A note on estimating eddy diffusivity for oceanic double-diffusive convection

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.

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 < R_ρ< 2, and DC is considered active when 0.5 < R_ρ< 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 K_S, heat K_T, and density K_ρ 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 J_b is as follows:

$$0 = \varepsilon + g\frac{{\overline{\rho 'w'} }}{{\bar{\rho }}} = \varepsilon + J_{b} .$$

(1)

Thus, J_b should be negative for DDC. From Eq. (82), and under the Boussinesq approximation ($\bar{\rho } = \rho_{0}$), J_b can be written as

$$J_{b} = g\frac{{\overline{{\rho^{\prime}w^{\prime}}} }}{{\rho_{0} }} = g\frac{{F_{\rho } }}{{\rho_{0} }} = g(\beta F_{S} - \alpha F_{T} ) = g\beta F_{S} \left( {1 - \frac{{\alpha F_{T} }}{{\beta F_{S} }}} \right),$$

(2)

Then, Eq. (1) can be rewritten as

$$\varepsilon = - g\beta F_{S} \left( {1 - \frac{{\alpha F_{T} }}{{\beta F_{S} }}} \right) = - g\beta F_{S} \left( {1 - \gamma^{\text{SF}} } \right),$$

(3)

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

$$N^{2} = - \frac{g}{{\rho_{0} }}\frac{{\partial {\kern 1pt} \bar{\rho }}}{{\partial {\kern 1pt} z}} = g\alpha \frac{{\partial {\kern 1pt} \bar{T}}}{{\partial {\kern 1pt} z}} - g\beta \frac{{\partial {\kern 1pt} \bar{S}}}{{\partial {\kern 1pt} z}} = - g\beta \frac{{\partial {\kern 1pt} \bar{S}}}{{\partial {\kern 1pt} z}}\left( {1 - R_{\rho } } \right).$$

(4)

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 $K_{S}^{\text{SF}}$:

$$K_{S}^{\text{SF}} = \frac{{R_{\rho } - 1}}{{1 - \gamma^{\text{SF}} }}\frac{\varepsilon }{{N^{2} }}.$$

(5)

From the definition of $R_{\rho }$, the vertical eddy diffusivity of heat for SF $K_{T}^{\text{SF}}$ is given by:

$$K_{T}^{\text{SF}} = \frac{{\gamma^{\text{SF}} }}{{R_{\rho } }}K_{S}^{\text{SF}} = \frac{{\gamma^{\text{SF}} \left( {R_{\rho } - 1} \right)}}{{R_{\rho } \left( {1 - \gamma^{\text{SF}} } \right)}}\frac{\varepsilon }{{N^{2} }}.$$

(6)

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):

$$- K_{\rho } \frac{g}{{\rho_{0} }}\frac{{\partial {\kern 1pt} \bar{\rho }}}{{\partial {\kern 1pt} z}} = g\alpha K_{T} \frac{{\partial {\kern 1pt} \bar{T}}}{{\partial {\kern 1pt} z}} - g\beta K_{S} \frac{{\partial {\kern 1pt} \bar{S}}}{{\partial {\kern 1pt} z}}$$

(7)

$$K_{\rho }^{\text{SF}} = \frac{{K_{T}^{\text{SF}} R_{\rho } - K_{S}^{\text{SF}} }}{{R_{\rho } - 1}}$$

(8)

From Eqs. (5, 6, 7, and 8), we have

$$K_{\rho }^{\text{SF}} = - \frac{\varepsilon }{{N^{2} }} < 0.$$

(9)

Eddy diffusivities for DC ($K_{S}^{\text{DC}}$, $K_{T}^{\text{DC}}$, and $K_{\rho }^{\text{DC}}$) are obtained in the same way:

$$K_{S}^{\text{DC}} = \gamma^{\text{DC}} R_{\rho } K_{T} = \frac{{\gamma^{\text{DC}} \left( {1 - R_{\rho } } \right)}}{{\left( {1 - \gamma^{\text{DC}} } \right)}}\frac{\varepsilon }{{N^{2} }}{\kern 1pt} {\kern 1pt}$$

(10)

$$K_{T}^{\text{DC}} = \frac{{\left( {1 - R_{\rho } } \right)}}{{R_{\rho } \left( {1 - \gamma^{\text{DC}} } \right)}}\frac{\varepsilon }{{N^{2} }},$$

(11)

$$K_{\rho }^{\text{DC}} = \frac{{K_{T}^{DC} R_{\rho } - K_{S}^{DC} }}{{R_{\rho } - 1}} = - \frac{\varepsilon }{{N^{2} }} < 0.$$

(12)

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.

Fig. 1

Eddy diffusivities calculated from flux laws created by Huppert (1971); Kunze (1987) and Kelley (1986, 1990), taking ε = 10⁻¹⁰ W kg⁻¹ and N = 5.2 × 10⁻³ s⁻¹ (mode values for both quantities obtained in NATRE, Gregg 1989)

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

$$\tau = B_{1} \frac{{\ell {\kern 1pt} }}{q}{\kern 1pt} = \frac{{q^{2} }}{\varepsilon } = \frac{2K}{\varepsilon },$$

(13)

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 K_S, temperature K_T, density K_ρ, and momentum $K_{\upsilon }$, defined as:

$$K_{S} = K\tau S_{S}$$

(14)

$$K_{T} = K\tau S_{T}$$

(15)

$$K_{\rho } = K\tau S_{\rho }$$

(16)

$$K_{\upsilon } = K\tau S_{\upsilon }$$

(17)

From Eq. (8) or Eq. (12), relations among $S_{S}$, $S_{T}$, and $S_{\rho }$ can be obtained:

$$S_{\rho } = \frac{{R_{\rho } S_{T} - S_{S} }}{{R_{\rho } - 1}}.$$

(18)

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 R_i, defined as Eq. (90), R_ρ and N:

$$\tau^{2} N^{2} \frac{{R_{\rho } }}{{\left( {R_{\rho } - 1} \right)}}\left[ {S_{\upsilon } \frac{{\left( {R_{\rho } - 1} \right)}}{{R_{\rho } R_{i} }} - \left( {S_{T} - \frac{{S_{S} }}{{R_{\rho } }}} \right)} \right] = 2.$$

(19)

Introduce the non-dimensional numbers, $G_{T}$ and $G_{\upsilon }$ such that

$${\kern 1pt} {\kern 1pt} G_{T} = \tau^{2} N^{2} ,$$

(20)

$$G_{\upsilon } = \tau^{2} \left( {\frac{{\partial {\kern 1pt} \bar{u}{\kern 1pt} }}{{\partial {\kern 1pt} z}}} \right)^{2} .$$

(21)

Using Eqs. (20 and 21), we have the ratio between $G_{T}$ and $G_{\upsilon }$ as follows

$$\frac{{G_{T} }}{{G_{\upsilon } }} = \frac{{N^{2} }}{{\left( {\frac{{\partial {\kern 1pt} \bar{u}}}{\partial z}} \right)^{2} }} = R_{i}$$

(22)

Using Eq. (22), Eq. (19) can be written as

$$S_{\upsilon } G_{\upsilon } - \frac{{G_{T} R_{\rho } }}{{\left( {R_{\rho } - 1} \right)}}\left( {S_{T} - \frac{{S_{S} }}{{R_{\rho } }}} \right) = 2$$

(23)

From Eqs. (13, 14, 15, 16, 17, 18, and 20), Eq. (23) becomes:

$$\frac{{K_{\upsilon } }}{{R_{i} }} - \frac{{K_{T} R_{\rho } - K_{S} }}{{R_{\rho } - 1}} = \frac{\varepsilon }{{N^{2} }}.$$

(24)

