Salt-Fingering Theory

The equations of motion for tall $(\partial/\partial z\ll\partial/\partial x,\partial/\partial y)$fingers were first derived by Stern (1960)

$$\begin{array}{l} w_t=g(\alpha T'- \beta S')+\nu \nabla^2w\\ ... ...abla^2T'\\ S'_t+w\overline{S}_z=\kappa_S\nabla^2S' \end{array}$$ (1)

where w, T' and S' are the vertical velocity, temperature and salinity perturbations (relative to a horizontal average) of the fingers, $\overline{T}_z$ and $\overline{S}_z$ are background vertical temperature and salinity gradients (possibly modified by the fingers), $\nabla^2=\partial^2/\partial x^2+\partial^2/\partial y^2$ the horizontal Laplacian, $\nu$ the molecular viscosity, $\kappa_T$ the molecular diffusivity of heat, and $\kappa_S$ the molecular diffusivity of salt. The quantity $g(\alpha T'-\beta S')$ is the buoyancy anomaly b' of the fingers which induces vertical velocities. The buoyancy anomaly is created by more rapid diffusion of heat than salt between adjacent fingers, creating relatively warm fresh anomalies that are lighter and relatively cool salty anomallies that are heavier than the surrounding fluid. For high Prandtl number, $\nu/\kappa_T\gg 1$ (=10 in the ocean), the viscous term in the vertical momentum equation greatly exceeds vertical acceleration $\partial w/\partial t$. The vertical acceleration can be neglected for all but density ratios $R_\rho-1=\alpha \overline{T}_z/\beta\overline{S}_z-1\ll\kappa_T^2/(4\nu^2)=0.0025$which are not realized in the ocean. Schmitt (1979b) got around the low density ratio limitation by expressing the problem in terms of flux ratio $R_F=\langle w\alpha T'\rangle/\langle w\beta S'\rangle$. For the oceanic range of density ratios $R_\rho$, the vertical momentum balance reduces to being between the buoyancy anomaly and viscous drag. For low Lewis number, $\kappa_S/\kappa_T\ll 1$ (= 0.01 in ocean), the diffusion term in the salinity equation can be neglected for density ratios $R_\rho\ll\kappa_T/\kappa_S=100$.

Solutions to the above equations can be found of square planform $e^{\sigma t}\sin(k_xx) \sin(k_yy),$ sheet planform $e^{\sigma t}\sin(k_xx)$or a combination of the two that can take the form of triangles, hexagons or asymmetric plumes (Schmitt 1994). Here, $\sigma$is the growth rate, and k_x and k_y are horizontal wavenumbers. Growth rates $\sigma$ are positive for density ratios $R_\rho <\kappa_T/\kappa_S=100$ (Huppert and Manins 1973) and horizontal wavenumbers k

$$k<\root 4 \of { \frac{(\kappa_T-R_\rho\kappa_S)g\beta\overline{S}_z} {4\nu\kappa_T\kappa_S} }.$$ (2)

(Fig. 1). Growth rates increase as density ratios $R_\rho$ approach 1. For any given density ratio $R_\rho$, maximum growth rates occur at intermediate wavelengths (dashed curve) due to competition between destabilizing heat diffusion and viscous drag both which become stronger at smaller scales.

Figure 2 displays the ratio of heat- to salt-flux contributions to buoyancy-flux $R_F=\langle w'T'\rangle /\langle w'S'\rangle =R_\rho \delta _T/\delta _S$as a function of density ratio $R_\rho$ and horizontal wavelength $\lambda$. Heat-salt laboratory experiments corrected for molecular diffusion of heat through the interface and tank sidewalls typically find flux ratios of 0.5-0.65 (Fig. 3; Turner 1967; Schmitt 1979a; Taylor and Bucens 1989; Shen 1993), consistent with fastest-growing finger flux ratios or fingers of maximum velocity, though McDougall and Taylor (1984) and Linden (1973) both reported lower values, more consistent with steady fingers. Schmitt et al. (1987) inferred flux ratios of 0.85 in the thermohaline staircase east of Barbados based on the lateral density ratio of the homogeneous layers. These were argued to be a combination of fastest- growing finger flux ratio and either (i) turbulence with $R_F=R_\rho$ (Marmorino 1990; Fleury and Lueck 1991) or nonlinearity in the seawater equation of state influencing the flux divergence (McDougall 1991).

