A Computational Model for Monitoring Water Quality and ... - CiteSeerX

A Computational Model for Monitoring Water Quality and Ecological Impacts in Marine Environments S. M. Wong ; , Y. C. Hon , S. L. Chung , K. C. Ho 12

2

1

1

Open Learning Institute of Hong Kong Department of Mathematics, City University of Hong Kong 1

2

E. J. Kansa

Lawrence Livermore National Laboratory, Livermore, U. S. A.

Abstract Multiquadric method is employed for solving the two dimensional depth-integrated water quality model and the eutrophication model in marine environments. The models simulate the spatial and temporal variations of nutrients in the Tolo Harbour of Hong Kong by taking considerations of the tidal current, wind stress, growth dynamics of phytoplankton and the transformation rate of nutrients. The phytoplankton biomass is measured indirectly by using the concentration of chlorophyll-a as an indicator for algae abundance. The Multiquadric scheme has shown to be comparably stable in computational results and eective in the elimination of the unexpected negative predictions of pollutants concentration in the boundary regions. Details of boundary loading of in ow pollutants, data requirements for the modelling set up and calibration are discussed. Agreement is exhibited between the simulated results and the eld data.

1 Introduction In consequent to the increases of human population and industrial development, the marine system, in particular the areas along the coastline are seriously impaired by the transmission of toxic materials, the abundant growth of phytoplankton species which generates red tide occurrence as well as oxygen depletion. All of these factors are harmful to the public health and aquatic life. While eutrophication is a necessary condition for red tide formation, there are other factors that will also affect the growth of algae. These factors include solar radiation, water temperature, 1

tidal current and wind stress. Blooms of algae are generally encountered at certain times of the year when all these conditions for growth are optimal [1] and [2]. In other words, high solar radiation level, warm seas, and high levels of nutrients are the prerequisite conditions for phytoplankton growth. Although eld sampling data for the water quality and meteorological data can be collected periodically, the comprehensive analysis of the eects of these factors on the growth dynamics of phytoplankton and subsequently the formation of red tide in the water region concerned are dicult and time consuming. It was recognized that the data are often insucient to answer all questions being posed. Particularly in a large marine area, monitoring programs will be limited in their extend of coverage both spatially and temporally. The availability of data for many substances and the knowledge of signi cant processes will be limited. In the light of the complicated interaction between the hydrological conditions and biochemical processes, mathematical modelling which is commonly used to cope with complicated systems is considered to be practically valid for the prediction of future water quality variations. It also helps to enhance the understanding of the physical and biological interrelationship between the eutrophication phenomenon and phytoplankton growth dynamic. From the water quality management point of view, the application of computational modelling has also been recognized to be an advance technology for simulation and a relatively economical device in comparison with others experimental methods. In this study, a mathematical model was developed by taking the topographical data, meteorological data and nutrients loading from various source points as input and to simulate the nutrients distribution and chlorophyll-a concentration which indicates the algal biomass variation over a speci c time period. The model is used to simulate the concentration of nutrients and monitor the eutrophication impacts within the Tolo Harbour of Hong Kong. The model consists of ve sub-models which are described by the hydrodynamic equations; the advective-dispersion transport equations for nutrients; the growth dynamics of phytoplankton, the transformation rate of nutrients and the advective transport of algae biomass. A new numerical scheme called the Multiquadric Method (MQ) was devised for the numerical computations setup. The computational results have shown that MQ scheme is a more stable method which reduces the unexpected negative prediction in regions of high concentration gradients and with complex boundary conditions. It also provides high exibility in the approximation of irregular coastal boundary in comparison 2

with other traditional methods such as the nite dierence and the nite element methods. Section 2 describes the details of the theory and the set up of the mathematical model. Section 3 gives the numerical scheme in Multiquadric Method which is followed by the numerical computations in Section 4. Finally, some discussions and conclusions are presented in Section 5.

2 Theory and Methods for Modelling A common diculty in assessing water environmental changes is the large natural variability that exists in the biological and chemical interactions processes. Many substances exhibit strong seasonal patterns as well as variations over a long period of time, due to wind, temperature, light conditions, circulation ow and other natural eects. This study is to set up a modelling framework that can bring together these factors in a more structural way. The modelling framework is divided into ve major model components. They are described in detail below.

