Abstract
In this note, we provide an overview of the theoretical, numerical, and observational studies focused on oceanic eddy diffusivity, with an emphasis on double-diffusive convection (DDC). DDC, when calculated using the turbulent kinetic energy (TKE) equation, produces a negative diffusion of density. A second-moment closure model shows that DDC is effective within a narrow range. Other parameterizations can use in the actual sea, but improvements are still needed. Mixing coefficients referring to mixing efficiency are key factors when distinguishing DDC from conventional turbulence. Here, we show that measurements involving the gradient Richardson number, the buoyancy Reynolds number, and density ratio play a crucial role in determining eddy diffusivity in the presence of DDC. Therefore, deployment of a microstructure profiler together with either an acoustic Doppler current profiler (ADCP), lowered ADCP, or electromagnetic current meter is essential when measuring eddy diffusivity in the ocean’s interior.
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1 Introduction
Microstructures resulting from conventional turbulence (CT) and double-diffusive convection (DDC) are among the many noteworthy physical processes occurring in the ocean. Although the rates of microstructure occurrence and their effects are gradually being revealed, more complete information about the occurrence of microstructures remains unknown. A better understanding of oceanic microstructures will provide value to multiple fields and may help in answering some outstanding questions in climatic modeling, water mass modification, and oceanic nutrient distribution processes.
CT and DDC are related to large-scale oceanic processes. For example, internal wave (IW) breaking can produce significant amounts of turbulence (e.g., Polzin et al. 1997). DDC occurring at ~ 400 db generates North Pacific Intermediate Water (Talley and Yun 2001) and leads to intrusions in the subsurface layer off the Sanriku Coast of Japan (e.g., Nagata 1970; Nagasaka et al. 1999). Taken together, studies on microscale mixing [~ O(10−2) m] are strongly correlated with large-scale processes (e.g., meridional circulation: ~ O(103) m, intrusion and IWs: ~ O(102) m; Munk 1966; Bryan 1987; Gargett and Holloway 1992; Karl 1999). Nonetheless, the effects of DDC have been historically ignored in scientific study.
One reason why DDC has been ignored is the shortage of empirical knowledge typically obtained through observation. The opportunity for observations is limited because DDC is known to occur in areas such as shallow regions with commercial usage or in polar regions (e.g., Hirano et al. 2010). Moreover, limited ship time for observation and high cost of the microstructure profiler interrupt microstructure observations. Difficulties in handling microstructure data also exist. In addition, the areas surveyed for the detection of microstructures have incomplete coverage because the spatiotemporal scales of microscale processes are smaller than those detected by routine observations.
In order to compensate for the difficulties mentioned above, parameterizations of eddy diffusivities and kinetic energy dissipation rates have been developed using the conductivity temperature depth (CTD) profiler, lowered ADCP (LADCP), and other common oceanic instruments for hydrographic data collection. However, at its current state, the parameterization is not completely developed, because the methods are based on certain limitations. Nearly all of the parameterization concerns deal with shear-driven turbulence (CT), which is due to IWs. When the velocity shear is superior (Kunze 1990), DDC coexists with CT; nevertheless, DDC has been ignored in the parameterization. Parameterizations of DDC are carried out using laboratory experiments and direct numerical simulations (DNS). This means that a comparison focusing on DDC with microstructure data is still required. Therefore, a more precise parameterization is required for future microstructure studies.
The rest of this overview is structured as follows. We summarize previous studies on eddy diffusivity and present the results of DDC parameterization in oceanic turbulent mixing. DDC in the turbulent kinetic energy (TKE) equation is discussed in Sect. 2. Parameterization of eddy diffusivity using a second-moment closure (SMC) model is described in Sect. 3. Other types of DDC parameterizations in numerical simulations are described in Sects. 4 and 5. Key points regarding the eddy diffusivity estimation with measurement data are described in Sect. 6. Finally, concluding remarks are presented in Sect. 7. Details regarding the turbulent kinetic energy (TKE) equation, laboratory flux laws, SMC model, and relevant terminologies are presented in Appendices A–D, respectively.
2 Eddy diffusivity with turbulent kinetic energy equation and flux laws
DDC has two forms of convection: salt finger convection (SF) and diffusive convection (DC). DDC is characterized by the density ratio \(R_{\rho } {{ = \alpha \frac{{\partial \bar{T}}}{\partial z}} \mathord{\left/ {\vphantom {{ = \alpha \frac{{\partial \bar{T}}}{\partial z}} {\beta \frac{{\partial \bar{S}}}{\partial z}}}} \right. \kern-0pt} {\beta \frac{{\partial \bar{S}}}{\partial z}}}\), which is the ratio of the background density gradient due to temperature to that due to salt, where α and β are the expansion and contraction coefficients for heat and salt, respectively (Eq. 77). \(\frac{{\partial \bar{T}}}{\partial z}\) and \(\frac{{\partial \bar{S}}}{\partial z}\) represent the background temperature and salt gradients, respectively. Generally, SF is considered active when 1 < Rρ< 2, and DC is considered active when 0.5 < Rρ< 1 (e.g., Inoue et al. 2007). When CT is weak and DDC is active, the density is transported downward because of the difference in molecular diffusivity for heat and salt; therefore, the eddy diffusivities for salt KS, heat KT, and density Kρ are not equal to one another (see Appendices A and B). This characteristic is unique to DDC.
Consider the steady-state TKE equation for SF without background velocity shear (refer to Eq. 67). The balance equation between the dissipation rate of the TKE ε (refer to kinetic energy dissipation rates) and the energy production via buoyancy flux Jb is as follows:
Thus, Jb should be negative for DDC. From Eq. (82), and under the Boussinesq approximation (\(\bar{\rho } = \rho_{0}\)), Jb can be written as
Then, Eq. (1) can be rewritten as
where \(\gamma^{\text{SF}}\) is the density flux ratio due to SF (see Appendix B). Here, the square of buoyancy frequency \(N\) is described as
From the definition of \(K_{S}\) and \(K_{T}\) in DDC (Eq. 101) with Eq. 4, we obtain an expression for the vertical eddy diffusivity of salt for SF \(K_{S}^{\text{SF}}\):
From the definition of \(R_{\rho }\), the vertical eddy diffusivity of heat for SF \(K_{T}^{\text{SF}}\) is given by:
Rewriting Eq. (82) as Eq. (7), the vertical eddy diffusivity of the density of SF \(K_{\rho }^{\text{SF}}\) can be written as Eq. (8):
From Eqs. (5, 6, 7, and 8), we have
Eddy diffusivities for DC (\(K_{S}^{\text{DC}}\), \(K_{T}^{\text{DC}}\), and \(K_{\rho }^{\text{DC}}\)) are obtained in the same way:
Note that \(K_{\rho }\) is negative in the presence of DDC, indicating that DDC reduces the potential energy of the system and intensifies density stratification. Using the flux laws created by Huppert (1971, Eq. 102), Kunze (1987, Eq. 93), and Kelley (1986, Eq. 94, Kelley 1990, Eq. 103), variations of the eddy diffusivity in DDC with inactive CT (taking ε = 10−10 W kg−1 and N = 5.2 × 10−3) are shown in Fig. 1. \(K_{S}^{\text{SF}}\) and \(K_{T}^{SF}\) take large values with active SF (1 < \(R_{\rho }\) < 2).\(K_{S}^{\text{DC}}\) and \(K_{T}^{\text{DC}}\) take large values with active DC (0.5 < \(R_{\rho }\) < 1). The validity of this range will be confirmed in the next section.
