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Turbulence in a sheared, salt-fingering-favorable environment:1

Anisotropy and effective diffusivities2

Satoshi Kimura ∗ William Smyth

College of Oceanic and Atmospheric Sciences, Oregon State University, Oregon

3

Eric Kunze

School of Earth and Ocean Sciences, University of Victoria, Canada

4

∗Corresponding author address: College of Oceanic and Atmospheric Sciences, Oregon State UniversityOcean Admin Bldg, Corvallis, OR 97331, USA.E-mail: skimura@coas.oregonstate.edu

1

ABSTRACT5

Direct numerical simulations (DNS) of a shear layer with salt-fingering-favorable stratifica-6

tion have been performed for different Richardson numbers Ri and density ratios Rρ. When7

the Richardson number is infinite (unsheared case), the primary instability is square plan-8

form salt-fingering, alternating cells of rising and sinking fluid. In the presence of shear,9

salt-fingering takes the form of salt sheets, planar regions of rising and sinking fluid, aligned10

parallel to the sheared flow. After the onset of secondary instability, the flow becomes11

turbulent; however, our results indicate that the continued influence of the primary insta-12

bility biases estimates of the turbulent kinetic energy dissipation rate. Thermal and saline13

buoyancy gradients become more isotropic than the velocity gradients at dissipation scales.14

Estimates of the turbulent kinetic energy dissipation rate by assuming isotropy in the ver-15

tical direction can underestimate its true value by a factor of 2 to 3, whereas estimates of16

thermal and saline dissipation rates are relatively accurate. Γ approximated by assuming17

isotropy agrees well with observational estimates, but is larger than the true value of Γ by18

approximated a factor of 2. The transport associated with turbulent salt sheets is quanti-19

fied by thermal and saline effective diffusivities. Salt sheets are ineffective at transporting20

momentum. Thermal and saline effective diffusivities decrease with decreasing Ri, despite21

the added energy source provided by background shear. Nondimensional quantities, such as22

thermal to saline flux ratio, are close to the predictions of linear theory.23

1

1. Introduction24

Salt-fingering-favorable stratification occurs when the gravitationally unstable vertical25

gradient of salinity is stabilized by that in temperature. In such conditions, the faster diffu-26

sion of heat than salt can generate cells of rising and sinking fluid known as salt fingers, which27

take a variety of planforms such as squares, rectangles, and sheets (Schmitt 1994b; Proctor28

and Holyer 1986). Salt-fingering-favorable stratification is found in much of the tropical29

and subtropical pycnocline (You 2002). The most striking signatures are in thermohaline30

staircases, which are stacked layers of different water-types separated by sharp thermal and31

saline gradients. These are found at several locations, such as in the subtropical confluence32

east of Barbados, under the Mediterranean and Red Sea salt tongues and in the Tyrrhenian33

Sea (Tait and Howe 1968; Lambert and Sturges 1977; Schmitt 1994a).34

Salt-fingering is subjected to vertically varying horizontal currents (background shear)35

(Gregg and Sanford 1987; St. Laurent and Schmitt 1999). Zhang et al. (1999) found a36

significant relationship between salt fingering and shear in a model of the North Atlantic.37

A theory for the possible relationship between salt fingering and shear has been introduced38

by Schmitt (1990). In the presence of shear, salt-fingering instability is supplanted by salt-39

sheet instability, alternating planar regions of rising and sinking fluid, aligned parallel to the40

sheared flow (Linden 1974). While theories for the initial growth of salt-fingering and salt-41

sheet instabilities are well-established (Stern 1960; Linden 1974; Schmitt 1979; Kunze 2003;42

Smyth and Kimura 2007), the vertical fluxes of momentum, heat, and salt in the non-linear43

regime are not well understood. We call this non-linear regime turbulent salt-fingering.44

Dissipation rates of property variances in double-diffusive turbulence can be measured45

in the ocean by microstructure profilers, but the interpretation of these results has relied46

on Kolmogorov’s hypothesis of isotropic turbulence (Gregg and Sanford 1987; Lueck 1987;47

Hamilton et al. 1989; St. Laurent and Schmitt 1999; Inoue et al. 2008). Kolmogorov (1941)48

2

proposed that small-scale turbulence statistics are universal in the limit of high Reynolds49

number. According to this hypothesis, anisotropy of the energy-containing scales is lost in50

the turbulent energy cascade, so that the small scales, where energy is finally dissipated,51

are statistically isotropic. The assumption of small-scale isotropy greatly simplifies both the52

theory and modeling of turbulence, as well as interpretation of microstructure measurements.53

Because the Reynolds number in oceanic salt-fingering systems is O(10) (McDougall and54

Taylor 1984), the validity of this assumption is questionable. We will examine both mixing55

rates and the isotropy of the dissipation range in turbulent salt-fingering by means of direct56

numerical simulations of diffusively unstable shear layers.57

Based on observations of shear-driven turbulence in flow over a sill, Gargett et al. (1984)58

concluded that the isotropy assumption was accurate as long as the separation between59

Ozmidov and Kolmogorov length scale was sufficiently large. Itsweire et al. (1993) tested60

isotropic approximations on flows with uniform shear and stratification for Richardson num-61

ber using DNS, and found that dissipation rates could be underestimated by factors of 2 to62

4 at low Reynolds number. Smyth and Moum (2000) extended the analysis to a localized63

shear layer and found similar results.64

Estimations of dissipation rates, combined with the Osborn and Cox (1972) diffusivity65

model, can furnish estimates of the effective diffusivities of heat, salt, and momentum. Ef-66

fective diffusivities are used to parameterize turbulent fluxes in order to model large-scale67

phenomena affected by mixing processes. These applications range from fine-scale thermo-68

haline intrusions (e.g. Toole and Georgi 1981; Walsh and Ruddick 1995; Smyth and Ruddick69

2010) to basin-scale circulations (e.g. Zhang et al. 1999; Merryfield et al. 1999).70

Effective diffusivities can be calculated directly from DNS. Thermal and saline effective71

diffusivities for 2D salt-fingering have been computed in previous studies (Merryfield and72

Grinder 2000; Stern et al. 2001; Yoshida and Nagashima 2003; Shen 1995). The effective73

diffusivities for 3D sheared, salt-fingering were first calculated by Kimura and Smyth (2007)74

3

for a single initial state.75

In this paper, we extend Kimura and Smyth (2007) to cover a range of oceanographically76

relevant initial background states. Although oceanic salt fingering does not always lead to77

staircases, that is the regime we focus on here as it is most amenable to direct simulation.78

Our objective is twofold. First, we explore the isotropy assumption for turbulent salt fingers79

and salt sheets. Second, we compute effective diffusivities to identify effects of anisotropy80

that could affect the accuracy of in situ estimates.81

In section 2, we describe our DNS model and initial conditions. An overview of salt sheet82

evolution is given in section 3. Section 4 discusses the nature of anisotropy at dissipation83

scales and its implications for interpreting profiler measurements. In section 5, we discuss84

the effective diffusivities of momentum, heat, and salt, and suggest a parameterization in85

terms of ambient property gradients. Our conclusions are summarized in section 6.86

