What controls the mixed-layer depth in deep-sea sediments? The importance of particulate organic carbon flux

Smith and Rabouille (2002; hereafter SR02) have examined the relationships between the mixed-layer depth in deep-sea sediments, L, as based on 210Pb and 14C profiles, and a set of environmental variables that include particulate organic carbon (POC) flux, oxygen penetration depth, and oceanographic depth. They have established the existence of a strong nonlinear correlation between their operationally defined mixed-layer depth, L, and the POC flux. Such a correlation is useful for predictive purposes, potentially suggests a cause and effect relationship, and satisfies the expectations of benthic biologists and geochemists, including this commentator. However, SR02 further attempted an evaluation of the model offered by Boudreau (1998; hereafter B98) for the relationship between the mean mixed-layer depth, the intensity of bioturbation, and the reactivity of the organic matter; in this latter matter the analysis in SR02 is in error, a judgment shared by two independent reviewers of this comment and the senior author of SR02. It is the aim of this note to explain that error and illustrate the correct implementation of the resource-feedback model contained in B98. In addition, this comment unreservedly acknowledges a notational shortcoming in B98 that contributed significantly to SR02’s misinterpretation of B98; specifically, B98 used the symbol L for both the site-specific and mean mixed-layer depths, with the result that SR02 used formulas for the calculation of site-specific mixed depths, L, that are only valid for the (interenvironmental) mean mixed depth, L̄. Before the discussion can begin, we need a means to avoid confusion about the numbering of equations, since equations appear in SR02, B98, and this comment. Thus, equations found in B98 will be designated Eq. B98 #, those referred to in SR02 will be noted as Eq. SR02 #, and equations in this comment will simply be given as Eq. #; Table 1 summarizes the equivalences between equations in the different papers. The resource-feedback model addresses the prevalent criticism that the amount (G) and lability (k) of the infaunal food resource and the intensity (DB) and the site-specific depth (L) of mixing in the widely used constant-coefficient diffusion model of bioturbation (largely a construct of geochemists) are all independent (unrelated) quantities. B98 argued that this independence was unlikely and that the intensity of mixing, as characterized by DB, should be a function of the amount of food at a given depth, at least for some types of mixing. In the absence of any data, B98 assumed a linear function between mixing intensity and the food resource, DB(x) 5 D*(G) (1)