Table 2-1: Characteristic Data for the Ocean2-1: Dynamic Box Model1Chapter 2: Mass Balance - The Cornerstone of Chemical Oceanography(10/15/01)The chemical distributions on the earth and in the ocean reflect transport andtransformation processes, many of which are cyclic. The cycling of water from the oceanto the atmosphere to land and back to the ocean via rivers is such an example. This basiccycling is often described in terms of the content of the various reservoirs (e.g., the ocean,the atmosphere, etc.) and the fluxes between them (e.g., evaporation, rivers, etc.). Afundamental question is how the rates of transfer between the reservoirs depends on thecontent of the reservoirs and on other external factors. The details of the distributionswithin the reservoirs is neglected.Most oceanographers construct simple models to test their understanding of the essentialelements of the system and to predict the response of a system to perturbations andforcings. The principal of Ockham's Razor has served oceanographers well. Thisprinciple states that: When seeking to explain phenomena, start with the simplest theory(see Safire, 1999, for the etymology).A few words about the scientific method. Some fields of science are advancing morerapidly than others. It is the contention of Platt (1964) that those rapidly advancing fieldsare those where the method of doing research called "strong inference" is systematicallyused and taught. Strong inference consists of formally and explicitly applying thefollowing steps to every problem.1) Devising alternative hypotheses2) Devising a crucial experiment (or experiments) with alternative possible outcomeseach of which will, as nearly as possible, exclude one of more of the hypotheses3) Carrying out the experiment so as to get a clean result.1') Recycling the procedure to refine the hypotheses that remain.Scientific hypotheses are most securely "validated" when (i) they make successfulpredictions; (ii) there are conceivable observations that could, in principle, refute them,but have not; and (iii) there is a comparably sensible competitor theory that is faringworse. Developing alternative simple models is part of this process.The purpose of this chapter is to introduce the tools necessary to develop the two maintypes of models used. These are:.-Box (or reservoir) Modelsand-Continuous Transport-reaction ModelsFirst some basic definitions related to models in general and box or reservoir models.- Model - A simplified or idealized description of a particular system or process that isput forward as a basis for calculations, predictions or further investigation. A model2should contain only those elements of reality that are needed to solve the problem.The least necessary model is the best possible model for the purpose. A model is animitation of reality which stresses those aspects that are assumed to be important andomits all properties considered to be nonessential. A model is like a caricature of areal system.- Parameter - A quantity which is constant (as distinct from ordinary variables) in aparticular case considered, but which varies in different cases. An independentvariable in terms of which each co-ordinate of a point is expressed.- Variable - A quantity or force which, throughout a mathematical calculation orinvestigation, is assumed to vary or be capable of varying in value.- Closure - Closure in a modeling sense usually means having the number of unknownsequal the number of equations. Often, closure is achieved by making simplifyingassumptions.- Reservoir (M)(also box or compartment) - The amount of material contained by adefined physical regime, such as the atmosphere, the surface ocean or the lithosphere.The size of the reservoirs are determined by the scale of the analysis as well as thehomogeneity of the spatial distribution. The units are usually in mass of moles.- Flux (F) - The amount of material transferred from one reservoir to another per unittime.- Source (Q) - A flux of material into a reservoir.- Sink (S) - A flux of material out of a reservoir- Budget - A balance equation of all sources and sinks for a given reservoir.- Turnover Time (ττττ) - The ratio of the content (M) of a reservoir divided by the sum ofits sources (ΣQ) or sinks (ΣS). Thus τ = M/ΣQ or τ = M/ΣS.- Cycle - A system consisting of two or more connected reservoirs where a largefraction of the material is transferred through the system in a cyclic fashion. Budgetsand cycles can be considered over a wide range of spatial scales from local to global.- Steady State - When the sources and sinks are in balance and do not change withtime.- Closed System - When all the material cycles within the system- Open System - When material exchanges with outside the system.A. Mass Balance - Simple Box ModelsMany processes may act to control the distributions of chemicals in the ocean. Themethod of putting these processes together in a model utilizes the principle of massbalance applied to the system as a whole or some parts of it (control volumes). Controlvolumes or boxes are connected by internal transport processes such as advection anddiffusion. The system as a whole is linked to the environment by external inputs andoutputs. Box models are especially useful for understanding geochemical cycles and theirdynamic response to change.The framework of box models for geochemical cycles is conceptually similar to chemicalkinetic reactions and similar equations are used to describe the stability of chemical and3biochemical systems (Prigogine, 1967). The Lotka-Voltera preditor-prey model ofpopulation dynamics is a classic example of the such equations.Describing a model first requires choosing a system, that is, the division between what is"in" and what is "out". The second step involves choosing the complexity of thedescription of the "internal" system.The goal in modeling is to analyze all the relevant processes simultaneously. The conceptof mass balance serves as a way to link everything together. To use the idea of a massbalance, the system is first divided into one or several "control volumes" which areconnected with each other and the rest of the world by mass fluxes. A mass balanceequation is written for each control volume and each chemicalThe Change in = Sum of + Sum of Internal - Sum of - Sum of allMass with Time all Inputs Sources Outputs Internal SinksSuch box models are used to determine the rates