## 1. Introduction

Double diffusion is a type of convection that may occur when two constituents of a stably stratified fluid have opposing density stratification and different molecular diffusivities (Radko 2013). The classic example is oceanic double diffusion that can be active for certain temperature and salinity stratifications because thermal diffusivity is approximately 100 times larger than the diffusivity of salt (e.g., Schmitt 1994). One mode of double diffusion, known as diffusive convection, can arise when temperature and salinity both increase with depth. Consider, for example, a relatively cold and fresh water layer overlying a (more dense) relatively warm and salty layer (Turner and Stommel 1964). Between the two layers the diffusive thermal interface grows faster than the haline interface. This sets up the formation of gravitationally unstable diffusive boundary layers at the edges of the interface (Fig. 1), and when conditions are favorable, convection is triggered. The subsequent vertical fluxes of heat and salt across the interface are of primary interest, and a number of parameterizations for diffusive–convective fluxes have been developed (e.g., Turner 1965; Linden and Shirtcliffe 1978; Kelley 1990; Flanagan et al. 2013). Note that the heat and salt of the oceanic setting can be replaced by any two fluid constituents (i.e., a two-component system) that have different diffusivities, and double diffusion has been studied in a variety of other settings, including planetary interiors and stellar evolution (see, e.g., Radko 2013).

The Arctic Ocean, characterized by relatively cool and fresh water overlying relatively warm and salty water, presents a well-recognized example of diffusive convection at middepth in its water column (e.g., Neal et al. 1969; Neshyba et al. 1971; Padman and Dillon 1987; Timmermans et al. 2008; Guthrie et al. 2015). The process manifests itself as stacked layers, each of uniform temperature and salinity, separated by relatively thin high-gradient interfaces—a diffusive–convective staircase. Diffusive convection is an important mechanism for transporting heat from the deeper, warmer ocean layers toward the surface ocean and overlying sea ice (e.g., Polyakov et al. 2012; Carmack et al. 2015). Reliable parameterized estimates of these heat fluxes are needed where turbulence observations are sparse and logistically challenging to acquire. Planetary rotation, however, is usually not accounted for in heat flux parameterizations, although its influence may be important in certain settings (Kelley 1987; Carpenter and Timmermans 2014). In particular, Carpenter and Timmermans (2014) found that the influence of planetary rotation may limit double-diffusive heat fluxes in some regions of the Arctic Ocean where staircase interfaces are relatively thick. In this study, we explore the influence of Earth’s rotation on diffusive–convective heat fluxes. For direct comparison with the results of Carpenter and Timmermans (2014), who considered the Arctic setting in particular, in our analysis we use parameters consistent with those that characterize the Arctic Ocean’s diffusive–convective staircase.

Past studies have shown that planetary rotation can inhibit pure thermal convection and the associated heat fluxes (e.g., Chandrasekhar 1953; Niiler and Bisshopp 1965; Rossby 1969; King et al. 2012).This is explained as a result of the suppression of vertical motion through the Taylor–Proudman effect (Chandrasekhar 2013). Considering measurements from rotating thermal convection experiments, Kelley (1987) derived an empirical criterion to indicate when diffusive–convective heat fluxes may be affected by rotation. Carpenter and Timmermans (2014) reformulated Kelley’s condition in terms of the relative thicknesses of the Ekman layer and the thermal interface, and tested this via direct numerical simulation (DNS) of diffusive convection. Their simulations provided support for Kelley’s results, demonstrating that heat fluxes can be strongly inhibited by rotation when the thickness of the thermal interface is around 5 times the thickness of the Ekman boundary layer. A precise physical interpretation of this transition remains to be described.

