1. Introduction

One of the most fundamental and long-standing challenges in physical oceanography and geophysical fluid dynamics is the explanation of the direct cascade of energy and thermal variance. The ocean circulation is driven by the mechanical and thermodynamic forcing at the sea surface, which occurs on the scales of ocean basins. However, the energy and thermal variance are ultimately dissipated by molecular processes acting on the centimeter scale, commonly referred to as the microstructure (Wunsch and Ferrari 2004; Merryfield 2005). It has become increasingly clear that the role of small-scale turbulence in the ocean is not limited to the passive dissipation of energy and heat, and it can actively interact with large-scale patterns of motion (e.g., Merckelbach et al. 2010; Friedrich et al. 2011). The seminal model of Munk (1966) advocates a direct link between the intensity of small-scale mixing and the global thermohaline circulation of the World Ocean. Munk’s proposition was based on an unrealistically high estimate of diapycnal diffusivity (∼10⁻⁴ m² s⁻¹), which has been subsequently updated (e.g., Ledwell et al. 1998, 2011). The latter measurements indicate that small-scale eddy diffusivity (∼10⁻⁵ m² s⁻¹) might be insufficient to balance the vertical upwelling at the base of the main subtropical thermoclines. Nevertheless, the ability of the microstructure-induced mixing to influence both global and regional flow patterns is undeniable (Kunze 2017). It is also generally accepted that small-scale turbulence can play a major role in the ocean’s ability to transport and sequester heat, nutrients, pollutants, and carbon dioxide (e.g., Thorpe 2005). One microscale process that plays a significant role is that of salt fingers, which are important sources of mixing in the ocean thermocline. Salt fingers are a double-diffusive process, driven by the difference in the diffusivities of two buoyant fields (typically temperature and chemical composition) that are oppositely stratified. This process is readily seen throughout the World Ocean in the thermocline, separating the warm and salty waters of the surface mixed layer from the cooler and fresher waters of the deeper ocean. Full details on double-diffusive convection can be found in the reviews by Schmitt (1994, 2003) and Radko (2013).

The sensitivity of the general circulation of the World Ocean to the intensity and, perhaps more importantly, to the spatial distribution of small-scale mixing demands the establishment of the detailed phenomenology of turbulent events and the development of accurate parameterizations. Often, it is assumed that the mixing efficiency of microstructure processes (the fraction of the energy expended during mixing events to changing the background potential energy) is well approximated by a constant of approximately 0.2 (Ellison 1957; Osborn 1980). Salt fingers have generally proven an exception to this relation, taking values of up to 0.6 as was shown in field measurements by St. Laurent and Schmitt (1999) in regions of salt fingering with weak shear. This value has now been shown to change by location or process, and the results compiled from a wide range of studies in the review by Gregg et al. (2018) show a range of this efficiency factor from 0.02 to 0.58. Theoretical and empirical advances are therefore needed to accurately represent microscale turbulence in the ocean to better represent ocean dynamics at large scales. Salt fingers serve as an important process in this regard as a well-documented instance of deviation from typical efficiency factors.

Numerical measurements of the turbulence and mixing caused by salt fingers in quiescent flows are prevalent in the literature (see, e.g., Radko and Stern 1999, 2000; Stern et al. 2001; Stern and Simeonov 2005; Traxler et al. 2011; Garaud and Brummell 2015; Yang et al. 2016a,b); however, the ocean is rarely quiescent, and simulations modeling the interactions of salt fingers with shear and internal waves are still relatively new. The general understanding from numerical simulations (such as Kimura and Smyth 2007, 2011; Smyth and Kimura 2011; Garaud et al. 2019; Li and Yang 2022) and field measurements (Kunze 1990) is that shear inhibits the fundamental salt-fingering instability in the direction of the shear, resulting in inclined salt sheets rather than conventional salt fingers. However, these effects have yet to be characterized to a degree where they could reasonably be implemented into large-scale climate models. Two notable recent studies deserve further mention: that of Garaud et al. (2019), who demonstrated an analytic dependence of salt-finger transport on shear for low-Prandtl-number flows, and Li and Yang (2022), who characterized the morphology and transport of sheared salt fingers and convection in a bounded system. Li and Yang (2022) found that salt fingers of density ratio 2 are affected even in the presence of relatively weak shear. We are interested in generalizing their result to unbounded systems and characterize their dependence for a range of density ratios.

