To efficiently simulate the advection-diffusion processes along and across density surfaces, we need to deal with a diffusivity tensor containing off-diagonal elements (Redi, J Phys Oceanogr, 12:1154-1158, 1982. In the present paper, the Lagrangian model, in case of a space-varying diffusivity tensor, is developed. This random walk model is applied for two idealized test cases for which the analytical solutions are known. Results of the testing show that the Lagrangian approach provides accurate and effective solutions of advection-diffusion problems for general diffusivity tensor.

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Abstract. 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 for two test cases with discontinuous diffusivity, highlighting the advantages of this method.

Abstract. 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 backwardÎto 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.

In this paper we study the mean and standard deviation of concentrations using random walk models. Two-particle models that take into account the space correlation of the turbulence are introduced and some properties of the distribution of the particle concentration are studied. In order to reduce the CPU time of the calculation a new estimator based on reverse time diffusion is applied. This estimator has been recently introduced by Milstein et al. (Bernoulli 10(2):281-312, 2004). Some numerical aspects of the implementation are discussed for relatively simple test problems. Finally, a realistic application to predict the spreading of the pollutant in the Dutch coastal zone is described.