3 DDC in SMC
When estimating the eddy diffusivity in the presence of DDC, the effect of velocity shear has been traditionally ignored. Linden (1974) experimentally showed that three-dimensional SF in the steady shear flow aligned with the velocity shear to form two-dimensional sheets, and with the resultant vertical transports of salt and heat remaining unchanged. However, Kunze (1990) analyzed C-SALT data and confirmed that oceanic SF should take the form of two-dimensional sheets due to velocity shear, leading to a reduction in the vertical buoyancy flux of SF. Wells et al. (2001) numerically and experimentally investigated the structure of SF in the presence of periodic shear flow, with the results revealing a reduced vertical buoyancy flux of SF. Therefore, we cannot neglect the shear effects on DDC.
For investigating the effect of shear on both DDC and CT, SMC was employed by Canuto et al. (2008), Kantha and Carniel (2009), and Kantha (2012). In this review, we follow the approach used in Kantha et al. (2011) and Kantha (2012), including the variances of both temperature and salinity in the steady-state energy equation (Eq. 79). The turbulent timescale τ is introduced as
where B1 is the coefficient for the turbulent timescale, q is the turbulence velocity scale, \(\ell\) is the turbulence length scale, and K is the TKE (= q2/2). The second-moment terms of transport for heat \(\overline{{w^{\prime}T^{\prime}}}\), salt \(\overline{{w^{\prime}S^{\prime}}}\), and momentum \(\overline{{u^{\prime}w^{\prime}}}\) are parameterized in Eqs. (80, 81 and 88) (the first-order closure), and the structure functions for the salt \(S_{S}\), heat \(S_{T}\), density \(S_{\rho }\), and momentum \(S_{\upsilon }\) are introduced with the eddy diffusivity for salt KS, temperature KT, density Kρ, and momentum \(K_{\upsilon }\), defined as:
From Eq. (8) or Eq. (12), relations among \(S_{S}\), \(S_{T}\), and \(S_{\rho }\) can be obtained:
This model is described in Appendix C. After a series of manipulations involving Eqs. (52, 71, and 72) using Eqs. (105, 106, 107, 108, 109, 110, 111, and 112), one can obtain the relations between the structure functions in the DDC as functions of the gradient Richardson number Ri, defined as Eq. (90), Rρ and N:
Introduce the non-dimensional numbers, \(G_{T}\) and \(G_{\upsilon }\) such that
Using Eqs. (20 and 21), we have the ratio between \(G_{T}\) and \(G_{\upsilon }\) as follows
Using Eq. (22), Eq. (19) can be written as
From Eqs. (13, 14, 15, 16, 17, 18, and 20), Eq. (23) becomes:
When shear is ignored (Ri ≫ 1, DDC only), Eq. (23) is reduced to
In this case, Eq. (24) becomes equivalent to Eqs. (8 and 12). Thus, negative diffusion of density is obtained.
Kantha (2012) obtained the density flux ratio as a function of Rρ such that
and obtained relations among the structure functions for DDC without shear for SF:
and for DC:
CSMC is a parameter to be determined. Here, we have used \(\gamma\) obtained by Kelley (1986, Eq. 94 for SF and Eq. 103 for DC) on the left-hand side of Eq. 26 to calculate CSMC, and then to calculate the structure functions (Eqs. 27, 28, 29, 30, 31, 32, Fig. 2). SS and ST steeply increased as Rρ approached unity, which means that mixing due to DDC was intensified. Negative Sρ for both SF and DC implies negative diffusion of density. These functions indicate that the effect of DDC is certainly important but is restricted to a narrow range of Rρ (0.8 ~ 1.2). This point should be investigated in greater detail in future modeling studies.
SMC theories continue to be developed; however, there is difficulty when it comes to observational usage. Therefore, other parameterizations, which are mentioned in Sects. 3 and 4, have been proposed.
4 K-profile parameterization with DDC
Large et al. (1994) simulated meridional ocean circulation (MOC) using K-profile parameterization (KPP) and considered three different mechanisms that contribute to eddy diffusivity, namely vertical shear instability, IW breaking, and DDC, providing a linear combination for eddy diffusivity: \(K_{\rho } = K_{\rho }^{\text{Shear}} + K_{\rho }^{\text{IW}} + K_{\rho }^{\text{DDC}}\). When active SF occurred (\(1 < R_{\rho } < 1.9\)), they used a constant value of 0.7 for \(\gamma^{\text{SF}}\), describing \(K_{S}^{\text{SF}}\) and \(K_{T}^{\text{SF}}\) as
When \(R_{\rho }\) was greater than 1.9, \(K_{S}^{\text{SF}} = 0\). In the case of active DC (\(0.5 < R_{\rho } < 1\)), \(K_{T}^{\text{DC}}\) is calculated using \(\gamma^{\text{DC}}\) as proposed by Marmorino and Caldwell (1976) and \(K_{S}^{\text{DC}}\) as proposed by Huppert (1971, Eq. 102):
If \(R_{\rho }\) was less than 0.5,
Zhang et al. (1998) also simulated the MOC using a parameterization considering DDC effects. They defined the background diffusivity as Kb= 3 × 10−5 m2/s and parameterized SF and DC eddy diffusivity. When SF occurred, they used a constant value of 0.7 for \(\gamma^{\text{SF}}\) and described \(K_{S}^{\text{SF}}\) and \(K_{T}^{\text{SF}}\) as
When DC occurred, they used the \(\gamma^{\text{DC}}\) presented by Kelley (1984), wherein the molecular heat diffusivity kT = 1.5 × 10−7 m2 s−1, and they described \(K_{S}^{\text{DC}}\) and \(K_{T}^{\text{DC}}\) as
For both treatments, \(K_{\rho }^{\text{DDC}}\) is taken as
A calculation of the eddy diffusivities in the range of 0.5 < Rρ< 2 is shown in Fig. 3.
The parameterization set by Zhang et al. (1998) has smaller values than that of Large et al. (1994). However, the absolute diffusivity values in both parameterizations increase as Rρ approaches unity. When Rρ becomes smaller than 1.7, \(K_{\rho }^{\text{SF}}\) becomes negative. The notable difference between Zhang et al. (1998) and Large et al. (1994) is the behavior around Rρ= 1. Both \(K_{\rho }^{\text{SF}}\) diverge negatively, but Large et al. (1994)’s \(K_{\rho }^{\text{SF}}\) rapidly diverges because of the relatively large differences between \(K_{S}^{\text{SF}}\) and \(K_{T}^{\text{SF}}\). As for \(K_{\rho }^{\text{DC}}\), when we take the limit of \(K_{\rho }^{\text{DC}}\) as Rρ approaches unity, \(K_{\rho }^{\text{DC}}\) diverges negatively for Zhang et al. (1998) while becoming nearly constant for Large et al. (1994).
