1. Introduction

The ocean’s transport and uptake of tracers, such as heat and carbon, is strongly dependent on mixing processes (Groeskamp et al. 2017) and strongly constrained by the structure of isopycnal surfaces (surfaces of constant density), which descend from their outcrop at the sea surface down into the ocean interior. Isopycnal mixing has been shown to influence the stability of the climate in the long term (Sijp et al. 2006) and is linked to large-scale climate variability (Busecke and Abernathey 2019). A greater ability to measure the rate of mixing along isopycnals will provide more accurate carbon feedback projections (Gnanadesikan et al. 2015) as the removal of carbon from the upper ocean into the deep ocean depends crucially upon its transport along isopycnals (Sallée et al. 2012). Of particular importance is the isopycnal mixing that takes place in the Southern Ocean, where strongly sloped isopycnals connect the atmosphere to the deep ocean. In this region the uptake of heat (Gregory 2000) and carbon from the atmosphere into the deep ocean will be strongly influenced by isopycnal mixing.

Mesoscale eddies and a host of flows at smaller scales stir fluid along isopycnals, stretching tracers into filaments and increasing the tracer gradient. Eventually, after the stirring has caused the tracer gradient to become sharp enough, molecular diffusion becomes effective and the tracer gets irreversibly mixed (Smith and Ferrari 2009; Dufour et al. 2015). The net effect of eddy stirring along isopycnals is effectively parameterized by along-isopycnal, downgradient diffusion with an isopycnal diffusivity that is orders of magnitude larger than the molecular diffusivity. Likewise, stirring by a 3D turbulent flow transports tracers across isopycnals, and is effectively parameterized by 3D isotropic, downgradient diffusion characterized by a diapycnal diffusivity. In the deep ocean, the canonical value of isopycnal diffusivity is eight orders of magnitude larger than that for diapycnal diffusivity (Waterhouse et al. 2014; Groeskamp et al. 2020). The diapycnal diffusivity being so much weaker than the isopycnal diffusivity makes isopycnal diffusion one of the important drivers of the transport and uptake of tracers. The focus of this paper is to estimate the isopycnal diffusivity from tracer release experiments that provide (typically sparse) information about the evolution of the tracer concentration through time.

For simplicity, ocean models typically parameterize isopycnal diffusion as being isotropic, despite evidence for isopycnal diffusivity being anisotropic (e.g., Ferrari and Nikurashin 2010; Rypina et al. 2012). Recent work by Bachman et al. (2020) has suggestions on how to include anisotropic effects in both the Redi and Gent–McWilliams parameterizations. However, increasing the accuracy of isotropic diffusivity closures is still of great importance due to their widespread implementation in ocean models; isotropic isopycnal diffusion is our focus.

Carrying out large-scale tracer release experiments to estimate isopycnal mixing is labor intensive and expensive, and the data gathered only provide a modest subsample of how tracers are transported. In the Diapycnal and Isopycnal Mixing Experiment in the Southern Ocean (DIMES; DIMES 2009) experiment, a passive tracer was released in 2009 in the eastern South Pacific. Over the next three years the tracer was transported by the Antarctic Circumpolar Current (ACC) into the Scotia Sea, where several cruises sampled its concentration in the water column. From these tracer concentration data, estimates of the isopycnal diffusivity were made (e.g., Klocker et al. 2012; Tulloch et al. 2014; Zika et al. 2020).

To estimate isopycnal diffusivity using only observational data requires practical diagnostics (Abernathey et al. 2013). One example of a practical diagnostic is the Lagrangian diffusivity, which is based on the mean squared separation of a cloud of tracer particles (Taylor 1922; Abernathey et al. 2013). Another example, more relevant to this project, is to use the growth of the width of a Gaussian that has been fit to the tracer concentration. However, to perform the latter method properly requires an ensemble of tracer release experiments, rather than a single release such as obtained by DIMES, to ensure a Gaussian distribution of tracer concentration over the domain (Abernathey et al. 2013).

The difficulties of releasing and tracking many tracer experiments means obtaining data for an ensemble of tracer release experiments is challenging. To make up for having data for a single tracer release, an ensemble of tracer release experiments inspired by observed data (or the salient aspects of observed data) can be carried out using tracer advection–diffusion simulations (see, e.g., Tulloch et al. 2014).

