Editorial Type:
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Article Type:
Research Article

Double-Diffusive Interleaving: Properties of the Steady-State Solution

Yuehua Li School of Mathematics and Statistics, University of New South Wales, Sydney, New South Wales, Australia

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Trevor J. McDougall School of Mathematics and Statistics, University of New South Wales, Sydney, New South Wales, Australia

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Print Publication:
01 Mar 2015
DOI:
https://doi.org/10.1175/JPO-D-13-0236.1
Page(s):
813–835
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Abstract

Double-diffusive interleaving is examined as it progresses from a linear instability toward finite amplitude. When the basic stratification is in the “finger” sense, the initial series of finger interfaces is unstable and one grows in strength at the expense of the others. At an intermediate stage of its development, the interleaving motions pass through a stage when every second interface in the vertical is stable to double diffusion. At a later time this interface turns into a “diffusive” double-diffusive interface. This study takes the fluxes of heat and salt across both the finger and diffusive interfaces to be given by the laboratory flux laws, and the authors ask whether a steady state is possible. It is found that the fluxes across the diffusive interfaces must be many times stronger relative to the corresponding fluxes across the finger interfaces than is indicated from existing flux expressions derived from laboratory experiments. The total effect of the interleaving motion on the vertical fluxes of heat and of salt are calculated for the steady-state solutions. It is found that both the fluxes of heat and salt are upgradient, corresponding to a negative vertical diffusion coefficient for all heat, salt, and density. For moderate to large Prandtl numbers, these negative effective diapycnal diffusivities of heat and salt are approximately equal so that the interleaving process acts to counteract some of the usual turbulent diapycnal diffusivity due to breaking internal waves.

Corresponding author address: Trevor J. McDougall, School of Mathematics and Statistics, University of New South Wales, NSW 2052, Australia. E-mail: trevor.mcdougall@unsw.edu.au

Abstract

Double-diffusive interleaving is examined as it progresses from a linear instability toward finite amplitude. When the basic stratification is in the “finger” sense, the initial series of finger interfaces is unstable and one grows in strength at the expense of the others. At an intermediate stage of its development, the interleaving motions pass through a stage when every second interface in the vertical is stable to double diffusion. At a later time this interface turns into a “diffusive” double-diffusive interface. This study takes the fluxes of heat and salt across both the finger and diffusive interfaces to be given by the laboratory flux laws, and the authors ask whether a steady state is possible. It is found that the fluxes across the diffusive interfaces must be many times stronger relative to the corresponding fluxes across the finger interfaces than is indicated from existing flux expressions derived from laboratory experiments. The total effect of the interleaving motion on the vertical fluxes of heat and of salt are calculated for the steady-state solutions. It is found that both the fluxes of heat and salt are upgradient, corresponding to a negative vertical diffusion coefficient for all heat, salt, and density. For moderate to large Prandtl numbers, these negative effective diapycnal diffusivities of heat and salt are approximately equal so that the interleaving process acts to counteract some of the usual turbulent diapycnal diffusivity due to breaking internal waves.

Corresponding author address: Trevor J. McDougall, School of Mathematics and Statistics, University of New South Wales, NSW 2052, Australia. E-mail: trevor.mcdougall@unsw.edu.au
Keywords: Atm/Ocean Structure/ Phenomena; Diapycnal mixing; Frontogenesis/frontolysis; Mixing; Thermohaline circulation

1. Introduction

Toole and Georgi (1981) built a theoretical model of double-diffusive interleaving using a linear stability analysis, and they showed that these quasi-horizontal intrusions are driven by the vertical buoyancy fluxes of double-diffusive convection. McDougall (1985a) showed that the same growing intrusions occurred whether the environment was rotating or whether it was not rotating, and McDougall (1985b) made a start at studying these interleaving motions at finite amplitude. He hypothesized that it may be possible that the growth of the intrusions might be arrested at finite amplitude when every second interface changes its nature from the “finger” type to the “diffusive” type. The reason for this possible steady state is that the ratio of the fluxes of heat and salt across the two types of double-diffusive interfaces is quite different. In a steady state there needs to be a three-way balance between three processes, 1) advection, 2) finger flux divergences, and 3) diffusive flux divergences, and this three-way balance needs to occur in both heat and salt.

While McDougall (1985b) showed that it was feasible that steady-state balances could be achieved for both heat and salt, it remained to be shown if such steady-state balances could be achieved with the fluxes across the double-diffusive interfaces taken from the laboratory flux laws, these fluxes having been measured in one-dimensional laboratory experiments.

In this paper, we form a finite-amplitude model of double-diffusive interleaving by integrating the temperature and salinity equations of each intrusion forward in time using the Runge–Kutta integration technique. Each layer is taken to be well mixed in the vertical, separated by relatively sharp interfacial regions where the double-diffusive fluxes originate. The vertical length scale of the intrusions, and their slope with respect to the isopycnals, are taken from the linear stability analysis. Following McDougall (1985b), there are three regimes as the intrusions evolve: first, each interface is of the finger type (see Fig. 1). In the linear stability analysis, each alternative finger interface grows at the expense of its neighbor. We start our model with a small (but finite) disturbance, and the model allows the interleaving motions to grow to finite amplitude, thereafter using realistic flux laws while still in this “finger–finger” regime. This stage is followed by a stage where each alternate interface ceases to be a finger interface and instead becomes stably stratified in both temperature and salinity. A third stage follows in which each alternative interface becomes of the diffusive type in which cool freshwater overlies warmer saltier fluid (see Figs. 2, 3).

Fig. 1.
Fig. 1.

The sketch at the left is a vertical cross section through the frontal region showing the direction of the cross-frontal motion of the intrusions and their slopes. The interfaces with vertical short lines represent the dominant finger interfaces. On the right-hand side, the two graphs show the Absolute Salinity and the density profiles at position A. The dashed lines indicate the initial state without perturbations, and the full lines show the profiles at a later stage.

Citation: Journal of Physical Oceanography 45, 3; 10.1175/JPO-D-13-0236.1

Fig. 2.
Fig. 2.

Conservative Temperature–Absolute Salinity diagram showing the evolution of the properties of the double-diffusive intrusions with time. The initial properties lie on the dashed line with slope . The arrowed lines connect the initial and final points of several layers.

Citation: Journal of Physical Oceanography 45, 3; 10.1175/JPO-D-13-0236.1

Fig. 3.
Fig. 3.

Absolute Salinity–Conservative Temperature diagram showing the evolution of the subservient finger interface between layers a and b. From a to 1, it is a finger interface and then from 1 to 2 is a nondouble-diffusive interface. At last, from 2 to 3 it is a diffusive interface. In the steady state, a diffusive interface exists between points 3 on this diagram.

Citation: Journal of Physical Oceanography 45, 3; 10.1175/JPO-D-13-0236.1

2. The model equations

For reference purposes, there is a list of symbols in appendix F. We write down the basic conservation equations of intrusion motion following Toole and Georgi (1981) and McDougall (1985a). We assume an initial state of rest with horizontal isopycnals. We take the salinity and temperature variables to be those of the International Thermodynamic Equation of Seawater—2010 (TEOS-10), namely, Absolute Salinity and Conservative Temperature (IOC et al. 2010; McDougall 2003; Graham and McDougall 2013). These variables vary along the initially horizontal isopycnal surfaces, with and increasing in the positive direction so that and
e1
The overbars indicate the state before interleaving sets in, and and are defined from the equation of state by
e2
The conservation equations for momentum, continuity, Absolute Salinity, and Conservative Temperature are [see Eqs. (3)–(8) of McDougall (1985a)]
e3
e4
e5
e6
e7
e8
where , and are the fluxes of x, y, and z momentum, and and are the fluxes of Absolute Salinity and Conservative Temperature caused by double-diffusive convection.