2. Methodology87

We employ the three-dimensional incompressible Navier-Stokes equations with the Boussi-88

nesq approximation. The evolution equations for the velocity field, ~u(x, y, z, t) = {u, v, w},89

in a nonrotating, Cartesian coordinate system, {x, y, z}, are90

[

D

Dt− ν∇2

]

~u = −∇π + bk̂ + ν∇2~u

∇ · ~u = 0. (1)

D/Dt = ∂/∂t+ ~u · ~∇ and ν are the material derivative and kinematic viscosity, respectively.91

The variable π represents the reduced pressure (pressure scaled by the uniform density ρ0).92

The total buoyancy is defined as b = −g(ρ−ρ0)/ρ0, where g is the acceleration due to gravity.93

Buoyancy acts in the vertical direction, as indicated by the vertical unit vector k̂. We assume94

4

that equation of state is linear, and therefore the total buoyancy is the sum of thermal and95

saline buoyancy components (bT and bS), each governed by an advection-diffusion equation:96

b = bT + bS;

DbT

Dt= κT∇2bT ; (2)

DbS

Dt= κS∇2bS. (3)

Molecular diffusivities of heat and salt are denoted by κT and κS, respectively.97

Periodicity intervals in streamwise (x) and spanwise (y) directions are Lx and Ly. The98

variable Lz represents the vertical domain length. Upper and lower boundaries, located at99

z = −Lz/2 and z = Lz/2, are impermeable (w = 0), stress-free (∂u/∂z = ∂v/∂z = 0), and100

insulating with respect to both heat and salt (∂bT /∂z = ∂bS/∂z = 0).101

To represent mixing in the high-gradient “interface” of a thermohaline staircase, we102

initialize the model with a stratified shear layer in which shear and stratification are concen-103

trated at the center of the vertical domain with a half-layer thickness of h:104

u

∆u=

bT

∆BT

=bS

∆BS

= tanh(z

h

)

.

The constants ∆u, ∆BT , and ∆BS represent the change in streamwise velocity, thermal buoy-105

ancy, and saline buoyancy across the half-layer thickness of h. For computational economy,106

we set h = 0.3 m. This is at the low end of the range of observed layer thickness (e.g. Gregg107

and Sanford 1987; Kunze 1994). The change in the total buoyancy is ∆B = ∆BT + ∆BS.108

In all the DNS experiments, the initial buoyancy frequency (√

∆B/h) is fixed at 1.5× 10−2109

rad s−1, a value typical of the thermohaline staircase east of Barbados (Gregg and Sanford110

1987).111

These constants can be combined with the fluid parameters ν, κT , and κS to form non-112

5

dimensional parameters, which characterize the flow at t = 0:113

Ri =∆Bh

∆u2;

Rρ = −∆BT

∆BS

;

Pr =ν

κT;

τ =κS

κT.

We have done 7 experiments with different Ri and Rρ (table 1). The bulk Richardson114

number, Ri, measures the relative importance of stratification and shear. If Ri < 0.25, the115

initial flow is subjected to shear instabilities (Miles 1961; Howard 1961; Hazel 1972). Here,116

we chose high enough Ri to ensure that shear instabilities do not disrupt the growth of117

salt-fingering modes. A typical range of Ri in a sheared, thermohaline staircase is ∼ 3− 100118

(Gregg and Sanford 1987; Kunze 1994). The Reynolds number based on the half-layer119

thickness and the half change in streamwise velocity is given by Re = h∆u/ν. Since ∆B120

and h are kept constant in our simulations, Re and Ri are not independent: Re = 1354Ri−1/2121

with Re = 0 and Ri = ∞ representing the unsheared case.122

The density ratio, Rρ, quantifies the stabilizing effect of thermal to destabilizing effect123

of saline buoyancy components; salt-fingering grows more rapidly as Rρ approaches unity.124

We varied Rρ between 1.2 and 2, which covers the range of observational data available for125

comparison (St. Laurent and Schmitt 1999; Inoue et al. 2008). The Prandtl number, Pr,126

and the diffusivity ratio, τ , represent ratios of the molecular diffusivities of momentum, heat,127

and salt. The Prandtl number was set to 7, which is a typical value for salt water. The128

diffusivity ratio in the ocean is 0.01, i.e., the heat diffuses two orders of magnitude faster than129

salt. The vast difference in diffusivity requires DNS to resolve a wide range of spatial scales,130

making it computationally expensive. In previous DNS of salt water, τ has been artificially131

increased to reduce required computational expenses (e.g. Stern et al. 2001; Gargett et al.132

6

2003; Smyth et al. 2005). Kimura and Smyth (2007) conducted the first 3D simulation with133

τ = 0.01 and found that increasing τ from 0.01 to 0.04 reduced thermal and saline effective134

diffusivities by one half. In the cases presented here, we set τ to 0.04 to allow an extended135

exploration of the Ri and Rρ dependence.136

The fastest-growing salt-sheet wavelength predicted by linear stability analysis, is λfg =137

2π(νκT 2h/∆B)1/4 (Stern 1975; Schmitt 1979). Our value matches the observed value, λ =138

0.032 m (Gregg and Sanford 1987; Kunze 2003). We accommodate four wavelengths of139

the fastest-growing primary instability in the spanwise direction, Ly = 4λfg. The vertical140

domain length, Lz, was chosen such that vertically propagating plumes reach statistical141

equilibrium. We found that Lz equal to six times h was sufficient. Lx was chosen to be large142

enough to accommodate subsequent secondary instabilities. After sensitivity tests, we chose143

Lx = 28λfg.144

The primary instability was seeded by adding an initial disturbance proportional to the145

fastest-growing mode of linear theory, computed numerically as described in Smyth and146

Kimura (2007). We seed square salt-fingering for Ri = ∞ and sheets in the presence of147

shear. The vertical displacement amplitude is set to 0.02h, and a random noise was added148

to the initial velocity field with an amplitude of 1× 10−2hσL to seed secondary instabilities.149

The variable, σL indicates the growth rate of the fastest growing linear normal mode.150

The numerical code used to solve (1) - (3) is described by Winters et al. (2004). The151

code uses Fourier pseudospectral discretization in all three directions, and time integration152

using a third-order Adams-Bashforth operator. A time step is determined by a Courant-153

Friedrichs-Lewy (CFL) stability condition. The CFL number is maintained below 0.21 for154

DNS experiments presented here. The code was modified by Smyth et al. (2005) to accom-155

modate a second active scalar, which is resolved on a fine grid with spacing equal to one half156

the spacing used to resolve the other fields. The fine grid is used to resolve salinity. The157

fine grid spacing is equal to 0.15λfg

√τ in all three directions, as suggested by Stern et al.158

7

(2001).159

3. Flow overview160

Figure 1 shows the salinity buoyancy field for the case Ri = 6, Rρ = 1.6 at selected times.161

The time is scaled by the linear normal growth rate of salt sheets, σL, described by Smyth162

and Kimura (2007). Figure 1a shows the salt-sheet instability; the planar regions of vertical163

motions oriented parallel to the background shear flow. Rising and sinking fluid are shown164

in green and blue, respectively. The computational domain accommodates four salt sheets.165

