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  • View in gallery

    The total horizontal heat flux (thin) and the maximum (thick) as a function of time illustrating the equilibration of the fastest-growing intrusion (n = 1) for a = 0.01, R = 0.6, and τ = 1/6.

  • View in gallery

    Vertical profiles of the (a) mean temperature, (b) salinity, (c) density, and (d) lateral velocity, averaged over four successive time intervals: 0 < t < 50 (light gray), 200 < t < 400 (dark gray), 500 < t < 1000 (thick black), and 2000 < t < 3000 (thin black) for the run in Fig. 1.

  • View in gallery

    Total salinity at (a) t = 1100 and (b) t = 2250 for the calculation in Fig. 1. The contour interval is 50 in the high-gradient region at the center and 5 in the two layers above and below that. There are two kinds of plumes in (a) fresh rising/salty sinking plumes near the bottom/top, which correspond to salt fingers, and salty rising/fresh sinking plumes above/below the high-gradient layer, which correspond to diffusive convection. Despite the weaker T–S inversions at t = 2250 (Fig. 2), there are fresh rising (lower right) and salty sinking (top center) plumes in (b).

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    Profiles of the time-averaged (2000 < t < 3000) small-scale (thick) and intrusion (thin) vertical fluxes of (a) heat and (b) salt for the calculation in Fig. 1 [see Eq. (5)].

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    Profiles of the (a) mean temperature, (b) salinity, (c) and lateral velocity in the calculation for a = 0.05 and τ = 1/6 with vertical domain size Lη = 334, which includes n = 3 fastest-growing wavelengths. The thin black profiles are time averages over successive intervals with a length of 100 time units. The thick black profiles are time averages in 1900 < t < 3000 and indicate that and become quasi steady at about t = 1700. The profiles are offset by (a) 80, (b) 100, and (c) 20. The thick gray lines in (a) and (b) show the undisturbed temperature and salinity for each profile.

  • View in gallery

    The domain-averaged lateral heat flux as a function of time in the calculation in Fig. 5.

  • View in gallery

    The maximum (a) lateral velocity and (b) the lateral heat flux as a function of the layer thickness Lη for different values of the lateral gradient a. The lines in (a) and (b) are straight lines passing through the origin with slopes as indicated on the graph. (c) The heat flux coefficient B [Eq. (6b)] as a function of the lateral gradient a; the straight line fitting the data is given by Eq. (6c). (d) The thickness of the diffusive interface hρ as a function of the layer thickness Lη from the DNS in Table 1. The straight line is a linear fit with a slope of 0.12.

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    The lateral heat flux (thin) and maximum lateral velocity (thick) in four calculations for a = 0.05, n = 1 and four different values of the Lewis number: τ = 1/6 (0 < t < 1500), τ = 1/12 (1500 < t < 3000), τ = 1/24 (3000 < t < 4500), and τ = 1/48 (4500 < t < 6000).

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Double-Diffusive Intrusions in a Stable Salinity Gradient “Heated from Below”

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  • 1 Department of Oceanography, The Florida State University, Tallahassee, Florida
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Abstract

Two-dimensional direct numerical simulations (DNS) are used to investigate the growth and nonlinear equilibration of spatially periodic double-diffusive intrusion for negative vertical temperature Tz < 0 and salinity Sz < 0 gradients, which are initially stable to small-scale double diffusion. The horizontal temperature Tx and salinity Sx gradients are assumed to be uniform, density compensated, and unbounded. The weakly sloping intrusion is represented as a mean lateral flow in a square computational box tilted with a slope equal to that of the fastest-growing linear theory mode; the vertical (η) domain size of the box L*η is a multiple of the fastest-growing wavelength. Solutions for the fastest-growing wavelength show that the intrusion growth is disrupted by salt fingers that develop when the rotation of the isotherms and isohalines by the intrusion shear results in temperature and salinity inversions; the thick inversion regions are separated by a thin interface supporting diffusive convection. These equilibrium solutions were always unstable to longer vertical wavelengths arising because of the merging of the inversion layers. The DNS predicts the following testable results for the maximum lateral velocity U* max = 0.13NSL*η, the lateral heat flux F* = 0.008ρCP(Sx/Sz)1/2(NS/KT)1/4NSL*η2.5(βSz/α), and the interface thickness hρ = 0.12L*η, where NS = , g is the gravity acceleration, ρ is the density, β/α is the haline contraction/heat expansion coefficient, and CP is the specific heat capacity. The results are compared with observations in the Arctic Ocean.