In some studies (e.g., Toole and Georgi 1981; Walsh and Ruddick 1998), these double-diffusive fluxes are assumed to be directed down the respective salinity and temperature gradients with eddy diffusion coefficients that depend on whether the vertical gradients are conducive to the salt-fingering type of double diffusion or to the diffusive type. In these studies, the vertical profiles are smooth, continuous functions of the vertical coordinate; often the vertical structure function is harmonic and the nonlinear advection terms turn out to be zero. We follow McDougall (1985a,b) and adopt a different strategy that we believe is more consistent with applications to the ocean and with comparisons to the laboratory-determined flux laws. We take the values of Absolute Salinity and Conservative Temperature and their respective perturbations to be constant in the vertical within each intrusion layer, with sharp property differences across the sheared interfaces that separate the quasi-horizontally moving intrusions. In this way we are able to apply the laboratory flux laws that describe the fluxes of heat and salt across sharp interfaces (as opposed to down smooth gradients).

Since we assume Absolute Salinity to be piecewise constant in the vertical direction, the vertical profile of perturbations in this study is a square wave. The vertical flux of salt due to finger double-diffusive convection is negative, which indicates a downward flux. During the linearly unstable growing solution, each interface is of the finger type, with each alternate interface being either a dominant or subservient finger interface.

In McDougall’s model, is taken as the proportionality constant between the salt flux across a finger interface and the salinity difference across the interface during the linearly unstable growth phase; that is, . During this initial exponential growth phase the fluxes are a linearization of the flux laws. Thereafter we adopt the finite-amplitude result that the fluxes are proportional to the power of the property contrasts instead of the power that has been used in previous studies. We consider the mean vertical stratification to be in the finger sense, that is, in this work we take (with height defined positive upward) and . Initially each interface is in the finger sense, and as the perturbations grow, the salinity contrast across one interface increases by and the salinity contrast across the other interface decreases by . The difference between the fluxes of salt across adjacent interfaces is then .

The vertical wavenumber of the intrusions is defined as , and the vertical wavelength is , with the height of an individual intrusion being . The right-hand side of the salt conservation Eq. (7) during this linearly unstable growth phase is then
e9
where we have defined for convenience. Similarly, the divergences of the fluxes of momentum in Eqs. (3), (4), and (5) are given by
e10
where is the flux coefficient of momentum, and we simply use for convenience.
This study starts with the exponentially growing linearly unstable solutions of McDougall (1985a), and the essential features of that study are summarized in appendix A. We follow McDougall (1985b) in assuming that the isopycnals do not slope so that the horizontal gradients of Absolute Salinity and Conservative Temperature are balanced in their effects on density, that is, , where is the horizontal cross-frontal coordinate. Consider one layer with thickness of , where is the vertical wavelength of the disturbance. In the present model, the flux divergences are interpreted in terms of lower and upper fluxes bounding an individual intrusion. Integrating the conservation Eq. (7) over the thickness of an individual layer gives
e11
Several properties of the growing interleaving motion are set during the exponentially growing phase, namely, the slope of the intrusion to the horizontal s, the dimensionless vertical wavenumber m, and the ratio of the perturbation horizontal and vertical velocities. The slope of the interleaving motions with respect to the isopycnals s is the solution of the following cubic equation (from McDougall 1985a):
e12
while the vertical wavenumber is given by
e13
and the vertical and horizontal velocity perturbations are related by
e14
Here, we have introduced the shorthand notations
eq1
and m, s, and the Prandtl number σ, are given by
eq2
where and are defined by
eq3
These relationships have been written in terms of the following properties of the background mean oceanic stratification, namely, the stability ratio Rρ and the square of the buoyancy frequency N:
eq4
and is the buoyancy–flux ratio of the fluxes across the salt-finger interfaces. Equations (12), (13), and (14) above are the finite Prandtl number solutions, and we will adopt this finite Prandtl number case in this work, as Ruddick Griffiths and Symonds (1989) and Smyth and Kimura (2007) considered that the appropriate value of Prandtl number is to be O(1) or less.
Equation (11) is now simplified by using the known ratio of the vertical and horizontal velocity perturbations [from Eqs. (12), (13), and (14) above], thus eliminating w′:
e15
Similarly, Eq. (8) becomes
e16
In what follows it proves convenient to use the following nondimensional salinity and temperature variables X and Y, defined by
e17
We now discuss the relationship between the perturbation horizontal velocity u′ and the density perturbation. During the exponentially growing stage, the momentum equation is deduced from Eqs. (11), (13), and (38) of McDougall (1985a), as follows (under the assumption of the slope of the interleaving motions s is small):
e18
and initially u′ is
e19
which is derived from the linearly unstable momentum equation [see Eq. (39) of McDougall 1985a]. If a steady state is reached, the rate of change of the horizontal velocity perturbation [corresponding to the term on the left-hand side of Eq. (18)] needs to vanish or otherwise the intrusion velocity will continue to accelerate. Under this assumption, the steady-state versions of the two terms on the right-hand side of Eq. (18) will balance each other, implying that the horizontal velocity perturbation and the density perturbation in the steady state are related by
e20
Beginning at Eq. (18), we have allowed the possibility that the value of the turbulent viscosity may increase from to as the steady state is approached. During the process of evolution, we assume that the ratio of the perturbation horizontal velocity to the density perturbation will change linearly with Y and is represented by
e21
where
e22
This rather arbitrary linear function of Y allows the velocity perturbation to be that of the exponentially growing solution at the initial condition , while ensuring that the ratio of the velocity and density perturbations becomes constant at well before any possible steady state is reached. That is, the linear function of of Eq. (21) provides a credible way of transitioning between Eqs. (19) and (20). For , we use Eq. (20).
Substituting Eq. (21) into Eq. (15) and dividing both sides by leads to
e23
where [from Eq. (21)]
eq5
Dividing both sides of Eq. (23) by and using nondimensional time , we find
e24
Similarly, the Eq. (16) becomes
e25
To choose the initial starting point for the model, we take a small value of , and use the relationship between and that applies during the exponentially growing linearly unstable solution [see Eq. (B1) in appendix B].

For the results we present in this paper we have taken the initial value of to be 0.1, but we have demonstrated that the results are insensitive to this starting value by also doing some cases with and . In summary, the cases with yielded values of and at the steady state that were typically larger than those obtained with by only , while the cases with yielded values of and at the steady state that were typically smaller than those obtained with by only . This demonstrates that the solutions are quite insensitive to our choice of as the initial condition where the growth of the interleaving changes from being the linearly unstable solution to one based on the finite-amplitude laboratory flux laws.

The sensitivity to the initial condition was tested in another way by deliberately disobeying the ratio of the initial values of and as given by Eq. (B2) in appendix B. We multiplied the right-hand side of Eq. (B2) by the factor 0.8 so that the initial value of the temperature perturbation was only 80% that of the linearly growing solution. This changed the steady-state values of and by only or less than 0.5% of these steady-state values. On the basis of these two types of test, we conclude that the model is quite insensitive to details of the initial conditions.

In this work, we adopt the finite-amplitude laboratory-based expressions for the double-diffusive fluxes of heat from Huppert (1971) and of salt from McDougall and Taylor (1984) at all stages of the numerical integration after the initial condition at . The laboratory flux laws for both finger interfaces and diffusive interfaces are described in appendix C. Note that in this entire study we take the flux ratio of salt fingers to be .

