## 1. Introduction

Salt-fingering-favorable stratification occurs when a gravitationally unstable vertical gradient of salinity is stabilized by that of temperature. In such conditions, the faster diffusion of heat relative to salt can generate cells of rising and sinking fluid known as salt fingers. Salt-fingering-favorable stratification is found in much of the tropical and subtropical pycnocline (You 2002). The most striking signatures are in thermohaline staircases, stacked layers of different water types separated by sharp thermal and saline gradient interfaces. These are found at several locations, such as in the subtropical confluence east of Barbados, under the Mediterranean and Red Sea salt tongues, and in the Tyrrhenian Sea (Tait and Howe 1968; Lambert and Sturges 1977; Schmitt 1994).

Salt-fingering-favorable conditions can be created by large-scale vertical shears associated with the subinertial circulation (Schmitt 1990). Zhang et al. (1998) found a significant relationship between the formation of fingering-favorable conditions and large-scale vertical shear in a model of the North Atlantic.

In the presence of finescale shear such as that due to high-wavenumber near-inertial waves in the ocean, salt-fingering instability is supplanted by salt-sheet instability, alternating planar regions of rising and sinking fluid, aligned parallel to the sheared flow (Linden 1974). Although theories for the initial growth of salt-fingering and salt-sheet instabilities are well established (Stern 1960; Linden 1974; Schmitt 1979b; Kunze 2003; Smyth and Kimura 2007), the vertical fluxes of momentum, heat, and salt in the nonlinear regime are not well understood.

Dissipation rates of property variances in turbulent salt fingering can be measured in the ocean by microstructure profilers, but interpretation of these results has historically relied on Kolmogorov’s hypothesis of isotropic turbulence (Gregg and Sanford 1987; Lueck 1987; Hamilton et al. 1989; St. Laurent and Schmitt 1999; Inoue et al. 2008). Kolmogorov (1941) proposed that small-scale turbulence statistics are universal in the limit of high Reynolds number. According to this hypothesis, anisotropy of the energy-containing scales is lost in the turbulent energy cascade so that the small scales, where energy is finally dissipated, are statistically isotropic. The assumption of small-scale isotropy greatly simplifies both the theory and modeling of turbulence, as well as interpretation of microstructure measurements. Because the Reynolds number in oceanic salt-fingering systems is *O*(10) (McDougall and Taylor 1984), the validity of this assumption is questionable. We will examine both mixing rates and the isotropy of the dissipation range in turbulent salt fingering by means of direct numerical simulations of diffusively unstable shear layers.

Estimates of dissipation rates, combined with the Osborn and Cox (1972) diffusivity model, furnish estimates of the effective diffusivities of heat, salt, and momentum. Effective diffusivities are used to parameterize turbulent fluxes in models of large-scale phenomena, ranging from finescale thermohaline intrusions (e.g., Toole and Georgi 1981; Walsh and Ruddick 1995; Smyth and Ruddick 2010) to basin-scale circulations (e.g., Zhang et al. 1998; Merryfield et al. 1999).

Effective diffusivities can be calculated directly from direct numerical simulations (DNS). Thermal and saline effective diffusivities for two-dimensional (2D) salt fingering have been computed in previous studies (W. J. Merryfield and M. Grinder 2000, personal communication; Stern et al. 2001; Yoshida and Nagashima 2003; Shen 1995). The effective diffusivities for 3D sheared salt fingering were first calculated by Kimura and Smyth (2007) for a single initial state.

In this paper and the companion paper (Smyth and Kimura 2011), we extend Kimura and Smyth (2007) to cover a range of oceanographically relevant initial background states. We focus on the case for which the Richardson number Ri exceeds the critical value ¼: that is, shear is too weak to produce Kelvin–Helmholtz instability (Miles 1961). The case Ri < ¼ is addressed separately (Smyth and Kimura 2011). Although oceanic salt fingering does not always lead to staircases; that is the regime we focus on here as it is most amenable to direct simulation.

In section 2, we describe our DNS model and initial conditions. An overview of salt-sheet evolution is given in section 3. Section 4 discusses the nature of anisotropy at dissipation scales and its implications for interpreting profiler measurements. In section 5, we discuss the effective diffusivities of momentum, heat, and salt and suggest a parameterization in terms of ambient property gradients. Our conclusions are summarized in section 6.

