Origin of Thermohaline Staircases

William J. Merryfield Institute of Ocean Sciences, Sidney, British Columbia, Canada

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Abstract

Hypotheses concerning the origin of thermohaline staircases in salt fingering regions are reviewed and assessed. One such hypothesis, that staircases arise from thermohaline intrusions, is developed into a quantitative theory. It is shown that growing intrusions evolve toward staircases when the background density ratio lies below a threshold value, and nonlinear computations confirm that staircases are viable intrusion equilibria. Staircase properties such as step heights, lateral density ratios, and layer slopes lie closest to observed values when salt fingers are assumed not to contribute to shear stress and when turbulent mixing rates are smaller than usual thermocline values.

Corresponding author address: Dr. William Merryfield, Institute of Ocean Sciences, Sidney, BC V8W 4B2, Canada.

Email: merryfieldw@dfo-mpo.gc.ca

Abstract

Hypotheses concerning the origin of thermohaline staircases in salt fingering regions are reviewed and assessed. One such hypothesis, that staircases arise from thermohaline intrusions, is developed into a quantitative theory. It is shown that growing intrusions evolve toward staircases when the background density ratio lies below a threshold value, and nonlinear computations confirm that staircases are viable intrusion equilibria. Staircase properties such as step heights, lateral density ratios, and layer slopes lie closest to observed values when salt fingers are assumed not to contribute to shear stress and when turbulent mixing rates are smaller than usual thermocline values.

Corresponding author address: Dr. William Merryfield, Institute of Ocean Sciences, Sidney, BC V8W 4B2, Canada.

Email: merryfieldw@dfo-mpo.gc.ca

1. Introduction

Thermohaline staircases are striking sequences of mixed layers, ranging from tens to hundreds of meters thick, separated by steep-gradient interfaces, which are typically several meters thick. Staircases are found where large-scale temperature T and salinity S both increase upward in a manner that favors vigorous salt fingering (Schmitt 1994).1

Well-developed staircases have been observed beneath the Mediterranean outflow (Tait and Howe 1968;Howe and Tait 1970; Zenk 1970; Elliott et al. 1974), in the Western Mediterranean (Johannessen and Lee 1974; Zodiatis and Gasparini 1996; Krahmann 1997) and northeast Caribbean (Lambert and Sturges 1977), and throughout the region of the C-SALT observing program in the western tropical North Atlantic (Mazeika 1974; Boyd and Perkins 1987; Schmitt et al. 1987; Boyd 1989). Less distinct examples have been noted in the Sargasso Sea (Cooper and Stommel 1968) and the eastern equatorial Pacific (Miller and Browning 1974).

Thermohaline staircases are distinguished by their extreme stability in space and time. Individual steps have been traced laterally for upwards of 100 km (Elliott et al. 1974; Johannessen and Lee 1974; Schmitt 1988), and can persist several years or more (Johannessen and Lee 1974; Molcard and Tait 1977).

The connection between staircases and salt fingering was suggested by Turner (1967) and Stern and Turner (1969), who noted that the vertical gradients in oceanic staircases favor fingering, and also that laboratory experiments can produce analogous structures in which salt fingers occupy the high-gradient interfaces, and drive convection in the adjoining mixed layers. The occurrence of salt fingers in the interfaces of oceanic staircases has been established by microstructure measurements (Magnell 1976; Lueck 1987; Marmorino et al. 1987; Fleury and Lueck 1991) and optical imaging (Williams 1975; Kunze et al. 1987). Observations also provide evidence of convective plumes in the mixed layers (Marmorino et al. 1987).

Several hypotheses have been put forth for how thermohaline staircases might develop from initially uniform gradients of T and S. This paper first reviews three such hypotheses: that staircases arise from an instability of salt fingers (section 2), that they are metastable equilibria induced by external disturbances (section 3), and that they arise because salt fingers effect negative density diffusion (section 4). In section 5, the hypothesis that staircases arise from thermohaline intrusions is developed into a quantitative theory. (Readers interested primarily in a proposed solution to the staircase problem should skip to this section.) From linear theory it is shown that intrusions can evolve toward a staircase configuration featuring alternating fingering and convective layers, provided the density ratio lies below a threshold value. An analytical criterion for this threshold that depends on mixing rates, property gradients, as well as the heights, slopes, and growth rates of the intrusions is derived. Nonlinear calculations confirm that intrusions can equilibrate as staircases and suggest that the relatively high vertical gradients in staircase interfaces are maintained by shearing of large-scale horizontal gradients. Assuming plausible salt-fingering flux laws, the theory best reproduces observed staircase properties when the turbulent mixing coefficient is small (<0.1 cm2 s−1), the turbulent Prandtl number is of order unity, and salt fingers do not contribute to viscous stress. A summary and conclusions are presented in section 6.

Before proceeding, a few facts about salt fingers and some terminology are briefly reviewed. (For a comprehensive review, see Schmitt 1994.) Salt fingering occurs when a stably stratified component (temperature T) diffuses more rapidly than an unstably stratified component (salinity S). Fingering is characterized by three dimensionless parameters: molecular Prandtl number σ = ν/κT, where ν is viscosity and κT is thermal diffusivity;inverse Lewis number τ = κS/κT, where κS is the diffusivity of salt; and density ratio Rρ = αTz/βSz, where Tz is the stabilizing vertical temperature gradient, Sz is the destabilizing vertical salinity gradient, α is the coefficient of thermal expansion, and β is the coefficient of saline contraction. Values σ = 7 and τ = 0.01 characteristic of seawater are assumed except where noted. Salt fingers are linearly unstable for 1 < Rρ < τ−1, and produce observable effects in the ocean primarily when Rρ ≲ 4. In the sections that follow, reference is occasionally made to laboratory experiments in which the slower and faster diffusing components are sugar and salt. In such instances, S is identified with sugar and T with salt.

2. Collective instability of salt fingers

In investigating the stability of steady, vertically uniform salt fingers at large Prandtl number, Stern (1969) found that collections of fingers should excite internal gravity waves when a dimensionless parameter (equivalent to a finger Reynolds number) exceeds a critical value of order unity. The instability criterion, as later refined by Holyer (1981), is
i1520-0485-30-5-1046-e21
where FT and FS represent vertical fluxes of temperature and salinity. This parameter and its variants have become known as the Stern number.2

Stern (1969) speculated that this collective instability of salt fingers might be instrumental in generating stepped structures from initially uniform gradients. In Stern’s scenario, growing salt fingers generate internal waves, which disrupt the fingers in regions of large vertical shear. Convection sets in, driven by buoyancy fluxes from adjoining interfaces in which salt fingers survive. Direct evidence for this scenario comes from qualitative observations of developing fingers. For example, Stern and Turner (1969), describing laboratory experiments in which a mixed layer of sugar solution was placed over a linearly stratified salt solution, reported that “Intermittent sideways oscillations of the fingers are followed by stronger horizontal motions until large-scale vertical convection is established. . . .”

Additional evidence is provided by the fact that many laboratory salt fingering interfaces exhibit Stern numbers of order the critical value (Linden 1973; Schmitt 1979; Taylor and Bucens 1989), although McDougall and Taylor (1984) reported Stern numbers as high as 5 at low Rρ. Experiments where salt and sugar replace heat and salt as the faster and slower diffusing components have Prandtl number (defined here as the ratio of viscosity to diffusivity of the faster-diffusing solute) σ ≈ 700 and hence should reflect predictions of the collective instability theory at least as well as heat–salt fingers (σ ≈ 7). Stern numbers in salt–sugar fingering interfaces have been found to range from 0.002 (Lambert and Demenkow 1971) to 0.006 (Griffiths and Ruddick 1980).3 The apparent discrepancy with (2.1) could indicate that the stability of growing and finite-length fingers has some further dependence on σ and τ or that the Stern (1969) scenario does not govern fingering interfaces under these conditions.

An issue connected with the Stern (1969) hypothesis concerns the nature of salt finger instability. Subsequent stability calculations for steady fingers have shown that small-scale zigzag or varicose modes generally grow faster than collectively excited gravity waves (Holyer 1984; Howard and Veronis 1992). Shen (1995) extended the analysis to growing fingers and showed that varicose modes grow faster than oscillatory modes and faster than the fingers themselves. In numerical experiments initialized with large-amplitude growing salt fingers, the varicose instability appears to govern finger breakdown (Whitfield et al. 1989; Shen 1995), although such simulations might not encompass enough fingers for collective instability to be adequately represented.

A further unresolved question concerns the conditions under which the Stern (1969) scenario operates. Stern and Turner (1969) reported that layers formed when a sharp T and S interface was present initially and when a mixed layer of sugar solution was placed over a linearly stratified salt solution as described above. However, when T and S gradients were both initially present, the outcome depended upon the care with which the initial stratification was established. Disturbances imposed deliberately during the filling process grew and triggered convection, whereas experiments that were set up carefully exhibited vertically uniform fingering. The latter outcome is typical of numerical experiments having initially uniform T and S gradients (Whitfield et al. 1989; Shen 1995; Merryfield and Grinder 2000, manuscript submitted to J. Phys. Oceanogr., hereafter MG).

An exception is the simulation of Özgökmen et al. (1998) in which the upper and lower boundaries are solid and insulating, rather than periodic as in other investigations. They chose system parameters so that the combined layer and interface thicknesses as predicted by Stern (1969) would fit within the computational domain. Additional requirements that fingers be adequately resolved and that interface thickness be an appreciable fraction of the layer thickness led them to adopt σ = 100 and τ = 1/30. In this simulation a well-defined fingering interface develops, with convective layers above and below. However (as the authors acknowledge), this appears most likely to be due to convergence of buoyancy flux at the solid boundaries.

