# Diapycnal Velocity in the Double-Diffusive Thermocline

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminary Considerations

#### 2.1. Turbulent Mixing

#### 2.2. Double-Diffusion: The Constant Flux Ratio Model

#### 2.3. Combined Effects of Double-Diffusion and Turbulence

## 3. Large-Scale Numerical Simulations

#### 3.1. Formulation

#### 3.2. Diapycnal Velocity

#### 3.3. Diapycnal Transport in the Double-Diffusive and in the Turbulent Ocean

#### 3.4. The Role of the Flux Ratio

#### 3.5. Forced Simulations

## 4. Theoretical Model of Double-Diffusive Insulation

## 5. Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Asymptotic Solutions for the Double-Diffusive Model with Weak Variation in the Flux Ratio

#### Appendix A.2. General Solutions

#### Appendix A.3. Specific Solutions

#### Appendix A.4. Constraints on the Diapycnal Velocity

**Figure A1.**The non-dimensional departure of temperature from the linear gradient ${T}^{\prime}(z)=T(z)-z$. ${T}^{\prime}$ is computed numerically (black curve) and compared with the calculation based on the asymptotic expansion in $\mathsf{\epsilon}$ truncated at the first, second, and third orders (blue, green, and red curves respectively). For $\mathsf{\epsilon}=0.32$ ($\mathsf{\delta}=0.1$), the numerical result is almost indistinguishable from the third order asymptotic prediction.

**Figure A2.**The range of diapycnal velocities permitted in the one-dimensional model with variable flux ratio ($\mathsf{\gamma}$). Parameter $\mathsf{\delta}$ measures the extent of variation in $\mathsf{\gamma}$, with positive (negative) values corresponding to the increasing (decreasing) $\mathsf{\gamma}({R}_{\rho})$relation. The numerical calculations resulting in regular solutions are indicated by heavy dots and light dots represent conditions under which no solutions were found.

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**Figure 1.**One-dimensional model. We search for the temperature and salinity profiles $T(z)$ and $S(z)$ satisfying the vertical advection-diffusion equations for given vertical velocity ($w$) and boundary conditions at the ends of the mixing zone ($-{H}_{bot}<z<-{H}_{top}$).

**Figure 2.**The range of diapycnal velocities permitted in the hybrid model which includes both double-diffusive and turbulent mixing. Parameter ${K}^{turb}/{K}_{T}^{dd}\left({R}_{0}\right)$ measures the relative contributions from turbulence and double-diffusion to the net mixing. The numerical calculations resulting in regular solutions are indicated by heavy dots and light dots represent conditions under which no solutions were found.

**Figure 3.**The meridional patterns of the model forcing fields. Thermohaline forcing is applied by relaxing the surface temperature and salinity to the target patterns shown in (

**a**) and (

**b**) respectively. The wind stress is shown in (

**c**). All forcing fields are zonally-uniform.

**Figure 4.**The final state ($t=200\text{}\mathrm{years}$) realized in the numerical experiments with double-diffusive mixing (the constant flux ratio case): (

**a**) The horizontal temperature distribution and velocity pattern at $z=-500\text{}\mathrm{m}$; (

**b**) The zonal section of temperature at $y=0.5{L}_{y}$. Clearly visible is a well-defined thermocline with relatively warm water extending several hundred meters downward from the surface.

**Figure 5.**Three-dimensional view of the average isopycnal surface $\rho ={\rho}_{av}$ defined in Equation (19). Color coding represents the density ratio distribution on this isopycnal. The observed range of density ratios $1<{R}_{\rho}<3$ indicates that fingering is active and that its intensity is relatively high.

**Figure 6.**Physical interpretation of diapycnal velocity. A Lagrangian particle initially located on a motionless tilted isopycnal surface is advected from A to B by a cross-isopycnal flow. Point C is the vertical projection of B onto the isopycnal surface. Diapycnal velocity represents the rate of the increase in vertical separation of the Lagrangian point from the isopycnal surface.

