1. Introduction
A fundamental result in the study of ocean mixing is that certain configurations of temperature T and salinity S are unstable to small perturbations, despite the fact that the water column is gravitationally stable. This instability is due to the differing rates of diffusion of T and S, and has come to be known as double-diffusive (DD) instability. The first stability analysis to demonstrate the basic mechanism of DD was performed by Stern (1960) using linear profiles of T and S, bounded above and below (see Fig. 1a, where T and S are plotted in density units). This analysis was subsequently extended by Veronis (1965), Nield (1967), and Baines and Gill (1969), keeping the same linear profiles. These studies found that unstable motions may develop from small perturbations of the T–S profiles, as long as at least one component is in a gravitationally unstable configuration, thus providing the energy source required to drive the instability. Two different cases exist depending on whether T (the faster diffusing component) or S is destabilizing: (i) the salt-fingering case where warm, salty water overlies cool, fresh water or (ii) the diffusive convection (DC) case with cool, fresh water over warm, salty water (Fig. 1).
(a) Linear profiles of the background 
Citation: Journal of Physical Oceanography 42, 5; 10.1175/JPO-D-11-0118.1
Conditions favoring DD are found over vast areas of the oceans (You 2002; Kelley et al. 2003), as well as in lakes (Hoare 1966; Schmid et al. 2004; Sánchez and Roget 2007; Boehrer et al. 2009; Schmid et al. 2010), and are often accompanied by a thermohaline staircase structure in the T and S fields. These staircases consist of a sequence of sharp, high-gradient interfaces of T and S surrounded by nearly homogeneous mixed layers. In this study, we shall focus on the stability properties of the “diffusive interface”: that is, with cool, fresh water overlying warm, salty water, as shown in Fig. 1b (for a recent examination of the salt-finger interface, see Smyth and Kimura 2007). We show that the presence of the staircase considerably modifies the linear stability properties of the water column from the linear representation. Whereas the stability analysis of the linear T–S profiles has been crucial in our understanding of the basic mechanism of instability, and recently in the process of staircase formation (Noguchi and Niino 2010), the stability analysis of the diffusive interface constitutes an important step in understanding the processes governing staircase evolution and maintenance. To our knowledge, the linear stability properties of the diffusive interface have not been studied previously.
Despite the clear mechanism of instability that has developed from analysis of the linear profiles, which consists of an exponentially growing oscillation of the water column, a different mechanism is often invoked for assessing the stability of a diffusive interface (Linden and Shirtcliffe 1978; Newell 1984; Padman and Dillon 1987, 1989; Worster 2004). In this case, the basic instability mechanism is thought to result from the formation of gravitationally unstable layers on either side of the interface (Fig. 1b). Because the T interface diffuses more rapidly than the S interface, molecular diffusion will tend to produce a relatively thicker T interface (i.e., hT > hS; Fig. 1b). Because the T profile is gravitationally unstable, this leads to a density profile that exhibits regions of gravitationally unstable fluid, which we will refer to as the diffusive boundary layer. It is the convective-type (nonoscillatory) instability of this boundary layer that is thought to be responsible for the maintenance of turbulence and enhanced mixing that occurs within the mixed layers of a thermohaline staircase. In keeping with the purely convective-type breakdown of the boundary layer, it has been assumed that a critical Rayleigh number that is characteristic of the boundary layer, of the order of 103, must be exceeded for the boundary layers to break away from the interface (Linden and Shirtcliffe 1978).