## 2. Methodology

**u**(

*x*,

*y*,

*z*,

*t*) = (

*u*,

*υ*,

*w*), in a nonrotating, Cartesian coordinate system (

*x*,

*y*,

*z*) areHere,

*D*/

*Dt*= ∂/∂

*t*+

**u**·

**∇**and

*ν*are the material derivative and kinematic viscosity, respectively. The variable

*π*represents the reduced pressure (pressure scaled by the uniform density

*ρ*

_{0}). The total buoyancy is defined as

*b*= −

*g*(

*ρ*−

*ρ*

_{0})/

*ρ*

_{0}, where

*g*is the acceleration due to gravity. Buoyancy acts in the vertical direction, as indicated by the vertical unit vector

**k**. We assume that the equation of state is linear, and therefore the total buoyancy is the sum of thermal and saline buoyancy components (

*b*and

_{T}*b*), each governed by an advection–diffusion equation,Molecular diffusivities of heat and salt are denoted by

_{S}*κ*and

_{T}*κ*, respectively.

_{S}Periodicity intervals in streamwise *x* and spanwise *y* directions are *L _{x}* and

*L*

_{y}_{.}The variable

*L*represents the vertical domain length. Upper and lower boundaries, located at

_{z}*z*= −

*L*/2 and

_{z}*z*=

*L*/2, are impermeable (

_{z}*w*= 0), stress free (∂

*u*/∂

*z*= ∂

*υ*/∂

*z*= 0), and insulating with respect to both heat and salt (∂

*b*/∂

_{T}*z*= ∂

*b*/∂

_{S}*z*= 0).

*h*,The constants Δ

*u*, Δ

*B*, and Δ

_{T}*B*represent the change in streamwise velocity, thermal buoyancy, and saline buoyancy across the half-layer thickness of

_{S}*h*. For computational economy, we set

*h*= 0.3 m, which is larger than the maximum finger height predicted by Kunze (1987). This is at the low end of the range of observed layer thickness (e.g., Gregg and Sanford 1987; Kunze 1994). The change in the total buoyancy is Δ

*B*= Δ

*B*+ Δ

_{T}*B*. In all the DNS experiments, the initial interface buoyancy frequency

_{S}^{−2}rad s

^{−1}, a value typical of the thermohaline staircase east of Barbados (Gregg and Sanford 1987). These constants can be combined with the fluid parameters

*ν*,

*κ*, and

_{T}*κ*to form the following nondimensional parameters, which characterize the flow at

_{S}*t*= 0:

We have done seven experiments with different Ri and *R _{ρ}* (Table 1). The bulk Richardson number Ri measures the relative importance of stratification and shear. If Ri < 0.25, the initial flow is subjected to shear instabilities (Miles 1961; Howard 1961; Hazel 1972; Smyth and Kimura 2011). Here, we chose high enough Ri to ensure that shear instabilities do not disrupt the growth of salt-fingering modes. A typical range of Ri in a sheared thermohaline staircase is ~(3–100) (Gregg and Sanford 1987; Kunze 1994). The Reynolds number based on the half-layer thickness and the half change in streamwise velocity is given by Re =

*h*Δ

*u*/

*v*. Because Δ

*B*and

*h*are kept constant in our simulations, Re and Ri are not independent: Re = 1354Ri

^{−1/2}with Re = 0 and Ri = ∞, representing a single unsheared case that was included for comparison.

Relevant parameters used in our DNS experiments. The wavenumber of the fastest-growing salt-fingering instability is determined by the magnitude of wavenumber, *k*^{2} + *l*^{2}, where *k* and *l* represent the streamwise and spanwise wavenumbers. In the case of salt sheets (all cases except DNS5), there is not streamwise dependence (*k* = 0), where the salt-fingering case (DNS5) has *k* = *l*. In our DNS experiments, *k*^{2} + *l*^{2} is kept constant.