**Figure 7.**Diapycnal transport in the double-diffusive and turbulent oceans. Diapycnal velocity $\left({w}^{*}\right)$ is evaluated at the average isopycnal surface $\rho ={\rho}_{av}$ in the ocean interior ($x>{L}_{\mathrm{int}}$) and shown for the double-diffusive (

**a**) and turbulent (

**b**) experiments. In the double-diffusive case, typical values of ${w}^{*}$ are on the order of $~\text{}5\times {10}^{-9}\text{}\mathrm{m}/\mathrm{s}$ or less with both positive and negative values observed. In the turbulent experiment, diapycnal velocities are mostly positive and larger by at least an order of magnitude ($~\text{}5\times {10}^{-8}\text{}\mathrm{m}/\mathrm{s}$).

**Figure 8.**The vertical distribution of diapycnal transport. (

**a**) The mean diapycnal velocity evaluated at various isopycnals (${w}_{iso}^{*}$) and plotted as a function of the average depth of those surfaces; (

**b**) The local vertical profile of diapycnal velocity (${w}_{loc}^{*}$) at $\left(x,y\right)=\left(0.5{L}_{x},0.5{L}_{y}\right)$. Both diagnostics indicate that diapycnal velocity in the turbulent case (indicated by the blue curves) substantially exceeds that in the double-diffusive ocean (green curves). The patterns of ${w}_{iso}^{*}$ and ${w}_{loc}^{*}$ are qualitatively similar.

**Figure 9.**The variation of the mean diapycnal velocity at the density surface $\rho ={\rho}_{av}$ (${w}_{av}^{*}$) as a function of ${K}^{turb}$. The diagnostics are based on a series of simulations which incorporate both double-diffusive and turbulent mixing. Note the monotonic increase in ${w}_{av}^{*}$ with increasing ${K}^{turb}$.

**Figure 10.**The assumed patterns of the flux ratio (

**a**) and salt diffusivity (

**b**) used for parameterization of double-diffusion in the numerical simulations (Section 3.4).

**Figure 11.**A series of large-scale simulations in which the variation in the flux ratio (as measured by the parameter $\mathsf{\delta}$) is systematically increased. For each experiment, the mean diapycnal velocity (${w}_{av}^{*}$) at the average isopycnal surface ${\rho}_{av}$ is plotted as a function of $\mathsf{\delta}$. Note the monotonic—nearly linear—increase in ${w}_{av}^{*}$ with $\mathsf{\delta}$.

**Figure 12.**Vertical T-S profiles obtained with the “forced” model, in which upward vertical velocity was prescribed below the surface and above the sea-floor of the model ocean. The solid (dashed) curves represent the salinity (temperature) profiles: (

**a**) The experiment in which vertical mixing is represented by turbulent diffusion with uniform and equal T-S diffusivities; (

**b**) The analogous experiment performed with double-diffusive mixing. Note the dramatic differences in the resulting stratification patterns.

**Figure 13.**Schematic diagram illustrating the analytical model of double-diffusive insulation. The model suggests that the average diapycnal velocity in the regions bounded by closed streamlines on the density surfaces in the ocean interior (indicated by grey shading) is zero if vertical mixing is double-diffusive but can be finite in the presence of mechanically generated turbulence.

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Radko, T.; Edwards, E.
Diapycnal Velocity in the Double-Diffusive Thermocline. *Fluids* **2016**, *1*, 25.
https://doi.org/10.3390/fluids1030025

**AMA Style**

Radko T, Edwards E.
Diapycnal Velocity in the Double-Diffusive Thermocline. *Fluids*. 2016; 1(3):25.
https://doi.org/10.3390/fluids1030025

**Chicago/Turabian Style**

Radko, Timour, and Erick Edwards.
2016. "Diapycnal Velocity in the Double-Diffusive Thermocline" *Fluids* 1, no. 3: 25.
https://doi.org/10.3390/fluids1030025