To examine the possibility that collective instability plays a role in forming the Özgökmen et al. (1998) layer–interface structure, a similar calculation was undertaken in which the solid upper and lower boundaries were replaced by periodic boundary conditions. Details of the pseudospectral numerical algorithm are described in MG. Figure 1 depicts salinity over the range of times recorded in the Özgökmen et al. Figs. 4, 5, and 6. As in Özgökmen et al., a large sinusoidal perturbation was imposed on the the initial salinity field. This perturbation has a longer horizontal wavelength than the fastest-growing salt fingers. At time t = 0.11 (Fig. 1a), the initial perturbation has developed into fingers, which have become “feathered” due to smaller-scale perturbations growing more rapidly. This phenomenon resembles the instability of wide fingers in a Hele Shaw cell subject to a sudden increase in gravity (cf. Fig. 2 of Taylor and Veronis 1996) and is evident in Fig. 4 of Özgökmen et al. At later times the most unstable scales dominate, and a statistical equilibrium is established in which fingering and mean solute gradients remain nearly uniform (Figs. 1b,c). Because fluxes of solute leaving the bottom of the domain are continually replenished at the top, the system does not subsequently “run down” as in Özgökmen et al.

The absence of layer formation in this calculation suggests that layer formation from initially uniform gradients as observed by Özgökmen et al. (1998) is strongly influenced by their choice of boundary conditions and that the Stern (1969) scenario requires vertical inhomogeneity to operate, if indeed it plays a role here.

3. Staircases as metastable equilibria

A second hypothesis for the formation of staircases from initially uniform T and S gradients is that permanent steps are induced by disturbances such as mechanical mixing or imposed flux imbalances. In this picture, staircases and statistically homogeneous salt fingering (such as that in Figs. 1b,c) both represent metastable equilibria. This notion is supported by the Stern and Turner (1969) experiments, described in the last section, in which either uniform fingering or layers develop depending on initial conditions.

The metastable equilibrium hypothesis implies that the stepped structures observed in the laboratory represent true equilibria and that interface thicknesses would persist indefinitely at equilibrium values if the property jumps across them remained constant. In laboratory experiments to date the property jumps always decrease over time; thus the equilibrium thickness prediction has been tested only indirectly thus far. As property jumps run down, the interfaces generally broaden, linearly with t when fingering is vigorous (Stern and Turner 1969; Linden 1973; Taylor and Veronis 1996) and as t1/2 when fingers are only marginally unstable (Taylor and Veronis 1996). In the former instance, the interface spreads at a speed that is at least an order of magnitude smaller than vertical velocities in the fingers (Linden 1973). A uniform spread rate is consistent with predictions which assume a critical Stern number and the so-called 4/3 law for solute flux (Stern and Turner 1969). This relatively slow rate of spread has been interpreted as evolution through a series of quasi equilibria (e.g., Stern and Turner 1969). Further evidence in favor of an equilibrium interface thickness is provided by the observation of Schmitt (1979) that interface spread can be reversed by replenishing the destabilizing property concentration in the upper layer.

The notion of an equilibrium interface thickness could be tested more directly by devising an experiment in which properties are extracted from the lower layer and replenished in the upper layer in such a manner that the property jumps are maintained.4 Such maintenance is achievable in numerical simulations having periodic boundaries that allow the fluxes exiting the lower boundary to reenter from above. In an attempt to model an equilibrium interface, a numerical simulation was undertaken using the two-dimensional pseudospectral code described in MG. Parameters were σ = 7 and τ = 0.01, characteristic of seawater, and Rρ = 2. The computational domain height L ≈ 42d, where d ≡ (νκT/gαTz)1/4, was spanned by 1024 grid points and the width L/2 by 512 grid points. Initial conditions consisted of a step in T and S, with property jumps smoothed across five or so grid points according to a tanh profile. Small random perturbations to S were imposed.

As in similar experiments (Shen 1993), a series of fingers initially developed at the interface (Fig. 2a). The subsequent spread of the interface is compared against the spreading of a purely diffusive interface governed by
tκΦzz
with
z,n0nLznL,n
The solution to (3.1)–(3.2) is
i1520-0485-30-5-1046-e33
(Elliott and Tait 1977). Interface sharpness for the modeled T and S fields and for Φ was quantified as the integrated difference between the horizontally averaged fields and constant-gradient diffusive equilibria:
i1520-0485-30-5-1046-e34
where Tz, Sz, and Φz are the equilibrium gradients, and the brackets denote horizontal averages. The evolution of the two-dimensional salinity field and of 〈T〉 and 〈S〉 is depicted in Figs. 2 and 3. In Fig. 4, evolution of ΛT and ΛS is compared against that of ΛΦ for different values of κΦ. Here T, S and Φ are scaled to have unit equilibrium gradients. Time is expressed in units of d2/κT. Because initial Φ is a perfect step, ΛΦ initially is slightly greater than ΛT and ΛS. For t ≲ 20, the fingering interface spreads significantly faster than by thermal diffusion alone. However, by t = 50 the interface is decaying at much the same rate as a thermal diffusive interface. Similar results were reported by Shen (1989) for σ = 1, τ = 0.1, and Rρ = 2. [At such a rate, 10-m steps would decay on a timescale of order 6 months and 50-m steps on a timescale of order 15 years. In addition, steps would lose their “sharpness” on timescales an order of magnitude shorter (Elliott and Tait 1977).]

The failure of this numerical experiment to attain a layered equilibrium should not be weighed as evidence against the multiple equilibrium hypothesis because it is two-dimensional and samples only a single set of parameters. It does, however, suggest the existence of a criterion dividing conditions under which true layered equilibria arise (if indeed they do) and do not arise.

Finally, a particularly effective way to produce multiple layers in the laboratory is to impose an S flux from above onto a stable T gradient (Stern and Turner 1969;Schmitt 1997). Experiments in which a destabilizing S gradient is also present have not been reported. However, in the case of diffusive convection, layers develop in an analogous fashion when a T flux is applied from below, whether a destabilizing T gradient is present (Linden 1976) or not (Turner 1968). In the diffusive case, layer thicknesses have been shown to collapse to a function of Rρ alone when scaled by κT/N, where N is the buoyancy frequency of the mean stratification (Kelley 1984). Such a relation has not yet been demonstrated for fingering staircases.

4. Negative density diffusion

Salt fingering has the distinctive property that, while vertical T and S fluxes are downgradient, vertical density flux is upgradient. This is because the potential energy released in transporting salt downward must exceed that expended in transporting heat upward, resulting in a net downward transport of mass.

Negative diffusion sharpens rather than smooths fluctuations. The negative diffusion of density associated with salt fingers thus suggests a means for generating and maintaining staircases (Schmitt 1994). This hypothesis is evaluated with the aid of some numerical experiments below.

a. Diffusion equation for density

Suppose that salt fingering causes slow negative diffusion of density and convection more rapid positive diffusion. The influence of fingering on horizontal-mean density ρ (averaged laterally over many finger widths) can then be represented by a simple one-dimensional diffusion equation,
ρtKρρzz
where z is the vertical coordinate and Kρ is an effective vertical diffusion coefficient specified by
i1520-0485-30-5-1046-e42
The value assigned to the negative fingering diffusivity Kfρ is comparable to estimates based on microstructure measurements and numerical simulations5 and is large enough to overcome the estimated smoothing effects of intermittent turbulence (Schmitt 1994). The large value assigned to the positive convective diffusivity Kcρ reflects the assumption that convective mixing occurs very rapidly compared to mixing by salt fingers. The results are insensitive to its exact value, provided Kcρ ≫ |Kfρ|.
Equations (4.1)–(4.3) were solved numerically using an implicit finite difference scheme,
i1520-0485-30-5-1046-e44
where superscripts denote time levels, subscripts denote grid points, and Δt and Δz denote time step and grid spacing. One hundred grid points were employed, with Δz = 20 cm and Δt = 400 s. Boundary conditions were periodic (modulo the density contrast across the domain), which required inverting a cyclic tridiagonal system at each time step. Initial conditions consisted of small-amplitude random noise superimposed on a uniform stable ρ gradient. Subsequent evolution of ρ(z, t) for a particular realization of initial conditions is shown in Fig. 5. Other realizations are qualitatively similar. Initially, the density fluctuations are minute. Because of negative diffusion, the fluctuations amplify, giving rise to numerous density inversions. Convection rapidly mixes across the inversions, yielding density steps between adjacent grid cells. Smaller steps weaken and merge successively with larger ones until a single step spans the entire domain. Merging occurs because layers having larger density jumps below than above transport more mass across the steeper density interface below than they receive from above. Density thus decreases until the layer merges with the one above it. (It should be noted that this process depends upon grid spacing because the upgradient density flux across a jump Δρ spanning one grid cell is KfρΔρz. Reducing Δz increases the upgradient fluxes and speeds the merging processes.) Although having a fixed interface thickness and flux proportional to Δρ seems artificial, such behavior is not unlike that of laboratory salt fingering interfaces, whose thicknesses are relatively insensitive to salinity jump ΔS, scaling as (βΔS)−1/3 (Stern and Turner 1969; Linden 1973). Similarly, fingering salt fluxes at a fixed Rρ are often cited as following βFS ∝ (βΔS)4/3 (e.g., Schmitt 1979; Taylor and Bucens 1989), which is not too different from Fρ ∝ Δρ.