The density ratio *R _{ρ}* quantifies the stabilizing effect of thermal to destabilizing effect of saline buoyancy components; salt fingering grows more rapidly as

*R*approaches unity. The value

_{ρ}*R*≈ 2 is characteristic of the main thermocline in the subtropical gyres (Schmitt et al. 1987; Schmitt 2003). Thermohaline staircases form when

_{ρ}*R*is less than 1.7 (Schmitt et al. 1987), and values as low as 1.15 have been reported (Molcard and Tait 1977). To cover this range of fingering regimes, we use the values 2, 1.6, and 1.2. The Prandtl number Pr and diffusivity ratio

_{ρ}*τ*represent ratios of the molecular diffusivities of momentum, heat, and salt. The Prandtl number was set to 7, which is typical for seawater. The diffusivity ratio in the ocean is 0.01: that is, the heat diffuses two orders of magnitude faster than salt. The vast difference in diffusivity requires DNS to resolve a wide range of spatial scales, making it computationally expensive. In previous DNS of seawater,

*τ*has been artificially increased to reduce computational expense (e.g., Stern et al. 2001; Gargett et al. 2003; Smyth et al. 2005). Kimura and Smyth (2007) conducted the first 3D simulation with

*τ*= 0.01 and found that increasing

*τ*from 0.01 to 0.04 reduced thermal and saline effective diffusivities by one-half. In the cases presented here, we set

*τ*to 0.04 to allow extended exploration of the Ri and

*R*dependence.

_{ρ}The fastest-growing salt-sheet wavelength predicted by linear stability analysis is *λ*_{fg} = 2*π*(*vk _{T}*2

*h*/Δ

*B*)

^{1/4}(Stern 1975; Schmitt 1979b). Our value of

*λ*

_{fg}= 0.032 m is similar to the observed wavelength (Gregg and Sanford 1987; Lueck 1987). We accommodate four wavelengths of the fastest-growing primary instability in the spanwise direction,

*L*= 4

_{y}*λ*

_{fg}. The vertical domain length

*L*was chosen such that vertically propagating plumes reach statistical equilibrium; we found that

_{z}*L*equal to 6 times

_{z}*h*was sufficient. Here,

*L*was chosen to be large enough to accommodate subsequent secondary instabilities. After sensitivity tests, we chose

_{x}*L*= 28

_{x}*λ*

_{fg}.

The primary instability was seeded by adding an initial disturbance proportional to the fastest-growing mode of linear theory, computed numerically as described in Smyth and Kimura (2007). We seed square planforms for the unsheared case and sheets in the presence of shear. The vertical displacement amplitude is set to 0.02*h*, and a random noise is added to the initial velocity field with an amplitude of 1 × 10^{−2} *hσ*_{fg} to seed secondary instabilities. The variable *σ*_{fg} indicates the growth rate of the fastest-growing linear normal mode.

The numerical code used to solve (1)–(3) is described by Winters et al. (2004). The code uses Fourier pseudospectral discretization in all three directions and time integration using a third-order Adams–Bashforth operator. A time step is determined by a Courant–Friedrichs–Lewy (CFL) stability condition. The CFL number is maintained below 0.21 for DNS experiments presented here. The code was modified by Smyth et al. (2005) to accommodate a second active scalar, which is resolved on a fine grid with spacing equal to ½ the spacing used to resolve the other fields. The fine grid is used to resolve salinity. The fine-grid spacing is equal to

## 3. Simulation overview

Figure 1 shows the saline buoyancy field for the case Ri = 6, *R _{ρ}* = 1.6 at selected times. The time is scaled by the linear normal growth rate of salt sheets

*σ*

_{fg}described by Smyth and Kimura (2007). Figure 1a shows the salt-sheet instability: the planar regions of vertical motions oriented parallel to the background shear flow. Rising and sinking fluid appear in green and blue, respectively.

When the salt sheets reach the edges of the interface (indicated by purple and red), they start to undulate in the spanwise direction (Fig. 1b). At the same time, the salt sheets develop streamwise dependence at the edges and center of the interface. At the edges of the interface, the streamwise dependence appears as ripples. At the center, tilted laminae are evident, reminiscent of those seen in shadowgraph images of optical microstructure (Kunze 1987, 1990; St. Laurent and Schmitt 1999). After the salt sheets are disrupted, the flow evolves into turbulent salt fingering (Fig. 1c).

## 4. Dissipation rates and isotropy

*b*′

*and*

_{T}*b*′

*are thermal and saline buoyancy perturbations,where*

_{S}*B*and

_{T}*B*are horizontally averaged

_{S}*b*and

_{T}*b*, respectively. The angle brackets indicate averaging over volume between −2

_{S}*h*<

*z*< 2

*h*, where turbulent salt fingering is most active.

*U*(

*z*,

*t*) is the streamwise velocity averaged over horizontal directions.