We approach this problem through the lens of numerical modeling by simulating salt fingers in the presence of shear. Using the recently developed Rocking Ocean Modeling Environment (ROME) model (Brown and Radko 2021), we are able to measure the microstructure in the presence of uniform shear in a periodic system. This permits simulations without the impact of nearby boundaries or the requirement for small local wavelengths of shear, approximations which have often been necessary in prior numerical work on this problem. We find that the microstructure of salt fingers is notably anisotropic in large regions of parameter space, and that even weak shear can substantially diminish the heat fluxes of these systems. To apply this work to larger-scale systems, we also include a model fit for the heat flux and thermal dissipation rate. We demonstrate that these results are remarkably consistent with observations by comparing with the North Atlantic Tracer Release Experiment (NATRE) dataset (St. Laurent and Schmitt 1999).

This manuscript is organized as follows. Section 2 summarizes the governing equations and numerical setup. Section 3 describes the direct results from the simulations, and section 4 describes the empirical fit. In section 5, we compare the simulation results to NATRE observations. We conclude with some final remarks in section 6.

2. Methodology

We simulate this system, changing the Richardson number and the density ratio, as described in Table 1. Density ratios of 1.25, 1.5, 2.0, 3.0, and 5.0 represent nearly the entire range of density ratios in the ocean. For each value of the density ratio, we consider a range of Richardson numbers from Ri = 0.5 to Ri → ∞. The simulations are characterized by a domain size of 50 length units in $\tilde{x}$ and $\tilde{y}$ and 100 length units in $\tilde{z}$. We have found that doubling the size of the domain in $\tilde{x}$ (as might be reasonable considering the extended finger morphology seen in Li and Yang 2022) has little impact on the final transport. For example, a simulation with Ri = 10 and R₀ = 2 (which we will demonstrate is an intermediate case where the structures are not yet horizontally uniform but are strongly inclined) shows deviations in transport comparable to temporal variations in a single simulation.

Table 1

Simulation parameters.

View Table

3. Numerical results

Each simulation begins from small random perturbations at the grid scale in the temperature and salinity fields. These initial perturbations grow exponentially at the wavenumbers associated with the fastest growing mode. These structures are typically elongated vertically and inclined in the direction of shear. Eventually, nonlinear interactions between adjacent structures cause the development of turbulence, which inhibits the initial instability growth. This results in a quasi-statistical equilibrium where the salt fingers or sheets maintain their strength indefinitely. This final equilibrated state is shown in Fig. 1 for simulations with R₀ = 2 and Ri = (∞, 20, 0.5). In the case without shear, the salt fingers are horizontally isotropic, which is the canonical behavior for the system (Radko 2013). However, shear causes the finger structures to become increasingly inclined in the streamwise direction, and by Ri = 0.5, the system is nearly homogenous in x. In the cross-stream direction, the fingers appear qualitatively consistent to cases without shear, though the intensity of the fingers decreases. This is consistent with the structures seen in prior work of sheared salt fingers (see, e.g., Kimura et al. 2011; Radko et al. 2015). This drastic change in structure also has consequences for the heat and salinity transport in the system.

View Full Size

The salinity perturbation field for simulations with R₀ = 2 and (a) Ri → ∞, (b) Ri = 20, (c) Ri = 0.5.

Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0049.1

• Download Figure
• Download figure as PowerPoint slide

Figure 2 shows the fluxes of temperature and salinity for the series of simulations with R₀ = 2. The irreversible fluxes of temperature and salinity are approximated using the dissipation of each quantity:

$F_T\equiv -⟨{(\nabla T)}^2⟩,$(17)

$F_T^{*}=-{\rho }_0^{*}{c}_p^{*}{F}_T\frac{[l]}{[t]}[T],$(18)

$F_S\equiv -R_0\tau ⟨{(\nabla S)}^2⟩,$(19)

$F_S^{*}=-{\rho }_0^{*}{F}_S\frac{[l]}{[t]}[S].$(20)

In general, the use of the dissipation as a proxy for the irreversible vertical heat and salt fluxes is justified by the balance of production and dissipation terms of the temperature and salinity equations (Osborn and Cox 1972), which has been recently confirmed for some double-diffusive systems (Hieronymus and Carpenter 2016). It is important to note that the dissipation can only be used as a proxy for the turbulent flux of the system and the (constant) diffusive flux is not included in these comparisons. The total temperature and salinity fluxes are given by

$F_T^{\text{total}}=F_T-1,$(21)

$F_S^{\text{total}}=F_S-\tau R_0^{-1}.$(22)