The stability properties of the diffusive interface, as opposed to the linear profiles, are fundamental to understanding the enhanced fluxes of heat and salt present in thermohaline staircases. The microstructure profiles of T and S taken in the thermohaline staircase of Lake Kivu by T. Sommer et al. (2012, unpublished manuscript) clearly illustrate the presence of diffusive boundary layers (an example is shown in Fig. 2). The present study is largely motivated by these observations and the need to provide a more relevant theoretical understanding of the conditions limiting the growth of the boundary layer. The results of this study provide a basis for developing simple phenomenological models for DC (Linden and Shirtcliffe 1978; Newell 1984; Fernando 1989; Kelley 1990; Worster 2004), as well as in the interpretation of laboratory (Shirtcliffe 1973; Marmorino and Caldwell 1976; Newell 1984; Fernando 1989) and field measurements of the diffusive interface and its stability (Padman and Dillon 1987, 1989; Sánchez and Roget 2007; T. Sommer et al. 2012, unpublished manuscript).
Example of interface profiles measured with a high-resolution microstructure profiler in the DD thermohaline staircase in Lake Kivu (for details of the measurements, see Sommer et al. 2012). The T, S, and ρ profiles are plotted in density units, as in Fig. 1. A gravitationally unstable boundary layer in the ρ profile, shown by the gray stripe, can be seen on the upper side of the T and S interfaces, which have different thicknesses.
Citation: Journal of Physical Oceanography 42, 5; 10.1175/JPO-D-11-0118.1
After formulating the basic linear stability problem and its solution in section 2, we present the results of the stability analysis (section 3). These results shed light on the assumption of a purely convective-type instability of the boundary layer. In particular, we find that the first mode to become unstable is oscillatory and is more akin to the DC modes found in linear gradients than to a purely convective mode. However, the assumption of steady T and S profiles that is used in the linear analysis is then tested in section 4 using direct numerical simulations (DNS) of the interface breakdown. The results of the DNS show that time-dependent effects are important for the type of unstable mode that develops (i.e., DC or convective), and the conditions under which the nonlinear breakdown of the interface takes place can exceed the linear predictions. By accounting for this diffusive growth of the interface in time, we propose that the dominant mode of instability for the diffusive interface is in fact of the convective type and is concentrated in the boundary layers. In section 5, we examine the applicability of a critical boundary layer Rayleigh number to describe the onset of instability and breakdown. Conclusions are stated in the final section.
2. Formulation and solution of the linear stability problem
a. Formulation
It is important to note that in arriving at this system of equations we have made a quasi-steady assumption for the evolution of the background profiles. This is equivalent to neglecting the molecular diffusion of these profiles in time and is necessary in order to carry out a standard stability analysis. This assumption is not necessary for the stability of the linear profiles (because they are steady solutions to the diffusion equation), and we will test the accuracy of this assumption using the simulations. In addition, we have limited the analysis to two dimensions only for simplicity. The laboratory experiment discussed in Linden and Shirtcliffe (1978) indicates that the instabilities at the diffusive interface can take three-dimensional forms, and a more complete analysis must be left for future study.
The equations in (6) describe an eigenvalue problem for the eigenvalue σ and the eigenfunctions 
In nondimensional terms, the vertical boundaries extend from ζ = −H/2 to ζ = H/2, where H ≡ Lz/L and L remains to be defined for the particular problem of interest. The set of equations in (7) is subject to the following conditions on these boundaries:
b. Numerical solution method
3. Linear stability properties
In this section, we present the results of the stability analysis that was formulated in the previous section. Before discussing new results for the diffusive interface, we shall briefly review some important results from the analysis of the linear profiles.
a. Linear profiles
The linear profiles in (12) considerably simplify the system of equations in (7), and an analytical solution is possible. The dispersion relation governing the eigenvalues can be found in Baines and Gill (1969), and the reader is referred to this work as well as Turner (1973) and Linden (2000) for further details.
The results of the stability analysis are most conveniently displayed on what we shall refer to as a stability diagram, which is shown in Fig. 3. It contours the Rρ–RaH plane with the positive unstable growth rates σr in thin contours and color filling and the frequency of oscillation σi in thick contours. At each point in the unstable region (i.e., for a fixed RaH and Rρ), there is generally a continuous band of unstable wavenumbers present. However, we plot only the mode with the largest growth rate, because it is this “most unstable mode” that is expected to dominate the initial linear growth phase and possibly the nonlinear transition to turbulent convection.