2.1 Hydrodynamics Model for Fluid Velocities The distribution and spatial variation of the marine water substances are highly related to the water ow, tidal action and wind stress. The governing equations are basically the depth-integrated version of three dierential equations arising from the principles of hydrodynamics, namely the continuity equation (2.1) and the momentum conservation equations (2.2) and (2.3) resolved into the x and y directions respectively. According to [3, 4], these equations are formulated as: @ + @ (uH ) + @ (vH ) = 0; (2.1) @t @x @y

p

@ (uH ) + @ (u H ) + @ (uvH ) + gH @ + g u u + v @t @x @y @x C  a ?fvH ?  CsWxWs = 0; 2

2

2

2

w

p

@ (vH ) + @ (uvH ) + @ (v H ) + gH @ + g v u + v @t @x @y @y C  +fuH ?  a Cs Wy Ws = 0; 2

2

(2.2)

2

2

w

3

(2.3)

where u; v are the depth-averaged advective velocities in x; y directions respectively;  is the sea water surface elevation; h is the mean sea level; H is the total depth of sea level, such that H = h +  ; Wx; Wy are the wind velocity components in x; y q directions respectively, and Ws is the wind speed given as Ws = Wx + Wy . The other parameters of the equations are C = Chezy bed roughness coecient; f = coriolis force parameter; g = gravitational acceleration; a = density of air; w = density of water; Cs = surface friction coecient. The water boundary and land boundary conditions are de ned as  (x; y; t) =   (x; y; t) and Q~
~n respectively, where  (x; y; t) is the speci c sea surface elevation level on the water boundary, Q~ represents the velocity vector (u; v), ~n is the direction of the outward normal vector on the land boundary. The initial conditions are Q~ (x; y; 0) = ~0,  (x; y; 0) = 0. 2

2

2.2 Advective-diusive Mass Transport Model The mass transport model is devised on the basis of a 2-dimensional depth averaged advective-diusion equation. As in the hydrodynamic model, the mass transport equation resolves the time and space into x, y directions. The variation of nutrients concentration can be modelled as @C @t

i + u @Ci + v @Ci = @x

@y

@ @x

 D

xx

@C



i + D @Ci + xy

@x

@y

@ @y



yx

D

@C



i + D @Ci + S B + S K  ; (2.4) yy i i

@x

@y

where Ci is the concentration of the nutrient i which are the inorganic phosphorus(P ), organic phosphorus(P~ ), ammonia nitrogen(NH 4), nitrate nitrogen(NO3) and organic nitrogen(N~ ), Dij 's are the components of diusion coecients, SiB is the source loading rate of nutrients and SiK is the rst-order kinetic transformation rate of nutrients, the kinetic transformation process is described in Section 2.4. In general, the diusive coecients Dij 's depend on the advective velocities and depth of sea level. However, for the sake of simplicity, they are considered to be constants in this study. The model is subject to the boundary conditions that the in ow and out ow concentrations are speci ed on both land and water boundary. At t = 0, we specify the initial condition to be the known euent concentration as

C (x; y; 0) = C (x; y; 0) 0

4

(2.5)

There is no advective ux of pollutants across the land boundaries, since the current eld ows are assumed to be parallel to the land boundary. Here the land boundary condition is de ned as @C (x; y; t) = 0; (2.6) @~n where ~n is the normal vector to the land boundary. On the water boundary, we assume the in owing concentration is known and speci ed as

C (x; y; t) = C (x; y; t);

(2.7)

whereas no speci c value is given on out owing cencentration.

2.3 Phytoplankton Growth Dynamics Model On the basis of the bottle assay experiments performed by Ho et al [5], the four most signi cant ecological limiting factors aecting the growth of phytoplankton in Tolo Harbour were found to be dissolved inorganic nitrogen, dissolved inorganic phosphorus, solar radiation and temperature. The eect of the inorganic nitrogen and phosphorus ratio and their relations to red tides were also discussed by Ho [1, 10]. The growth kinetic model is formulated in a similar manner follow the approach by DiToro [6] and WASP5 [7]. The solutions of equation (2.4) yield the values of ammonia nitrogen(NH4), nitrate nitrogen(NO3) and inorganic phosphorus(PO4) concentrations at each point of the Harbour. The population of plankton biomass can be determined by a growth dynamic model which relates its growth and death rate to the above limiting factors. The underlying equation is of the form: @'(x; y) = (G ? D ? V )'(x; y); (2.8) ' @t where '(x; y) is the population of plankton biomass at point (x; y), G and D are respectively the growth and death rate of phytoplankton. V' is the phytoplankton settling rate. Equation (2.8) depicts the sole contribution of nutrients and energy factors to the growth rate of phytoplankton at a point without the eect of advection and dispersion. The growth rate G of phytoplankton depends jointly on nutrients, photosynthesis and respiration which in turns are governed by the uptake rate of inorganic nitrogen 5

and inorganic phosphorus and the variation of solar radiation and temperature. It is described mathematically as ! !   I e ?ke H ) I) N P  ? ? ( ( I I G = k +N k +P k H e s ? e s G GT ? ; (2.9) e N P where N; P are respectively the concentration of total inorganic nitrogen, and inorganic phosphorus, H is the total height of the water level, I is the solar radiation intensity, and T is the water temperature (C ), kN and kP are half-saturation coef cients for uptaking nitrogen and phosphorus, ke is the light extinction coecient,  is the fraction of day with daylight, Is is the optimal light intensity for phytoplankton growth, G is the maximum growth rate of phytoplankton at 20C and G is the growth temperature coecient. In order to reduce the complexity in formulation, the grazing rate by zooplankton is assumed to be relative small in the considered region within the study period and therefore excluded in this study. The death of phytoplankton depends on the respiration rate at which the phytoplankton is oxidizing their organic carbon and it is temperature dependent given as 1

(

)

1

D = kD DT ? ; (

20)

(

20)

(2.10)

where kD is the phytoplankton respiration coecient at 20 C and D is the phytoplankton mortality temperature coecient. The mentioned growth kinetic are adopted from various reference [7, 8, 9] for the computations.

