Generated using version 3.0 of the official AMS L A T E X template Turbulence in a sheared, salt-fingering-favorable environment: 1 Anisotropy and effective diffusivities 2 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 University Ocean Admin Bldg, Corvallis, OR 97331, USA. E-mail: [email protected]1
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Generated using version 3.0 of the official AMS LATEX template
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: [email protected]
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
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
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.
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