Stability diagram of the classical linear profiles. Colored contours are of the growth rate σr on the logarithmic scale shown, whereas the thick dark contour lines are of frequency σi, which are also on a logarithmic scale. The DC unstable modes with σi > 0 occur in the region with thick dark contour lines, and convective-type unstable modes are found in the higher RaH and Rρ < 1 region as labeled. No instability is present when Rρ > 1.16, no matter how large RaH.
Citation: Journal of Physical Oceanography 42, 5; 10.1175/JPO-D-11-0118.1
Three different regions of the diagram in Fig. 3 may be identified:
A stable region exists for either sufficiently low RaH or sufficiently high Rρ. The lowest RaH that may be unstable occurs for Rρ = 0, in which no gravitationally stabilizing S stratification is present, and we recover the classical case of pure thermal convection in a linear gradient (Linden 2000; Kundu et al. 2004). This has the well-known solution that instability is triggered once a critical RaH,cr = 27π4/4 = 658 is exceeded (though this value depends on the form of the boundary conditions). The resulting unstable modes are of the convective type and form the second region of the diagram.
In the large RaH, small Rρ portion of the diagram, instability is due to a convective-type instability; the unstable modes are nonoscillatory with σi = 0. The physical mechanism of growth for this convective type of instability is straightforward: the perturbations are able to extract the potential energy available in the gravitationally unstable (top heavy) density field to directly increase their amplitude over time, once the stabilizing effects of viscosity and diffusion are overcome. These convective-type unstable modes exist only when the density field is gravitationally unstable and so are limited to the region 0 ≤ Rρ < 1, assuming that RaH is sufficiently large.
The third portion of the diagram consists of DC-type unstable modes that have an oscillating growth with σi > 0 (region of thick dark contour lines in Fig. 3). These DC-type modes are the first to become unstable as RaH is increased and are the only type of instability that is possible when the density stratification is gravitationally stable (i.e., Rρ > 1). The physical mechanism of instability can be understood as follows: After any small displacement from its equilibrium level, a fluid parcel will find itself in a different T–S state than it originated from. Because of the differing rates of molecular diffusion, the parcel exchanges T more rapidly than S with its surrounding environment. Because S is stably stratified, the parcel either sinks or rises back to its original level, but it overshoots, because it carries with it a density anomaly. The density anomaly grows because the parcel is now exchanging T in more or less buoyant surroundings. The oscillation can therefore grow in time. However, the DC instability is very restricted in that it cannot operate if Rρ is too large, and the flow is found to be stable for Rρ > (Pr + 1)/(Pr + τ) ≈ 1 + Pr−1 ≈ 1.16. Also, the transition between the DC-type modes and the convective-type modes occurs at Rρ → 1 as RaH → ∞.
Because most geophysical observations of DC unstable profiles have a mean Rρ > 1.16 (Padman and Dillon 1987; Kelley et al. 2003; Schmid et al. 2004; Timmermans et al. 2008; Schmid et al. 2010), it is unlikely that the instability of the linear profiles plays a role in these systems. Therefore, in order to assess DC processes in thermohaline staircases, we now turn to the stability of the diffusive interface.
b. The diffusive interface
Stability diagrams using the tanh interface model are shown in Fig. 4 for various r. Beginning with r = 1 (Fig. 4a), the profiles of T and S have the same interface thicknesses. In this case, no boundary layers exist, and for Rρ > 1 the background density field is gravitationally stable for all ζ. We should intuitively expect that the stability diagram show similar behavior to the linear gradient results in Fig. 3. This is because approximately linear gradients are present within the interfaces and only the length scale has changed, along with the proximity of the vertical boundaries. In comparing Figs. 4a and 3, this is indeed what is observed. In both cases, there is a stable region at low Ra, and the cutoff in Rρ occurs at the same value of 1.16, above which the profiles are stable.