In shear-driven turbulence, the degree of isotropy is predicted by the buoyancy Reynolds number, Re* _{b}* =

*ϵ*/

*vN*

^{2}, where

*N*

^{2}= 〈

*B*

_{T}_{,z}+

*B*

_{S}_{,z}〉. Subscripts preceded by commas indicate partial differentiation. As Re

*increases, the dissipation range separates from the energy-containing scales and become increasingly isotropic. Gargett et al. (1984) concluded from observations of shear-driven turbulence that*

_{b}*ϵ*can be accurately estimated based on isotropy when Re

*> 2 × 10*

_{b}^{2}. Itsweire et al. (1993) found that the assumption of isotropy is invalid for Re

*< 10*

_{b}^{2}using DNS of a uniformly stratified shear flow. A similar result holds in DNS of a localized stratified shear layer (Smyth and Moum 2000). The skill of Re

*as a predictor of isotropy in turbulent salt fingering remains to be assessed.*

_{b}In fingering-favorable stratification with turbulence, laboratory experiments have shown that finger structures have Re* _{b}* ~

*O*(10) for

*R*< 2 (McDougall and Taylor 1984). Oceanic observations also suggest that Re

_{ρ}*~*

_{b}*O*(10) (St. Laurent and Schmitt 1999; Inoue et al. 2008). The Re

*from our simulation increases until salt sheets start to break up (*

_{b}*σ*

_{fg}

*t*≈ 5) and reaches a maximum value of Re

*= 10.8 at*

_{b}*σ*

_{fg}

*t*= 8 (Fig. 2). After

*σ*

_{fg}

*t*> 8, the flow becomes quasi steady with Re

*≈ 10.*

_{b}In our DNS, Re* _{b}* is comparable to the observed values quoted above; therefore, we may use our results to quantify the bias inherent in one-dimensional observational estimates of dissipation rates. Although the criterion found for shear-driven turbulence that dissipation-scale isotropy occurs above Re

*~ 200 does not apply directly to turbulent salt fingering, the relatively small value for Re*

_{b}*found in both DNS and observations suggests that the dissipation scales could be significantly anisotropic.*

_{b}### a. Isotropic approximations for χ_{S} and χ_{T}

*i*ranges from 1 to 3 and is not summed over. Each of three different forms has the same magnitude for isotropic turbulence (e.g.,

*χ*=

_{S}*~*

_{b}*O*(10) for observed salt fingering, observations and numerical simulations support the thermal buoyancy field being nearly isotropic. Two-dimensional numerical simulations of Shen (1995) showed that the thermal spectral variance is distributed approximately equally in both vertical and horizontal wavenumbers. Lueck (1987) found that the magnitude of the vertical thermal buoyancy gradient was similar to the horizontal gradient in a thermocline staircase east of Barbados. Because the measurements were taken at sites with Ri ~ 10 and never less than unity, Lueck (1987) argued that the isotropic structure is not likely the result of shear-driven turbulence. Shadowgraph images of salt fingering showed coherent tilted laminae (Kunze 1987). Because shadowgraph images tend to emphasize the dissipative scales of the salinity field, the shadowgraph images are thought to reflect anisotropic salinity structures at the salinity Batchelor scale (Kunze 1987, 1990; St. Laurent and Schmitt 1999).

In our simulations, signatures of salt-sheet anisotropy decrease with time in both saline and thermal buoyancy fields (Figs. 3a,b). In the linear regime (0 < *σ*_{fg}*t* < 3), virtually all salinity and thermal buoyancy variance dissipation comes from spanwise derivatives (indicated by solid lines in Figs. 3a,b), consistent with the geometry of salt sheets. With the onset of secondary instability, contributions from streamwise and vertical derivatives increase. The ratios *χ _{S}* and

*χ*become quasi steady in the turbulent regime (

_{T}*σ*

_{fg}

*t*> 8), by which time temperature and salinity structure are nearly isotropic. The slight (~10%) anisotropy that remains in the turbulent state is consistent with that of the initial instability, with the contribution from the spanwise derivative being the largest. Even after transition, salt-fingering-favorable mean stratification is maintained and salt-sheet instability continues to influence small-scale anisotropy.

We now turn to the effects of Ri and *R _{ρ}* on isotropy, focusing on the thermal buoyancy because it is easiest to measure in the ocean. Time averages of the ratios

*σ*

_{fg}

*t*> 8 succinctly represent the Ri and

*R*dependences of anisotropy in the turbulent state (Fig. 4). As in Fig. 3, anisotropy characteristics of the linear instability are evident in the turbulent regime (Fig. 4a). In the unsheared case Ri = ∞, the contributions from the horizontal derivatives are approximately equal (

_{ρ}*χ*increases with decreasing Ri, resembling the characteristics of shear-driven turbulence (Itsweire et al. 1993; Smyth and Moum 2000).