Corresponding author address: Julian Simeonov, Department of Oceanography, The Florida State University, Tallahassee, FL 32306-4320. Email: simeonov@ocean.fsu.edu

Abstract

Two-dimensional direct numerical simulations (DNS) are used to investigate the growth and nonlinear equilibration of spatially periodic double-diffusive intrusion for negative vertical temperature Tz < 0 and salinity Sz < 0 gradients, which are initially stable to small-scale double diffusion. The horizontal temperature Tx and salinity Sx gradients are assumed to be uniform, density compensated, and unbounded. The weakly sloping intrusion is represented as a mean lateral flow in a square computational box tilted with a slope equal to that of the fastest-growing linear theory mode; the vertical (η) domain size of the box L*η is a multiple of the fastest-growing wavelength. Solutions for the fastest-growing wavelength show that the intrusion growth is disrupted by salt fingers that develop when the rotation of the isotherms and isohalines by the intrusion shear results in temperature and salinity inversions; the thick inversion regions are separated by a thin interface supporting diffusive convection. These equilibrium solutions were always unstable to longer vertical wavelengths arising because of the merging of the inversion layers. The DNS predicts the following testable results for the maximum lateral velocity U* max = 0.13NSL*η, the lateral heat flux F* = 0.008ρCP(Sx/Sz)1/2(NS/KT)1/4NSL*η2.5(βSz/α), and the interface thickness hρ = 0.12L*η, where NS = , g is the gravity acceleration, ρ is the density, β/α is the haline contraction/heat expansion coefficient, and CP is the specific heat capacity. The results are compared with observations in the Arctic Ocean.

Corresponding author address: Julian Simeonov, Department of Oceanography, The Florida State University, Tallahassee, FL 32306-4320. Email: simeonov@ocean.fsu.edu

1. Introduction

The fine structure thermohaline steps and inversions observed at oceanic fronts are usually attributed to isopycnal mixing by double-diffusive intrusions (Ruddick 1992; Quadfasel et al. 1993; Carmack et al. 1997; Joyce et al. 1978). This notion is supported by recent high-resolution seismic images (Holbrook et al. 2003; Tsuji et al. 2005; Nakamura et al. 2006) of the two-dimensional structure of thermohaline steps. These novel observations correlate well with hydrographic measurements and will hopefully encourage a wider acceptance of the importance of double diffusion.

The isopycnal mixing by intrusions probably plays an important role in water mass formation, especially in the Arctic Ocean where the thermohaline steps are coherent over hundreds of kilometers (Walsh and Carmack 2003) and persist on the time scale of a decade. It is unclear, however, what produces the layers in the central parts of the Arctic, where the horizontal gradients are much weaker than those near the basin lateral boundaries (Rudels et al. 1999a) resulting from the inflow of warm and salty Atlantic water. Besides, Timmermans et al. (2003) point out that in the Canada Basin the interfaces are too thick to support the observed large geothermal heat flux. One possibility is that the intrusions emerging from the Atlantic boundary current propagate to large distances by means of weak vertical fluxes that maintain the intrusion velocity against viscosity.

In this paper, we consider vertical gradients of temperature Tz < 0 and salinity Sz < 0, such that no small-scale double-diffusive convection is initially present (cf. Simeonov and Stern 2007); such conditions exist in the Arctic Ocean above the Atlantic water. In the presence of horizontal gradients, this basic state is unstable to lateral intrusions driven by the molecular diffusivities (Holyer 1983). The amplifying intrusion shear will rotate the undisturbed isotherms (isohalines), thereby generating an interface of increased (negative) vertical gradients. As the molecular heat flux at the interface exceeds the corresponding salt flux, a thermal (diffusive) convection (Linden and Shirtcliffe 1978) would result that may further increase the buoyancy fluxes driving the intrusion. The intrusion shear may also produce salinity inversions unstable to salt fingers. These nonlinear effects and the intrusion equilibration will be investigated here with direct numerical simulations (DNS) for uniform density-compensated horizontal gradients. We also consider whether subharmonic instabilities of the fastest-growing intrusion (Holyer 1983) would produce lateral flows with stronger amplitudes; for example, longer wavelengths may be generated because double-diffusive layers have a tendency to merge (Huppert 1971; Radko 2007).

Laboratory experiments (Ruddick et al. 1999; Bormans 1992) and direct numerical simulations (Simeonov and Stern 2007) for finger favorable vertical gradients of temperature Tz > 0 and salinity Sz > 0 suggest that the intrusion velocity is proportional to the layer thickness. This empirical relation and the thickness of the diffusive interface will be determined here for Arctic Ocean conditions and the results will be compared with measurements in the Canadian (Timmermans et al. 2003) and the Eurasian (Walsh and Carmack 2003) basins. We also determine the dependence of the lateral and vertical convective fluxes on the magnitude of the horizontal gradients. Because a full parameter exploration is not currently feasible in 3D, the dependence of the intrusion amplitude on the horizontal gradients and layer thickness will be investigated in 2D. The assumption of two-dimensionality is partly justified by linear theory (Holyer 1983), which shows that the most unstable intrusion is independent of the alongfront (y) direction. While the small “turbulent” scales that appear at finite amplitude would generally be three-dimensional, Stern et al. (2001) have shown that the three-dimensional salt finger fluxes are no more than 3 times larger than the two-dimensional ones. We thus expect that our results will be qualitatively similar to those in a three-dimensional model.

