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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


Eric Kunze

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


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



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 R. 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. approximated by assuming17

isotropy agrees well with observational estimates, but is larger than the true value of 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. 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 Kolmogorovs 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


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


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



Dt 2


~u = + bk + 2~u

~u = 0. (1)

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

The variable represents the reduced pressure (pressure scaled by the uniform density 0).92

The total buoyancy is defined as b = g(0)/0, where g is the acceleration due to gravity.93

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


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;


= T2bT ; (2)DbSDt

= S2bS. (3)

Molecular diffusivities of heat and salt are denoted by T and S, 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-