SIMULATION OF OBSERVED COASTAL DISPERSION USING ACCELERATED FBM PARTICLE TRACKING METHOD

Bo Qu¹ , Paul S. Addision² and Christopher T. Mead³

¹Civil Engineering Department, Hong Kong University, Hong Kong, China

E-mail:quba@hkucc.hku.hk.

²The School of the Built Environment, Napier University, UK.

E-mail: p.addison@napier.ac.uk

³HR Wallingford Ltd. Howbery Park, Wallingford, UK.

E-mail: ctm@hrwallingford.co.uk

* Corresponding Author: Tel:(852)28578470, Fax: (852) 2559 5337.

E-mail: quba@hkucc.hku.hk.  

Abstract:In recent year, the particle tracking technique has received more and more attention in modelling pollutant dispersion in fluids. The most popular modelling technique is traditional particle tracking method that only could model Fickian diffusion cloud. However, in oceanic surface, non-Fickian diffusion dispersion is observed by some researchers. A new particle tracking technique has been studied and used in practice (Ref[2]-[9]). However, recently by using real data in coastal water, the authors found in coastal water or oceanic surface, the Hurst exponent H is not restricted within the range 0 ~ 1. A new particle tracking method that generates non-Fickina diffusion by employing accelerated fractional Brownian motion (AFBM) has been studied and introduced in this paper. The work based on the idea suggested by Sanderson and Booth ([16]). This can produce superdiffusive behaviour for H > 1.0. The numerical techniques to model the non-Fickian spreading observed in the real coastal data has been attempted successfully and the simulation results are encouraging comparing with the observed data and Wallingford simulation results. 

Keywords: FBM-fractional Brownian motion, non-Fickian, AFBM- Accelerated fractional Brownian motion

1    INTRODUCTION

In the past, Particle Tracking models presented in the literature employ random Brownian motion to simulate turbulent diffusion. Such models assume that particle tracks are neutrally persistent, i.e. the particle executes a simple random walk. However, some researchers ([14], [15], [16]) strongly indicate that particle movements in turbulent fluids are persistent, where the Lagrangian memory of the particle plays an important role in indicating the future direction of the particle. To simulate persistent motion, one must resort to the engineering field of fractal statistics. It was found by Mandelbrot ([13]) that fractional Brownian motion (fBm) is a random fractal function that can simulate persistent motion. A new accelerated fBm that based on original fBm will be introduced in this paper. 

2    FRACTIONAL BROWNIAN MOTION (FBM) AND ACCELERATED FBM (AFBM)

Brownian motion B(t) can be expressed as a continuous-time random function, which is the integral of a Gaussian white noise W(s),

              B(t) =                            (1)

The random variables W(s) are uncorrelated and have the same Gaussian distribution with zero mean and unit standard deviation. However, a simpler distribution, constant distribution or delta function, can also be used in practice ([1])

The generation of fBm is not as simple as generating Brownian motion, because an fBm trace does not take statistically independent steps (as Brownian motion does), but rather each point on an fBm trace depends upon the whole of the history of the fBm previous to that point. In other words, an fBm has a long-term memory associated with it. Mandelbrot and Van Ness ([13]) defined the random function B (t) with zero mean roughly as a moving average of dB (t), in which past increments of B (t) are weighted by the kernel (t-s) , as

                                (2)

Here (x) is gamma function, H is the Hurst exponent of the trace. This definition states that the value of the random function at time t depends on all previous increments dB(s) at time s < t of a Gaussian random process B(t) with average zero and unit variance.

The authors have proposed a more practical fractional Brownian motion model ([1]-[9]) that developed from the original definition of Mandelbrot and Van Ness ([13]). It is

                  (3)

                         (4)

Equation (3) and (4) form the two-step fBm model developed by the authors ([1]). Here  is the i’th discrete approximation to the fBm at time  and  is the discrete time step used. M is a limited memory used in the approximation of the fBm. R(i) are random steps discretely sampled from a Gaussian probability distribution.

Osborn et al (1989) and Sanderson & Booth (1991) have found that the trajectories of satellite tracked ocean surface drifters may be described as fractional Brownian motion with non-Fickian scaling properties. The Hurst exponent H is around 0.8 as average value for the trajectories of satellite-tracked ocean surface drifters ([14], [15], [16]).

The standard deviation of a diffusing cloud of fBm particles follows following relationship:

                                             (5)

Note that when H = 1/2, (5) becomes regular Brownian motion.

Figure 1 shows the difference between Brownian motion and fractional Brownian motion trajectories in two dimensions. Notice that the space filling properties of the particle paths reduce with increasing H. The fractal dimension is

                                  (6)

Which indicate the ability of the fBm trajectory to fill up the space. Hence, 1< .

