A Three-Dimensional Tidal Model of the Great Australian Bight

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 coordinates and an oblique piece wise-linear coastal boundary representation is applied 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?, constituents 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 movements in this area although there is a small amount of observational tide-

height data at coastal stations. For these reasons a three-dimensional (3-

D), spherical coordinate tidal model, using finite differences and an oblique 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 floorbelo w 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

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 directions, 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

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 horizontal 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.

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\ ^ 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 rectangular 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 technique 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

x Fowlers Bay -

iElliston

Figure 2: The computational coastline (dotted line) and open sea boundary (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, ) 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 obtained 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. Small amplitudes progressing from 0.2cm on the western side to 0.4cm on the eastern side of the Bight and phases which increase progress- ively from 26* in the west to 190* in the east are used for MS tide-height open boundary data. This input gives predictions at the coast which are in good agreement with observations. Parameters associated with the eddy viscosity coefficients are obtained by a calibration which minimises differences between predicted and observed tide-heights and tidal currents at stations within the GAB.

4.1 Sea surface elevation

Predictions for the semi-diurnal M2 and ter-diurnal Ma amplitude and phase of sea surface elevation are shown in Figure 3. These agree well with depth- averaged predictions[7] and with tide-height observations in the GAB [10].

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0.10 0.12 0.14 Eucla 0.16 Fowlers Bay 0.18 Twilight Cove 0.20 * iThevenard 0.22 0.24

Esperance Elliston

60 (a) M2 amplitude (m) 63 66 69 72 75 78 81 84

(b) Mz phase ("GMT) 0.01 0.02 0.03 0.04 0.05 0.06 0.07

20 (a) Ma amplitude (m) 40 60 80 100 120 140 160 180 200 A "'ft" A

(b) Mg phase ("GMT)

Figure 3: Amplitude and pha.se predictions for the M2 and Ma tidal constituents of sea surface elevation.

Advances in Fluid Mechanics 273

Eucla Fowlers Bay -, Thevenard

Figure 4: Predicted tidal ellipses (m s~*) at depth level 1 for the constituents indicated.

The M2 constituent of tide-height is steadily amplified as it propagates across the continental shelf near Thevenard and the head of the Bight until it is doubled at the coast. The phase is a minimum in the mid Bight region and increases in both westerly and easterly directions. There is a strong amplification (by a factor of six at the coast) of the MS component of tide-height across the continental shelf near Thevenard and the phase steadily increases from west to east. Shallow water interactions are insufficient to generate the MS constituent recorded on the coast in the GAB. When there is no Mg tide-height input to the open boundary the overall Ma tide-height amplitudes are less than 20% of that observed.

Clearly the MS tide-height constituent must be specified as open boundary input. Davies and Lawrence[12] also found it necessary to prescribe Mi tide-height and tidal current data on the open boundary of their Irish Sea tidal model.

4.2 Tidal currents

The semi-diurnal M2 and ter-diurnal MS tidal current ellipse plots for the

GAB are given in Figure 4 for depth level 1, which is 5% of the total depth below the sea surface. Ellipse predictions agree well with observations[13] in the southeast of the Bight. At each computational point these ellipses show the behaviour of velocity of a specified constituent, over a complete cycle. The start of the ellipse is denoted by a vector (without the arrowhead) which indicates the direction and strength of the current, then the path of endpoints of the

274 Advances in Fluid Mechanics

Twilight Cove

Esperance //////-.-/////////////\"V/.:v/// ///^

(a) depth level 1

o.oio ' ';- -

(b) depth level 9

Figure 5: Predicted residual velocities (m s~*) for the depth levels indicated.

velocity vector is traced. In order to identify the direction of rotation of the ellipse the endpoints of the last few of the twenty time increments used are excluded. In general the ellipses rotate in an anticlockwise direction. The M2 ellipses are consistently largest nea.r the edge of the continental shelf where the major axis is orientated in a cross shelf direction. Ellipses for the M] constituent of tidal current are also more circular near the edge of the continental shelf and become smaller and more rectilinear nearer to the coast. The Mg ellipses are similar to the M2 ellipses but are of reduced magnitude, have major axes lengths which are only 20% of the length of Ma, and are more rectilinear. Near the shelf edge the Ma ellipses are slightly rounded, have major axes aligned in a cross shelf direction, and rotate in an anticlockwise direction. They get smaller and more rectilinear half way between the shelf edge and the coast. Close to the coast, between the head of the Bight and Theveiiard, the ellipses are more rounded, their major axes are parallel to the coast, and their rotation is in a clockwise direction. The ellipse plots for the other depth levels show similar characteristics but their overall magnitude is reduced at lower depth levels. For example, the length of the major axes of the Ms ellipses at depth level 9, which is 5% of the total depth above the sea floor, are half that for depth level 1.

Residual velocities are obtained by removing the five major tidal constituents from the predicted signal and averaging over the 29 days involved. Residual velocities are shown in Figure 5 for depth levels 1 and 9. Near the western open boundary there are westward flowing residual velocities close to the sea floor but this feature is not observed near the sea surface. At the

Advances in Fluid Mechanics 275

western end of the Bight there is an anticlockwise rotating eddy between

Esperance and Twilight Cove. This is observed both near the surface and near the sea floor, but is of reduced magnitude and displaced to the east for the lower depth level. An anticlockwise rotating eddy below Twilight

Cove is seen at depth level 1. Further northeast a similar eddy of reduced magnitude is seen at depth level 9. The direction of the residual velocities is significantly different for these two depth levels on the north and eastern continental shelf. In the central Bight region, on the continental shelf below Eucla, residual velocities are directed north to northeast near the surface but near the sea floor are directed east to southeast. In the deep water in the southeast of the Bight there is an anticlockwise rotating eddy. This is observed at all depth levels and its magnitude remains the same. In the southeast corner of the GAB near the open boundary there are large residual velocities. In future work, it is intended to move the open boundary further south to determine whether these residual velocities are affected by boundary location.

5 Conclusions

Results from the spherical coordinate three-dimensional tidal model of the GAB predict some interesting tidal movements in the region, namely amplification of the semi-diurnal and ter-diurnal constituents, and residual currents on the shelf edge directed east to southeast. These agree with observations which indicate that the entire shelf waters south of Australia tend to run eastwards along the shelf, that the tidal signal is dominated by the residual stream, and that stronger currents exist near the shelf edge[14]. They also agree with observational tidal current characteristics in the southeast of the GAB which show an elliptic motion with the main axis aligned in the cross shelf direction [13].

Acknowledgements

This work was supported by funding from an Australian Postgraduate

Award and a supplementary scholarship from the Defence Science and Tech- nology Organisation.

References

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276 Advances in Fluid Mechanics

[2] Noye, B.J., May. R.., & Teubner. M. A three-dimensional tidal model for a shallow gulf, in (ed B.J. Noye). pp. 417-436, Numerical Solutions

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[3] Sommerfeld, A. Partial Differential Equations, Academic Press. 1949.

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[13] Hahn, S.D. Physical Structure of the Waters of the South Australian Continental Shelf, TR. 45, Flinders Inst. for Atmos. and Marine Science, The Flinders University of South Australia, Australia, 1986.

[14] Provis, D.G. & Lennon, G.W. Some oceanographic measurements in

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