2.4 Nutrients Transformation Rate Model Basic researches on the nitrogen and phosphorus cycle in estuaries have already been conducted intensively. The process of nitri cation and denitri cation ux depend on both oxygen concentration and temperature. The particulate organic matters which settle to the bottom sediment will re-suspend to the water column through the denitri cation and mineralization processes. Due to the complexity of the biological and chemical interaction process in the nutrients regeneration cycle, we only consider two forms of nutrients compounds, namely the dissolved inorganic nitrogen and dissolved inorganic phosphorus absorbed by phytoplankton which are the most important limiting factors for algae growth. Dead phytoplankton cells also release a fraction of organic material which are then converted to dissolved inorganic nutrients through mineralization. By adopting WASP5 water quality model [7] 6

developed by the US Environmental Protection Agency, the process of nutrients transformation is described mathematically as follows: Inorganic Phosphorus ! @C ' K P T ? SP = @t = D(') P (1 ? RP ) ' ? G(') P ' + P P hP + ' CP ;(2.11) (

20)

~

Organic Phosphorus

SK P~

@C = @tP = D(') P RP ' ? P PT ? ~

(

!

' hP + ' CP ? VP CP ;

20)

~

(2.12)

~

Ammonia Nitrogen K = @CNH = D(')  (1 ? R ) ' ? G(')  F SNH N N N NH ' ? KNH CNH + @t ! (2.13) N NT ? h '+ ' CN ; 4

4

4

(

4

4

20)

~

N

Nitrate Nitrogen K = @CNO = K SNO NH CNH ? G(') N (1 ? FNH ) ' ? KNO CNO ; (2.14) @t 3

4

3

4

4

3

3

Organic Nitrogen !

@C SNK = @tN = D(') N RN ' ? N NT ? h '+ ' CN ? VN CN : (2.15) N Following the idea of WASP5, FNH stands for the ammonia preference factor, which can be described mathematically as " # C h C NH N NO FNH = (h + C ) (h + C ) + (C + C ) ; (2.16) N NO N NH NO NH ~

(

20)

~

~

~

4

4

3

4

3

4

3

4

where CP , CNH , CNO , CN and CP are respectively the concentration of the inorganic phosphorus; ammonia nitrogen; nitrate nitrogen; organic nitrogen and organic K , S K , S K are the nutrients kinetic transformation rate phosphorus, SPK , SPK , SNH NO N which will be input into the mass transport equation, ' is the population of phytoplankton biomass, N is the nitrogen to carbon ratio, P is the phosphorus to carbon ratio, RN is the fraction of dead phytoplankton recycled to organic nitrogen, RP is the fraction of dead phytoplankton recycled to organic phosphorus, both RN and RP are assumed to be 0.5, N is the organic nitrogen mineralization coecient 7 4

3

~

~

~

4

3

~

Topological Data, Wind Data, Tide Data

Hydrodynamic

Nutrients Source Loading

u, v

Submodel

u, v

Algae Biomass Transport Submodel

Nutrients Mass Transport Submodel

Nutrients Transformation Rate

N, P G, D

ϕ

G, D

Eutrophication Submodel

Temperature

Solar Radiation

Nutrients Transformation Submodel

ϕ

Figure 1: The System of the Water Quality Modelling Showing the Interrelation Between submodels at 20C , P is the organic phosphorus mineralization coecient at 20 C , N and P are the temperature coecient for nitrogen and phosphorus respectively, hN is the half saturation constant for phytoplankton nitrogen recycle to the water pool, hP is the half saturation constant for phytoplankton phosphorus recycle to the water pool, KNH is the nitri cation rate, KNO is the denitri cation rate, VN , VP are the settling rate of the nitrogen and phosphorus. The transformation ratios and parameters used in the computational models are taken from reference [7, 8, 9]. 4

3

2.5 Advective Transport Plankton Biomass model The aggregation of algal patch often brings together by the water velocity and tidal action, this results in intensive phytoplankton concentrations in dierent locations of a marine region, the blooms of phytoplankton which in turn trigger o red tide will then occur coincide with a optimal growth condition. This model is used to predict the concentrations of plankton biomass at any speci c grid points of the study region in terms of concentration of chlorophyll-a. The model puts together the advective transport forces from the neighbouring grid points due to water currents and the phytoplankton growth dynamic equation (2.8). The process is given mathematically 8

by the following dierential equation @' + u @' + v @' ? (G ? D ? V ) ' = 0: (2.17) ' @t @x @y where the death rate D and growth rate G are the results of the Equations (2.9) and (2.10). The above ve submodels which are described in Section 2.1 to 2.5 formed a system of equations and their interrelationships are illustrated in Figure 1.

