<?xml version="1.0" encoding="utf-8" standalone="no"?>
<!DOCTYPE article SYSTEM "http://www.ocean-sci.net/inc/os/copernicus.dtd">
<article language="en">
	<journal>
		<journal_title>Ocean Science</journal_title>
		<journal_url>www.ocean-sci.net</journal_url>
		<issn>1812-0784</issn>
		<eissn>1812-0792</eissn>
		<volume_number>3</volume_number>
		<issue_number>4</issue_number>
		<publication_year>2007</publication_year>
	</journal>
	<doi>10.5194/os-3-525-2007</doi>
	<article_url>http://www.ocean-sci.net/3/525/2007/</article_url>
	<abstract_html>http://www.ocean-sci.net/3/525/2007/os-3-525-2007.html</abstract_html>
	<fulltext_pdf>http://www.ocean-sci.net/3/525/2007/os-3-525-2007.pdf</fulltext_pdf>
	<start_page>525</start_page>
	<end_page>535</end_page>
	<publication_date>2007-12-20</publication_date>
	<article_title content_type="html">The backward Îto  method for the Lagrangian simulation of transport processes with large space variations of the diffusivity</article_title>
	<authors>
		<author numeration="1" affiliations="1">
			<name>D. Spivakovskaya</name>
			<email>d.spivakovskaya@ewi.tudelft.nl</email>
		</author>
		<author numeration="2" affiliations="1">
			<name>A. W. Heemink</name>
		</author>
		<author numeration="3" affiliations="2">
			<name>E. Deleersnijder</name>
		</author>
	</authors>
	<affiliations>
		<affiliation numeration="1" content_type="html">Department of Mathematical Physics, Delft Institute of Applied Mathematics (DIAM), Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands</affiliation>
		<affiliation numeration="2" content_type="html">Université catholique de Louvain, G. Lemaitre Institute of Astronomy and Geophysics (ASTR) &amp; Centre for Systems Engineering and Applied Mechanics (CESAME), Avenue G. Lemaitre 4, 1348 Louvain-la-Neuve, Belgium</affiliation>
	</affiliations>
	<abstract content_type="html">Random walk models are a powerful tool for the investigation of transport
processes in turbulent flows. However, standard random walk methods are
applicable only when the flow velocities and diffusivity are sufficiently
smooth functions. In practice there are some regions where the rapid but
continuous change in diffusivity may be represented by a discontinuity. The
random walk model based on backward Îto calculus can be used for these
problems. This model was proposed by LaBolle et al. (2000). The latter is best suited to the problems
under consideration. It is then applied to two test cases with discontinuous
diffusivity, highlighting the advantages of this method.</abstract>
	<references>
		<reference numeration="1" content_type="text"> Bj\oork, T.: Arbitrage Theory in Continuous Time, Oxford University Press, 1998. </reference>
		<reference numeration="2" content_type="text"> Bolin, B. and Rodhe, H.: A note on concepts of age distribution and transit time in natural reservoirs, Tellus, 25, 58&amp;ndash;62, 1973. </reference>
		<reference numeration="3" content_type="text"> Braunschweig, F., Martins, F., Chambel, P., and Neves, R.: A methodology to estimate renewal time scales in estuaries: the Tagus Estuary case, Ocean Dynam., 53, 137&amp;ndash;145,2003. </reference>
		<reference numeration="4" content_type="text"> Celia, M A., Russell, T F., Herrera, I., and Ewing, R E.: An Eulerian-Lagrangian localized adjoint method for an advection-diffusion equation, Adv. Water Resour., 13(4), 187&amp;ndash;206, 1990. </reference>
		<reference numeration="5" content_type="text"> Costa, M. and Ferreira, J S.: Discrete particle distribution model for advection-diffusion transport, J. Hydraul. Eng., 126(7), 525&amp;ndash;532, 2000. </reference>