Using simulations of passive tracer release experiments also allows the use of perfect diagnostics, which require full knowledge of the flow, as is available from numerical simulations but not from observational campaigns (Abernathey et al. 2013). Perfect diagnostics to measure mixing include Nakamura’s effective diffusivity (Nakamura 1996), which has been implemented by, for example, Marshall et al. (2006), Abernathey et al. (2010), and Boland et al. (2012). On the average of an ensemble of tracer release experiments, perfect diagnostics and practical diagnostics give broadly consistent estimates for isopycnal diffusivity (Abernathey et al. 2013).

In this project, a diagnostic for the area of tracer patches is proposed and used to estimate lateral diffusivity from simulations of passive tracer release experiments. Early work tracking the area of tracer patches used curves of isoconcentration to define the equivalent radius for a circle which enclosed some fraction of tracer concentration (Joseph and Sendner 1958; Okubo 1971). An equivalent radius variance can be derived from the tracer-weighted average of the equivalent radius squared, whose growth can be related to an effective diffusivity (Okubo 1971; Lee et al. 2009). Forming a circle with the equivalent radius squared gives the tracer-weighted average area that is enclosed by tracer contours (Lee et al. 2009). We derive this same tracer-weighted average area in a different manner, without conceptually redistributing the tracer concentration to be radially symmetric, and propose an efficient numerical method to calculate its evolution in time, which we relate to an isotropic lateral diffusivity.

We use our area diagnostic to investigate whether the diffusivity of a single passive tracer can be estimated based on the growth of its area. In section 2, the diagnostic for estimating the growth of the area of tracer patches is developed and linked to lateral diffusivity. Section 3 describes the model used to simulate passive tracer release experiments. The output of these experiments is then used to estimate a “true” lateral diffusivity based on the growth of the ensemble mean concentration field in section 4. We then compute an estimate of lateral diffusivity for each ensemble member and use the root-mean-square error (RMSE) relative to the true lateral diffusivity to see if accurate estimates of diffusivity can be obtained from a single tracer release and how lower spatial, temporal, and spatiotemporal resolution affects the estimates of diffusivity. Section 5 concludes.

2. Lateral diffusivity and an area diagnostic for tracer patches

d. Numerical methods and example

Numerical simulations provide the tracer concentration in each of a discrete number of grid cells. To generate the functional relationship between tracer concentration and area, we begin by sorting the concentration data at time t from highest to lowest. This gives an array of concentrations (c₁,…, c_N) such that $c_1>\cdots >c_N$, and an array of grid areas (a₁, …, a_N) in the corresponding order (a_n is the area of the grid cell with concentration c_n). We are here handling time implicitly, suppressing the time index for brevity. The discrete version of (1) is

$A(c_n)={\displaystyle \sum _{m=1}^n{a}_m}$(8)

with inverse

$C[A(c_n)]=c_n$(9)

for n ∈ [1, …, N]. With this ordering, the average area (2) discretizes to

$⟨A⟩\approx \frac{{\displaystyle \sum _{n=1}^{N}A}(c_n)c_n{a}_n}{{\displaystyle \sum _{n=1}^N{c}_n}{a}_n}.$(10)

All simulations in this paper use a uniform grid wherein all grid cells have the same area, a_n = a, hence also A(c_n) = na.

The method of sorting the data, calculating the average area, and relating the average area growth rate to a diffusion coefficient is next carried out for an isotropic diffusion experiment without an advecting flow and with the tracer concentration initialized as a two-dimensional Gaussian (henceforth, a “blob”) centered at the origin with a standard deviation of 1. Note that the simulation in this section is carried out in the nondimensional space described in section 3. This means all plots shown in this section are in the nondimensional space. Figure 1 plots the sorted concentration as a function of cumulative area, C(A, t) (Figs. 1a,c) and maps the concentration c(x, y, t) (Figs. 1b,d) for the initial and final time of a diffusion experiment. We compute 〈A〉 from (10) at each time step, observing that 〈A〉 grows almost perfectly linearly with time (not shown). To quantitatively verify this, we fit a linear model to 〈A〉 with time as the predictor, finding a Pearson correlation coefficient (R²) of 0.999 87, indicating an extremely tight linear growth of 〈A〉, as expected from (7). By equating the slope α of this linear relationship with d〈A〉/dt = 4πK from (7), we estimate the diffusivity to be

$\widehat{K}=\frac{\alpha }{4\pi }.$(11)

The diffusivity computed using (11) is $\widehat{K}=0.246$ which is a good estimate to the parameterized diffusion coefficient which was set at K = 0.25.