Last, we present the salinity and temperature contrasts across finger and diffusive interfaces in terms of nondimensional variables and . For the dominant finger interfaces, the property contrasts across them (see Fig. 1, adopted from McDougall 1985b) are
e26
and the finger stability ratio is
e27
while for the other interfaces, the “subservient” finger interfaces across which the salinity contrast decreases with time:
e28
e29
After sufficient time we anticipate that these subservient finger interfaces will become stably stratified with respect to both temperature and salinity, and after more time has elapsed, these interfaces will be stratified in the diffusive sense of double-diffusive convection. After this time, the property contrasts across them are
e30
and the diffusive stability ratio is
e31

3. The transition to finite amplitude in the finger–finger regime

The evolution of double-diffusive interleaving goes through three regimes: the finger–finger (FF) regime with finger interfaces at each interface, the finger–nondouble-diffusive (FN) regime with nondiffusive interfaces as each alternate interface, and the finger–diffusive (FD) regime with diffusive interfaces alternating with finger interfaces (see Fig. 3). As explained above, we transition from the linearly unstable growing solution to having the interfacial fluxes determined by the laboratory flux laws at a very early part of the FF stage when . We ensure that the laboratory flux law expressions are joined in a continuous fashion to the exponentially growing linearly unstable solutions that are used to initialize our model intrusions. In this study, the value of the finger flux ratio is fixed at 0.5 throughout the three regimes.

The integration remains in the FF regime until reaches 1 [see Eq. (29)]. In this regime, both upper and lower interfaces are finger interfaces. More specifically, in the intrusion we are considering the lower finger interface is dominant and the upper finger interface is subservient. The evolution equations for the nondimensional variables and in this finite-amplitude FF regime are
e32
and
e33
where
e34
Equations (32)–(34) are derived in appendix D. The nondimensional Eqs. (32) and (33) are used in regime FF, that is, from the initial value of until reaches 1.

4. The integration in the finger–nondouble-diffusive regime

In the FN regime, the upper interface is not double diffusive so that the fluxes of heat and salt across it are set to zero. In this FN regime, the lower interface remains a finger interface. This regime lasts while and , and the evolution Eqs. (24) and (25) become [see the first part of Eq. (34) and Eqs. (32) and (33) for motivation]
e35
e36

5. The integration in the finger–diffusive regime

In the FD regime, the lower interface remains a finger interface and is parameterized exactly as in the FN regime. The fluxes across the upper interface are obtained by using laboratory diffusive flux laws (see appendix C). Combining the contributions from the lower and upper interfaces, Eq. (24) now becomes
e37
where the flux ratio of the diffusive interface is given by Eqs. (C2) and (C3) in appendix C, and the 0.7674 number is actually [from Eqs. (C1) and (C4)]
eq6
which is the numerical factor that arises from substituting the laboratory diffusive flux of heat into Eq. (24), where ν is the viscosity. This same term gives rise to the following term of the right-hand side of Eq. (25):
e38
and hence, in the FD regime, the evolution equation for [Eq. (25)] is
e39

Last, we mention that if the density difference across the diffusive interface goes to zero, the integration cannot be continued since the layers above and below this interface would physically homogenize, and this is not part of our model. This occurs when . If this condition is detected, the numerical integration is stopped.

6. The feasibility of the steady state

McDougall (1985b) suggested that a steady state would be possible once the fluxes across each alternate interface are in the diffusive sense, and it is the primary purpose of this paper to find out the conditions under which such a steady state is achieved. In the steady state, the double-diffusive fluxes across the finger and the diffusive interfaces must work together to balance the advective fluxes of both heat and salt so that the temporal derivatives and are both equal to zero. However, McDougall (1985b) only demonstrated the feasibility of such a steady state. In this paper, we incorporate the laboratory flux laws and we investigate the conditions under which a steady state is actually achieved.

Figure 1 (adapted from McDougall 1985b) shows a sketch of a series of interleaving layers, in which the dominant finger interfaces are indicated by the short vertical lines. The two panels on the right show the Absolute Salinity and the density profiles at position A. The dashed lines represent the initial state in which the perturbations are zero, and the full lines indicate a later state when the perturbations have grown to finite amplitude and the flow is in the FD regime. Above the dominant finger interfaces, the salt-finger salt flux reduces the salinity of the intrusion layer; however, the horizontal advection of salt dominates so that actually the salinity of this layer increases with time in the growing solution. In McDougall (1985a), it is shown that the perturbation Conservative Temperature is greater than the perturbation Absolute Salinity (both expressed in terms of density) during the initial growth phase. This implies that the density contrast across the dominant salt-finger interface increases with time, while that across the subservient finger interface decreases with time in the FF regime.

Figure 2 (adapted from McDougall 1985b) is a diagram that shows the evolution of the layer properties. The layer properties initially lie on a line with slope . The Conservative Temperature and Absolute Salinity change in the ratio of Eq. (40) of McDougall (1985a). Each successive layer has the opposite sign of change. Figure 3 (adapted from McDougall 1985b) shows the evolution of the subservient finger interface between layers with properties a and b. Starting as finger interfaces (from points a to 1), the subservient interfaces evolve to be not double-diffusive interfaces (between points 1 and 2) when the interface is stable in both temperature and salinity contrasts. After the cold freshwater overlies the warm salty water, the interfaces become diffusive interfaces (between points 2 and 3). Eventually, the finger fluxes, the diffusive fluxes, and the advective fluxes may reach a steady-state balance for both temperature and salinity. This stage is called steady state because the temporal rates of change of both temperature and salinity are zero.

As stated at the beginning of this section, the steady state occurs when the sum of the finger flux divergence term, the diffusive flux divergence term, and the advective flux divergence term is zero, which implies that . In the FF regime, the diffusive flux does not appear, and during this stage, the growing solution is shown in Fig. 4a with both the temporal derivative terms and the advective terms being larger than the finger flux divergence term. This balance applies until point 1 in Fig. 3, and we should be mindful that in this study, even in this FF regime, the fluxes obey the finite-amplitude laboratory-determined flux laws. Figure 4b corresponds to the period between points 1 and 2 of Fig. 3, where the rate of increase of the finger flux divergence is approximately halved because each alternate interface is stably stratified with respect to both and . During this FN regime it is clear that a steady state is not possible between just a finger flux divergence and the advective term; the temporal term and the advective term are observed to be larger than the finger flux divergence.

Fig. 4.
Fig. 4.

Contributions to the temporal derivative vector from the double-diffusive fluxes and the advective terms. (a) The stage from points a to 1 in Fig. 3; this is the growing solution of the linear stability analysis of McDougall (1985a). (b) A sketch of the terms applying at the points 2 of Fig. 3. (c) Beyond the points 2, the diffusive fluxes begin to grow and allow the possibility of a steady-state solution in which is (0, 0).

Citation: Journal of Physical Oceanography 45, 3; 10.1175/JPO-D-13-0236.1

The feasibility of achieving a steady state is shown in Fig. 4c. Diffusive fluxes emerge as the evolution reaches point 2, and these diffusive fluxes increase with time thereafter. Together with the finger fluxes, it is possible to reach a balance. We will now examine this process of reaching a steady state with our model.

We started at , with and with determined from Eq. (40) of McDougall (1985a). The differential equations for and are then integrated forward in time. We have done this for a range of values of the stability ratio of the water column . To our surprise, we found that a steady state was not possible with the laboratory flux laws. Rather, the subservient interface was always driven to be statically unstable in the FD regime. If this occurred in practice, the two layers bounding this interface would overturn and mix, resulting in a doubling of the vertical wavelength, as sometimes occurred in Ruddick (1984). To achieve a steady state, it became clear that the fluxes across the diffusive interface needed to be stronger relative to those across the finger interface (or equivalently, the fluxes across the finger interfaces needed to be weaker with respect to those across the diffusive interface). For this reason we ran the model for a range of values of an “exaggeration factor,” where we replaced the 0.7674 number that represents the strength of the diffusive interfaces in Eqs. (37)–(39) with 0.7674 multiplied by an exaggeration factor that we varied from 5 to 40. One way to rationalize such an exaggeration factor is that the oceanic interfaces have much smaller property contrasts across them than do double-diffusive interfaces in the laboratory and perhaps this affects the strengths of the finger and diffusive interfaces in different ways.