3 Numerical Scheme using Multiquadric Method Multiquadric method was rst developed by Hardy [11] to approximate two dimensional geographical surfaces. It had been modi ed by Kansa [12] and [13] recently for solving physical equations over irregular regions. The method has its advantages of higher stability, mesh free for implementing and allowing high exibility in the approximation of irregular coastline. On the basis of its excellent computational performance, it was employed for numerical solutions of the dierential equations derived from the model in this study. The idea of MQ method is to interpolate an unknown function f (P ) where P 2 Rn at N distinct xed points Pj 2 Rn by the following radial basis functions:

f (P ) ' where

N X j =1

j qj (P )

(3.18)

qj (P ) = (k P ? Pj k +r ) = (3.19) with the free parameter r > 0 referred as the shape parameter because the basis functions qj change its shape by varying the magnitude of r. The performance of MQ depends heavily on the choice of a user-input parameter r. In our computations, the model adopted and modi ed the suggestion from Hardy [11] by choosing r to be 0:815 dmin , where dmin is the minimum distance between any two interpolation points in the region. k P ? Pj k is the Euclidean distance and j is a set of unknown coecients. In two dimensional problem, the basis function qj (P ) becomes 2 1 2

qj (xi; yi) = ((xi ? xj ) + (yi ? yj ) + r ) = : 2

2

2 1 2

(3.20)

To solve the time-dependent numerical computations of equations (2.1), (2.2), (2.3), (2.4), (2.9) and (2.17), the numerical discretization for the time derivative of these 9

equations are derived using central nite dierence method and the corresponding spatial derivative are solved by the Multiquadric scheme. The resulting equations are written as follows: ( )n @uH @vH n n ?  =  ? 2
t @x + @y ; (3.21) +1

un+1

vn+1

C n+1

1

= un?1 ? 2
t

= vn?1 ? 2
t

@u + v @u + g @ + g u u + v u @x @y @x HC ) n  a ?fv ? H Cs WxWs ; w 2

2

2

p

(

(3.22)

@v + v @v + g @ + g v u + v u @x @y @y HC ) n  a +fu ? Hw CsWy Ws ; 2

2

2

(

"

(3.23) #

@C ? @ D @C + D @C u @C + v xy @x" @y @x #xx @x @y) @ D @C + D @C + hS B + S k i n ; ? @y yx yy @x @y

= C n?1 ? 2
t

'n+1

p

(

= 'n?1 ? 2
t

(

(3.24)

n @' @' +u @x + v @y ? (G ? D ? V' ) ' ; )

(3.25)

where
t is the time step and  n , un , vn ,C n and 'n denote the (n +1)th iterates solution at the time t+
t. Applying the Multiquadric approximation scheme, we chose N data points (xj ; yj )Nj within the domain of which (xj ; yj )Nj are on the water boundary; (xj ; yj )Nj N are on the land boundary and (xj ; yj )Nj N are interior points. To illustrate the application of the Multiquadric method to the model, the numerical computation of the mass transport equation (3.24) was demonstrated as an example. We rstly determine the values of C and C by using the central nite dierence techniques, then apply MQ method to interpolate the concentration C n (x; y) by assuming 10 +1

+1

+1

+1

+1

1

=1

=1

2

=

1 +1

=

1

+2

2

2 +1

C n (x; y) = +2

N X j =1

nj qj (x; y):

(3.26)

+1

Collocating at the same points (xi; yi)Ni , for i = 1; 2; : : : ; N , we obtain, =1

C n (xi; yi) = +1

N X j =1

nj qj (xi; yi);

(3.27)

+1

which gives the following system of linear equations A~ = C~ ;

(3.28)

i h where A = [Aij ] = [qj (xi; yi)] is a N  N matrix; ~ = nj and C~ = [C n (xi; yi)] are N  1 matrices. Micchelli [14] has proven that in the case of interpolation, the matrix A is always conditionally positive de nite for distinct points (xi; yi), and hence the matrix A is invertible. Although this condition is still needed to be further veri ed, one can argue that the collocation method for solving PDEs is a linear combination of the interpolants and partial derivatives, and that matrix should also be conditionally positive de nite, hence invertible for distinct points. The unknown coecients nj can now be determined by using Gaussian elimination. The partial derivatives of C n (xi; yi) can then be approximated by +1

+1

+1

+1

@C n (xi; yi) = @x +1

@C n (xi; yi) = @y +1

@ C n (xi; yi) = @x 2

+1

2

@ C n (xi; yi) = @y 2

+1

2

N X j =1 N X j =1 N X j =1 N X j =1

11

x i ; yi ) ; nj @qj (@x

(3.29)

x i ; yi ) ; nj @qj (@y

(3.30)

nj @ qj@(xxi; yi) ;

