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[1] Based on a Lagrangian description of fluid particle dispersion we suggest that there is a single expression for the vertical eddy diffusivity for all scalars following fluid particles in stably stratified flows. This expression is the same as the Osborn-Cox diffusivity for buoyancy. To test this hypothesis we carry out turbulence simulations with stable background stratification by solving the Boussinesq equations with random forcing together with the equation for a passive scalar with an initial vertical Gauss profile. The development of the mean scalar concentration is studied for three different values of the width of the profile, σ. It is found that the passive scalar diffuses in very good agreement with the classical diffusion equation if the ratio between σ and a turbulent length scale is large enough. The associated eddy diffusivity agrees exactly with the Osborn-Cox diffusivity for buoyancy.

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[2]Osborn and Cox [1972] derived an expression for the vertical eddy diffusivity for temperature in the ocean. Apart from a small correction term, associated with the compressible properties of sea water, the expression reads

where 〈T〉 is the mean temperature profile, ε_{T} = κ_{T}〈∇T · ∇T〉 is the destruction or dissipation of mean square temperature fluctuations and κ_{T} is the molecular diffusivity for temperature. The expression (1) is derived by assuming that there is a balance between production and dissipation in the equation for the mean square temperature fluctuations. For any scalar, ϕ, satisfying such a balance the eddy diffusivity can be obtained as K_{ϕ} = ε_{ϕ}/(∂_{z}〈ϕ〉)^{2}, that is by replacing T by ϕ in (1) [Winters and D'Asaro, 1996]. As pointed out by Davis [1994] balance between production and dissipation is rarely completely satisfied and in many cases there may be leading order deviations from such a balance. In cases where the mean scalar profile has maxima or minima the expression (1) obviously cannot be valid, since the eddy diffusivity would be infinitely large at such points.

[3] In this paper, we will argue that the Osborn-Cox model, nevertheless, in a certain sense is a more general and exact model than what is suggested by the original derivation. We will present an argument as well as a numerical experiment suggesting that a good model for vertical diffusion of any scalar following fluid particles in strongly stratified turbulence is obtained by replacing temperature by buoyancy in the expression (1).

2. Gradient-Diffusion

[4] Consider a scalar, ϕ, which is being mixed by incompressible turbulence. Defining 〈〉 as a mean value over horizontal planes, the equation for 〈ϕ〉 can be written as

where w is the turbulent vertical velocity and ϕ′ = ϕ − 〈ϕ〉 is the fluctuating part of ϕ. Molecular diffusion has been neglected in (2) since it can generally be assumed that it is much weaker than turbulent diffusion. The gradient-diffusion hypothesis states that the turbulent scalar flux is proportional to the scalar mean gradient, that is

where the constant of proportionality, K_{ɛ}, is the eddy diffusivity. If (3) is valid, 〈ϕ〉 satisfies the classical diffusion equation

The gradient-diffusion hypothesis is rarely fulfilled in any exact sense [Tennekes and Lumley, 1972]. The main reason for this is that turbulence, in contrast to a random walk of molecules, in most cases cannot be regarded as a microscopic process and therefore does not have characteristic length and time scales which are much smaller than macroscopic scales.

[5] Recently, Lindborg and Brethouwer [2008] derived an expression for the growth of the mean square of vertical fluid particle displacements, δz, in stationary stratified turbulence,

Here, the quantity b = −gρ′〈/(Nρ_{o}) is the reduced gravity divided by the Brunt-Väisälä frequency, henceforth referred to as the buoyancy. Defined in this way the buoyancy has the dimension of velocity rather than acceleration. Other quantities are defined as follows: ρ_{o} is the background density, ρ′ is the fluctuating density measured in an Eulerian frame, g is the acceleration due to gravity, N = is the Brunt-Väisälä frequency, ε_{P} = κ_{ρ}〈∇b · ∇b〉 is the dissipation of potential energy per unity mass and δb = b(t) − b(0) is the difference between the buoyancy measured at time t at a fluid particle and the buoyancy measured at the same fluid particle at time zero. The average, 〈〉_{L}, is a Lagrangian average over fluid particles. The formula (5) was derived under the assumption that N and ε_{P} are constant over typical turbulent time and length scales and that the buoyancy Reynolds number, R = ε/N^{2}ν, is large. Here, ε is the mean dissipation of kinetic energy and v is the molecular viscosity. G. Brethouwer and E. Lindborg (Numerical study of vertical dispersion by stratified turbulence, submitted to Journal of Fluid Mechanics, 2008) confirmed the formula (5) by direct numerical simulations (DNS) of stratified turbulence as well as simulations using hyperviscocity. The first term on the right hand side of (5) goes to the finite limit 4E_{P}/N^{2} in a time proportional to E_{P}/ε_{P}, where E_{P} = 〈b^{2}〉_{L}/2 is the mean available potential energy per unit mass of fluid particles. At later times, the second term in (5) becomes dominant and 〈δz^{2}〉_{L} starts to grow linearly. This linear growth allows us to define the eddy diffusivity in a way which is analogous to the definition of molecular diffusivity. According to the analysis of Einstein [1956], the molecular diffusivity can be defined from the growth of the mean square displacement of molecules. Analogously, the vertical eddy diffusivity can be calculated from (5) as