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Isotropic diffusion experiment of initial concentration set to a two-dimensional Gaussian centered at the origin with a standard deviation of 1. (a),(c) The sorted concentration data as a function of cumulative area, as defined by (9), at the initial and final times, respectively. (b),(d) The initial and final concentration, respectively, over the two-dimensional grid in dimensionless space. All figures for this paper are produced using Makie.jl (Danisch and Krumbiegel 2021).

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-22-0145.1

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3. Model

We estimate the lateral diffusivity in numerical simulations of a two layer quasigeostrophic (QG) flow by applying our diagnostics to the concentration field of a passive tracer that is advected and diffused in the QG system. The lateral tracer diffusivity in our layered QG model is analogous to the isopycnal tracer diffusivity in a 3D stratified fluid. As our model is eddy resolving, we are diagnosing a Redi (1982) diffusivity, rather than an eddy–advective transport (Gent and McWilliams 1990; Gent et al. 1995), which is not considered in this project. In this section we describe how the QG model is set up for advecting and diffusing passive tracers.

a. Two-layer quasigeostrophic model

We simulate two layer QG flow on a β plane (Phillips 1954). This model generates a turbulent flow capturing the essential elements of baroclinic instability driven by sloping isopycnals. To numerically simulate the QG system, we use the GeophysicalFlows.jl package (Constantinou et al. 2021), which solves the equations of motion using spectral methods on a doubly periodic domain.

We use the same background height H for each layer and take the background zonal velocity in the upper layer, U, to be equal but opposite to that in the bottom layer, −U. With these choices, the equations of motion for the streamfunction ψ_i in each layer are

$\left(\frac{\partial }{\partial t}+U\frac{\partial }{\partial x}\right)q_1+J({\psi }_1,q_1)+\frac{\partial {\psi }_1}{\partial x}\left(\beta +\frac{4U}{L_d^2}\right)=\nu {\nabla }^2{q}_1,$(12a)

$\left(\frac{\partial }{\partial t}-U\frac{\partial }{\partial x}\right)q_2+J({\psi }_2,q_2)+\frac{\partial {\psi }_2}{\partial x}\left(\beta -\frac{4U}{L_d^2}\right)=\nu {\nabla }^2{q}_2-\mu {\nabla }^2{\psi }_2,$(12b)

where the Jacobian operator J is defined by

$J(f,g)=({\partial }_{x}f)({\partial }_{y}g)-({\partial }_{y}f)({\partial }_{x}g)$(13)

and all model variables and parameters are detailed in Table 1. The equations of motion, (12), are nondimensionalized using the background lateral velocity U and the Rossby deformation radius L_d, as done by Boland et al. (2012). With these two parameters only, the nondimensionalization is uniquely determined, resulting in the nondimensional equations of motion:

$\left(\frac{\partial }{\partial \widehat{t}}+\frac{\partial }{\partial \widehat{x}}\right){\widehat{q}}_1+\widehat{J}({\widehat{\psi }}_1,{\widehat{q}}_1)+\frac{\partial {\widehat{\psi }}_1}{\partial \widehat{x}}(\widehat{\beta }+4)=\widehat{\nu }{\widehat{\nabla }}^2{\widehat{q}}_1,$(14a)

$\left(\frac{\partial }{\partial \widehat{t}}-\frac{\partial }{\partial \widehat{x}}\right){\widehat{q}}_2+\widehat{J}({\widehat{\psi }}_2,{\widehat{q}}_2)+\frac{\partial {\widehat{\psi }}_2}{\partial \widehat{x}}(\widehat{\beta }-4)=\widehat{\nu }{\widehat{\nabla }}^2{\widehat{q}}_2-\widehat{\mu }{\widehat{\nabla }}^2{\widehat{\psi }}_2,$(14b)

where a hat indicates a nondimensional quantity and, again, all terms are detailed in Table 1. In particular, $\widehat{J}$ is the Jacobian operator from (13) using partial derivatives with respect to the nondimensional coordinates $\widehat{x}$ and $\widehat{y}$.