When the salt sheets reach the edges of the interface (indicated by purple and red), they166

start to undulate in the spanwise direction (figure 1b). At the same time, the salt sheets167

develop streamwise dependence at the edges and center of the interface. At the edges of the168

interface, the streamwise dependence appears as ripples. At the center, tilted laminae are169

evident, reminiscent of those seen in shadowgraph images of optical microstructure (Kunze170

1987, 1990; St. Laurent and Schmitt 1999). After the salt sheets are disrupted, the flow171

evolves into turbulent salt-fingering (figure 1c).172

4. Dissipation rates and isotropy173

We will explore the geometry of scalar (thermal and saline buoyancy) and velocity gra-174

dient fields. The isotropy of scalar fields is diagnosed in terms of the thermal and saline175

buoyancy variance dissipation rates:176

χT = 2κT

⟨

|∇b′T |2⟩

; χS = 2κS

⟨

|∇b′S|2⟩

. (4)

8

Variables, b′T and b′S are thermal and saline buoyancy perturbations:177

b′T (x, y, z, t) = bT (x, y, z, t) − BT (z, t);

b′S(x, y, z, t) = bS(x, y, z, t) − BS(z, t),

where BT and BS are horizontally averaged bT and bS, respectively. The angle brackets178

indicate averaging over volume between −2h < z < 2h, where turbulent salt-fingering is179

most active.180

Turbulent kinetic energy dissipation rate, ǫ, is used to quantify the anisotropy of velocity181

fields:182

ǫ =ν

2

⟨

(

∂u′i

∂xj+

∂u′j

∂xi

)2⟩

.

The primes indicate the perturbation velocity fields:183

~u′(x, y, z, t) = ~u(x, y, z, t) − {U(z, t), 0, 0}, (5)

where U(z, t) is the streamwise velocity averaged over horizontal directions.184

In shear-driven turbulence, the degree of isotropy is predicted by the buoyancy Reynolds185

number, Reb = ǫ/νN2, where N2 = 〈BT ,z + BS,z〉. As Reb increases, the dissipation range186

separates from the energy-containing scales and become increasingly isotropic. Gargett et al.187

(1984) concluded from observations of shear-driven turbulence that ǫ can be accurately188

estimated from a single term when Reb > 2 × 102. Itsweire et al. (1993) found that the189

assumption of isotropy is invalid for Reb < 102 using DNS of an uniform stratified-shear190

layer. A similar result holds in DNS of localized stratified-shear layer (Smyth and Moum191

2000).192

In fingering-favorable stratification with turbulence, laboratory experiments showed that193

the finger structures have Reb ∼ O(10) for Rρ < 2 (McDougall and Taylor 1984). Oceanic194

9

observations of a thermohaline staircase also suggest that Reb ∼ O(10) (St. Laurent and195

Schmitt 1999; Inoue et al. 2008). Reb from our simulation increases until salt sheets start to196

break up (σLt ≈ 5) and reaches a maximum value of Reb = 10.8 at σLt = 8 (figure 2). After197

σLt > 8, the flow becomes quasi-steady with Reb ≈ 10.198

In our DNS, Reb is comparable to that observed Reb; therefore, we may use our results199

to quantify the bias inherent in observational estimates of dissipation rates. Although the200

criterion Reb ∼ 200 found in shear-driven turbulence does not apply directly to turbulent201

salt-fingering, the relatively small value found in both DNS and observations suggests that202

the dissipation scales could be significantly anisotropic.203

a. Expressions of χS and χT204

For isotropic turbulence, the saline and thermal variance dissipation rates can be ex-205

pressed in three different forms as206

χSi = 6κS

⟨

(

∂b′S∂xi

)2⟩

and (6)

χT i = 6κT

⟨

(

∂b′T∂xi

)2⟩

(7)

where i ranges from 1 to 3 and is not summed over. Each of three different forms takes207

exactly the same value for isotropic turbulence (e.g. χS = χS1 = χS2 = χS3). Despite208

Reb ∼ O(10) in salt-fingering system, observations and numerical simulations support the209

thermal buoyancy field being nearly isotropic. Two-dimensional numerical simulations of210

Shen (1995) showed that the thermal spectral variance is distributed approximately equally211

in both vertical and horizontal wavenumbers. Lueck (1987) found that the magnitude of the212

vertical thermal buoyancy gradient was similar to the horizontal gradient in a thermocline213

staircase east of Barbados. Since the measurements were taken at sites with Ri ∼ 10 and214

10

never less than unity, Lueck (1987) argued that the isotropic structure is not likely the result215

of shear-driven turbulence. Shadowgraph images of salt-fingering showed coherent tilted216

laminae (Kunze 1987). Because shadowgraph images tend to emphasize the dissipative217

scales of the salinity field, the shadowgraph images are thought to reflect anisotropic salinity218

structures (Kunze 1987, 1990; St. Laurent and Schmitt 1999).219

Signatures of salt-sheet anisotropy decrease with time in both saline and thermal buoy-220

ancy fields (figure 3a, b). In the linear regime (0 < σLt < 3), all salinity and thermal221

buoyancy variance dissipation rates come from spanwise derivatives (indicated by solid lines222

in figure 3a, b), consistent with the motion of salt sheets. With the onset of secondary223

instability, the contribution from spanwise derivative decreases, while contributions from224

streamwise and vertical derivatives increase. These ratios become quasi-steady in the tur-225

bulent regime (σLt > 8), by which time these fields are nearly isotropic. The slight (∼ 10%)226

anisotropy that remains in the turbulent state is consistent with that of the initial instability,227

with the contribution from the spanwise derivative being the largest. Even after transition,228

salt-fingering-favorable stratification is maintained, and salt-sheet instability continues to229

influence the anisotropy at small scales.230

We turn now to the effects of Ri and Rρ on anisotropy in the turbulent regime, focusing on231

the thermal buoyancy as it is easiest to measure. Time averages of the ratios χT i over σLt > 8232

succinctly represent the Ri and Rρ dependences of anisotropy in the turbulent state (figure 4).233

As in figure 3, anisotropy characteristics of the linear instability are evident in the turbulent234

regime (figure 4a). In the unsheared case Ri = ∞, the contributions from the horizontal235

derivatives are approximately equal (χT 1 ≈ χT 2), indicating that the thermal buoyancy236

gradient is statistically axisymmetric about the vertical direction. As Ri decreases (i.e., as237

background shear is increased from zero), the contribution from χT 2increases, while the238

contribution from χT 1 decreases. The contribution from χT 3 to χT increases with decreasing239

Ri, resembling the characteristics of shear-driven turbulence (Itsweire et al. 1993; Smyth and240

11

Moum 2000).241

As Rρ increases, the geometric characteristics of salt sheets become more dominant (figure242

4b), i.e., the contribution from χT 2increases with increasing Rρ, while the contribution from243

χT 1 decreases. Salt sheets become more horizontally isotropic (χT 1 ≈ χT 2) with decreasing244

Rρ (figure 4b), as observed in laboratory experiments of salt-fingering by Taylor (1992).245

b. Expressions for ǫ246

In isotropic turbulence, ǫ can be represented by any one of the nine expressions:247