3    FBM PARTICLE TRACKING MODEL

The pollutant dispersion in fluids is determined by both advection and diffusion. The total particle displacement at time step i is the summation of the advective component and the diffusive component (i.e. the increment of fBm) over the time interval:

                                              (7)

                                               (8)

The diffusion displacements of each time step in both x and y diections are  and , they are two independent fBm increments defined by (4).

The location of the particle at the next time step (i+1) is then,

                                       (9)

                                       (10)

In a large water body, such as open sea, the Hurst exponent of trajectory of two particles even reach a value higher than unity (Ref [16]). An accelerated fBm model (AFBM) is suggested by Sanderson and Booth ([15]). They suggested that a modified fractional Brownian motion model could still be used. The only difference is that the time needs to be rescaled. The rescaling of time produces trajectories on the x, y plane that still have scaling parameter H (herein still set at H=0.8). However, the relative speed along trajectory is no longer stationary, rather it is accelerated. The mean square particle separation will grow as

                                             (11)

Where B > 0, when the time rescaled, the correspondent parameters (such as memory, diffusion coefficient, time intervals, etc) will also be rescaled (see [1] for the details).

The new AFBM model is the extension of fBm model. While B = 0, the AFBM model becomes the standard fBm model.

4    SIMULATION THE OBSERVED COASTAL DISPERSION

During 1995 and 1996, several dye released near the Northumbrian coast were carried out on behalf of Northumbrian Water Ltd in UK at various locations. HR Wallingford’s midfield water quality model (Ref [18]) was calibrated using the observations of the resulting dye patches (Figure 2, two dye patches B and C only shown here). The model simulates turbulent dispersion using a random walk technique.

The authors attempted to reproduce spatial concentration data sets from the contour data supplied by Northumbrian Water. The Hurst exponent calculated is in the range of 0.56~1.2326 (for the two data sets B and C only). The AFBM particle tracking models need to be used. Figure 3 contain plots of the simulation results for the particle clouds of the two data sets B, C, 2000 particles are shown in the plots.

Figure 4 contains the contour plots for the author’s simulation of the two dye patches (B, C) using the AFBM model. 4000 particles are used in the calculation. The authors compared their simulation results with observed data sets in two ways: One was to compare their standard deviations (see Table 1) and another was to compare the isoconcentration contour areas (Table 2). Okubo (Ref [14]) pointed out that the standard deviation (or variance) is one of the most stable parameters to characterise diffusion. He also stated that the concentration in a patch of dye may not be a good measure of diffusion due to the sensitivity of the peak concentration to the decay of the dye, hence causing greater uncertainty. Hence, the standard deviation results are listed in Table 1 for comparisons. The ratios for standard deviation between the simulated and observed results ( and ) are calculated in the table. The ratios fell in the range 0.56-1.85 with most reasonably close to unity. This is a good result considering that a single H value was used over the whole time scale of the diffusing patches.

Table 1    Comparison of the simulation results with the observed data Sets for standard devistions in u and v directions (for B and   C patch)

Table 2    Area comparison (between fickian and AFBM) within each contour level

(for concentration value c), where concentration unit is /Litre and the area unit is m².

The concentration area inside each contour level was also calculated (not show) in order to compare with the ratios obtained from the original HR Wallingford model. It is noticeable that some results of the simulation using the authors’ AFBM models are closer to the observed data, i.e. the ratio is closer to unity, especially for data set B. Some of the authors’ simulation results tend to over predict lower concentration levels while some of the simulation results from HR Wallingford tend to under predict the same concentration levels. It is difficult to give a general conclusion for which is better and which is poorer. Within a dye patch, an over predicted lower concentration level will cause its higher concentration level to be under predicted, due to the nature of the decay from a dye patch. Hence, the ratios of the area are not very useful in providing an obvious comparison of the spreading between model and reality.

5    CONCLUSION

A new accelerated fBm Particle Tracking model (AFBM) has been described for the generation of non-Fickian diffusion to predict particle cloud dispersion using observed coastal data. The simulation method for the real data using AFBM (fBm as a special case of AFBM) was approached and simulated particle clouds and concentration plots were generated. It was found that the simulated patches (by the authors) spread out slightly more than the original observed dye patches (see Table1 and 2). It is encouraging that the authors’ results are reasonable close to the observed data.

The authors wish to thank Northumbrian Water Ltd, for permission to use the data sets and simulation results used in this paper.

