Satoshi Kimura William Smyth Eric Kunze - Bill ?· Satoshi Kimura ∗ William Smyth College of Oceanic…

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<p>Generated using version 3.0 of the official AMS LATEX template</p> <p>Turbulence in a sheared, salt-fingering-favorable environment:1</p> <p>Anisotropy and effective diffusivities2</p> <p>Satoshi Kimura William Smyth</p> <p>College of Oceanic and Atmospheric Sciences, Oregon State University, Oregon</p> <p>3</p> <p>Eric Kunze</p> <p>School of Earth and Ocean Sciences, University of Victoria, Canada</p> <p>4</p> <p>Corresponding author address: College of Oceanic and Atmospheric Sciences, Oregon State UniversityOcean Admin Bldg, Corvallis, OR 97331, USA.E-mail:</p> <p>1</p> <p>ABSTRACT5</p> <p>Direct numerical simulations (DNS) of a shear layer with salt-fingering-favorable stratifica-6</p> <p>tion have been performed for different Richardson numbers Ri and density ratios R. When7</p> <p>the Richardson number is infinite (unsheared case), the primary instability is square plan-8</p> <p>form salt-fingering, alternating cells of rising and sinking fluid. In the presence of shear,9</p> <p>salt-fingering takes the form of salt sheets, planar regions of rising and sinking fluid, aligned10</p> <p>parallel to the sheared flow. After the onset of secondary instability, the flow becomes11</p> <p>turbulent; however, our results indicate that the continued influence of the primary insta-12</p> <p>bility biases estimates of the turbulent kinetic energy dissipation rate. Thermal and saline13</p> <p>buoyancy gradients become more isotropic than the velocity gradients at dissipation scales.14</p> <p>Estimates of the turbulent kinetic energy dissipation rate by assuming isotropy in the ver-15</p> <p>tical direction can underestimate its true value by a factor of 2 to 3, whereas estimates of16</p> <p>thermal and saline dissipation rates are relatively accurate. approximated by assuming17</p> <p>isotropy agrees well with observational estimates, but is larger than the true value of by18</p> <p>approximated a factor of 2. The transport associated with turbulent salt sheets is quanti-19</p> <p>fied by thermal and saline effective diffusivities. Salt sheets are ineffective at transporting20</p> <p>momentum. Thermal and saline effective diffusivities decrease with decreasing Ri, despite21</p> <p>the added energy source provided by background shear. Nondimensional quantities, such as22</p> <p>thermal to saline flux ratio, are close to the predictions of linear theory.23</p> <p>1</p> <p>1. Introduction24</p> <p>Salt-fingering-favorable stratification occurs when the gravitationally unstable vertical25</p> <p>gradient of salinity is stabilized by that in temperature. In such conditions, the faster diffu-26</p> <p>sion of heat than salt can generate cells of rising and sinking fluid known as salt fingers, which27</p> <p>take a variety of planforms such as squares, rectangles, and sheets (Schmitt 1994b; Proctor28</p> <p>and Holyer 1986). Salt-fingering-favorable stratification is found in much of the tropical29</p> <p>and subtropical pycnocline (You 2002). The most striking signatures are in thermohaline30</p> <p>staircases, which are stacked layers of different water-types separated by sharp thermal and31</p> <p>saline gradients. These are found at several locations, such as in the subtropical confluence32</p> <p>east of Barbados, under the Mediterranean and Red Sea salt tongues and in the Tyrrhenian33</p> <p>Sea (Tait and Howe 1968; Lambert and Sturges 1977; Schmitt 1994a).34</p> <p>Salt-fingering is subjected to vertically varying horizontal currents (background shear)35</p> <p>(Gregg and Sanford 1987; St. Laurent and Schmitt 1999). Zhang et al. (1999) found a36</p> <p>significant relationship between salt fingering and shear in a model of the North Atlantic.37</p> <p>A theory for the possible relationship between salt fingering and shear has been introduced38</p> <p>by Schmitt (1990). In the presence of shear, salt-fingering instability is supplanted by salt-39</p> <p>sheet instability, alternating planar regions of rising and sinking fluid, aligned parallel to the40</p> <p>sheared flow (Linden 1974). While theories for the initial growth of salt-fingering and salt-41</p> <p>sheet instabilities are well-established (Stern 1960; Linden 1974; Schmitt 1979; Kunze 2003;42</p> <p>Smyth and Kimura 2007), the vertical fluxes of momentum, heat, and salt in the non-linear43</p> <p>regime are not well understood. We call this non-linear regime turbulent salt-fingering.44</p> <p>Dissipation rates of property variances in double-diffusive turbulence can be measured45</p> <p>in the ocean by microstructure profilers, but the interpretation of these results has relied46</p> <p>on Kolmogorovs hypothesis of isotropic turbulence (Gregg and Sanford 1987; Lueck 1987;47</p> <p>Hamilton et al. 1989; St. Laurent and Schmitt 1999; Inoue et al. 2008). Kolmogorov (1941)48</p> <p>2</p> <p>proposed that small-scale turbulence statistics are universal in the limit of high Reynolds49</p> <p>number. According to this hypothesis, anisotropy of the energy-containing scales is lost in50</p> <p>the turbulent energy cascade, so that the small scales, where energy is finally dissipated,51</p> <p>are statistically isotropic. The assumption of small-scale isotropy greatly simplifies both the52</p> <p>theory and modeling of turbulence, as well as interpretation of microstructure measurements.53</p> <p>Because the Reynolds number in oceanic salt-fingering systems is O(10) (McDougall and54</p> <p>Taylor 1984), the validity of this assumption is questionable. We will examine both mixing55</p> <p>rates and the isotropy of the dissipation range in turbulent salt-fingering by means of direct56</p> <p>numerical simulations of diffusively unstable shear layers.57</p> <p>Based on observations of shear-driven turbulence in flow over a sill, Gargett et al. (1984)58</p> <p>concluded that the isotropy assumption was accurate as long as the separation between59</p> <p>Ozmidov and Kolmogorov length scale was sufficiently large. Itsweire et al. (1993) tested60</p> <p>isotropic approximations on flows with uniform shear and stratification for Richardson num-61</p> <p>ber using DNS, and found that dissipation rates could be underestimated by factors of 2 to62</p> <p>4 at low Reynolds number. Smyth and Moum (2000) extended the analysis to a localized63</p> <p>shear layer and found similar results.64</p> <p>Estimations of dissipation rates, combined with the Osborn and Cox (1972) diffusivity65</p> <p>model, can furnish estimates of the effective diffusivities of heat, salt, and momentum. Ef-66</p> <p>fective diffusivities are used to parameterize turbulent fluxes in order to model large-scale67</p> <p>phenomena affected by mixing processes. These applications range from fine-scale thermo-68</p> <p>haline intrusions (e.g. Toole and Georgi 1981; Walsh and Ruddick 1995; Smyth and Ruddick69</p> <p>2010) to basin-scale circulations (e.g. Zhang et al. 1999; Merryfield et al. 1999).70</p> <p>Effective diffusivities can be calculated directly from DNS. Thermal and saline effective71</p> <p>diffusivities for 2D salt-fingering have been computed in previous studies (Merryfield and72</p> <p>Grinder 2000; Stern et al. 2001; Yoshida and Nagashima 2003; Shen 1995). The effective73</p> <p>diffusivities for 3D sheared, salt-fingering were first calculated by Kimura and Smyth (2007)74</p> <p>3</p> <p>for a single initial state.75</p> <p>In this paper, we extend Kimura and Smyth (2007) to cover a range of oceanographically76</p> <p>relevant initial background states. Although oceanic salt fingering does not always lead to77</p> <p>staircases, that is the regime we focus on here as it is most amenable to direct simulation.78</p> <p>Our objective is twofold. First, we explore the isotropy assumption for turbulent salt fingers79</p> <p>and salt sheets. Second, we