This expression can also be written as ε_{ρ}/(∂_{z}ρ_{o})^{2}, where ε_{ρ} is the dissipation of mean square density fluctuations. Comparing this with (1), we see that (6) is the Osborn-Cox diffusivity for buoyancy. The expression (6) is also equivalent to the Osborn [1980] expression, Γε/N^{2}, for the eddy diffusivity of buoyancy, where Γ = ε_{P}/ε is the flux coefficient.

[6] The condition on the mean scalar, 〈ϕ〉, to satisfy the classical diffusion equation (4) with eddy diffusivity (6) is that 〈ϕ〉 diffuses over a time scale which is much larger than the time at which 〈δz^{2}〉_{L} reaches its asymptotic linear growth. If σ is the typical length scale of the variation of 〈ϕ〉 its diffusive time scale is equal to T_{K} = σ^{2}/K_{ɛ}. Thus the condition is T_{K} ≫ E_{P}/ε_{P}, giving the length scale condition

Since the mean available potential energy, E_{P}, always is limited, l_{v} will decrease with increasing N. For a given macroscopic length scale, σ, the diffusion equation (4) will thus become a better and better approximation with increasing degree of stratification. Therefore, gradient-diffusion with eddy diffusivity (6) may be an exact model for turbulent diffusion of scalars in strongly stratified flows. The reason is that strong stratification imposes the scale separation which is often absent in other flows.

[7] The original derivation of Osborn and Cox [1972] suggests that there are different expressions for the eddy diffusivity of different scalars, each including the mean gradient of the particular scalar being diffused. The derivation based on the expression (5), on the other hand, suggests that the expression (6) is the eddy diffusivity for any scalar which is conserved at a fluid particle. In contrast to the derivation of Osborn and Cox we don't make the assumption that the scalar field is approximately stationary or that the main balance in the equation for the scalar variance is between production and destruction. Therefore, the model should be valid for a freely diffusing passive scalar. In this study, we carry out a numerical experiment to test this. The simulated turbulent field is statistically homogeneous and stationary. It is true that these conditions are very ideal. Nevertheless, the experiment will be a sensitive test of the model since the scalar is allowed to diffuse freely over a long period of time.

3. Numerical Experiment

[8] The Boussinesq equations are solved together with the equation for a passive scalar using a standard pseudo-spectral code. The equations that are solved reads

Here, u is the velocity field, D_{u}, D_{b} and D_{ϕ} are diffusion operators, p is the pressure and e_{z} is the vertical unit vector. The diffusion operators are chosen as

where ν_{h} and ν_{v} are horizontal and vertical hyper-diffusion coefficients, respectively and Δ_{h} is the horizontal Laplace operator. The use of hyper-diffusion ensures that fluctuations of velocity, buoyancy and the passive scalar are dissipated at the very smallest resolved scales. The formula (5) is only valid when the buoyancy Reynolds number, R, is large. In a DNS, the value of ν must be sufficiently large for the Kolmgororov scale, η = ν^{3/4}ε^{−1/4}, to be resolved. For fixed ν and η, R decreases with increasing N and therefore it is extremely demanding to obtain (5) in a DNS with a high degree of stratification. However, if the Navier-Stokes diffusion operator is replaced by a higher order diffusion operator it is not that demanding to obtain (5) in a flow with very strong stratification (Brethouwer and Lindborg, submitted manuscript, 2008).