Theoretical fingering studies have focussed on fastest-growing, steady and spectral descriptions of finger dynamics. Fastest-growing investigations (Stern 1975; Schmitt 1979b; Kunze 1987) have been justified by the close match between the fastest-growing flux ratio and that found in the laboratory and numerical simulations (Fig. 3) as well as the expected long-time dominance of fastest-growing fingers. The spectral approach recognizes that a range of scales is likely to be present, and that fingers may break down due to secondary instability before fastest-growing fingers dominate, particularly at low density ratio (Gargett and Schmitt 1982; Shen and Schmitt 1995).

Steady models have been applied to thin interfaces sandwiched between homogeneous layers on the grounds that, because of the interface's finite thickness l_i and salinity step $\Delta S$, these are expected to evolve quickly to steady state (Huppert and Manins 1973; Stern 1976; Joyce 1982; Howard and Veronis 1987; 1992). Stern (1976) examined the solution to the steady equations, choosing the wavenumber $[g\alpha \overline{T}_z/(4\nu \kappa _T)]^{1/4}$that produced the maximum buoyancy-flux for a given $\Delta S$. The associated flux ratio was 0.2-0.25. Howard and Veronis (1987) examined the structure of steady fingers of maximum buoyancy-flux extending between two homogeneous reservoirs in the limit of very small salt diffusivity $\kappa_S/\kappa_T\ll 1$, obtaining similarity solutions for the boundary layers between adjacent fingers. They again found flux ratios of $\sim 0.25$ associated with the wavenumber of maximum buoyancy-flux. Howard and Veronis (1992) examined the stability of these fingers as a function of a nondimensionalized salinity step, or finger aspect ratio, $Q=\beta \Delta S/(\lambda _b\alpha \overline{T}_z)=l_i/(\lambda _bR_\rho)$ where $\lambda _b=[4\nu \kappa _T/(g\alpha \overline{T}_z)]^{1/4}$is the buoyancy-layer scale. They found that the dominant instability switched from being oscillatory involving viscosity and heat diffusivity for $Q<\nu /\kappa _T$ to pure real and due to shear between adjacent fingers $\nabla w$ at higher Q.

Relaxing the condition that the fingers be tall and thin (e.g., Howard and Veronis 1987), additional terms must be included such as the vertical pressure gradient p_z in the vertical momentum equation, vertical diffusion $\nu \partial ^2/\partial z^2$, and horizontal advection of temperature and salinity, particularly at the finger tips. These additional terms could play an important role at low density ratios, where fingers are short and stubby rather than tall and thin (Howard and Veronis 1987), or for fingers evolving in a thin high-gradient interfaces sandwiched between two homogeneous layers as characterizes most lab and numerical experiments. Numerical simulations under these conditions show formation of bulbous tips where the fingers intrude into the adjacent homogenous layers (Fig. 4; Shen 1989; Shen and Veronis 1996). The resulting structure depends on the destabilizing salinity step across the interface $\Delta S$ as well as the density ratio $R_\rho$. Presumably, there is also a dependence on the thickness of the high-gradient interface l_i.