The rest of the paper is organized as follows. The model assumptions and the model equations are discussed in section 2, and the numerical results are presented in section 3. The calculations first consider the amplification and equilibration of the fastest-growing vertical wavelength (section 3a). Because it is not a priori clear that this wavelength dominates, the vertical domain size is then increased (section 3b) to allow the development of longer vertical wavelengths. The oceanic implications of the results are discussed in section 4.

2. Model formulation and assumptions

Following previous models, we assume here lateral temperature Tx and salinity Sx gradients that are density compensated, αTx = βSx < 0, horizontally uniform, and unbounded. The density ratio RαTz/βSz will not be varied and is set to R = 0.6 (see Timmermans et al. 2003; Walsh and Carmack 2003); the basic state is therefore stable to the oscillatory instability of Veronis (1965). We also neglect turbulent diffusivities due to breaking waves, which are generally weak in the Arctic Ocean interior (Padman 1994). Due to the horizontal gradients, the basic state will be unstable to intrusions driven by the molecular diffusivities of heat KT and salt KS (Holyer 1983; Stern 2003).

In the assumed uniform gradients, the y-independent normal-mode intrusions take the form of plane waves with a slope s, which is a fraction of the small absolute isohaline slope a ≡ |Sx/Sz|. Because DNS that resolve the large horizontal (x) intrusion wavelength are very expensive, we tilt the computational box such that the slope of the rotated across-front axis (ξ) equals that of the plane wave under consideration. Thus, the intrusions in our calculations will be represented by a mean lateral flow whose slope remains fixed throughout the calculation; this is an intrinsic limitation of the model considered here.

The finite-amplitude evolution of the amplifying intrusion will be studied using the Navier–Stokes equations. The equations are formally nondimensionalized using the time scale N−1S ≡ (gβ|Sz|)−1/2, length scale d ≡ (KT/NS)1/2, temperature scale β|Sz|d/α, and salinity scale |Sz|d; here, NS is the buoyancy frequency due to the mean salinity gradient. The velocity scale is KT/d. Typical ocean values of d and N−1S are, respectively, 1 cm and 10 min. For the sake of generality, the numerical results in section 3 will be presented in nondimensional units; dimensional estimates for given assumed |Sz| will be discussed in section 4.

Using primes to denote the deviations from the undisturbed basic state, the nondimensionalized equations in the rotated reference frame are
i1520-0485-38-10-2271-e1a
i1520-0485-38-10-2271-e1b
i1520-0485-38-10-2271-e1c
i1520-0485-38-10-2271-e1d
i1520-0485-38-10-2271-eq1
where in Eqs. (1c) and (1d) the coefficients in front of u′ and w′ are the undisturbed across-front (ξ) and vertical (η) temperature and salinity gradients. The vector
i1520-0485-38-10-2271-eq2
is a unit vector antiparallel to gravity, Pr ≡ ν/KT = 7 is the Prandtl number based on molecular viscosity ν, and τKS/KT is the Lewis number.

The smallest scales in the DNS will be determined by the salinity field, because sharp gradients produced by convective stirring decay very slowly due to the small salt diffusivity KSKT/100. To reduce the range of molecular dissipation scales that needs to be resolved in the calculations, most of our DNS will use a hypothetical salt diffusivity, which is larger than that of seawater and such that the Lewis number is τ ≡ 1/6. A limited number of calculations (section 3c) will use smaller τ so that we can make an extrapolation relevant to seawater.

Assuming periodic boundary conditions in (ξ, η), Eqs. (1) will be solved with the pseudospectral Fourier method. The integration in time is performed with a fourth-order Runge–Kutta scheme (RK-4). The along-intrusion Lξ, and vertical Lη domain sizes are chosen to be a multiple (n) of the fastest-growing intrusion wavelength h*. Calculations based on Holyer’s (1983) linear theory for R = 0.6, Pr = 7, and τ = 1/6 give the following for the largest growth rate and the corresponding vertical wavelength and slope:
i1520-0485-38-10-2271-e2
for given a, the latter will determine the tilt and the size of the computational box.

3. Numerical results

a. The lateral velocity of the fastest-growing wavelength (n = 1) for a = 0.01

Our investigation of the large-amplitude intrusion dynamics begins with the simplest case in which a two-dimensional numerical domain includes a single vertical wavelength of the fastest-growing intrusion, namely, Lη = h*. For the assumed a = 0.01, Eq. (2) suggests a numerical domain with a tilt of s = 0.0029 and a height of Lη = h* = 249. It is further assumed that the lateral scale of the important small-scale perturbations (eddies) is comparable to the intrusion thickness (Simeonov and Stern 2007); thus, we also set the along-intrusion domain size Lξ equal to h*.

For the corresponding square computational grid we use 768 grid nodes in each direction, resulting in a grid step of Δx = Δz = 0.324. We verify below that this grid step is sufficient in resolving the dissipation scales. The time step for the RK-4 time integration is Δt = 0.005. The initial condition consisted of the fastest-growing plane wave intrusion
i1520-0485-38-10-2271-e3
where U0 = 10, T0 = 55.2, and S0 = 73.6 are determined from the linear theory solution. Here, and in all that follows, we use an overbar to denote ξ average and angle brackets to denote vertical average. To the above initial condition we also added random noise temperature T ′(ξ, η) and salinity S′(ξ, η) distributions with a maximum value of 0.02.