compute effective diffusivities to identify effects of anisotropy80</p> <p>that could affect the accuracy of in situ estimates.81</p> <p>In section 2, we describe our DNS model and initial conditions. An overview of salt sheet82</p> <p>evolution is given in section 3. Section 4 discusses the nature of anisotropy at dissipation83</p> <p>scales and its implications for interpreting profiler measurements. In section 5, we discuss84</p> <p>the effective diffusivities of momentum, heat, and salt, and suggest a parameterization in85</p> <p>terms of ambient property gradients. Our conclusions are summarized in section 6.86</p> <p>2. Methodology87</p> <p>We employ the three-dimensional incompressible Navier-Stokes equations with the Boussi-88</p> <p>nesq approximation. The evolution equations for the velocity field, ~u(x, y, z, t) = {u, v, w},89</p> <p>in a nonrotating, Cartesian coordinate system, {x, y, z}, are90</p> <p>[</p> <p>D</p> <p>Dt 2</p> <p>]</p> <p>~u = + bk + 2~u</p> <p> ~u = 0. (1)</p> <p>D/Dt = /t+ ~u ~ and are the material derivative and kinematic viscosity, respectively.91</p> <p>The variable represents the reduced pressure (pressure scaled by the uniform density 0).92</p> <p>The total buoyancy is defined as b = g(0)/0, where g is the acceleration due to gravity.93</p> <p>Buoyancy acts in the vertical direction, as indicated by the vertical unit vector k. We assume94</p> <p>4</p> <p>that equation of state is linear, and therefore the total buoyancy is the sum of thermal and95</p> <p>saline buoyancy components (bT and bS), each governed by an advection-diffusion equation:96</p> <p>b = bT + bS;</p> <p>DbTDt</p> <p>= T2bT ; (2)DbSDt</p> <p>= S2bS. (3)</p> <p>Molecular diffusivities of heat and salt are denoted by T and S, respectively.97</p> <p>Periodicity intervals in streamwise (x) and spanwise (y) directions are Lx and Ly. The98</p> <p>variable Lz represents the vertical domain length. Upper and lower boundaries, located at99</p> <p>z = Lz/2 and z = Lz/2, are impermeable (w = 0), stress-free (u/z = v/z = 0), and100</p> <p>insulating with respect to both heat and salt (bT /z = bS/z = 0).101</p> <p>To represent mixing in the high-gradient interface of a thermohaline staircase, we102</p> <p>initialize the model with a stratified shear layer in which shear and stratification are concen-103</p> <p>trated at the center of the vertical domain with a half-layer thickness of h:104</p> <p>u</p> <p>u=</p> <p>bTBT</p> <p>=bS</p> <p>BS= tanh</p> <p>(z</p> <p>h</p> <p>)</p> <p>.</p> <p>The constants u, BT , and BS represent the change in streamwise velocity, thermal buoy-105</p> <p>ancy, and saline buoyancy across the half-layer thickness of h. For computational economy,106</p> <p>we set h = 0.3 m. This is at the low end of the range of observed layer thickness (e.g. Gregg107</p> <p>and Sanford 1987; Kunze 1994). The change in the total buoyancy is B = BT + BS.108</p> <p>In all the DNS experiments, the initial buoyancy frequency (</p> <p>B/h) is fixed at 1.5 102109</p> <p>rad s1, a value typical of the thermohaline staircase east of Barbados (Gregg and Sanford110</p> <p>1987).111</p> <p>These constants can be combined with the fluid parameters , T , and S to form non-112</p> <p>5</p> <p>dimensional parameters, which characterize the flow at t = 0:113</p> <p>Ri =Bh</p> <p>u2;</p> <p>R = BTBS</p> <p>;</p> <p>Pr =</p> <p>T;</p> <p> =ST</p> <p>.