(3.31)

2 n +1 @ qj (xi ; yi ) j @ 2y ;

(3.32)

+1

+1

+1

2

2

N @ C n (xi; yi) = X n @ qj (xi ; yi) ;  (3.33) j @x@y @x@y j where the rst and second partial derivatives of qj with respect to x, and y are given respectively as follows: @qj = (x ? x ) q? (x ; y ); (3.34) i j j i i @x 2

+1

+1

2

=1

1

@qj = (y ? y ) q? (x ; y ); i j j i i @y

(3.35)

1

@ qj = h1 ? (x ? x ) q? (x ; y )i q? (x ; y ); i j j i i j i i @x

(3.36)

@ qj = h1 ? (y ? y ) q? (x ; y )i q? (x ; y ); i j j i i j i i @y

(3.37)

2

2

2

1

2

2

2

2

1

2

@ qj = ?(x ? x )(y ? y ) q? (x ; y ): (3.38) i j i j j i i @x@y The next iterates C n can be determined by substituting the partial derivatives (3.29) to (3.33) into the Equation (3.24) together with the given boundary conditions. Apply the similar approach as described above, the numerical computations for the derivatives of current velocities u; v, the surface elevation  , the phytoplankton biomass ' can be determined by letting 2

3

+2

un (xi; yi) = +1

vn (xi; yi) = +1

 n (xi; yi) = +1

'n (xi; yi) = +1

N X

j =1 N X j =1 N X j =1 N X j =1

jn qj (xi; yi);

(3.39)

jn qj (xi; yi);

(3.40)

jn qj (xi; yi);

(3.41)

nj qj (xi; yi):

(3.42)

+1

+1

+1

+1

For boundary conditions, the estimated surface elevation  n on water boundary and the current velocities un , vn on land boundary are computed as  n (xi; yi; t) =  (t + TCORi) + HCORi (3.43) 12 +1

+1

+1

+1

for i = 1; 2;


; N ; (xi; yi) sin(i) cos(i) + 1; N + 2;


; N ; (xi; yi) sin(i) cos(i) + 1; N + 2;


; N ; 1

un (xi; yi) = u~n (xi; yi) sin (i) ? v~n for i = N vn (xi; yi) = v~n (xi; yi) cos (i) ? u~n for i = N +1

+1

2

+1

1

+1

+1

2

+1

1

1

(3.44)

2

1

(3.45)

2

where  (xi; yi) is the actual tide data measured at tide gauge, TCORi is the time correction parameter, HCORi is the tide level correction parameter, u~n (xi; yi) and v~n (xi; yi) are the values computed from the Multiquadric interpolant and i is the outward normal angles at the land boundary points which are computed by taking the average of the vectors joining the neighbouring points. For the nutrients transport equation, the derivative normal of land boundary condition is given as ! ~ !n ~ !n @C n = @C @C sin (i) ? @y sin(i) cos(i) (3.46) @xi @xi i for i = 1; 2;


; N ; +1

+1

+1

+1

+1

2

1

~ !n ~ !n @C @C = @x cos (i) ? @y sin(i) cos(i) (3.47) i i for i = 1; 2;


; N ;  n n  ~ ~ @C @C where the partial derivatives @x and @y are the values approximated i i from the multiquadric interpolant. The source loading rate S B (xi; yi; t) is the measured values from rivers, streams and sewage at a set of given points. The rst order kinetic transformation of nutrients S k (t) at time t which are described in equations (2.11) to (2.15) can also be computed simultaneously using Multiquadric method.

@C @yi

!n+1

+1

+1

2

1

+1

+1

4 Numerical Computations We verify the proposed model by computing the water currents, tide level, distribution of nutrients concentrations and algal biomass re ecting the chlorophyll-a concentration in the Tolo Harbour for a period between 6 February 1991 to 30 April 1991. Tolo Harbour is situated in the north-east New Territories of Hong Kong. The embayment of the Tolo Harbour consists of a shallow inner basin (Inner Tolo Harbour) and a narrow and deep Channel Tolo which extends into Mars Bay (Outer 13

Tolo Harbour) in the South China Sea as shown in Figure 2. The embayment occupies an area of 50 km and is 16 km long. The width of the embayment varies from 5 km in the inner basin to just over 1 km at the mouth of the Harbour. The Inner Tolo Harbour is shallow and is less than 10 m deep, while the Tolo channel is more than 20 m deep. The location of the Harbour is situated approximately 114 100 to 114 270 longitude and 22220 to 22 320 latitude. The tide in Tolo Harbour is a mixed semi-diurnal type with a tidal period of 24.5 hours. The overall range of the tidal level is around 0:1 m to 2:7 m. The measurement of the current ows is recorded as an average 10 cm=sec in the channel of the harbour. 2

30 1 : Lam Tsuen River 2 : Tai Po River 3 : Tai Po Kau Stream 4 : Shing Mun River 5 : Tung Tze Stream 6 : Shan Liu Stream