In the remainder of this section on salt-finger theory, attention will be focussed on fastest-growing fingers because the fastest-growing flux ratio is found in lab and numerical simulations (Fig. 3), because thin interfaces for which fingers might be expected to extend from one reservoir to the next do not appear to be realized in the ocean, and because numerical models do not achieve a steady state except in a statistical sense (Shen 1993; 1995). Shen (1993) demonstrated that steady fingers of maximum-buoyancy-flux wavenumber could not reproduce the lab and numerical flux ratios, regardless of the degree of salt diffusion between adjacent fingers. He showed that choosing the wavenumber associated with maximum velocity across an interface produced flux ratios in the observed range. Maximum velocity fingers closely resemble fastest-growing.

a. fastest-growing fingers

Theory readily identifies the fastest-growing salt-fingering wavenumber

$$k=\frac{2\pi}{\lambda}=\root 4 \of { \frac{g\beta\overline{S}_z(R_\rho 1)}{\nu \kappa _T}}$$ (3)

(Fig. 5), associated growth rate

$$\sigma_{\mbox{\rm max}}={1 \over 2}\sqrt{\frac{(\kappa_T-R_\r... ...ine{S}_z}{\nu } }\left( \sqrt{R_\rho}-\sqrt{R_\rho-1} \right),$$ (4)

(Fig. 6) and flux ratio

$$R_F=\frac{\alpha F_T}{\beta F_S}= \frac{\alpha \langle w'T'\r... ...a_S}=\sqrt {R_\rho}\left( \sqrt{R_\rho}-\sqrt{R_\rho-1}\right)$$ (5)

(Fig. 3) as functions of salinity gradient $\overline{S}_z=\partial S/\partial z$ and density ratio $R_\rho=\alpha\overline{T}_z/\beta\overline{S}_z$ (Stern 1960; 1975; Schmitt 1979b; Kunze 1987). Flux ratios and wavelengths observed in the laboratory (Turner 1967; Linden 1973; Schmitt 1979a; McDougall and Taylor 1984; Taylor and Bucens 1989) and in numerical simulations (Shen 1993; 1995) are consistent with dominance of fastest-growing fingers (Fig. 3).

b. fluxes

Quantifying the fingering heat- and salt-fluxes of relevance to the general oceanographic community is more difficult, requiring knowledge of what limits the growth of fingers to finite amplitude. This was originally attempted through laboratory experiments in which the heat- and salt-fluxes were measured between two well-mixed layers separated by a thin fingering-favorable interface (Turner 1967; Linden 1973; Schmitt 1979a; McDougall and Taylor 1984; Taylor and Bucens 1989). The heat- and salt-fluxes were found to depend only on the salinity step across the interface $\Delta_S$ and the density ratio $R_\rho$

$$\begin{array}{l} g\beta F_S=c(g\beta\Delta S)^{4/3}f(R_\rho) \\ g\alpha F_T=R_F\cdot g\beta F_S \end{array}$$ (6)

where the flux ratio R_F was in agreement with the theoretical expression for fastest-growing fingers (3) in the Turner (1967), Schmitt (1979a) and Taylor and Bucens (1989) work but lower in the Linden (1973) and McDougall and Taylor (1984) experiments. The $\Delta S^{4/3}$ flux law (6) is consistent with dimensional reasoning under the assumption that the interface thickness is unimportant (Turner 1967). Application of this $\Delta S^{4/3}$ flux law is hampered in most of the fingering-favorable ocean by the absence of well-defined layers and interfaces so that the salinity step $\Delta S$ cannot be quantified. Even in well-defined staircases, the lab $\Delta S^{4/3}$ flux laws overestimate fluxes by an order of magnitude (Gregg and Sanford 1987; Lueck 1987; Kunze 1987; Hebert 1988).

Dimensional arguments were also used by Stern (1969) to argue that the ratio of the buoyancy-flux $\langle w'b'\rangle$ to $\nu N^2$ is limited to a maximum value

$$\frac{\langle w'b'\rangle}{\nu N^2}\sim O(1)$$ (7)

where $\nu$ is the molecular viscosity and N the buoyancy frequency. This nondimensional collective instability criterion for the magnitude of the finger fluxes has come to be known as the Stern number. More rigorously, Holyer (1981) demonstrated secondary instability of steady $(\sigma= 0)$ fingers to oscillatory long internal-wave oscillations of very low aspect ratio K_x/K_z if $\langle w'b'\rangle/\nu N > 1/3$. In Holyer (1984), she confirmed that collective instability is the fastest-growing provided the Prandtl number $\nu/\kappa_T$ is very large, $(K_x^2/K^2)(\nu\kappa_S/\kappa_T^2)[(R_\rho-1)/(1-R_\rho\kappa_S/\kappa_T)]\gg 1$ and $K^2\ll k^2$. However, for the heat-salt system, she identified a different nonoscillatory instability with K_x=0, K_z=0.3k_x that grew ten times more rapidly than collective instability.