Stability properties of the diffusive interface at various r for the tanh interface model with H = Lz/hT = 10, Pr = 6, and τ = 0.01. All notations as in Fig. 3. (c) The vertical pink strip indicates the location of the plot in Fig. 5.
Citation: Journal of Physical Oceanography 42, 5; 10.1175/JPO-D-11-0118.1
As r is increased from unity, diffusive boundary layers develop above and below the gravitationally stable interface center (Fig. 1b). These boundary layers can be seen to modify the stability properties by destabilizing the profiles at larger Rρ values. As is to be expected, the larger r becomes—and therefore the thicker and more gravitationally unstable the boundary layers become—the lower is the RaI required to achieve instability. Despite the changes that occur in the stability properties once diffusive boundary layers are present (i.e., for r > 1), each of the panels in Fig. 4 shows a transition from stability to DC-type unstable modes to convective-type unstable modes as RaI is increased. The DC-type modes are always found to be adjacent to the stability boundary; however, the width of the DC-type unstable region is found to shrink as r increases, and the boundary layers grow in size and strength. At r = 10 (not shown), the expected upper limit, the DC-type unstable region is still present but confined to a very small region close to the stability boundary.
To emphasize the distinction between the convective-type modes and the DC-type unstable modes, we plot in Fig. 5 a cut along Rρ = 3 while varying RaI. This represents the cut in Fig. 4c indicated in pink with r = 1.5. The plot shows the growth rate σr, frequency σi, and the wavenumber of maximum growth αmax for the most amplified mode. A jump in αmax and σi are present at the transition between the DC- and convective-type unstable modes, as well as a change in the trend of σr. It should be noted that the transition from a DC- to convective-type unstable mode occurs because the growth rate of the convective mode becomes larger than that of the DC mode, not because the mode ceases to become unstable.
Cut along Rρ = 3.0 for varying RaI from Fig. 4c with r = 1.5 showing σr, σi, and αmax. The three distinct regions of stable, DC unstable, and convective unstable modes can clearly be seen with a jump in αmax and σi.
Citation: Journal of Physical Oceanography 42, 5; 10.1175/JPO-D-11-0118.1
Another feature that is apparent in Fig. 4 is the similarity, in all panels, of the Rρ < 1 portion of the diagrams. This suggests that, in this region, the stability of the profiles is governed by the entire interface, opposed to the boundary layers at larger Rρ, because no boundary layers are present for r = 1. This behavior can be seen by comparing the eigenfunctions from these regions of the diagram because the eigenfunctions give the vertical structure of the unstable mode. In Fig. 6, we fix r = 1.25 and RaI = 104 but vary Rρ in order to cross from an interface-dominated instability at Rρ = 0.5 to boundary layer instabilities at Rρ = 1.5. This can be seen in the amplitudes of both the vertical velocity and temperature perturbation eigenfunctions, 
(b),(c) Amplitudes of the eigenfunctions 
Citation: Journal of Physical Oceanography 42, 5; 10.1175/JPO-D-11-0118.1
We now compare these results for the tanh interface to the same analysis using the erf interface model. Although the two different models show the same qualitative behavior, there are differences in the location of the stability boundary. As an example, we plot the RaI at marginal stability for two fixed values of Rρ for various r in Fig. 7. This shows that the exact form of the profiles is important in determining the stability boundary, particularly when r is close to unity. However, the stability diagrams (as in Fig. 4) show the same structure in each case, with both interface-centered and boundary layer–centered regions and DC unstable modes composing the stability boundary.
Location of the stability boundary in the r–RaI plane for Rρ = 2 and 5, comparing the tanh interface with the erf interface.
