Oceanic thermohaline intrusions: theory

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Abstract

This is a review of theories governing growth and evolution of thermohaline intrusive motions. We discuss theories based on eddy coefficients and salt finger flux ratios and also on molecular Fickian diffusion, drawing relationships and parallels where possible. We discuss linear theories of various physical configurations, effects of rotation and shear, and nonlinear theories. A key requirement for such theories to become quantitatively correct is the development and field testing of relationships between double-diffusive fluxes and average vertical gradients of temperature and salinity. While we have some ideas about the functional dependencies and rough observational constraints on the magnitudes of such flux/gradient relationships, many questions will not be answered until usable ‘flux laws’ exist. Furthermore, numerical experiments on double-diffusive intrusions are currently feasible, but will have more quantitative meaning when fluxes are parameterised with such laws. We conclude that more work needs to be done in at least two areas. Firstly, tests of linear theory against observations should continue, particularly to discover the extent to which linear theories actually explain the genesis of intrusions. Secondly, theoretical studies are needed on the nonlinear effects that control the evolution and finite amplitude state of intrusions, since these determine the lateral fluxes of salt, heat, and momentum.

Introduction

Inversions in temperature and salinity occur in most oceanic CTD casts, and are a signature of thermohaline intrusions, produced by lateral sheared advection across lateral water mass gradients. They are typically ‘thermohaline’ in origin—self-driven by the release of potential energy via vertical double-diffusion, and cause lateral mixing that is slow and steady, but comparable to the stirring by baroclinic eddies (cf. Joyce, Zenk, & Toole, 1978).

The dynamic mechanism behind thermohaline intrusions is simple but subtle, and was first elucidated by Stern (1967) and later in laboratory experiments by Turner (1978). Consider a situation with lateral gradients of temperature and salinity (Fig. 1), and a vertical stratification that supports salt fingering. If a perturbation consisting of alternating shear zones is superposed, the lateral advection and lateral T/S gradients act to produce alternating vertical T and S gradients that will alternately enhance and weaken the existing salt fingers. This produces flux convergences that tend to reduce the T and S perturbations. However, because the buoyancy flux for salt fingers is up-gradient (a downward density flux), the fluxes will make the warm, salty perturbations become less dense but will make the cool, fresh perturbations more dense. If the initial perturbation has a slight slope (as shown) such that the warm, saline perturbations slope upwards from the warm, salty side, then the density perturbations will act to reinforce the initial motion. The warm salty layers thus become anomalously light because of the flux convergence, and ‘slide upwards’ from the warm salty side, whereas the converse occurs to the cool fresh layers. The linear instability works via a positive feedback loop:

  • 1.

    Lateral, along-intrusion, sheared advection;

  • 2.

    Alternately strengthened and weakened gradients and salt finger fluxes;

  • 3.

    Alternately positive and negative density perturbations;

  • 4.

    Sloping density perturbations create pressure perturbations;

  • 5.

    Pressure perturbations accelerate the original advective motions.

Since velocity is anti-correlated with the S and T perturbations, the lateral intrusive heat and salt fluxes are down-gradient. Since there is a systematic density perturbation, there is an along-intrusion (and slightly downwards) density flux towards the warm, salty side. If the diffusive fluxes dominate, the slope and along-intrusion density flux are reversed.

Intrusions cause significant lateral fluxes, and a main goal is to provide a parameterisation for these fluxes. The aim of most intrusion theories is to understand the mechanisms of formation, growth, evolution, and eventual finite-amplitude limitation, since these affect the lateral fluxes created.

The main factor preventing a quantitative understanding is the fact that we cannot yet successfully either predict or parameterise the vertical fluxes in a double-diffusive oceanic environment. In an environment with sharp interfaces and well-formed layers, laboratory laws for double-diffusive and turbulent entrainment fluxes are well-established (cf. Turner, 1973), although the observational tests conducted during the C-SALT experiment have proved problematic (Schmitt, 1994). However, layer formation and growth in a turbulent, sheared oceanic environment, as well as double-diffusive fluxes in more uniform gradients, are poorly understood; it is fair to say that no fully tested parameterisations exist to predict average vertical fluxes in terms of average gradients. Despite this, a great deal of progress has been made using eddy diffusivity parameterisations building on that of Stern (1967) as well as molecular diffusivities. In the oceans the use of molecular diffusivities to analyse the development of intrusions that are of order 10 m thick is certainly inappropriate. So why should oceanographers be interested in this body of work? One reason is that the results from the two approaches often have strong parallels, with the results in one field having direct analogues in the other. Another reason is that the sequence of instabilities leading to intrusions begins with those involving molecular fluxes. In the following sections we summarize many of the laboratory experiments that have motivated the studies, then look at linear theory based on eddy diffusivities, comparing it with theory based on molecular diffusivities, look at the effects of rotation and baroclinic shear and finally discuss finite amplitude theories.

Section snippets

Uniform gradient configurations

Linearized instability theories predict exponential growth, and therefore provide no inherent limitations on the eventual amplitude or lateral fluxes. However, intrusion scales, slopes, physics and growth rates can be predicted, provided the vertical fluxes, double-diffusive and otherwise, can be accurately parameterised. Stern’s (1967) vertical diffusivity parameterisation incisively captured the major effect of salt fingers. He parameterised the vertical salt flux βFS by an eddy diffusivity, K

Finite-amplitude theories

Linear intrusion theories predict fastest-growing scales and slopes, but give no information on either the secondary instabilities or the processes that guide the evolution of the intrusions and serve to limit their growth at finite amplitude. Since typical growth rates (O(day)) are much shorter than the lifetime of fronts, intrusions may spend most of their lifetime in a finite-amplitude state, and so fluxes are probably limited by the processes that control intrusion amplitude.

We know that at

The cornerstone

What are appropriate vertical gradient flux laws for a vertical double-diffusive stratification in which staircases have not formed (i.e. the gradient or ‘irregular steppy’ situation)? Do they depend on the background level of turbulence, shear, or fine-scale internal waves? Once this difficult question is answered, it will be straightforward to model intrusive situations numerically. So far, it has been argued on dimensional grounds (Kelley, 1984) and assumed by those attempting to model the

Acknowledgements

We thank Bill Merryfield, Trevor McDougall, and two excellent reviewers for very helpful suggestions.

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