The equations apply to any finite Prandtl number, and we show results for values of Prandtl number equal to 0.3, 1, and 10, but we mainly illustrate our results using . To begin with, Fig. 5 shows contours of the steady-state values of the nondimensional salinity and temperature perturbations, denoted , plotted as a function of and the exaggeration factor for the three selected values of Prandtl number . The similarity in those three pairs of figures is obvious. The values of and depend more on the environmental stability ratio than on the exaggeration factor. In each of these panels the lower-left corners are blank, which indicates that no steady states were found; in these regions, the diffusive flux divergence is unable to balance the advective and finger terms. With increasing , when the basic stratification is less fingering favorable, the exaggeration factor required to achieve a steady state appears to be smaller. The results throughout this paper for are almost identical to those for an infinite Prandtl number.

Fig. 5.
Fig. 5.

Contours of the nondimensional salinity and temperature variables and of the steady-state points with respect to the exaggeration factor and . Three values for Prandtl number are selected: (top two) , (middle two) , and (bottom two) .

Citation: Journal of Physical Oceanography 45, 3; 10.1175/JPO-D-13-0236.1

The evolution of and toward the steady-state values and is illustrated in Fig. 6a. Once the steady state had been reached, the solution was perturbed a little in and , and the integration was continued. The stable spiraling toward the steady-state values and as illustrated in Fig. 6b was a characteristic feature of all the steady states.

Fig. 6.
Fig. 6.

(top) A typical evolution of the nondimensional salinity and temperature variables and toward the steady-state values at the end of the “hook” near the upper right-hand of the figure. (bottom) The loci of seven artificially perturbed points around the steady-state point. These two panels are for ef = 25, , and .

Citation: Journal of Physical Oceanography 45, 3; 10.1175/JPO-D-13-0236.1

The stability ratios of both the finger and diffusive interfaces at steady state can be calculated from Eqs. (27) and (31), and the values for are shown in Fig. 7. The stability ratio of the diffusive interfaces are mostly near 2 or greater than 2, and this can be understood from Eqs. (C2) and (C3) in appendix C, which describe the flux ratio across the diffusive interfaces. If the stability ratio across these diffusive interfaces becomes much less than 2 and approaches 1, then the flux ratio increases from 0.15 and approaches 1. In this limit it is clear from the angle of the diffusive flux vector on Fig. 4c that a steady-state balance is not possible as the flux of density across the finger interfaces is too strong. Note that when a steady state is not possible and the diffusive interface overturns, its stability ratio passes through 1.0 at that time. It appears that the only steady states that are possible have the stability ratio of the diffusive interfaces substantially greater than 1.0, with numbers greater than 1.6 being apparent from Fig. 7a.

Fig. 7.
Fig. 7.

The stability ratios of the (top) diffusive and (bottom) finger interfaces at the steady state for σ = 1.

Citation: Journal of Physical Oceanography 45, 3; 10.1175/JPO-D-13-0236.1

7. The relations among variables in the steady state

Notice that, in the steady state, the expressions of the vertical fluxes of salt across the finger and diffusive interfaces respectively correspond to the first and second right-hand side term of Eq. (37). The absolute value of the ratio of the vertical flux of Absolute Salinity across the diffusive interface to that across the finger interface in steady state is labeled and is given by
e40
where is the exaggeration factor. Similarly, the expressions of the vertical fluxes of Conservative Temperature across the finger and the diffusive interfaces in the steady state can be calculated from the two terms on the right-hand side of Eq. (39), which gives
e41
The values of and at each steady-state point are shown in Fig. 8 as a function of the exaggeration factor and . It is easy to notice that different values of the Prandtl number does not have a significant impact on the values of and . For most of the steady-state solutions, the ratio of the diffusive to finger salt flux is about 0.1, while the ratio of the diffusive to finger temperature flux is about 1.0.
Fig. 8.
Fig. 8.

The upper panel of each pair shows the absolute value of the ratio of the vertical flux of salt across the diffusive interface to that across the finger interface in the steady state. The lower panel of each pair shows the absolute value of the ratio of the vertical flux of temperature across the diffusive interface to that across the finger interface in the steady state. Three values for Prandtl number are selected: (a) , (b) , and (c) .

Citation: Journal of Physical Oceanography 45, 3; 10.1175/JPO-D-13-0236.1

From Eq. (41) of McDougall (1985a), we have
e42
where and are the gradients of Conservative Temperature and Absolute Salinity, respectively, in the direction of motion of an intrusion (or leaf). The ratio of these along-leaf gradients is set in the exponentially growing solution and remains the same in each of our three regimes. In the steady state, define by
e43
where is the salinity contrast across the finger interface in the steady state.
With the shorthand notations above, at steady state Eqs. (37) and (39) are written as
e44
and
e45
Taking the ratio of Eqs. (44) and (45) gives a relation between and , which is,
e46
Figure 9 shows the values of , when , at a range of steady-state points as a function of the exaggeration factor and . From Eq. (C3) in appendix C, we see that is constant when , explaining why there are no contours in the upper-right region in Fig. 9.
Fig. 9.
Fig. 9.

Values of for the case of .

Citation: Journal of Physical Oceanography 45, 3; 10.1175/JPO-D-13-0236.1

Equation (44) can be rewritten using Eq. (42) to give an expression that relates and the steady-state values of and to the ratio of to , namely,
e47
which is shown in Fig. 10. From the above equation, , and the environmental stability ratio all contribute to this ratio, but from Fig. 10 we see that is more sensitive to the environmental stability ratio than to the exaggeration factor. The use of the finite-amplitude laboratory flux laws has led to an increase of over the value used in the linearly growing solution of between 20% and 50%. Another relationship that applies in the steady state, this time between the steady-state values of , and , is given by Eq. (E2).
Fig. 10.
Fig. 10.

As in Fig. 9, but for the ratio .

Citation: Journal of Physical Oceanography 45, 3; 10.1175/JPO-D-13-0236.1

In the above development we have allowed for the possibility that the turbulent eddy viscosity may change as the interleaving motions grow to finite amplitude. We have investigated whether this is a significant issue by doing some cases with and the results of and were no more than 10% different to those using . This value of was chosen as being approximately equal to the corresponding ratio of the change in the coefficient of the finger salt flux at finite amplitude (see Fig. 10). Since the results were rather insensitive to the value of , all the results in this paper have used .

8. The diapycnal fluxes at steady state

One of the most important features of interleaving motions is their ability to transport heat and salt across isopycnals. In the steady state there are two different types of contributions to the mean diapycnal flux of both salinity and temperature. First there is the spatial average of the double-diffusive fluxes of salt and temperature across the finger and diffusive interfaces, and this relevant average flux is simply the average of the finger and diffusive fluxes. Second, there is the area average on an isopycnal surface of the spatial correlation between the vertical velocity perturbation and the temperature (and salinity) perturbations. In turn, this advective contribution has two components: one being due to the correlated nature of the vertical velocity and the temperature and salinity perturbations at any location in space and the other due to the spatial correlation across the front of the vertical velocity and the cross-front salinity and temperature differences. This last aspect was missing from McDougall’s (1985b) work, and we will find that this spatial correlation is significantly larger than the corresponding advective correlation at a fixed point in space (by a factor of about 4).