Citation: Journal of Physical Oceanography 42, 5; 10.1175/JPO-D-11-0118.1
4. Simulating interface instability
In our linear stability analysis of the diffusive interface in the previous section we identified two principle modes of instability: the oscillating DC type and the purely convective type. Furthermore, depending primarily on the density ratio Rρ, these modes can grow from within the interface or the boundary layers. In this section, we test these predictions using DNS. In particular, the DNS allows us to examine the instability of diffusive interfaces that are growing in time due to molecular diffusion. This time dependence was not explicitly accounted for in the linear stability analysis. A central question that we pursue is whether the boundary layer DC unstable modes that compose the stability boundary at r > 1 are significant in the breakdown of the boundary layers when a time-dependent interface is considered.
a. Description of the simulations
The DNS is performed using the code originally described in Winters et al. (2004), which has been modified to carry a second scalar by Smyth et al. (2005). The second scalar field is resolved on a grid that has a spacing twice as fine as the other fields and so is especially suitable to study the low τ that is present in the heat–salt system. The addition of the second scalar field, however, is restricted only to three-dimensional rectangular domains. We therefore choose a relatively small scale for the third dimension Ly—compared to the horizontal Lx and vertical Lz—scales in testing the two-dimensional linear predictions. All of the parameters chosen in the DNS are summarized in Table 1.
Values of various important parameters for the simulations performed. The gridpoint resolution listed for the simulations {Nx, Ny, Nz} is that of the S field. The T, velocity, and pressure fields are resolved on a grid with half this resolution. In the case of the nonstationary simulation (V), the dimensionless parameters listed are evaluated using the initial conditions. In all cases, Pr = 6, τ = 0.01, and Ly/hT = 0.66. Also included is the total number of waves (in the horizontal x direction) predicted in the domain n. Here, Lx and Nx have been doubled for simulation II to ensure two wavelengths of the most unstable mode are possible. Note that the value of H in these simulations is different from that of Fig. 4, and they cannot be directly compared.
Of the five simulations that were performed, four are used to directly test the linear predictions by “turning off” the molecular diffusion of the background T–S profiles (simulations I–IV in Table 1). In each of these cases, which will be referred to as “stationary,” there is a well-defined σ and αmax that can be compared with the linear predictions. It is also of interest to observe the finite-amplitude motions of these instabilities before proceeding to the time-dependent nonstationary case where molecular diffusion is acting on the background profiles.
Each simulation is initialized with background 
b. Testing the linear predictions: The stationary interface
As an initial test of the linear analysis, we have chosen four simulations (I–IV in Table 1) that are predicted to produce a stable mode (I), a DC-type unstable boundary layer mode (II), a convective-type unstable boundary layer mode (III), and a DC-type unstable interface mode (IV).
The growth of the perturbations in time is best seen by plotting the volume-averaged kinetic energy. In the case of simulation I, the kinetic energy simply decayed in time (not shown). The other simulations, II–IV, are shown in Figs. 8a–c, respectively. Because the vertical axis is a logarithmic scale, a strictly exponentially growing perturbation will appear as a straight line, as is predicted by the linear theory. This growth rate prediction (dashed lines) is found to be in agreement with the DNS results, despite the fact that the growth rates each differ by nearly an order of magnitude in each case. Oscillations in the kinetic energy are observed only in the DC-type unstable modes (simulations II and IV in Figs. 8a,c), and the predicted period of oscillation, given by the scale indicated by the arrows, is also in agreement with the simulations.
Three simulations (II–IV) designed to test the linear predictions. These correspond to (a) a DC unstable boundary layer mode (II), (b) a convective unstable boundary layer mode (III), and (d) a DC unstable interface mode (IV). The time evolution of the volume-averaged kinetic energy is shown by the solid lines. The slope of the curves is a measure of the growth rate σr and can be compared with the predicted σr from the linear theory shown as dashed lines. The time interval indicated by arrows in (a),(c) indicates the predicted oscillation period given by 4π/σi (the kinetic energy has twice the oscillation frequency of other fields). A time scale of hT2/κT has been used to nondimensionalize t.
