
A common approach for simulating the fate and transport of MeBr for saturated and unsaturated water flow conditions, with consideration of variable soil temperature, includes descriptions for at least three governing processes: water flow, heat transport and fate and movement of MeBr. Programs exist that will numerically solve the nonlinear partial differential equations for one and twodimensional systems, nonequilibrium coupled transport of water, heat, and solute (in both liquid and gaseous phases) in a variably saturated porous medium. Degradation is usually described using a firstorder decay reaction and, often, the degradation rate in each phase (liquid, vapor and solid) can be specified. The governing transport equations can be written as (Šimůnek and van Genuchten, 1994): 

Water Transport 


where θ is the volumetric water content [L^{3}L^{3}], h is the pressure head [L] and K_{ij} are components of the unsaturated hydraulic conductivity tensor [L t^{1}], and S is a sink term [t^{1}]; t is time, x is distance [L], and indices i and j represent the horizontal and vertical directions. 

Heat Transport 


where C_{h} and C_{w} respectively, are the volumetric heat capacity for the porous media [Jm^{3} K^{1}], liquid and λ_{ij}, is the apparent thermal conductivity [Wm^{1}K^{1}]. 

Solute Transport 


where C_{L} [M L^{3}], C_{S} [M M^{1}], and C_{g} [M L^{3}] are solute concentrations for the liquid, solid, and gaseous phases, respectively; q is the volumetric flux density; μ_{w}, μ_{s}, and μ_{g} are firstorder rate constants [t^{1}] for solutes in the liquid, solid, and gas phases, respectively; θ is the volumetric water content, ρ is the soil bulk density, η is the soil air content, S is the sink term in the water flow equation [t^{1}], C_{r} is the concentration of the sink term, D_{ij}^{w} is the dispersion coefficient tensor for the liquid phase [L^{2} t^{1}], and D_{ij}^{g} is the diffusion coefficient tensor for the gas phase. 

Numerous computer programs have been developed to evaluate the effects of interacting processes and factors on pesticide movement through the root zone and to the groundwater. The approach used in developing the programs varies with the intended use of the model. Some of these include GLEAMS (Leonard et al., 1987); LEACHM (Wagenet and Hutson, 1987); PRZM (Carsel et al., 1985; 1998); PESTAN (Enfield et al., 1982); and SESOIL (Bonazountas and Wagner, 1984). Some of these models are not capable of predicting pesticide movement when water is applied in a controlled manner by furrow or subsurface drip irrigation systems. This has led to the development of processbased models which can be used to predict the transport in irrigated agriculture: CHAIN2D (Šimůnek and van Genuchten, 1994), HYDRUS2D (Šimůnek et al., 1996), and PESTLA (van den Berg and Boesten, 1997). 

Volatilization Boundary Condition 
For methyl bromide, critical processes affecting the fate and transport in soils are vapor diffusion and volatilization. Volatilization is an especially important route of dissipation due to MeBr's large vapor pressure and Henry's Law constant as demonstrated in recent field experiments (Yagi et al., 1995; Majewski et al., 1995; Yates et al., 1996b). Excessive volatilization is associated with many problems, such as a reduction in the amount of material available to control of pests and increased potential for contamination of the atmosphere. Emission losses to the atmosphere pose an increased risk to persons living near treated fields. When simulating MeBr emissions to the atmosphere, the approach used to describe the soil surfaceatmospheric boundary condition strongly affects the simulated emission response. 

The most common volatilization boundary condition used in current models is based on stagnant boundary layer theory (Jury et al., 1983). For this formulation, the volatilization rate, f_{s}(t), is 


where C_{L} and C_{g}, respectively, are the liquidphase and gasphase concentrations (M L^{3}) at the soil surface; C_{atm} is the concentration in the atmosphere above the boundary layer; D_{E} is the effective soil diffusion coefficient (L^{2}/t); q is the Darcian water flux (L/t); h is the mass transfer coefficient (L/t); D_{g}^{air} is the methyl bromide diffusion in air; and d is the boundary layer thickness (L). This approach assumes that a thin stagnant air layer occurs at the soilatmosphere interface and chemical movement across the layer is due to vapor diffusion. The controlling parameter is the mass transfer coefficient, which is expressed as the ratio of the binary diffusion coefficient (i.e., air and MeBr) to the boundary layer thickness (Jury et al., 1983). A limitation of this approach is the estimation of the thickness of the stagnant boundary layer. Further, for some atmospheric conditions (e.g., changes in barometric pressure), it is likely that chemical transport occurs by both advection and diffusion. For these situations, assuming a stagnant boundary layer is inappropriate and more complex boundary conditions are required (Massmann and Farrier, 1992; Chen et al., 1995). An advantage of the stagnant boundary layer approach is that information about atmospheric conditions is unnecessary. However, adopting this boundary condition produces MeBr emission histories that are very regular and often do not resemble the erratic behavior commonly observed in the field (Majewski et al., 1995; Yates et al., 1996b; Yates et al., 1997). 

A more accurate description of the volatilization process requires the coupling of soilbased processes with those operating in the atmosphere. This has led Baker et al. (1996) to develop an alternate formulation for the masstransfer coefficient that includes atmospheric resistance terms. 



where R_{e} and S_{c}, respectively, are the roughness Reynolds and Schmidt numbers, u* is the friction velocity (L t^{1}), U_{r} is the wind speed at the measurement height (L t^{1}) and Φ_{m} is an atmospheric stability correction. This boundary condition depends on several aerodynamic parameters, such as the roughness Reynolds number, the Schmidt number, the friction velocity, the wind speed and an atmospheric stability term. The atmospheric resistance to diffusion near the soil surface and aerodynamic resistance from the diffusive layer to the measurement height affects the predicted emission rate. Further research is needed to evaluate the effectiveness of this approach in simulating the volatilization boundary condition, especially for agricultural fumigation. 

Several studies have been conducted to determine whether conventional modeling approaches can accurately predict the rate of MeBr volatilization from bare soils (Wang et al., 1997b; Yates et al., 2002). Wang et al. (1997b) used CHAIN2D to simulate methyl bromide emissions from a 3.5 ha field and compared the simulation results to emissions measured in a field experiment (Yates et al., 1996c). They found that the model simulated the total emission within a few percent of the measured value but the pattern of instantaneous emission rate was much more regular than the measured values and, at times, a value of the predicted volatilization rate could be very different from the measured value. Yates et al. (2002) conducted a similar study using the same experimental data and the volatilization boundary condition of Baker et al. (1996). They found that the predicted emissions had a more realistic temporal pattern compared a simulation based on the stagnant boundary layer. The total emissions were also within a few percent of the measured value. 

When discrepancies occur, it cannot be determined whether the model or the measured values were in error since the measured volatilization rate is also subject to uncertainty (Majewski, 1997). Further research is needed to improve the accuracy of volatilization measurements and simulation models. Research is also needed to develop and test methods for coupling atmospheric and soil processes in models so that more accurate predictions of the volatilization rate can be obtained. 

