I was financed by a Commonwealth Postgtaduate Research Award' M.M. Rietrecker - 1 CHAPTER 1 INTRODUCTION The Gulf of Carpentaria, located in tl're North-Eastern part of Australia, is an area of difficult access by either land or water and hence remains relatively uncharted and unstudied. However, over the past few years, there has been an increase in the nuirber of research programs in the area as its potential for the lucrative rnining and fishing industries has been recognized. Lirnited sectors, such as at lrleipa, PorÈ McArthur, Groote Eylandt and Gove Peninsular, have been surveyed as port facilities l{ere required by mining companies. The C.S.I.R.O. has reported on the hydrology of the region (Rochford (1966), Newell (1973)) and has made a preliminary study of the circulation in the Gulf (Cresswell (1971)). Further work by this organízatíoa is currently under way. The Gulf r^ras also chosen by Teleki et al (1973), as the test site for assessing the usefulness of satellite irnagery to the rnapping of hydrological parameters in areas of difficult access. Tides and wind are the only two mechanisms which generate Èhe currenÈs in the atea. For several months of the year it is the influence of the strong, sÈeady trade winds which drive the circulation. However, overall, the circulation and mixing of waters ín the Gulf are governed by the clockwise motíon of tides and by density gradients resulting from the stratification of the water masses. Stratification is partly induced by differential evaporation rates betvreen the Northern and Southern halves of the bay (Teleki et al (1973)). According to Newell (1973), "the annual evaporatiorr/precipitation budget of Èhe Gulf forms a very sma1l part of ?- the total water exchange but is of great importance in influencing water movement.tt One examPle of the effect, in the Gulf, of winds and precipitation/ evaporation is the striking feature of the annual cycle of about .7m rn the tides, the lowesttídes being r:ecorded at the end of the drl' season and the highest during the wet season' It is during the wet season, which occurs in the summer months, that the activity of cyclones sometimes has a disastrous effect, causing large surges. some description of this is given by Easton (1970) who outlines the general tidal features of the Gulf. Realistic and accurate modelling of the waÈer movement in the Gulf of Carpentaria is obviously an intricate affair, it being necessary to incorporate the effects of tide, wind, stratification, precipitation/ evaporation and Pressure surges, not to mention ríver run-off and sediment transport. However, whereas the effects of the other mechanisms diminish at certain times of the year, Eidal forcing is always Present; and it is only the response of the Gulf to tidal forcing which is investigated in this thesis. Once the Èi-dal response is understood, it is easier to sËudy the coupling of the tide with other effsu[s' The tides are caused by the movement of the sun and the moon and their changing gravitational pull on the $Iater of the earth; however, in coastal areas, such as gulfs and estuaries, a\¡Iay from the deep oceans, astronomical tidal forcing can usually be neglected compared to the direct forcing from the motion of adjacent r¡raters. From the results of Hamblin (1976) for different size basins, the maximum amplitude of the resPonse of the Gulf of Carpentaría to direct astronomical forcing could be expected to be only about 3 cm, a very sma1l contribution to the response as a whole In this thesis, Eidal propagation in Gulf systems is investigated by finding the resPonse of Ëhe system Èo tidal forcing on an open boundary' 3 One of Èhe main problems in ascertaining the accuracy or limitations of any model of an area is that reliable input data across the open boundary is rarely available, especially if the boundary is wide. Data is usually available from coastal areas, but this may be disturbed by local effects and is not always representative for the open sea. Values may be inter- polated between coastal areas, but this does not normally take into account the disÈribution of depth (see Hansen (1962)). A more accurate input may be obtained from the results of an analytic model which produces the input from the solution to the model equations. This still does not account for the true distribution of depth, but avoids the need to interpolate over wide areas. Analytic models have their limiEations in that they provide solutions only for simplified situations; however, they can be very useful in providing an insight into the important feaÈures of a model area. As well as providing the tidal forcing daÈa along an open boundary for numerical models, they may also act as a guideline to the accuracy or validity of these more complicated models. lüith this in mind, two analytic models of the tidal propagation in the Gulf of Carpentaria are developed. They a, e essentially extensions of Taylorrs (I92Oa) problem of the reflection of a Kelvin wave by an end barrier in a semi-infinite channel. tr{illiams (L972) modelled the Gulf as a rectangular resonator on a serni-infinite channel. His first model neglects the effect of the Coriolis force; his second considers the frequency of rotation of the earEh to be small compared to the frequency of the tidal motion. The models in this thesis are extensions of his work, incorporating, amongst other things, the dissipation of energy by bottom friction.