One-Dimensional Lagrangian Models

Our previous discussion regarding the treatment of uneven terrain and boundary conditions is directly applicable to one-dimensional (ID) atmospheric models. For example, it is often difficult to specify appropriate initial conditions for a one-dimensional column model. In this modeling framework the characteristic species decay time is much longer than the residence time xr, because the only loss mechanism of inert chemical species is dry deposition [see (25.24) and (25.25)]. The characteristic time for a Lagrangian model can then be defined as... 

Natural attenuation is controlled by numerous processes, which include sorption, intraparticle diffusion as weU as biological and chemical degradation. In order to be able to quantify respectively predict the fate and transport of contaminants, appropriate models that are able to deal with the complexity and interactions of the involved processes need to be developed. Due to insufficient information on the spatial distribution of transport parameters in the subsurface, stochastic methods are a preferred alternative to deterministic approaches. In the present paper a one-dimensional Lagrangian streamtube model is used to describe the reactive transport of acenaphthene as a sample organic compoimd at field scale. As the streamtube model does not consider the heterogeneity of hydrogeochemical parameters but only hydraubc heterogeneity, model results from the streamtube model are compared in a Monte Carlo approach to results of a two-dimensional Eulerian model. 

In order to simulate the SDEs, we will introduce a large ensemble of notional particles that move through the simulation domain according to the Lagrangian PDF models. As an example, we will consider a single inert-scalar field in a one-dimensional domain. The position and composition of the th notional particle can be denoted by X n t) and 4i(n)(f), respectively. The SDEs for the Lagrangian composition PDF (with closures) become... 

Ma.s,small, W. J. and Weil, 1. C. (1999). An analytical one-dimensional second-order closure model of turbulence statistics and the Lagrangian time scale within and above plant canopies of arbitrary structure. Botindary-Layer Meteorol. 91, 81-107. 

The model is established on a mixture of Eulerian and Lagrangian coordinate systems. The river is approximated as a series of completely mixed cells (typically 10-1,000 m in length) fixed in position, as shown in Fig. 20.1. The slick is approximated as a series of completely mixed cells that move across the water surface in a Lagrangian coordinate system. This treatment of the slick as a series of moving cells allows for spatial variation in the concentration of the slick. The application of the model, per se, is to situations where both flow and slick can be described as one-dimensional. This occurs when the slick is spread completely across the river, as with relatively narrow streams. The length of the river from the spill site before the one-dimensional assumption can be applied is approximately ... 

The acoustic response of resonant viscoelastic fluid structures to a pressure wave may be simulated by a four-dimensional calculation, three dimensions in space and one in time. The Lagrangian, primitive finite element and Eulerian finite difference schemes form the basis for two models presented in this paper which are able to simulate a wide range of fluid structures containing inclusions of arbitrary spacing, shape and composition. 

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