When shear is ignored (R_i ≫ 1, DDC only), Eq. (23) is reduced to

$$S_{S} G_{T} - R_{\rho } S_{T} G_{T} = 2\left( {R_{\rho } - 1} \right).$$

(25)

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 R_ρ such that

$$\gamma = \frac{{R_{\rho } \left\{ {\lambda_{9} + \left[ {\lambda_{11} \left( {\frac{1}{{R_{\rho } }} + 1} \right) - \lambda_{10} \frac{1}{{R_{\rho } }}} \right]C_{\text{SMC}} } \right\}}}{{\left\{ {\lambda_{5} + \left[ {\lambda_{8} - \lambda_{11} \left( {\frac{1}{{R_{\rho } }} + 1} \right)} \right]C_{\text{SMC}} } \right\}}},$$

(26)

and obtained relations among the structure functions for DDC without shear for SF:

$$S_{T} = \frac{{2\gamma^{\text{SF}} }}{{C_{\text{SMC}} \left( {1 - \gamma^{\text{SF}} } \right)}},$$

(27)

$$S_{S} = \frac{{R_{\rho } }}{{\gamma^{\text{SF}} }}S_{T} ,$$

(28)

$$S_{\rho } = - \frac{{R_{\rho } \left( {1 - \gamma^{\text{SF}} } \right)}}{{\gamma^{\text{SF}} \left( {R_{\rho } - 1} \right)}}S_{T} ,$$

(29)

and for DC:

$$S_{T} = - \frac{2}{{C_{\text{SMC}} \left( {1 - \gamma^{\text{DC}} } \right)}},$$

(30)

$$S_{S} = R_{\rho } \gamma^{\text{DC}} S_{T} ,$$

(31)

$$S_{\rho } = - \frac{{R_{\rho } \left( {1 - \gamma^{\text{DC}} } \right)}}{{1 - R_{\rho } }}S_{T} .$$

(32)

C_SMC 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 C_SMC, and then to calculate the structure functions (Eqs. 27, 28, 29, 30, 31, 32, Fig. 2). S_S and S_T steeply increased as R_ρ approached unity, which means that mixing due to DDC was intensified. Negative S_ρ 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 R_ρ (0.8 ~ 1.2). This point should be investigated in greater detail in future modeling studies.

Fig. 2

Dependence of structure functions for (top) heat S_T, (middle) salt S_S, and (bottom) density S_ρ on $R_{\rho }$

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

$$K_{S}^{\text{SF}} = \left[ {1 - \left( {\frac{{R_{\rho } - 1}}{0.9}} \right)^{2} } \right]^{3} \times 10^{ - 3} ,$$

(33)

$$K_{T}^{\text{SF}} = \frac{{\gamma^{\text{SF}} }}{{R_{\rho } }}K_{S}^{\text{SF}} .$$

(34)

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):

$$K_{T}^{\text{DC}} = 0.909 \times 1.5 \times 10^{ - 6} \exp \left[ {4.6\exp \left( { - 0.54\left( {R_{\rho }^{ - 1} - 1} \right)} \right)} \right],$$

(35)

$$K_{S}^{\text{DC}} = (1.85 - 0.85R_{\rho }^{ - 1} )R_{\rho } K_{T}^{\text{DC}} .$$

(36)

If $R_{\rho }$ was less than 0.5,

$$K_{S}^{\text{DC}} = 0.15R_{\rho } K_{T}^{\text{DC}} .$$

(37)

Zhang et al. (1998) also simulated the MOC using a parameterization considering DDC effects. They defined the background diffusivity as K_b= 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

$$K_{S}^{\text{SF}} = \frac{{1 \times 10^{ - 4} }}{{1 + \left( {\frac{{R_{\rho } }}{1.6}} \right)^{6} }} + K_{b} ,$$

(38)

$$K_{T}^{\text{SF}} = \frac{{\gamma^{\text{SF}} }}{{R_{\rho } }}\left( {K_{S}^{\text{SF}} - K_{b} } \right) + K_{b} .$$

(39)

When DC occurred, they used the $\gamma^{\text{DC}}$ presented by Kelley (1984), wherein the molecular heat diffusivity k_T = 1.5 × 10⁻⁷ m² s⁻¹, and they described $K_{S}^{\text{DC}}$ and $K_{T}^{\text{DC}}$ as

$$K_{S}^{\text{DC}} = R_{\rho } \gamma^{\text{DC}} (K_{T}^{\text{DC}} - K_{b} ) + K_{b} ,$$

(40)

$$K_{T}^{\text{DC}} = 0.0032\exp \left( {4.8R_{\rho }^{0.72} } \right) \cdot (0.25 \times 10^{9} R_{\rho }^{ - 1.1} )^{1/3} \cdot k_{T} + K_{b} .$$

(41)

For both treatments, $K_{\rho }^{\text{DDC}}$ is taken as

$$K_{\rho }^{\text{DDC}} = \frac{{K_{T}^{\text{DDC}} R_{\rho } - K_{S}^{\text{DDC}} }}{{R_{\rho } - 1}}.$$

(42)

A calculation of the eddy diffusivities in the range of 0.5 < R_ρ< 2 is shown in Fig. 3.

Fig. 3

Eddy diffusivities employed for numerical simulations by Large et al. (1994) and Zhang et al. (1998)

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 R_ρ approaches unity. When R_ρ 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 R_ρ= 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 R_ρ 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 R_ρ (< 2.0, active SF). The study showed that when SF develops, both $K_{S}^{\text{SF}}$ and $K_{T}^{\text{SF}}$ increase as R_i increases, which is an unexpected result. In typical cases, a shear instability (energy source) should be inactive as R_i increases, with both $K_{S}^{\text{SF}}$ and $K_{T}^{\text{SF}}$ increasing as R_ρ decreases. This result agrees with previous theoretical, observational, and situational estimations. The result follows the functional dependency of diffusivity on R_i and R_ρ:

$$K_{S}^{\text{SF}} = 4.38 \times 10^{ - 5} R_{\rho }^{ - 2.7} R_{i}^{0.17} ,$$

(43)

$$K_{T}^{\text{SF}} = 3.07 \times 10^{ - 5} R_{\rho }^{ - 4.0} R_{i}^{0.17} .$$

(44)

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 R_eb and R_i (both at 10 m scale) as the distinguishing parameters between CT and DDC. They obtained the following new relationship between R_eb and R_i:

$$R_{eb} = 19.5R_{i}^{ - 1.03} .$$

(45)

From this relation, we can obtain critical values for R_eb from R_i such that:

$$\left( {R_{eb} ,R_{i} } \right){ = }\left( { 80,0. 2 5} \right) ,\left( { 20, 1} \right).$$

The value of R_i = 1 is the stability criterion of the water column, and if R_i < 0.25, the water column can become unstable and turbulent. Therefore, values of R_eb= 20 and 80 corresponding to R_i, which indicate that R_eb< 80 and R_i> 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 R_i and R_ρ (Kimura et al. 2011, Eqs. 43 and 44), improving their functional dependency using the following equations:

$$\left. {\begin{array}{*{20}c} {K_{S}^{\text{SF}} = 9.35 \times 10^{ - 5} R_{\rho }^{ - 2.7} R_{i}^{0.17} } \\ {K_{T}^{\text{SF}} = 7.61 \times 10^{ - 5} R_{\rho }^{ - 2.7} R_{i}^{0.17} } \\ \end{array} } \right\},R_{i} > 0.25\left( {R_{eb} < 80} \right).$$

(46)

The estimated average diffusivities of salt and heat are 2.2 × 10⁻⁵ m²/s and 3.5 × 10⁻⁵ m²/s (R_ρ= 1.25), and 3.5 × 10⁻⁵ m²/s and 1.1 × 10⁻⁴ m²/s (R_ρ= 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 R_ρ= 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 R_ρ, which agrees fairly well with the laboratory prediction:

$$F_{S}^{\text{SF}} = \frac{135.7}{{\sqrt {R_{\rho } - 1} }} - 62.75,$$

(47)

$$\gamma^{\text{SF}} = 2.709\exp \left( { - 2.513R_{\rho } } \right) + 0.5128,$$

(48)

$$F_{T}^{\text{SF}} = \gamma^{\text{SF}} F_{S}^{\text{SF}} .$$

(49)

