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We consider the tracer equations that are part of the primitive equations used in ocean modeling. These equations describe the transport of tracers, such as temperature, salinity or chemicals, in the ocean. Depending on the number of tracers considered, several equations may be added to and coupled to the dynamics system. In many relevant situations, the time-step requirements of explicit methods caused by the transport and mixing in the vertical direction are more restrictive than the ones for the horizontal. We propose an exponential time differencing (ETD) solver where the vertical transport is treated with a matrix exponential, whereas the horizontal is dealt with in an explicit way. We investigate numerically the accuracy and computational speed-ups that can be obtained over an explicit or a fully exponential method. Results for the complete set of primitive equations are also shown, where a tracer system is coupled to the dynamics. Here, the tracer and dynamics equations are decoupled in each time-step, and a second-order ETD solver is employed for the dynamics system.
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