Table 1

Dimensional and nondimensional model variables for each layer i ∈ {1, 2} (upper group) and model parameters (lower group). The model parameters are fixed for all experiments.

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Numerical values for all model parameters are listed in Table 1. With the Coriolis parameter set for a reference latitude of 60°N, we chose the reference layer height to obtain a Rossby deformation radius around 30 km: specifically, we have L_d = 29 862 m. The other notable controlling parameter is the background mean flow of U = 0.02 m s⁻¹, which provides the same shear as used by Sundermeyer and Price (1998). Other parameters, listed in Table 1, are chosen so that the simulated flow reaches a statistically steady state and approximates a realistic eddying oceanic zonal jet at midlatitudes.

c. An ensemble of tracer mixing simulations

Our main results are based on an ensemble of 50 tracer release simulations, each with the same QG parameters (Table 1). The ensemble members differ only in their initial condition, which is a small-amplitude random noise set in the QGPV anomaly q_i for both layers. The background shear then amplifies this random initial condition through baroclinic instability, generating a turbulent flow in each layer. This flow is spun up for 518 days of dimensional time, after which the flow is statistically steady. At this time, we inject the passive tracer, initialized as a two-dimensional Gaussian (or a Gaussian blob) centered at the origin with a nondimensional standard deviation of unity, approximating a point-source injection of anthropogenic tracer into the ocean (see Fig. 2a). The random initial condition ensures that each ensemble member has a different realization of the turbulent advecting flow. The simulation is then integrated for 1555 days (4.3 years) of dimensional time. The tracer field in both layers for each ensemble member is saved every 100 time steps (approximately every 8.64 days). For illustration, Fig. 2 shows snapshots of the tracer concentration at various times during one simulation.

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Snapshots of a tracer experiment using the model described in section 3. These plots are from the upper layer where the background lateral flow is positive, hence the tracer moves to the right. Note the color bar changes between panels to highlight the spreading of the tracer.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-22-0145.1

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d. Domain size

The choice of domain size affects how long the area of the tracer patch is able to realistically grow for. With a finite computational domain and double periodic boundary conditions, the area of the tracer patch asymptotes to the area of the domain as the tracer approaches a uniform concentration. To inspect how the domain size affects the length of time that a simulation is realistically capturing tracer stirring prior to this asymptotic state, single release tracer experiments were carried out on various domain sizes.

Figure 3 shows the average area of the tracer, computed from (10), for four different square domain sizes in the upper layer of a tracer simulation. The two smallest domain sizes do not capture much of the linear trend before the concentration begins to saturate the domain and the growth of the area plateaus. The domain size of $L\widehat{x}=L\widehat{y}=128$ has a longer period of linear growth, but noticeably starts plateauing around $\widehat{t}=30$, approximately 518 days. The domain size of $L\widehat{x}=L\widehat{y}=256$ grants high-fidelity simulations of tracer mixing until $\widehat{t}=90$ (approximately 4.3 years), before the linear growth stage begins to plateau. To optimize computational efficiency, a zonally elongated domain could be used, as the tracer mixes slightly faster zonally than meridionally due to the β plane. Note that such anisotropic diffusion cannot be represented by our isotropic diffusion framework, as discussed in section 1, and our estimate of κ reflects the isotropic diffusivity that best matches the actual anisotropic diffusion of the tracer patch. However, we hereafter use the square domain with $L\widehat{x}=L\widehat{y}=256$, this being sufficient for our purposes while still computationally fast.