In this study, we examine the linear stability properties of a double-diffusive system, with a focus on the diffusive–convective mode, to understand the transition to rotationally controlled heat transfer. Instability of a double-diffusive system was first shown via linear stability analysis (LSA) applied to a fluid layer characterized by linear background temperature and salinity stratification (Stern 1960; Veronis 1965; Nield 1967; Baines and Gill 1969; Pearlstein 1981). Compared to this system, diffusive–convective staircases, characterized by sharp interfaces in temperature and salinity that are bounded by two mixed layers (Fig. 1), have been shown to have considerably different linear stability properties (Carpenter et al. 2012b; Smyth and Carpenter 2019). Here, we extend the analysis of Carpenter et al. (2012b) to include the effects of planetary rotation in a LSA, and examine for what regime and how rotation influences diffusive convection. We show that physical insights into the effects of rotation on diffusive–convective heat fluxes can be gained through examination of the LSA solutions.

The paper is organized as follows. In the next section, we formulate and solve the LSA, showing consistency with the Kelley condition for the transition from the nonrotating to the rotationally controlled regime for diffusive–convective heat fluxes. We then examine the LSA solutions in the context of the vorticity balance to demonstrate the mechanism by which the influence of rotation inhibits the vertical transfer of properties. In section 3, we validate our LSA-based transition criterion using DNS, and summarize and discuss our results in section 4. While we explore a parameter regime relevant to diffusive convection in the Arctic Ocean, our approach and findings may be generalized to other diffusive–convective settings.

## 2. Linear stability analysis

### a. Formulation and solution

*f*plane (i.e., with a fixed rate of rotation). Let

*T*and

*S*be temperature and salinity, respectively, such that density

*ρ*may be written as

*ρ*

_{0},

*T*

_{0}, and

*S*

_{0}are reference values for density, temperature, and salinity, respectively, and thermal expansion coefficient

*α*and haline contraction coefficient

*β*are assumed to be constant. Conservation of momentum and mass are given by

*p*is pressure,

*ν*is molecular viscosity, and

*f*is the Coriolis parameter. Advection–diffusion equations for

*T*and

*S*are given by

*κ*

_{T}and

*κ*

_{S}are the molecular diffusivities of temperature and salinity, respectively.

The vertical structure of the

*h*

_{T}and the haline interfacial thickness

*h*

_{S}

*z*= 0 indicates that the derivative is calculated at the center of the interface (see Fig. 1) and | | indicates the absolute value. Temperature and salinity scales Δ

*T*and Δ

*S*are the temperature and salinity differences across the diffusive interface, respectively.

*h*

_{T}. Nondimensionalizing (8)–(11) yields (asterisks denote nondimensional variables):

*R*

_{ρ}=

*β*Δ

*S*/(

*α*Δ

*T*) are diffusivity and density ratios.

*r*> 1), and

**X**of length 4

*N*created by concatenating the Fourier coefficients

*N*× 4

*N*, and

_{I}, Ra

_{I},

*R*

_{ρ},

*r*, and

*H*. The complex growth rate

_{I}, Ra

_{I},

*R*

_{ρ},

*r*, and

*H*) by solving the eigenvalue problem (we take

*N*= 100). We take Pr = 6.25 and

*τ*= 0.01, which we use later in the DNS for comparison with Carpenter and Timmermans (2014).