ǫ =15ν

2 − δi,j

⟨

(

∂u′i

∂xj

)2⟩

, (8)

with no summation over i and j (Taylor 1935). The variable, δi,j, represents the Kronecker248

delta function. In general, these expressions are unequal, and their differences reflect the249

degree of anisotropy of the velocity gradients at dissipation scales.250

We look first at expressions involving the three derivatives of the vertical velocity com-251

ponents, as these will prove to dominate. In the unsheared case (figure 5a), the contribution252

from w′,y

2 and w′,x

2 are the same, reflecting the geometry of salt-fingering instability. In the253

sheared case (figure 5b), the contributions from w′,y

2 to ǫ exceeds these of w′,x

2 and w′,z

2,254

reflecting the geometry of salt sheet instability. As the flow becomes turbulent, all the ratios255

reach steady values between 1.5 and 3 at σLt ≈ 8 (figures 5a and 5b); however, none of the256

ratios converge to unity. Anisotropy of salt-fingering instability continues to influence the257

geometry of the dissipation scales in the turbulent state (figure 5a). In contrast, anisotropy258

of salt-sheet instability is disrupted by the increase in contribution from w′,x in the turbulent259

state (figure 5b).260

To quantify Ri and Rρ dependence, the nine approximations in (8) were averaged over261

σLt > 8 and normalized by the true value of ǫ in the turbulent regime (figure 6). As already262

12

seen (figure 5), approximations involving the vertical velocity overestimate ǫ, whereas those263

involving horizontal velocities underestimate ǫ. This is consistent with the dominance of ver-264

tical motions in turbulent salt fingers. In the absence of background shear, the contribution265

from w′,x

2 and w′,y

2 are the largest, which is consistent with the axisymmetry of salt-fingering266

about the vertical direction. In the sheared cases, the contribution from w′,y

2 is the largest,267

consistent with salt-sheet geometry, but unlike in the linear regime, the contribution from268

w′,x

2 is significant.269

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

2.270

This vertical divergence acts to squeeze the fluid vertically at the tips of rising and sinking271

plumes. This vertical compression is compensated by normal strains in horizontal direc-272

tions(

〈w′,z

2〉 ≈ 〈u′,x

2〉, 〈v′,y

2〉)

. Because of the difference in geometry between sheared and273

unsheared cases, the contributions from u′,x

2 and v′,y

2 to balance w′,z

2 are different. In the274

unsheared case, the vertically squeezed fluid at the edges of plumes is displaced equally in the275

streamwise and spanwise directions(

〈u′,x

2〉 ≈ 〈v′,y

2〉)

, resembling salt-fingering. In contrast,276

the geometry of the sheared case displaces more fluid in spanwise direction(

〈u′,x

2〉 < 〈v′,y

2〉)

.277

The approximations using horizontal shear strain rates also show the influence of the278

linear instabilities. The balances shown in the unsheared case, 〈u′,y

2〉 ≈ 〈v′,x

2〉, 〈u′,x

2〉 ≈ 〈v′,y

2〉,279

and 〈u′,z

2〉 ≈ 〈v′,z

2〉, indicate axisymmetry about the vertical. As Ri decreases, contributions280

from u′,z

2 and v′,z

2 increase. This indicates that the flow is developing characteristics of281

shear-driven turbulence as described by Itsweire et al. (1993) and Smyth and Moum (2000).282

c. Impact of anisotropy on ǫ and χT from vertical profilers283

Observational estimates of dissipation rates are often based on data from vertical profilers.284

These profilers measure the vertical change of horizontal velocities and temperature for which285

ǫ and χT are estimated by286

13

ǫz =15ν

4

(⟨

(

∂u′

∂z

)2⟩

+

⟨

(

∂v′

∂z

)2⟩)

; (9)

287

χzT = 6κT

⟨

(

∂b′T∂z

)2⟩

. (10)

Though Reb associated with fingering is ∼ O(10), these isotropic approximations have288

been justified by observations (Lueck 1987) and numerical simulation (Shen 1995), both289

showing nearly isotropic thermal buoyancy field in the turbulent regime. But, isotropy in290

temperature gradient does not guarantee isotropy in velocity gradients. For isotropic flows,291

these approximations are exact, i.e. ǫ = ǫz and χT = χzT . Estimates of the salinity vari-292

ance dissipation rate are relatively scarce (e.g. Nash and Moum 1998), because conductivity293

sensors need to resolve finer scales than temperature. This justification is not consistent for294

the early stage of the flow when salt sheets or salt fingers are active. The geometry of salt295

fingers and salt sheets led theoretical models to utilize the “tall fingers” (TF) approximation296

in an unbounded salt fingers and salt sheets (Stern 1975; Kunze 1987; Smyth and Kimura297

2007). Since salt fingers and salt sheets are tall and narrow, the TF approximation assumes298

that the vertical derivative is negligible relative to horizontal derivatives (ǫz = χzT = 0).299

The ǫz gives a poor estimate of ǫ, even in the turbulent regime (figures 7a and 7b). As300

the flow evolves, ǫz/ǫ increases from near zero but does not converge to unity. Instead, the301

ratio becomes quasi-steady, ranging between 0.32 and 0.52 for σLt > 8 (figure 7a and 7b). In302

contrast, the value of χzT /χT becomes quasi-steady between 0.8 and 1.2 (figures 7c and 7c).303

These results suggest that, in the presence of turbulent salt fingers, χzT is an appropriate304

approximation, but ǫz underestimates ǫ by a factor of 2 to 3.305

14

5. Turbulent fluxes and diffusivities306

Of primary interest to the oceanographic community is understanding the turbulent fluxes307

associated with turbulent salt fingers. Measurements of ǫ and χT allow indirect estimates308

of turbulent fluxes via the dissipation ratio (also called “mixing efficiency”; e.g. McEwan309

1983):310

Γ =N2χT

2ǫ⟨

BT,z

2⟩ . (11)

In turbulent mixing, mechanical energy that goes into mixing can be expended by raising311

the mass of fluid or kinetic energy dissipation by molecular viscosity. Γ approximates the312

fraction of the turbulent kinetic energy that is irreversibly converted to potential energy due313

to mixing. In shear-driven turbulence, Γ can be used to estimate the effective diffusivity of314

heat and salt as KT = KS = Γ ǫN2 (Osborn 1980).315

The effective diffusivities of heat, salt, and momentum are defined via the standard flux-316

gradient relations:317

KT = −〈w′b′T 〉⟨

∂BT

∂z

⟩ ; KS = −〈w′b′S〉⟨

∂BS

∂z

⟩ ; KU = −〈u′w′〉⟨

∂U∂z

⟩ . (12)

Relationships between the thermal and saline buoyancy and momentum fluxes can be ex-318

pressed using the heat-salt flux ratio and the Schmidt number:319

γs = −〈w′b′T 〉〈w′b′S〉

; Sc =KU

KS

.