_{T}As *R _{ρ}* increases, the geometric characteristics of salt sheets dominate (Fig. 4b); that is, the contribution from

*R*, whereas the contribution from

_{ρ}*χ*

_{T}_{1}decreases. Salt sheets become more horizontally isotropic (

*R*(Fig. 4b), as observed in laboratory salt-fingering experiments (Taylor 1992). Furthermore, isotropy approximations become more accurate with decreasing

_{ρ}*R*(Figs. 4a–c), which also holds in DNS of three-dimensional unbounded salt fingers (Caplan 2008).

_{ρ}### b. Isotropic approximations for ϵ

*ϵ*can be represented by any one of the nine expressions,with no summation over

*i*or

*j*(Taylor 1935). The variable

*δ*represents the Kronecker delta function. In general, these expressions are unequal, and their differences reflect the degree of anisotropy of the velocity gradients at dissipation scales.

_{ij}We look first at expressions involving the three derivatives of the vertical velocity, because these will prove to dominate. In the unsheared case (Fig. 5a), the contribution from *ϵ* exceed those of along-sheet *σ*_{fg}*t* ≈ 8 (Figs. 5a,b); however, none of the ratios converge to unity. The axisymmetry of the primary salt-fingering state continues to influence the geometry of the dissipation scales in the turbulent state (Fig. 5a). In contrast, the geometry of the salt-sheet instability is disrupted by the increase in contribution from

To quantify Ri and *R _{ρ}* dependence, the nine approximations in (8) were averaged over

*σ*

_{fg}

*t*> 8 and normalized by the true value of

*ϵ*in the turbulent regime (Fig. 6). Approximations involving the vertical velocity all overestimate

*ϵ*, as illustrated previously in Fig. 5, whereas those involving horizontal velocities underestimate

*ϵ*. This is consistent with the dominance of vertical motions in turbulent salt fingers and salt sheets. In the absence of background shear, the contributions from

In both sheared and unsheared cases, the second largest contribution comes from

The approximations using horizontal shears also show the influence of the linear instabilities. The balances in the unsheared case,

### c. Impact of anisotropy on ϵ and χ_{T} from vertical profilers

*ϵ*and

*χ*are estimated by

_{T}For isotropic flows, these approximations are exact: that is, *ϵ* = *ϵ ^{z}* and

*χ*=

_{T}*associated with fingering is ~(10), isotropic approximations have been justified by observations (Lueck 1987) and numerical simulation (Shen 1995), both showing nearly isotropic thermal buoyancy field in the turbulent regime. However, isotropy in temperature gradient does not guarantee isotropy in velocity gradients. Furthermore, this justification does not hold for the initial stage of the flow when salt sheets or salt fingers are active. The geometry of salt fingers and salt sheets led theoretical models to utilize the “tall fingers” (TF) approximation in an unbounded salt fingers and salt sheets (Stern 1975; Kunze 1987; Smyth and Kimura 2007). Because salt fingers and salt sheets are tall and narrow, the TF approximation assumes that the vertical derivative is negligible relative to horizontal derivatives (*

_{b}*ϵ*=

^{z}The approximation (9) gives a poor estimate of *ϵ*, even in the turbulent regime of fingering (Figs. 7a,b). As the flow evolves, *ϵ ^{z}*/

*ϵ*increases from near zero but never converges to unity. Instead, at quasi-steady state, the ratio ranges between 0.32 and 0.52 for

*σ*

_{fg}

*t*> 8 (Figs. 7a,b). In contrast, the quasi-steady value of

*χ*ranges between 0.8 and 1.2 (Figs. 7c,d). These results suggest that, in the presence of turbulent salt fingering,

_{T}*ϵ*underestimates

^{z}*ϵ*by a factor of 2–3.

