Scaling of a system for budgets

One of the important considerations in picking a system to budget is the issue of scale: What are the spatial boundaries of the system to be budgeted? There are some interesting tradeoffs here. Recall that the reason to develop these materials is to calculate fluxes of carbon, nitrogen, and phosphorus (CNP) which do not simply follow the flow of water through the system. These fluxes are said to be "nonconservative" with respect to the water and salt budgets. In general, large systems are more likely to show nonconservative fluxes of CNP than are small systems. However, the larger the system, the less the ability to resolve what components of the system account for these nonconservative changes. Thus, it is fairly easy to obtain a budget for nonconservative CNP fluxes for the whole ocean: As long as we assume that the masses of CNP in ocean water are not changing, all of those materials added to the system must either be transferred to the sediments or transferred to the atmosphere. However, this scale of analysis does not tell us where, within the entire ocean, the reactions occur to remove these materials from the water. For smaller systems such as individual bays and estuaries, there are likely to be hydrographic fluxes which add and remove materials. These hydrographic fluxes complicate the calculation of nonconservative fluxes. Whatever nonconservative fluxes are calculated to occur, however, can be attributed to processes within those bays and estuaries. As the scale gets smaller—perhaps individual shoal areas within the bay—hydrographic fluxes are likely to become so dominant that nonconservative fluxes cannot be calculated.

We find it useful to approach the issue of scale from two points of view: The first and simplest point of view is that it "makes sense" to work with physiographically defined units. Whole estuaries are examples of such units; sometimes there are obvious breaks such as individual basins within an estuary. Relatively enclosed bays may be physiographically obvious units to budget; bays which are little more than slight coastal indentations may not be. Individual shelf seas (such as the North Sea or the East China Sea) are physiographically relatively easily defined and should be amenable to budgeting; stretches of open shelf (such as much of the Atlantic coast of the Americas) may be more challenging. In general, it makes sense to budget a whole system first, and then consider the possibility of budgeting smaller portions of the system.

The second point of view for approaching budgets is that if water exchange is to be estimated by a salt and water budget, there must be a difference in salinity between the system and the oceanic water with which the system changes. Such a salinity difference is fundamental to using the salt and water budget. Alternative methods of estimating nonconservative material fluxes include measurement of water composition of an isolated water mass over time (for example, in some sort of incubation chamber); following a water mass through time with dyes, again, measuring water composition changes; either direct physical observations or numerical models of water flow to calculate water residence time. These methods are discussed in Gordon et al. (1996) and elsewhere.

The next consideration is that the water has to stay in the system long enough to have its water composition potentially modified by net biogeochemical reactions. This criterion is at once tricky and important. In general, systems with water residence time of weeks to months are easier to budget than those with residence times of minutes to hours. But we can refine this somewhat, based on the following reasoning. Material turnover rate within a system is defined as the absolute value of the sum of rates adding material to the system divided by the mass of material in the system (typically the concentration multiplied by the water volume). We can discuss turnover rates due to only some things. For example, the turnover rate of Y due to inputs is |S Inputs_Y/Mass_Y|; that due to the hydrographic outputs is |S Outputs_Y/Mass_Y|. Turnover due to internal reactions (|D Y|) may be expressed either of two ways: either |S Inputs_Y-S Outputs_Y)/Mass_Y|, or |D Y|/Mass_Y. One can then usefully look at scaling ratios; a useful ratio is the ratio of reaction turnover to input turnover. Note that the mass of material drops out of this ratio: |D Y/S Inputs_Y|.

So one approach to choosing systems of sufficient scale to be able to measure D Y becomes choosing systems in which D Y is reasonably large relative to S Inputs_Y. We suggest that a good rule of thumb is that this ratio needs to be at least 0.25. We assume, for this analysis, that a local estimate can be made for S Inputs_Y. These will in general be local forcing related to freshwater inflow (both river flow and groundwater; in the case of N, possibly precipitation) and inputs perhaps associated with little freshwater flow (e.g., sewage). Dry deposition may also be important, especially for N.

Some guesses can be made for the rates for the D Y’s. Primary production is a piece of information which is often available for many ecosystems and which can provide some scaling insight. The first part of the scaling insight is to convert the rates (typically expressed per unit area or per unit volume) to rates over the entire system of interest. Thus, for example:

If local data are not available, then the following range of primary production rates is probably not a bad guess: Low nutrient water such as the oligotrophic open ocean typically has primary production rates of about 10 mol C m^-2 yr^-2 (i.e., around 120 g C m^-2 yr^-2). More fertile systems such as inshore waters may have rates of 3 or more times this rate. The budgeting approach described in these web pages results in estimates of net production (or production - respiration), not primary production (i.e., in (p-r), not p). This quantity can be either positive (a "net autotrophic system") or negative ("net heterotrophic system") but usually |(p-r)| is less than 10% of p. It follows that:

This value, in carbon units expressed for the whole system, can be scaled to nitrogen or phosphorus units. For this analysis, see the discussion of (p-r) in the stoichiometry section. In addition, net denitrification (or, in some cases, nitrogen fixation) may be important. A conservative estimate for the absolute difference between these two rates (i.e., |(nfix-denit)|) is likely to be at least 0.15 mol N m^-2 yr^-1, if the rate is important in the system (also see stoichiometry); again, this needs to be scaled as above, back to the size of the system.

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Last Updated 12 May 2009 by FW