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 χ_T (and thus, K_T 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:

$$K_{\rho }^{\text{CT}} = \varGamma^{\text{CT}} \frac{\varepsilon }{{N^{2} }},$$

(50)

where Γ^CT 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. Γ^CT is the result of the observed values of $\varepsilon$, χ_T, density stratification, and temperature stratification (see Eq. 86), but Γ^CT 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 Γ^CT (= 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 R_ρ, and R_i measured by the vertical velocity shear $\frac{{\partial \bar{u}}}{\partial z}$, N and R_eb (see Eq. 92) have been used to distinguish between CT and DDC. Also, the value of Γ for DDC (Γ^DDC) is a key factor in distinguishing DDC from CT. The definition of Γ^DDC is the same as Γ^CT via observation (right-hand side of Eq. 86):

$$\varGamma^{\text{DDC}} = \frac{{\chi_{T} N^{2} }}{{2\varepsilon \left( {\frac{{\partial \,\overline{T} }}{\partial \,z}} \right)^{2} }}.$$

(51)

Historically, Γ^DDC has been investigated separately from SF (Γ^SF) or DC (Γ^DC). St. Laurent and Schmitt (1999) surveyed the distributions of Γ^SF and Γ^DC with R_ρ and R_i and found that Γ^SF and Γ^DC 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 R_eb< 20 in the perturbed region. Inoue et al. (2008) revisited North Atlantic Tracer Release Experiment (NATRE) data, adding the R_i criterion to restrict their attention to cases when CT was not active (R_i> 0.25 and R_eb< 20). They found that Γ^SF decreased when R_ρ increased:

$$\begin{aligned} \left( {\varGamma^{\text{SF}} ,R_{\rho } } \right) & = \left( { 1.0, 1. 3} \right),\left( {0. 6, 1. 9} \right), \\ \left( {\varGamma^{\text{SF}} ,R_{i} } \right) & = \left( {0. 6,0. 4} \right),\left( {0. 9, 1.0} \right),\left( { 1. 3, 10} \right). \\ \end{aligned}$$

Fig. 4

a Occurrence of diffusive convection. (a) Initially warm/salty layer is above, and the cold/fresh layer is below. The separating interface is initially at rest. (b) The interface becomes unstable because of differences in molecular diffusivity of heat and salt ($k_{T} \approx 100k_{S}$). The upward portion of the lower layer (w′ > 0, vertical blue arrow) has both negative temperature and salt anomalies due to a surrounding warm and salty layer (T′ < 0 and S′ < 0). The downward portion (w′ < 0, vertical red arrow) has positive temperature and salt anomalies due to a surrounding cold fresh layer (T′ > 0 and S′ > 0). Lateral molecular diffusion of heat is greater than that of salt. Therefore, the upward portion is warmed and the downward portion is cooled. (c) Consequently, the upward portion attains a positive buoyancy force and keeps ascending upward (vertical blue arrow), whereas the downward portion attains a negative buoyancy force, causing its descent downward (vertical red arrow). The motions are aligned horizontally to form a salt finger cell. b Occurrence of diffusive convection. (a) Initially, cold/fresh layer is above, and the warm/salty layer is below. The separating interface is initially at rest. (b) The interface becomes unstable to be wavy because of differences in molecular diffusivity of heat and salt ($k_{T} \approx 100\;k_{S}$). The upward portion from the lower layer (w′ > 0, vertical red arrow) has positive temperature and salt anomalies from a surrounding cold and fresh layer (T′ > 0 and S′ > 0). The negative portion from the lower layer (w′ < 0, vertical blue arrow) has negative temperature and salt anomalies from a surrounding warm and salty layer (T′ > 0 and S′ > 0). Lateral molecular diffusion of heat is greater than that of salt. Therefore, the upward portion is cooled and the downward portion is warmed. (c) Consequently, the upward portion receives negative buoyancy force and descends downward (vertical blue arrow), whereas the downward portion obtains positive buoyancy force and ascends upward (vertical red arrow). These upward and downward portions lose or gain heat repeatedly from the surrounding water. These upward and downward motions repeat to produce mixed layers separated by the interface to form a clear diffusive interface

Nakano (2016) analyzed the TurboMAP and CTD/LADCP data at 10 m scale and surveyed Γ^DDC for wide ranges in R_i and R_eb values, showing that Γ^DDC became large as R_i increased and R_eb decreased (Fig. 5, also see Eq. 45). Large values of Γ^DDC apparently stem from the large values of χ_T (Eq. 75) in DDC. Previous investigations cited above also showed low ε and high χ_T values in DDC layers, resulting in large values of Γ^DDC. The observed values of Γ are summarized in Table 1. Taken together, it is certain that Γ^DDC takes a large value. Thus, in evaluating the eddy diffusivity in the presence of DDC, the use of Γ^CT (~ 0.2) should be avoided.

Fig. 5

Γ plots on Log($R_{eb}$) − Log(R_i) plane. Data were obtained from the western North Pacific Ocean (Nakano et al. 2014)

Table 1 Examples of direct estimation of Γ in the presence of DDC

6.2 Practical eddy diffusivity estimation

St. Laurent and Schmitt (1999) calculated K_T (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.

Table 2 Examples of direct estimation of eddy diffusivity in the presence of DDC

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 R_ρ (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 R_ρ and R_i 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 R_ρ and R_i. The details of this relationship require further scientific study.

Therefore, measurements of R_i, R_eb, and R_ρ 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.

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:

$$\begin{aligned} \bar{\rho }\frac{{\partial {\kern 1pt} }}{{\partial {\kern 1pt} t}}\left( {\frac{1}{2}\overline{{u^{\prime}_{i} u^{\prime}_{i} }} } \right) + \bar{\rho }\bar{u}_{j} \frac{{\partial {\kern 1pt} {\kern 1pt} }}{{\partial {\kern 1pt} x_{j} }}\left( {\frac{1}{2}\overline{{u^{\prime}_{i} u^{\prime}_{i} }} } \right) & = - \frac{\partial }{{\partial {\kern 1pt} x_{j} }}\left( {\overline{{p^{\prime}u^{\prime}_{j} }} + \frac{1}{2}\bar{\rho }\overline{{u^{\prime}_{i} u^{\prime}_{i} {\kern 1pt} u^{\prime}_{j} }} - \bar{\rho }\upsilon \frac{\partial }{{\partial {\kern 1pt} x_{j} }}\left( {\overline{{u^{\prime}_{i} \left( {\frac{{\partial {\kern 1pt} u^{\prime}_{i} }}{{\partial {\kern 1pt} x_{j} }} + \frac{{\partial {\kern 1pt} u^{\prime}_{j} }}{{\partial {\kern 1pt} x_{i} }}} \right)}} } \right)} \right) \\ & \quad - \frac{1}{2}\bar{\rho }\overline{{u^{\prime}_{i} u^{\prime}_{j} }} \left( {{\kern 1pt} \frac{{\partial {\kern 1pt} \bar{u}_{i} {\kern 1pt} }}{{\partial {\kern 1pt} x_{j} }} + \frac{{\partial {\kern 1pt} \bar{u}_{j} {\kern 1pt} }}{{\partial {\kern 1pt} x_{i} }}} \right) - \overline{{u^{\prime}_{i} \rho^{\prime}g\delta_{i3} }} - \frac{{\bar{\rho }\upsilon }}{2}\overline{{\left( {\frac{{\partial {\kern 1pt} u^{\prime}_{i} }}{{\partial {\kern 1pt} x_{j} }} + \frac{{\partial {\kern 1pt} u^{\prime}_{j} }}{{\partial {\kern 1pt} x_{i} }}} \right)^{2} }} , \\ \end{aligned}$$