### b. Stability diagrams

The stability diagrams of Fig. 2 show positive unstable growth rates, *R*_{ρ}–Ra_{I} plane, for fixed values of *r* = 2 and *H* = 10. In addition, the oscillation frequency *r* of thermal to haline interfacial thicknesses, and the dimensionless domain height *H*, are taken to be consistent with the observations and simulations of Sommer et al. (2013, 2014) and Carpenter et al. (2012a). The stability diagrams for nonrotating (Ek_{I} = ∞) and rotating (Ek_{I} = 0.1, 0.01) cases share some general features: For any fixed *R*_{ρ}, the system is stable at low Ra_{I}, then becomes oscillatory unstable (i.e., _{I}. That the system is oscillatory unstable at the stability boundary (*R*_{ρ} into two regions: for the nonrotating case (Fig. 2a) for example, one region for _{I}, and the other for _{I}. Carpenter et al. (2012b) showed that the two regions have unstable modes with different vertical structures: the region of smaller *R*_{ρ} (i.e., *R*_{ρ} (i.e., *R*_{ρ} region is the interfacial analog of the instability of linear profiles (e.g., Stern 1960; Baines and Gill 1969) acting within the interface center, and a convective-type instability will only grow for sufficiently large Ra_{I} such that viscosity and diffusion can be overcome; whereas the larger *R*_{ρ} region is associated with gravitationally unstable boundary layers and requires *r* > 1. It is this large *R*_{ρ} unstable region that is relevant for Arctic Ocean diffusive convection, since Arctic Ocean staircases typically have 2 < *R*_{ρ} < 10 (Shibley et al. 2017), and we expect this boundary layer unstable mode to trigger instability for staircases with and without the effects of rotation. We therefore confine our attention to the boundary layer modes throughout this paper.

With respect to the influence of rotation, it can be seen that the value of Ra_{I} at the stability boundary is significantly larger, and the growth rate at the same Ra_{I} is smaller, in the rotating (Figs. 2b,c) versus the nonrotating (Fig. 2a) case. This suggests that rotation inhibits the onset of diffusive convection and stabilizes the system. Finally, it is of note that nonrotating and rotating cases are indistinguishable for Ra_{I} > 10^{6}, indicating that for large Ra_{I}, weak rotation does not affect the linear stability properties of a staircase. Next, we examine how a critical interfacial Rayleigh number (characterizing the onset of instability) depends upon the strength of rotational effects, characterized by the interfacial Ekman number.

### c. Quantifying the effects of rotation

In quantifying the effects of rotation on diffusive–convective fluxes, we seek a critical interfacial Rayleigh number, _{I} for different combinations of *H*, *R*_{ρ}, and *r* (Fig. 3). It can be seen that the _{I} relationship is similar among each group of parameters: When the influence of rotation is sufficiently small (_{I}). In a rotationally controlled regime (_{I}. For each set of parameters considered, a least squares fit yields _{I} relationship follows the −4/3 power law.

We note that for rotating thermal convection with linear background temperature profiles, the most unstable normal mode of each perturbation variable consists of only the fundamental *n* = 1 Fourier component (i.e., the perturbation is strongest at the center of the interface; see Chandrasekhar 2013). In the limit Ek_{I} → 0, Chandrasekhar (2013) found analytically that the asymptotic dependence of _{I} yields the −4/3 power law. In the setting described here however, the most unstable normal mode consists of more than one vertical Fourier component, and the eigenvalue problem cannot be easily simplified. Still, the same dependence of _{I} is found, suggesting that the −4/3 power law may hold for a range of background density profiles under the influence of rotation, and not just for linear profiles. We can also compute the critical horizontal wavenumber

We can calculate a value of Ek_{I} characterizing the transition between nonrotating and rotationally controlled regimes by intersecting the fit in the rotationally controlled regime with the nonrotating value of *r*, *R*_{ρ}, and *H* (Fig. 4). Recall that

Kelley (1987) analyzed heat flux measurements from the rotating thermal convection experiments of Rossby (1969) in the context of a Taylor number *r* and *R*_{ρ}. However, we note that the salinity stratification does affect stability properties of the staircase, as *R*_{ρ} (Fig. 3b) and smaller *r* (Fig. 3c). We return to a comparison of Kelley’s condition and our LSA-based condition for transition to rotationally controlled convection in section 3.

The LSA solutions can further provide physical insight into how rotation inhibits convection, which we describe next.