[9] The equations are solved in a box with horizontal sides L_{x} = L_{y} = 2π for three different box-heights, L_{z} = L_{x}/32, L_{x}/16 and L_{x}/8, using periodic boundary conditions in all three directions. The horizontal resolution is L_{x}/Δx = 256 and the ratio between the vertical and the horizontal grid spacing is Δz/Δx = 1/8. The turbulence is forced by introducing a random forcing term [Lindborg, 2006] into the momentum equation (8), injecting energy into vortical modes at a rate P = 1. The injection is concentrated around horizontal wave number k_{f} = 4, corresponding to a horizontal length scale l_{f} = π/2. Defining a Froude number as F_{h} = P^{1/3}/Nl_{f}^{2/3}, we use the very low value F_{h} = 7.7 × 10^{−3}, which means that the stratification is very strong. The values of the diffusion coefficients are chosen so that turbulent energy is dissipated at the very smallest scales in the box, a requirement leading to the formula ν_{h} = (aΔx)^{22/3}P^{1/3}, where a is parameter of order unity and a corresponding formula for ν_{v}. We use a = 0.7. The time scale over which 〈ϕ〉 is vertically diffused by the hyper-diffusion term is larger than T_{K} by at least a factor of 10^{5}. The diffusion of 〈ϕ〉 is therefore totally dominated by the turbulent process. Initially, the velocity and buoyancy fields are zero, but eventually a statistically stationary state is reached where the energy injected at large scales cascades to the smallest scales where it is dissipated. As described by Lindborg [2006] the flow structure in the fully developed state is characterised by highly elongated “pancake eddies” subjected to shear instabilities. Internal gravity waves are also present.

[10] The passive scalar equation is solved starting at a time when the turbulence has reached a stationary state. The initial profile is a truncated cosine series

of a Gauss profile

where m = 31. It is worth pointing out that an eddy diffusion model with diffusivity of the Osborn-Cox form, K_{ϕ} = ε_{ϕ}/(∂_{z}〈ϕ〉)^{2}, would never be able to describe the diffusion process of a scalar with such initial profile, since ∂_{z}〈ϕ〉 = 0 at the boundaries as well as the centreline. This will be true at all times, since the homogeneous turbulent field cannot break the symmetry of the mean scalar profile.

[11] We carry out three different simulations, successively doubling the value of σ, keeping the ratio σ/L_{z} constant and equal to 0.127. This means that we successively double the box height, starting with L_{z}/L_{x} = 1/32 in run 1 and ending with L_{z}/L_{x} = 1/8 in run 3. The background turbulence field has the same characteristics in each of the runs which means that all mean quantities, such as ε_{P} and E_{P}, have nearly the same value in each run. Thus, the ratio σ/l_{v} is approximately increased by a factor of two each time L_{z} is doubled, while the value of K_{ɛ} is approximately the same in all three runs. The parameters which are varied between the three runs are listed in Table 1. In Table 1 we have also included the ratio between the diffusive time scale T_{K} and the buoyancy time scale N^{−1}. In all three simulations this ratio is much larger than unity, reflecting the strong degree of stratification. The time step is set according to a CFL condition, which gave us ΔtN ≈ 0.1 in each simulation, ensuring that pancake eddies and gravity waves are well resolved. To simulate the relatively slow diffusion process with such a small time step requires very long simulations. Note that run 3, including 256^{3} grid points, was running for more than two hundred thousand time steps, each step including four iterations, since we use a four step Runge-Kutta method. In all three simulations the scalar field evolved in a truly turbulent way, with a normalised variance, 〈ϕ′ϕ′〉/〈ϕ〉^{2}, varying from around 0.1 at the centreline to 0.8 at the boundaries.

Table 1. Simulation Parameters

Number of time

Run

L_{x}/L_{z}

steps (×10^{−4})

T_{K}N

σ/l_{v}

1

32

4.125

2.2 × 10^{3}

4.8

2

16

9.06

8.1 × 10^{3}

9.1

3

8

21.375

32 × 10^{3}

18.5

[12] The solution to the diffusion equation (4) with the initial condition (13) is

To calculate K_{ɛ} we need to calculate the mean dissipation of potential energy. Its spatial mean is calculated as

where is the Fourier transform of b, the star denotes the complex conjugate and the summation is over all wave numbers k = (k_{x}, k_{y}, k_{z}). In the fully developed state, there are short time fluctuations of a few percent of ε_{P}, why we also calculate the mean value with respect to time. The high accuracy of the spectral method together with the very long simulation ensure that we obtain a very accurate value of K_{ɛ}.

