One-dimensional, Mass Conservative, Spatially- Dependent Transport Equation: New Analytical Solution

The transport equation (ADE) is one of the pivotal equations in atmospheric sciences and surface/subsurface water quality models. Since analytical methods are at the heart of the verification process in geophysical and environmental fluid mechanics, several analytical solutions have been already derived for this equation. Those previous exact solutions mostly refer to the local mass conservation of a constituent (in an infinitesimal volume); however, there is a need of verifying models which use integrated versions of the ADE, and which at the same time satisfy the integrated mass conservation for water. This work derives a new analytical solution for the cross-sectional-integrated ADE. Starting from the analytical solution by Ogata and Banks, we use the Method of Undetermined Coefficients, and modify the ADE so that its coefficients are varying with space while the mass continuity of water is preserved. At the end, we solve a hypothetical problem numerically and we use the new analytical solution for numerical verification. The new analytical solution offers a better probing capability within existing benchmarks for verification of transport solvers.