The results shown in Fig. 5 superficially appear to support the notion that negative density diffusion is responsible for staircases. However, at least two considerations cast doubt on this interpretation. The first is that the physical scale of the steps increases with time and is limited only by the height of the computational domain. Thus there is no obvious mechanism for establishing a preferred scale. The second is that (4.1)–(4.3) do not provide a consistent description of the mixing process. This is because countergradient transport of ρ is a consequence of the combined transports of T and S, which should be treated separately. Such a treatment is considered next.

b. Diffusion equations for temperature and salinity

Consider horizontal-mean T and S, which are governed by
i1520-0485-30-5-1046-e45
and together determine horizontal-mean density through
ρρ0αTT0βSS0
where T0, S0 and ρ0 denote constant reference values and a linear equation of state is assumed. Equations (4.5)–(4.7) were solved using
i1520-0485-30-5-1046-e48
where KfS and KfT have been assigned values similar to those found in a Rρ = 1.5 numerical simulation (MG). When initial T, S consist of small fluctuations about uniform gradients, large-amplitude steps did not spontaneously arise. Convective layers did form when departures from uniform gradients were sufficiently large because of the more rapid decay of S fluctuations than T fluctuations. The scale of these layers is determined by that of the imposed disturbances. A sequence of layers such as a staircase thus might be generated by a train of internal waves, although one would need to show how such a process could produce coherent steps hundreds of kilometers in lateral extent. Zhurbas et al. (1987) obtained similar results using Rρ-dependent KT and KS. They suggested that regular disturbances in T and S might be produced by thermohaline intrusions (see sec. 5), but made no attempt to account for crucial lateral advection effects.

5. Staircases as intrusions

This section develops the hypothesis that thermohaline staircases arise from double-diffusive intrusions. Such a connection was proposed by Zhurbas and Ozmidov (1983), who found that steps in the C-SALT region develop T and S inversions near their peripheries.

After a brief review of some properties of intrusions, linear and nonlinear calculations are presented which support this hypothesis. Observational evidence is then discussed.

a. Nature of intrusions

Salt fingers occur because diffusive transport of T exceeds that of S. However, salt fingers themselves transport S more effectively than T, and this unequal transport, in conjunction with lateral T and S gradients, gives rise to inclined, fingerlike intrusions. Stability when vertical T and S gradients favor fingering and lateral T and S gradients are density-compensating has been explored by Stern (1967), Toole and Georgi (1981), McDougall (1985), Niino (1986), and Walsh and Ruddick (1995, 2000). Kuzmina and Rodionov (1992) and May and Kelly (1997) consider the influence of baroclinicity. Under oceanic conditions, predicted vertical wavelengths are of order tens of meters, and slopes are less than one degree or so from horizontal. A distinctive feature of intrusions driven by salt fingering is that warm and salty intrusions rise across isopycnals, whereas cool and fresh intrusions sink.

Because double-diffusive intrusions grow most rapidly where lateral T and S gradients are large, they are expected to be most conspicuous at fronts. Observations show that many fronts indeed harbor intrusive features (e.g., Gregg 1975; Horne 1978; Ruddick 1992), although intrusions have also been found where lateral gradients are spatially extended (Rudels et al. 1999).

A somewhat standard characterization of fully developed intrusions is that they feature prominent T and S inversions so that regions of salt fingering (T and S increasing upward) alternate in depth with regions of diffusive convection (T and S increasing downward).

b. Linear theory

In this subsection, linear theory is employed to argue that conventional intrusions featuring prominent T and S inversions are not the only possible outcome of intrusive instability and that staircaselike configurations as are possible as well. A criterion dividing these two outcomes is derived. As in recent studies by Walsh and Ruddick (1995, 1998, 2000, henceforth WR95, WR98, WR00, respectively), attention is focused on effects of Rρ-dependent fingering fluxes and turbulent mixing, whereas effects of rotation are excluded. This can partially be justified by the fact that rotation does not affect the growth rate, wavelength, or cross-front slope of the fastest-growing intrusions (McDougall 1985; Kerr and Holyer 1986). Such modes also have vanishing alongslope velocity, but do possess an alongslope component of slope when rotation is accounted for. Baroclinic effects (Kuzmina and Rodionov 1992; May and Kelly 1997) are also excluded.

To motivate the discussion, consider the initial development of intrusions at a particular horizontal location. Assuming that initial gradients are uniform and that a single Fourier mode dominates, temperature and salinity evolve according to
i1520-0485-30-5-1046-e51
where m is vertical wavenumber, λ is growth rate, and Ŝ are the (possibly complex) intrusion amplitudes (e.g., Toole and Georgi 1981), and Tz and Sz are the background vertical gradients.

It is convenient to represent this evolution on the (αTz, βSz) plane as in Fig. 6, where approximate boundaries dividing regions of salt fingering, convection, diffusive convection and stability are indicated. Initial amplitudes are assumed arbitrarily small, so (5.1)–(5.2) represent essentially a point on this plane (Fig. 6a). As the intrusions develop, (5.1)–(5.2) describe line segments having a slope (αT̂/βŜ)−1 (Fig. 6b), which must be <1 for intrusions to grow. If (T, S) continue to be described by their linear eigenfunctions, one of two possible developments must occur. The first, labeled “I,” is that Sz reverses sign before Tz and a portion of the intrusion becomes stable (Fig. 6c). At still later times, Tz also reverses sign, and intrusions consist of a cyclical arrangement of fingering, stable, diffusively convective, stable, and fingering layers (Fig. 6d). The second possible development, labeled “S,” is that evolution occurs toward αTz = βSz, the locus of neutral stability to convection (Fig. 6b). At later times this criterion is exceeded (Figs. 7c,d), and intrusions consist of alternating fingering and convective layers—a thermohaline staircase.

It remains to establish whether evolution toward staircase configurations can be realized. Referring again to Fig. 6, growing intrusions governed by (5.1)–(5.2) will evolve toward staircases if
αT̂βŜαTzβSzRρ
Is it straightforward to express this criterion in terms of system parameters. The linearized equations governing small departures T and S from a state of uniform T and S gradients and no motion are
i1520-0485-30-5-1046-e54
(WR95) where u and w are horizontal and vertical velocity and F(T) and F(S) are vertical diffusive fluxes of temperature and salt. (The momentum equations need not be considered here.) The fluxes are assumed, as in WR00, to consist of Rρ-dependent fingering contributions, which differ for T and S, together with a turbulent flux that mixes T and S equally:
i1520-0485-30-5-1046-e56
where KfT and KfS are effective fingering diffusivities for T and S, and Kturb is an effective turbulent diffusivity. The fingering diffusivities are related by KfT = γfKfS/Rρ, where flux ratio γf is a relatively slowly varying function of Rρ, taken to be constant. Substituting (5.6)–(5.7) into (5.4)–(5.5) and considering perturbations (T, S) = (T̂, Ŝ)eikx+imz+λt yields
i1520-0485-30-5-1046-e58
where KS is the derivative of KfS with respect to Rρ, ψ̂ is streamfunction, for which (u, w) = (−ψ̂z, ψ̂x), and KS and KS are evaluated at Rρ. Straightforward evaluation of αT̂/βŜ from (5.8)–(5.9) and substitution into (5.3) yields the proposed criterion for evolution of intrusions into thermohaline staircases:
i1520-0485-30-5-1046-e510
where ΓSSx/Sz is the background isohaline slope, and sk/m, the cross-frontal intrusion slope, is positive for growing intrusions.
Implications of (5.10) are now examined. In this discussion, density ratio Rρ and turbulent mixing coefficient Kt, which depend upon local conditions, are allowed to vary. The buoyancy flux ratio for fingering is set to γf = 0.6, a value fairly well constrained by laboratory and numerical experiments (e.g., Schmitt 1979;MG). In specifying KfS(Rρ), three functional forms are considered (Fig. 7). The first two are based on the ad hoc formula KfS(Rρ) = KmaxRnfρ of WR98, choosing Kmax = 1 cm2 s−1 and nf = 0 (case WR0) or nf = 2 (WR2). The third form is obtained from the Merryfield and Grinder simulations of salt fingering subject to uniform background gradients: KfS = 0.17(1 − τRρ)/(Rργf) cm2 s−1 (case MG). The growth rate λ is determined by m, s and the parameters through a linear eigenvalue problem (WR00). When the fastest-growing mode is considered, m and s acquire specific values. Finally, the effective Prandtl number, Pr ≡ A(Rρ)/[KfS(Rρ) + Kturb] where A is effective viscosity, appears in the linear eigenvalue problem. The proper choice for this parameter is not well known, and a range of choices is thus considered. Following Ruddick (1985), WR98 take momentum flux to be proportional to the fingering buoyancy flux plus a turbulent flux so that
i1520-0485-30-5-1046-e511
with Pr0 = 10. Conversely, laboratory experiments of Ruddick et al. (1989) suggest that fingering does little to enhance viscous stress. Assuming that viscous stress is enhanced by turbulence alone implies
i1520-0485-30-5-1046-e512
where the molecular viscosity ν ≈ 0.01 cm2 s−1 has been introduced to ensure a physical lower bound.

Several properties of criterion (5.10) merit discussion.

  1. Staircases tend to be favored at low values of Rρ because the right-hand side of (5.10) increases more rapidly than growth rate λ as Rρ → 1. This tendency is illustrated in Fig. 8, where λ and the right-hand side of (5.10) are plotted against Rρ for KfS given by WR0, Kturb = 0.01 cm2 s−1, Pr given by (5.11) with Pr0 = 10, and m and s characterizing the fastest-growing mode. According to criterion (5.10), staircases are favored when Rρ is less than a critical value Rstairρ, which in this instance is about 1.12.