Note that the gradients and spatial derivatives here remain defined in the inertial system. For the manner of simulation described here, these proxies yield nearly identical results to time- and domain-averaged turbulent fluxes but exhibit less temporal variability (Brown and Radko 2022). The nondimensional ratio of the turbulent flux proxies is defined as the flux ratio γ:

$\gamma \equiv \frac{F_T}{F_S}.$(23)

For ease of comparison between different cases, for which the growth from the initial perturbations vary slightly, we define a shifted time coordinate t′, which is 0 when −F_T first exceeds unity. All cases show nearly identical growth rates in the early, exponential stage of development, but near F_T = 20, the sheared simulations start to show signs of nonlinear effects, where the case without shear grows for another factor of 3 before the exponential growth phase ends. This is likely due to the physical structure of the linear instability: even a small amount of shear severely inhibits the development of three-dimensional salt fingers, which instead form salt sheets (Kimura and Smyth 2007). The mean fluxes of the simulations vary monotonically with Richardson number, with stronger shear more easily disrupting fingers and reducing both thermal and haline fluxes. This brings the sheared 3D salt-finger fluxes closer to unsheared 2D salt-finger fluxes (Radko et al. 2015), which is consistent with the quasi-two-dimensional structure of T–S fields in shear (e.g., Fig. 1c).

View Full Size

The fluxes of (top) heat and (bottom) salt for systems with R₀ = 2 and varying shear with respect to the shifted time coordinate t′.

Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0049.1

• Download Figure
• Download figure as PowerPoint slide

Unfortunately, the irreversible fluxes are difficult to measure directly in the ocean, but other metrics have been used as proxies to quantify the mixing in these regions. We calculate some of these metrics of turbulence in Fig. 3, including the turbulent dissipation rate ϵ, thermal dissipation rate χ, effective turbulent diffusivity K_T, and mixing efficiency factor Γ_e. The turbulent dissipation rate requires some particular discussion. For an incompressible system, the mean turbulent dissipation rate is (Taylor 1935)

${ϵ}^{*}\equiv {⟨2{\nu }^{*}{\displaystyle \sum _{i=1}^3{\displaystyle \sum _{j=1}^3{s}_{ij}^{*}{s}_{ij}^{*}}}⟩}_t,$(24)

where $s_{ij}^{*}\equiv \frac{1}{2}(\partial u_i^{*}/\partial x_j^{*}+\partial u_j^{*}/\partial x_i^{*})$ is the strain-rate tensor. For a triply periodic incompressible system, integration by parts reveals

$2⟨\frac{\partial u^{*}}{\partial y^{*}}\frac{\partial {\upsilon }^{*}}{\partial x^{*}}⟩+2⟨\frac{\partial u^{*}}{\partial z^{*}}\frac{\partial w^{*}}{\partial x^{*}}⟩+2⟨\frac{\partial y^{*}}{\partial z^{*}}\frac{\partial w^{*}}{\partial y^{*}}⟩=-⟨{\left(\frac{\partial u^{*}}{\partial x^{*}}\right)}^2⟩-⟨{(\frac{\partial {\upsilon }^{*}}{\partial y^{*}})}^2⟩-⟨{\left(\frac{\partial w^{*}}{\partial z^{*}}\right)}^2⟩,$(25)

which permits us to express the turbulent dissipation rate as $ϵ={\nu }^{*}{⟨{\nabla }^{*}{\mathbf{u}}^{*}:{\nabla }^{*}{\mathbf{u}}^{*}⟩}_t$, where the “:” symbol indicates taking the inner product of the gradient of velocity tensor with itself. Thus, our turbulence metrics are measured and defined as follows:

$\begin{array}{cc}{ϵ}^{*}={\nu }^{*}{⟨{\nabla }^{*}{\mathbf{u}}^{*}:{\nabla }^{*}{\mathbf{u}}^{*}⟩}_t,& ϵ=\text{Pr}{⟨\nabla \mathbf{u}:\nabla \mathbf{u}⟩}_t,\end{array}$(26)

$\begin{array}{cc}{\chi }^{*}\equiv {\kappa }_T^{*}{⟨{({\nabla }^{*}{T}^{*})}^2⟩}_t,& \chi ={⟨{(\nabla T)}^2⟩}_t,\end{array}$(27)

$\begin{array}{cc}{K}_T^{*}\equiv -\frac{{⟨w^{*}{T}^{*}⟩}_t}{\frac{\partial {\overline{T}}^{*}}{\partial z^{*}}},& K_T=-{⟨wT⟩}_t,\end{array}$(28)