(52)

$$\frac{d}{{d{\kern 1pt} {\kern 1pt} t}}\left( {\frac{1}{2}\overline{{u^{\prime}_{i} u^{\prime}_{i} }} } \right) = - \frac{\partial }{{\partial {\kern 1pt} x_{j} }}\left( {D_{ij} } \right) - \frac{1}{2}\overline{{u^{\prime}_{i} u^{\prime}_{j} }} \left( {{\kern 1pt} \frac{{\partial {\kern 1pt} \bar{u}_{i} {\kern 1pt} }}{{\partial {\kern 1pt} x_{j} }} + \frac{{\partial {\kern 1pt} \bar{u}_{j} {\kern 1pt} }}{{\partial {\kern 1pt} x_{i} }}} \right) - \frac{{\overline{{u^{\prime}_{i} \rho^{\prime}g\delta_{i3} }} }}{{\bar{\rho }}} - \frac{1}{2}\upsilon \overline{{\left( {\frac{{\partial {\kern 1pt} u^{\prime}_{i} }}{{\partial {\kern 1pt} x_{j} }} + \frac{{\partial {\kern 1pt} u^{\prime}_{j} }}{{\partial {\kern 1pt} x_{i} }}} \right)^{2} }}$$

(53)

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). δ_ij 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

$$K = \frac{1}{2}q^{2} = \frac{1}{2}\left( {\overline{{{u^{\prime }}^{2} }} + \overline{{{v^{\prime }}^{2} }} + \overline{{{w^{\prime }}^{2} }} } \right),$$

(54)

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, D_ij 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

(55)

(56)

Therefore, the second term on the right-hand side of Eq. (53) is

$$- \frac{1}{2}\overline{{u^{\prime}_{i} u^{\prime}_{j} }} \left( {{\kern 1pt} \frac{{\partial {\kern 1pt} \bar{u}_{i} {\kern 1pt} }}{{\partial {\kern 1pt} x_{j} }} + \frac{{\partial {\kern 1pt} \bar{u}_{j} {\kern 1pt} }}{{\partial {\kern 1pt} x_{i} }}} \right) = - \overline{{u^{\prime}w^{\prime}}} \frac{{\partial {\kern 1pt} \bar{u}{\kern 1pt} }}{{\partial {\kern 1pt} z}} = P.$$

(57)

The third term on the right-hand side of Eq. (53) is

$$- \frac{{\overline{{u^{\prime}_{i} \rho^{\prime}g\delta_{i3} }} }}{{\bar{\rho }}} = - \frac{{\overline{{\rho^{\prime}w^{\prime}}} }}{{\bar{\rho }}}{\kern 1pt} g{\kern 1pt} {\kern 1pt} {\kern 1pt} = - J_{b} \quad {\kern 1pt} {\kern 1pt} (i = 3).$$

(58)

.

The details of the fourth term on the right-hand side of Eq. (53) are described as

$$\begin{aligned} i \ne j: - \frac{1}{2}\upsilon \left[ {\overline{{\left( {\frac{{\partial {\kern 1pt} u^{\prime}}}{{\partial {\kern 1pt} y}} + \frac{{\partial {\kern 1pt} v^{\prime}}}{{\partial {\kern 1pt} x}}} \right)^{2} + \left( {\frac{{\partial {\kern 1pt} v^{\prime}}}{{\partial {\kern 1pt} z}} + \frac{{\partial {\kern 1pt} w^{\prime}}}{{\partial {\kern 1pt} y}}} \right)^{2} + \left( {\frac{{\partial {\kern 1pt} w^{\prime}}}{{\partial {\kern 1pt} x}} + \frac{{\partial {\kern 1pt} u^{\prime}}}{{\partial {\kern 1pt} z}}} \right)^{2} }} } \right] \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = - \frac{1}{2}\upsilon \left[ {\overline{{\left( {\frac{{\partial {\kern 1pt} u^{\prime}}}{{\partial {\kern 1pt} y}}} \right)^{2} }} + \overline{{\left( {\frac{{\partial {\kern 1pt} v^{\prime}}}{{\partial {\kern 1pt} x}}} \right)^{2} }} + \overline{{\left( {\frac{{\partial {\kern 1pt} v^{\prime}}}{{\partial {\kern 1pt} z}}} \right)^{2} }} + \overline{{\left( {\frac{{\partial {\kern 1pt} w^{\prime}}}{{\partial {\kern 1pt} y}}} \right)^{2} }} + \overline{{\left( {\frac{{\partial {\kern 1pt} w^{\prime}}}{{\partial {\kern 1pt} z}}} \right)^{2} }} + \overline{{\left( {\frac{{\partial {\kern 1pt} u^{\prime}}}{{\partial {\kern 1pt} z}}} \right)^{2} }} } \right. \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + 2\left. {\overline{{\left( {\frac{{\partial {\kern 1pt} u^{\prime}}}{{\partial {\kern 1pt} y}} \cdot \frac{{\partial {\kern 1pt} v^{\prime}}}{{\partial {\kern 1pt} x}}} \right)}} + 2\overline{{\left( {\frac{{\partial {\kern 1pt} v^{\prime}}}{{\partial {\kern 1pt} z}} \cdot \frac{{\partial {\kern 1pt} w^{\prime}}}{{\partial {\kern 1pt} y}}} \right)}} + 2\overline{{\left( {\frac{{\partial {\kern 1pt} w^{\prime}}}{{\partial {\kern 1pt} x}} \cdot \frac{{\partial {\kern 1pt} u^{\prime}}}{{\partial {\kern 1pt} z}}} \right)}} } \right] \hfill \\ \end{aligned}$$

(59)

$$i = j: - \frac{1}{2}\upsilon \left[ {4\left\{ {\overline{{\left( {\frac{{\partial {\kern 1pt} u^{\prime}}}{{\partial {\kern 1pt} x}}} \right)^{2} + \left( {\frac{{\partial {\kern 1pt} v^{\prime}}}{{\partial {\kern 1pt} y}}} \right)^{2} + \left( {\frac{{\partial {\kern 1pt} w^{\prime}}}{{\partial {\kern 1pt} z}}} \right)^{2} }} } \right\}} \right]$$

(60)

Assuming the isotropic turbulence (e.g., Yih 1979, Eqs. 61, 62 and 63), we obtain Eq. (64).

$$\overline{{\left( {\frac{{\partial {\kern 1pt} u^{\prime}}}{{\partial {\kern 1pt} x}}} \right)^{2} }} = \overline{{\left( {\frac{{\partial {\kern 1pt} v^{\prime}}}{{\partial {\kern 1pt} y}}} \right)^{2} }} = \overline{{\left( {\frac{{\partial {\kern 1pt} w^{\prime}}}{{\partial {\kern 1pt} z}}} \right)^{2} }} = \frac{1}{2}\overline{{\left( {\frac{{\partial {\kern 1pt} u^{\prime}}}{{\partial {\kern 1pt} z}}} \right)^{2} }}$$

(61)

$$\overline{{\left( {\frac{{\partial {\kern 1pt} u^{\prime}}}{{\partial {\kern 1pt} y}}} \right)^{2} }} = \overline{{\left( {\frac{{\partial {\kern 1pt} u^{\prime}}}{{\partial {\kern 1pt} z}}} \right)^{2} }} = \overline{{\left( {\frac{{\partial {\kern 1pt} v^{\prime}}}{{\partial {\kern 1pt} x}}} \right)^{2} }} = \overline{{\left( {\frac{{\partial {\kern 1pt} v^{\prime}}}{{\partial {\kern 1pt} z}}} \right)^{2} }} = \overline{{\left( {\frac{{\partial {\kern 1pt} w^{\prime}}}{{\partial {\kern 1pt} x}}} \right)^{2} }} = \overline{{\left( {\frac{{\partial {\kern 1pt} w^{\prime}}}{{\partial {\kern 1pt} y}}} \right)^{2} }}$$