The need for the spatial correlation along the horizontal isopycnals can be understood from Fig. 1. The relevant salinity perturbations that should multiply the vertical velocity perturbations of the intrusions are the ones at the large dots of the left-hand panel of Fig. 1. The total relevant salinity perturbation is then the sum of the perturbation at a fixed point and a contribution because the two dots are separated in space in the across-front direction. The distance from the middle to the right-hand dot is , that is, half the thickness of a single intrusion divided by the magnitude of the cross-front slope of the intrusions with respect to the density surfaces. The positive salinity perturbation at this right-hand dot is then plus (and is positive for the intrusions that move upward from right to left and that have a positive ). The average vertical advective salinity flux across isopycnals is then . The total diapycnal flux of salt (TDFS) is then
e48
where the first term is the average of the salt fluxes across the finger and diffusive interfaces [with the finger flux of salt being negative (downward) and the diffusive flux of salt being positive (upward)].
Combining Eqs. (14) and (20), we find the following expression in the steady state for the vertical velocity in terms of the density perturbation:
e49
and after a series of substitutions involving Eqs. (13) and (47), we obtain
e50
From the definition of , we have
e51
and substituting this back into Eq. (48), using Eq. (50), we find that TDFS becomes
e52
For the total diapycnal flux of Conservative Temperature , a similar analysis applies, giving
e53
The total temperature perturbation at the right-hand big dot of Fig. 1 is
e54
and substituting this into Eq. (53) and using Eq. (50), we find
e55
Recall that the vertical flux of salt at steady state across the finger interface is
e56
and it is convenient to express the total diapycnal fluxes in terms of , so that Eqs. (52) and (55) become
e57
e58

Figure 11 shows these two measures of the total diapycnal fluxes of salt and temperature as a function of the exaggeration factor and the stability ratio for Prandtl numbers of 0.3, 1, and 10. For , the values of the nondimensionalized total diapycnal fluxes of salt and temperature are both significantly less than their counterparts when or . For , in the case of salt, this nondimensional measure varies from −0.48 to −0.28, while in the case of temperature, the nondimensional ratio varies from −1 to −0.7. Concentrating on the particular case when the exaggeration factor is 25 and the stability ratio is 2, Eq. (57) is about −0.29 and Eq. (58) is about −0.75. At these values of the exaggeration factor and the stability ratio, we see from Fig. 8 that and so that the leading terms on the right-hand sides of Eqs. (57) and (58) are and . These are the average fluxes across the double-diffusive interfaces, and they are positive. It follows that the dominant contributions to the total diapycnal fluxes of heat and salt in these steady-state intrusions come from the advection of salt and heat, that is, from the spatial correlations of for salt and of for temperature [see Eqs. (48) and (53)]. The first of the spatial correlation terms, and represent the contribution from the correlations at a given point in space, while the second terms and represent the correlations arising from doing the spatial average in the horizontal direction (see Fig. 1), and these are the dominant terms.

Fig. 11.
Fig. 11.

The values of the total diapycnal fluxes in terms of . These panels are plots of Eqs. (57) and (58), and the values are substantially negative. Three values for Prandtl number are selected: (a) , (b) , and (c) .

Citation: Journal of Physical Oceanography 45, 3; 10.1175/JPO-D-13-0236.1

The numbers that appear in the second line of Eqs. (57) and (58) represent the values of the terms above them when the exaggeration factor is 25 and the stability ratio is 2. It is seen that the salt flux across the finger interfaces dominates that across the diffusive interfaces, and the resulting average salt flux is directed downward (0.47) in the downgradient direction. It is the vertical advection of the salinity perturbations in the intrusions (−0.76) that overpowers this downgradient salt diffusion, and most of the negative contribution of this advection comes from the horizontal spatial salinity correlation (−0.61), not the salinity perturbation at a fixed location (−0.15).

The corresponding situation for temperature is apparent from the numbers on the second line of Eq. (58). It is seen that the temperature flux across the finger interfaces slightly exceeds that across the diffusive interfaces, and the resulting average salt flux is directed downward (+0.04) in the downgradient direction. It is the vertical advection of the temperature perturbations in the intrusions (−0.79) that counteracts this small downgradient temperature diffusion, and most of the negative contribution of this advection comes from the horizontal spatial correlation (−0.61), not from the temperature and vertical velocity correlations at a fixed location (−0.18).

These results for the dominance of the upgradient advection of both salt and heat are in agreement with the inferences made by McDougall (1985b), but it must be said that that paper ignored the dominant advective correlation, namely, that between the vertical velocity and the horizontally varying salinity and temperature fields. The take-home message from this analysis of the total diapycnal flux of salt and heat in double-diffusive intrusions at finite-amplitude steady state is that the total diapycnal flux of salt is ~38% smaller (−0.29/47 ≈ −0.62) than one might deduce from purely knowledge of the fluxes across the finger and diffusive interfaces, and, most importantly, the total salt flux is going in the opposite direction to these driving interfacial double-diffusive fluxes. That is, the total diapycnal flux of salt is upgradient in the sense of a negative diffusion coefficient. The corresponding take-home message for the total diapycnal flux of temperature is that it is 19 times (−0.75/0.04 = −18.75) what would be deduced from only knowledge of the fluxes across the finger and diffusive interfaces. Again, this flux of temperature is upgradient in the sense of a negative diapycnal diffusivity of temperature.

9. Discussion

McDougall (1985b) gave a plausible analysis of the feasibility of the existence of a steady-state for finite-amplitude double-diffusively driven intrusions by analyzing how the double-diffusive fluxes and advective fluxes evolve and finally balance in both temperature and salinity, and he developed expressions for some properties of the steady state such as the vertical velocity and the total diapycnal fluxes.

The present paper extends McDougall’s (1985b) approach, and consequently the basic assumptions of this model are the same as that of McDougall (1985a,b). We take the buoyancy–flux ratio of salt fingers as the fixed value 0.5, and we consider only the case where the salinity and temperature of the ocean increase with height so that the basic stratification is salt-fingering favorable. Moreover, we take the vertical structure of the property perturbations to be square waved since we believe this is appropriate when considering laboratory-based flux laws that vary as the four-thirds power of property contrasts. The laboratory flux laws are taken from McDougall and Taylor (1984) for finger interfaces and from Huppert (1971) for diffusive interfaces.

This study complements those of Walsh and Ruddick (1998), Merryfield (2000), and Mueller et al. (2007) who have studied the formation of intrusions in a continuously stratified fluid, specifying turbulent diffusivities for salt, heat, and momentum as functions of the stability ratio and the Froude number. Walsh and Ruddick (1998) confirmed that a steady state is possible and that this is achieved after each alternate interface is in the diffusive sense. These studies had the double-diffusive fluxes as general power laws of the stability ratio, whereas we have adopted the laboratory-based flux laws, albeit with an extra multiplicative exaggeration factor for the diffusive flux law. In addition, in contrast to these papers, we concentrate on the magnitude and signs of the total diapycnal fluxes of heat and salt in the steady-state intrusions. In this study, we made a linear transitioning from the initial state to the steady state [referring to Eqs. (19)–(22)] based on the momentum equation. This fundamental assumption is made because it is inappropriate to apply the momentum equation of the exponentially growing solution through all three stages, and we certainly know that if a steady state is achieved, the growth rate with respect to time has to be zero, which gives another equation for the horizontal velocity [Eq. (20)]. This linear transitioning of the momentum equation is an assumption that is not needed at large values of the turbulent Prandtl number.