Merryfield et al. (1999) used parameterization similar to that of Zhang et al. (1998), which changed the background diffusivity. Following studies by following Gargett (1984) and Gargett and Holloway (1984), they defined the background diffusivity as proportional to \(N^{ - 1}\), and found that relatively minor changes occurred in the global circulation (mass transport) even when DDC was present. Nevertheless, there were substantial changes in the local temperature and salt distributions: the lower layer became saltier because of the efficient salt transport resulting from SF. Inoue et al. (2007) analyzed turbulence data observed in a perturbed region off Sanriku Coast, Japan, and compared their observed diffusivity values with those of Zhang et al. (1998, Eqs. 38, 39, and 40). This comparison showed a fairly good agreement for SF, but not for DC.
5 Direct numerical simulation of DDC
Recent developments in computer power have enabled us to conduct DNS of DDC. Such studies have the advantage of directly estimating the vertical fluxes and diffusivities. Kimura et al. (2011) conducted DNS at low Rρ (< 2.0, active SF). The study showed that when SF develops, both \(K_{S}^{\text{SF}}\) and \(K_{T}^{\text{SF}}\) increase as Ri increases, which is an unexpected result. In typical cases, a shear instability (energy source) should be inactive as Ri increases, with both \(K_{S}^{\text{SF}}\) and \(K_{T}^{\text{SF}}\) increasing as Rρ decreases. This result agrees with previous theoretical, observational, and situational estimations. The result follows the functional dependency of diffusivity on Ri and Rρ:
This parameterization was verified and improved by Nakano et al. (2014), who analyzed the microstructure and CTD/LADCP results in the perturbed region off the Sanriku Coast, Japan, and western North Pacific Ocean. They also employed the buoyancy Reynolds number Reb and Ri (both at 10 m scale) as the distinguishing parameters between CT and DDC. They obtained the following new relationship between Reb and Ri:
From this relation, we can obtain critical values for Reb from Ri such that:
The value of Ri = 1 is the stability criterion of the water column, and if Ri < 0.25, the water column can become unstable and turbulent. Therefore, values of Reb= 20 and 80 corresponding to Ri, which indicate that Reb< 80 and Ri> 0.25, are suitable as criteria for the onset of DDC. Taking into account this criteria, Nakano et al. (2014) applied a DNS parameterization of diffusivity as the functions of Ri and Rρ (Kimura et al. 2011, Eqs. 43 and 44), improving their functional dependency using the following equations:
The estimated average diffusivities of salt and heat are 2.2 × 10−5 m2/s and 3.5 × 10−5 m2/s (Rρ= 1.25), and 3.5 × 10−5 m2/s and 1.1 × 10−4 m2/s (Rρ= 1.75), respectively. It was considered that the difference in coefficients between DNS (Eqs. 43, 44) and observation (Eq. 46) was caused by vertical scale difference.
Radko and Smith (2012) conducted fine-grid simulations and non-dimensional analyses of typical SF width and length scales at Rρ= 1.9. They produced vertically aligned fingers disturbed by a secondary instability. In their calculations, fluxes become almost constant after a secondary instability became comparable to the elevator mode. They obtained γ as a function of Rρ, which agrees fairly well with the laboratory prediction:
As mentioned above, although parameterizations will continue to be refined with increasing computer machine power, verification of parameterization with observational data is still required.
6 Key points of eddy diffusivity estimation with measurement data
6.1 Mixing coefficients and distinguishing DDC from CT
Most microstructure observations aimed at evaluating eddy diffusivity in the presence of DDC have been based on observations of the dissipation rate of temperature variance χT (and thus, KT estimation by Eq. 86) and mixing efficiency Γ.
To elucidate the effects of microstructures, eddy diffusivity of density for CT generated by shear \(K_{\rho }^{\text{CT}}\) is parameterized as follows:
where ΓCT is the mixing coefficient for CT, which can be regarded as the mixing efficiency. A detailed derivation of Eq. (50) is presented in Appendix A. ΓCT is the result of the observed values of \(\varepsilon\), χT, density stratification, and temperature stratification (see Eq. 86), but ΓCT has been considered to have a constant value of 0.2 or 0.25 (e.g., Osborn 1980; Oakey 1982). Thus, \(K_{\rho }^{\text{CT}}\) is calculated using ε and N. However, the results discussed in the previous studies cast doubt on the validity of using a constant value for ΓCT (= 0.2) when estimating the eddy diffusivity in the presence of DDC.
When estimating eddy diffusivity in the presence of DDC, non-dimensional parameters, such as Rρ, and Ri measured by the vertical velocity shear \(\frac{{\partial \bar{u}}}{\partial z}\), N and Reb (see Eq. 92) have been used to distinguish between CT and DDC. Also, the value of Γ for DDC (ΓDDC) is a key factor in distinguishing DDC from CT. The definition of ΓDDC is the same as ΓCT via observation (right-hand side of Eq. 86):
Historically, ΓDDC has been investigated separately from SF (ΓSF) or DC (ΓDC). St. Laurent and Schmitt (1999) surveyed the distributions of ΓSF and ΓDC with Rρ and Ri and found that ΓSF and ΓDC increased substantially because of DDC. This is one of the current key issues in microstructure studies (e.g., de Lavergne et al. 2016). This is readily understood because DDC can efficiently diffuse temperature fluctuations and create a large diffusion of temperature (Fig. 4). Inoue et al. (2007) proposed that DDC is effective in mixing when Reb< 20 in the perturbed region. Inoue et al. (2008) revisited North Atlantic Tracer Release Experiment (NATRE) data, adding the Ri criterion to restrict their attention to cases when CT was not active (Ri> 0.25 and Reb< 20). They found that ΓSF decreased when Rρ increased:
Nakano (2016) analyzed the TurboMAP and CTD/LADCP data at 10 m scale and surveyed ΓDDC for wide ranges in Ri and Reb values, showing that ΓDDC became large as Ri increased and Reb decreased (Fig. 5, also see Eq. 45). Large values of ΓDDC apparently stem from the large values of χT (Eq. 75) in DDC. Previous investigations cited above also showed low ε and high χT values in DDC layers, resulting in large values of ΓDDC. The observed values of Γ are summarized in Table 1. Taken together, it is certain that ΓDDC takes a large value. Thus, in evaluating the eddy diffusivity in the presence of DDC, the use of ΓCT (~ 0.2) should be avoided.
6.2 Practical eddy diffusivity estimation
St. Laurent and Schmitt (1999) calculated KT (as shown in Table 2 together with other estimations). They separated the observed layers as favorable to either DDC or CT in order to calculate the percentages of DDC and CT layers. In addition, they obtained weighted averages of diffusivities at almost 100 m depth intervals. Relatively high values (~ 10−4 m2/s) were obtained at 90 m depth, but values were generally lower below the thermocline. Inoue et al. (2007) presented four scenarios for estimating diffusivity and vertical buoyancy flux: (1) CT (2), DDC, (3) a simple average of CT and DDC, and (4) weighted average of CT and DDC. They concluded that scenario (4) provided the best estimation for diffusivity due to DDC and CT (1.56 × 10−5 m2/s for heat, 1.85 × 10−5 m2/s for salt). Nakano et al. (2014) also obtained a relatively small diffusivity value (10−5 m2 /s). Schmitt et al. (2005) estimated a relatively high diffusivity value for salt (> 10−4 m2/s) in the western Tropical Atlantic Ocean using Eq. (5). Ishizu et al. (2012) and Nagai et al. (2015) obtained a high diffusivity value (> 10−4 m2/s) under the Soya Current and the Kuroshio Extension, respectively.