(62)

$$\overline{{\left( {\frac{{\partial {\kern 1pt} u^{\prime}}}{{\partial {\kern 1pt} y}} \cdot \frac{{\partial {\kern 1pt} v^{\prime}}}{{\partial {\kern 1pt} x}}} \right)}} = \overline{{\left( {\frac{{\partial {\kern 1pt} v^{\prime}}}{{\partial {\kern 1pt} z}} \cdot \frac{{\partial {\kern 1pt} w^{\prime}}}{{\partial {\kern 1pt} y}}} \right)}} = \overline{{\left( {\frac{{\partial {\kern 1pt} w^{\prime}}}{{\partial {\kern 1pt} x}} \cdot \frac{{\partial {\kern 1pt} u^{\prime}}}{{\partial {\kern 1pt} z}}} \right)}} = \frac{1}{2}\overline{{\left( {\frac{{\partial {\kern 1pt} u^{\prime}}}{{\partial {\kern 1pt} z}}} \right)^{2} }}$$

(63)

$$- \frac{1}{2}\upsilon \overline{{\left( {\frac{{\partial {\kern 1pt} u^{\prime}_{i} }}{{\partial {\kern 1pt} x_{j} }} + \frac{{\partial {\kern 1pt} u^{\prime}_{j} }}{{\partial {\kern 1pt} x_{i} }}} \right)^{2} }} = - \frac{15}{2}\upsilon \overline{{\left( {\frac{{\partial {\kern 1pt} u^{\prime}}}{{\partial {\kern 1pt} z}}} \right)^{2} }} = - \varepsilon$$

(64)

Therefore, we can sum Eqs. (54, 57, 58, and 64) into Eq. (65).

$$\frac{{d{\kern 1pt} K}}{{d{\kern 1pt} {\kern 1pt} t}} = - \overline{u'w'} \frac{{\partial {\kern 1pt} {\kern 1pt} \bar{u}}}{\partial \,z} - g\frac{{\overline{\rho 'w'} }}{{\bar{\rho }}} - \varepsilon = P - J_{b} - \varepsilon ,$$

(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 J_b 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 J_b 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):

$$\varepsilon = \frac{15}{2}\upsilon \overline{{\left( {\frac{{\partial \bar{u}}}{\partial z}} \right)^{2} }}$$

(66)

$\overline{{\left( {\frac{{\partial {\kern 1pt} {\kern 1pt} u^{\prime}}}{{\partial {\kern 1pt} {\kern 1pt} z}}} \right)^{2} }}$ 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 J_b and ε; thus, the TKE equation can be presented as

$$0 = - \overline{u'w'} \frac{{\partial \,\bar{u}}}{\partial \,z} - g\frac{{\overline{\rho 'w'} }}{{\bar{\rho }}} - \varepsilon = P - J_{b} - \varepsilon .$$

(67)

The ratio between P and J_b is the flux Richardson number:

$$R_{f} = \frac{{\left( {\frac{g}{{\bar{\rho }}}} \right)\overline{{\rho^{ \prime}} u^{\prime }} }}{{ - \overline{u^{\prime} w^{\prime} } \frac{{\partial \bar{u}}}{\partial z}}} = \frac{{J_{b} }}{P}.$$

(68)

In stably stratified fluids ($\frac{{\partial \,\bar{\rho }}}{\partial \,z} < 0$), R_f indicates how much TKE (K) is consumed to mix the stably stratified fluid (J_b). The remainder of the term P is dissipated by viscosity.

1.2 A.2 Eddy diffusivity

Vertical eddy diffusivity of density K_ρ is used in the calculation of vertical density flux F_ρ.

$$F_{\rho } = \overline{{\rho^{ \prime} w^{\prime} = - {\rm K}_{\rho } \frac{{\partial \bar{\rho }}}{\partial z},}}$$

(69)

where $\frac{{\partial \,\bar{\rho }}}{\partial \,z}$ is the background density gradient. Using Eqs. (67, 68, and 69), we can obtain an expression for K_ρ under a steady-state condition as follows (Osborn 1980):

$$K_{\rho } = \varGamma^{\text{CT}} \frac{\varepsilon }{{N^{2} }},\;{\text{where}}\;\varGamma^{\text{CT}} = \frac{{R_{f} }}{{1 - R_{f} }} = \frac{{J_{b} }}{{P - J_{b} }} = \frac{{J_{b} }}{\varepsilon },\;N = \sqrt { - \frac{g}{{\rho_{0} }}\left( {\frac{{\partial \bar{\rho }}}{\partial z}} \right)} .$$

(70)

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 R_f, we can estimate K_ρ accurately. However, it is difficult to measure R_f. Osborn (1980) proposed 0.2 as a value for Γ^CT on the grounds that the critical value of $R_{f}$ is about 0.15 in the Kelvin–Helmholtz billow.

Γ^CT 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 R_f is the rate of conversion of turbulent energy produced (from various energy sources) to buoyancy energy needed to mix stratification layers, and Γ^CT is simply the ratio of consumed buoyancy energy to energy dissipation by viscosity. Hereafter, Γ^CT is called the mixing coefficient (Gregg et al. 2018; Kantha and Luce 2018).

Temperature and salinity variance equations are given by

$$\begin{aligned} & \frac{{\text{d}}}{{{\text{d}}t}}\left( {\overline{{T^{{\prime 2}} }} } \right) = - \frac{\partial }{{\partial x_{j} }}\left[ {\overline{{\left( {u_{j}^{\prime } T_{j}^{{\prime 2}} } \right)}} - \left( {k_{T} \overline{{\frac{{\partial T^{{\prime 2}} }}{{\partial x_{j} }}}} } \right)} \right] - 2\overline{{u_{j}^{\prime } T^{\prime } }} \frac{{\partial \bar{T}}}{{\partial x_{j} }} - 2k_{T} \overline{{\frac{{\partial T^{\prime } }}{{\partial x_{j} }}\frac{{\partial T^{\prime } }}{{\partial x_{j} }}}} , \\ & \frac{{\text{d}}}{{{\text{d}}t}}\left( {\overline{{T^{{\prime 2}} }} } \right) = - \frac{\partial }{{\partial x_{j} }}(D_{T} ) - 2\overline{{u_{j}^{\prime } T^{\prime } }} \frac{{\partial \bar{T}}}{{\partial x_{j} }} - 2k_{T} \overline{{\frac{{\partial T^{\prime } }}{{\partial x_{j} }}\frac{{\partial T^{\prime } }}{{\partial x_{j} }}}} . \\ \end{aligned}$$

(71)

$$\begin{aligned} & \frac{{\text{d}}}{{{\text{d}}t}}\left( {\overline{{S^{{\prime 2}} }} } \right) = - \frac{\partial }{{\partial x_{j} }}\left[ {\overline{{\left( {u_{j}^{\prime } S_{j}^{{\prime 2}} } \right)}} - \left( {k_{S} \overline{{\frac{{\partial S^{{\prime 2}} }}{{\partial x_{j} }}}} } \right)} \right] - 2\overline{{u_{j}^{\prime } S^{\prime } }} \frac{{\partial \bar{S}}}{{\partial x_{j} }} - 2k_{S} \overline{{\frac{{\partial S^{\prime } }}{{\partial x_{j} }}\frac{{\partial S^{\prime } }}{{\partial x_{j} }}}} , \\ & \frac{{\text{d}}}{{{\text{d}}t}}\left( {\overline{{S^{{\prime 2}} }} } \right) = - \frac{\partial }{{\partial x_{j} }}(D_{S} ) - 2\overline{{u_{j}^{\prime } S^{\prime } }} \frac{{\partial \bar{S}}}{{\partial x_{j} }} - 2k_{S} \overline{{\frac{{\partial S^{\prime } }}{{\partial x_{j} }}\frac{{\partial S^{\prime } }}{{\partial x_{j} }}}} . \\ \end{aligned}$$