  2. Staircases are favored by a low turbulent mixing coefficient Kturb. Figure 9 illustrates how the predicted threshold Rstairρ varies with Kturb for KfS given by WR0 (solid curves) and WR2 (dashed curves), with Pr specified by (5.11) (thick curves) and (5.12) (thin curves). In each instance, threshold Rstairρ grows larger as Kturb decreases.

  3. Threshold Rstairρ is considerably larger when salt fingering is presumed not to contribute to viscous stress, so that Pr is specified by (5.12). For example, for flux law WR2 and Kturb = 0.01 cm2 s−1 (Fig. 9a), Rstairρ increases from 1.08 when Pr is specified by (5.11) to 1.44 when Pr is specified by (5.12), assuming Pr0 = 10. For (5.11) and flux law MG, (5.10) is not satisfied for any Pr0 ≥ 1, Rρ > 1.

  4. Threshold Rstairρ becomes still larger as the coefficient Pr0 in (5.12) decreases from 10 to 1 (Fig. 9b).

  5. Criterion (5.10) tends to be satisfied for sufficiently large m even when it is not for the fastest-growing mode. In Fig. 10, contours depict linear growth rate versus m and s. Values of (m, s) where evolution is toward staircases are indicated by “+” and values where evolution is toward conventional intrusions are indicated by “−.” With Rρ = 1.1, flux law WR2 and Pr given by (5.12) with Pr0 = 10, (5.10) is satisfied for the fastest-growing mode (Fig. 10a). For Rρ = 1.6, the fastest-growing mode evolves toward conventional intrusions, but modes having comparable s and m ≳ 30% larger evolve toward staircases (Fig. 10b).

  6. Criterion (5.10) is independent of ΓS provided s ≪ 1 because λ, s, and m2 are then proportional to ΓS (see appendix).

  7. In the limit of large Prandtl number, αT̂/βŜ can be obtained analytically (McDougall 1985; WR95). When KS = 0 and Kturb = 0, this leads to Rstairρ = 1 + γ/3, or Rstairρ = 1.133 when γ = 0.6.

c. Nonlinear calculations

In the preceding subsection, linear theory was employed to determine the conditions under which intrusions evolve toward staircase configurations rather than toward conventional intrusions. To assess whether these tendencies are realized in equilibrated intrusions, nonlinear calculations are needed.

Equilibration of intrusions has been examined previously by WR98, who solved the nonlinear equations of motion in a tilted reference frame aligned with the intrusions. Because the ambient T and S gradients were assumed uniform, the governing equations of WR98 depend only on the spatial coordinate normal to the intrusions. All numerical solutions shown by WR98 equilibrated as conventional intrusions. This is not in contradiction with the present results, as discussed below.

1) Formulation

The approach adopted here for computing the nonlinear evolution of intrusions is similar to that of WR98, although the equations solved and numerical method employed are different. As a starting point, consider the Boussinesq equations of motion in two dimensions,
i1520-0485-30-5-1046-e513
together with equation of state (4.7). Here ψ is streamfunction, defined as in section 5b, and J(ψ, · ) = ψxzψzx is the Jacobian operator. Time has been expressed in units of inverse buoyancy frequency N−1 of the undisturbed state, diffusivity in units of Kmax = 1.0 cm2 s−1, length in units of d ≡ (Kmax/N)1/2, temperature in units of N2 d/gα, and salinity in units of N2 d/gβ. The effective viscosity A, like KT and KS, is assumed to operate on vertical gradients only.

To complete (5.13)–(5.15), effective viscosity A and diffusivities KS and KT must be specified as functions of the vertical gradients of temperature and salinity. In a nonlinear calculation, vertical gradients may depart from the fingering regime, and transports under convective, diffusive convective and stable conditions must be specified as well. These transports are somewhat uncertain, and the functional forms suggested by WR98 are adopted here (Table 1), except in instances where cases MG for fingering diffusivities and (5.12) for fingering-regime viscosity are considered.

Two-dimensional equations (5.13)–(5.15) were solved using a pseudospectral algorithm much like that described in MG. Such computations pose significant challenges because the growing intrusions are subject to convective and stratified-shear instabilities, which excite motions on temporal and spatial scales much smaller than those of the intrusions. By using very high resolution together with hyperviscosity and hyperdiffusion to damp the smallest scales, numerically stable solutions were obtained. However, because of the significant computing resources required, it was not feasible to follow many such runs to equilibrium. In addition, explicit convective and gravity-wave motions complicate the interpretation of these runs. Therefore, results of these computations are not described here. Instead, a one-dimensional problem was solved, as described below.

Consider a coordinate system (χ, ζ) that is rotated with respect to (x, z) by −θ, where θ = −tan−1s (Fig. 11). Equations (5.13)–(5.15) can be expressed with respect to the new coordinates by applying the transformation
i1520-0485-30-5-1046-e516
If ambient T and S gradients are uniform so that intrusions are invariant with respect to the along-intrusion coordinate χ, (5.13)–(5.15) become
i1520-0485-30-5-1046-e518
where Z = ψζζ is vorticity, U = −ψζ is along-intrusion velocity, and
i1520-0485-30-5-1046-e521
are the background gradients. As throughout this paper, it is assumed that horizontal temperature and salinity gradients are density compensating.

For each choice of parameters, slope s is that of the fastest-growing mode, and the range in ζ of the computational domain is equal to the wavelength of the fastest-growing mode. Boundary conditions with respect to ζ are periodic, and (5.18)–(5.22) are solved numerically by means of a Fourier pseudospectral scheme, with dealiasing according to the 2/3 rule (Canuto et al. 1988). Between 64 and 512 Fourier modes are considered, as dictated by requirements of resolution. Initial conditions consist of density compensating perturbations to T and S, constructed by assigning Gaussian-distributed random values to the Fourier coefficients and scaling by inverse wavenumber squared. The initial perturbations were assigned dimensionless variances of 10−2.

2) Results

The nonlinear evolution of intrusions is in accord with tendencies deduced from linear theory: in all instances considered, intrusions equilibrated as staircases when criterion (5.10) was satisfied and as conventional intrusions when it was not. An example is shown in Fig. 12, where the critical value Rstairρ dividing staircase and conventional-intrusion regimes is traced by the heavy curve. Flux law MG and Pr given by (5.12) with Pr0 = 1 are considered, and ambient conditions are characteristic of the C-SALT region (except for Rρ; see Table 2). When (Rρ, Kturb) lies to the left of this curve, the numerical solutions equilibrate as staircases (filled circle; inset a). Conventional intrusions are realized to the right of this curve (open circles; insets b, c, d).

A notable feature of the numerical solutions is that for staircases the along-intrusion velocity U(ζ) consists of pairs of narrow, oppositely directed jets centered on the fingering interface (Fig. 12, inset a). The resultant shear, coupled with along-intrusion property variations, maintains the sharp interfacial T and S gradients. This is in marked contrast to conventional intrusions for which U(ζ) has broad extrema nearly spanning the depth range of each intrusion (Fig. 12; inset b).

The development of laminae in staircases and conventional intrusions is illustrated in Fig. 13. For staircases, convection sets in at some critical intrusion amplitude and rapidly grows in extent (Fig. 13a). This is exactly the scenario suggested by Fig. 6. Conventional intrusions first develop stable layers, which rapidly give way to layers of diffusive convection and convection (Fig. 13b). Such behavior was also found by WR98, and prior to the onset of convection it matches the “I” scenario of Fig. 6.

From the numerical solutions, depth-averaged diapycnal and isopycnal fluxes of T and S can be calculated. (These are defined with respect to mean isopycnals, which here are level.) Results are expressed in terms of effective diffusion coefficients, defined as (mean flux)/(mean gradient). Three types of coefficients are considered: diapycnal coefficients KdifT and KdifS describing mixing by fingering, diffusive convection, and convection;diapycnal coefficients KadvT and KadvS describing diapycnal advection by intrusions; and isopycnal coefficients KisoT and KisoS describing isopycnal advection by intrusions. Results for the model runs in Fig. 12 and others are summarized in Table 3, where dimensional values are assigned based on estimated C-SALT conditions (Table 2). The diapycnal coefficients are independent of ΓS, whereas the isopycnal coefficients scale as Γ−2S (see appendix).

Several key aspects of these results are now noted.

  1. Mixing coefficients KdifT and KdifS are larger than coefficients KT and KS describing mixing by spatially uniform salt fingering. For the staircase run in Fig. 12, for example, KdifT = 0.359 cm2 s−1 and KdifS = 0.793 cm2 s−1, whereas KT = 0.091 cm2 s−1 and KS = 0.212 cm2 s−1.

  2. The advective diapycnal fluxes are upgradient.

  3. Total diapycnal heat fluxes, proportional to KdifT + KadvT, are sometimes upgradient for conventional intrusions, though not for staircases. Upgradient vertical fluxes of heat also were found in the analytical intrusion model of Garrett (1982). Total diapycnal salt fluxes are consistently downgradient.

  4. Isopycnal mixing coefficients KisoT and KisoS are one to two orders of magnitude smaller for staircases than for conventional intrusions, perhaps because along-intrusion velocity is concentrated in the fingering interfaces.

It is evident from these results and from Fig. 9 that all of the numerical solutions of WR98 equilibrated as conventional intrusions because they considered Rρ = 1.6 and Pr given by (5.11), choices for which staircases are excluded regardless of flux law and Kturb.