7 Concluding remarks
In oceanic regions susceptible to DDC, parameterizations of \(K_{S}^{\text{DDC}}\) and \(K_{T}^{\text{DDC}}\) have been carried out under the assumption that velocity shear is negligible. However, CT is a common feature in the Global Ocean and can coexist with DDC. Therefore, in this note, parameterizations of DDC in oceanic mixing processes are reviewed and their applicability assessed.
The notion of representing DDC in TKE with an inactive CT variable was introduced. The applicability of DDC was investigated using an SMC model. In cases where DDC and CT coexist, the effect of DDC is certainly important but is restricted to a narrow range of Rρ (0.8–1.2). Some DDC parameterizations used in numerical simulations were reviewed in terms of physical empirical validity and applicability. An approximation can be made by combining Rρ and Ri to roughly estimate the eddy diffusivity for SF, but these parameterizations are currently being verified. A mixing coefficient is required to distinguish DDC from CT and is related to Rρ and Ri. The details of this relationship require further scientific study.
Therefore, measurements of Ri, Reb, and Rρ are essential for determining the intensity of mixing due to DDC. When measuring the eddy diffusivity in the ocean interior, it is thus necessary to deploy an ADCP/LADCP or electromagnetic current meter, along with a microstructure profiler. The accumulation of observations gained by these instruments will improve the ability to map eddy diffusivity in the Global Ocean, potentially leading to better parameterization of eddy diffusivity in numerical modeling.
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Acknowledgements
This work is part of Haruka Nakano’s PhD thesis (Nakano 2016). The manuscript was prepared under the guidance of Prof. Kantha (University of Colorado). The work is supported by MEXT KAKENHI grant number JPH05817.
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Appendices
Appendix A Parameterization of eddy diffusivity in a turbulent, non-double-diffusive system
1.1 A.1 TKE equation
To parameterize eddy diffusivity in a CT system without DDC, we use the TKE equation derived from the momentum equation (e.g., Kantha 2012), as follows:
Here, variables (u: velocity, p: pressure, T: temperature, S: salinity and ρ: density) are divided into mean and fluctuation (turbulence) components as \(u_{i} = \bar{u}_{i} + u^{\prime}_{i}\), \(p = \bar{p} + p^{\prime}\), \(T = \bar{T} + T^{\prime}\), \(S = \bar{S} + S^{\prime}\), and \(\rho = \bar{\rho } + \rho^{\prime}\). Indices (i, j) take the values 1, 2, and 3, which correspond to the x-, y-, and z-direction; g is the gravitational acceleration. Einstein’s law of summation is applied, in which a summation is made over three values repeated in the expression for the general term (Hinze 1975, p. 774). δij is the Kronecker delta. Overbars denote the ensemble averages. \(\upsilon\) is the kinematic molecular viscosity (~ 1.05 × 10−6 m2/s at 20 °C and 34 PSU). Note that \(\upsilon\) varies with temperature and salinity. The TKE K (in the blanket on the left-hand side of Eq. 53) is
where q is the turbulence velocity scale, and \(u^{\prime}\), \(v^{\prime}\), and \(w^{\prime}\) are x, y, and z components of turbulence velocity (fluctuation components in Eq. 53).
Here, Dij indicates the energy transport via the fluctuation components. \(\overline{{p^{\prime}u^{\prime}_{j} }}\) is due to the correlation between pressure and velocity fluctuation. \(\frac{1}{2}\bar{\rho }\overline{{u^{\prime}_{i} u^{\prime}_{i} {\kern 1pt} u^{\prime}_{j} }}\) is produced by the triple correlation of the velocity fluctuation. \(- \bar{\rho }\upsilon \frac{\partial }{{\partial {\kern 1pt} x_{j} }}\left( {\overline{{u^{\prime}_{i} \left( {\frac{{\partial {\kern 1pt} u^{\prime}_{i} }}{{\partial {\kern 1pt} x_{j} }} + \frac{{\partial {\kern 1pt} u^{\prime}_{j} }}{{\partial {\kern 1pt} x_{i} }}} \right)}} } \right)\) is viscous dissipation. The term \(- \frac{\partial }{{\partial {\kern 1pt} x_{j} }}\left( {D_{ij} } \right)\) represents the diffusion of energy transport; this is considered to be small and is traditionally neglected.
Considering the isotropy of turbulence in three dimensions, mean velocity (also called the background velocity) in the x-direction \(\bar{u}\), and its vertical variation, components of the second term on the right-hand side in Eq. (53) are described as
Therefore, the second term on the right-hand side of Eq. (53) is
The third term on the right-hand side of Eq. (53) is
.
The details of the fourth term on the right-hand side of Eq. (53) are described as
Assuming the isotropic turbulence (e.g., Yih 1979, Eqs. 61, 62 and 63), we obtain Eq. (64).
Therefore, we can sum Eqs. (54, 57, 58, and 64) into Eq. (65).
The z-axis is taken to be positive upward. The left term of Eq. (65) is the time variation of TKE (K). The term P is the energy production of the Reynolds stress \(\overline{{u^{\prime}w^{\prime}}}\) against background velocity shear (\(\frac{{\partial \,\bar{u}}}{\partial \,z}\)). \(\overline{{u^{\prime}w^{\prime}}}\) is the turbulence momentum transport created by the correlation (via eddy motion) between \(u^{\prime}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{and}}{\kern 1pt} {\kern 1pt} {\kern 1pt} w^{\prime}\). It is negative if \(\frac{{\partial \,\bar{u}}}{\partial \,z} > 0\), and positive if \(\frac{{\partial \,\bar{u}}}{\partial \,z} < 0\). Thus, the term P is always positive, and acts as a source of TKE. The term Jb is the energy production or dissipation by the turbulent density flux \(\overline{\rho 'w'}\), which is created by the correlation (and also by the eddy motion) between \(\rho^{\prime}\) and \(w^{\prime}\). If the density stratification is stable, \(\overline{\rho 'w'}\) is positive and the term Jb acts as a sink for TKE. If the density stratification is unstable, \(\overline{\rho 'w'}\) is negative and acts as a source for TKE. The last term ε is the TKE dissipation rate defined from the isotropic turbulence and is presented as follows (e.g., Osborn 1980):
\(\overline{{\left( {\frac{{\partial {\kern 1pt} {\kern 1pt} u^{\prime}}}{{\partial {\kern 1pt} {\kern 1pt} z}}} \right)^{2} }}\) is the variance of turbulent velocity shear. If the turbulent field is not isotropic, \(\varepsilon = \frac{15}{4}\upsilon \left[ {\overline{{\left( {\frac{{\partial {\kern 1pt} {\kern 1pt} u^{\prime}}}{{\partial {\kern 1pt} {\kern 1pt} z}}} \right)^{2} }} + \overline{{\left( {\frac{{\partial {\kern 1pt} {\kern 1pt} v^{\prime}}}{{\partial {\kern 1pt} {\kern 1pt} z}}} \right)^{2} }} } \right]\) is defined as the dissipation rate (e.g., Lozovatsky and Fernando 2012). The dissipation term acts as a sink for TKE. Taken together, the terms on the right side of Eq. (65) determine whether the total TKE increases (\(\frac{{d{\kern 1pt} {\kern 1pt} K}}{d\,t} > 0\)) or decreases (\(\frac{{d{\kern 1pt} {\kern 1pt} K}}{d\,t} < 0\)). Traditionally, to obtain turbulent diffusivity, a steady state of turbulence (\(\frac{d}{d\,t} \approx 0\), the tendency and advection terms are neglected) is assumed to exist. In steady state, the production term P is divided into Jb and ε; thus, the TKE equation can be presented as
The ratio between P and Jb is the flux Richardson number:
In stably stratified fluids (\(\frac{{\partial \,\bar{\rho }}}{\partial \,z} < 0\)), Rf indicates how much TKE (K) is consumed to mix the stably stratified fluid (Jb). The remainder of the term P is dissipated by viscosity.