(72)

Here, k_T is the molecular diffusivity of heat (= 1.5 × 10⁻⁷ m²/s at 20 °C and 34 PSU), and k_S is the molecular diffusivity of salt (= 1.5 × 10⁻⁹ m²/s at 20 °C and 34 PSU). Note that k_T and k_S vary with temperature and salinity. The terms D_T and D_S are defined as

$$D_{T} = \begin{array}{*{20}c} {\overline{{\left( {u_{j} ^{\prime} T_{j}^{\prime 2} } \right)}} } & { - k_{T} \overline{{\frac{{\partial T^{\prime 2} }}{{\partial x_{j} }}}} } \\ {{\text{Transport of temperature}}\;{\text{variance}}} & {{\text{Molecular diffusion}}\;{\text{of temperature variance}}} \\ \end{array} .$$

(73)

$$D_{S} = \begin{array}{*{20}c} {\overline{{\left( {u_{j}^{{\prime }} S^{{\prime _{j}^{2} }} } \right)}} } & { - k_{S} \overline{{\frac{{\partial S^{{{\prime }2}} }}{{\partial x_{j} }}}} } \\ {{\text{Transport of salinity}}\;{\text{variance}}} & {{\text{Molecular diffusion}}\;{\text{of salinity variance}}} \\ \end{array} .$$

(74)

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

$$0 = - \overline{w'T'} \frac{{\partial \,\overline{T} }}{\partial \,z} - \frac{1}{2}\chi_{T} \therefore \chi_{T} = 6k_{T} \overline{{\left( {\frac{\partial \,T'}{\partial \,z}} \right)^{2} }} \left[ {{}^{ \circ }{\text{C}}^{2} /\text s } \right],$$

(75)

$$0 = - \overline{{w^{\prime}S^{\prime}}} \frac{{\partial {\kern 1pt} \overline{S} }}{{\partial {\kern 1pt} z}} - \frac{1}{2}\chi_{S} \therefore \chi_{S} = 6k_{S} \overline{{\left( {\frac{\partial \,S'}{\partial \,z}} \right)^{2} }} \left[ {{\text{PSU}}^{2} /\text s } \right].$$

(76)

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

$$\rho = \rho_{0} \left\{ {1 - \alpha \left( {\bar{T} - T_{0} + T^{\prime}} \right) + \beta \left( {\bar{S} - S_{0} + S^{\prime}} \right)} \right\},$$

(77)

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

$$\rho^{\prime} = \rho - \bar{\rho } = - \rho_{0} \alpha T^{\prime} + \rho_{0} \beta S^{\prime}\therefore \overline{{\rho^{\prime}w^{\prime}}} = \rho_{0} \left( { - \alpha \overline{{w^{\prime}T^{\prime}}} + \beta \overline{{w^{\prime}S^{\prime}}} } \right).$$

(78)

Putting Eq. (78) into Eq. (67), we obtain

$$0 = - \overline{u'w'} \frac{{\partial \,\bar{u}}}{\partial \,z} + g\left( {\alpha \overline{{w^{\prime}T^{\prime}}} - \beta \overline{{w^{\prime}S^{\prime}}} } \right) - \varepsilon .$$

(79)

The vertical fluxes of heat and salt (F_T and F_S) can then be written as

$$F_{T} = \overline{{w^{\prime}T^{\prime}}} = - K_{T} \frac{{\partial \,\bar{T}}}{\partial \,z},$$

(80)

$$F_{S} = \overline{{w^{\prime}S^{\prime}}} = - K_{S} \frac{{\partial \,\bar{S}}}{\partial \,z},$$

(81)

where K_S and K_T are the eddy diffusivity of salt and heat, respectively. From Eqs. (69, 78, 80, and 81), F_ρ is

$$F_{\rho } = - \rho_{0} \alpha F_{T} + \rho_{0} \beta F_{S} .$$

(82)

For a fully developed CT, K_T, K_S, and K_ρ must be equal to one another:

$$K_{T} = K_{S} = K_{\rho } .$$

(83)

K_T is derived using Eqs. (75 and 80) such that

$$K_{T} = \frac{{\chi_{T} }}{{2\left( {\frac{{\partial \,\overline{T} }}{\partial \,z}} \right)^{2} }} = \frac{{3k_{T} \overline{{\left( {\frac{\partial \,T'}{\partial \,z}} \right)^{2} }} }}{{\left( {\frac{{\partial \,\overline{T} }}{\partial \,z}} \right)^{2} }}\therefore K_{T} = k_{T} C_{x} = K_{\rho } ,$$

(84)

where

$${{C_{x} = 3\overline{{\left( {\frac{\partial \,T'}{\partial \,z}} \right)^{2} }} } \mathord{\left/ {\vphantom {{C_{x} = 3\overline{{\left( {\frac{\partial \,T'}{\partial \,z}} \right)^{2} }} } {\left( {\frac{{\partial \,\overline{T} }}{\partial \,z}} \right)^{2} }}} \right. \,\kern-0pt} {\left( {\frac{{\partial \,\overline{T} }}{\partial \,z}} \right)^{2} }}$$

(85)

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 K_T is known as the Osborn-Cox method.

Under the assumption of equality among all eddy diffusivities (Eq. 83), Γ^CT can be expressed using Eqs. (70 and 84) as follows:

$$\varGamma^{\text{CT}} = \frac{{R_{f} }}{{1 - R_{f} }} = \frac{{\chi_{T} N^{2} }}{{2\varepsilon \left( {\frac{{\partial \,\overline{T} }}{\partial \,z}} \right)^{2} }}.$$

(86)

The quantities on the right-hand side of Eq. (86) can be measured by a microstructure profiler. Therefore, it is possible to estimate Γ^CT, with its estimated value being 0.265 (Oakey 1982, 1985). Moum (1996) obtained a value for Γ^CT in the range of 0.25–0.33. Thus, R_f 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), R_f can be written as

$$R_{f} = \frac{{\chi_{T} N^{2} }}{{2\varepsilon \left( {\frac{{\partial \,\overline{T} }}{\partial \,z}} \right)^{2} + \chi_{T} N^{2} }}.$$

(87)

Thus, R_f can be estimated from microstructure measurements. Eddy diffusivity of momentum $K_{\upsilon }$ is defined as

$$- \overline{{u^{\prime}w^{\prime}}} = K_{\upsilon } \frac{{\partial \,\bar{u}}}{\partial \,z}.$$

(88)

Using Eq. (88), R_f can also be written as (St. Laurent and Schmitt 1999)

$$R_{f} = \frac{{g\overline{\rho 'w'} }}{{ - \rho_{0} \overline{u'w'} \frac{{\partial \,\bar{u}}}{\partial \,z}}} = \frac{{ - K_{\rho } \frac{g}{{\rho_{0} }}\frac{{\partial {\kern 1pt} \bar{\rho }}}{{\partial {\kern 1pt} z}}}}{{K_{\upsilon } \left( {\frac{{\partial \,\bar{u}}}{\partial \,z}} \right)^{2} }} = \frac{{K_{\rho } }}{{K_{\upsilon } }}\frac{{N^{2} }}{{\left( {\frac{{\partial \,\bar{u}}}{\partial \,z}} \right)^{2} }} = \frac{{K_{\rho } }}{{K_{\upsilon } }}R_{i} ,$$

(89)

where R_i is the gradient Richardson number:

$$R_{i} = {{N^{2} } \mathord{\left/ {\vphantom {{N^{2} } {\left( {\frac{{\partial \,\bar{u}}}{\partial \,z}} \right)^{2} }}} \right. \kern-0pt} {\left( {\frac{{\partial \,\bar{u}}}{\partial \,z}} \right)^{2} }}.$$

(90)

Note that when R_i < 0.25, the fluid layer can become turbulent. Equation 89 can be rewritten as

$$\frac{{R_{i} }}{{R_{f} }} = \frac{{K_{\upsilon } }}{{K_{\rho } }} = \Pr_{t} ,$$

(91)

where Pr_t is the turbulent Prandtl number. If R_f is determined from Eq. (87) and R_i is measured from background shear and stratification, from the diffusivity of momentum, $K_{\upsilon }$ can be estimated if K_ρ is known. In any case, it is important to recognize that R_f and the resultant Γ are not constants but depend on the prevailing stratification, more specifically as a function of R_i.

