Transactions on Engineering Sciences vol 9, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533 A three-dimensional tidal model of the Great Australian Bight using a new coastal boundary procedure BJ. Noye," K. Matthews* "Department of Applied Mathematics, The University of Adelaide, Adelaide, Australia, 5005 ^Aeronautical and Maritime Research Laboratories, PO Box 1500, Salisbury, Australia, 5108 Abstract A three-dimensional tidal model using finite differences, spherical coordin- ates and an oblique piece wise-linear coastal boundary representation is ap- plied to the Great Australian Bight. The Bight is the site of a national marine park and various fishing industries. This region has not yet been the subject of a three-dimensional numerical model and little observational data is available so depth-dependent numerical predictions will provide a valuable resource. Computed sea surface elevations and current patterns at different depth levels is provided for the M2 and unusually large M?, con- stituents along with comments on some unique flow features of the area. 1 Introduction The Great Australian Bight (GAB) of southern Australia is rather unique in a world-wide context. It has a continental shelf which in some places is over 200km wide, has a long ice-free east-west extent (2000km), and is adjacent to the only circumpolar ocean. It also supports a large fishing industry, has prospects of oil exploration and, more recently, has become the site of a national marine park. For these reasons it is a region of academic interest and economic importance. There is still minimal understanding of the tide-induced water move- ments in this area although there is a small amount of observational tide- Transactions on Engineering Sciences vol 9, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533 268 Advances in Fluid Mechanics height data at coastal stations. For these reasons a three-dimensional (3- D), spherical coordinate tidal model, using finite differences and an ob- lique piecewise-linear coastal boundary representation has been applied to the GAB. Spherical coordinates were chosen because of the large spatial extent of the region. A discussion of predicted sea, surface elevations and depth-dependent current patterns is provided along with comments on some unique flow features of the area. 2 Mathematical model The mathematical formulations used to predict sea surface elevation and depth-dependent velocity are very complicated. Earth curvature is accoun- ted for through the spherical form of the equations of motion, Earth rotation is incorporated by the Coriolis terms, advective and eddy viscosity terms are included, as are effects of sea level changes due to tides at the open boundary. A sigma transformation of the vertical coordinate is carried out where the new vertical coordinate r/ (directed downward) is defined by The variable ( is sea surface elevation above mean sea level (m), z is the vertical coordinate-directed upward (m), h is depth of sea floor below mean sea level (m), and H is total depth, so that H = h -f- C (m)- The depth-averaged continuity equation, in spherical coordinates, is given by dt R cos c where t is time (s), A and <p are latitude (°E) and longitude (°N), respectively, U and V are depth-aver aged velocities in the A and 0 directions, respectively (ms~*), and R is Earth's mean radius (m). The depth-aver aged velocities in the vertically transformed coordinates are represented by rl rl = a dr\ and V = v drj. (3) Jo Jo where u, v, and w are depth-dependent velocities in the A, (/>, and z direc- tions, respectively (ms~*). The 3-D, depth transformed, spherical A-momentum equation is 3u u 3u v d(ucoscj)) du % 72 cos (60 A #cos(^> cW> 07? 90 ., ' . ?; s Transactions on Engineering Sciences vol 9, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533 Advances in Fluid Mechanics 269 z\ tan 0 dv HT^ d^u [AT* tan (p du and the 3-D, depth transformed, spherical o-momentum equation is dv u dv tan (p 2 v dv dv ~ + R " " ^TA tanp du dv //T» tan (p dv p #2 COS tan Rm* [pcos*2 o p J H* drj [ p where LU is depth-dependent velocity in the 77 direction (m s~*). fl is Earth's angular rotation speed (s~^). p is sea water density (kgm~^), g is accel- eration due to gravity (ms~^), and /^, //^ and ^/^ are eddy viscosity coefficients in the A, 6, and 77 directions, respectively (kgm~^s~^). A 3-D form of the Schwiderskifl] formulation is used to model the hori- zontal eddy viscosity coefficients and an asymmetric parabolic vertical eddy viscosity coefficient is prescribed [2]. The transformed vertical velocity at depth 77 is given by a - HV(n) cos*} (6) where dr}', and V(??) = Ti'cf)?'. (7) JO The upward velocity in the vertical is given by ..