The diffusive regime of double-diffusive convection
Introduction
Of the two types of double-diffusive convection (DDC, henceforth) that may occur if large-scale gradients of salinity S and temperature T point in the same vertical direction, the salt-finger (SF, henceforth) mode has been studied more extensively than the diffusive-layer (DL, henceforth) mode. There appear to be three broad reasons for this focus, not all of which are compelling.
First, there are historical reasons. DDC research began with the SF mode, in the guise of the ‘perpetual salt fountain’ (Stommel, Arons, & Blanchard, 1956). This may explain why the seminal and defining paper of Stern (1960) relegated the DL mode to a footnote, and why followup studies dealt mainly with the SF mode.
Second, there are geographical reasons. SF is commonly found at locations that are easily accessible to research vessels, whereas DL is prevalent at high-latitude locations where the logistics of sampling are very demanding. Also, the geographical expanse of SF-susceptible waters exceeds that of DL-susceptible waters. This in itself says little, however, since the high-latitude DL zones may be especially important to the global climate system. Besides, DL may be important to interleaving across the globe.
Third, there are reasons that stem from the difficulty of extrapolating laboratory measurements to the ocean. In the laboratory, the SF mode is more vigorous than the DL mode, e.g. causing larger buoyancy fluxes across interfaces with analogous T and S steps. However, this implies nothing about the relative strength of SF and DL fluxes in the ocean, since the S and T steps need not be analogous, even if the background gradients are analogous. (Indeed, relating S and T steps to background gradients is a key goal discussed at some length below.) Another factor is the response of SF and DL to shear, which is common in the ocean but not in laboratory models. While SF transports are apparently inhibited by shear (Linden, 1974b, Kunze, 1994), it has been speculated that DL transports may not be (Padman, 1994). Given such things, it seems that the dominance of the SF mode in laboratory may not be relevant to the ocean.
Reasons such as these may explain the historical concentration on the SF mode, but they should not preclude future DL work. This case is underlined by the current interest in the role of high-latitude oceanography in the climate system (Walsh & Crane, 1992). A key climate component is watermass formation in northern seas, and DL fluxes may play an important role in this process (McDougall, 1983). Further evidence is provided by decades of observations of robust DL signatures in the Arctic, going back to Neal, Neshybya and Denner (1969). DL is also thought to be the main DDC driving agency of Arctic intrusions (May & Kelley, 2002). These intrusions, considered to be a key aspect of Arctic thermodynamics, have remarkably coherent intrusive signatures, traceable over basin scales (Perkin & Lewis, 1984) and perhaps over decades (cf. Carmack et al., 1997). They have also been implicated in the large changes currently taking place in the Arctic (Carmack, Macdonald, Perkin, McLauglin and Pearson, 1995a, Carmack, Aagaard, Swift, Perkin, McLauglin, Macdonald and Jones, 1995b). Taken together, these things suggest that DL may be an important element of the global climate system.
With this potential importance in mind, we turn next to a discussion of DL signatures in the world ocean. After presenting an overview of susceptible ocean regions (Section 2.1) and some thoughts on methods of detecting staircases (Section 2.2), we offer an abbreviated list of some prominent DL examples that have been well sampled (Section 2.3). Then, in the second half of the paper, we turn to matters of modelling DL. We start with the issue of parameterizing fluxes between a pair of DL layers in the laboratory (Section 3.1) and then move on to the related, but more difficult, matter of parameterizing fluxes in oceanic DL staircases (Section 3.2). This foundation is used for a cursory outline of some unresolved questions regarding oceanic staircase formation (Section 3.3) and evolution (Section 3.4).
Our focus is entirely on the DL case, and mostly on the staircase mode. Readers with wider interests may consult other contributions to this volume, along with various reviews (Turner, 1974, Huppert and Turner, 1981, Turner, 1985, Schmitt, 1994, Fernando and Brandt, 1994), and themed collections (Brandt & Fernando, 1995), as well as recent GCM sensitivity studies exploring the role of DDC under a range of surface boundary conditions (Zhang, Schmitt and Huang, 1998, Merryfield, Holloway and Gargett, 1999, Zhang and Schmitt, 2000). Although aspects of the DL mode appear to be simpler than those of the SF mode, it is not clear whether this is because the DL mode is actually simpler or because the DL has been so little studied that contradictory information has not been uncovered. We take up such issues at the end.
Section snippets
Susceptible regions
Laboratory studies indicate that DDC produces coherent structures, e.g. staircases and intrusions (together with mixed modes) that may be recognized from examination of the horizontal and vertical variation of S and T. Unfortunately, the signatures have length scales (meters to tens of meters) that are below the resolution of ocean atlases. Therefore, we cannot easily construct maps of global staircase occurrence, and are left to estimate the world-wide prevalence of DDC by mapping
Fluxes between two layers
Early DDC laboratory work with thin interfaces between well-mixed layers suggested that vertical heat fluxes are determined not by the details of the interfaces, but rather by the temperature and salinity differences, δT and δS, between the two layers (Turner, 1965). (For brevity, we focus here on the heat flux. Salt fluxes are typically assumed to be given by the product of the heat flux and a dimensionless function, the ‘flux ratio’, that depends on layer-based density ratio and the
Summary
In many ways, the DL case seems simpler than the SF case. For example, consider fluxes. For decades, oceanographers estimated SF fluxes in the ocean using layer-based flux laws based on laboratory experiments, but the C-SALT program (Gregg and Sanford, 1987, Lueck, 1987) revealed this to be in error by well over an order of magnitude. This may be the result of SF disruption by internal-wave shear (Linden, 1974b, Kunze, 1994). In the DL case, whether by virtue of weaker shears or differences in
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