The nonlinear time evolution of the intrusion and its equilibration is illustrated by the (ξ, η)-averaged horizontal heat flux 〈〉 (Fig. 1, thin line) and the maximum ξ-averaged lateral velocity Umax ≡ max{(η)} (Fig. 1, thick line). Figures 2a,b (light gray lines) show that the assumed initial and are relatively large and result in TS inversions near η = 0. In this particular calculation (a = 0.01), the large initial amplitude is chosen to avoid a lengthy (and trivial) constant growth rate stage; Eq. (2) suggests a large e-folding time λ−1 = 1164. Calculations with a much smaller initial amplitude show that the intrusion grows as in linear theory until it produces similar TS inversions. For example, in a run for a = 0.05 initialized with U0 = 1.5, T0 = 8.3, and S0 = 11.0 (no inversions initially) the maximum velocity U0 = 4.35, temperature T0 = 24.1, and salinity S0 = 32.1 at t = 250 differed by less than 1% from their linear theory values.

The diffusive intrusion growth is interrupted very early (t = 100) in the calculation of Fig. 1 due to linearly growing salt fingers (not shown) in the TS inversion zones; the large heat and salt finger fluxes result in a slight decrease of the intrusion TS amplitude (Figs. 2a,b, dark gray lines). Although both molecular diffusion and salt fingers tend to decrease the intrusion temperature and salinity, their effect on the buoyancy anomaly is different—the density of hot salty intrusion is increased (decreased) by molecular diffusion (salt fingers). After the fingers break into disorganized convection (Stern and Simeonov 2005) their amplitude decreases and is less disruptive to the intrusion. The intrusion flux then increases again until t = 1000 when the bottom/top TS inversions develop for a second time (Figs. 2a,b, thick black line). The small-scale perturbations at the time of the second TS inversions (t = 1100) are illustrated in Fig. 3a by the total salinity. The plumes in the TS inversions are of the salt finger type, as verified by the vertical velocity and temperature fields (not shown) corresponding to Fig. 3a, which indicate that the fresh plumes near the bottom are also colder and rising, while the salty ones near the top are warmer and sinking. In addition to the salt fingers, Fig. 3a also shows plumes that detach from a central region of increased (stable) salinity gradient. Unlike the salt fingers, the salty plumes in the central region are rising, propelled by positive heat anomaly (not shown) and indicative of the diffusive convection mechanism. At this time (t = 1100), the largest vertical velocities O(8) are associated with the small-scale plumes—of both the salt finger type and the diffusive convection type.

By comparing the time-averaged and at the end of the calculation (2000 < t < 3000; Figs. 2a,b, thin black line) with those averaged in 500 < t < 1000 (Fig. 2a,b, thick black line) we conclude that the mean intrusion temperature and salinity amplitudes stop growing at about t = 1000. The intrusion lateral velocity (Figs. 1, 2d), however, continues to increase beyond t = 1000 and equilibrates much later at about t = 2200 with a time-averaged Umax = 32.5 ± 2.1. Thus, the secondary increase of the lateral heat flux (Fig. 1, thin line) at t = 1800 and its subsequent equilibration is related to the evolution of Umax. This equilibration scenario differs from the Tz > 0, Sz > 0 scenario (Simeonov and Stern 2007), wherein Umax equilibrates first (before and ) due to strong Reynolds stress damping by mixed layer eddies. In the present calculation, there is a weak Reynolds stress forcing (negative eddy viscosity) and the main balance in the lateral momentum equation is between buoyancy and viscous dissipation (not shown). Ultimately, however, the intrusion equilibrates because double diffusion reduces the vertical divergence of the vertical buoyancy flux due to molecular diffusion.

Figures 2a–c (thin black line) also show that the final equilibrium state consists of a single diffusive interface (with large stable salinity gradient) separating two nearly uniform layers. The quasi-steady interface thickness hρ, defined as the region in which the vertical density gradient ( + R − 1) exceeds its undisturbed value R − 1 = −0.4, is hρ = 24.3, and the corresponding velocity and density jumps are ΔUd = 20.4 and Δρd = 104.6, respectively. Based on these values, the bulk Richardson number of the diffusive interface
i1520-0485-38-10-2271-e4
is Rid = 6.1, which suggests that there is no significant interfacial mixing due to shear instabilities (cf. Smyth et al. 2007). The intrusion Reynolds number Re = UmaxLη/ Pr = 1174 is also quite small. However, the gravitationally unstable edge of the thermal boundary layer of the diffusive interface has a density anomaly Δρ ≅ 10, which gives rise to upward small-scale fluxes of heat FH and salt FS (Figs. 4a,b, thick black line):
i1520-0485-38-10-2271-e5
where
i1520-0485-38-10-2271-eq3
is the perturbation vertical velocity in the nonrotated reference frame. Note that the small-scale fluxes are determined by subtracting the mean flow vertical fluxes −s() and −s() from the respective total fluxes and . In the small-scale fluxes, we also include the relatively small molecular heat and salt fluxes, − and −τ, respectively. The thickness of the gravitationally unstable edge is about 1/3 of hρ. Because of the space–time average in Fig. 2c (thin black line), this thin unstable density layer is smeared across the whole depth of the mixed layer.