In salt-fingering and salt-sheet instabilities, an unstable distribution of mean saline buoyancy320

drives salt and heat fluxes downward. This implies that thermal buoyancy flux is working321

against the gravity, w′b′T < 0. Schmidt number quantifies the relative importance of effective322

diffusivity of momentum to that of salt. Next, we will quantify the Ri and Rρ dependences323

of Γ, the effective diffusivities, γs, and Sc.324

15

a. Estimation of Γ325

In homogeneous, stationary isotropic turbulence,326

Γz =N2χz

T

2ǫz⟨

BT,z

2⟩ . (13)

In shear-driven turbulence, measurements indicate Γz ≈ 0.2 (Osborn 1980; Oakey 1982;327

Moum 1996). This value is accurate for Reb > 2×102 (Gargett et al. 1984). In salt-fingering328

systems, fastest-growing finger theory suggests 0.25 < Γz < 0.5, while measurements suggest329

that Γ can have higher values: 0.4 < Γz < 2 (St. Laurent and Schmitt 1999; Inoue et al.330

2008). At steady state, balances of turbulent kinetic energy and scalar variance imply331

Γ =Rρ − 1

Rρ

γs

1 − γs(14)

in the absence of shear (Hamilton et al. 1989; McDougall and Ruddick 1992). For fastest-332

growing fingers, Γ is a function of Rρ by substituting γs = (Rρ)1/2[(Rρ)

1/2 − (Rρ − 1)1/2]333

(Stern 1975; Kunze 1987). In the presence of shear, for the fastest-growing salt sheets,334

Γ =Rρ − 1

Rρ

PrRi

PrRi + 1(15)

(Smyth and Kimura 2007). In the present DNS, Γ lies below 0.6 (figure 8a). In the linear335

regime (σLt < 2), Γ is consistent with (15). Γ decays slowly as the flow becomes unstable336

to secondary instability (σLt ∼ 3) then becomes quasi-steady between 0.3 and 0.4 in the337

turbulent regime.338

Γz is generally larger than Γ (figure 8b), consistent with ǫz > ǫ. For Ri > 6, Γz can be339

30 times larger than Γ in the pre-turbulent state (0 < σLt < 3). A local peak of Γz/Γ is340

formed during the secondary instability (σLt ∼ 5). As the flow becomes turbulent, the ratio341

16

decreases and approaches a quasi-steady value ∼ 2, i.e., Γz overestimates Γ by a factor of 2342

in the turbulent regime.343

Since both Γ and Γz become quasi-steady after σLt = 8, each of Γ and Γz is averaged344

over σLt > 8 to quantify Ri and Rρ dependences in the turbulent regime. In all cases,345

Γz coincides approximately with observations (figure 9). Both observed and simulated Γz346

increase with increasing Ri. Similarly, Γ increases with increasing Ri. The Rρ dependence of347

the observed Γz and the DNS result have different trends (figure 10). Observed Γz decreases348

with increasing Rρ, while both Γz and Γ from DNS increase with increasing Rρ. The linear349

result (15) predicts Γ from DNS remarkably well (figures 9 and 10).350

Our results cast doubt on the isotropy assumption used to infer Γ from microstructure351

profiles. This has implications for recent attempts to extend the k − ǫ formalism to include352

double diffusion (Canuto et al. 2008), where observed Γ estimates were used to validate353

model parameters. Also, Inoue et al. (2008) used Γz from St. Laurent and Schmitt (1999) to354

suggest that the fingers must have larger wavelengths than the fastest-growing mode (their355

figure 5). In contrast, we find Γ consistent with linear fastest-growing fingers (figures 9 and356

10).357

b. Effective diffusivities358

Effective diffusivities are often used in large-scale models in order to represent small-scale359

physics (e.g. Bryan 1987; Gargett and Holloway 1992; Walsh and Ruddick 1995). Gargett360

and Holloway (1992) found that steady-state solutions of low-resolution general circulation361

models (GCMs) were sensitive to the ratio of KT to KS. More recent modeling studies have362

found that regional circulations are significantly altered when parameterizations of KT and363

KS due to fingering are introduced (Merryfield et al. 1999; Zhang et al. 1999).364

In our DNS, KS increases exponentially until disruption of the primary instability at365

17

σLt ≈ 5 (figure 11). After a period of slow decline, KS approaches a quasi-steady state at366

σLt ≈ 8. KS is reduced in the presence of the background shear (figure 11a). For Ri = 0.5,367

KS is smaller by a factor of 4 compared to Ri = ∞ case. This finding was unexpected,368

and has important implications for turbulence modeling, where it is often assumed that,369

since background shear represents an energy source for mixing, its introduction will increase370

effective diffusivities (e.g. Large et al. 1994; Zhurbas and Oh 2001; Smyth and Ruddick 2010).371

We also find that KS increases with decreasing Rρ (figure 11b). This Rρ dependence372

had been reported by the previous DNS of two-dimensional salt-fingering (Merryfield and373

Grinder 2000; Stern et al. 2001). The flux ratio γs remains within ∼ 10% of 0.6 (figure 12a),374

which agrees with its linear value. In the the turbulent regime, γs is about 8% higher than375

its linear value in the unsheared case, but fluctuates within 3% of its linear value in the376

sheared cases.377

Ruddick (1985) suggested that individual salt sheets rapidly lose their momentum via378

lateral diffusion and therefore Sc < 1. A laboratory experiment to confirm this hypothesis379

was inconclusive (Ruddick et al. 1989). Linear stability analysis of salt sheets (Smyth and380

Kimura 2007), and subsequent DNS (Kimura and Smyth 2007) for a single initial case381

confirmed that Sc < 1 for Ri = 2. Sc < 1 holds in Ri ranging from 0.5 to 20 (figure 12b).382

Effective diffusivity of the momentum is an order of magnitude smaller than that of salt,383

confirming that salt sheets are inefficient in transporting momentum over the observed range384

of Ri and Rρ.385

In two-dimensional unsheared, turbulent salt fingers, Stern et al. (2001) showed that386

decrease in τ from 0.04 to 0.01 increased the heat flux by 15%. In 3D sheared, double-387

diffusive turbulence, Kimura and Smyth (2007) found that the decrease in τ from 0.04 to388

0.01 increases the KT and KS by a factor of 2. The effective diffusivities of heat and salt for389

the oceanic water may therefore be 2 times larger than our DNS results presented here.390

Both KT and KS decrease with increasing Rρ (figures 13 b,d), consistent with DNS results391

18

from Merryfield and Grinder (2000), despite the difference in Ri, τ , and computational392

domain dimensions. Estimates for three-dimensional salt-fingering by Stern et al. (2001)393

are twice as large; this could be due the presence of the shear and difference in τ in our394

simulations.395

Parameterizations of these diffusivities are essential in modeling large-scale flows where396

double-diffusive turbulence controls small-scale fluxes of heat and salt, such as thermohaline397

interleaving. Walsh and Ruddick (2000) employed a parameterization of KS for pure salt-398

fingering: KS = KS0R−nρ . Smyth (2008) fitted this simple model to DNS results from Stern399

et al. (2001) and obtained n = 2 and KS0 = 10−4. To add dependence on Ri, we compute a400

least squares fit of KT and KS, averaged over σLt > 8401

KT (Rρ, Ri) = 3.07 × 10−5R−4.0ρ Ri0.17 (16)