Kunze (1987) suggested that finger growth could be limited by secondary instability when a finger Froude number

$$Fr_f =\frac{\vert\nabla w\vert}{N} < O(1)$$ (8)

in which the usual vertically-sheared horizontal velocity |V_z| is replaced by horizontally-sheared vertical velocity $\vert\nabla w\vert$. As in Miles and Howard's (1961), this identifies when the shear contains sufficient kinetic energy to overcome the potential energy of the stratification. For fastest-growing fingers, this nondimensional constraint is virtually identical to the Stern number of collective instability except very near density ratios $R_\rho = 1$. A Reynolds number constraint $w\lambda/\nu$ yields a similar constraint (Stern 1969). Applied to the gradients in oceanic fingering-favorable regions, the Stern, or finger Froude, number constraint predicts fluxes of comparable magnitude to those estimated from microstructure observations (Schmitt and Gargett 1982; Gregg and Sanford 1987; Lueck 1987; Marmorino et al. 1987) and largescale budgets (Hebert 1988)

$$\begin{array}{l} g\alpha F_T = g\alpha\langle w'T'\rangle = ... ...\left(\sqrt{R_\rho}+\sqrt{R_\rho-1}\right)^2\right] \end{array}$$ (9)

where Fr_c is the critical finger Froude number, C_w, C_T and C_S are geometric constants associated with the horizontal planform structure (= 0.25 for square planform fingers, 0.5 for sheets), and the time-average factor

$$\sigma t_{\mbox{\scriptsize {\rm max}}}=ln\left[ \sqrt{\frac{... ...-1}} \left(\sqrt{R_\rho}+ \sqrt{R_\rho-1}\right)^{3/2}\right].$$ (10)

Assuming that the salinity step $\Delta S$ is defined by the maximum attained height $h_{\rm max}$ of the fingers, that is, equating the high-gradient interface thickness l_i and maximum finger height $h_{\rm max}$, the Stern or finger Froude number constraints reproduce laboratory $\Delta S^{4/3}$ flux laws except for density ratios $R_\rho < 2$ where fluxes are underpredicted by factors of 2-3 (Figs. 7 and 8; Kunze 1987; Shen 1993). This criterion implies interface thicknesses $l_i\sim O$(0.1 m), an order of magnitude smaller than the 2-5 m thickness typically observed in oceanic staircases. Shen (1993) argued that comparison of Kunze's gradient formulation with lab measurements was inappropriate because lab fingers extend across the entire interface so are fed by homogeneous reservoirs. However, his numerical simulations showed little sensitivity to whether the horizontal salinity structure was a square wave or a sine wave.

c. staircase lengthscales

With a dependence on the salinity gradient $\overline{S}_z$ across the interface (9), oceanic fingering fluxes depend both on the layer thickness L_o, which establishes the interfacial salinity step $\Delta S = \langle \overline{S}_z\rangle (L_o + l_i)$ given a largescale smoothed salinity gradient $\langle \overline{S}_z\rangle$, and the interface thickness l_i which determines the interfacial gradient $\overline{S}_z = \Delta S/l_i =\langle \overline{S}_z\rangle(L_o + l_i)/l_i [\simeq\langle \overline{S}_z\rangle L_o/l_i$ for $L_o\gg l_i$.