### d. Physical process of stabilization by rotation

*y*direction

*y*component of the horizontal vorticity

The vertical structures of perturbation quantities _{I}, *R*_{ρ}, *H*, and *r*, are shown in Figs. 5a,c. The density perturbation *ρ*_{0}*α*Δ*T* and first appears in (14). The mixed layer is in the upper part of the domain (only the top half of the domain is shown), and we approximate the unstable boundary layer to be the region below this whose upper and lower boundaries have vertical density gradients of 0.01 and 0, respectively (indicated by the gray bar in Fig. 5); the stable interface core is below the unstable boundary layer. The maximum magnitude of each nondimensional perturbation variable is set to 3; phase relationships between perturbation variables are not retained. The corresponding linear growth rates

In both rotating and nonrotating cases (Figs. 5a,c, respectively) we find that the vertical velocity perturbation

*y*component of the horizontal vorticity perturbation

*n*is small); the first term on the rhs of (27) is negligibly small, and

The growth of

In summary, the growth of horizontal vorticity (

## 3. Direct numerical simulations

In this section, we perform two-dimensional (2D) DNS as well as a number of 3D DNS to explore the applicability of the linear stability transition criterion to finite amplitude turbulent convection. Beyond the transition, we expect the suppression of convection by rotation to yield smaller diffusive–convective heat fluxes compared to the nonrotating case. Although double-diffusive convection is three-dimensional, 2D experiments have been shown to accurately simulate heat fluxes compared to their 3D counterparts (Flanagan et al. 2013; Hieronymus and Carpenter 2016). Our simulations solve the Boussinesq momentum equations, continuity equation and advection–diffusion equations assuming a linear equation of state. For the 2D experiments, variations in the *y* direction are set to 0. Periodic boundary conditions are applied at the sidewalls of the domain. Periodic boundary conditions are also applied at the bottom and top boundaries, except with restoring such that the mean temperature and salinity differences across the domain do not change. The domain width to height ratio is set to 2 for 2D experiments and 1 or 0.5 for 3D experiments (we refer to this later); and the resolution is chosen such that the grid size is smaller than or approximately equal to twice the Batchelor scale given by *ϵ* is the volume averaged kinetic energy dissipation rate (Carpenter et al. 2012a; Carpenter and Timmermans 2014); this ensures that the molecular diffusion of salt is properly simulated. For our simulations, we use the DEDALUS package which utilizes a spectral numerical method (Burns et al. 2020). Carpenter and Timmermans (2014) used a different numerical code (see Winters et al. 2004), and only conducted 2D experiments.

To begin, we perform a set of 2D experiments using the same parameters as the DNS experiments of Carpenter and Timmermans (2014) for comparison with their results. While for the LSA, we used interfacial Ekman and Rayleigh numbers (i.e., that depended on the thermal interfacial thickness), for the DNS we use Rayleigh and Ekman numbers that depend on the domain height *h*_{T} varies over the course of a simulation. Following Carpenter and Timmermans (2014), we choose *R*_{ρ} = (2, 5), *R*_{ρ} = 5 and ^{−3}. For these experiments, we use a 1152 × 576 grid in the horizontal and vertical, respectively. We conduct an additional set of experiments with Ra = 1 × 10^{8} and a broader range of *R*_{ρ} = (3, 5, 7). A resolution of 2880 × 1440 is used for the *R*_{ρ} = 3 simulation and 2048 × 1024 for *R*_{ρ} = (5, 7). Note that in the Arctic Ocean, Ra ≈ 10^{8}–10^{9}, and *O*(1–5) m (Shibley et al. 2017), so that Ek ≈ 10^{−2}– 4 × 10^{−4}, taking *δ*_{E} ≈ 10 cm for laminar flows in the Arctic Ocean. Therefore, our simulations may be at least generally representative of the Arctic diffusive–convective staircases.

Finally, we perform a number of complimentary 3D experiments for *R*_{ρ} = 5, Ra = 6.5 × 10^{5}, and *R*_{ρ} = 5, Ra = 3.3 × 10^{6}, and

All experiments have Pr = 6.25 and *τ* = 0.01 (as used in the LSA). These values differ from those characterizing the Arctic Ocean (13 and 0.005, respectively) for which the simulations would be too computationally intensive for large Ra (1 × 10^{8}). While the simulated heat fluxes will differ from the Arctic setting, the choice of parameters will not affect validation of the LSA results.