The instantaneous salinity field at t = 2250 (Fig. 3b) suggests that the mixed layer convection is driven by plumes that are much larger than those in Fig. 3a; the vertical velocity (Fig. 3b) in the resulting mixed layer eddies is O(30) and is comparable to the mean intrusion velocity (Fig. 1a, thick line). It should be mentioned here that the nearly uniform layers (Figs. 2 a–c, thin black line) have weak but positive temperature and salinity gradients with a density ratio of R = 1.1 at η = 0. Because of this temperature gradient, the heat anomaly that initially drives the diffusive convection plumes is lost faster than it would be due to molecular diffusion alone. The plumes are subsequently driven by a salinity anomaly (Fig. 3b), which they acquire due to the variation of the background salinity in the mixed layer. This modifies the purely diffusive convection and results in vertical heat and salt fluxes near η = 0, which are downward and have a flux ratio 8/12, typical for salt fingers. Because of these downward mixed layer fluxes, the total (vertically averaged) small-scale fluxes 〈FH〉 = 1.9 and 〈FS〉 = −3.4 are much smaller than the maximum upward fluxes at the diffusive interface (Figs. 4a,b, thick line). The velocity and temperature in (3) are positively correlated, and the vertical fluxes by the mean intrusion −s′〉 = −2.3 and −s′〉 = −3.8 are also downward in Figs. 4a,b (thin line). This suggests that the overall vertical heat and salt fluxes are downward (upgradient). The same result was previously obtained by McDougall (1985) using a quite different intrusion model with ad hoc parameterizations of the small-scale fluxes.

To check whether the small scales are properly resolved, we have repeated the calculation of Fig. 1, from t = 2347 to t = 2517 with twice smaller Δx = Δz = 0.162 and Δt = 0.001. During the first 40 time units, the time variations of the averaged lateral heat flux (Fig. 1, thin line) were indistinguishable in the two runs. After that the time variations of the lateral flux in the two runs diverged, but the time-averaged flux remained approximately the same (less than 0.3% difference).

b. Dependence of the intrusion amplitude on the layer thickness and the lateral gradient

There are two reasons why vertical wavelengths longer than the fastest-growing one should be considered in our model with uniform unbounded gradients. First, using Holyer’s (1983) linear theory it can be shown that the growth rate of wavelengths twice the fastest-growing one is only 10% smaller than the maximum growth rate [Eq. (2)]. Such wavelengths will compete with the fastest-growing one and will reach larger amplitudes at the time when inversions form (maximum perturbation gradient becomes equal to the mean); note that longer wavelengths require larger amplitude to produce the same maximum gradient. The second reason is that double-diffusive layers are unstable to a merging instability (Huppert 1971; Merryfield 2000; Radko 2007) that doubles the layer thickness. The purpose of this section is to determine the dependence of the intrusion amplitude on the layer thickness for different values of the horizontal gradient a. For computational efficiency, the procedure adopted here is to use previously obtained one-layer “equilibrium” solutions where the layer thickness equals the fastest-growing wavelength (e.g., section 3a) and consider their stability in the presence of longer vertical wavelengths. Accordingly, we use a larger computational box with Lξ = Lη = nh* and an initial condition, which is an n-fold periodic extension (in both ξ and η) of the one-layer (h*) solution; vertical wavelengths longer than h* are initialized as small random noise.

The instability will be illustrated here starting with a one-layer steady-state solution corresponding to the fastest-growing wavelength for a = 0.05, h* = 111.3. In this a = 0.05 run (not shown), the equilibrium lateral heat flux and maximum velocity were 〈〉 = 142 ± 8 and Umax = 10.5, respectively; the numerical grid had 256 nodes in ξ and ηx = Δz = 0.435), and the time step was Δt = 0.01. This equilibrium solution was then continued in a 3 times (n = 3) larger computational box with Lξ = Lη = 333.9, using a numerical grid with the same Δx and Δz (768 × 768 grid nodes) and smaller Δt = 0.005.

A sequence of the mean temperature and salinity profiles (Figs. 5a,b, thin black line) shows that the equilibrium solution corresponding to the fastest-growing wavelength is unstable. The instability consists of two consecutive merger events in which the top and bottom interfaces erode and disappear while the middle interface becomes stronger. Thus, the merging of layers seems to continue until there is only one large step in the computational domain. Although this process is very similar to the merging instability described by Radko (2007), the evolution of and is determined by the lateral advection of Tx and Sx rather than by the convergence of the vertical fluxes. For example, the convergence of the upward heat flux in layer 1(Fig. 5), due to weakening of the top interface and strengthening of the middle interface, would raise the temperature of this layer; the temperature of layer 1, however, decreases due to increased advection (u profiles) of cold water from the right. Figures 5a,b (thick black and the rightmost thin profiles) also indicate that the mean TS reaches a steady state by t = 1800. The final equilibrium state is characterized with a clearly defined TS inversion where the salt finger fluxes are an order of magnitude larger than those in the initial three-layer state. The new longer wavelength solution also has an order of magnitude larger lateral heat flux (Fig. 6) and 3 times larger Umax (Fig. 5c).