${\Gamma }_e\equiv \frac{N^{*2}{\chi }^{*}}{{ϵ}^{*}}=\frac{\text{Pr}(1-R_0^{-1})\chi }{ϵ}.$(29)

We note here that u and T (and their dimensional equivalents) are perturbations away from the background state and—as such—are appropriate for defining these turbulence metrics. Generally, Fig. 3 shows that the turbulent dissipation rate, thermal dissipation rate, and effective diffusivity decrease as the strength of shear increases. This demonstrates that double-diffusive turbulence is most intense when the magnitude of shear is weak, and the turbulence gradually weakens as shear intensifies and disrupts the salt-finger structures in the streamwise direction. The Richardson number at which shear becomes substantial (i.e., halves the turbulence metrics) depends on the density ratio, with higher density ratios showing much greater sensitivity to shear. This effect arises based on the transition between salt-sheet and salt-finger structures of the primary salt-fingering instability, which is outlined in more detail in section 4. In addition, the mixing efficiency is also strongly dependent on the density ratio but appears to be largely insensitive to the presence of shear.

View Full Size

Various turbulence metrics for the simulations. For ease of comparison with observations, these are presented in dimensional units, assuming $l^{*}=0.01\mathrm{m}$, ${\kappa }_T^{*}=1.4\times {10}^{-7}{\mathrm{m}}^2{\mathrm{s}}^{-1}$, and $\partial {\overline{T}}^{*}/\partial z^{*}=0.01°\mathrm{C}{\mathrm{m}}^{-1}$.

Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0049.1

• Download Figure
• Download figure as PowerPoint slide

A major hurdle in converting these fundamental properties to ocean observations is that observations are typically limited to gradients in the vertical direction, and some assumption must be made regarding the isotropy of the flow. In simulations, we can directly measure the anisotropy of the turbulence in the presence of shear by presenting the Cox numbers for the system in Fig. 4. We define the Cox numbers as

${\text{Cx}}_i\equiv {⟨{\left(\frac{\partial }{\partial x_i}T\right)}^2⟩}_t,$(30)

where the i subscript indicates the dimension, and the total Cox number is defined as $\text{Cx}={\displaystyle {\sum }_i{\text{Cx}}_i}=\chi =-F_T$. For simulations without shear or with weak shear, the system is laterally isotropic, but as the shear becomes more substantial, the structures become more extended in x, and Cx_x can become orders of magnitude smaller than Cx_y. Comparable to the other turbulence metrics, the value of the Richardson number at which shear substantially affects the Cox number is dependent on R₀. The y-directed Cox number is always larger than the z-directed value, consistent with the understanding that fingers are generally more extended in z. We can ascertain the anisotropy directly by measuring the ratio of χ to 3Cx_z in Fig. 4d. Were the turbulence perfectly isotropic, this would yield a value of 1 for all simulations, but Fig. 4d shows that the true values depend on both density ratio and Richardson number. The turbulence at high density ratios is substantially underpredicted by assuming isotropy if only the vertical gradients are known. Shear is also significant, with stronger shear typically decreasing the horizontal Cox number, which (typically but not always) improves the performance of isotropy-based models.

View Full Size

The mean Cox numbers measured for each simulation. (a) The Cox number for variations in x. (b) The Cox number for variations in y. (c) The Cox number for variations in z. (d) The measurement of anisotropy by comparing the total dissipation (χ = Cx_x + Cx_y + Cx_z) to an assumption of isotropy (3Cx_z. The dashed line indicates the prediction for isotropic turbulence. The dotted line represents a reasonable fit to moderate Ri values (see section 4).

Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0049.1

• Download Figure
• Download figure as PowerPoint slide

We can also investigate the anisotropy of the flow and compare with observations of this quantity by comparing the components of the strain-rate tensor:

$s_{ij}\equiv {⟨{\left(\frac{1}{2}\frac{\partial u_i}{\partial x_j}+\frac{1}{2}\frac{\partial u_j}{\partial x_i}\right)}^2⟩}_t.$(31)

For incompressible isotropic turbulence, Taylor (1935) demonstrated that the turbulent dissipation rate reduces to