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The average area diagnostic (10) as a function of time for single tracer release experiments in the upper layer over varying sizes of square domains.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-22-0145.1

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4. Results

We run an ensemble of 50 passive tracer release experiments using the numerical model described in section 3, each with a different realization of a statistically steady turbulent flow at the time that the tracer is injected as a two-dimensional Gaussian approximating a point source release. Using the area diagnostic, Eq. (10) presented in section 2, we calculate the growth rate of the average area of the ensemble mean concentration field and whence estimate the diffusivity. We then repeat this for individual ensemble members, as well as for ensemble members that have been spatially or temporally subsampled. At each of these steps toward sparser data, we ask whether we can accurately estimate the true diffusivity obtained from the full ensemble mean tracer concentration. In all experiments, the tracer is released from the same, fixed point in the domain, and thus our diagnostic provides the effective diffusivity appropriate for a region around this release point. If the turbulence were statistically inhomogeneous in space, then ideally one would repeat this method by releasing more tracers in different locations to sample the spatially inhomogeneous effective diffusivity field. Note, however, that since our method focuses on the third stage of growth identified by Garrett (1983), it inherently cannot identify inhomogeneity on a scale smaller than the mesoscale.

a. Diffusivity estimate from the ensemble mean concentration field

For both model layers, Fig. 4 shows the growth of the average area of the 50 passive tracers, the growth of the average area of the ensemble mean concentration field, and a linear fit to the average area of the ensemble mean concentration field as a function of time. While individual ensemble members exhibit a rich and filamentary tracer concentration field, the ensemble mean concentration field grows approximately as a 2D Gaussian. Moreover, the average area of the ensemble mean concentration field grows almost perfectly linearly in time: the Pearson correlation coefficient is 0.9998 and 0.999 994 in the upper and lower layer, respectively. The lateral diffusivity associated with the turbulent flow in this QG system is then estimated from (11), taking α as the slope of this linear fit from the ensemble mean concentration field. This procedure estimates the diffusivity at ${\mathcal{K}}_u=5477{\mathrm{m}}^2{\mathrm{s}}^{-1}$ in the upper layer and ${\mathcal{K}}_{\ell }=5633{\mathrm{m}}^2{\mathrm{s}}^{-1}$ in the lower layer.

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Growth of the average area of the concentration field in the upper and lower layers (a),(c) for the whole simulation of 4.3 years, $\widehat{t}=0\text{–}90$, and (b),(d) for the first year, $\widehat{t}=0\text{–}20$, of the simulation. The average area is shown for individual ensemble members (gray) and the ensemble mean concentration (blue). A linear fit to the average area of the ensemble mean concentration is also shown (dashed orange).

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-22-0145.1

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For comparison, we also fit a symmetric, two-dimensional Gaussian to the ensemble mean concentration field at each saved snapshot and estimate the diffusivity using the linear growth of the width from the fitted Gaussians (Abernathey et al. 2013). Using this established method, we obtain diffusivity estimates of 5559 m² s⁻¹ in the upper layer and 5666 m² s⁻¹ in the lower layer, which are similar to the values that we estimate using our method. Note that these values pertain only to our particular QG system and should not be taken as an accurate estimate of isopycnal diffusion in the real ocean. Rather, the important point is that using our method, the diffusivity values are robustly estimated from the almost perfectly linear growth rate of the ensemble mean concentration’s average area.

We obtain statistics on the above diffusivities by bootstrapping, as follows. Thirty unique ensemble members are selected and a diffusivity is calculated, as above, from the average concentration field of these members. Repeating this 1000 times gives 1000 estimates of the diffusivity, the standard deviation of which provides a measure of the error. The result of this bootstrapping is that ${\mathcal{K}}_u$ and ${\mathcal{K}}_{\ell }$ are estimated to be (5477 ± 5) and (5633 ± 8) m² s⁻¹, respectively. Figure 5 also shows the histogram of these bootstrapped diffusivity estimates.