References

[1]    Qu Bo (1999). The Use of Fractional Brownian Motion in the Modelling of the Dispersion of Contaminants in Fluids’, PhD thesis, Napier University, UK.

[2]    Addison P.S. & Qu B. (1996). Modelling Fractal Diffusion on the Ocean Surface.  The 6th International Symposium on Flow Modelling and Turbulence Measurements. 1996, September 8-10. Tallahassee, Florida-U.S.A. in 'Flow Modelling and Turbulence  Measurements VI', Balkema, Rotterdam, p703-709.

[3]    Addison P.S. & Qu B., Nisbet A. and Pender G. (1997). A Non-Fickian, Particle-Tracking Diffusion Model Based on Fractional Brownian Motion.  International Journal for Numerical Methods in Fluids, Vol. 25, p1373-1384.

[4]    Addison P. S., Qu B., Ndumu A. S. and Pyrah I. C. (1998). A Particle Tracking Model for Non-Fickian Subsurface Diffusion. Mathematical Geology,  Vol. 30, No. 6, p695-716.

[5]    Addison, P.S., Qu B., Pender G. and Sloan S. (1998). Coastal Transport Modelling Using Fractal Geometry. Engineering Mechanics: A Force for the 21th Century. proceedings of the 12th Engineering Mechanics Conference. la Jolla, California, H. Murakami and J. E. Luco (Editors), @ASCE, reston, VA. p1673-1676.

[6]    Addison P. S., Qu B., Nisbet A. and Pender G. (1997). Modelling Contaminant Spread on the Ocean Surface and Within Soils Using fBm's: Two Civil Engineering Applications. Fractals in Engineering, Springer, p375-382.

[7]    Addison P. S. & Qu B. (1997). Non-Fickian Random Walk Diffusion Models. Environmental and Coastal Hydraulics: Protecting the Aquatic Habitat. ASCE, XXVII IAHR Congress, San Francisco, Vol. 1, p45-50.

[8]    Addison P. S., Qu B., Mead C. D. and Pender G. (1999). Non-Fickian Dispersion in Coastal Water. 13th ASCE Eng. Mech. Conference, Baltimore, USA, June 13-16 (in CD-Rom).

[9]    Addison P. S., Ndumu A. S. and Qu B. (2000). A Fast Non-Fickian Particle Tracking Diffusion Simulator and the Effect of Shear on the Pollutant Diffusion Process. International Journal for Numerical Methods in Fluids. Vol.34, pp 145-166.

[10]    Chatwin P. C. and Allen C. M. (1985). Mathematical Models of Dispersion in Rivers     and Estuaries. Annual Review of Fluid Mechanics, Vol. 17, p119-149.

[11]    Feder J. (1988). Fractals. Plenum Press. New York and London.

[12]    Hunter J. R., Craig P. D. and Phillips H. E. (1993). On the Use of Random Walk Models with Spatially Variable Diffusivity. Journal of Computational Physics, Vol. 106. p366-376.

[13]    Mandelbrot B. B. and Van Ness J. W. (1968). Fractional Brownian Motions, Fractal Noises and Applications. SIAM review, Vol. 10, No 4, p422-437.

[14]    Okubo A. (1971). Oceanic Diffusion Diagrams. Deep-sea Research, Vol. 18, p789-802.

[15]    Osborne A. R. , Kirwan A. D., Provenzale A. and Bergamasco L. (1989). Fractal Drifter Trajectories in the Kuroshio Extension. Tellus, Vol. 41A,  p416-435.

[16]     Sanderson B. G. and Booth D. A. (1991). The Fractal Dimension of Drifter Trajectories and Estimates of Horizontal Eddy-Diffusivity. Tellus, Vol. 43A, p334-349.

[17]    Smith R. W. (1989). Review of Recent Developments in Mixing and Dispersion. Hydraulic and Environmental Modelling of Coastal, Estuarine and River Waters. Proceedings of the International Conference held at the University of Bradford, p277-290.

[18]    HR Wallingford (1996). Northumbrian Coastal Modelling System––Theoretical Reference Manual. Report EX 3358. p8-14, p39-44.

  (1) Sub-diffusive fBm H = 0.2     (2) Brownian Motion H = 0.5    (3)  Super-diffusive fBm H = 0.8

Fig. 1    Comparisons between brownian motion (H=0.5) and fractional brownian motion (H=0.2 for sub-diffusion, H=0.8 for super-diffusion) trajectories in two dimensions.

Fig. 2    Observed dye patches B, C – by northumbrian water ltd.

Fig. 3    Simulation results for a cloud of 2000 particles spreading

Fig. 4    Simulation results for the data sets B and C concentration contour plots (for 4000 particles) –by the authors