1.2 A.2 Eddy diffusivity
Vertical eddy diffusivity of density Kρ is used in the calculation of vertical density flux Fρ.
where \(\frac{{\partial \,\bar{\rho }}}{\partial \,z}\) is the background density gradient. Using Eqs. (67, 68, and 69), we can obtain an expression for Kρ under a steady-state condition as follows (Osborn 1980):
ε can be measured by microstructure profilers such as the TurboMAP (e.g., Nakano et al. 2014), and N can be estimated using CTD measurements. If we know Rf, we can estimate Kρ accurately. However, it is difficult to measure Rf. Osborn (1980) proposed 0.2 as a value for ΓCT on the grounds that the critical value of \(R_{f}\) is about 0.15 in the Kelvin–Helmholtz billow.
ΓCT is traditionally called the mixing efficiency for CT (e.g., Oakey 1985), but it is actually a mixing coefficient (Gregg et al. 2018; Kantha and Luce 2018). From Eqs. (68 and 67), it is determined that Rf is the rate of conversion of turbulent energy produced (from various energy sources) to buoyancy energy needed to mix stratification layers, and ΓCT is simply the ratio of consumed buoyancy energy to energy dissipation by viscosity. Hereafter, ΓCT is called the mixing coefficient (Gregg et al. 2018; Kantha and Luce 2018).
Temperature and salinity variance equations are given by
Here, kT is the molecular diffusivity of heat (= 1.5 × 10−7 m2/s at 20 °C and 34 PSU), and kS is the molecular diffusivity of salt (= 1.5 × 10−9 m2/s at 20 °C and 34 PSU). Note that kT and kS vary with temperature and salinity. The terms DT and DS are defined as
The terms \(- \frac{{\partial {\kern 1pt} }}{{\partial {\kern 1pt} x_{j} }}\left( {D_{T} } \right)\) and \(- \frac{{\partial {\kern 1pt} }}{{\partial {\kern 1pt} x_{j} }}\left( {D_{S} } \right)\) are also the diffusion of temperature and salinity variances and are considered to be small and negligible in a steady-state condition. Under the isotropic condition, we sum Eqs. (71 and 72), and obtain
Here, \(\frac{{\partial \,\overline{T} }}{\partial \,z}\) and \(\frac{{\partial \,\overline{S} }}{\partial \,z}\) are the background temperature and salt gradients, respectively, and \(\chi_{T}\) and \(\chi_{S}\) are the dissipation rate of variances of turbulence temperature and salt gradients diffused by the molecular process, respectively. From the equation of state, we define
where a subscript 0 indicates a reference value. α and β take values as α = 2.62 × 10−4/ °C and β = 7.62 × 10−4/PSU, 20 °C, and 34PSU. Note that α and β vary with temperature and salinity. The density flux \(\overline{{\rho^{\prime}w^{\prime}}}\) can then be written as
Putting Eq. (78) into Eq. (67), we obtain
The vertical fluxes of heat and salt (FT and FS) can then be written as
where KS and KT are the eddy diffusivity of salt and heat, respectively. From Eqs. (69, 78, 80, and 81), Fρ is
For a fully developed CT, KT, KS, and Kρ must be equal to one another:
KT is derived using Eqs. (75 and 80) such that
where
is the Cox number (Osborn and Cox 1972), which represents the ratio of the variance of temperature gradient fluctuations to the square of the mean temperature gradient. The method by which one estimates KT is known as the Osborn-Cox method.
Under the assumption of equality among all eddy diffusivities (Eq. 83), ΓCT can be expressed using Eqs. (70 and 84) as follows:
The quantities on the right-hand side of Eq. (86) can be measured by a microstructure profiler. Therefore, it is possible to estimate ΓCT, with its estimated value being 0.265 (Oakey 1982, 1985). Moum (1996) obtained a value for ΓCT in the range of 0.25–0.33. Thus, Rf is found to range between 0.2 and 0.25 when using these specified values. Using these values, one-fifth to one-fourth of TKE is converted into potential energy of the system. Also, CT changes the prevailing stratification. From Eq. (86), Rf can be written as
Thus, Rf can be estimated from microstructure measurements. Eddy diffusivity of momentum \(K_{\upsilon }\) is defined as
Using Eq. (88), Rf can also be written as (St. Laurent and Schmitt 1999)
where Ri is the gradient Richardson number:
Note that when Ri < 0.25, the fluid layer can become turbulent. Equation 89 can be rewritten as
where Prt is the turbulent Prandtl number. If Rf is determined from Eq. (87) and Ri is measured from background shear and stratification, from the diffusivity of momentum, \(K_{\upsilon }\) can be estimated if Kρ is known. In any case, it is important to recognize that Rf and the resultant Γ are not constants but depend on the prevailing stratification, more specifically as a function of Ri.
Reb is defined as
From Eq. 66, Reb represents the ratio of the variance of velocity gradient fluctuation to the stabilizing stratification. It is derived from the typical length and velocity scales based on ε and N as \(L_{B} = \left( {\varepsilon /N^{3} } \right)^{1/2}\), \(U_{B} = \left( {\varepsilon /N} \right)^{1/2}\). Reb can be defined as \(R_{eb} = \frac{{U_{B} L{}_{B}}}{\upsilon } = \frac{\varepsilon }{{\upsilon N^{2} }}\) (Gregg and Sanford 1988). Inoue et al. (2007) used Reb for discriminating DDC from turbulence (Reb < 20, CT is depressed, a nd DDC prevails, from Yamazaki, 1990). See Kantha and Luce (2018) for the significance of Reb.
Appendix B Laboratory flux law
2.1 B.1 Salt finger convection (SF)
Salt finger convection (SF) can effectively transport salt and heat downward. The net downward density flux due to salt βFS is larger than the net downward density flux due to the heat αFT (FS: vertical salt flux, FT: vertical heat flux). The results show a decrease in total potential energy in the SF layer. This is in contrast to CT, in which the total potential energy increased. A threshold in the existence of SF is defined as 1 < Rρ<100 (Turner 1967; Baines and Gill 1969).