The dependence of Umax and 〈〉 on the layer thickness will be determined here using a suite of similar calculations for a = 0.01, 0.02, and 0.05 (Table 1) in computational domains with n = 1, 2, and 3. In calculations with n ≥ 2, the merging of layers produces a single large intrusion as in Fig. 5, with vertical wavelength equal to Lη. Thus, in all calculations (including n = 1) the final layer thickness equals the vertical domain size. In Fig. 7, we therefore plot the lateral heat flux and Umax as a function of Lη.

The results in Table 1 suggest that there is a significant increase of the lateral heat flux and Umax as Lη increases. Figures 7a,b show that the heat flux is proportional to L2.5η and that Umax varies linearly with the layer thickness Lη. While the latter agrees qualitatively with previous laboratory intrusion experiments (Bormans 1992, Ruddick et al. 1999), it will be shown that our results predict much stronger velocities. In addition, there has been no previous systematic investigation of how the dependence of Umax on Lη varies with the frontal gradient a. Figure 7a shows that Umax as a function of Lη is independent of the horizontal gradient a and is given approximately by the straight line
i1520-0485-38-10-2271-e6a
On the other hand, Fig. 7b suggests for the heat flux
i1520-0485-38-10-2271-e6b
where the coefficient B(a) decreases with a (Fig. 7c):
i1520-0485-38-10-2271-e6c
There seems to be no simple explanation of the functional dependencies (6a) and (6b). For example, assuming that the intrusion equilibrates once it produces TS inversions with salt fingers would suggest a temperature amplitude T0 proportional to Lη. Likewise, advective–diffusive balance with vertical eddy diffusivity independent of Lη and a implies aU0T0/L2η or U0 ∼ (aLη)−1. The latter is clearly inconsistent with the numerical result (6a). We believe that the dependences (6a) and (6b) are not valid for very large Lη because such wavelengths may not be observable due to very small growth rates; note that in Fig. 5 intermediate wavelengths develop faster than the longest wavelength Lη.

Our DNS also suggests that the thickness of the thermal boundary layer hρ increases approximately linearly with the intrusion thickness Lη (Fig. 7d). Because the intrusion amplitude and the temperature jump across the interface ΔTd are approximately proportional to Lη, Fig. 7d also suggests that hρ increases with ΔTd. This result differs from models of purely diffusive convection (Kelley 1990; Fernando 1989), where hρ decreases with ΔTd. Because both hρ and Lη are easily observed, Fig. 7d provides a straightforward test of our results. For example, the observed interface and layer thickness in the Canada Basin (Timmermans et al. 2003) are 2–16 m and 10–60 m, respectively. In the Eurasian basin (Walsh and Carmack 2003) the interfaces are 5–10 m thick and the layers are 30–40 m thick. These observations suggest a ratio hρ/Lη between 1/5 and 1/6, which is comparable to the 0.12 slope of the linear fit in Fig. 7d.

Table 1 also shows that the small-scale heat flux (5) remains bounded as Lη increases and becomes positive for small a. The corresponding vertical fluxes by the mean flow −s() increase with Lη and decrease with a. These results suggest that at large Lη and intermediate a, the total vertical flux will be dominated by the downward intrusion flux. Applied to the subsurface Atlantic water in the Arctic, this suggests that the main effect of the intrusions is to warm the surrounding cold waters lying at the same level or below the Atlantic temperature maximum. On the other hand, further in the interior where the horizontal gradient a is very weak, the intrusion flux (6b) and its vertical projection will be negligible and the total heat flux would be upward.

c. Dependence on the Lewis number

For the purpose of extrapolating the above results for τ = 1/6 to seawater, we next extend our calculation for a = 0.05 and n = 1 (Table 1) by systematically decreasing the Lewis number τ (Fig. 8) and keeping the rest of the parameters the same. For the calculation with τ = 1/12, we use the same grid resolution (256 × 256 grid nodes) as in the τ = 1/6 run. For the runs with τ = 1/24 and τ = 1/48, we double the grid resolution (512 × 512 grid nodes). Figure 8 indicates that as the Lewis number is decreased to τ = 1/48, Umax increases by 35% and the lateral heat flux by only 10%. We therefore conclude that the formulas given by (6a)(6c) can be applied to seawater with less than 50% error. These formulas will be used in section 4 to obtain a dimensional estimate of Umax and the lateral flux for smaller values of the parameter a relevant to the Arctic.