The main result of the present paper is that we have been able to achieve steady-state double-diffusively driven intrusions by using the laboratory flux laws, but in order to find these steady-state solutions we have found that the strength of the fluxes across the diffusive interfaces need to be increased significantly (by at least an order of magnitude) above the laboratory-determined values, relative to the laboratory-based finger fluxes. One way of rationalizing this need for an “exaggeration ratio” is that the laboratory experiments are performed with interfacial property contrasts that are much larger than oceanic ones. Another explanation might be that when an interface is near to becoming statically unstable, convectively driven turbulence is established, and this will change the effective flux ration of heat and salt across the interface. This effect was parameterized in the study of Mueller et al. (2007) but has not been included in the present study.

We have also quantified the total diapycnal fluxes of heat and salt, taking into account the advection of the perturbations in the steady state as well as the interfacial double-diffusive fluxes themselves. An important new aspect of this study is the realization that it is the spatial correlations of the diapycnal velocity of the intrusions with the temperature and salinity perturbations that make the largest contribution to the upgradient fluxes of heat and salt [see the large −0.61 numbers in Eqs. (57) and (58)]; this important feature was missed by previous studies, including that of McDougall (1985b), and it is the spatial correlation that makes the total diapycnal flux of salt be upgradient. We have shown that the total diapycnal fluxes of both Absolute Salinity and Conservative Temperature are upgradient, that is, both are in the sense of a negative vertical diffusion coefficient.

Perhaps the largest assumption that we have made is that if a steady state is achieved in salinity and temperature, then a steady state will also be achieved in momentum so that the lateral velocity of the interleaving motions becomes constant rather than continuing to accelerate. This assumption is not needed for very large Prandtl number [since the left-hand side of Eq. (18) tends to zero at infinite Prandtl number], but it was invoked via our Eqs. (21)–(22), which ensure that if or when the salt and heat equations evolve to a steady state then the interleaving velocity will also approach a constant value at this stage. As a partial test of the sensitivity of our results to this assumption, we have performed some runs where was simply put equal to the initial value . In practice, this means that at all stages of the process the velocity of the intrusions will be accelerating since Eq. (18) indicates that the dimensional growth rate would remain positive. For the case , we find that a steady state in temperature and salinity is still reached in this situation with , albeit at values of and that are reduced by approximately 15% and 5%, respectively. In this situation, the minimum exaggeration factor required to reach a steady state in temperature and salinity is larger by approximately 30%, and, interestingly, the ratio of the negative diffusivities of temperature and salinity is still close to unity, with the figure corresponding to Fig. 12b showing a rather uniform increase of only 0.05. This indicates that our conclusions are not very sensitive to the proportionality factor between the horizontal velocity and the density perturbation of each intrusion [see Eqs. (21)–(22)]. However, if a steady state is to be achieved then the horizontal velocity also needs to converge to a constant and for noninfinite Prandtl numbers, this aspect of this study remains an assumption.

Fig. 12.
Fig. 12.

The ratio , being the ratio of the negative effective vertical diffusion coefficient for temperature to that for salt. This is the ratio of the numbers in the two panels in Fig. 11 divided by . Three values of the Prandtl number are selected: (a) , (b) , and (c) .

Citation: Journal of Physical Oceanography 45, 3; 10.1175/JPO-D-13-0236.1

These effective negative diffusion coefficients for both temperature and salinity (and density) when considering the total effect of both double diffusion and advection will surely have implications for how these interleaving motions can or should be parameterized in intermediate-scale and large-scale ocean models. For example, we suggest that it is appropriate to take the vertical eddy diffusivity of temperature and salinity in an ocean model (which represents regular small-scale turbulent mixing) and to reduce it differently for temperature and for salinity to account for double-diffusive interleaving while running an ocean model. It would be wise of course to not let the total eddy diffusivity of either temperature or salinity to go negative or the numerical model will surely exhibit instabilities.

If a certain negative vertical diffusivity for salinity were decided upon to represent the effects of double-diffusive interleaving in an ocean model, then the magnitude of the appropriate negative vertical diffusivity for temperature would be larger by the ratio , and we plot this ratio in Fig. 12. It is seen that when the Prandtl number is not less than 1, this ratio is not a particularly strong function of the stability ratio and that it is not very different from unity over much of Fig. 12, especially for exaggeration ratios less than 20. This suggests that for finite-amplitude double-diffusively driven interleaving has the net effect of simply reducing the vertical turbulent diffusivity of both temperature and salinity and therefore of density, thus making it harder to balance the diapycnal transport implied by the production rate of bottom water. In other words, since double-diffusively driven interleaving motion acts like a negative diffusivity of heat, salt, and density with almost equal negative vertical diffusion coefficients, a given amount of upward diapycnal motion requires larger turbulent diapycnal mixing in order to combat the effects of the interleaving process. Thus, we see that double-diffusively driven interleaving acts in a similar fashion to thermobaricity and cabbeling (Klocker and McDougall 2010) in that the presence of these three processes means that more intense mixing from the breaking of internal gravity waves is required in order to achieve a given amount of dianeutral advection.

Acknowledgments

We have benefited from many constructive comments from Dr. Barry Ruddick. YL acknowledges the support of a University of New South Wales International Postgraduate Award and partial scholarship support from the Australian Research Council Centre of Excellence for Climate System Science and the UNSW School of Mathematics and Statistics.

APPENDIX A

The Linearly Unstable Solutions

The Prandtl number is the ratio of the flux coefficient of momentum to that of salt [see Eqs. (9)–(10)].

The disturbances of Eqs. (3)–(8) are taken to be proportional to with amplitudes for Conservative Temperature, for Absolute Salinity, for cross-front velocity, for alongfront velocity, for vertical velocity, and for pressure, so that Eqs. (3)–(8) become
ea1
ea2
ea3
ea4
ea5
ea6
The isopycnal surfaces are initially assumed to be flat, and the coordinates are chosen so that . The acceleration terms that are in brackets in Eqs. (A1)–(A3) are neglected for large turbulent Prandtl numbers.
Nondimensional variables are defined by
ea7
ea8
ea9
and for finite turbulent Prandtl number, the nondimensional growth rate satisfies
ea10
The growth rate is now maximized with respect to , and . In practice, the vertical wavenumber is much larger than the two horizontal wavenumbers and , and so . By differentiating Eq. (A10) with respect to both r and s and setting them equal to zero, it can be shown that
ea11
Substituting Eq. (A11) into Eq. (A10) and dividing both sides of Eq. (A10) by gives
ea12
which, as noted by McDougall (1985a), is independent of and so the growth rate and the vertical and horizontal wavenumbers are all independent of the Coriolis parameter. Equations (12) and (13) in the body of this paper are derived from this equation, being the values of and that maximize the growth rate of the intrusions.