KS(Rρ, Ri) = 4.38 × 10−5R−2.7ρ Ri0.17. (17)

This Ri dependence differs significantly from the prediction of Kunze (1994), KS ∝ Ri1.0,402

but is closer to that of Inoue et al. (2008), KS ∝ Ri0.403

c. Observational estimates of effective diffusivities404

In interpretation of observational data, the Osborn and Cox (1972) diffusivity model in405

conjunction with the isotropy assumption gives406

Kχz

T =χz

T

2⟨

(

∂BT

∂z

)2⟩ ; Kχz

S =χz

S

2⟨

(

∂BS

∂z

)2⟩ , (18)

based on property variance dissipations, where χzS = 6κS

⟨

(

∂b′S∂z

)2⟩

. These estimates are407

exact for stationary, homogeneous, isotropic turbulence. Here, we compare these estimates408

19

with the true diffusivities KT and KS, computed directly using fluxes (12).409

The isotropic Osborn-Cox model (18) captures the Rρ dependence of KT and KS (figure410

14a and 14b). Both Kχz

T and Kχz

S decrease with increasing Rρ, consistent with the trends411

of KT and KS from DNS and observations by St. Laurent and Schmitt (1999). However,412

both Kχz

T and Kχz

S are smaller than KT and KS for the range of Rρ discussed here. The413

Osborn-Cox diffusivity model (1972) can therefore underestimate KT and KS by up to a414

factor of 3 for salt fingers at steady state.415

6. Conclusions416

We have simulated sheared, double-diffusive turbulence in a thermohaline staircase, using417

DNS. The following caveats should be noted:418

i. the ratio of molecular diffusivity of salt to heat is 4 times larger than the real ocean;419

ii. the interface thickness is at least 3 times smaller than observed thickness of thermoha-420

line staircases. In much thicker layers and continuous stratification, salt-finger flux laws421

change because the fingers no longer extend across the interface from one homogeneous422

layer to the other (Kunze 1987);423

iii. the equation of state is linear.424

Our main findings are:425

i. Velocity gradients are not isotropic, but scalar gradients are nearly isotropic. The426

approximations of ǫ based on vertical shears, which are often used in interpretations427

of microstructure data, underestimate its value by a factor of 2 to 3. This suggests428

that the rate of dissipation by sheared, double-diffusive turbulence can be 2 to 3 times429

larger than measured inferences.430

20

ii. The geometry of the primary instability is often (though not always) reflected in the431

anisotropy of the turbulent flow.432

iii. The isotropy assumption can lead to overestimation of the dissipation ratio Γ by a433

factor of 2 to 3. This will impact estimates of correlation time scales in the k−ǫ model434

of double-diffusive turbulence (e.g. Canuto et al. 2008). It also caused Inoue et al.435

(2008) to incorrectly ascribe the signals in St. Laurent and Schmitt (1999) to fingers436

with wavelengths larger than those of the fastest-growing mode.437

iv. A decrease in Ri or an increase in Rρ reduces the effective diffusivities of heat and438

salt. Empirical fits give KT (Rρ, Ri) = 3.07×10−5R−4.0ρ Ri0.17 and KS(Rρ, Ri) = 4.38×439

10−5R−2.7ρ Ri0.17. Kimura and Smyth (2007) showed that decrease in τ from 0.04 to 0.01440

increase KT and KS by a factor of 2. The dependence on Ri differs significantly from441

the theoretical prediction of Kunze (1994).442

v. Effective diffusivities estimated by the Osborn-Cox model (1972) produced consistent443

Rρ dependence but are up to a factor of three too small.444

vi. Values of nondimensional mixing efficiency Γ, flux ratio γ, and Schmidt number Sc in445

the turbulent regime are predicted by linear theory with remarkable accuracy.446

Acknowledgments.447

Computer time was provided by the National Center for Atmospheric Research (NCAR).448

The graphics are produced using VAPOR with help from VisLab at NCAR. The work was449

supported by the National Science Foundation under Grant No. 0453140.450

21

451

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27

List of Tables564

1 Relevant parameters used in our DNS experiments. The wave number of the565

fastest growing salt-fingering instability is determined by the magnitude of566

wave number, k2 + l2, where k and l represent the streamwise and spanwise567

wave numbers. In the case of salt sheets (all cases except DNS5), there is568

not streamwise dependence (k = 0), where the salt-fingering case (DNS5) has569

k = l. In our DNS experiments, k2 + l2 is kept constant. 29570

28

DNS1 DNS2 DNS3 DNS4 DNS5 DNS6 DNS7Ri 0.5 2 6 20 ∞ 6 6Rρ 1.6 1.6 1.6 1.6 1.6 1.2 2.0Pr 7 7 7 7 7 7 7Re 1914 957 553 302 0 553 553τ 0.04 0.04 0.04 0.04 0.04 0.04 0.04λfg[m] 0.031 0.031 0.031 0.031 0.044 0.031 0.032Lx[m] 0.9 0.9 0.9 0.9 1.2 0.9 0.9Ly[m] 0.12 0.12 0.12 0.12 0.18 0.12 0.12Lz [m] 1.8 1.8 1.8 1.8 1.8 1.8 1.8nx 1024 1024 1024 1024 1024 1024 1024ny 144 144 144 144 144 144 144nz 2048 2048 2048 2048 1538 2048 2048

Table 1. Relevant parameters used in our DNS experiments. The wave number of thefastest growing salt-fingering instability is determined by the magnitude of wave number,k2 + l2, where k and l represent the streamwise and spanwise wave numbers. In the case ofsalt sheets (all cases except DNS5), there is not streamwise dependence (k = 0), where thesalt-fingering case (DNS5) has k = l. In our DNS experiments, k2 + l2 is kept constant.

29

List of Figures571

1 Evolution of salinity buoyancy field for Ri = 6, Rρ = 1.6 in an interface572

sandwiched between two homogeneous layers with respect to the scaled time573

σLt. The interface occupies one third of the domain height. Homogenous574

regions above and below the interface are rendered transparent. Inside the575

interface, the lowest (−7.15 × 10−5m2s−1) and highest (7.15 × 10−5m2s−1)576

salinity buoyancy are indicated by purple and red, respectively. 33577

2 Evolution of volumed-averaged Reb for Ri = 6, Rρ = 1.6. 34578

3 (a) Saline (a) and thermal (b) buoyancy variance dissipation rates computed579

from individual derivatives and represented as fractions of the true value for580

the case, Ri = 6, Rρ = 1.6. Thin solid horizontal lines in (a) and (b) indicate581

unity (the ratio for isotropic turbulence). 35582

4 Approximate thermal variance dissipation rate from derivatives of squared583

perturbations as a fraction of its true values for different Ri and Rρ. Each of584

the three different forms in (6) and (7) is normalized by its true value, χS or585