</p> <p>We have done 7 experiments with different Ri and R (table 1). The bulk Richardson114</p> <p>number, Ri, measures the relative importance of stratification and shear. If Ri &lt; 0.25, the115</p> <p>initial flow is subjected to shear instabilities (Miles 1961; Howard 1961; Hazel 1972). Here,116</p> <p>we chose high enough Ri to ensure that shear instabilities do not disrupt the growth of117</p> <p>salt-fingering modes. A typical range of Ri in a sheared, thermohaline staircase is 3 100118</p> <p>(Gregg and Sanford 1987; Kunze 1994). The Reynolds number based on the half-layer119</p> <p>thickness and the half change in streamwise velocity is given by Re = hu/. Since B120</p> <p>and h are kept constant in our simulations, Re and Ri are not independent: Re = 1354Ri1/2121</p> <p>with Re = 0 and Ri = representing the unsheared case.122</p> <p>The density ratio, R, quantifies the stabilizing effect of thermal to destabilizing effect123</p> <p>of saline buoyancy components; salt-fingering grows more rapidly as R approaches unity.124</p> <p>We varied R between 1.2 and 2, which covers the range of observational data available for125</p> <p>comparison (St. Laurent and Schmitt 1999; Inoue et al. 2008). The Prandtl number, Pr,126</p> <p>and the diffusivity ratio, , represent ratios of the molecular diffusivities of momentum, heat,127</p> <p>and salt. The Prandtl number was set to 7, which is a typical value for salt water. The128</p> <p>diffusivity ratio in the ocean is 0.01, i.e., the heat diffuses two orders of magnitude faster than129</p> <p>salt. The vast difference in diffusivity requires DNS to resolve a wide range of spatial scales,130</p> <p>making it computationally expensive. In previous DNS of salt water, has been artificially131</p> <p>increased to reduce required computational expenses (e.g. Stern et al. 2001; Gargett et al.132</p> <p>6</p> <p>2003; Smyth et al. 2005). Kimura and Smyth (2007) conducted the first 3D simulation with133</p> <p> = 0.01 and found that increasing from 0.01 to 0.04 reduced thermal and saline effective134</p> <p>diffusivities by one half. In the cases presented here, we set to 0.04 to allow an extended135</p> <p>exploration of the Ri and R dependence.136</p> <p>The fastest-growing salt-sheet wavelength predicted by linear stability analysis, is fg =137</p> <p>2(T 2h/B)1/4 (Stern 1975; Schmitt 1979). Our value matches the observed value, =138</p> <p>0.032 m (Gregg and Sanford 1987; Kunze 2003). We accommodate four wavelengths of139</p> <p>the fastest-growing primary instability in the spanwise direction, Ly = 4fg. The vertical140</p> <p>domain length, Lz, was chosen such that vertically propagating plumes reach statistical141</p> <p>equilibrium. We found that Lz equal to six times h was sufficient. Lx was chosen to be large142</p> <p>enough to accommodate subsequent secondary instabilities. After sensitivity tests, we chose143</p> <p>Lx = 28fg.144</p> <p>The primary instability was seeded by adding an initial disturbance proportional to the145</p> <p>fastest-growing mode of linear theory, computed numerically as described in Smyth and146</p> <p>Kimura (2007). We seed square salt-fingering for Ri = and sheets in the presence of147</p> <p>shear. The vertical displacement amplitude is set to 0.02h, and a random noise was added148</p> <p>to the initial velocity field with an amplitude of 1 102hL to seed secondary instabilities.149</p> <p>The variable, L indicates the growth rate of the fastest growing linear normal mode.150</p> <p>The numerical code used to solve (1) - (3) is described by Winters et al. (2004). The151</p> <p>code uses Fourier pseudospectral discretization in all three directions, and time integration152</p> <p>using a third-order Adams-Bashforth operator. A time step is determined by a Courant-153</p> <p>Friedrichs-Lewy (CFL) stability condition. The CFL number is maintained below 0.21 for154</p> <p>DNS experiments presented here. The code was modified by Smyth et al. (2005) to accom-155</p> <p>modate a second active scalar, which is resolved on a fine grid with spacing equal to one half156</p> <p>the spacing used to resolve the other fields. The fine grid is used to resolve salinity. The157</p> <p>fine grid spacing is equal to 0.15fg</p> <p> in all three directions, as suggested by Stern et al.158</p> <p>7</p> <p>(2001).159</p> <p>3. Flow overview160</p> <p>Figure 1 shows the salinity buoyancy field for the case Ri = 6, R = 1.6 at selected times.161</p> <p>The time is scaled by the linear normal growth rate of salt sheets, L, described b...</p>