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, L_η = h_*. For the assumed a = 0.01, Eq. (2) suggests a numerical domain with a tilt of s = 0.0029 and a height of L_η = 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 L_ξ 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

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 U_max ≡ max{(η)} (Fig. 1, thick line). Figures 2a,b (light gray lines) show that the assumed initial and are relatively large and result in T–S 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 T–S 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 T–S inversion zones; the large heat and salt finger fluxes result in a slight decrease of the intrusion T–S 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 T–S inversions develop for a second time (Figs. 2a,b, thick black line). The small-scale perturbations at the time of the second T–S inversions (t = 1100) are illustrated in Fig. 3a by the total salinity. The plumes in the T–S 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 U_max = 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 U_max. This equilibration scenario differs from the T_z > 0, S_z > 0 scenario (Simeonov and Stern 2007), wherein U_max 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 h_ρ, defined as the region in which the vertical density gradient ( − + R − 1) exceeds its undisturbed value R − 1 = −0.4, is h_ρ = 24.3, and the corresponding velocity and density jumps are ΔU_d = 20.4 and Δρ_d = 104.6, respectively. Based on these values, the bulk Richardson number of the diffusive interface

is Ri_d = 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 = U_maxL_η/ 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 F_H and salt F_S (Figs. 4a,b, thick black line):

where

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 h_ρ. 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 〈F_H〉 = 1.9 and 〈F_S〉 = −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 L_ξ = L_η = 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 U_max = 10.5, respectively; the numerical grid had 256 nodes in ξ and η (Δx = 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 L_ξ = L_η = 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 T_x and S_x 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 T–S reaches a steady state by t = 1800. The final equilibrium state is characterized with a clearly defined T–S 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 U_max (Fig. 5c).

The dependence of U_max 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 L_η. 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 U_max as a function of L_η.

The results in Table 1 suggest that there is a significant increase of the lateral heat flux and U_max as L_η increases. Figures 7a,b show that the heat flux is proportional to L^2.5_η and that U_max varies linearly with the layer thickness L_η. 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 U_max on L_η varies with the frontal gradient a. Figure 7a shows that U_max as a function of L_η is independent of the horizontal gradient a and is given approximately by the straight line

On the other hand, Fig. 7b suggests for the heat flux

where the coefficient B(a) decreases with a (Fig. 7c):

There seems to be no simple explanation of the functional dependencies (6a) and (6b). For example, assuming that the intrusion equilibrates once it produces T–S inversions with salt fingers would suggest a temperature amplitude T₀ proportional to L_η. Likewise, advective–diffusive balance with vertical eddy diffusivity independent of L_η and a implies aU₀ ∼ T₀/L²_η or U₀ ∼ (aL_η)⁻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 L_η 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 L_η.

Our DNS also suggests that the thickness of the thermal boundary layer h_ρ increases approximately linearly with the intrusion thickness L_η (Fig. 7d). Because the intrusion amplitude and the temperature jump across the interface ΔT_d are approximately proportional to L_η, Fig. 7d also suggests that h_ρ increases with ΔT_d. This result differs from models of purely diffusive convection (Kelley 1990; Fernando 1989), where h_ρ decreases with ΔT_d. Because both h_ρ and L_η 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 h_ρ/L_η 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 L_η increases and becomes positive for small a. The corresponding vertical fluxes by the mean flow −s() increase with L_η and decrease with a. These results suggest that at large L_η 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, U_max 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 U_max 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.