What controls layer and interface staircase thicknesses in the ocean is not known. While staircase formation has traditionally been thought of as a 1-D instability induced by the countergradient double-diffusive buoyancy-fluxes based on lab studies (Stern and Turner 1969), it may also arise from horizontal interleaving processes (ref. appropriate section) which may also explain the alternating diffusive and fingering layers that extend across entire basins in the Arctic Ocean, crossing different water- masses and water ages with impunity (Carmack et al. 1998).

Given the persistence and invariance of oceanic thermohaline staircase T, S structures over at least 25 years (Schmitt 1995), staircases have more than enough time to establish thin $\Delta S^{4/3}$ flux law interfaces. A range of interface thicknesses were found in the thermohaline staircase east of Barbados, including a few of O(0.1 m). However, the bulk of the interfaces were 2-5 m thick, an order of magnitude too large for $\Delta S^{4/3}$flux laws to apply.

Kelley's (1984) scaling for layer thickness $H =\sqrt{\kappa_T/N\cdot f} (R_\rho, \nu/\kappa_T, \kappa_T/\kappa_S)$ and diffusivity $K_T = cfRa^{1/3}\kappa_T$appears to work well for diffusively-unstable staircases. One might blindly replace the molecular diffusivity of heat $\kappa_T$ with the fingering salt diffusivity $F_S/\langle \overline{S}_z\rangle$. However, using observed microstructure estimates of the salt-flux (Gregg and Sanford 1987; Lueck 1987) then underestimates the layer thicknesses. Interestingly, a laboratory $\Delta S^{4/3}$ salt-flux (6) produces layer thicknesses of the right order of magnitude.

d. differing fluxes in the lab and ocean

Why are oceanic fluxes so much weaker, or equivalently, oceanic interfaces so much thicker, than those found in laboratory experiments and numerical simulations? For one thing, most lab and numerical experiments have been initialized as two homogeneous layers separated by a very thin interface which is allowed to grow under the influence of fingers. In the ocean, the initial state is better-described as one of continuous stratification. However, even when experiments are initialized with thick gradient regions, they evolve toward thinner interfaces with $\Delta S^{4/3}$ fluxes across them (Stern and Turner 1969; Linden 1978). This suggests that there is some additional process in the ocean not present in these idealized experiments which prevents thin interfaces from forming. Possible candidates are finescale internal wave shear and strain fluctuations, and intermittent shear-driven turbulence.

e. interaction with internal-wave strain $\xi_z$

Internal-wave vertical divergence $\partial w/\partial z=\partial(\xi_z)/\partial t$would act to thicken and thin interfaces, modulating the fingering environment and possibly leading to a rectified effect on fluxes, depending on the relative timescales of the strain $\xi_z$ and finger adjustment. Stamp et al. (1998) found a feedback that augmented the internal waves. However, this process has not been explored an oceanic parameter regime.

f. interaction with internal-wave vertical shear ${\bf V}_z$

Vertical shear ${\bf V}_z = (U_z, V_z)$ should act to tilt square planform salt fingers. Linden (1974) demonstrated in the lab and analytically (see also Thangam et al. 1984) that, in steady shear U_z, fingers formed vertical sheets $\sin(k_yy)$ aligned with the shear. Linden reported the fluxes to be unaltered by the presence of vertical shear.

However, finescale vertical shear in the ocean is dominated by O(N) near-inertial internal wave fluctuations. These rotate clockwise in time on a timescale of f^­1 where $f = 2\Omega\sin({\rm lat}^\circ)$ is the Coriolis frequency, and both clockwise and counterclockwise with depth. Near-inertial shear will turn out of alignment with initially-aligned sheets, causing them to tilt over on timescales comparable to the finger growth rate (Kunze 1990). Shear- tilting may explain the nearly-horizontal 0.5-cm laminae consistently observed with a shadowgraph (Laplacian of index of refraction $\nabla^2_\eta$) in fingering-favorable parts of the ocean (Kunze et al. 1987; St. Laurent and Schmitt 1997). These laminae appear to have horizontal scales consistent with fastest-growing scales. Their small vertical scale would diffuse away molecularly in $\sim 4$ minutes if it was temperature, and in $\sim 8$ h if it was salinity, indicating that it must be continuously produced in order to be present in the ocean. Shear-tilting of growing fingers was argued to be a plausible explanation. Kunze (1990) showed that the observed structure could represent remnant shear-tilted salt microstructure just before it is molecularly diffused away using the WKB wavenumber equation

$$\frac{Dk_z}{Dt}=-k_xU_z.$$ (11)