$ϵ=\frac{15}{2}\text{Pr}{⟨{\left(\frac{\partial u_i}{\partial x_j}\right)}^2⟩}_t,$(32)

if i ≠ j and

$ϵ=15\text{Pr}{⟨{\left(\frac{\partial u_i}{\partial x_i}\right)}^2⟩}_t,$(33)

otherwise. We explore how fingering convection adheres to this approximation in Fig. 5. In general, the components that are most readily observed in the ocean (ϵ_xz and ϵ_yz) are displayed in Figs. 5c and 5f. These components would underpredict the results from assuming isotropy by approximately a factor of 2. In other regions of the parameter space, Eq. (32) leads to major errors—by as much as an order of magnitude—in the estimates of ϵ. The physical reasoning here is that stronger shear inclines the finger structures, which reduces their vertical extent to be more comparable to their lateral extent. This is true even for particularly weak values of oceanic shear. The likelihood of extreme inaccuracy during oceanographic measurements is therefore low, requiring a vertical probe traversing directly along a finger structure, and the factor of 2 correction is most likely sufficient in most cases.

View Full Size

The estimates of the total dissipation rate from each component of the dissipation tensor, scaled by the total dissipation rate.

Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0049.1

• Download Figure
• Download figure as PowerPoint slide

4. Microstructure–shear interaction model

For implementation in climate modeling and comparison with observations, it is useful to develop an empirical model of the microstructure and fluxes. To do so, we define a critical Richardson number, Ri_c, as the interpolated Richardson number where the thermal dissipation rate (and hence the irreversible thermal flux) has fallen to half of the value from the case without shear (Ri → ∞). We define the thermal dissipation rate without shear for a given value of R₀ as χ_∞ (see Table 2). These critical Richardson numbers are then fitted to a power law in density ratio:

${\text{Ri}}_{\mathrm{c}}=a{R}_0^b,$(34)

where a = 0.4626 and b = 5.839. Though the exponent here seems large, it is readily explained by linear theory in appendix section as the Richardson number at which the system transitions from salt fingers to salt sheets. It is worth comparing this to the results of Garaud et al. (2019), who find a similar relationship for low-Prandtl-number flows. In their work, this critical Richardson number is not directly a function of the density ratio but rather the Richardson number at which the shearing rate is twice the linear growth rate of the unsheared system. In the high-Prandtl-number limit, the relationship between the critical Richardson number and the linear growth rate is more complicated, as will be discussed in section appendix. As in Garaud et al. (2019), χ/χ_∞ collapses when calculated as a function of Ri/Ri_c, and the theory from Garaud et al. (2019) predicts an expression of the form

$\chi ={\chi }_{\infty }{\left(1+c\frac{{\text{Ri}}_c}{\text{Ri}}\right)}^{-1},$(35)

where c is unknown a priori, but is expected to be of order unity and comes out to 0.902, which is shown in Fig. 6. Also as in Garaud et al. (2019), the low-Richardson-number limit is not well described by a function of this form, and this fit is only appropriate for Ri/Ri_c > 0.1. As is common in studies of salt fingers (see, e.g., Radko 2013), the flux ratio is well approximated by a constant. Of particular note, there is a well-studied weak and nonmonotonic dependence of the flux ratio on the density ratio in the case of infinite Richardson number, γ_∞, which has been characterized in Radko and Smith (2012), for example. This variation in γ is evident for Richardson numbers above 10, but as the Richardson number approaches 1, the flux ratio approaches a roughly constant value of 0.69 with no notable dependence on density ratio. The values of χ_∞(R₀) and γ_∞(R₀) are well studied, and the reader is referred to Table 3.1 of Radko (2013) for a collection of empirical and theoretical fits to these.

Table 2

Fitting parameters.

View Table

View Full Size

(a) The scaled thermal dissipation of the simulations. The solid curve indicates the best fit to a hyperbolic tangent (b) The flux ratio from the simulations. The solid line represents γ = 0.69. (c) The critical Richardson number as a function of density ratio. The solid line indicates the best polynomial fit to the simulations, and the crosses indicate the predictions from the theory in the appendix.

Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0049.1

• Download Figure
• Download figure as PowerPoint slide

As can be seen in the appendix, the spectral linear equations in the sheared ( $\tilde{x}$) coordinate system are dependent both on wavenumber and time. This results in a time-dependent growth rate of the primary instability ${\lambda }_p(\tilde{k},\tilde{t})$, where $\tilde{k}$ is the wave vector in the sheared coordinate system. The fastest growing modes at the initial time are vertical, as in the unsheared system, but as these wave modes become increasingly inclined due to shear, the diffusion in the inertial z direction becomes significant. Thus, the growth rate decreases with $\tilde{t}\hspace{0.17em}$ for any mode with a nonzero value of ${\tilde{k}}_x$, as can be seen in Fig. 7a. We can define the time required for the growth rate to become 0 as $\mathrm{\Delta }\tilde{t}\hspace{0.17em}$. These linear modes are continuously being generated at a range of wavenumbers, but only modes that survive to the nonlinear regime will become dynamically significant. As such, we assert that modes for which $\mathrm{\Delta }\tilde{t}\lesssim 1/{\lambda }_p$ will not survive long enough to contribute. This comparison marks the transition between salt sheets—for which modes with ${\tilde{k}}_x$ ≠ 0 will decay rapidly—and salt fingers. We plot $\mathrm{\Delta }\tilde{t}{\lambda }_p$ for the salt-finger modes as a function of Richardson number in Fig. 7b in addition to the critical Richardson number values that were measured from the simulations. A criterion of $\mathrm{\Delta }\tilde{t}{\lambda }_p\sim 1.2$ yields excellent agreement with simulation measurements. This implies that the conditions under which salt fingers are sheared before they can develop are the same as the measured Richardson numbers for which the thermal transport is halved. We take this as evidence that it is shear preventing salt-finger development as the reason that the thermal transport decreases.

View Full Size

(a) The growth rate of the cross-stream primary instability for R₀ = 2 and a range of Ri as a function of time in the sheared system. (b) The product of the initial primary instability growth rate and the cross-stream decay time for varying values of Ri and R₀. The crosses show the values of Ri_c from the simulations, which reasonably are predicted by $\mathrm{\Delta }\tilde{t}{\lambda }_p=1.2$.

Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0049.1

• Download Figure
• Download figure as PowerPoint slide

A thorough growth-rate-balance analysis requires the computation of the growth rates for the full three-dimensional sheared system in both the salt-sheet and salt-finger geometries. The fluxes through the system are then estimated by the amplitudes of the thermal and vertical velocity fields when the growth-rate-balance condition is met. Full details of this analysis are included in the appendix, and the results are depicted in Fig. 8. As has already been characterized in prior studies, growth-rate balance characterizes the limit of weak shear considerably well. We demonstrate the scaling from Eq. (35) using the unsheared three-dimensional flux predictions from Radko and Smith (2012) to model the system for weak shear (Ri > 0.1 Ri_c). The final predicted fluxes are shown in Fig. 8a, which agree with the simulations. We also attempt a more thorough analysis, calculating the growth rates and flux predictions from the complete sheared system in Fig. 8b. This figure presents three distinct growth-rate-balance curves for each density ratio: the prediction for salt sheets [Eq. (A20)], the prediction for salt fingers [Eq. (A21)], and the maximum of the two [Eq. (A19)]. The general behavior is consistent, and so it is reasonable to believe that the dynamics are largely captured. For values where $\mathrm{\Delta }\tilde{t}{\lambda }_p<1$, the results from growth-rate balance for a salt-sheet geometry agree well for C = 2.5. The prediction from growth-rate-balance for a salt-finger geometry predicts that the fingers should be disrupted at larger Richardson numbers than is seen. We attribute this to the nature of the setup in the intermediate regime: the primary instability during transition is not as well represented by either salt fingers or salt sheets. Despite this, Eq. (35) predicts and explains the dynamics of the system accurately for Ri/Ri_c > 0.1, and the salt-sheet prediction from growth-rate balance follows the behavior for Ri/Ri_c < 0.1.

View Full Size

(a) The thermal fluxes from the simulations (circles) compared with the model from Fig. 6. (b) The same comparison but using the complete growth-rate-balance model [Eq. (A19)]. In these calculations, C = 2.5. The dashed lines indicate the prediction from the salt-sheet growth-rate-balance analysis [Eq. (A20)], and the dotted lines indicate the same from the salt-finger analysis [Eq. (A21)]. The complete growth-rate-balance model is the maximum of these two solutions.

Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0049.1

• Download Figure
• Download figure as PowerPoint slide

5. Comparison with observations

We are interested in comparing these scaled dissipation measurements with the NATRE dataset (St. Laurent and Schmitt 1999). We narrow the dataset to include only regions where active fingering convection is the primary form of transport. Observations outside of the fingering-favorable regime (R₀ < 1 or R₀ > 6) or with mixing too weak to reliably detect ( ${ϵ}_{z,s}^{*}<{10}^{-11}{\mathrm{m}}^2{\mathrm{s}}^{-3}$ or ${\chi }_{z,s}^{*}<{10}^{-11}°{\mathrm{C}}^2{\mathrm{s}}^{-1}$) are excluded from analysis. In addition, the data in this range include two primary forms of turbulent mixing: salt fingers and larger-scale shear. We can distinguish between these by recognizing that salt fingers are typically characterized by a weak buoyancy Reynolds number (Re_b < 20). Discussion of these traits can be found in St. Laurent and Schmitt (1999). The buoyancy Reynolds number is defined as

${\text{Re}}_b=\frac{{ϵ}_{z,s}^{*}}{{\nu }^{*}{N}^{*2}}.$(41)

We show the buoyancy Reynolds number from several simulations in Fig. 9, which demonstrates that this quantity is largely independent of Richardson number but decreases sharply with density ratio. In general, Re_b is less than 20 for salt fingers, and this has been used historically to distinguish salt fingers from other forms of stratified microstructure (Inoue et al. 2007). To effectively compare the observations of salt fingers without much other turbulence present, we filter the field measurements in this analysis by excluding observations where Re_b > 20, the same condition as used by Inoue et al. (2007).

View Full Size

The buoyancy Reynolds number for several select simulations, plotted as functions of density ratio. (a) Simulations with Ri = 0.5. (b) Simulations with Ri = 1. (c) Simulations with Ri = 10. (d) Simulations with Ri → ∞.

Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0049.1

• Download Figure
• Download figure as PowerPoint slide

We compare the scaled dissipation to the measurements from NATRE in Fig. 10. To most effectively compare these, we present the NATRE data in terms of a probability distribution plot, binned according to dissipation rate and density ratio. The data then indicate the typical range of salt-finger dissipation measurements for a given density ratio. As expected, at low density ratios, the dissipation rates are typically large, and they gradually decrease in both magnitude and prevalence as the density ratio increases. We see remarkable agreement between these observations and the simulations from this study. For a given density ratio, varying the shear magnitude in the simulations results in approximately an order-of-magnitude difference in the dissipation rates. A similar range of measurements is evident in the observations, which would accurately reflect that a range of external shear values are present in the ocean. At higher density ratios, the simulation measurements fall below the detection threshold of the instrument, but such miniscule transport is likely not of substantial interest.

View Full Size

(a) The turbulent dissipation rate for the simulations (symbols) and NATRE observations (color). The symbols indicate the strength of the imposed shear by the size of the arrow. The cross symbols represent simulations without externally imposed shear. The observations are presented as a probability distribution plot indicating the number of observations in the dataset that occur with the matching density ratio and turbulent dissipation rate. (b) The observed thermal dissipation rate presented in the same manner.

Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0049.1

• Download Figure
• Download figure as PowerPoint slide

In addition, we plot the observed mixing efficiency Γ_z from the simulations and NATRE measurements in Fig. 11:

${\Gamma }_z\equiv \frac{N^{*2}{\chi }_{z,s}^{*}}{{ϵ}_{z,s}^{*}{\left(\frac{\partial {\overline{T}}^{*}}{\partial z^{*}}\right)}^2}.$(42)

For more conventional forms of ocean turbulence, this quantity is typically observed to be approximately 0.2 (e.g., Gregg et al. 2018; Monismith et al. 2018). We focus on the NATRE data in distinct ranges of the Richardson number from low values (0.2 < Ri < 1 and 0.5 < Ri < 2) to high values (5 < Ri < 20 and Ri > 1000). Generally, the mixing efficiency only depends weakly on the Richardson number in both the simulations and field measurements. However, the behavior of the mixing efficiency differs with respect to density ratio between the two datasets. Simulations typically report low (∼0.5) values for small density ratios, and the mixing efficiency increases monotonically. The field measurements begin with considerably larger values at low density ratios (∼1.5), decrease to a minimum of just below unity for R₀ ∼ 2, and increase past that value. The values are comparable in the R₀ > 1 range, where most of the observations occur. It is likely that the disagreement arises on the high R₀ end from contamination in the observations by non-double-diffusive phenomena. Such effects are difficult to filter out due to the weakness of double-diffusion in that regime. Disagreement at low values of R₀ may arise instead from instances of convection. However, the agreement in the 1.5 < R₀ < 2 range covers most of the observed instances of fingering convection in the dataset (see Fig. 10).