Citation: Journal of Physical Oceanography 42, 5; 10.1175/JPO-D-11-0118.1
It is also possible to test the linear predictions of whether the instability will be centered in the boundary layers or the interface. This can be seen in Figs. 9a,b, where contours of the T field are shown for simulations III and IV once the instability has developed a finite amplitude. In the case of Fig. 9a, the central core of the interface is left virtually undisturbed, and the displacements are focused in the boundary layers at the interface edges. In contrast, the predicted DC interface-centered instability of Fig. 9b shows displacements that are focused within the interface itself. In this case, the interface performs exponentially growing standing oscillations.
The T field at the final times of simulations (a) III and (b) IV. The contours are equally spaced between −0.45ΔT and 0.45ΔT. The wavelengths (2πhT/αmax) predicted by linear theory are shown by the length of the horizontal scales. In the case of (b), only part of the domain is plotted.
Citation: Journal of Physical Oceanography 42, 5; 10.1175/JPO-D-11-0118.1
Because it is necessary to choose a finite horizontal domain size Lx in the DNS, it is not possible to represent a continuous range of α. Instead, the periodic boundary conditions allow only discrete values of α = 2πnhT/Lx, where n = 1, 2, … is the number of waves in the domain. The wavenumber that is expected to emerge from the DNS is that with the largest growth rate, and this is not necessarily the wavenumber of maximum growth αmax. In Table 1, the total number of waves n predicted by the linear theory is listed. For each of the T fields plotted in Figs. 9a,b, the linear predictions agree with the number of waves observed. The wavelengths predicted by the theory, once corrected for the periodic boundary conditions, correspond to the length of the scales shown in Figs. 9a,b.
c. Breakdown of the time-dependent diffusive interface
Trajectory of the interface for simulation V in the r–RaI plane (thick dark line with arrows) up to the time of interface breakdown at t = tb (to be defined more precisely in the text below). The changing position of the stability boundary is given by the dashed line initially and the thin solid line at t = tb.
Citation: Journal of Physical Oceanography 42, 5; 10.1175/JPO-D-11-0118.1
At the initial time t = 0, the interface has a position on the diagram corresponding to the initial values of r and RaI and is located in the stable region (Fig. 10). The position of this boundary, however, is changing in time because of its dependence on H(t). At a later time, both r and RaI have increased. The location of the interface therefore moves upward and rightward on the trajectory shown in Fig. 10, whereas the position of the stability boundary moves away as H decreases. During the simulation the interface crosses into the unstable portion of the diagram, thus initiating instability. A complete breakdown of the boundary layers occurs at the final position shown in Fig. 10.
The growth and eventual breakdown of the instability on the nonstationary interface may be seen by looking at a plot of the time evolution of hT(t). This is shown in Fig. 11a, where we have normalized by the initial thickness h0 = hT(0). As the instability develops, a range of different hT are present in the domain at a given t [note that hT is measured using Eq. (14) at each x and y in the domain]. We therefore plot the mean, as well as the 10th and 90th percentiles of hT, given by the gray shading. At early times, these curves all fall on one another as hT grows by molecular diffusion, in accordance with (17). The interface enters the unstable region predicted by the linear theory soon after the simulation begins, indicated by the vertical gray strip denoting the time spent inside the DC unstable region. The interface does not experience a growth in the kinetic energy during this time, which can only be seen well after the interface enters the convective unstable region of the diagram (right of the vertical gray strip in Fig. 11b). In other words, the kinetic energy only starts to grow in the convective-type region. The increase in kinetic energy indicates an increasing growth rate over time and eventually saturates after the boundary layers have broken away. The time of the interface breakdown tb is defined as the time of the greatest mean hT, which is indicated by the vertical dashed line at t ≈ 2.