[13] In Figures 1, 2 and 3we see the evolution of 〈ϕ(z, t)〉 in run 1, 2 and 3, respectively, together with the analytical solution (15). As can be seen in Figure 1, the turbulent diffusion process follows the analytical solution only qualitatively in run 1, where σ/l_{v} = 4.8. In particular, at t^{☆} = 1.35, where t^{☆} = t/T_{K}, there is a considerable disagreement between the simulated curve and the analytical solution. We would not be able to improve the agreement very much by choosing another value of the eddy diffusivity in the analytical solution. Therefore, we have to conclude that the evolution of 〈ϕ〉 is not very well described by the diffusion equation (4) in run 1, even though the diffusivity (6) seems to provide a reasonably good prediction of the time scale of its evolution. In run 2, on the other hand, where σ/l_{v} is about twice as large as in run 1, there is a good agreement between the analytical and experimental curves, which can be seen in Figure 2. It is possible to discern a minor disagreement, as can be seen around the centreline of the curve for t^{☆} = 0.759. Nevertheless, 〈ϕ〉 evolves in good agreement with (4) and it would be difficult to choose a better value of the eddy diffusivity than the value given by (6). Moving to Figure 3, we see that there is a very good agreement between the experimental and the analytical curves in run 3 where σ/l_{v} = 18.5. At the latest time, t^{☆} = 0.659, the mean absolute difference between the numerical and analytical solution is 0.8% of the centre line value 〈ϕ(L_{z}/2)〉. The relative difference between the curves is less than 2% in the centre region 0.2 < z/L_{z} < 0.8 and takes its largest value of 12% at the boundaries, where the magnitude of 〈ϕ(z)〉 is small. As a direct test of the gradient-diffusion hypothesis (3) we have also calculated the ratio −〈ϕ′w〉/∂_{z}〈ϕ〉 from run 3, normalised by K_{ɛ}. To obtain good statistics we have averaged the curve over time and with respect to the mirror symmetry given by the centreline. Note that it is impossible to obtain converged statistics at the centreline and at the boundaries, since ∂_{z}〈ϕ〉 = 0 at these points. As seen in Figure 4, the curve displays only minor deviations from unity, of the order of one percent, and there is no systematic deviation. Gradient-diffusion with eddy diffusivity given by (6) is therefore very well satisfied.

4. Conclusion

[14] Based on a Lagrangian description of vertical dispersion of fluid particles we have defined the vertical eddy diffusivity in a similar way as the molecular diffusivity is defined in statistical mechanics. The expression (6) which we derive is equivalent to the Osborn-Cox eddy diffusivity for buoyancy. The vertical eddy diffusivity is in (6) defined as a property of the turbulent field, that determines its ability to mix a scalar which is following fluid particles. According to such a definition, different scalars following fluid particles in the same turbulent field should be influenced by the same eddy diffusivity. To test this, we have performed a numerical experiment in which a passive scalar diffuses freely in stratified turbulence. The result of the numerical experiment supports the hypothesis that there is a single expression for the vertical eddy diffusivity for all scalars in stratified turbulence.

[15] The length scale condition (7) will generally be fulfilled for most applications in the oceans. Using typical values of the mean potential energy and the Brunt-Väisäilä frequency given by Munk [1981] we find that l_{v} ≈ 5 m in the main thermocline. According to this study it is therefore sufficient that the vertical scale of variation of the scalar is of the order of one hundred metres for gradient-diffusion with K_{ɛ} = ε_{P}/N^{2} to be a good model. In reality, there are, of course, spatial and temporal variations of N as well as ε_{P} which are not taken into account in this study. Spatial variations that are small on the length scale l_{v} can be taken into account by replacing K_{ɛ}∂_{zz} by the operator ∂_{z}K_{ɛ}∂_{z}. If the temporal variation of K_{ɛ} also is small on the time scale E_{P}/ε_{P}–in the thermocline of the order of one week–then the model is in all likelihood a very accurate model.

Acknowledgments

[16] We thank two anonymous reviewers for constructive criticism. Erik Lindborg acknowledges financial support from the Swedish Research Council.