R_eb is defined as

$$R_{eb} = \frac{\varepsilon }{{\upsilon N^{2} }}.$$

(92)

From Eq. 66, R_eb 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 $L_{B} = \left( {\varepsilon /N^{3} } \right)^{1/2}$, $U_{B} = \left( {\varepsilon /N} \right)^{1/2}$. R_eb 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 R_eb for discriminating DDC from turbulence (R_eb < 20, CT is depressed, a nd DDC prevails, from Yamazaki, 1990). See Kantha and Luce (2018) for the significance of R_eb.

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 βF_S is larger than the net downward density flux due to the heat αF_T (F_S: vertical salt flux, F_T: 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 < R_ρ<100 (Turner 1967; Baines and Gill 1969).

From the linear stability treatment of SF, Stern (1975) and Kunze (1987) obtained the density flux ratio γ^SF = αF_T/βF_S (< 1) for the fastest-growing SF as

$$\gamma^{\text{SF}} = \sqrt {R_{\rho } } \left( {\sqrt {R_{\rho } } - \sqrt {R_{\rho } - 1} } \right)$$

(93)

Kelley (1986) compiled γ^SF as a function of R_ρ from laboratory data on SF:

$$\gamma^{\text{SF}} = 0.35\exp \left[ {1.05\exp \left( { - 2.16\left( {R_{\rho } - 1} \right)} \right)} \right]$$

(94)

The laboratory flux ratios and the numerically and theoretically determined ratios are shown in Fig. 6; γ^SF asymptotes to a constant value as R_ρ becomes large (~ 0.5: Eq. 93, ~ 0.35: Eq. 94) (together with Polzin et al. 1995; Shen 1993, 1995; Taylor and Buscens 1989).

Fig. 6

DDC flux ratio dependence on R_ρ in which solid lines are theoretically or laboratory determined flux laws. Some laboratory and experimental numerical data are also shown for comparison

Buoyancy fluxes of salt and heat for SF are summarized by Kelley (1986):

$$g\beta F_{S} = C_{1} k_{T}^{{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0pt} 3}}} \left( {g\beta \Delta S} \right)^{{\frac{4}{3}}} ,C_{1} = 0.04 + 0.327R_{\rho }^{ - 1.91} ,$$

(95)

$$g\beta F_{T} = \gamma^{\text{SF}} g\beta F_{S} ,$$

(96)

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):

$$\begin{aligned} g\beta F_{S} = 2\upsilon g\beta \frac{{\partial {\kern 1pt} {\kern 1pt} \bar{S}}}{{\partial {\kern 1pt} {\kern 1pt} {\kern 1pt} z}}\left[ {R_{\rho }^{{\frac{1}{2}}} + \left( {R_{\rho } - 1} \right)^{{\frac{1}{2}}} } \right]^{2} , \hfill \\ g\beta F_{T} = \gamma^{\text{SF}} g\beta F_{S} , \hfill \\ \end{aligned}$$

(97)

and for thin interfaces (< 1 m):

$$\begin{aligned} g\beta F_{S} = \frac{1}{8}k_{T}^{1/3} \left( {g\beta \Delta S} \right)^{{\frac{4}{3}}} , \hfill \\ g\alpha F_{T} = \gamma^{\text{SF}} g\beta F_{S} . \hfill \\ \end{aligned}$$

(98)

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, S_t:

$$S_{t} = \frac{{(\beta F_{S} - \alpha F_{T} )}}{{\upsilon (\alpha {\kern 1pt} {{\partial {\kern 1pt} \bar{T}} \mathord{\left/ {\vphantom {{\partial {\kern 1pt} \bar{T}} {\partial {\kern 1pt} z}}} \right. \kern-0pt} {\partial {\kern 1pt} z}}{\kern 1pt} {\kern 1pt} - {\kern 1pt} \beta {\kern 1pt} {{\partial {\kern 1pt} \bar{S}} \mathord{\left/ {\vphantom {{\partial {\kern 1pt} \bar{S}} {\partial {\kern 1pt} z}}} \right. \kern-0pt} {\partial {\kern 1pt} z}})}} = \frac{{g(\beta F_{S} - \alpha F_{T} )}}{{\upsilon N^{2} }}{\kern 1pt} {\kern 1pt} {\kern 1pt} = \frac{{\beta F_{S} (1 - \gamma^{\text{SF}} )}}{{\upsilon \beta {\kern 1pt} \bar{S}_{z} ({\kern 1pt} R_{\rho } - 1)}} \approx 1.$$

(99)

$\frac{{\partial {\kern 1pt} \bar{T}}}{{\partial {\kern 1pt} z}}$ is the vertical salt gradient. If S_t 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:

$$\beta F_{S} = \frac{{S_{t} \upsilon ({\kern 1pt} R_{\rho } - 1)}}{{(1 - \gamma^{\text{SF}} )}}\beta {\kern 1pt} \frac{{\partial {\kern 1pt} {\kern 1pt} \bar{S}}}{{\partial {\kern 1pt} {\kern 1pt} {\kern 1pt} z}}.$$

(100)

As determined in the laboratory, the value of S_t 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:

$$\begin{aligned} & F_{S} = - K_{S}^{{{\text{SF}}}} \frac{{\partial \bar{S}}}{{\partial z}}\therefore K_{S}^{{{\text{SF}}}} = - \frac{{F_{S} }}{{\partial \bar{S}/\partial z}}, \\ & F_{T} = - K_{T}^{{{\text{SF}}}} \frac{{\partial \bar{T}}}{{\partial z}}\therefore K_{T}^{{{\text{SF}}}} = - \frac{{F_{T} }}{{\partial \bar{T}/\partial z}}. \\ \end{aligned}$$

(101)

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 αF_T is larger than that due to βF_S. The density flux ratio for DC is defined as γ^DC = βF_S/αF_T (< 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 < R_ρ<1. Huppert (1971) and Kelley (1990) obtained the following relations for γ^DC as a function of R_ρ for DC using laboratory data: Huppert (1971) introduced

$$\gamma^{\text{DC}} = \left\{ {\begin{array}{*{20}c} {1.85 - 0.85R_{\rho }^{ - 1} {\kern 1pt} {\kern 1pt} {\kern 1pt} } & {{\text{for}}\,\,\,0.5 < R_{\rho } < 1} \\ {0.15} & {{\text{for}}\,\,\,R_{\rho } < 0.5} \\ \end{array} } \right.,$$

(102)

and Kelley (1990) introduced

$$\gamma ^{{{\text{DC}}}} = \frac{{R_{\rho }^{{ - 1}} + 1.4\left( {R_{\rho }^{{ - 1}} - 1} \right)^{{3/2}} }}{{1 + 14\left( {R_{\rho }^{{ - 1}} - 1} \right)^{{3/2}} }}.$$