		<reference numeration="6" content_type="text">Deleersnijder E., Beckers J M., and Delhez E J M.: The residence time of settling in the surface mixed layer, Environ. Fluid Mech., 6(1), 25&amp;ndash;42, 2006a. </reference>
		<reference numeration="7" content_type="text">Deleersnijder E., Beckers J M., and Delhez E J M.: On the behavior of the residence time at the bottom of the mixed layer, Environ. Fluid Mech., 6, 541&amp;ndash;547, 2006b. </reference>
		<reference numeration="8" content_type="text"> Delhez, E J M., Heemink, A W., and Deleersnijder, E.: Residence time in a semi-enclosed domain from the solution of an adjoint problem, Estuarine, Coastal and Shelf Science, 61, 691&amp;ndash;702, 2004. </reference>
		<reference numeration="9" content_type="text"> Delhez, E J M. and Deleersnijder, E.: The boundary layer of the residence time field, Ocean Dynam., 56, 139&amp;ndash;150, 2006. </reference>
		<reference numeration="10" content_type="text"> Dimou, K N. and Adams, E E.: A random-walk, particles tracking models for well-mixed estuaries and coastal waters, Estuarine, Coastal and Shelf Science, 37, 99&amp;ndash;110, 1993. </reference>
		<reference numeration="11" content_type="text"> Heemink, A W.: Stochastic modeling of dispersion in shallow water, Stochastic Hydrol. Hydraul., 4, 161&amp;ndash;174, 1990. </reference>
		<reference numeration="12" content_type="text"> Hunter, J R.: The application of Lagrangian particle-tracking technique to modelling of dispersion in the sea, in: Numerical Modelling: Applications to Marine Systems, edited by: Noye, J., 257&amp;ndash;269, 1987. </reference>
		<reference numeration="13" content_type="text"> Hunter, J R., Craig P D., and Phillips H E.: On the use of random walk models with spatially variable diffusivity, J. Comput. Phys., 106, 366&amp;ndash;376, 1993. </reference>
		<reference numeration="14" content_type="text"> Karatzas, I. and Shreve, S. E.: Brownian motion and stochastic calculus, Springer, New York, 1998. </reference>
		<reference numeration="15" content_type="text"> Kloeden, P E. and Platen, E.: Numerical Solution of Stochastic Differential Equations, Springer, New York, 1999. </reference>
		<reference numeration="16" content_type="text"> LaBolle, E M., Fogg, G E., and Tompson, A F B.: Random-walk simulation of transport in heterogeneous porous media: Local mass-conservation problem and implementation methods, Water Resour. Res., 32(3), 583&amp;ndash;593, 1996. </reference>
		<reference numeration="17" content_type="text"> Konikow, L F. and Bredehoeft, J D.: Computer model of two-dimensional solute transport and dispersion in ground water, U.S. Geological survey water-resources investigations book 7, Chapter C2, 1978. </reference>
		<reference numeration="18" content_type="text"> LaBolle, E M., Quastel, J., Fogg, G E., and Granver, J.: Diffusion processes in composite porous media and their numerical integration by random walks: Generalized stochastic differential equations with discontinuous coefficients, Water Resour. Res., 36(3), 651&amp;ndash;662, 2000. </reference>
		<reference numeration="19" content_type="text"> Oksendal, B.: Stochastic differential equations, Springer, New-York, 1985. </reference>
		<reference numeration="20" content_type="text"> Proehl, J A., Lynch, D E., McGillicuddy Jr., D J., and Ledwell, J R.: Modelling turbulent dispersion of the North Flank of Georges Bank using Lagrangian Methods, Cont. Shelf Res., 25, 875&amp;ndash;900, 2005. </reference>
		<reference numeration="21" content_type="text"> Ross, O N. and Sharples, J.: Recipe for 1-D Lagrangian particle tracking models in space-varying diffusivity, Limnol. Oceanogr.: Methods 2, 289&amp;ndash;302, 2004. </reference>