From the linear stability treatment of SF, Stern (1975) and Kunze (1987) obtained the density flux ratio γSF = αFT/βFS (< 1) for the fastest-growing SF as
Kelley (1986) compiled γSF as a function of Rρ from laboratory data on SF:
The laboratory flux ratios and the numerically and theoretically determined ratios are shown in Fig. 6; γSF asymptotes to a constant value as Rρ becomes large (~ 0.5: Eq. 93, ~ 0.35: Eq. 94) (together with Polzin et al. 1995; Shen 1993, 1995; Taylor and Buscens 1989).
Buoyancy fluxes of salt and heat for SF are summarized by Kelley (1986):
where \(\Delta S\) is the salinity difference across the SF interface. Equations (95 and 96) are called Turner’s 4/3 flux law. Kunze (1987) presented another set of flux laws for SF which depend on whether SF developed in thick or thin interfaces; for thick interfaces (> 1 m):
and for thin interfaces (< 1 m):
Another estimate of buoyancy flux comes from the “collective instability of SF” argued by Stern (1969). The author considered the interaction of SF with a large-scale IW that resulted in the tilting of SF due to vertical velocity shear. As a result, vertical fluxes change their direction, causing a divergence or convergence of fluxes, changes in density and velocity fields, and a collapse of SF. The critical condition of collapse is presented by the non-dimensional Stern number, St:
\(\frac{{\partial {\kern 1pt} \bar{T}}}{{\partial {\kern 1pt} z}}\) is the vertical salt gradient. If St becomes larger than unity, the transport of energy to large-scale IW overcomes viscous dissipation, and the SF collapses. From this equation, the vertical transport of salt is estimated as:
As determined in the laboratory, the value of St varies from 1 (Schmitt 1979) to 4 (McDougall and Taylor 1984). Based on flux estimation, \(K_{S}^{\text{SF}}\) and \(K_{T}^{\text{SF}}\) for SF can be obtained as:
2.2 B.2 Diffusive convection (DC)
For diffusive convection (DC), salt and heat are transported upward. Moreover, the net downward density flux due to αFT is larger than that due to βFS. The density flux ratio for DC is defined as γDC = βFS/αFT (< 1) (Turner, 1965). For DC, net density transport is downward and increases the density in the lower layer. The threshold of existence for DC is 0 < Rρ<1. Huppert (1971) and Kelley (1990) obtained the following relations for γDC as a function of Rρ for DC using laboratory data: Huppert (1971) introduced
and Kelley (1990) introduced
Thus, γDC becomes 1 for Rρ= 1, and becomes a constant (~ 0.15: Eq. 102, ~ 0.13: Eq. 103, see Fig. 6) as Rρ decreases. Individual fluxes of salt and heat for DC were summarized by Kelley (1986, 1990) as the following:
\(\Delta T\) is the temperature difference across the DC interface. Eddy diffusivities for salt and heat for DC (\(K_{S}^{\text{DC}}\) and \(K_{T}^{\text{DC}}\)) are formulated in the same way as Eq. (101).
Appendix C SMC model (Kantha 2012)
Kantha (2012) and Kantha et al. (2011) introduced conservation equations for the TKE, temperature, and salinity variances \(\overline{{u^{\prime}v^{\prime}}} = 0\), \(\overline{{v^{\prime}w^{\prime}}} = 0\), \(\overline{{v^{\prime}T^{\prime}}} = 0,\overline{{v^{\prime}S^{\prime}}} = 0\) as follows:
Using closure modeling developed by Galperin et al. (1988), one can obtain estimates for variances of temperature and salinity as well as covariance between temperature and salinity as follows:
where λs are closure constants \(\lambda_{1} = 0.1239\), \(\lambda_{2} = \lambda_{3} = \lambda_{4} = 0.1050\), \(\lambda_{5} = \lambda_{9} = 8.9209\), \(\lambda_{6} = \lambda_{7} = 0.5709\), \(\lambda_{8} = \lambda_{10} = 0.5801\), and \(\lambda_{11} = 0.27\). Closure constants for CT are
Those for DDC are defined as
Those for the combination of CT and DDC are defined as
Appendix D Terminology
4.1 D.1 Acronyms and abbreviations
- ADCP:
-
Acoustic Doppler current profiler
- C-SALT:
-
Caribbean sheets and layers transect
- CT:
-
Conventional turbulence
- CTD:
-
Conductivity temperature depth profiler
- DC:
-
Diffusive convection
- DDC:
-
Double-diffusive convection
- DNS:
-
Direct numerical simulations
- EM-APEX:
-
Electromagnetic autonomous profiling explorer
- HRP:
-
High-resolution profiler
- IW:
-
Internal wave
- KPP:
-
K-profile parameterization
- LADCP:
-
Lowered ADCP
- Meddy:
-
Mediterranean eddy
- MOC:
-
Meridional overturning circulation
- NATRE:
-
North Atlantic Tracer Release Experiment
- PSU:
-
Practical salinity unit
- SF:
-
Salt finger
- SMC:
-
Second-moment closure
- TKE:
-
Turbulent kinetic energy
- TurboMAP:
-
Turbulence ocean microstructure acquisition profiler
4.2 D.2 Symbols
- α :
-
Expansion coefficient due to heat [= 2.62 × 10−4/ °C, 20 °C and 34 PSU]
- α \( F_{T} \) :
-
Density flux of heat [\( {\text{m}}/\text s \)]
- \( \alpha \frac{{\partial {\kern 1pt} \bar{T}}}{{\partial {\kern 1pt} z}} \) :
-
Background density gradient due to temperature
- β :
-
Contraction coefficient due to salinity [= 7.62 × 10−4/PSU, 20 °C, 34 PSU]
- β \( F_{S} \) :
-
Density flux of salt [\( {\text{m}}/\text s \)]
- \( \beta \frac{{\partial {\kern 1pt} \bar{S}}}{{\partial {\kern 1pt} z}} \) :
-
Background vertical density gradient due to salt [\( 1 / {\text{m}} \)]
- ΓCT:
-
Mixing coefficient for CT [non-dimensional]
- ΓDDC:
-
Mixing coefficient for DDC [non-dimensional]
- ΓSF:
-
Mixing coefficient for SF [non-dimensional]
- ΓDC:
-
Mixing coefficient for DC [non-dimensional]
