Large-scale free-surface ocean models designed to run over climatic time scales are required to globally conserve the volume and any tracer up to machine precision. In addition, local consistency is critical and requires that the discrete tracer equation preserve constants in a closed domain and if there is no tracer source or sink. Local consistency, together with monotonicity, will ensure that no spurious tracer extrema occur. A three-dimensional, finite-element, shallow-water model is presented. The mesh is unstructured in the horizontal, extruded in the third dimension, and made up of multiple layers of prisms. In addition, the mesh is allowed to move in the vertical and adapts itself to the free-surface motions. It is shown that achieving consistency requires a discrete compatibility between the tracer and continuity equations. In addition, to ensure global tracer conservation in a consistent way, a discrete compatibility between the tracer, continuity, and free-surface elevation equations must be fulfilled. It is suggested that this compatibility constraint, together with the use of a numerically stable scheme, severely restricts the choice of usable finite-element spatial discretizations. A consistent and conservative time-stepping algorithm is described for which a unique time step is used. It is suggested that future research is needed in order to design a consistent and conservative split-explicit algorithm. Some illustrative test cases are presented in which the method is shown to satisfy all conservation properties. A few experiments in which consistency breaks down are carried out, and the consequences of this breakdown process are investigated. |