View Full Size

The mixing efficiency for the simulations (black circles) and binned NATRE data (orange crosses) as a function of density ratio. The NATRE measurements are binned by density ratio, permitting 1000 individual measurements in each density ratio bin. The density ratio and dissipation rates are then averaged for each bin to generate the plot. (a) Only data with Richardson numbers ranging from 0.2 to 1 are presented. (b) Only data with Richardson numbers ranging from 0.5 to 2 are presented. (c) Only data with Richardson numbers from 5 to 20 are presented. (d) Only data with Richardson numbers greater than 1000 are presented. The error bars are determined using Eq. (8) from St. Laurent and Schmitt (1999).

Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0049.1

• Download Figure
• Download figure as PowerPoint slide

6. Conclusions

Through an expansive series of numerical simulations, we have approached the topic of sheared salt fingers in oceanographic contexts. The analysis was made possible through the implementation of the spectral algorithm ROME (Brown and Radko 2021), which permits microstructure modeling in effectively unbounded shear flows. The results of these simulations compare well with measurements in the Atlantic thermocline and have been compiled into a single empirical relation for diapycnal heat and salt fluxes that can be implemented into general circulation models. These simulations demonstrate that shear reduces the fluxes of salt fingers by up to an order of magnitude, bringing these results closer to two-dimensional fluxes. The anisotropy of the finger structures is also affected: salt fingers without shear show stronger horizontal gradients, and become more anisotropic at higher R₀, though these effects are small. Shear reduces horizontal gradients, which typically makes salt fingers more isotropic. The mixing efficiency is not substantially affected by shear but remains a strong function of density ratio. In particular, the mixing efficiency is typically larger than vertical measurements would suggest, further distinguishing salt fingers from other forms of turbulence, for which Γ_e ∼ 0.2. The flux ratio is only weakly affected by shear but tends to converge in density ratio as shear increases.

Together, these effects could potentially alter typical behavior in oceanographic contexts, especially as these pertain to climate modeling. For moderate density ratios, fluxes of salt fingers in the ocean are a factor of 2–3 smaller than previously predicted by models not taking external shear into account. For larger density ratios, however, the effect of shear will likely be less. These changes in fluxes are likely to have global consequences for surface heat fluxes and outgassing as reported by Glessmer et al. (2008), including an increase in the predicted outgassing of CO₂ into the atmosphere. In addition, these results will likely be critical in understanding the development of larger double-diffusive structures. In particular, the lack of variation in flux ratio for large values of shear will likely result in an inability for strongly sheared systems to develop thermohaline staircases via the γ instability (Radko 2003). Shear can also dramatically impact the development of fingering-favorable intrusions, which may contribute substantially to later transport in the ocean across fronts and around eddies.

Finally, we systematically explore the parameter space to characterize the traditionally used microstructure and mixing characteristics of salt fingers in shear. We present this analysis in terms of a critical Richardson number, the value at which the salt-finger fluxes fall to half their maximum value. Normalizing by Ri_c, the thermal dissipation rate (and hence the irreversible heat flux) for all simulations collapses onto a single curve which is well approximated by a rational expression. As in the work of Li and Yang (2022), we find that a remarkably weak shear is capable of causing substantial changes to the transport properties of salt fingers. The results of this study show that unbounded finger studies appear to be affected much more strongly than the bounded calculations of Li and Yang (2022), where the shear can concentrate at the boundaries. The critical Richardson number measured in this work also scales reasonably well with the density ratio, with higher density ratios being more sensitive to shear. Anisotropy is also reasonably well addressed with a piecewise power-law in Ri, though the assumption of isotropy holds within a factor of two for χ and ϵ. We are able to explain the fluxes and anisotropy accurately using growth-rate balance theory.

The work here represents a critical building block to realistic microstructure modeling by the addition of external shear to double-diffusive mixing. However, the analysis here has been restricted to steady flows, which are only rarely observed in the ocean. The next important stage of this research is then to use oscillating and time-dependent flows such as full internal wave spectra (Garrett and Munk 1972). In addition, the parameterized model presented here creates an opportunity to address larger-scale phenomena that depend on sheared salt-finger fluxes, such as intrusions, both numerically and through observations.

Data availability statement.

The numerical code used to generate the data in this study is described in Brown and Radko (2021) and can be found at Brown (2021). Limited records from the numerical simulations can be found at Brown (2023). These records include the spatially averaged quantities discussed here and limited snapshots of the full fields from select simulations due to limited storage constraints. More detailed outputs are archived on NSF systems and can be made available upon reasonable request.