All numerical experiments are initialized with ^{8} and Ek = 1 × 10^{−4}, and refer to this later). Representative snapshots of the density field for a nonrotating and rotating simulation are shown in Fig. 6. The effect of rotation on the structure of double diffusive convection is qualitatively clear in these examples: when rotation is absent (Fig. 6a), there appears to be a single large convection cell on either side of the diffusive core. Under the influence of rotation (Fig. 6b), however, a number of distinct plumes are observed rising and sinking from the interface. It is clear that the most unstable mode has a larger horizontal wavenumber in the presence of rotation.

*F*

_{H}, and estimates of heat fluxes will allow us to test whether the nonrotating and rotating regimes found in the LSA apply. Here we calculate heat fluxes across the central isotherms of interfaces in the same manner as Winters and D’Asaro (1996) and Carpenter and Timmermans (2014). This diascalar heat flux (in dimensional form) is estimated by

*c*

_{p}is the specific heat capacity of water,

**n**is the unit vector normal to the central isotherm,

*ds*is the length of integration along the central isotherm, and

*A*is the horizontal width of the simulated domain. The central isotherm temperature is taken to be the average temperature of the entire system. For the 3D experiments,

*F*

_{H}is first calculated in each vertical (

*x*–

*z*direction) plane and averaged along the

*y*direction. Note that considering averages of convective heat fluxes in the mixed layer returns values that only differ by about 1% from estimates inferred from (29) (see Carpenter and Timmermans 2014), although the convective heat fluxes have much larger temporal variation. Finally, it is useful to introduce the Nusselt number (Nu) which is a measure of the diascalar heat flux relative to the value for pure conduction:

*T*is the temperature jump across the interface (equivalent to the temperature difference between the top and bottom of the domain).

Previous studies have suggested that when

Average Nu versus Ek^{−1/2} for all experiments are plotted in Fig. 7; for comparison, the corresponding simulation results of Carpenter and Timmermans (2014) are also shown (asterisks). For the three experiments with Ra = 1 × 10^{8} and Ek = 1 × 10^{−4}, values of Nu did not reach quasi-equilibrium and were continuing to decrease at the end of the simulations. Open circles are used to indicate the final values of Nu attained for these experiments (Fig. 7), which represent only upper bound values. Two transition lines, indicating the Kelley and LSA transition conditions, are also plotted. The Nu remains largely unchanged as Ek is decreased from the nonrotating case (on the left side of the plot) toward Ek = 0.01 (i.e., Ek^{−1/2} = 10), in agreement with the LSA that weak rotation does not change the stability properties of the diffusive boundary layer. When Ek is reduced further (i.e., enhanced rotation), the results approach the two transition lines. Near this point, there is a relatively sharp increase in Nu, which has been suggested to be caused by nonlinear Ekman transport in the thermal boundary layer forced by cyclonic circulation of convective plumes (e.g., Rossby 1969; Julien et al. 1996; King et al. 2012). After the transition lines are crossed, further reducing Ek completely shuts down convection, until heat transport is only pure conduction (Nu = 1).

To the left of the transition, the simulated Nu agrees reasonably well between 2D and 3D experiments (Fig. 7). The 3D-simulated Nu is generally larger than the 2D value, in part due to the reduced width to height ratio of the 3D domain. For the parameters of the simulation, we find that reducing the 2D domain width to height ratio by a factor of 1/2 or 1/4 yields an increase in Nu by ~10%. After accounting for this effect, we find the 3D Nu to be ~10% larger than the 2D values, which may result from better resolving the convective cells. Closer to the transition, the simulated Nu is found to differ between the 2D and 3D experiments; after convection has completely shut down (Nu = 1) in the 2D experiments, convection remains active in the 3D experiments (Nu > 1). This may be due to the fact that the 3D-simulated Nu is generally larger than the 2D values (i.e., stronger convection) for the reasons noted above. Further reducing Ek yields Nu = 1 for both the 2D and 3D experiments.

