## 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 _{z} < 0 and salinity _{z} < 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 _{z} > 0 and salinity _{z} > 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 _{x} and salinity _{x} gradients that are density compensated, *α*_{x} = *β*_{x} < 0, horizontally uniform, and unbounded. The density ratio *R* ≡ *α*_{z}/*β*_{z} 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 *K*_{T} and salt *K*_{S} (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* ≡ |_{x}/_{z}|. 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*^{−1}_{S} ≡ (*g**β*|_{z}|)^{−1/2}, length scale *d* ≡ (*K*_{T}/*N*_{S})^{1/2}, temperature scale *β*|_{z}|*d*/*α*, and salinity scale |_{z}|*d*; here, *N _{S}* is the buoyancy frequency due to the mean salinity gradient. The velocity scale is

*K*

_{T}/

*d*. Typical ocean values of

*d*and

*N*

^{−1}

_{S}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 |

S

_{z}| will be discussed in section 4.

*u*′ and

*w*′ are the undisturbed across-front (

*ξ*) and vertical (

*η*) temperature and salinity gradients. The vectoris a unit vector antiparallel to gravity, Pr ≡

*ν*/

*K*

_{T}= 7 is the Prandtl number based on molecular viscosity

*ν*, and

*τ*≡

*K*

_{S}/

*K*

_{T}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 *K*_{S} ≈ *K*_{T}/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.

*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

*L*, and vertical

_{ξ}*L*domain sizes are chosen to be a multiple (

_{η}*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: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, *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*

_{*}.

*x*= Δ

*z*= 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 Δ

*t*= 0.005. The initial condition consisted of the fastest-growing plane wave intrusionwhere

*U*

_{0}= 10,

*T*

_{0}= 55.2, and

*S*

_{0}= 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 〈*ξ*-averaged lateral velocity *U*_{max} ≡ max{*η*)} (Fig. 1, thick line). Figures 2a,b (light gray lines) show that the assumed initial *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 *U*_{0} = 1.5, *T*_{0} = 8.3, and *S*_{0} = 11.0 (no inversions initially) the maximum velocity *U*_{0} = 4.35, temperature *T*_{0} = 24.1, and salinity *S*_{0} = 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 *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 _{z} > 0, _{z} > 0 scenario (Simeonov and Stern 2007), wherein *U*_{max} equilibrates first (before

*h*, defined as the region in which the vertical density gradient (

_{ρ}*R*− 1) exceeds its undisturbed value

*R*− 1 = −0.4, is

*h*= 24.3, and the corresponding velocity and density jumps are Δ

_{ρ}*U*= 20.4 and Δ

_{d}*ρ*= 104.6, respectively. Based on these values, the bulk Richardson number of the diffusive interfaceis 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 =*

_{d}*U*

_{max}

*L*

_{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 Δ

*ρ*≅ 10, which gives rise to upward small-scale fluxes of heat

*F*and salt

_{H}*F*(Figs. 4a,b, thick black line):whereis 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}*s*(

*τ*

*h*. 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.

_{ρ}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*〈*s*〈

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 Δ*x* = Δ*z* = 0.162 and Δ*t* = 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 〈*U*_{max} = 10.5, respectively; the numerical grid had 256 nodes in *ξ* and *η* (Δ*x* = Δ*z* = 0.435), and the time step was Δ*t* = 0.01. This equilibrium solution was then continued in a 3 times (*n* = 3) larger computational box with *L _{ξ}* =

*L*= 333.9, using a numerical grid with the same Δ

_{η}*x*and Δ

*z*(768 × 768 grid nodes) and smaller Δ

*t*= 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 _{x} and _{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 〈*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}.

*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 lineOn the other hand, Fig. 7b suggests for the heat fluxwhere 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

*T*proportional to

_{0}*L*

_{η}. Likewise, advective–diffusive balance with vertical eddy diffusivity independent of

*L*

_{η}and

*a*implies

*aU*

_{0}∼

*T*

_{0}/

*L*

^{2}

_{η}or

*U*

_{0}∼ (

*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 Δ

*T*are approximately proportional to

_{d}*L*

_{η}, Fig. 7d also suggests that

*h*increases with Δ

_{ρ}*T*. This result differs from models of purely diffusive convection (Kelley 1990; Fernando 1989), where

_{d}*h*decreases with Δ

_{ρ}*T*. Because both

_{d}*h*and

_{ρ}*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*(*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.

## 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 *T*–*S* 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 *L*_{η} = 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 *L*_{η} = 333 wavelength in the *a* = 0.05, *n* = 3 run (Fig. 6).

*L**

_{η}, as well as the isohaline slope

*a*≡

S

_{x}/

S

_{z}. Equations (6) suggest the following dimensional velocity

*U**

_{max}and heat flux

*F** (W m

^{−2}):where

*N*

_{S}=

*L**

_{η}

^{2}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—|_{z}| = 5 × 10^{−3} PSU m^{−1} and *a* = 10^{−4} in Walsh and Carmack (2003), while |_{z}| = 5 × 10^{−4} PSU m^{−1} and *a* = 2 × 10^{−3} in Rudels et al. (1999a). Here, we will use the intermediate values |_{z}| = 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 ≡ 10^{6} m^{3} s^{−1}) Atlantic inflow as it travels from Svalbard to Severnaya Zemlya (Rudels et al. 1999b); the latter represents a heat loss of 2 × 10^{12} W. Using a distance of 2000 km (between Svalbard and Severnaya Zemlya), the lateral area of the 500-m-thick Atlantic layer is 10^{9} m^{2}. 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 *K _{H}* = 50 m

^{2}s

^{−1}, our results predict a 4-times larger

*K*

_{H}≡ 〈

*K*

_{T}|

S

_{z}|/|

S

_{x}| ≅ 200 m

^{2}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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The maximum nondimensional lateral velocity *U*_{max} and the corresponding horizontal heat flux 〈*F _{H}* and

*F*, and the vertical fluxes by the mean flow in DNS for different

_{S}*a*≡

S

_{x}/

S

_{z}and different vertical domain size. Also given is the diffusive interface Richardson number Ri

*(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.*

_{d}