		<reference numeration="22" content_type="text"> Sawford, B L: Recent developments in the Lagrangian stochastic theory of turbulent dispersion, Bound.-Lay. Meteorol., 62, 197&amp;ndash;215, 1993. </reference>
		<reference numeration="23" content_type="text"> Spivakovskaya, D., Heemink, A W., Milstein, G N., and Schoenmakers, J G M.: Simulation of the transport of particles in coastal waters using forward and reverse time diffusion, Adv. Water Resour., 28, 927&amp;ndash;938, 2005. </reference>
		<reference numeration="24" content_type="text"> Spivakovskaya, D., Deleersnijder, E., and Heemink, A W.: Lagrangian modelling of multi-dimensional advection-diffusion with space-varying diffusivities: theory and idealized test cases, Ocean Dynam., 57, 189&amp;ndash;203, 2007. </reference>
		<reference numeration="25" content_type="text"> Stijnen, J W., Heemink A W., and Lin, H X.: An efficient 3D particle model for use in stratified flow, Int. J. Numer. Meth. Fluids 51, 331&amp;ndash;350, 2006. </reference>
		<reference numeration="26" content_type="text"> Takeoka, H.: Fundamental concepts of exchange and transport time scales in a coastal sea, Cont. Shelf Res., 3, 311&amp;ndash;326, 1984. </reference>
		<reference numeration="27" content_type="text"> Thomson, D J.: Criteria for the selection of stochastic models of particles trajectories in turbulent flow, J. Fluid Mech., 180, 529&amp;ndash;556, 1987. </reference>
		<reference numeration="28" content_type="text"> Thomson, D J., Physick, W L., and Maryon, R H.: Treatment of Interfaces in Random Walk Dispersion Models, J. Appl. Meteorol., 36, 1284&amp;ndash;1295, 1997. </reference>
		<reference numeration="29" content_type="text"> Visser, A W.: Using random walk models to simulate the vertical distribution of particles in a turbulent water column, Mar. Ecol. Prog. Ser., 158, 275&amp;ndash;281, 1997. </reference>
		<reference numeration="30" content_type="text"> Yeh, G T.: A Lagrangian-Eulerian method with zoomable hidden fine-mesh approach to solving advection-dispersion equations, Water Resour. Res., 26(6), 1133&amp;ndash;1144, 1990. </reference>
		<reference numeration="31" content_type="text"> Zhang, R., Huang, K., and van Geruchten, M T.: An efficient Eulerian-Lagrangian method for solving solute transport problems in steady and transient flow fields, Water Resour. Res., 28(12), 4131&amp;ndash;4138, 1993. </reference>
		<reference numeration="32" content_type="text"> Zheng, C. and Bennett, G D.: Applied contaminant transport modeling, Wiley, 2002. </reference>
		<reference numeration="33" content_type="text"> Zheng, C. and Wang, P P.: MT3DMS: A Modular Three-Dimensional Multispecies Transport Model for Simulation of Advection, Dispersion and Chemical Reactions of Contaminants in Groundwater Systems; Documentation and User&apos;s Guide, Contract Report SERDP-99-1, U.S. Army Engineer Research and Development Center, Vicksburg, MS, 1999. </reference>
		<reference numeration="34" content_type="text"> Zimmerman, J F T.: Mixing and flushing of tidal embrayments in the Western Dutch Wadden Sea, Part I: distribution of salinity and calculation of mixing time scales, Neth. J. Sea Res., 10, 149&amp;ndash;191, 1976. </reference>
		<reference numeration="35" content_type="text"> Zimmerman, J F T.: Estuarine residence time, in: Hydrodynamics of estuaries, edited by: Kjerve, B., 1, 75&amp;ndash;84, CRC Press, Boca Raton, FL, 1988 </reference>
		<reference numeration="36" content_type="text"> Zimmermann, S., Koumoutsakos, P., and Kinzelbach, W.: Simulation of pollutant transport using a particle method, J. Comput. Phys., 173(1), 322&amp;ndash;347, 2001. </reference>
	</references>
</article>