25

20

e LO

6 EL

5 15

2

c

d

CH

INNER TOLO a

10

5

5

Tide Gauge a : Tai Po Kau b : Ko Lau Wan

Station c : TM3 d : TM6 e : TM9

4

0 0

b

N AN

1 3

TO

10

15

20

25

30

35

40

45

Figure 2: A map of Tolo Harbour of Hong Kong showing the locations of the major rivers and streams, two tide gauges and three water sampling stations

The main source of pollutants loading input into the model included three major rivers, three streams and two sewage treatment works along the coastal line of the Inner Harbour as indicated in Figure (2). Tolo Harbour is one of the marine water control zones in the coastal region of Hong Kong. The Hong Kong government have set up a marine water monitoring programme since 1986, nine water quality moni14

toring stations were designated within the harbour. Three of these stations namely TM3, TM6 and TM9 are selected showing in Figure (2). The water quality data are the biweekly measurement which are collected at each of the monitoring stations by the Environmental Protection Department of Hong Kong. The measurement was carried out in the morning around 10:00am to 12:00am, water sample was taken from three dierent layers. The surface layers is 1m below the water surface; the middle layer and the bottom layer is 1m above the bottom of the sea. The tide and wind stress data to be input into the hydrodynamic model were obtained from the Royal Observatory of Hong Kong. Both tide and wind data were the average hourly monitoring results measured at two tide gauges which are located at Ko Lau Wan in the Channel Tolo Harbour and Tai Po Kau near in the Inner Tolo Harbour. The discharge of daily sewage be used for the loading source was provided by the Drainage Services of the Hong Kong Government. In order to enhance the physical understanding of the environmental conditions and the marine pollution behaviour, the most important environmental variables and the collected water quality data at each sampling stations are analyzed and compared statistically. The analysed results of two most important environmental variables which are the water temperature and solar radiation are displayed in Figure 3. For the season of spring, the variation of water temperature often shows an Weekly Maximum

Weekly Mean

25

5

Bottom Layer

Surface Layer

Middle Layer

Middle Layer

Bottom Layer

Surface Layer

10

Middle Layer

10

15 Bottom Layer

15

20

Surface Layer

Water Temperature oC

20

5

84

77

70

63

56

49

42

35

28

21

7

0 14

Daily Solar Radiation (MJ/m2)

Weekly Minimum

25

0 Feb 91

Days

Mar 91

Apr 91

Figure 3: Monthly Average Water Temperature and Weekly Range of Daily Solar Radiation during the Computational periods

insigni cant dierence between the surface and bottom layers as can be veri ed by the Figure 3. The temperature range from 17 C in February to 23 C in April with a gentle vertical temperature gradients. The mean values of the chlorophyll-a level in the Tolo Harbour varies from 6.4 g=M in the surface layer to 1.6 g=M in the middle layer during the simulation period. The concentration of chlorophyll-a tends 15

5.8 5.6 5.4 5.2 5 4.8 4.6 4.4 4.2 4 3.8 2

2.5

3

3.5

4

4.5

5

Figure 4: Structure of Multiquadric Interpolation points for the Tolo Harbour of Hong Kong to be higher in the surface water layer due to the phytoplankton grow primarily at the top few meters below the water surface. The data also shows that an increase in surface chlorophyll-a level between February to April coincide with the increase in temperature and radiation. Lam and Ho [15] reported that the phytoplankton species of Tolo Harbour are dominated by diatoms and dino agellates. There was no de ned seasonal bloom observed due to the temporal variations of both diatoms and dino agellates uctuating irregularly during the year. There was no strati cation appeared due to the well vertical mixing condition along the water column during spring. The warm and well mixed water temperature support the growth of phytoplankton, and therefore a drastic algal blooms often occurred during spring time. The formation of the multiple functional phytoplankton groups and the temperature strati cation are not considered in the model in spite of their signi cant impact on the ecological process. A depth averaged 2-dimensional model is assumed to be feasible for the predictions for the present study. In the computation, the entire Harbour region is interpolated by 245 interpolation points as shown in Figure 4. Numerical computations are carried out using FORTRAN executed on a Sun Sparc workstation with time step of 30 seconds. The 16

output results are taken in every 30 minutes interval. The results have demonstrated that the Multiquadric Method can eectively eliminate the unexpected negative predictions in regions with high gradient of nutrient concentration, in particular the areas close or opposite to the pollutant discharge points. The simulated results of the boundary points are also shown to be relatively stable, while the nite dierence and nite element methods often encounter unexpected negative prediction in the water areas with complex boundary conditions and source loading points. 260

Station: Ko Lau Wan 240

Tidal Level (m)

220

200

180

160

Predicted Tidal Level

140

Observed Tidal Level

82

73

64

55

Days

46

37

28

19

10

1

120

Figure 5: Comparsion of the High Tide level at Ko Lau Wan in Tolo Harbour between 6 Feb 91 to 30 April 91