(103)

Thus, γ^DC becomes 1 for R_ρ= 1, and becomes a constant (~ 0.15: Eq. 102, ~ 0.13: Eq. 103, see Fig. 6) as R_ρ decreases. Individual fluxes of salt and heat for DC were summarized by Kelley (1986, 1990) as the following:

$$\begin{aligned} g\beta F_{S} = \gamma^{\text{DC}} g\alpha F_{T}, \hfill \\ g\alpha F_{T} = C_{2} \left( {{\raise0.7ex\hbox{${k_{T} }$} \!\mathord{\left/ {\vphantom {{k_{T} } \upsilon }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\upsilon $}}} \right)^{{\frac{1}{3}}} \left( {g\alpha \Delta T} \right)^{{\frac{4}{3}}} ,C_{2} = 0.0032\exp \left( {4.8R_{\rho }^{0.72} } \right). \hfill \\ \end{aligned}$$

(104)

$\Delta T$ 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:

$$\overline{{u^{\prime } w^{\prime } }} = - \frac{\tau }{2}\frac{{\partial \bar{u}}}{{\partial z}}\left[ {\frac{1}{2}\left( {\lambda _{1} - \frac{4}{3}\lambda _{2} } \right)q^{2} + \left( {\lambda _{2} - \lambda _{3} } \right)\overline{{u^{{\prime 2}} }} + \left( {\lambda _{2} + \lambda _{3} } \right)\overline{{w^{{\prime 2}} }} } \right]{\text{ }} + \lambda _{4} \tau g\alpha \overline{{u^{\prime } T^{\prime } }} - \lambda _{4} \tau g\beta \overline{{u^{\prime } S^{\prime } }} ,$$

(105)

$$\overline{{u^{\prime } T^{\prime } }} = - \frac{\tau }{{\lambda _{5} }}\left( {\overline{{u^{\prime } w^{\prime } }} \frac{{\partial \bar{T}}}{{\partial z}} + \frac{1}{2}\left( {\lambda _{6} + \lambda _{7} } \right)\overline{{w^{\prime } T^{\prime } }} \frac{{\partial \bar{u}}}{{\partial z}}} \right)$$

(106)

$$\overline{{u^{\prime } S^{\prime } }} = - \frac{\tau }{{\lambda _{9} }}\left( {\overline{{u^{\prime } w^{\prime } }} \frac{{\partial \bar{S}}}{{\partial z}} + \frac{1}{2}\left( {\lambda _{6} + \lambda _{7} } \right)\overline{{w^{\prime } S^{\prime } }} \frac{{\partial \bar{u}}}{{\partial z}}} \right)$$

(107)

$$\overline{{w^{\prime } T^{\prime } }} = - \frac{\tau }{{\lambda _{5} }}\left\{ {\overline{{w^{{\prime 2}} }} \frac{{\partial \bar{T}}}{{\partial z}} - g\left[ { - \lambda _{8} \tau \overline{{w^{\prime } T^{\prime } }} \alpha \frac{{\partial \bar{T}}}{{\partial z}} + \lambda _{{11}} \tau \left( {\overline{{w^{\prime } T^{\prime } }} \beta \frac{{\partial \bar{S}}}{{\partial z}} + \overline{{w^{\prime } S^{\prime } }} \beta \frac{{\partial \bar{T}}}{{\partial z}}} \right)} \right]} \right\}$$

(108)

$$\overline{{w^{\prime } S^{\prime } }} = - \frac{\tau }{{\lambda _{9} }}\left\{ {\overline{{w^{{\prime 2}} }} \frac{{\partial \bar{S}}}{{\partial z}} - g\left[ { - \lambda _{{11}} \tau \left( {\overline{{w^{\prime } T^{\prime } }} \alpha \frac{{\partial \bar{S}}}{{\partial z}} + \overline{{w^{\prime } S^{\prime } }} \alpha \frac{{\partial \bar{T}}}{{\partial z}}} \right) + \lambda _{{10}} \tau \overline{{w^{\prime } S^{\prime } }} \beta \frac{{\partial \bar{S}}}{{\partial z}}} \right]} \right\}$$

(109)

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:

$$\overline{{T^{\prime2} }} = - \frac{{\lambda_{8} \tau }}{{\lambda_{0} }}\overline{{w^{\prime}T^{\prime}}} \frac{{\partial \bar{T}}}{\partial z},$$

(110)

$$\overline{{S^{\prime2} }} = - \frac{{\lambda_{10} \tau }}{{\lambda_{0} }}\overline{{w^{\prime}S^{\prime}}} \frac{{\partial \bar{S}}}{\partial z},$$

(111)

$$\overline{{T^{\prime}S^{\prime}}} = - \frac{{\lambda_{10} \tau }}{{\lambda_{0} }}\left[ {\overline{{w^{\prime}T^{\prime}}} \frac{{\partial {\kern 1pt} {\kern 1pt} \overline{S} }}{{\partial {\kern 1pt} {\kern 1pt} z}} + \overline{{w^{\prime}S^{\prime}}} \frac{{\partial {\kern 1pt} {\kern 1pt} \overline{T} }}{{\partial {\kern 1pt} {\kern 1pt} z}}} \right].$$

(112)

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

$$\begin{aligned} \widetilde{{\lambda_{5} }} = \lambda_{5} \left[ {1 + R_{i} } \right], \hfill \\ \widetilde{{\lambda_{9} }} = 0.02\lambda_{9} \left[ {1 + R_{i} } \right]. \hfill \\ \end{aligned}$$

(113)

Those for DDC are defined as

$$\begin{aligned} \widehat{{\lambda_{5} }} = 0.02\lambda_{5} \left[ {1 + 6.5\left( {R_{\rho }^{ - 1} } \right)^{{{\raise0.7ex\hbox{$5$} \!\mathord{\left/ {\vphantom {5 4}}\right.\kern-0pt} \!\lower0.7ex\hbox{$4$}}}} } \right], \hfill \\ \widehat{{\lambda_{9} }} = 0.02\lambda_{9} \left[ {1 + 6.5\left( {R_{\rho } } \right)^{{{\raise0.7ex\hbox{$5$} \!\mathord{\left/ {\vphantom {5 4}}\right.\kern-0pt} \!\lower0.7ex\hbox{$4$}}}} } \right], \hfill \\ \widehat{{\lambda_{11} }} = \lambda_{11} \left( {\frac{2}{{R_{\rho } + R_{\rho }^{ - 1} }}} \right). \hfill \\ \end{aligned}$$

(114)

Those for the combination of CT and DDC are defined as

$$\begin{aligned} \overline{{\lambda_{5} }} = \widetilde{{\lambda_{5} }}\left[ {1 - f\left( {R_{i} } \right)} \right] + \widehat{{\lambda_{5} }}\left[ {f\left( {R_{i} } \right)} \right], \hfill \\ \overline{{\lambda_{9} }} = \widetilde{{\lambda_{9} }}\left[ {1 - f\left( {R_{i} } \right)} \right] + \widehat{{\lambda_{9} }}\left[ {f\left( {R_{i} } \right)} \right], \hfill \\ \overline{{\lambda_{11} }} = \widetilde{{\lambda_{11} }}\left[ {1 - f\left( {R_{i} } \right)} \right] + \widehat{{\lambda_{11} }}\left[ {f\left( {R_{i} } \right)} \right]. \hfill \\ \end{aligned}$$

(115)

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 [$1 / {\text{m}}$]

Γ^CT:

Mixing coefficient for CT [non-dimensional]

Γ^DDC:

Mixing coefficient for DDC [non-dimensional]

Γ^SF:

Mixing coefficient for SF [non-dimensional]

Γ^DC:

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 [${\text{W/kg}}$]

Pr_t:

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 (δ_ij = 1 when i = j, δ_ij= 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