## 5. Turbulent fluxes and diffusivities

*ϵ*and

*χ*allow indirect estimates of turbulent fluxes via the dissipation ratio (also called “mixing efficiency”; e.g., McEwan 1983),In turbulence, mechanical energy can be expended by raising the mass of fluid through mixing or kinetic energy dissipation by molecular viscosity. The quantity Γ approximates the fraction of the turbulent kinetic energy that is irreversibly converted to potential energy due to mixing. In shear-driven turbulence, Γ can be used to estimate the effective diffusivity of heat and salt as

_{T}*K*=

_{T}*K*= Γ(

_{S}*ϵ*/

*N*

^{2}) (Osborn 1980).

*w*′

*b*′

*< 0. The turbulent Schmidt number quantifies the relative importance of effective diffusivity of momentum to that of salt. In the following subsections, we will quantify the Ri and*

_{T}*R*dependences of Γ, the effective diffusivities,

_{ρ}*γ*, and Sc.

_{s}### a. Estimation of Γ

*,In homogeneous, stationary isotropic turbulence, Γ = Γ*

^{z}*. In shear-driven turbulence, measurements indicate Γ*

^{z}*≈ 0.2 (Osborn 1980; Oakey 1982; Moum 1996). This value is accurate for Re*

^{z}*> 2 × 10*

_{b}^{2}(Gargett et al. 1984). Fastest-growing salt-finger theory suggests 0.2 < Γ

*< 0.8 (Hamilton et al. 1989; Inoue et al. 2008), whereas measurements suggest that Γ can have higher values: 0.4 < Γ*

^{z}*< 2 (St. Laurent and Schmitt 1999; Inoue et al. 2008). At statistical steady state, balances of turbulent kinetic energy and scalar variance implyin the absence of shear (Hamilton et al. 1989; McDougall and Ruddick 1992). For fastest-growing fingers, Γ can be expressed as a function of*

^{z}*R*by substituting

_{ρ}*γ*= (

_{s}*R*)

_{ρ}^{1/2}[(

*R*)

_{ρ}^{1/2}− (

*R*− 1)

_{ρ}^{1/2}] (Stern 1975; Kunze 1987). In the presence of shear, for the fastest-growing salt sheets,(Smyth and Kimura 2007). In the present DNS for

*R*= 1.6, Γ lies below 0.6 (Fig. 8a). In the linear regime (

_{ρ}*σ*

_{fg}

*t*< 2), Γ is consistent with (15). The value of Γ decreases slowly as the flow becomes unstable to secondary instability (

*σ*

_{fg}

*t*~ 3) and then becomes quasi steady between 0.3 and 0.4 in the turbulent salt-fingering regime.

The value of Γ* ^{z}* is generally larger than Γ (Fig. 8b), consistent with

*ϵ*<

^{z}*ϵ*. For Ri > 6, Γ

*can be 30 times larger than Γ in the preturbulent state (0 <*

^{z}*σ*

_{fg}

*t*< 3). A local peak of Γ

*/Γ occurs during the secondary instability (*

^{z}*σ*

_{fg}

*t*~ 5). As the flow becomes turbulent, the ratio decreases and approaches a quasi-steady value of ~2: that is, Γ

*overestimates Γ by a factor of 2 in the turbulent salt-fingering regime.*

^{z}Because both Γ and Γ* ^{z}* become quasi steady after

*σ*

_{fg}

*t*= 8, both Γ and Γ

*are averaged over*

^{z}*σ*

_{fg}

*t*> 8 to quantify Ri and

*R*dependences in the turbulent regime. In all cases, Γ

_{ρ}*coincides approximately with observations (Fig. 9). Both observed and simulated Γ*

^{z}*increase with increasing Ri. Similarly, Γ increases with increasing Ri. The*

^{z}*R*dependence of the observed Γ

_{ρ}*and the DNS result exhibit different trends (Fig. 10). Observed Γ*

^{z}*decreases with increasing*

^{z}*R*, whereas both Γ

_{ρ}*and Γ from DNS increase with increasing*

^{z}*R*. The difference in

_{ρ}*R*dependence of the observations and our simulation can be attributed to Ri. Our simulation has uniform Ri, whereas the Ri in the observation lies between 3 and 7. The linear result (15) from Smyth and Kimura (2007) predicts Γ from DNS remarkably well (Figs. 9, 10).