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Histogram of diffusivity estimates for each ensemble member (blue) and bootstrapped diffusivity estimates for each of 1000 concentration fields obtained by averaging a subset of 30 ensemble members (orange) normalized by probability in the (a) upper and (b) lower layer. The mean and ±1 standard deviation of the diffusivity estimates are shown as red and green circles, respectively, and the true diffusivity is shown by the magenta diamond.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-22-0145.1

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As discussed by Garrett (1983) and in section 2, the best estimate of the diffusivity is obtained using the ensemble mean concentration, rather than any individual ensemble member’s concentration. Thus, we take the values ${\mathcal{K}}_u$ and ${\mathcal{K}}_{\ell }$ as the true diffusivities of our QG system in its statistical equilibrium. Next, we investigate how well these true diffusivities can be estimated using only one ensemble member.

c. Diffusivity estimates from spatially and temporally subsampled data

When considering estimates of diffusivity from a single tracer release experiment in section 4b, we used the full spatiotemporal resolution of the simulations’ archived data: approximately 14.9 km between data points both zonally and meridionally, and approximately 8.64 days between data points in time. This sampling resolution is higher than would likely be used in real ocean tracer release experiments. For example, the tracer released as part of the DIMES experiment was surveyed approximately one year after the release date between 57° and 62°S, at increments of 0.5°–2°, and between 255° and 275°E at increments of approximately 3°–5°. At these latitudes, 1° zonally is approximately 56 km so the zonal distance between samples is approximately 168–280 km, while the meridional distance is approximately 55–220 km.

To investigate how the estimates of diffusivity are affected by lower sampling resolution, we calculate the diffusivity for each ensemble member using a spatial, temporal, or spatiotemporal subsample of the simulation data. Then, we report on the RMSE of these diffusivities from the true diffusivity. We only present the results from the upper layer as the results for the lower layer are similar.

1) Varying spatial resolution

The full spatial resolution of the simulation data from the tracer release experiments is a 512 × 512 grid. We take subsets of this data at varying spatial resolution. The largest subset that we examine divides the domain into four “super” cells, each of which occupies 256 × 256 grid cells from the finest grid. The concentration value assigned to each of these super cells is the concentration from the finest grid cell whose upper left vertex lies at the center of the super cell. The diffusivity of the ensemble members is then calculated as in section 4b for each spatial subset, and the RMSE from ${\mathcal{K}}_u$ is calculated. For now, we use the full temporal resolution, to focus on how the spatial resolution affects estimates of diffusivity.

Figure 6a shows the RMSE of the diffusivity estimates $K_{u,i}$ from ${\mathcal{K}}_u$ as a percentage of ${\mathcal{K}}_u$. As the spatial resolution decreases, the RMSE increases as expected. For all the spatial subsets considered up to zonal and meridional distance of 1910 km, this RMSE as a percentage of ${\mathcal{K}}_u$ does not exceed 13.3%. Once the zonal and/or meridional distance between data points is greater than 1910 km the RMSE increases, peaking at 85.1% of ${\mathcal{K}}_u$. This analysis suggests that at full temporal resolution, it is only once there is a significant distance between data points (greater than 1910 km) that we see a substantial increase in the RMSE as a percentage of ${\mathcal{K}}_u$ and a correspondingly decreased confidence in the accuracy of the diffusivity estimate from a single ensemble member.

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The relative RMSE from the true diffusivity ${\mathcal{K}}_u$ for (a) spatial, (b) temporal, and (c) spatiotemporal subsets of the tracer concentration data. The color bars of (a) and (c) and the left y axis of (b) display the base-10 logarithm of this relative RMSE, ${\log }_{10}(\text{RMSE}/{\mathcal{K}}_u)$. The red text in (a) and (c) and the right y axis in (b) display the percentage RMSE, $100(\text{RMSE}/{\mathcal{K}}_u)$.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-22-0145.1

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2) Varying temporal resolution

The diffusivity estimate from the individual ensemble members was calculated, in section 4b, using simulation time $\widehat{t}=16.5$ (0.8 years) to $\widehat{t}=80$ (3.8 years), containing 128 snapshots (taken every 8.64 days) of archived concentration data. We take temporal subsets by successively halving the temporal resolution, yielding ${\log }_2(128)=7$ diffusivity estimates for each ensemble member, including the one at full temporal resolution. For now, we use the full spatial resolution to isolate the effect that lower temporal resolution has on the accuracy of the diffusivity estimate. Figure 6b shows the RMSE of the diffusivity of the ensemble members from ${\mathcal{K}}_u$ as a percentage of ${\mathcal{K}}_u$, as a function of the 7 temporal subsets. At full spatial resolution, sampling the tracer concentration every 553 days (using two snapshots in our simulations) increases the RMSE as a percentage of ${\mathcal{K}}_u$ to 2.6%, up from 1.1% when sampling every 8.64 days (the full archived resolution). Thus, sampling the tracer concentration infrequently has little effect on the diffusivity estimates of an ensemble member, so long as the spatial resolution is high.

