A novel computational framework is introduced for the efficient simulation of chemical and biological tracers in ocean models. The framework is based on the "transport matrix'' formulation, a scheme for capturing the complex three-dimensional transport of tracers in a general circulation model (GCM) as a sparse matrix, thus reducing the task of simulating tracers to a sequence of simple matrix-vector products. The principal advantages of this formulation are efficiency and convenience. It is many orders of magnitude more efficient than GCMs, allowing us to address problems that are currently either difficult or unaffordable with GCMs. The scheme also allows us to quickly "prototype'' new biogeochemical parameterizations or "plug in'' existing ones. This paper describes the key features and advantages of the transport matrix method, and illustrates its application to a series of realistic problems in chemical and biological oceanography. The examples range from simulation of a transient tracer (SF6) to adjoint sensitivity of a complex coupled biogeochemical model. Finally, the paper describes an efficient, portable, and freely available implementation of this computational scheme that provides the necessary framework for simulating any biogeochemical tracer.
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