- \( \gamma^{\text{SF}} \) :
-
Density flux ratio of SF [non-dimensional]
- \( \gamma^{\text{DC}} \) :
-
Density flux ratio of DC [non-dimensional]
- \( \Delta S \) :
-
Salinity difference across SF interface [PSU]
- \( \Delta T \) :
-
Temperature difference across DC interface [\( {}^{ \circ }{\text{C}} \)]
- ε :
-
Kinetic energy dissipation rate [\( {\text{W/kg}} \)]
- λ1 ~ λ11:
-
Closure constants
- \( \upsilon \) :
-
Kinematic molecular viscosity [~ 1.05 × 10−6 m2/s at 20 °C and 34 PSU]
- ρ :
-
Density [\( {\text{kg/m}}^{ 3} \)]
- \( \rho^{\prime} \) :
-
Fluctuation density [\( {\text{kg/m}}^{ 3} \)]
- \( \bar{\rho } \) :
-
Mean density [\( {\text{kg/m}}^{ 3} \)]
- \( \rho_{0} \) :
-
Reference density [\( {\text{kg/m}}^{ 3} \)]
- τ :
-
Timescale of turbulence dissipation [s]
- \( \chi_{S} \) :
-
Dissipation rate of salt variance [\( {\text{PSU}}^{2} /\text s \)]
- \( \chi_{T} \) :
-
Dissipation rate of temperature variance [\( {}^{ \circ }{\text{C}}^{2} /\text s \)]
- B 1 :
-
Coefficient for turbulent timescale [non-dimensional]
- C 1 :
-
Coefficient of Turner’s 4/3 flux law [non-dimensional]
- C SMC :
-
Parameter used in second closure constants
- C x :
-
Cox number [\( \left[{{ = 3\overline{{\left( {\frac{\partial \,T'}{\partial \,z}} \right)^{2} }} } \mathord{\left/ {\vphantom {{ = 3\overline{{\left( {\frac{\partial \,T'}{\partial \,z}} \right)^{2} }} } {\left( {\frac{{\partial \,\overline{T} }}{\partial \,z}} \right)^{2} }}} \right. \kern-0pt} {\left( {\frac{{\partial \,\overline{T} }}{\partial \,z}} \right)^{2} }}\right] \), non-dimensional]
- D ij :
-
Energy transport by triple-correlation components [\( {\text{m}}^{3} / {\text{s}}^{3} \)]
- D S :
-
Diffusion of salt by triple-correlation components [\( {\text{PSU}}^{2} {\text{m}}/{\text s} \)]
- D T :
-
Diffusion of temperature by triple-correlation components [\( {}^{ \circ }{\text{C}}^{2} \, {\text{m}}/\text s \)]
- \( F_{S} \) :
-
Vertical salt flux [\( {\text{PSU}} \cdot {\text{m}}/\text s \)]
- \( F_{T} \) :
-
Vertical heat flux [\( {}^{ \circ }{\text{C}} \, {\text{m}}/\text s \)]
- F ρ :
-
Vertical density flux [\( {\text{kg}} \, {\text{m}}^{2} /\text s \)]
- \( G_{T} \) :
-
Square of the ratio of the turbulent timescale to the buoyancy timescale [non-dimensional]
- \( G_{\upsilon } \) :
-
Square of the ratio of the turbulent timescale to the shear timescale [non-dimensional]
- g :
-
Gravitational acceleration [\( {\text{m/s}}^{ 2} \)]
- i, j :
-
Indices take the values 1, 2, and 3, which correspond to the x-, y-, and z-direction
- J b :
-
Energy production or dissipation via the turbulent density flux [\( {\text{W/kg}} \)]
- K :
-
Turbulent kinetic energy (= q2/2) [\( {\text{m}}^{ 2} / {\text{s}}^{2} \)]
- K b :
-
Background eddy diffusivity [\( {\text{m}}^{ 2} / {\text{s}} \)]
- \( K_{S} \) :
-
Vertical eddy diffusivity of salt [\( {\text{m}}^{ 2} / {\text{s}} \)]
- \( K_{S}^{\text{DC}} \) :
-
Vertical eddy diffusivity of salt for DC [\( {\text{m}}^{ 2} / {\text{s}} \)]
- \( K_{S}^{\text{SF}} \) :
-
Vertical eddy diffusivity of salt for SF [\( {\text{m}}^{ 2} / {\text{s}} \)]
- \( K_{T} \) :
-
Vertical eddy diffusivity of heat [\( {\text{m}}^{ 2} / {\text{s}} \)]
- \( K_{T}^{\text{SF}} \) :
-
Vertical eddy diffusivity of heat for SF [\( {\text{m}}^{ 2} / {\text{s}} \)]
- \( K_{T}^{\text{DC}} \) :
-
Vertical eddy diffusivity of heat for DC [\( {\text{m}}^{ 2} / {\text{s}} \)]
- \( K_{\upsilon } \) :
-
Vertical eddy diffusivity of momentum [\( {\text{m}}^{ 2} / {\text{s}} \)]
- \( K_{\rho } \) :
-
Vertical eddy diffusivity of density [\( {\text{m}}^{ 2} / {\text{s}} \)]
- \( K_{\rho }^{\text{CT}} \) :
-
Vertical eddy diffusivity of density for CT [\( {\text{m}}^{ 2} / {\text{s}} \)]
- \( K_{\rho }^{\text{DC}} \) :
-
Vertical eddy diffusivity of density for DC [\( {\text{m}}^{ 2} / {\text{s}} \)]
- \( K_{\rho }^{\text{DDC}} \) :
-
Vertical eddy diffusivity of density for DDC (indicates both \( K_{\rho }^{\text{SF}} \) and \( K_{\rho }^{\text{DC}} \)) [\( {\text{m}}^{ 2} / {\text{s}} \)]
- \( K_{\rho }^{\text{IW}} \) :
-
Eddy diffusivities due to internal wave breaking [\( {\text{m}}^{ 2} / {\text{s}} \)]
- \( K_{\rho }^{\text{SF}} \) :
-
Vertical eddy diffusivity of density for SF [\( {\text{m}}^{ 2} / {\text{s}} \)]
- \( K_{\rho }^{\text{Shear}} \) :
-
Eddy diffusivities due to vertical shear instability [\( {\text{m}}^{ 2} / {\text{s}} \)]
- \( k_{S} \) :
-
Molecular diffusivity of salt (= 1.5 × 10−9 m2/s at 20 °C and 34 PSU)
- \( k_{T} \) :
-
Molecular diffusivity of temperature (= 1.5 × 10−7 m2/s at 20 °C and 34 PSU)
- L B :
-
Typical length scale of turbulence [m]
- \( \ell \) :
-
Turbulence length scale [m]
- N :
-
Buoyancy frequency [\( 1 / {\text{s}} \)]
- P :
-
Energy production of Reynolds stress against mean shear [\( {\text{W/kg}} \)]
- Prt:
-
Turbulent Prandtl number [\( = \frac{{K_{\upsilon } }}{{K_{\rho } }} \) non-dimensional]
- p :
-
Pressure [\( {\text{kg/}}\left( {{\text{m}} \,{\text{s}}^{2} } \right) \)]
- \( \bar{p} \) :
-
Mean pressure [\( {\text{kg/}}\left( {{\text{m}} \,{\text{s}}^{2} } \right) \)]
- \( p^{\prime} \) :
-
Fluctuation pressure [\( {\text{kg/}}\left( {{\text{m}}\,{\text{s}}^{2} } \right) \)]
- q :
-
Turbulence velocity scale [\( {\text{m/s}} \)]
- R eb :
-
Buoyancy Reynolds number [\( = \frac{\varepsilon }{{\upsilon N^{2} }} \), non-dimensional]
- \( R_{f} \) :
-