3) Varying spatial and temporal resolution

Lastly, we consider how varying both the spatial and temporal resolution affects the estimates of diffusivity from the ensemble members. To limit the number of subsampling parameters to two, we now take spatial subsets having equal zonal and meridional distances between sampled grid cells. The temporal subsets are as in section 4c(2) above, though we now compute a temporal subset for each spatial subset.

Figure 6c shows the RMSE of the diffusivity estimates for the varying spatiotemporal subsets as a percentage of ${\mathcal{K}}_u$. We see that it is the combination of temporal and spatial subsampling of the tracer concentration that causes the greatest increase in the error of a diffusivity estimate from a single ensemble member. While the meridional and zonal distance between samples is 956 km or less, the RMSE as a percentage of ${\mathcal{K}}_u$ is 11.1% or less for all temporal resolutions considered. As the spatial distance between samples increases beyond 956 km, the RMSE as a percentage of ${\mathcal{K}}_u$ rises rapidly as the temporal resolution decreases, peaking at 154% for the coarsest subsampling considered here.

The DIMES tracer was surveyed after approximately one year at meridional distances between 168 and 280 km and zonal distances between 55 and 220 km. Making a mostly conservative approximation of the DIMES sampling strategy as being every 239 km and every 553 days, we find that, on this spatiotemporal resolution, the RMSE of the diffusivities of the ensemble members as a percentage of ${\mathcal{K}}_u$ is 3.0%. Based on the results from our simulations, this resolution of spatiotemporal sampling has little effect on the RMSE as a percentage of ${\mathcal{K}}_u$.

Our results indicate that the RMSE is affected most when the spatial and temporal resolution are decreased together. This suggests that the error in estimating the diffusivity from a single tracer release experiment may be kept acceptably low if either the temporal or spatial resolution is high. It may also be possible to find an optimal balance between spatial and temporal sampling of tracer experiments that could keep the RMSE of any “observed” diffusivities relative to the true diffusivity to a minimum. Such an optimal sampling strategy could be decided upon in the design stages of a tracer release experiment by running simulations similar to ours, with bespoke parameters for the study region.

5. Conclusions

We have presented a definition for the average area of a tracer concentration field. Further, we used the average area to develop a diagnostic for the effective diffusivity of a turbulent flow field, which is proportional to the linear growth rate of the average area of tracer concentration, following the theory of Garrett (1983).

We first tested this procedure for estimating the diffusivity in a diffusion-only tracer release simulation, finding excellent agreement between the estimated diffusivity and the explicit diffusivity set in the model. We then applied this procedure in an eddy-resolving, two-layer quasigeostrophic model to diagnose the effective diffusivity caused by a turbulent flow. The turbulent flow stirs the tracer into filaments while a relatively weak diffusivity, parameterizing the subgrid-scale motions, ultimately leads to tracer diffusion. We ran an ensemble of such simulations differing only in the particular realization of the turbulent flow field when the tracer is injected. Applying our diagnostic for the average area occupied by a tracer, we find that the area of the ensemble mean tracer concentration increases remarkably linearly with time, as in the diffusion-only experiment. For each of the two layers, we diagnose the effective diffusivity for this particular quasigeostrophic system as proportional to this linear growth rate of the average area of the ensemble mean concentration. We take this effective diffusivity as the “true” effective diffusivity, since the ensemble mean concentration, rather than that of a single ensemble member, provides the most accurate estimate of the system’s effective diffusivity, as shown by Garrett (1983).

Recognizing that one rarely has access to an ensemble, we then examined how accurately the true effective diffusivity can be diagnosed when only one ensemble member is available. In our experiments, the variability of the effective diffusivity from the ensemble members (the mean diffusivity plus/minus one standard deviation) enveloped the true effective diffusivity. The relative RMSE of the effective diffusivity estimates from the ensemble members compared to the true diffusivity was 1.14% in the upper layer and 1.43% in the lower layer. The low relative RMSE (in both layers) suggests that a highly accurate estimate of effective diffusivity can be obtained based on the spreading of a single ensemble member.