Finally, it was found that higher resolution was required for staircases than for conventional intrusions. Table 4 shows KdifT and KdifS versus resolution for the Rρ = 1.4 runs in Fig. 12. For the staircase run (Kturb = 0.1 cm2 s−1), the coefficients appear just to be converging when 512 Fourier modes are employed. However, the conventional intrusion run (Kturb = 0.1 cm2 s−1) appears nearly converged when just 64 Fourier modes are employed.

d. Comparison with observations

1) Threshold Rρ

An oft-noted property of oceanic staircases is that they appear only in locations having particularly low values of Rρ. For example, the steps in the Western Mediterranean are characterized by Rρ ≈ 1.1–1.2 (Schmitt 1981; Krahmann 1997) and those beneath the Mediterranean outflow by Rρ ≈ 1.3. In the northwestern tropical Atlantic, the locale of the C-SALT experiment, steps are confined to regions where Rρ ≲ 1.7 (Schmitt et al. 1987). There are no known instances of staircases having Rρ ≳ 1.7.

The existence of staircases only in regions of sufficiently low Rρ is in accord with the results in sections 6b,c. The threshold Rstairρ predicted from linear theory ranges from about 1.1 to more than 2, depending on turbulence levels and assumed laws for salt flux and viscous stress (Fig. 9). For Rstairρ to approach the observed threshold of 1.7, it is necessary to assume that turbulent mixing is extremely small (Kturb ≲ 0.01 cm2 s−1), and that salt fingering does not contribute to viscous stress, that is, effective Prandtl number is given by (5.12).

The threshold Rstairρ also becomes larger when Pr0 characterizing the ratio of effective viscosity to salt diffusivity is smaller than the value 10 used by WR98. For Pr0 = 1 and fingering flux law MG, arguably the most plausible because it is based on numerical simulations, Rstairρ reaches 1.52 when Kturb = 0.01 cm2 s−1 and 1.76 when Kturb = 0.001 cm2 s−1.

2) Height of steps

According to the theory in this section, thermohaline step height (mixed layer plus interface) can be identified with intrusion wavelength. To evaluate the latter requires a knowledge of N, Rρ, and ΓS. For at least two staircases, these quantities can be estimated from published data, as summarized in Table 2. Evaluation of wavelength also requires specification of fingering and turbulent fluxes, and as throughout this section a range of choices is considered.

The mean step height in the core of the western tropical North Atlantic (C-SALT) staircase appears to be 30 or 40 m (Schmitt et al. 1987), as indicated by the vertical dashed lines in Figs. 14a,b. Predicted step heights tend to exceed the observed values, but fall within the observed range for flux law MG when Pr is specified by (5.12) with Pr0 < 10, and turbulence is weak (Kturb ≲ 0.02 cm2 s−1).

In the Algerian and Provencal Basins of the western Mediterranean, observed step heights are typically 60 to 100 m (Krahmann 1997). In this case (Figs. 14c,d), predicted step heights fall within the observed range for all flux laws provided Pr is specified by (5.12), and Kturb ≲ 0.05 cm2 s−1 (Pr0 = 10) or 0.05 cm2 s−1Kturb ≲ 0.2 cm2 s−1 (case MG; Pr0 = 1).

3) Lateral property variations

When followed laterally, staircase layers are found to lie across isopycnals, and to become warmer and saltier as their density increases. Such a trend, documented for the C-SALT staircase by Schmitt et al. (1987) and for the western Mediterranean by Zodiatis and Gasparini (1996) and Krahmann (1997), is identical to that exhibited by salt finger-driven intrusions (WR95): cool and fresh intrusions sink across isopycnals, becoming warmer and saltier as they mix diffusively with warmer, saltier intrusions above and below (Fig. 11).

4) Lateral density ratio

In the C-SALT staircase, lateral variations of layer temperature and salinity are related by αTχ/βSχ = 0.85 ± 0.03 (Schmitt 1994). In the western Mediterranean, this ratio is 0.9 ± 0.06 (Krahmann 1997).

For intrusions driven by uniform background gradients, lateral density ratio depends on the gradients and intrusion slope, and is given by
i1520-0485-30-5-1046-e523
Figure 15 compares the observed lateral density ratios with (5.23) evaluated for the fastest-growing mode. For the C-SALT staircase, predicted ratios lie within the error bars for all three flux laws provided that Pr is given by (5.12), and Kturb ≲ 0.01 cm2 s−1 (Figs. 15a,b). (For case MG, Pr0 ≲ 2 is required as well.) For the Western Mediterranean staircase, predicted ratios lie within the error bars under all conditions shown, although optimal agreement with the centroid value 0.92 again is realized when Kturb is small and Pr is given by (5.12) with Pr0 < 10 (Figs. 15c,d).

Other proposed explanations for the observed lateral density ratios include modification of water properties by combined effects of salt fingering and turbulence (Marmorino 1990; Schmitt 1994), and interfacial advection due to nonlinearities in the equation of state (McDougall 1991).

5) Layer slopes

As mentioned above, staircase mixed layers slope across isopycnals in the same sense as intrusions. Slopes were estimated for the C-SALT and Mediterranean steps using
ρχθρzsρz
The lateral density gradient ρχ was estimated from Fig. 5 of Schmitt et al. (1987) and Fig. 4 of Krahmann (1997) by dividing total lateral density contrast within a given layer by the approximate horizontal extent of the staircase. These estimates are somewhat uncertain because they assume that ρχ is uniform.

For the C-SALT staircase, estimated slope exceeds the theoretical slopes (Figs. 16a,b). However, the disagreement is least (≲20%) for Kturb ≲ 0.01 cm2 s−1 and Pr law (5.12) with Pr0 < 10. For the Western Mediterranean staircase, estimated slope again exceeds the theoretical slopes (Figs. 16c,d). Disagreement is again minimized by Kturb ≲ 0.01 cm2 s−1 and Pr law (5.12) with Pr0 < 10, although this time the remaining discrepancy is approximately a factor of 3.

6) Shear in fingering interfaces

The numerical solutions described above indicate that, for staircases, intrusion shear is highly concentrated in the fingering interface. In the staircase illustrated in Fig. 12a, shear attains 0.8 × 10−2 s−1 in the center of the fingering interface (Fig. 17a), and gradient Richardson number Ri = N2/u2z reaches a minimum of 0.9 (Fig. 17d).

Observations indicate that shear indeed is concentrated in fingering interfaces. In the C-SALT staircase (Rρ ≲ 1.6), interface shears averaged over 0.5-m intervals sometimes exceed 10−2 s−1 (Gregg and Sanford 1987; Gregg 1988). While such values are in accord with Fig. 17a, the double-jet structure seen in Fig. 12a is not obviously present in the observed horizontal velocities. However, other sources of shear are present which could mask such signals in an instantaneous profile. Richardson numbers computed from average gradients across interfaces range between 1 and 5.

Shears in the staircase beneath the Mediterranean outflow (Rρ ≈ 1.3) were examined by Simpson et al. (1979). Shears evaluated from differences over 1 m again were highest in the interfaces. Interface shear profiles typically were multipronged as in Fig. 17b, although interpretation as a signature of double jets is tempered by the relatively low resolution and the presence of layered substructure in some of the interfaces. In profiles of velocity there is some suggestion of ∼1 cm s−1 interfacial jets oriented NE–SW, but too little data is shown for a definitive assessment. Interfacial gradient Richardson numbers were between 0.3 and 1, with uncertainties of up to 30%.

Shear measurements also were taken by Evans (1981) for a rather weak staircase embedded in a Mediterranean salt lens north of Haiti (Rρ ≈ 1.5 to 1.7). Shear computed on 2-m intervals is again much higher in interfaces than in layers, although shears within the staircase are much smaller than in waters above and below.

7) Coexistence of steps and intrusions

In thermohaline staircases, prominent T and S inversions that resemble conventional intrusions can appear in place of steps. Such features sometimes replace individual steps near the top of an otherwise normal staircase. This is seen in the C-SALT region (Mazeika 1974, Fig. 2; Marmorino et al. 1987, Fig. 6), beneath the Mediterranean outflow (Tait and Howe 1968, Fig. 2; Tait and Howe 1971, Fig. 2), and in the Tyrrhenian Sea (Zodiatis and Gasparini 1996, Fig. 5). Observations show that Rρ tends to increase upward near the tops of staircases (e.g., Schmitt 1994). The appearance of inversions instead of steps thus may result from Rρ locally exceeding Rstairρ. This conjecture is supported by data of Zodiatis and Gasparini (1996), who show two temperature profiles taken 5 months apart. In the later profile, the uppermost large step has become inverted, while Rρ has increased from 1.26 to 1.6. (An inversion is also developing at the top of the second large step, for which Rρ has increased from 1.19 to 1.25.)

Staircases also can evolve laterally into intrusions, especially in the presence of steepening horizontal gradients. In the C-SALT area, for example, staircases develop inversions near thermohaline fronts associated with eddies (Zhurbas et al. 1988; Marmorino 1991) and the staircase edge (Zhurbas and Ozmidov 1983; Marmorino et al. 1987). In the Western Mediterranean, such a transition occurs in the proximity of higher horizontal gradients where Levantine Intermediate Water enters the Algerian Basin (Krahmann 1997).

Assuming Rρ is horizontally uniform, at least two possible reasons are evident for why staircases develop inversions near thermohaline fronts. The first, for which there is little evidence, is that turbulent mixing is systematically larger within such fronts. This would increase Rstairρ (Fig. 9). If the elevated threshold exceeded ambient Rρ, conventional intrusions instead of steps would develop within the front.