We find the overall Nu–Ek^{−1/2} relationship resembles the study of Carpenter and Timmermans (2014), which is now examined in a larger parameter space. Note the simulated Nu for *R*_{ρ} = 5, Ra = 3.3 × 10^{6}, and Ek = 1 × 10^{−3} in this study is different from that of Carpenter and Timmermans (2014). This result is close to the transition lines and may be sensitive to initial random noise and numerical methods. We find that the LSA-based condition gives a good estimation of the transition to a rotationally controlled regime, except when *R*_{ρ} = 7 and Ra = 1 × 10^{8}. For this pair of parameters, Nu continues to increase after the LSA condition is crossed and does not decrease until the Kelley condition is crossed. This may be partly due to the fact that larger *R*_{ρ} and smaller *H* correspond to somewhat larger

## 4. Summary and discussion

We have investigated how the linear stability properties, and heat fluxes, of a diffusive–convective staircase are affected by planetary rotation. Via LSA we have shown that rotation stabilizes convection by increasing the critical interfacial Rayleigh number, *H*, *R*_{ρ}, and *r*. A transition from nonrotating to rotating diffusive convection is shown to occur for

Despite the generally good agreement between the Kelley and LSA transition criteria, there is no reason to expect they should be equal given the approximations and assumptions that have been made. For the LSA-based condition, we describe the background temperature and salinity profiles using error functions, which are an idealization of the actual structure of diffusive–convective interfaces. This is important because the stability boundary has been shown to be sensitive to the exact form of background density stratification (Carpenter et al. 2012b). Further, *π*. While this is only an approximate criterion; for the LSA to be valid, *π* if neglecting the time evolution of the background profiles by diffusion (Smyth and Carpenter 2019). Different thresholds of

We have then investigated the physical mechanisms for the suppression of diffusive convection by rotation through an examination of the horizontal vorticity budget. In the absence of rotation, the horizontal density gradient effectively generates horizontal vorticity and drives vertical motion in the unstable boundary layer. In the presence of strong rotation, however, the tilting of planetary vorticity by vertical shear of the horizontal velocity strongly counteracts baroclinic production. As a result, the vertical motion of convection is strongly inhibited. Previous studies have usually referred to the Taylor–Proudman theorem as the mechanism for the stabilizing effects of rotation (Chandrasekhar 1953, 2013; Niiler and Bisshopp 1965). The current analysis, however, offers a new perspective in understanding the effect of rotation.

To validate our LSA-based transition criterion, we have conducted a series of 2D DNS experiments as well as a number of complimentary 3D experiments using a range of parameters consistent with past experiments and broadly applicable to diffusive–convective staircases in the Arctic Ocean. For sufficiently large Ek (

Although we have used parameters in the LSA and DNS that resemble those of the Arctic Ocean’s diffusive–convective staircases, the analysis method and revealed physical processes may be generalized to other diffusive–convective settings. The LSA can serve as an important tool to understand the effects of rotation on diffusive convection, which is especially useful for those systems characterized by large Rayleigh numbers, as it will require exceptional computing resources to conduct direct numerical simulations.

## Acknowledgments

Y. Liang acknowledges discussions with Daniel Lecoanet, Zhongtian Zhang, and Nicole Shibley. M.-L. Timmermans acknowledges support from the National Science Foundation Division of Polar Programs under Award 1950077. J. R. Carpenter acknowledges the Helmholtz Association funding through the POF IV programme. We acknowledge high-performance computing support from the Center for Research Computing at Yale.

## Data availability statement

Output for all DNS experiments is being deposited in the Dryad Digital Repository (https://doi.org/10.5061/dryad.8sf7m0cmp). Code for the linear stability analysis is available upon request.

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