„ --,-+---,_. (8) In order to solve equations (2)-(8), appropriate initial and boundary conditions must be prescribed. The coastal boundary condition requires that there is no flow across the coastal boundary. The no-slip condition is applied to the sea bed, the free surface condition is applied at the sea surface, and gradients of the horizontal velocity components at the surface in the 77 direction are zero. Numerical calculations start from an initial condition of zero elevation and motion, and circulation is forced by input on the open boundary of the sum of the contributions of the five dominant astronomical tidal constituents Oi. KI. Mg, 82. and Ma. This open boundary condition is used in conjunction with the one-dimensional Sommerfeld[3] and Orlanski[4] radiation condition, which was adapted for spherical coordinates. Transactions on Engineering Sciences vol 9, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533 270 Advances in Fluid Mechanics \ ^ J 32- ^ — ^L 7^\ A Australia y / Great \X? / 34 *— -~«. 36- -^ Australian g)^ / 38 Bight ^^ySr ^-^ *?10E 115E 120E 125E 130E 135E 140E USE 150E 15 Figure 1: A map of southern Australia showing the location of the GAB. 3 Numerical model In most finite difference models of tidal flow the Arakawa C grid, a rect- angular staggered grid, is used in the horizontal plane. Here the simplest way to incorporate a fixed coastal boundary is to allow the boundary to be approximated by grid line segments. This produces a stepped boundary and as a result, velocity components which lie on and are normal to the model coastal boundary are automatically zero because of the impermeab- ility boundary condition. A major disadvantage of representing the coastal boundary in this way is the low order of accuracy which results when the finite difference formu- lae are applied at interior computational points adjacent to the boundary. An alternative technique which retains the rectangular grid but approxim- ates the coastal boundary by oblique piecewise-linear segments which slice through grid elements is used in this work. Velocity information along the segmented boundary is computed using a slip boundary condition and is subsequently interpolated to nearby computational points. This novel tech- nique for representing the coa-stal boundary is described in detail in Mat- thews et al.[5] and has been applied in Matthews et al.[6] and Matthews and Noye[7]. The finite difference form of the continuity and momentum equations along with the initial and boundary conditions use centred time and space differences where possible, giving rise to a solution which is second order accurate in space and almost second order accurate in time[8]. 4 Application to the Great Australian Bight The location of the GAB is shown on the map of southern Australia in Figure 1. The model covers the region 120°-136° E and 31°-35° S. Figure 2 shows the computational boundary used along with the bathymetry of the region and the positions of some tidal recording stations. Hydrographic maps of the GAB were used to obtain the position of the coastline and bathymetric data for the continental shelf were obtained from the RAN Hydrographic Office[9]. Deep sea bathymetric data were obtained from the Transactions on Engineering Sciences vol 9, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533 Advances in Fluid Mechanics 271 x Fowlers Bay -<sft "A - •f•c iElliston Figure 2: The computational coastline (dotted line) and open sea bound- ary (dashed line) for the GAB model. Some tidal stations are named and bathymetry (m) is shown. ETOP05 data set from the National Tidal Facility[10]. The continental shelf is widest in the northeast of the Bight near Thevenard and narrowest in the southwest near Esperance. The computational grid used has a horizontal element dimension of 5' x 5'. The top left and bottom right corners of the grid are at 31*20' S 120*50' E and 35*10' S 136* E. respectively. In (A, <f>) space the grid consists of 47 rows and 181 columns, a total of 8507 elements, of which only two thirds are used in computations. The Kappa method[ll] is used to set the spacing of the 11 depth levels so that the resulting vertical grid is fine near the sea surface and bed, and coarse at mid depths. The model is run for a simulated time of 30 days, starting at 0:00 hr on January 1 1900, using a 20 s time step. Because of contamination by starting transients, due to the initial condition, the first day of data is discarded. The amplitudes and phases for the four major tide-height constituents Oi, KI, M2, and 82 along the open sea boundary of the model are ob- tained from the Schwiderski l*xl* global ocean model data set[10]. Open boundary data for Ms, the most significant higher order tidal species at Thevenard, are estimated from a consideration of observations along the coastline.