A second possibility is related to factors determining the vertical scale of intrusions near fronts. Suppose first that a region of low horizontal gradients contains fully developed steps and that a higher-gradient region develops adjacent to it, perhaps at the staircase edge or within an eddy. Such a situation is illustrated in Fig. 18a, where growth rates are contoured for low-gradient [ΓS = (ΓS)0] and higher-gradient [ΓS = 2(ΓS)0] regions. Plus signs denote where staircases are preferred, and minus signs where conventional intrusions are preferred. If the existing large-amplitude steps in the low-gradient region (left panel) induce growth at like wavenumber m in the higher-gradient region, the latter disturbances will develop as conventional intrusions (right panel, solid dot).

Conversely, it seems plausible that under some conditions intrusions established at a front will propagate away from the front without altering their vertical scale (Fig. 18b). In this case, if conventional intrusions are preferred in the high-gradient region (right panel), intrusions having like m in the low-gradient region will develop as staircases (left panel, solid dot). Both of these scenarios replicate the observed tendency for conventional intrusions at a front to evolve into staircases away from the front.

e. Discussion

The theory for thermohaline staircases developed in this section appears able to account for a cutoff value for Rρ, as well as observed staircase properties. Predictions and observations tend to be in closest agreement when salt fingers are assumed not to appreciably contribute to shear stress, when the turbulent Prandtl number is of order unity, and when turbulent mixing is very weak (Kturb ≲ 0.01 cm2 s−1).

Such low turbulent mixing may not be implausible. In the C-SALT staircase, turbulence due to breaking internal waves is even weaker than the generally low levels found elsewhere in the thermocline (Schmitt 1994), which can be <0.1 cm2 s−1 throughout large regions (e.g., Polzin et al. 1997). Marmorino (1990) concludes that turbulent mixing in this region increases buoyancy flux 10%–20% over that of fingers acting alone. The buoyancy mixing coefficient for fingering is
Kfρ = (KfSRρKfT)/(1 − Rρ)
(Shen 1995). For KfT = 0.063 cm2 s−1 and KfS = 0.167 cm2 s−1 (flux law MG with Rρ = 1.6), Kfρ = −0.112 cm2 s−1. Because Kρ = Kturb under the assumed turbulent flux law (equal mixing of T and S), Marmorino’s estimate yields Kturb = 0.01–0.02 cm2 s−1. Turbulent mixing coefficients as low as 0.01 cm2 s−1 have been found in the core of the Pacific Equatorial Undercurrent (Peters et al. 1988).

While the comparisons in section 5c are encouraging, it should be emphasized that they are based upon crude estimates of hydrographic parameters from published data. Estimates of ΓS and layer slope are especially uncertain because they suppose uniform lateral gradients across the staircase region and ignore possible alongfront contributions to slope. Gradients certainly are not uniform in the Western Mediterranean (Krahmann 1997, his Fig. 11), which may account for the relatively large discrepancy between observed and predicted slopes (Figs. 16c,d).

It should be emphasized that the above theory does not account for baroclinic effects and that their likely importance in the C-SALT region and Western Mediterranean has not been assessed. A stability theory for intrusions that accounts for baroclinic shear has been developed by May and Kelly (1997), and it should not be too difficult to extend the procedure described in section 5b to the baroclinic case.

Finally, the theory developed here does not explain at least one observed property of staircases, the presence of sublayering within fingering interfaces (Williams 1975; Marmorino 1989).

6. Summary and conclusions

This paper has sought to assess four hypotheses concerning the origin of thermohaline staircases. The hypothesis that staircases arise from double-diffusive intrusions was developed into a quantitative theory, consistent with a number of observed staircase properties. Using the stability theory of WR98, it was shown that intrusions can evolve toward staircase configurations if Rρ and turbulent mixing coefficient Kturb are sufficiently small. This is in accord with observations showing that Rρ ≲ 1.7 and Kturb is unusually small in staircase regions.

In considering a range of parameter values and flux prescriptions, turbulent diffusivities Kturb ≲ 0.01 cm2 s−1 and effective viscosities (5.12) that include no contribution from fingering tended to yield closest agreement with observations. Quantities then predicted with reasonable accuracy included step heights, lateral density ratios, layer slopes, and the threshold Rstairρ, although such comparisons were based on rough estimates of lateral gradients and slopes and could be improved.

Numerical solutions confirmed that staircases are viable equilibria of intrusions. The equilibrated solutions exhibit pronounced shears within the fingering interfaces (Fig. 12). These shears, coupled with the lateral gradients in T and S, provide a means for maintaining the large interfacial gradients, and might be observable.

The remaining hypotheses for staircase formation differ from the intrusion hypothesis in that they require only vertical gradients to operate. The collective instability hypothesis of Stern (1969) and the multiple equilibrium hypothesis are in accord with aspects of step formation observed in the laboratory, but remain to be linked convincingly to the fingering staircases observed in the oceans. The negative density diffusion hypothesis in its more realistic formulation (sec. 4b) resembles the multiple equilibrium hypothesis in that finite-amplitude disturbances are necessary to generate steps.

These one-dimensional mechanisms could be responsible for the observed tendency of staircase interfaces to form sublayers (Williams 1975; Linden 1978; Marmorino 1989), a phenomenon for which the intrusion hypothesis as formulated here offers no explanation.

Although the intrusion hypothesis for staircases appears promising, more accurate and detailed observational comparisons should be undertaken. Further work is also needed to improve prescriptions for fluxes and to incorporate baroclinic effects, which have been neglected here. Finally, it would be worthwhile to explore the applicability of the intrusion hypothesis to staircases in regions of diffusive convection.

Acknowledgments

This paper owes much to recent work by David Walsh and Barry Ruddick (WR95, WR98, WR00), who provided preprints and helpful advice. Matt Grinder coded and carried out the calculations in section 4. Greg Holloway, Ann Gargett, Eric Kunze, and two anonymous referees provided helpful comments that led to improvements to the manuscript. This work was supported by the Office of Naval Research (N00014-96-1-0518).

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APPENDIX

Similarity Relations for Intrusions

It is shown here that when slope is small, solutions to intrusion evolution equations (5.18)–(5.20) obey simple scaling relations with respect to changes in isohaline tilt ΓS = Sx/Sz. In the limit s ≪ 1, (5.18)–(5.20) become
i1520-0485-30-5-1046-ea1
to O(θ2). Consider the transformation
i1520-0485-30-5-1046-ea4
which implies ΓSCΓS, and to which (A.1)–(A.3) are invariant. Intrusion growth rate, slope, and wavenumber thus obey
i1520-0485-30-5-1046-ea15
Similarly, the isopycnal flux FisoT and mixing coefficient KisoT for heat obey
i1520-0485-30-5-1046-ea18
likewise for salinity. By contrast, the diapycnal fluxes and mixing coefficients are independent of ΓS:
i1520-0485-30-5-1046-ea20
(likewise for S), where “adv” denotes the effect of along-intrusion advection, and “dif” the effect of small-scale vertical mixing.

Fig. 1.
Fig. 1.

Evolution of “salinity” (slower-diffusing solute) in a numerical experiment having the same parameter values and initial conditions as that of Özgökmen et al. (1998), but with periodic rather than solid upper and lower boundaries. Lighter shades represent higher concentrations, and darker shades lower concentrations. (a) At t = 0.11 initial sinusoidal perturbations have grown to become fingers, which in turn are becoming unstable to smaller, faster-growing fingers. (b) The smaller fingers soon dominate, and by t = 0.50 a statistically uniform equilibrium has been established. (c) This state persists, and at later times shows no signs of developing convective layers.

Citation: Journal of Physical Oceanography 30, 5; 10.1175/1520-0485(2000)030<1046:OOTS>2.0.CO;2

Fig. 2.
Fig. 2.

Salinity field in a numerical simulation of fingers developing at an initially sharp interface for (a) t = 6.13, (b) t = 9.19, (c) t = 12.25, and (d) t = 84.18. In each frame, lighter shades represent higher salinities, and darker shades lower salinities.

Citation: Journal of Physical Oceanography 30, 5; 10.1175/1520-0485(2000)030<1046:OOTS>2.0.CO;2

Fig. 3.
Fig. 3.

Horizontally-averaged temperature in numerical simulation of fingers developing at an initially sharp interface, at the same times represented in Fig. 2: (a) t = 6.13, (b) t = 9.19, (c) t = 12.25, (d) t = 84.18. Instead of equilibrating at a finite thickness, the simulated interface decays toward a constant-gradient equilibrium, represented by the dashed lines. The interface sharpness measure ΛT corresponds to the area between the dashed lines and solid curves.

Citation: Journal of Physical Oceanography 30, 5; 10.1175/1520-0485(2000)030<1046:OOTS>2.0.CO;2

Fig. 4.
Fig. 4.

Evolution of interface sharpness measures ΛT, ΛS, and ΛΦ for temperature, salinity, and a hypothetical scalar Φ having diffusion coefficient κΦ. Thick solid curve: ΛT; thick dashed curve: ΛS; thin curves: ΛΦ, for indicated κΦ. A boxcar filter 7 time units in width has been applied to ΛT and ΛS.

Citation: Journal of Physical Oceanography 30, 5; 10.1175/1520-0485(2000)030<1046:OOTS>2.0.CO;2

Fig. 5.
Fig. 5.

Evolution of density ρ(z, t) from small fluctuations imposed on an initially uniform gradient, according to negative diffusion equations (4.1)–(4.3). Density is expressed in arbitrary units.

Citation: Journal of Physical Oceanography 30, 5; 10.1175/1520-0485(2000)030<1046:OOTS>2.0.CO;2

Fig. 6.
Fig. 6.