It is also interesting to compare the small-scale salt finger and diffusive convection heat flux with previous flux laws (Kelley 1990; Stern et al. 2001); the present model differs from the latter in two important aspects—there is a sinusoidal shear flow and our stratification includes coupled salt finger and diffusive convection regions. Our results indicate that previous flux laws overestimate the small-scale fluxes in the intrusion by a factor of 2 to 3 (see also Simeonov and Stern 2007). One possible reason for the reduction of the double-diffusive fluxes is that in the presence of shear, double diffusion becomes essentially two-dimensional as its downstream variation is suppressed by the shear flow (Linden 1974; Smyth and Kimura 2007). Because the shear in this study is relatively weak (Table 1), a more likely explanation of the flux discrepancy is that the flux laws based on local properties cannot be applied to a coupled system of salt fingers and diffusive convection.

4. Discussion

Here we have used direct numerical simulations to investigate finite-amplitude, double-diffusive interleaving across uniform fronts, stratified with cold and freshwater above warm and salty water (e.g., the Arctic halocline). DNS for the fastest-growing wavelength (section 3a) show that the growth of the intrusion temperature and salinity anomaly is arrested as the intrusion produces TS inversions with strong downgradient fluxes of heat and salt due to salt fingers. The present investigation, however, is not limited to the fastest-growing wavelength of linear theory and includes longer vertical wavelengths. The results show that the fastest-growing wavelength gives way to longer vertical wavelengths through a continuous subharmonic instability, which is limited here only by the finite domain size. While this process is similar to the well-known merging of layers (Huppert 1971), the calculations suggest that the instability is strongly influenced by the horizontal gradients. For example, starting at the same noise level, the Lη = 351 wavelength in the a = 0.02 (n = 2; Table 1) run reaches finite amplitude in 3000 time units (not shown), compared to only 1500 time units for the Lη = 333 wavelength in the a = 0.05, n = 3 run (Fig. 6).

Because our model does not give a preferred wavelength (or slope), the main objective has been obtaining the dependence of the maximum intrusion velocity, the interface thickness, and the lateral flux on the dimensional layer thickness L*η, as well as the isohaline slope aSx/Sz. Equations (6) suggest the following dimensional velocity U*max and heat flux F* (W m−2):
i1520-0485-38-10-2271-e7a
i1520-0485-38-10-2271-e7b
where NS = . It is surprising that the coefficient in (7a) is approximately the same as that obtained in our previous DNS (Simeonov and Stern 2007) for undisturbed gradients that support salt fingers. The coefficient, however, is much larger than the one in the laboratory experiments of Bormans (1992). Another difference from the laboratory experiments is that the flux in (7b) is not proportional to L*η2 as in Bormans (1992).

One difficulty in applying our results (7) to the ocean is the large variation in the values of a and the vertical salinity gradient relevant to intrusions in the Eurasian basin—|Sz| = 5 × 10−3 PSU m−1 and a = 10−4 in Walsh and Carmack (2003), while |Sz| = 5 × 10−4 PSU m−1 and a = 2 × 10−3 in Rudels et al. (1999a). Here, we will use the intermediate values |Sz| = 10−3 PSU m−1 and a = 5 × 10−4. Assuming also α = 7 × 10−5 °C−1, β = 8 × 10−4 PSU−1, and L*η = 50 m, our results (7a) and (7b) predict U* max = 1.8 cm s−1 and F* = 4900 W m−2. This will be compared here with the observed 0.5°C cooling of the 1 Sv (1 Sv ≡ 106 m3 s−1) Atlantic inflow as it travels from Svalbard to Severnaya Zemlya (Rudels et al. 1999b); the latter represents a heat loss of 2 × 1012 W. Using a distance of 2000 km (between Svalbard and Severnaya Zemlya), the lateral area of the 500-m-thick Atlantic layer is 109 m2. Then the observed heat loss per unit area is 2000 W m−2 or about half our prediction. Compared to Walsh and Carmack (2003), who estimate a horizontal diffusivity KH = 50 m2 s−1, our results predict a 4-times larger KH ≡ 〈KT|Sz|/|Sx| ≅ 200 m2 s−1 for the parameters used above.

Our prediction for the thickness of the thermal boundary layer was shown to be in good agreement with observations (section 3b). The present estimate of the lateral velocity, however, is an order of magnitude larger than the 2 mm s−1 obtained by Walsh and Carmack (2003) using a one-dimensional diffusion model of the heat loss of the Atlantic core. One possible reason for the discrepancy in the predicted intrusion amplitude is observational uncertainties in the spatial and temporal e-folding scales for the erosion of the Atlantic water. Our two-dimensional numerical model also has a number of limitations that can alter our estimates. For example, the small-scale convection and Reynolds stress effects are expected to be stronger in a three-dimensional model where double-diffusive modes with no downstream dependence would be unaffected by the shear (Smyth and Kimura 2007); this would result in stronger damping of the intrusion amplitude. Our extrapolation most likely underestimates the eddy viscosity at lower values of a and longer wavelengths where small Richardson number (Table 1) and large Reynolds number effects could also become important. The present results are further limited by the assumption of a fixed intrusion slope equal to that of linear theory with molecular diffusivities; both the slope and the thickness of intrusions may evolve in time in a larger-scale model. Finally, it would be desirable to consider the effects of finite frontal width, planetary rotation, and baroclinicity.