Future work could run similar simulations on larger domains with finer resolutions and with other flows (the flow used in this project is arbitrary, which makes comparisons of estimated diffusivities not appropriate). Replicating a flow like that used by Klocker et al. (2012) or Tulloch et al. (2014) would allow the diffusivities estimated from the area diagnostic to be compared to established results. Extending our method for estimating diffusivity to account for anisotopy and tracers with sources/sinks are other avenues for future work.

Real-world experiments in which passive tracers are released into the ocean must rely on measuring the tracer sparsely in space and time. To investigate how the estimated diffusivity is affected by the sparseness of the available data, we applied our diffusivity diagnostic to various spatial and/or temporal subsets of our simulation data. At full spatial resolution, the RMSE as a percentage of the true diffusivity (from the ensemble mean at full resolution) is only 2.6% at the coarsest temporal subsampling we tested (553 days), indicating that if the data can be sampled at high spatial resolution then the temporal resolution does not have a significant effect on the estimates of diffusivity.

At full temporal resolution, the RMSE increases as the spatial resolution decreases, though as a percentage of the true diffusivity, it does not exceed 3.1% when the zonal and meridional distance between data points is 956 km or less. This indicates that, at full temporal resolution, the diffusivity estimated from a single ensemble member is only marginally affected by spatial subsampling. When the zonal and meridional distance between data points is 1910 km or greater, the RMSE as a percentage of the true diffusivity does increase rapidly, weakening confidence in estimates of diffusivity from single ensemble members. The most significant increase in RMSE of the diffusivity from individual ensemble members comes from the combination of lower spatial and temporal resolution, with the RMSE relative to the true diffusivity reaching 154% when samples are taken every 553 days and separated by 3820 km both zonally and meridionally. Using the way the DIMES (DIMES 2009) tracer was surveyed after the first year as a guide for the spatial subsampling, we found that the RMSE as a percentage of the true diffusivity was 3%.

Implementing our method on observational data presents two additional but tractable challenges. First, our method requires knowledge of the tracer concentration in terms of accumulated area, which in turn requires having an area associated with each tracer concentration measurement. While this is naturally provided in ocean model data, for observational campaigns such areas could be generated by the Voronoi tessellation or other methods according to one’s data. Second, observational datasets—as well as realistic ocean general circulation models—must reckon with the nonlinear equation of state, which causes tracers to spread predominantly along neutral surfaces that are mathematically ill defined, rather than along well-defined isopycnal layers as in our QG model. Fortunately, serviceable approximations to neutral surfaces exist, along which the tracer concentration can be mapped and analyzed. For observational datasets, the pressure-invariant form of neutral density (Lang et al. 2020) is recommended over the original neutral density (Jackett and McDougall 1997) as the former filters out the effect of adiabatic heaving of the water column between sampling times and locations. For ocean model datasets, ω surfaces (Stanley et al. 2021) are recommended as the currently most accurate approximately neutral surfaces. Subsequent improvements to the Python ω-surface code¹ also enables their use with observational datasets, requiring only a list of pairs of observational casts that are considered adjacent. This list of pairs can be automatically generated by the Delaunay triangulation, which is the dual graph of the Voroni tesselation that provides the area of these tracer cells. Thus, our diagnostic can be applied to realistic ocean model data and to observational data from real-world tracer release experiments.

These results indicate that it may be possible to design spatial and temporal sampling strategies for oceanographic research cruises that allow for the diffusivity error to be reasonably managed.

Data availability statement.

The code used to generate the simulation data, and all data analysis, can be found and run in the public code repository (Bisits and Constantinou 2021). An example of the saved output from a single tracer release can be accessed at https://figshare.com/articles/dataset/Ensemble_member_simulation_data/20188739, and the saved data analysis of the individual ensemble members and ensemble mean concentration fields can be accessed at https://figshare.com/articles/dataset/data_analysis_TracerMixing_jld2/20188670.