The simulated result from the hydrodynamics model is veri ed using the the tide data which was measured at the two tide gauges in Tolo Harbour. The comparison of the high tide levels at Ko Lau Wan is depicted in Figures 5. The mass transport model produced the spatial and temporal variations of the total inorganic nitrogen(NH4+NO3); inorganic phosphorus(PO4) and the surface chlorophyll-a representing the algal biomass. These three water quality parameter are considered to be the primary indicators of eutrophication in the waters of Tolo Harbour. The weekly ranges of the simulated inorganic nitrogen, PO4 and surface chlorophyll-a level are shown in Figures 6 and 7. For each sampling station, the available eld data during the simulation period were indicated in the gures for comparison. 17

0.1

Computed Weekly Max. Values Computed Weekly Min. Values Field Data at Sampling Station TM3

PO4 Concentration (mg/L)

0.3

0.25

0.2

Computed Weekly Max. Value Computed Weekly Min. Value Field Data at Sampling Station TM3

0.09 0.08 0.07 0.06

Days

80

71

63

54

0.095 Field Data at Sampling Station TM6

PO4 Concentration (mg/L)

Field Data at Sampling Station TM6

0.3 0.25 0.2 0.15 0.1

0.09 0.085 0.08 0.075

80

71

63

54

45

36

28

19

10

1

80

71

63

54

45

36

28

19

10

0.07 1

0.05

Days

Days 0.05

0.015 Field Data at Sampling Satation TM9

Field Data at Sampling Station TM9

PO4 Concentration (mg/L)

0.04

0.03

0.02

0.01

0.012 0.009 0.006 0.003

Days

80

71

63

54

45

36

28

19

10

80

71

63

54

45

36

28

19

10

1

0 1

Total Inorganic Nitrogen Concentration (mg/L)

45

Days

0.35 Total Inorganic Nitrogen Concentration (mg/L)

36

28

1

80

71

63

54

45

36

28

19

10

1

19

0.05

0.15

10

Total Inorganic Nitrogen Concentration (mg/L)

0.35

Days

Figure 6: Comparison of the Weekly Range of the Simulated Total Inorganic Nitrogen and

Inorganic Phosphorus Variation Compared to the Field data at Three Sampling Stations of the Tolo Harbour During the Periods 6 Feb 91 to 30 April 91

18

0.002 Computed Weekly Max. Value Computed Weekly Min. Vlaue Field Data at Sampling Station TM6

Chlorophyll-a Concentration (mg/L)

0.006

0.004

0.002

Field Data at Sampling Station TM9

0.0016 0.0012 0.0008 0.0004

80

71

63

54

45

36

28

19

80

71

63

54

45

36

28

19

10

1

10

0

0

1

Chlorophyll-a Concentration (mg/L)

0.008

Days

Days

Figure 7: Comparison of the Weekly Range of the Simulated Surface Chlorophll-a level to the

Field Data at Three Sampling Stations of the Tolo Harbour During the Periods 6 Feb 91 to 30 April 91

5 Conclusions and Discussions A system of mathematical models was used to simulate the water quality and the eutrophication process for marine water was constructed. Computational scheme using a new kind of numerical method, the Multiquadric method, was employed to carry out the computations of the model. The model was applied to simulate the nutrients and phytoplankton concentration variations in the Tolo Harbour of Hong Kong for a period of three months from 6 February 91 to 30 April 91. The simulated results of tide level is highly agreed with those actual data measured at the tide gauge. The eld data which were measured biweekly at each sampleing station were used to verify the accuracy of the simulated results. The scarcity of eld data in open sea is one of diculties for model veri cation, the Reasonable agreement between the two sets of data can be observed. In this respect, we feel con dent that the model and the related computational scheme can serve as a useful prototype predictive tool in water quality management and control. In order for the model to be developed into a full-scale tool for use in reallife prediction, a number of enhancements and modi cations have to be made. It is argued that inputs relevant to phytoplankton growth dynamic cannot be speci ed on too ne a time scale (minutes-to-minute or even hours-to-hours) and phytoplankton population does not signi cantly change from hour to hour [16]. Also parameters of the growth dynamic model which represent biochemical and biological interaction may vary with time or seasons. Accordingly, a steady-state model may not be 19