Schematic depiction of intrusion growth on the (αTz, βSz) plane, according to (5.1)–(5.2). Long-dashed lines denote constant N, and short-dashed lines constant Rρ. Heavy lines separate fingering, convective, diffusive convective and stable regimes. (a) Initial conditions, lying close to background gradients, are denoted by the dot. (b) Growing intrusions describe line segments which lengthen in time. A line segment whose slope is less than R−1ρ is labeled by “S,” and one whose slope is greater than R−1ρ is labeled by “I.” (For intrusions to grow, this slope must lie between 0 and 1.) (c) As growth continues, I begins to penetrate the stable regime, and S the convective regime. (d) Still later, I penetrates the diffusive–convective regime as well. Intrusion I then consists of sequences of fingering, stable, diffusive convective, convective and fingering layers, whereas S consists of alternating fingering and convective layers.

Citation: Journal of Physical Oceanography 30, 5; 10.1175/1520-0485(2000)030<1046:OOTS>2.0.CO;2

Fig. 7.
Fig. 7.

Functional forms considered for KfS(Rρ), the effective vertical salt diffusivity due to salt fingering.

Citation: Journal of Physical Oceanography 30, 5; 10.1175/1520-0485(2000)030<1046:OOTS>2.0.CO;2

Fig. 8.
Fig. 8.

Example of the dependence on Rρ of λ, the intrusion growth rate and rhs of (5.10). Intrusions evolve toward staircase configurations when λ < rhs, which occurs when Rρ is less than a critical value Rstairρ, which in this instance ≈1.12. Parameters: WR0 flux law, ΓS = 5 × 10−4, Kturb = 0.1 cm2 s−1, Pr given by (5.11). The vertical axis is labeled in units of buoyancy frequency N.

Citation: Journal of Physical Oceanography 30, 5; 10.1175/1520-0485(2000)030<1046:OOTS>2.0.CO;2

Fig. 9.
Fig. 9.

Critical value Rstairρ, below which staircases form, as a function of Kturb for (a) flux laws WR0 and WR2, Pr given by (5.11) and (5.12) and Pr0 = 10, and (b) flux law MG, Pr given by (5.12) with Pr0 = 10, 2, and 1.

Citation: Journal of Physical Oceanography 30, 5; 10.1175/1520-0485(2000)030<1046:OOTS>2.0.CO;2

Fig. 10.
Fig. 10.

Linear growth rate (units of N) as a function of intrusion slope s and dimensionless vertical wave number m (units of [Kmax/N]1/2). Plus signs indicate where criterion (5.10) for evolution toward staircases is satisfied. Minus signs indicate where evolution is toward conventional intrusions. In both cases, Kturb = 0.01 cm2 s−1, ΓS = 5 × 10−4 and the WR2 flux law is employed, with Pr given by (5.12) and Pr0 = 10: (a) Rρ = 1.1, maximum growth rate 2.66 × 10−4; fastest-growing mode evolves toward a staircase; (b) Rρ = 1.6, maximum growth rate 0.75 × 10−4; fastest-growing mode evolves toward conventional intrusions.

Citation: Journal of Physical Oceanography 30, 5; 10.1175/1520-0485(2000)030<1046:OOTS>2.0.CO;2

Fig. 11.
Fig. 11.

Schematic illustration of coordinate system (χ, ζ), tilted with respect to (x, z) by an angle −θ = tan−1s, where s > 0 is the intrusion slope. Also shown is the manner in which warm, salty intrusions rise across isopycnals (which here are horizontal), and cool, fresh intrusions sink.

Citation: Journal of Physical Oceanography 30, 5; 10.1175/1520-0485(2000)030<1046:OOTS>2.0.CO;2

Fig. 12.
Fig. 12.

Equilibria of intrusions under C-SALT conditions with flux law MG, Pr given by (5.12) with Pr0 = 1, and Rρ and Kturb as indicated. The heavy solid curve represents Rstairρ. Insets show profiles of S (solid curves in the left panels), T (dashed curves), and horizontal velocity u for equilibrated intrusions as functions of z. Conventional intrusions (open circles) are found where Rρ > Rstairρ, whereas staircases (filled circle) are found where Rρ < Rstairρ. For clarity, two repetitions of the periodic numerical solutions are shown.

Citation: Journal of Physical Oceanography 30, 5; 10.1175/1520-0485(2000)030<1046:OOTS>2.0.CO;2

Fig. 13.
Fig. 13.

Evolution of intrusion structure for the Rρ = 1.4 cases shown in Fig. 12, and conditions otherwise as in the C-SALT region (see Table 1): (a) Kturb = 0.01 cm2 s−1, intrusions evolve into staircase consisting of alternating fingering and convective layers; (b) Kturb = 0.1 cm2 s−1, equilibration is as conventional intrusions featuring a cyclical arrangement of fingering, convective, and diffusive convective layers. As in Fig. 12, the MG flux law is employed and Pr is given by (5.12) with Pr0 = 1. Two repetitions of the periodic solutions are shown.

Citation: Journal of Physical Oceanography 30, 5; 10.1175/1520-0485(2000)030<1046:OOTS>2.0.CO;2

Fig. 14.
Fig. 14.

Predicted vs observed step heights for the C-SALT (a,b) and Western Mediterranean (c,d) staircases. The curves denote predicted step heights for the choices of flux law and Pr considered in Fig. 9. Dotted vertical lines indicate ranges of observed step heights.

Citation: Journal of Physical Oceanography 30, 5; 10.1175/1520-0485(2000)030<1046:OOTS>2.0.CO;2

Fig. 15.
Fig. 15.

Predicted vs observed lateral density ratios αTχ/βSχ for the C-SALT (a,b) and Western Mediterranean (c,d) staircases. The curves denote predicted ratios for the choices of flux law and Pr considered in Fig. 9. Heavy dotted lines denote observed values and light dotted lines the uncertainties, as reported by Schmitt (1994) and Krahmann (1997).

Citation: Journal of Physical Oceanography 30, 5; 10.1175/1520-0485(2000)030<1046:OOTS>2.0.CO;2

Fig. 16.
Fig. 16.

Predicted vs observed cross-isopycnal slope for the C-SALT (a,b) and Western Mediterranean (c,d) staircases. The curves indicate predicted ratios for the choices of flux law and Pr considered in Fig. 9. Dotted vertical lines denote estimated cross-isopycnal slopes from data of Schmitt (1994) and Krahmann (1997).

Citation: Journal of Physical Oceanography 30, 5; 10.1175/1520-0485(2000)030<1046:OOTS>2.0.CO;2

Fig. 17.
Fig. 17.

For a staircase step depicted in Fig. 12a, vertical profiles of (a) shear uz; (b) shear squared; (c) buoyancy frequency N (dotted line indicates background value); (d) gradient Richardson number (dotted line indicates critical value 0.25). Except for Rρ = 1.4, C-SALT conditions are assumed (Table 2).

Citation: Journal of Physical Oceanography 30, 5; 10.1175/1520-0485(2000)030<1046:OOTS>2.0.CO;2

Fig. 18.
Fig. 18.

Schematic illustration of two possible scenarios by which steps could evolve laterally into conventional intrusions near thermohaline fronts. Here the solid contours denote positive growth rates, and dashed contours negative growth rates. Plus signs indicate (m, s) where steps are favored on the basis of (5.10), and minuses where conventional intrusions are favored. (a) Step size set by low-gradient region. Suppose, for example, that an eddy leads to increased horizontal gradients (increased ΓS). The existing steps in the low-gradient region (left) might lead to intrusions at like m in the high-gradient region (right; solid dot) even though growth rate is not maximized. In this example, the intrusions in the high-gradient region would develop as conventional intrusions. (b) Step size set by high-gradient region. Suppose that the low-gradient region initially does not contain intrusions, and that growing intrusions in the high-gradient region (right) retain their characteristic m as they propagate into the low-gradient region (left; solid dot). In this example, conventional intrusions in the high-gradient region would become steps in the low-gradient region.

Citation: Journal of Physical Oceanography 30, 5; 10.1175/1520-0485(2000)030<1046:OOTS>2.0.CO;2

Table 1.

Specification of effective diffusivities KS and KT and effective viscosity A (units of cm2 s−1). In all cases γf = 0.6 and γd = 0.2.

Table 1.
Table 2.

Approximate conditions in C-SALT and Western Mediterranean staircases.

Table 2.
Table 3.

Effective diffusion coefficients (in cm2s−1) for equilibrated model runs, assuming ΓS = 5 × 10−4. Superscript “dif” applies to diapycnal diffusive mixing, “adv” to diapycnal advective mixing, and “iso” to isopycnal advective mixing. The first four runs correspond to those shown in Fig. 12. Resolution: 512 Fourier modes for staircases, and 128 Fourier modes for conventional intrusions.

Table 3.
Table 4.

Dependence of effective diffusion coefficients on resolution for Rp = 1.4 runs shown in Fig. 12 (units of cm2s−1).

Table 4.

1

Sequences of steps which are typically an order of magnitude thinner appear where T and S both increase downward, conditions that favor diffusive convection (e.g., Neshyba et al. 1971; Padman and Dillon 1987; Muench et al. 1990.) In this paper, attention is confined primarily to the salt fingering case.

2

The result (2.1) applies to two-dimensional (planar) salt fingers. An extension of the stability analysis to three-dimensional fingers having rectangular planforms yields a critical Stern number of 2/3 (Holyer 1985). Kunze (1987, 1995) considered a similar instability criterion in terms of a finger Richardson or Froude number.