χT , to quantify the degree of anisotropy. Each ratio is averaged for σLt > 8586

to represent the geometry in the turbulent state. Ratios below unity are blue,587

above unity red, and equal unity black. 36588

5 Evolution of7.5ν〈w′

,x2〉

ǫ,

7.5ν〈w′

,y2〉

ǫ, and

15ν〈w′

,z2〉

ǫfor (a) unsheared case (Ri =589

∞, Rρ = 1.6) and (b) sheared case (Ri = 6, Rρ = 1.6) . Thin solid horizontal590

lines in (a) and (b) indicate the ratio for isotropic turbulence. 37591

30

6 Approximations of ǫ from each of the squared perturbation velocity derivatives592

as a fraction of its value, ǫ for different Ri and Rρ. Each ratio is averaged for593

σLt > 8 to represent the geometry in the turbulent state. Ratios below unity594

are blue, above unity red, and equal unity black. 38595

7 Approximations of ǫ as a fraction of its true value with respect to σLt for (a)596

different Richardson number Ri and (b) different density ratio Rρ. Approxi-597

mations of χT as a fraction of its true value with respect to σLt for (c) different598

Richardson number Ri and (d) different density ratio Rρ. A solid horizontal599

line on each panel indicates the ratio for isotropic turbulence. 39600

8 Evolution of (a) Γ and (b) Γz normalized by its true value Γ for different Ri.601

These ratios are unity for isotropic turbulence as indicated by a thin solid line. 40602

9 Γ and Γz for different Ri compared to observations from Inoue et al (2008).603

Vertical bars denote 95% confidence limits (Inoue et al. 2008). Mean Rρ604

is nearly constant around 1.65 in Inoue et al. (2008). Smyth and Kimura605

(2007) calculated Γ using linear stability analysis. Here we showed their Γ for606

Rρ = 1.6. 41607

10 Γ and Γz for different Rρ compared to observations from Inoue et al. (2008).608

Vertical bars denote 95% confidence limits (Inoue et al. 2008). Mean Ri609

ranges between 0.4 and 10 for Inoue et al. (2008). Smyth and Kimura (2007)610

calculated Γ using linear stability analysis. Here we plot their Γ for Ri = 6. 42611

31

11 Effective diffusivity of salt, KS with respect to scaled time for (a) different Ri612

keeping Rρ = 1.6 and (b) different Rρ keeping Ri = 6. 43613

12 Evolution of (a) flux ratio, γs, and Schmidt number, Sc, with respect to scaled614

time for different Ri with keeping Rρ = 1.6. 44615

13 Effective diffusivity of heat with respect to Ri (a) and Rρ (b), and of salt with616

respect to Ri (c) and Rρ (d). Circles in (a) and (b) indicate diffusivities from617

three-dimensional DNS with molecular diffusivity ratio τ = 0.01 (Kimura and618

Smyth 2007); DNS results presented here have τ = 0.04. Downward triangles619

in (b) and (d) indicate two- dimensional DNS results with Ri = ∞ and620

τ = 0.01 (Merryfield and Grinder 2000). Squares in (b) and (d) are effective621

diffusivities for Ri = ∞ (Stern et al. 2001) estimated from the ratio of 2- to 3-622

D fluxes using accessible values of τ , then multiplying the ratio onto directly623

computed 2-D fluxes with τ = 0.01. 45624

14 Comparisons of effective diffusivities of heat (a) and salt (b) with respect625

to density ratio Rρ. Because of small scale structures pertained in salinity,626

estimation of χS from observations is difficult. Thus, KS in (b) is estimated627

as KobsS = Rρ

γsKχz

T in the interpretations of observations. 46628

32

(a) σLt = 3.1 (b) σLt = 5 (c) σLt = 8.7

Lx = 0.9m

Ly = 0.12m

Lz = 1.8m

Fig. 1. Evolution of salinity buoyancy field for Ri = 6, Rρ = 1.6 in an interface sandwichedbetween two homogeneous layers with respect to the scaled time σLt. The interface occupiesone third of the domain height. Homogenous regions above and below the interface arerendered transparent. Inside the interface, the lowest (−7.15 × 10−5m2s−1) and highest(7.15 × 10−5m2s−1) salinity buoyancy are indicated by purple and red, respectively.

33

0 1 2 3 4 5 6 7 8 9 10 110

2

4

6

8

10

12

σLt

Re b

Fig. 2. Evolution of volumed-averaged Reb for Ri = 6, Rρ = 1.6.

34

0

0.5

1

1.5

2

2.5

3

3.5

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5

χS1/χS

χS2/χS

χS3/χS

χT 1/χT

χT 2/χT

χT 3/χT

(a)

(b)

σLt

Fig. 3. (a) Saline (a) and thermal (b) buoyancy variance dissipation rates computed fromindividual derivatives and represented as fractions of the true value for the case, Ri = 6,Rρ = 1.6. Thin solid horizontal lines in (a) and (b) indicate unity (the ratio for isotropicturbulence).

35

0.54 0.77 0.92 1 1.1

0.96

0.78

1.2 1.2 1.2 1.2 1.1

1

1.3

1.3 1.1 0.9 0.81 0.82

1

0.88

1.2

1.6

2

0.50.50.5 222 666 202020 ∞∞∞

Rρ

RiRiRi

(a)6κT 〈b′

T2,x〉

χT(b)

6κT 〈b′T2,y〉

χT(c)

6κT 〈b′T2,z〉

χT

Fig. 4. Approximate thermal variance dissipation rate from derivatives of squared pertur-bations as a fraction of its true values for different Ri and Rρ. Each of the three differentforms in (6) and (7) is normalized by its true value, χS or χT , to quantify the degree ofanisotropy. Each ratio is averaged for σLt > 8 to represent the geometry in the turbulentstate. Ratios below unity are blue, above unity red, and equal unity black.

36

10−2

10−1

100

101

0 1 2 3 4 5 6 7 8 9 10 1110

−2

10−1

100

101

7.5ν〈w′

,x2〉

ǫ

7.5ν〈w′

,y2〉

ǫ

15ν〈w′

,z2〉

ǫ

(a)

(b)

σLt

Fig. 5. Evolution of7.5ν〈w′

,x2〉

ǫ,

7.5ν〈w′

,y2〉

ǫ, and

15ν〈w′

,z2〉

ǫfor (a) unsheared case (Ri = ∞, Rρ =

1.6) and (b) sheared case (Ri = 6, Rρ = 1.6) . Thin solid horizontal lines in (a) and (b)indicate the ratio for isotropic turbulence.

37

0.54 0.62 0.62 0.61 0.61

0.67

0.6

0.5 0.24 0.16 0.14 0.18

0.23

0.15

0.48 0.41 0.35 0.33 0.34

0.41

0.36

0.2 0.18 0.15 0.14 0.17

0.21

0.14

1 0.84 0.7 0.63 0.62

0.69

0.77

0.56 0.43 0.35 0.32 0.34

0.41

0.36

1.4 1.8 2.2 2.4 2.5

2.3

1.9

2.7 2.7 2.8 2.7 2.5

2.4

3

1.7 1.7 1.6 1.6 1.5

1.6

1.7

1.2

1.2

1.2

1.6

1.6

1.6

2

2

2

0.50.50.5 222 666 202020 ∞∞∞

Rρ

Rρ

Rρ

RiRiRi

(a)15ν〈u′

,x2〉

ǫ (b)7.5ν〈u′

,y2〉

ǫ(c)

7.5ν〈u′

,z2〉

ǫ

(d)7.5ν〈v′,x

2〉

ǫ (e)15ν〈v′,y

2〉

ǫ(f)

7.5ν〈v′,z2〉

ǫ

(g)7.5ν〈w′

,x2〉

ǫ (h)7.5ν〈w′

,y2〉

ǫ(i)

15ν〈w′

,z2〉

ǫ

Fig. 6. Approximations of ǫ from each of the squared perturbation velocity derivatives as afraction of its value, ǫ for different Ri and Rρ. Each ratio is averaged for σLt > 8 to representthe geometry in the turbulent state. Ratios below unity are blue, above unity red, and equalunity black.

