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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 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

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

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 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

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;

DbTDt

= 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-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=

bTBT

=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 102109

rad s1, a value typical of the thermohaline staircase east of Barbados (Gregg and Sanford110

1987).111

These constants can be combined with the fluid parameters , T , and S to form non-112

5

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

Ri =Bh

u2;

R = BTBS

;

Pr =

T;

=ST

.

We have done 7 experiments with different Ri and R (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 = hu/. Since B120

and h are kept constant in our simulations, Re and Ri are not independent: Re = 1354Ri1/2121

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

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

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

We varied R 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, , 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, 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

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

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

exploration of the Ri and R dependence.136

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

2(T 2h/B)1/4 (Stern 1975; Schmitt 1979). Our value matches the observed value, =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 = 4fg. 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 = 28fg.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 102hL to seed secondary instabilities.149

The variable, L 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 0.15fg

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, R = 1.6 at selected times.161

The time is scaled by the linear normal growth rate of salt sheets, L, described b...