3

Based on Rρ = 1.7, βFS = 1.0 × 10−6 cm s−1, αFT = 0.88βFS, ν = 0.01 cm2 s−1 and β = 0.68 for Griffiths and Ruddick (1980) experiment 2, assuming Sz = ΔS/h, with interfacial contrast ΔS = 1.4% and thickness h = 3.5 cm. For Rρ = 1.03, βFS = 1.6 × 10−4 cm s−1, αFT = 0.93βFS, ΔS = 7.8%, and h < 0.1 cm yields an upper limit for Stern number of 0.007.

4

More simply, one could examine the effect of layer depth, which determines the rate of run down, on the rate of interface spread.

5

In terms of effective salt diffusivity KS and temperature diffusivity KT, Kρ = (RρKTKS)/(Rρ − 1) (Shen 1995). On the basis of microstructure measurements, Schmitt (1988) estimates effective salt diffusivities of 0.28 to 0.74 cm2 s−1 (assuming buoyancy flux ratio αKTTz/βKSSz ≈ 0.5) for fingering interfaces in the western tropical North Atlantic (Rρ ≈ 1.6). These estimates are based on ratios of flux to mean gradient averaged over layers and interfaces, however. In the present context, what is needed is the ratio of flux to gradient within an interface. The latter is roughly ten times the mean gradient, implying KS ≈ 0.03 to 0.07 cm2 s−1, and Kρ ≈ −0.02 to −0.06 cm2 s−1. Merryfield and Grinder, in numerical simulations of heat/salt fingers in an infinitely thick interface, find Kρ ≈ −0.13 cm2 s−1 at Rρ = 1.5.

Save
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  • Fig. 1.

    Evolution of “salinity” (slower-diffusing solute) in a numerical experiment having the same parameter values and initial conditions as that of Özgökmen et al. (1998), but with periodic rather than solid upper and lower boundaries. Lighter shades represent higher concentrations, and darker shades lower concentrations. (a) At t = 0.11 initial sinusoidal perturbations have grown to become fingers, which in turn are becoming unstable to smaller, faster-growing fingers. (b) The smaller fingers soon dominate, and by t = 0.50 a statistically uniform equilibrium has been established. (c) This state persists, and at later times shows no signs of developing convective layers.

  • Fig. 2.

    Salinity field in a numerical simulation of fingers developing at an initially sharp interface for (a) t = 6.13, (b) t = 9.19, (c) t = 12.25, and (d) t = 84.18. In each frame, lighter shades represent higher salinities, and darker shades lower salinities.

  • Fig. 3.

    Horizontally-averaged temperature in numerical simulation of fingers developing at an initially sharp interface, at the same times represented in Fig. 2: (a) t = 6.13, (b) t = 9.19, (c) t = 12.25, (d) t = 84.18. Instead of equilibrating at a finite thickness, the simulated interface decays toward a constant-gradient equilibrium, represented by the dashed lines. The interface sharpness measure ΛT corresponds to the area between the dashed lines and solid curves.

  • Fig. 4.

    Evolution of interface sharpness measures ΛT, ΛS, and ΛΦ for temperature, salinity, and a hypothetical scalar Φ having diffusion coefficient κΦ. Thick solid curve: ΛT; thick dashed curve: ΛS; thin curves: ΛΦ, for indicated κΦ. A boxcar filter 7 time units in width has been applied to ΛT and ΛS.

  • Fig. 5.

    Evolution of density ρ(z, t) from small fluctuations imposed on an initially uniform gradient, according to negative diffusion equations (4.1)–(4.3). Density is expressed in arbitrary units.

  • Fig. 6.

    Schematic depiction of intrusion growth on the (αTz, βSz) plane, according to (5.1)–(5.2). Long-dashed lines denote constant N, and short-dashed lines constant Rρ. Heavy lines separate fingering, convective, diffusive convective and stable regimes. (a) Initial conditions, lying close to background gradients, are denoted by the dot. (b) Growing intrusions describe line segments which lengthen in time. A line segment whose slope is less than R−1ρ is labeled by “S,” and one whose slope is greater than R−1ρ is labeled by “I.” (For intrusions to grow, this slope must lie between 0 and 1.) (c) As growth continues, I begins to penetrate the stable regime, and S the convective regime. (d) Still later, I penetrates the diffusive–convective regime as well. Intrusion I then consists of sequences of fingering, stable, diffusive convective, convective and fingering layers, whereas S consists of alternating fingering and convective layers.

  • Fig. 7.

    Functional forms considered for KfS(Rρ), the effective vertical salt diffusivity due to salt fingering.

  • Fig. 8.

    Example of the dependence on Rρ of λ, the intrusion growth rate and rhs of (5.10). Intrusions evolve toward staircase configurations when λ < rhs, which occurs when Rρ is less than a critical value Rstairρ, which in this instance ≈1.12. Parameters: WR0 flux law, ΓS = 5 × 10−4, Kturb = 0.1 cm2 s−1, Pr given by (5.11). The vertical axis is labeled in units of buoyancy frequency N.

  • Fig. 9.

    Critical value Rstairρ, below which staircases form, as a function of Kturb for (a) flux laws WR0 and WR2, Pr given by (5.11) and (5.12) and Pr0 = 10, and (b) flux law MG, Pr given by (5.12) with Pr0 = 10, 2, and 1.

  • Fig. 10.

    Linear growth rate (units of N) as a function of intrusion slope s and dimensionless vertical wave number m (units of [Kmax/N]1/2). Plus signs indicate where criterion (5.10) for evolution toward staircases is satisfied. Minus signs indicate where evolution is toward conventional intrusions. In both cases, Kturb = 0.01 cm2 s−1, ΓS = 5 × 10−4 and the WR2 flux law is employed, with Pr given by (5.12) and Pr0 = 10: (a) Rρ = 1.1, maximum growth rate 2.66 × 10−4; fastest-growing mode evolves toward a staircase; (b) Rρ = 1.6, maximum growth rate 0.75 × 10−4; fastest-growing mode evolves toward conventional intrusions.

  • Fig. 11.

    Schematic illustration of coordinate system (χ, ζ), tilted with respect to (x, z) by an angle −θ = tan−1s, where s > 0 is the intrusion slope. Also shown is the manner in which warm, salty intrusions rise across isopycnals (which here are horizontal), and cool, fresh intrusions sink.

  • Fig. 12.

    Equilibria of intrusions under C-SALT conditions with flux law MG, Pr given by (5.12) with Pr0 = 1, and Rρ and Kturb as indicated. The heavy solid curve represents Rstairρ. Insets show profiles of S (solid curves in the left panels), T (dashed curves), and horizontal velocity u for equilibrated intrusions as functions of z. Conventional intrusions (open circles) are found where Rρ > Rstairρ, whereas staircases (filled circle) are found where Rρ < Rstairρ. For clarity, two repetitions of the periodic numerical solutions are shown.

  • Fig. 13.

    Evolution of intrusion structure for the Rρ = 1.4 cases shown in Fig. 12, and conditions otherwise as in the C-SALT region (see Table 1): (a) Kturb = 0.01 cm2 s−1, intrusions evolve into staircase consisting of alternating fingering and convective layers; (b) Kturb = 0.1 cm2 s−1, equilibration is as conventional intrusions featuring a cyclical arrangement of fingering, convective, and diffusive convective layers. As in Fig. 12, the MG flux law is employed and Pr is given by (5.12) with Pr0 = 1. Two repetitions of the periodic solutions are shown.

  • Fig. 14.

    Predicted vs observed step heights for the C-SALT (a,b) and Western Mediterranean (c,d) staircases. The curves denote predicted step heights for the choices of flux law and Pr considered in Fig. 9. Dotted vertical lines indicate ranges of observed step heights.

  • Fig. 15.

    Predicted vs observed lateral density ratios αTχ/βSχ for the C-SALT (a,b) and Western Mediterranean (c,d) staircases. The curves denote predicted ratios for the choices of flux law and Pr considered in Fig. 9. Heavy dotted lines denote observed values and light dotted lines the uncertainties, as reported by Schmitt (1994) and Krahmann (1997).

  • Fig. 16.

    Predicted vs observed cross-isopycnal slope for the C-SALT (a,b) and Western Mediterranean (c,d) staircases. The curves indicate predicted ratios for the choices of flux law and Pr considered in Fig. 9. Dotted vertical lines denote estimated cross-isopycnal slopes from data of Schmitt (1994) and Krahmann (1997).

  • Fig. 17.

    For a staircase step depicted in Fig. 12a, vertical profiles of (a) shear uz; (b) shear squared; (c) buoyancy frequency N (dotted line indicates background value); (d) gradient Richardson number (dotted line indicates critical value 0.25). Except for Rρ = 1.4, C-SALT conditions are assumed (Table 2).

  • Fig. 18.

    Schematic illustration of two possible scenarios by which steps could evolve laterally into conventional intrusions near thermohaline fronts. Here the solid contours denote positive growth rates, and dashed contours negative growth rates. Plus signs indicate (m, s) where steps are favored on the basis of (5.10), and minuses where conventional intrusions are favored. (a) Step size set by low-gradient region. Suppose, for example, that an eddy leads to increased horizontal gradients (increased ΓS). The existing steps in the low-gradient region (left) might lead to intrusions at like m in the high-gradient region (right; solid dot) even though growth rate is not maximized. In this example, the intrusions in the high-gradient region would develop as conventional intrusions. (b) Step size set by high-gradient region. Suppose that the low-gradient region initially does not contain intrusions, and that growing intrusions in the high-gradient region (right) retain their characteristic m as they propagate into the low-gradient region (left; solid dot). In this example, conventional intrusions in the high-gradient region would become steps in the low-gradient region.

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