Time evolution of parameters in the nonstationary simulation (V) as a function of time. The vertical gray strip indicates the time interval of DC instability with stability at earlier times and convective instability at later times. The time at which σr = 2π (i.e., when the instability growth rate equals the growth rate of the T interface) is indicated by the vertical solid line. (a) The T interface thickness hT, normalized by its initial value h0, with the upper and lower lines with gray shading indicating the 90th and 10th percentiles found within the domain and the central line indicating the mean. (b) Volume-averaged kinetic energy. (c) Rabl. In (a)–(c), the time axis has been nondimensionalized by 103κT/Lz2, and the vertical dashed line indicates the time of peak hT, which is denoted t = tb ≈ 2.
Citation: Journal of Physical Oceanography 42, 5; 10.1175/JPO-D-11-0118.1
The instability can be seen in the T field at the time of breakdown (t = tb), where a series of plumes have formed in the boundary layers and are in the process of detaching from the interface to be mixed in the upper and lower layers (Fig. 12). Note that the central core of the interface is left undisturbed, because the instability is centered in the boundary layers, as expected.
The T field for the entire domain in simulation V at t = tb, with contour intervals as in Fig. 9.
Citation: Journal of Physical Oceanography 42, 5; 10.1175/JPO-D-11-0118.1
The results of the nonstationary simulation show that the time-dependent growth of the interface is significant in determining the development of the instability. In the case of simulation V, the interface moves rapidly through the DC-type unstable region composing the stability boundary, which does not contribute to the eventual mode of interface breakdown. We can understand this behavior by requiring that the instability growth rate exceed that of the interface. This predicts that the breakdown is of the convective type and is concentrated in the boundary layers, in agreement with the interpretation of previous investigators (Linden and Shirtcliffe 1978; Newell 1984; Worster 2004). We next investigate whether the transition to instability in the boundary layers can be described by an appropriately defined boundary layer Rayleigh number.
5. Critical boundary layer Rayleigh number
As discussed in the introduction, numerous investigators assumed that the nonlinear breakdown of the diffusive interface is governed by a convective-type instability of the unstable boundary layers. In analogy with single-component convection, the criterion for breakdown was taken to be determined by a critical Rayleigh number that is appropriate for the boundary layers, Rabl = O(103) (Linden and Shirtcliffe 1978). Furthermore, comparisons with field measurements based on this model have also been attempted (Padman and Dillon 1989; Sánchez and Roget 2007). In this section, we test the use of Rabl in describing the linear stability properties, as well as the nonlinear state of the diffusive interface once convection has begun.
We are now in a position to ask the question, how applicable is Rabl for describing the onset of instability at a diffusive interface? This question is addressed in Fig. 13, where we have plotted the location of the stability boundary in the r–RaI plane from the linear stability analysis for different values of the domain height H, at a representative value of Rρ = 3. Two cases are examined: (i) where the transition to the DC-type unstable modes occurs at σr = 0 (Fig. 13a) and (ii) where the linear instability growth rate exceeds the growth rate due to diffusion of the interface at σr = 2π (Fig. 13b). In the case of (i), Fig. 13a shows that, by varying H, there are substantial shifts in the stability boundary, even at values of H as large as 40. It is interesting that this is not so for the convective-type modes that compose the boundary in Fig. 13b, whose curves all collapse.
The boundary between stable and unstable modes as a function of r and RaI for Rρ = 3. (a) The colored curves are plotted for different values of H, which can be seen to influence the conditions required for instability. (b) The boundary σr = 2π denoting the dominance of the linear instability growth over the growth of the interface by molecular diffusion. Each value of H from (a) is also plotted in (b); however, the curves are indistinguishable from each other. In each of (a),(b), the gray contours give the value of the boundary layer Rayleigh number Rabl.