Flux Richardson number \( {\left[= \frac{{\left( {\frac{g}{{\bar{\rho }}}} \right)\overline{\rho 'w'} }}{{ - \overline{{u^{\prime}w^{\prime}}} \frac{{\partial {\kern 1pt} {\kern 1pt} \bar{u}}}{{\partial {\kern 1pt} {\kern 1pt} z}}}}\,, \text{non-dimensional}\,\right]} \)
- R i :
-
Gradient Richardson number \( \left[ {{{ = N^{2} } \mathord{\left/ {\vphantom {{ = N^{2} } {\left( {\frac{{\partial \bar{u}}}{{\partial z}}} \right)^{2} ,{\text{non-dimensional}}}}} \right. \kern-\nulldelimiterspace} {\left( {\frac{{\partial \bar{u}}}{{\partial z}}} \right)^{2} ,{\text{non-dimensional}}}}} \right] \)
- \( R_{\rho } \) :
-
Density ratio, the ratio of the background density gradient due to temperature to that of salt [\( = {{\alpha \,\bar{T}_{z} } \mathord{\left/ {\vphantom {{\alpha \,\bar{T}_{z} } {\beta \,\bar{S}_{z} }}} \right. \kern-0pt} {\beta \,\bar{S}_{z} }} \) non-dimensional]
- S :
-
Salinity [PSU]
- \( \bar{S} \) :
-
Mean salinity [PSU]
- \( S^{\prime} \) :
-
Salinity fluctuation [PSU]
- \( \overline{{{S^{\prime }}^{2} }} \) :
-
Variance of salt fluctuation [\( {\text{PSU}}^{2} \)]
- S t :
-
Stern number [\( = \frac{{(\beta F_{S} - \alpha F_{T} )}}{{\upsilon (\alpha {\kern 1pt} {{\partial {\kern 1pt} \bar{T}} \mathord{\left/ {\vphantom {{\partial {\kern 1pt} \bar{T}} {\partial {\kern 1pt} z}}} \right. \kern-0pt} {\partial {\kern 1pt} z}}{\kern 1pt} {\kern 1pt} - {\kern 1pt} \beta {\kern 1pt} {{\partial {\kern 1pt} \bar{S}} \mathord{\left/ {\vphantom {{\partial {\kern 1pt} \bar{S}} {\partial {\kern 1pt} z}}} \right. \kern-0pt} {\partial {\kern 1pt} z}})}} \), non-dimensional]
- \( S_{S} \) :
-
Structure function for salt diffusivities [non-dimensional]
- \( S_{T} \) :
-
Structure function for heat diffusivities [non-dimensional]
- \( S_{\upsilon } \) :
-
Structure function for the momentum diffusivities [non-dimensional]
- \( S_{\rho } \) :
-
Structure function for density diffusivities [non-dimensional]
- T :
-
Temperature [\( {}^{ \circ }{\text{C}} \)]
- \( T^{\prime} \) :
-
Temperature fluctuation [\( {}^{ \circ }{\text{C}} \)]
- \( \bar{T} \) :
-
Mean temperature [\( {}^{ \circ }{\text{C}} \)]
- \( \overline{{{T^{\prime }}^{2} }} \) :
-
Variance of temperature fluctuation [\( {}^{ \circ }{\text{C}}^{2} \)]
- \( \overline{{T^{\prime}S^{\prime}}} \) :
-
Covariance between temperature and salinity fluctuations [\( {}^{ \circ }{\text{C}} \, {\text{PSU}} \)]
- t :
-
Time [s]
- U B :
-
Typical turbulence velocity scale [\( {\text{m}}/\text s \)]
- u i :
-
Velocity [\( {\text{m/s}} \)]. i takes the vales 1, 2, and 3, which correspond to the x-, y-, and z-direction. (u1, u2, u3) = (u, v, w)
- \( \bar{u} \) :
-
Mean velocity in x-direction [\( {\text{m/s}} \)]
- \( u^{\prime} \) :
-
Turbulence velocity in x-direction [\( {\text{m/s}} \)]
- \( \overline{{u^{\prime}w^{\prime}}} \) :
-
Turbulence momentum transport [\( {\text{m}}^{2} /{\text s^{2}} \)]
- \( \bar{v} \) :
-
Mean velocity in y-direction [\( {\text{m/s}} \)]
- \( v^{\prime} \) :
-
Turbulence velocity in y-direction [\( {\text{m/s}} \)]
- \( \bar{w} \) :
-
Mean velocity in z-direction [\( {\text{m/s}} \)]
- \( w^{\prime} \) :
-
Turbulence velocity in z-direction [\( {\text{m/s}} \)]
- \( \overline{{w^{\prime}S^{\prime}}} \) :
-
Turbulence salt transport [\( {\text{PSU}} \, {\text{m}}/\text s \)]
- \( \overline{{w^{\prime}T^{\prime}}} \) :
-
Turbulence heat transport [\( {}^{ \circ }{\text{C}} \, {\text{m}}/\text s \)]
- \( x \) :
-
Horizontal coordinate positive eastward
- \( y \) :
-
Horizontal coordinate positive northward
- \( z \) :
-
Vertical coordinate positive upward
- δ ij :
-
Kronecker delta (δij = 1 when i = j, δij= 0 when \( i \ne j \))
- \( \frac{{\partial {\kern 1pt} {\kern 1pt} \bar{S}}}{{\partial {\kern 1pt} {\kern 1pt} {\kern 1pt} z}} \) :
-
Background salt gradient [PSU/m]
- \( \frac{{\partial {\kern 1pt} {\kern 1pt} \bar{T}}}{{\partial {\kern 1pt} {\kern 1pt} {\kern 1pt} z}} \) :
-
Background temperature gradient [\( {}^{ \circ }{\text{C}}/{\text{m}} \)]
- \( \frac{{\partial {\kern 1pt} {\kern 1pt} \bar{u}}}{{\partial {\kern 1pt} {\kern 1pt} {\kern 1pt} z}} \) :
-
Background velocity shear [\( 1/{\text{s}} \)]
- \( \overline{{\left( {\frac{{\partial {\kern 1pt} {\kern 1pt} u^{\prime}}}{{\partial {\kern 1pt} {\kern 1pt} z}}} \right)^{2} }} \) :
-
Variance of turbulence velocity shear [\( 1/{\text{s}}^{2} \)]
- \( \overline{{\left( {\frac{{\partial {\kern 1pt} {\kern 1pt} T^{\prime}}}{{\partial {\kern 1pt} {\kern 1pt} z}}} \right)^{2} }} \) :
-
Variance of turbulence temperature gradient [\( \left( {{}^{ \circ }{\text{C}}/{\text{m}}} \right)^{2} \)]
- \( \overline{{\left( {\frac{{\partial {\kern 1pt} {\kern 1pt} S^{\prime}}}{{\partial {\kern 1pt} {\kern 1pt} z}}} \right)^{2} }} \) :
-
Variance of turbulence salt gradient [\( \left( {{\text{PSU}}/{\text{m}}} \right)^{2} \)]
- \( \frac{{\partial \bar{\rho }}}{{\partial {\kern 1pt} {\kern 1pt} z}} \) :
-
Background vertical density gradient [\( {\text{kg/m}}^{ 4} \)]
- \( \frac{{\partial^{2} \rho }}{{\partial {\kern 1pt} {\kern 1pt} z^{2} }} \) :
-
The second derivative of density [\( {\text{kg/m}}^{ 5} \)]
- [-]:
-
Denotes ensemble average of turbulence component
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Nakano, H., Yoshida, J. A note on estimating eddy diffusivity for oceanic double-diffusive convection. J Oceanogr 75, 375–393 (2019). https://doi.org/10.1007/s10872-019-00514-9
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DOI: https://doi.org/10.1007/s10872-019-00514-9