APPENDIX B

Initial Conditions

To choose the initial starting point for the model, we take a small value of and and use the a relationship between and that applies during the exponentially growing linearly unstable solution, namely, Eq. (40) of McDougall (1985a) (which is here expressed in terms of and ):
eb1
which gives
eb2

APPENDIX C

The Laboratory Flux Laws

A finger interface has warm salty water above cold freshwater, and for this type of double-diffusive interface we use the laboratory results of McDougall and Taylor (1984), where from their Fig. 2 the salt flux across the finger interface can be represented by the function
ec1
As for diffusive interfaces that have cool freshwater above warm salty water, from Fig. 2 of Huppert (1971), we have
ec2
and
ec3
From Eqs. (1)–(4) and Fig. 1 of Huppert (1971), we find that the flux across a diffusive interface is given by
ec4

APPENDIX D

The Finger Flux Divergence in the FF Regime

Based on the laboratory-based flux laws, the right-hand side of Eq. (24) is given by
ed1
As the evolution transition to finite amplitude at the beginning point, the expression for the FF regime is obtained by the linearly growing analysis of the physical basis of this integration. In the linear growth case, let , such that
ed2
and the right-hand side of Eq. (24) then becomes
ed3
and correspondingly, Eq. (25) is
ed4
Now requiring that Eqs. (32) and (34) be the same at implies that is given by
ed5
For simplicity, let
ed6
so that
ed7

APPENDIX E

The Relationship between the Steady-State Values , and

The ratio of the salt flux coefficients at steady state to that in the linearly unstable growth phase can also be obtained by using the definition [Eq. (43)] of together with the finite-amplitude finger flux, Eq. (C1) in appendix C, giving [using Eq. (27) at steady state]
ee1
Equating Eqs. (47) and (E1) leads to the following equation that is obeyed by the steady-state values of , and (we have checked that our solutions obey this equation):
ee2

APPENDIX F

List of Symbols

 The vertical wavelength of the intrusions, being twice the thickness of a single intrusion

 Thermal expansion coefficient of seawater defined by

 Saline contraction coefficient of seawater defined by

 Conservative Temperature (IOC et al. 2010)

 The gradient of Conservative Temperature in the direction of the motion of an intrusion (or leaf)

 Buoyancy flux ratio of finger interfaces, fixed at 0.5 throughout this paper

 Buoyancy flux ratio of diffusive interfaces

 Buoyancy flux ratio of diffusive interfaces at steady state

 The absolute value of the ratio of the vertical flux of salt across the diffusive interface to that across the finger interface in steady state

 The proportionality constant between the salt flux across a finger interface and the salinity difference across the interface

 The proportionality constant between the salt flux across a finger interface and the salinity difference across the interface at steady state

 Environmental stability ratio

 Stability ratio of finger interface

 Stability ratio of diffusive interface

 Absolute Salinity (IOC et al. 2010)

 The gradient of Absolute Salinity in the direction of the motion of an intrusion (or leaf)

 Nondimensional salinity perturbation variable defined as

 Nondimensional temperature perturbation variable defined as

REFERENCES

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    • Search Google Scholar
    • Export Citation

  • Huppert, H. E., 1971: On the stability of a series of double-diffusive layers. Deep-Sea Res. Oceanogr. Abstr., 18, 1005–1021, doi:10.1016/0011-7471(71)90005-2.

    • Search Google Scholar
    • Export Citation

  • IOC, SCOR, and IAPSO, 2010: The International Thermodynamic Equation of Seawater—2010: Calculation and use of thermodynamic properties. Intergovernmental Oceanographic Commission, Manuals and Guides 56, 220 pp. [Available online at www.teos-10.org/pubs/TEOS-10_Manual.pdf.]

  • Klocker, A., and T. J. McDougall, 2010: Influence of the nonlinear equation of state on global estimates of dianeutral advection and diffusion. J. Phys. Oceanogr., 40, 1690–1709, doi:10.1175/2010JPO4303.1.

    • Search Google Scholar
    • Export Citation

  • McDougall, T. J., 1985a: Double-diffusive interleaving. Part I: Linear stability analysis. J. Phys. Oceanogr., 15, 1532–1541, doi:10.1175/1520-0485(1985)015<1532:DDIPIL>2.0.CO;2.

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  • McDougall, T. J., 1985b: Double-diffusive interleaving. Part II: Finite amplitude, steady state interleaving. J. Phys. Oceanogr., 15, 1542–1555, doi:10.1175/1520-0485(1985)015<1542:DDIPIF>2.0.CO;2.

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  • McDougall, T. J., 2003: Potential enthalpy: A conservative oceanic variable for evaluating heat content and heat fluxes. J. Phys. Oceanogr., 33, 945–963, doi:10.1175/1520-0485(2003)033<0945:PEACOV>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • McDougall, T. J., and J. R. Taylor, 1984: Flux measurements across a finger interface at low values of the stability ratio. J. Mar. Res., 42, 1–14, doi:10.1357/002224084788506095.

    • Search Google Scholar
    • Export Citation

  • Merryfield, W. J., 2000: Origin of thermohaline staircases. J. Phys. Oceanogr., 30, 1046–1068, doi:10.1175/1520-0485(2000)030<1046:OOTS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Mueller, R. D., W. D. Smyth, and B. Ruddick, 2007: Shear and convective turbulence in a model of thermohaline intrusions. J. Phys. Oceanogr., 37, 2534–2549, doi:10.1175/JPO3137.1.

    • Search Google Scholar
    • Export Citation

  • Ruddick, B. R., 1984: The life of a thermohaline intrusion. J. Mar. Res., 42, 831–852, doi:10.1357/002224084788520729.

  • Ruddick, B. R., R. W. Griffiths, and G. Symonds, 1989: Frictional stress at a sheared double-diffusive interface. J. Geophys. Res., 94, 18 161–18 173, doi:10.1029/JC094iC12p18161.

    • Search Google Scholar
    • Export Citation

  • Smyth, W. D., and S. Kimura, 2007: Instability and diapycnal momentum transport in a double-diffusive, stratified shear layer. J. Phys. Oceanogr., 37, 1551–1565, doi:10.1175/JPO3070.1.

    • Search Google Scholar
    • Export Citation

  • Toole, J. M., and D. T. Georgi, 1981: On the dynamics and effects of double-diffusively driven intrusions. Prog. Oceanogr., 10, 123–145, doi:10.1016/0079-6611(81)90003-3.

    • Search Google Scholar
    • Export Citation

  • Walsh, D., and B. R. , 1998: Nonlinear equilibration of thermohaline intrusions. J. Phys. Oceanogr., 28, 1043–1070, doi:10.1175/1520-0485(1998)028<1043:NEOTI>2.0.CO;2.

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    • Export Citation
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Journal of Physical Oceanography

  • Graham, F. S., and T. J. McDougall, 2013: Quantifying the nonconservative production of Conservative Temperature, potential temperature, and entropy. J. Phys. Oceanogr., 43, 838–862, doi:10.1175/JPO-D-11-0188.1.

    • Search Google Scholar
    • Export Citation

  • Huppert, H. E., 1971: On the stability of a series of double-diffusive layers. Deep-Sea Res. Oceanogr. Abstr., 18, 1005–1021, doi:10.1016/0011-7471(71)90005-2.

    • Search Google Scholar
    • Export Citation

  • IOC, SCOR, and IAPSO, 2010: The International Thermodynamic Equation of Seawater—2010: Calculation and use of thermodynamic properties. Intergovernmental Oceanographic Commission, Manuals and Guides 56, 220 pp. [Available online at www.teos-10.org/pubs/TEOS-10_Manual.pdf.]

  • Klocker, A., and T. J. McDougall, 2010: Influence of the nonlinear equation of state on global estimates of dianeutral advection and diffusion. J. Phys. Oceanogr., 40, 1690–1709, doi:10.1175/2010JPO4303.1.

    • Search Google Scholar
    • Export Citation

  • McDougall, T. J., 1985a: Double-diffusive interleaving. Part I: Linear stability analysis. J. Phys. Oceanogr., 15, 1532–1541, doi:10.1175/1520-0485(1985)015<1532:DDIPIL>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • McDougall, T. J., 1985b: Double-diffusive interleaving. Part II: Finite amplitude, steady state interleaving. J. Phys. Oceanogr., 15, 1542–1555, doi:10.1175/1520-0485(1985)015<1542:DDIPIF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • McDougall, T. J., 2003: Potential enthalpy: A conservative oceanic variable for evaluating heat content and heat fluxes. J. Phys. Oceanogr., 33, 945–963, doi:10.1175/1520-0485(2003)033<0945:PEACOV>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • McDougall, T. J., and J. R. Taylor, 1984: Flux measurements across a finger interface at low values of the stability ratio. J. Mar. Res., 42, 1–14, doi:10.1357/002224084788506095.