He suggested that near-inertial shear might also alter the finger Froude number criterion as a means to explain towed microstructure measurements finding Cox numbers $\langle(\nabla T)^2\rangle/(\langle\overline{T}_z\rangle 2)$ that depended linearly on background temperature gradient in the staircase east of Barbados (Fig. 9; Marmorino et al. 1987; Fleury and Lueck 1991). This observation implies fluxes independent of interface thickness while the Stern or finger Froude number constraint (8) predicts Cox numbers independent of interface thickness, and fluxes inversely dependent on interface thickness (9). Kunze showed that a criterion

$$\frac{U_zw_y}{N^2}\sim O(1)$$ (12)

reproduced the observed Cox numbers, where $U_z = \Delta U/l_i$ is the background shear, w_y the finger shear and the velocity step $\Delta U$ was found to be independent of interface thickness, but his arguments were not rigorous. Clearly, a better understanding of how near-inertial shear modifies finger dynamics will be needed to quantify salt-finger fluxes of heat, salt and momentum in the ocean. A synergy of theory, laboratory and numerical experiments will likely be needed to resolve this problem.

g. interaction with intermittent shear-driven turbulence

Turbulence produced by internal wave shear is an intermittent process found in 5-10% of the stratified ocean interior. Linden (1971) demonstrated that even very weak turbulence completely disrupts finger fluxes. This was used by Kunze (1995) to argue that rapidly-growing fingers at density ratios $R_\rho < 2.0$ will be able to grow to their maximum height (as determined by a finger Froude number-like constraint) without being disturbed by turbulence. At higher density ratios, however, intermittent turbulence arising every 10-20 buoyancy periods would limit finger growth. This results in dramatically reduced fingering fluxes for density ratios $R_\rho > 2$ (Fig. 10) and may explain why thermohaline staircases are found in the ocean only for density ratios less than two. The intermittency of turbulence following a water parcel is not well-known in the ocean.

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FIGURE CAPTIONS

Figure EK1: Contours of salt-finger growth rate normalized by buoyancy frequency $\sigma/N$ for parameter values typical of the high-gradient interfaces in the thermohaline staircase east of Barbados (Table EK1) as a function of finger wavelength $\lambda$ and background density ratio $R_\rho=\alpha\overline{T}_z/ (\beta\overline{S}_z)$. Growth rates are negative (decay) for wavelengths $\lambda < 0.8$ cm and positive at larger wavelengths. They exceed the buoyancy frequency $N = \sqrt{g\beta\overline{S}_z(R_\rho-1)}$ only at very low density ratios. The dotted curve displays the wavelength of maximum growth rate.

Figure EK2: Contours of flux ratio $R_F=\langle w\alpha T'\rangle/\langle w\beta S'\rangle$ as a function of finger wavelength $\lambda$ and density ratio $R_\rho$. The dotted curve shows the wavelength of maximum growth rate as a function of density ratio. The flux ratio R_F increases from 0 for vanishing growth rate to 0.85 for density ratios $R_\rho < 1.10$ and wavelengths $\lambda > 7$ cm.

Figure EK3: Salt-finger flux ratio $R_F=\langle w\alpha T'\rangle/\langle w\beta S'\rangle$ vs density ratio $R_\rho$. Symbols are from laboratory and numerical simulation estimates, the solid curve for theoretical fastest-growing fingers and the dashed curve for steady $(\sigma= 0)$ fingers.