Acknowledgments

We gratefully acknowledge the support of the National Science Foundation (Grant OCE-0236304). This work was partially supported by the FSU School for Computational Science and Information Technology, by a grant of resources on the IBM pSeries690 Power4-based supercomputer Eclipse. Acknowledgement is also made to the National Center for Atmospheric Research, which is sponsored by the National Science Foundation, for computing time used in this research. We thank William Smyth and the anonymous reviewer for their helpful comments.

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Fig. 1.
Fig. 1.

The total horizontal heat flux (thin) and the maximum (thick) as a function of time illustrating the equilibration of the fastest-growing intrusion (n = 1) for a = 0.01, R = 0.6, and τ = 1/6.

Citation: Journal of Physical Oceanography 38, 10; 10.1175/2008JPO3913.1

Fig. 2.
Fig. 2.

Vertical profiles of the (a) mean temperature, (b) salinity, (c) density, and (d) lateral velocity, averaged over four successive time intervals: 0 < t < 50 (light gray), 200 < t < 400 (dark gray), 500 < t < 1000 (thick black), and 2000 < t < 3000 (thin black) for the run in Fig. 1.

Citation: Journal of Physical Oceanography 38, 10; 10.1175/2008JPO3913.1

Fig. 3.
Fig. 3.

Total salinity at (a) t = 1100 and (b) t = 2250 for the calculation in Fig. 1. The contour interval is 50 in the high-gradient region at the center and 5 in the two layers above and below that. There are two kinds of plumes in (a) fresh rising/salty sinking plumes near the bottom/top, which correspond to salt fingers, and salty rising/fresh sinking plumes above/below the high-gradient layer, which correspond to diffusive convection. Despite the weaker T–S inversions at t = 2250 (Fig. 2), there are fresh rising (lower right) and salty sinking (top center) plumes in (b).

Citation: Journal of Physical Oceanography 38, 10; 10.1175/2008JPO3913.1

Fig. 4.
Fig. 4.

Profiles of the time-averaged (2000 < t < 3000) small-scale (thick) and intrusion (thin) vertical fluxes of (a) heat and (b) salt for the calculation in Fig. 1 [see Eq. (5)].

Citation: Journal of Physical Oceanography 38, 10; 10.1175/2008JPO3913.1

Fig. 5.
Fig. 5.

Profiles of the (a) mean temperature, (b) salinity, (c) and lateral velocity in the calculation for a = 0.05 and τ = 1/6 with vertical domain size Lη = 334, which includes n = 3 fastest-growing wavelengths. The thin black profiles are time averages over successive intervals with a length of 100 time units. The thick black profiles are time averages in 1900 < t < 3000 and indicate that and become quasi steady at about t = 1700. The profiles are offset by (a) 80, (b) 100, and (c) 20. The thick gray lines in (a) and (b) show the undisturbed temperature and salinity for each profile.

Citation: Journal of Physical Oceanography 38, 10; 10.1175/2008JPO3913.1

Fig. 6.
Fig. 6.

The domain-averaged lateral heat flux as a function of time in the calculation in Fig. 5.

Citation: Journal of Physical Oceanography 38, 10; 10.1175/2008JPO3913.1

Fig. 7.
Fig. 7.

The maximum (a) lateral velocity and (b) the lateral heat flux as a function of the layer thickness Lη for different values of the lateral gradient a. The lines in (a) and (b) are straight lines passing through the origin with slopes as indicated on the graph. (c) The heat flux coefficient B [Eq. (6b)] as a function of the lateral gradient a; the straight line fitting the data is given by Eq. (6c). (d) The thickness of the diffusive interface hρ as a function of the layer thickness Lη from the DNS in Table 1. The straight line is a linear fit with a slope of 0.12.

Citation: Journal of Physical Oceanography 38, 10; 10.1175/2008JPO3913.1

Fig. 8.
Fig. 8.

The lateral heat flux (thin) and maximum lateral velocity (thick) in four calculations for a = 0.05, n = 1 and four different values of the Lewis number: τ = 1/6 (0 < t < 1500), τ = 1/12 (1500 < t < 3000), τ = 1/24 (3000 < t < 4500), and τ = 1/48 (4500 < t < 6000).

Citation: Journal of Physical Oceanography 38, 10; 10.1175/2008JPO3913.1

Table 1.

The maximum nondimensional lateral velocity Umax and the corresponding horizontal heat flux 〈〉, the small-scale vertical fluxes FH and FS, and the vertical fluxes by the mean flow in DNS for different aSx/Sz and different vertical domain size. Also given is the diffusive interface Richardson number Rid (4). The variation of the grid step in different calculations (less than 25%) results from using grid sizes with low prime factors for numerical efficiency.

Table 1.
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