_{ρ}Our results contradict the isotropy assumption used to infer Γ from microstructure profiles. This has implications for recent attempts to extend the *k*–*ϵ* formalism to include double diffusion (Canuto et al. 2008), where Γ* ^{z}* was used to validate model parameters. Also, Inoue et al. (2008) used Γ

*from St. Laurent and Schmitt (1999) to suggest that the wavelength of oceanic fingers is larger that that of the fastest-growing mode (their Fig. 5). In contrast, we find Γ consistent with linear fastest-growing fingers (Figs. 9, 10).*

^{z}### b. Effective diffusivities

Effective diffusivities are often used in large-scale models to represent small-scale physics (e.g., Bryan 1987; Gargett and Holloway 1992; Walsh and Ruddick 1995). Gargett and Holloway (1992) found that steady-state solutions of low-resolution general circulation models (GCMs) were sensitive to the ratio of *K _{T}* to

*K*

_{S}_{.}More recent modeling studies have found that regional circulations are significantly altered when parameterizations of

*K*and

_{T}*K*due to fingering are introduced (Merryfield et al. 1999; Zhang et al. 1998).

_{S}*K*for pure salt fingering,

_{S}*n*= 2 and

^{−4}. To add dependence on Ri, we compute a least squares fit of

*K*and

_{T}*K*, averaged over

_{S}*σ*

_{fg}

*t*> 8 for all simulations,

Both *K _{T}* and

*K*decrease with decreasing Ri (Figs. 11a,c), which is also true in the salt-fingering system with Ri < 0.25 described by Smyth and Kimura (2011). This finding was unexpected and has important implications for turbulence modeling, where it is often assumed that, because background shear represents an energy source for mixing, its introduction will increase effective diffusivities (e.g., Large et al. 1994; Zhurbas and Oh 2001; Smyth and Ruddick 2010). The Ri dependence differs significantly from the prediction of Kunze (1994),

_{S}*K*∝ Ri

_{S}^{1.0}, but is closer to that of Inoue et al. (2008),

*K*∝ Ri

_{S}^{0}(Fig. 11c).

We also find that *K _{T}* and

*K*increase with decreasing

_{S}*R*(Figs. 11b,d), which has been reported in previous DNS of two-dimensional salt fingering (W. J. Merryfield and M. Grinder 2000, personal communication; Stern et al. 2001) and laboratory experiments (Turner 1967; Linden 1973; McDougall and Taylor 1984; Taylor and Bucens 1989). Our

_{ρ}*K*and

_{T}*K*are consistent with DNS results of W. J. Merryfield and M. Grinder (2000, personal communication) (Figs. 11b,d). This agreement should be interpreted with caution because of the difference in Ri,

_{S}*τ*, and computational domain dimensions. Estimates for three-dimensional salt fingering by Stern et al. (2001) are twice as large; this could be due to the presence of the shear and difference in

*τ*in our simulations (Figs. 11b,d).

In the absence of shear, our *K _{S}* (open triangle) is consistent with laboratory experiments (Turner 1967; Linden 1973; Schmitt 1979a) and theoretical prediction (Kunze 1987) (Fig. 12, small symbols and dashed–dotted curve). Likewise, when

*R*is sufficiently small, the effect of shear is secondary and our results agree with those from the laboratory experiments and theoretical prediction (large solid triangle at

_{ρ}*R*= 1.2). However, when Ri is finite and

_{ρ}*R*is substantially greater than 1, shear may be expected to have a significant effect. Our DNS results show an overall reduction in

_{ρ}*K*in the sheared cases; moreover, the decrease in

_{S}*K*with increasing

_{S}*R*is amplified. Simulations with higher

_{ρ}*R*are needed to verify (17) with the laboratory experiments and theoretical prediction (Fig. 12, solid curve).

_{ρ}In two-dimensional unsheared, turbulent salt fingers, Stern et al. (2001) showed that decrease in *τ* from 0.04 to 0.01 increased the heat flux by 15%. In 3D sheared, turbulent salt fingering, Kimura and Smyth (2007) found that the decrease in *τ* from 0.04 to 0.01 increases the *K _{T}* and

*K*by a factor of 2. The effective diffusivities of heat and salt for the ocean may therefore be up to 2 times larger than (16) and (17).

_{S}The flux ratio *γ _{s}* remains within ~10% of the linear value 0.6 (Fig. 13a). In the sheared, turbulent salt-fingering regime,

*γ*is about 8% higher than its linear value for the unsheared case. Ruddick (1985) suggested that individual salt sheets rapidly lose their momentum via lateral diffusion and therefore Sc < 1. A laboratory experiment to confirm this hypothesis was inconclusive (Ruddick et al. 1989). Linear stability analysis of salt sheets (Smyth and Kimura 2007) and subsequent DNS (Kimura and Smyth 2007) for a single initial case confirm that Sc < 1 for Ri = 2. The value of Sc < 1 holds for Ri ranging from 0.5 to 20 (Fig. 13b). The effective diffusivity of the momentum (viscosity) is an order of magnitude smaller than that of salt, confirming (over the simulated range of Ri and

_{s}*R*) that salt sheets are inefficient at transporting momentum.