Citation: Journal of Physical Oceanography 42, 5; 10.1175/JPO-D-11-0118.1
In each case, it is possible to compare these stability boundaries with contours of Rabl to assess the appropriate critical Rabl. (We will restrict ourselves to H ≥ 5 in order to avoid the dependence of F on H.) It can be seen in both cases that the Rabl values that represent the stability boundaries most closely are generally more than two orders of magnitude smaller than the value of O(103) previously suggested for the nonlinear breakdown. Furthermore, a single Rabl value is not able to adequately capture the behavior of the stability boundary for the range of r shown and may change by an order of magnitude over 1 < r < 3, as can be seen in Figs. 13a,b. We can therefore conclude that a critical Rabl ≈ 103 does not accurately describe the linear stability boundary in the boundary layers of a diffusive interface.
That the Rabl required for instability is much less than 103 can also be seen in the time-dependent simulation (V) shown in Fig. 11c. The instability growth, seen in the kinetic energy plot of Fig. 11b, is initiated at a Rabl ≪ 103. It is interesting, however, that the Rabl at the point of the initial breakdown (t ≈ 2) is on the order of 103, which is what has previously been assumed. At later times (t > 3) during active convection, the interface appears to reach a quasi-steady state where Rabl ≈ 50. The linear stability threshold for time-dependent background profiles, given by σr = 2π, is Rabl ≈ 5.
6. Conclusions
In this study, we have utilized linear stability analysis and direct numerical simulations (DNS) to examine the conditions under which a diffusive interface will become unstable to small perturbations. It may be thought of as an extension of the classic work on linear profiles of T and S to include the geophysically relevant case of a diffusive interface. Similarity between the linear profile results and the diffusive interface is found when T and S have the same interface thicknesses (r = 1). In this case, the instability is centered within the interface and has similar stability properties to the linear profiles.
It has been found, however, that the T interface has a greater thickness than S (r > 1), because of the higher T molecular diffusivity (Turner 1973; Marmorino and Caldwell 1976; Linden and Shirtcliffe 1978; Fernando 1989), and this has recently been found in the observations of T. Sommer et al. (2012, unpublished manuscript), as shown in Fig. 2. When this is the case, we have shown that the instability will take place within the diffusive boundary layers on either side of the gravitationally stable interface core, and instability is then possible at the large Rρ values typically observed in the staircases of oceans and lakes (i.e., 2 < Rρ < 6, with Rρ = 1.5 typically the lowest observed; Padman and Dillon 1987; Timmermans et al. 2008; Schmid et al. 2010).
The stability analysis revealed that, within the gravitationally unstable boundary layers, it is possible for instability to take place as either an oscillating DC-type unstable mode or a convective-type unstable mode. The marginally unstable modes that compose the stability boundary are always of DC type. However, the DNS show that the time-dependent growth of the interface is significant in altering the position of the stability boundary. If these time-dependent effects are accounted for by requiring that the instability growth rate exceed that of the interface, then we find that the instability of the boundary layers is of convective type. This finding adds theoretical support to this idea, which has often been assumed.
The boundary layer Rayleigh number Rabl is not found to be a useful parameterization of the stability criterion but may be a good description of the time-dependent conditions under which the breakdown of the interface takes place. Further analysis is required to assess this possibility, and a series of fully three-dimensional DNS have been conducted to examine this issue (Carpenter et al. 2012, manuscript submitted to J. Fluid Mech.). The time dependence of the instability shows that it is not a simple matter to directly apply the stability conditions found here to geophysical observations. However, the findings provide insight into the instability mechanisms, as well as lead to bounds on the parameters governing interface stability.
Acknowledgments
We thank Marco Toffolon, Martin Schmid, and Mary-Louise Timmermans for comments on this manuscript, as well as Bill Smyth and Kraig Winters for providing the DNS code for the simulations. This project has been supported by the Swiss National Science Foundation Grant 200021–122183 (Lake Kivu—Turbulence and double diffusion in permanent stratification). The Swiss National Science Foundation is also acknowledged for partly financing the IPAZIA computational cluster (Project 206021–128754). Part of the salary of the first author was made possible by the EU Project Grant 263287 (FreshMon—High Resolution Freshwater Monitoring: FreshMon GMES Downstream Services).
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