appropriate when the simulation period is long enough that time eects on the parameters cannot be ignored. Hence, a time-variable model with sucient long simulation period, say, over one year, will de nitely give more realistic prediction results. The present study considered only a 2-dimensional depth-averaged model. Variations in dierent layers of marine water were averaged and were taken to be constant. In most marine waters system, eutrophication problem is often associated with temperature and nutrients interactions. Both water quality parameters and algae growth dynamics will vary signi cantly on dierent layers of water in parallel with the change in temperature. This phenomenon, known as strati cation, will occur in summer. In this respect, a 3-dimensional or layered model should be used to give better numerical prediction, providing the eld data are available which is dicult. Sediment processes may have profound eects on nutrients concentrations and subsequently will aect the algae growth mechanism. In marine water systems, the impact of sediment nutrients release can be signi cant and result in continuing eutrophication problems even after source loading have substantially reduced through control measures [16]. Although the processes are complicated, the eect of sediment on the eutrophication should be incorporated into the model. For the Multiquadric Method that we employed for the numerical computations, it has been shown that the method exhibit superiority over the traditional nite dierence and nite element method in that it performs better in terms of accuracy and stability. It also has the de nite advantage of avoiding element meshing which may become tedious in handling irregular coastline. The computations in this method involves the handling of full coecient matrix which is computationally intensive and will hinder the application of the method to solve large scale problems. Fortunately these problems have been solved by using The employment advanced numerical techniques, such as Laplace Transform [17] and domain decomposition [18]. Moreover, the domain decomposition method can easily be parallelized which enables the solving of large scale problem. The possible outbreak of red tide does not depend solely on the changes of the nutrients level. The force of meteological conditions may have a more signi cant impact on the bloom of phytoplankton. The above-mentioned full-scale model should take into consideration the seasonal eects on dierent species of phytoplankton. 20

Acknowledgment This research is supported by the Research and Develop-

ment Fund of the Open Learning Institute of Hong Kong and the Strategic Grant No. 7000473 of the City University of Hong Kong. The authors wish to thank the Environmental Protection Department, the Royal Observatory and the Drainages Service Department of Hong Kong Government for providing data for numerical computation.

References [1] K. C. Ho and I. J. Hodgkiss, \Macro nutrients in Tolo Harbour and their Relations to Red Tides," EMECS Newsletter, pp. 10-11, 1992. [2] \Marine Water Quality in Hong Kong", Report No. EP/TR2/92, Monitoring Section, Water Policy Group, Environment Protection Department, Hong Kong Government, 1993. [3] Y. C. Hon, \Typhoon surge in the Tolo Harbour of Hong Kong - An approach using Finite Element Method with quadrilateral elements and parallel processing technique," Chinese Journal of Numerical Mathematics and Applications, Vol. 15:4, pp. 21-33, 1993. [4] Roger A. Falconer, \Research developments of ow and water quality modelling in coastal and estuarine waters," Water Quality Modelling, pp. 81-95, 1993. [5] K. C. Ho and I. J. Hodgkiss, \Assessing the limiting factors of red tide by bottle bioassay," Asian Marine Biology, Vol. 10, pp. 77-94, 1993. [6] D. M. DiToro, \Estimate phytoplankton biomass models - veri cation analyzes and preliminary applications," The sea, Marine Modelling, 6, E. D. Goldbery, ed., Wiley, New York, N.Y., pp. 969-1019, 1977. [7] The Environmental Protection Agency of the United States, \WASP5: Water Quality Analysis Simulation Program Modelling System, version 5.10", 1993. [8] J. H. W. Lee, R. S. S. Wu, Y. K. Cheung, \Forecasting dissolved oxygen in marine sh culture zone," Journal of Environmental Engineering, Vol. 117, No. 6, pp. 816-833, 1991. 21

[9] J. H. W. Lee, R. S. S. Wu, Y. K. Cheung, \Dissolved Oxygen Variations in marine Fish Culture Zone," Journal of Environmental Engineering, Vol. 117, No. 6, pp. 799-815, 1991. [10] Catherine W. Y. Lam, K. C. Ho, \Red tides in Tolo Harbour, Hong Kong." Elsevier Science Publicating Co., Inc., pp. 49-52, 1989. [11] R. L. Hardy, \Multiquadric equations of topography and other irregular surfaces", J. Geophys, Res. 176, pp. 1905-1915, 1971. [12] E. J. Kansa, \Multiquadric - a scattered data approximation scheme with applications to computational uid dynamics - I", Computers Math, Applic. Vol. 19, No. 8/9, pp. 127-145, 1990. [13] E. J. Kansa, \A strictly conservative spatial approximation scheme for the governing engineering and physics equations over irregular regions and inhomogeneously scattered nodes", Computers Math. Applic., Vol. 24, No. 5/6, 169-190, 1992. [14] Micchelli C.A. (1986), "Interpolation of scattering data: distance matrices and conditionally positive de nite functions", Constr. Approx. 2, 11-22. [15] Catherine W. Y. Lam, K. C. Ho, \Phytoplankton Characteristics of Tolo Harbour", Asian Marine Biology 6, pp. 5-18, 1989. [16] Wu-seng Lung (1993), \Water Quality Modelling", CRC Press. [17] G. J. Moridis, E. J. Kansa, \The Laplace Transform Multiquadric method: A highly accurate scheme for the numerical solution of linear partial dierential equations", Journal of Applied Science & Computation , Vol. 1, No. 2, 375-407, 1994. [18] Mark R. Dubal, \Domain decomposition and local re nement for multiquadric approximations. I: Second-order equations in one-dimension", Journal of Applied Science & Computation , Vol. 1, No. 1, 146-171, 1994.

22