_{ρ}### c. Observational estimates of effective diffusivities

*κ*〈(∂

_{S}*b*′

*/∂*

_{S}*z*)

^{2}〉. These estimates are exact for stationary, homogeneous, isotropic turbulence. Here, we compare these estimates with the true diffusivities

*K*and

_{T}*K*, computed directly using fluxes (12).

_{S}The isotropic Osborn–Cox model (18) captures the *R _{ρ}* dependence of

*K*and

_{T}*K*(Figs. 14a,b). Both

_{S}*R*, consistent with the trends of

_{ρ}*K*and

_{T}*K*from DNS and observations by St. Laurent and Schmitt (1999). However, both

_{S}*K*and

_{T}*K*for the range of

_{S}*R*discussed here (Figs. 14a,b). The isotropy assumption alone cannot explain this difference, because ratios

_{ρ}*χ*and

_{T}*χ*range between 0.8 and 1.2 in section 5. The Osborn and Cox (1972) diffusivity model combined with isotropy assumption can underestimate

_{S}*K*and

_{T}*K*by up to a factor of 3 for turbulent salt fingering.

_{S}## 6. Conclusions

We have simulated sheared, turbulent salt fingering in a sharp high-gradient interface characteristic of laboratory studies (Turner 1967; Schmitt 1979a; McDougall and Taylor 1984; Taylor 1992). Although these simulations are intended to shed light on oceanic thermohaline staircases, the following caveats should be noted:

- The ratio of molecular diffusivity of salt to heat is 4 times larger than in the real ocean.
- The interface thickness is at least 3 times smaller than observed thickness of thermohaline staircases. In much thicker layers and continuous stratification, salt-finger flux laws change because fingers no longer extend across the interface from one homogeneous layer to the other (Kunze 1987).
- The equation of state is linear.

- Scalar gradients are nearly isotropic in the dissipation range, but velocity gradients are not. The approximations of
*ϵ*based on vertical shears, which are often used in interpretations of microstructure data, underestimate its value by a factor of 2–3. This suggests that the rate of dissipation by sheared, turbulent salt fingering can be 2–3 times larger than measured inferences. - The geometry of the primary instability is often (though not always) reflected in the anisotropy of the turbulent salt-fingering regime.
- The isotropy assumption can lead to overestimation of the dissipation ratio Γ by a factor of 2 to 3. This will impact estimates of correlation time scales in the
*k*–*ϵ*model of turbulent salt fingering (e.g., Canuto et al. 2008). It also caused Inoue et al. (2008) to incorrectly ascribe the signals in St. Laurent and Schmitt (1999) to fingers with wavelengths larger than those of the fastest-growing mode. - Effective diffusivities of heat and salt can be reduced by either an increase in
*R*or (surprisingly) a decrease in Ri. Empirical fits give_{ρ}*K*(_{T}*R*, Ri) = 3.07 × 10_{ρ}^{−5}*R*_{ρ}^{−4.0}Ri^{0.17}and*K*(_{S}*R*, Ri) = 4.38 × 10_{ρ}^{−5}*R*_{ρ}^{−2.7}Ri^{0.17}. Kimura and Smyth (2007) showed that a decrease in*τ*from 0.04 to 0.01 increased*K*and_{T}*K*by a factor of 2. The dependence on Ri differs significantly from the theoretical prediction of Kunze (1994)._{S} - Effective diffusivities estimated by the Osborn and Cox (1972) model with isotropy assumption produced consistent
*R*dependence but are up to a factor of 3 too small._{ρ} - Values of nondimensional mixing efficiency Γ, flux ratio
*γ*, and Schmidt number Sc in the turbulent salt-fingering regime are predicted by linear theory with remarkable accuracy.

## Acknowledgments

Comments from two anonymous reviewers improved the manuscript. Computer time was provided by the National Center for Atmospheric Research (NCAR). The graphics were produced using VAPOR with help from VisLab at NCAR. The work was supported by the National Science Foundation under Grant 0453140.

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