38

10−3

10−2

10−1

100

101

0 2 4 6 8 1010

−3

10−2

10−1

100

101

0 2 4 6 8 10

(a) ǫz

ǫ for different Ri

Ri = 0.5, Rρ = 1.6Ri = 20, Rρ = 1.6

Ri = 0.5, Rρ = 1.6Ri = 20, Rρ = 1.6

(b) ǫz

ǫ for different Rρ(b) ǫz

ǫ for different Rρ

Ri = 6, Rρ = 1.2Ri = 6, Rρ = 2

Ri = 6, Rρ = 1.2Ri = 6, Rρ = 2

(c)χz

T

χTfor different Ri(c)

χzT

χTfor different Ri

σLt

(d)χz

T

χTfor different Rρ

σLt

Fig. 7. Approximations of ǫ as a fraction of its true value with respect to σLt for (a)different Richardson number Ri and (b) different density ratio Rρ. Approximations of χT asa fraction of its true value with respect to σLt for (c) different Richardson number Ri and(d) different density ratio Rρ. A solid horizontal line on each panel indicates the ratio forisotropic turbulence.

39

0

0.1

0.2

0.3

0.4

0.5

0.6

0 1 2 3 4 5 6 7 8 9 10 110

2

4

6

8

Γ, Ri = 0.5, Rρ = 1.6

Γ, Ri = 6, Rρ = 1.6

Γ, Ri = ∞, Rρ = 1.6

Γz/Γ, Ri = 0.5, Rρ = 1.6

Γz/Γ, Ri = 6, Rρ = 1.6

Γz/Γ, Ri = ∞, Rρ = 1.6

ΓΓ

z/Γ

σLt

(a)

(b)

Fig. 8. Evolution of (a) Γ and (b) Γz normalized by its true value Γ for different Ri. Theseratios are unity for isotropic turbulence as indicated by a thin solid line.

40

100

101

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Γ from DNS Rρ = 1.6Γz from DNS Rρ = 1.6Γz from Inoue et al. (2008)Γ from Smyth and Kimura (2007)

Γ

Ri

Fig. 9. Γ and Γz for different Ri compared to observations from Inoue et al (2008). Verticalbars denote 95% confidence limits (Inoue et al. 2008). Mean Rρ is nearly constant around1.65 in Inoue et al. (2008). Smyth and Kimura (2007) calculated Γ using linear stabilityanalysis. Here we showed their Γ for Rρ = 1.6.

41

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Γ from DNS Ri = 6Γz from DNS Ri = 6Γz from Inoue et al. (2008)Γ from Smyth and Kimura (2007)

Γ

Rρ

Fig. 10. Γ and Γz for different Rρ compared to observations from Inoue et al. (2008).Vertical bars denote 95% confidence limits (Inoue et al. 2008). Mean Ri ranges between 0.4and 10 for Inoue et al. (2008). Smyth and Kimura (2007) calculated Γ using linear stabilityanalysis. Here we plot their Γ for Ri = 6.

42

10−7

10−6

10−5

10−4

0 1 2 3 4 5 6 7 8 9 10 1110

−7

10−6

10−5

10−4

Ri = 0.5, Rρ = 1.6

Ri = 20, Rρ = 1.6

Ri = ∞, Rρ = 1.6

Ri = 6, Rρ = 1.2

Ri = 6, Rρ = 1.6

Ri = 6, Rρ = 2

σLt

KS

[m

2s−

1]

KS

[m

2s−

1]

(b)

(a)

Fig. 11. Effective diffusivity of salt, KS with respect to scaled time for (a) different Rikeeping Rρ = 1.6 and (b) different Rρ keeping Ri = 6.

43

0.5

0.55

0.6

0.65

0.7

0 1 2 3 4 5 6 7 8 9 10 110

0.05

0.1

Ri = 0.5, Rρ = 1.6Ri = 20, Rρ = 1.6Ri = ∞, Rρ = 1.6

Ri = 0.5, Rρ = 1.6Ri = 2, Rρ = 1.6Ri = 20, Rρ = 1.6

σLt

γs

Sc

(b)

(a)

Fig. 12. Evolution of (a) flux ratio, γs, and Schmidt number, Sc, with respect to scaledtime for different Ri with keeping Rρ = 1.6.

44

10−6

10−5

100

101

10−6

10−5

1.2 1.4 1.6 1.8 2

KT[m

2s−

1]

(a)

KT (Rρ = 1.6, Ri)

DNS

Kimura and Smyth (2007), τ = 0.01

(b)

KT (Rρ, Ri = 6)

DNS

DNS, Ri = ∞

Merryfield and Grinder (2000), τ = 0.01

Estimate of Stern et al. (2001), τ = 0.01

KS[m

2s−

1]

(c)

KS(Rρ = 1.6, Ri)

DNS

Kimura and Smyth (2007), τ = 0.01

(d)

KS(Rρ, Ri = 6)

DNS

DNS, Ri = ∞

Merryfield and Grinder (2000), τ = 0.01

Estimate of Stern et al. (2001),τ = 0.01

Ri Rρ

Fig. 13. Effective diffusivity of heat with respect to Ri (a) and Rρ (b), and of salt with re-spect to Ri (c) and Rρ (d). Circles in (a) and (b) indicate diffusivities from three-dimensionalDNS with molecular diffusivity ratio τ = 0.01 (Kimura and Smyth 2007); DNS results pre-sented here have τ = 0.04. Downward triangles in (b) and (d) indicate two- dimensionalDNS results with Ri = ∞ and τ = 0.01 (Merryfield and Grinder 2000). Squares in (b) and(d) are effective diffusivities for Ri = ∞ (Stern et al. 2001) estimated from the ratio of 2- to3- D fluxes using accessible values of τ , then multiplying the ratio onto directly computed2-D fluxes with τ = 0.01.

45

10−6

10−5

10−4

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.110

−6

10−5

10−4

(a)

(b)

Rρ

KT[m

2s−

1]

KS[m

2s−

1]

KT , Ri = 6

Kχz

T , Ri = 6

Kχz

T from St. Laurent and Schmitt (1999)

KS , Ri = 6

Kχz

S , Ri = 6

KobsS from St. Laurent and Schmitt (1999)

Fig. 14. Comparisons of effective diffusivities of heat (a) and salt (b) with respect todensity ratio Rρ. Because of small scale structures pertained in salinity, estimation of χS from

observations is difficult. Thus, KS in (b) is estimated as KobsS = Rρ

γsKχz

T in the interpretationsof observations.

46