    • Search Google Scholar
    • Export Citation

  • Merryfield, W. J., 2000: Origin of thermohaline staircases. J. Phys. Oceanogr., 30, 1046–1068, doi:10.1175/1520-0485(2000)030<1046:OOTS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Mueller, R. D., W. D. Smyth, and B. Ruddick, 2007: Shear and convective turbulence in a model of thermohaline intrusions. J. Phys. Oceanogr., 37, 2534–2549, doi:10.1175/JPO3137.1.

    • Search Google Scholar
    • Export Citation

  • Ruddick, B. R., 1984: The life of a thermohaline intrusion. J. Mar. Res., 42, 831–852, doi:10.1357/002224084788520729.

  • Ruddick, B. R., R. W. Griffiths, and G. Symonds, 1989: Frictional stress at a sheared double-diffusive interface. J. Geophys. Res., 94, 18 161–18 173, doi:10.1029/JC094iC12p18161.

    • Search Google Scholar
    • Export Citation

  • Smyth, W. D., and S. Kimura, 2007: Instability and diapycnal momentum transport in a double-diffusive, stratified shear layer. J. Phys. Oceanogr., 37, 1551–1565, doi:10.1175/JPO3070.1.

    • Search Google Scholar
    • Export Citation

  • Toole, J. M., and D. T. Georgi, 1981: On the dynamics and effects of double-diffusively driven intrusions. Prog. Oceanogr., 10, 123–145, doi:10.1016/0079-6611(81)90003-3.

    • Search Google Scholar
    • Export Citation

  • Walsh, D., and B. R. , 1998: Nonlinear equilibration of thermohaline intrusions. J. Phys. Oceanogr., 28, 1043–1070, doi:10.1175/1520-0485(1998)028<1043:NEOTI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    The sketch at the left is a vertical cross section through the frontal region showing the direction of the cross-frontal motion of the intrusions and their slopes. The interfaces with vertical short lines represent the dominant finger interfaces. On the right-hand side, the two graphs show the Absolute Salinity and the density profiles at position A. The dashed lines indicate the initial state without perturbations, and the full lines show the profiles at a later stage.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    Conservative Temperature–Absolute Salinity diagram showing the evolution of the properties of the double-diffusive intrusions with time. The initial properties lie on the dashed line with slope . The arrowed lines connect the initial and final points of several layers.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    Absolute Salinity–Conservative Temperature diagram showing the evolution of the subservient finger interface between layers a and b. From a to 1, it is a finger interface and then from 1 to 2 is a nondouble-diffusive interface. At last, from 2 to 3 it is a diffusive interface. In the steady state, a diffusive interface exists between points 3 on this diagram.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    Contributions to the temporal derivative vector from the double-diffusive fluxes and the advective terms. (a) The stage from points a to 1 in Fig. 3; this is the growing solution of the linear stability analysis of McDougall (1985a). (b) A sketch of the terms applying at the points 2 of Fig. 3. (c) Beyond the points 2, the diffusive fluxes begin to grow and allow the possibility of a steady-state solution in which is (0, 0).

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    Contours of the nondimensional salinity and temperature variables and of the steady-state points with respect to the exaggeration factor and . Three values for Prandtl number are selected: (top two) , (middle two) , and (bottom two) .

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    (top) A typical evolution of the nondimensional salinity and temperature variables and toward the steady-state values at the end of the “hook” near the upper right-hand of the figure. (bottom) The loci of seven artificially perturbed points around the steady-state point. These two panels are for ef = 25, , and .

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    The stability ratios of the (top) diffusive and (bottom) finger interfaces at the steady state for σ = 1.

  • <sc>Fig</sc>. 8.
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    Fig. 8.

    The upper panel of each pair shows the absolute value of the ratio of the vertical flux of salt across the diffusive interface to that across the finger interface in the steady state. The lower panel of each pair shows the absolute value of the ratio of the vertical flux of temperature across the diffusive interface to that across the finger interface in the steady state. Three values for Prandtl number are selected: (a) , (b) , and (c) .

  • <sc>Fig</sc>. 9.
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    Fig. 9.

    Values of for the case of .

  • <sc>Fig</sc>. 10.
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    Fig. 10.

    As in Fig. 9, but for the ratio .

  • <sc>Fig</sc>. 11.
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    Fig. 11.

    The values of the total diapycnal fluxes in terms of . These panels are plots of Eqs. (57) and (58), and the values are substantially negative. Three values for Prandtl number are selected: (a) , (b) , and (c) .

  • <sc>Fig</sc>. 12.
    View in gallery
    Fig. 12.

    The ratio , being the ratio of the negative effective vertical diffusion coefficient for temperature to that for salt. This is the ratio of the numbers in the two panels in Fig. 11 divided by . Three values of the Prandtl number are selected: (a) , (b) , and (c) .

Fig. 1.

The sketch at the left is a vertical cross section through the frontal region showing the direction of the cross-frontal motion of the intrusions and their slopes. The interfaces with vertical short lines represent the dominant finger interfaces. On the right-hand side, the two graphs show the Absolute Salinity and the density profiles at position A. The dashed lines indicate the initial state without perturbations, and the full lines show the profiles at a later stage.
Fig. 2.

Conservative Temperature–Absolute Salinity diagram showing the evolution of the properties of the double-diffusive intrusions with time. The initial properties lie on the dashed line with slope . The arrowed lines connect the initial and final points of several layers.
Fig. 3.

Absolute Salinity–Conservative Temperature diagram showing the evolution of the subservient finger interface between layers a and b. From a to 1, it is a finger interface and then from 1 to 2 is a nondouble-diffusive interface. At last, from 2 to 3 it is a diffusive interface. In the steady state, a diffusive interface exists between points 3 on this diagram.
Fig. 4.

Contributions to the temporal derivative vector from the double-diffusive fluxes and the advective terms. (a) The stage from points a to 1 in Fig. 3; this is the growing solution of the linear stability analysis of McDougall (1985a). (b) A sketch of the terms applying at the points 2 of Fig. 3. (c) Beyond the points 2, the diffusive fluxes begin to grow and allow the possibility of a steady-state solution in which is (0, 0).
Fig. 5.

Contours of the nondimensional salinity and temperature variables and of the steady-state points with respect to the exaggeration factor and . Three values for Prandtl number are selected: (top two) , (middle two) , and (bottom two) .
Fig. 6.

(top) A typical evolution of the nondimensional salinity and temperature variables and toward the steady-state values at the end of the “hook” near the upper right-hand of the figure. (bottom) The loci of seven artificially perturbed points around the steady-state point. These two panels are for ef = 25, , and .
Fig. 7.

The stability ratios of the (top) diffusive and (bottom) finger interfaces at the steady state for σ = 1.
Fig. 8.

The upper panel of each pair shows the absolute value of the ratio of the vertical flux of salt across the diffusive interface to that across the finger interface in the steady state. The lower panel of each pair shows the absolute value of the ratio of the vertical flux of temperature across the diffusive interface to that across the finger interface in the steady state. Three values for Prandtl number are selected: (a) , (b) , and (c) .
Fig. 9.

Values of for the case of .
Fig. 10.

As in Fig. 9, but for the ratio .
Fig. 11.

The values of the total diapycnal fluxes in terms of . These panels are plots of Eqs. (57) and (58), and the values are substantially negative. Three values for Prandtl number are selected: (a) , (b) , and (c) .
Fig. 12.

The ratio , being the ratio of the negative effective vertical diffusion coefficient for temperature to that for salt. This is the ratio of the numbers in the two panels in Fig. 11 divided by . Three values of the Prandtl number are selected: (a) , (b) , and (c) .
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