Figure EK4: An example of numerical simulation of growing salt fingers in a thin interface (Shen 1993). The resulting structure does not resemble the tall thin fingers of theoretical treatments but quickly becomes nearly isotropic and blob-like.

Figure EK5: Theoretical wavelengths $\lambda$ as a function of density ratio $R_\rho$ for fastest-growing (solid) and steady (dashed) fingers using typical properties of interfaces in the thermohaline staircase east of Barbados (Table EK1).

Figure EK6: Maximum theoretical finger growth rates $\sigma$ (solid) as a function of density ratio $R_\rho$ for interfaces in the thermohaline staircase east of Barbados (Table EK1). Also shown are buoyancy frequencies N (dashed) and molecular viscous and diffusive timescales (dotted).

Figure EK7: Theoretical heat (dashed), salt (dotted) and total (solid) buoyancy-fluxes as a function of density ratio $R_\rho$. The maximum finger amplitude is assumed to be constrained by a critical finger Froude number $Fr_c = \vert\nabla w\vert/N =2.0$ where the vertically-sheared horizontal velocity of the usual gradient Froude number has been replaced with the horizontally- sheared vertical velocity between adjacent fingers. The negative of the total buoyancy-flux is shown. The upper panel assumes an interface thickness l_i of 2 m, consistent with observed values. The central panel assumes that the interface thickness l_i is identical to the maximum finger height $h_{\rm max}$, producing interface thicknesses of $\sim O$(10 cm) and higher fluxes as a result. The bottom panel normalizes these fluxes by the lab $\Delta S^{4/3}$ flux laws and compares the salt-flux (dotted) with values from laboratory and numerical experiments (symbols). The model reproduces the observed values for density ratios $R\rho_ > 2$ but underestimates fluxes at low density ratios $R_\rho$.

Figure EK8: Flux ratio R_f (a), $\Delta S^{4/3}$ flux law coefficient c (b) and Stern number A (c) as a function of density ratio from numerical simulations (solid diamonds) and lab experiments (other symbols) (from Shen 1993). The laboratory and numerical numbers are consistent with each other and indicate that the Stern number is not an invariant.

Figure EK9: Temperature Cox number C_T versus vertical temperature- gradient $\overline{T}_z$ in interfaces of the thermohaline staircase east of Barbados. Data are from (a) a towed microscale conductivity cell (Marmorino 1989) and (b) a towed microthermiistor (Fleury and Lueck 1991). The solid dots in (a) denote the mean of the distribution, the open circles the mode. Only the means are displayed in (b). Both data sets display mean Cox numbers $C_T\alpha\overline{T}_z$. The Stern or finger Froude number predicts a Cox number of $\sim 8$, independent of temperature-gradient.

Figure EK10: Theoretical finger salt diffusivities K_S as a function of density ratio $R_\rho$ in the thermohaline staircase east of Barbados. The solid curve assumes a finger Froude number constraint $Fr_c = \vert\nabla w\vert/N =2.0$, the thin dashed curve a mixed finger/wave Froude number U_zw_y/N² = 2.0 with wave shear U_z = 0.6N, and the thick dashed curve the appropriate diffusivity for gradients smoothed over a staircase. The dotted curves are diffusivities where finger growth is halted by intermittent internal-wave- driven turbulence every 10 buoyancy periods. A plausible scenario for the ocean is that (i) at high density ratios $R_\rho > 2.4$, finger growth is inhibited by turbulence rather than self secondary instability (dotted curves) while, (ii) for low density ratios $R_\rho < 1.6$, finger countergradient buoyancy-fluxes overcome turbulent downgradient buoyancy-fluxes (stippling) and staircases form, producing high-gradient interfaces and amplifying the fluxes (thick dashed). At intermediate density ratios, finger diffusivities will lie between the thick solid and thin dashed curves depending on the strength of the internal wave shear U_z. The resulting curve resembles Fig. 5 of Schmitt (1981), albeit a factor of 30 lower.