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    The shading gives the mean eddy kinetic energy (cm2 s−1) at 314-m depth (model level 25) for year 1995 from the 1/12° run. The highest eddy kinetic energy (indicated by darker shading) is in the region of the Gulf Stream. The contour (in units of 104 Pa) gives the mean pressure field at the same depth for the same year. The triangle indicates the site of tracer release in the model and the rectangular box corresponds to the plots shown in Figs. 2 –5.

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    Five-day mean fields from the 1/12° run. (a), (b) The pressure fields at 314 m. The color scale is on the left with the units of 104 Pa and contour interval is 80 Pa. (a) The time of tracer release for EXPl/12test. (b) The time of tracer release for EXP1/12. (c),(d) The tracer concentration on the target density layer after the 180 days [both (c) and (d) correspond to the release in (a) and (b), respectively]. The tracer concentration is normalized to have the maximum value of 1. The color scale is on the right with the magenta color showing the negative tracer value. The release site is marked by the triangle. For comparison, the domain of plots is also drawn in Fig. 1.

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    (a)–(d) As in Fig. 2, but for the 1/4° run. The tracer fields in (c),(d) are 360 days after releases.

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    For EXP1/12. (top) The normalized tracer concentration on the target density layer (color scale on the left). (bottom) The effective eddy diffusivity κe (m2 s−1; color scale is on the right). (left) 180, (middle) 365, and (right) 730 days.

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    As in Fig. 4, but for EXP1/4.

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    The 3-yr time series of areas from EXP1/12: from the equivalent variance, πσe2 (thick solid line), from the conventional distance variance, πσr2 (dashed line), the tracer area containing 65% tracer load (dotted line), the tracer area containing the 95% tracer load (dot–dashed line), and the tracer area, Γ (thin solid line).

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    (a),(b) The time series of the equivalent variance, σe2 (thick solid lines), in log–log scale for the run of (a) 1/12° and (b) 1/4°. The three triangles mark 180 days for 1 and 2 yr. The thin solid lines show σe2(0) + 4kt, where k = 15 and 60 m2 s−1 in the 1/12° and 1/4° run, respectively (thin solid lines). The σe2(0) is the variance at the time of release. Note that the thin and solid lines almost overlap. Also plotted are curves of the second power law, λkt2 (dashed lines), and the exponential law, σe2(T)e0.2η(tT) (dotted lines), relevant to the stirring-dominated stage. σe2(T) is the value of σe2 at the time T (70 and 360 days for the 1/12° and 1/4° runs, respectively). See text for the values of straining rate λ and stretching rate η. (c),(d) The conventional distance variance, σr2 (thick solid line). The superimposed line in EXP1/12 is 12kht, where kh = 1000 m2 s−1 (thin solid line).

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    The time series of the equivalent deviation σe from EXP1/12 (solid line) and EXP1/4 (dashed line).

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    The time series of the apparent diffusivity κa (solid line, smoothed with a 30-day filter) and the mean effective diffusivity κe (dashed line) from EXP1/12.

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    The time series of the apparent diffusivity κa (m2 s−1) plotted as a function of the equivalent deviation σe for EXP1/12 (solid line) and EXP1/4 (dashed line).

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    The diffusivity (m2 s−1) over 3 yr: (left) the 1/12° run and (right) the 1/4° run. The background diffusivity k is given by the dashed lines and the mean effective diffusivity κe is given by the solid lines. The thick lines are from the main runs and the thin lines are from the test runs. The dotted lines are κeadjust = κe + κeS.

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    The diffusivity (m2 s−1) as a function of equivalent radius γe (km). The dashed lines are the background diffusivities. The dotted and the solid lines are the effective diffusivities calculated from (8) and (10), respectively. (left) EXP1/12 at day 365 and (right) EXP1/4 at day 730.

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    Length for the contours of tracer from EXP1/12 at day 365 as a function of equivalent radius γe plotted in log–log scale: the equivalent length Le (thick solid line), the actual length L (dashed line), and the minimal length L0 = 2πγe (thick dotted line). The superimposed lines are γe2 (thin solid line) and Le/L0 (thin dotted line).

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    The eddy efficiency ζ plotted against the equivalent deviation σe. The solid line is from EXP1/12 and the dashed line is from EXP1/4.

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    Quantities contribute to mean diffusivities at day 730 of EXPl/12: (a) Mean effective diffusivity κe, (b) κs = ∫ κeC da/∫C da, and (c) apparent diffusivity κa. The x axis is the normalized tracer load encompassed by the tracer contours with 1 corresponding to the total tracer load enclosed by the lowest tracer concentration. In (a), κe (solid line), weighting |∂Ĉ/∂γe|2da/∫ |∂Ĉ/∂γe|2da (dashed line), and the multiplication of the two (dotted line). In (b), κe (solid line), weighting C da/∫C da (dashed line), and the multiplication of the two (dotted line). In (c), ∂/∂A(κeA) (solid line), weighting Cda/∫C da (dashed line), and the multiplication of the two (dotted line).

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Effective Eddy Diffusivities Inferred from a Point Release Tracer in an Eddy-Resolving Ocean Model

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  • 1 National Oceanography Centre, Southampton, Southampton, United Kingdom
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Abstract

This study uses tracer experiments in a global eddy-resolving ocean model to examine two diagnostic methods for inferring effective eddy isopycnic diffusivity from point release tracers. The first method is based on the growth rate of the area occupied by the tracers (the equivalent variance). During the period when tracer dispersion is dominated by stirring, the equivalent variance is found to increase at a rate between the second power law (for a pure shearing flow regime) and the exponential law (for a pure stretching flow regime). The second method is based on the length of the tracer contours. In the framework of equivalent radius, the two methods of inferring eddy diffusivity can be understood as two different averagings over the tracer patch. Over a shorter period of tracer dispersion the two methods give different eddy diffusivities, and only over a longer time when tracer dispersion approaches the final stage of diffusion do they give a similar value of diffusivity. A new diagnostic quantity called stirring efficiency is introduced to indicate different flow regimes by measuring the efficiency of stirring against mixing. The new diagnostic quantity has the advantage that it can be calculated directly from the gradients of tracer distribution without needing to estimate strain rate or background diffusivity.

Corresponding author address: Mei-Man Lee, National Oceanography Centre, Southampton, Empress Dock, European Way, Southampton SO14 3ZH, United Kingdom. Email: mmlee@noc.soton.ac.uk

Abstract

This study uses tracer experiments in a global eddy-resolving ocean model to examine two diagnostic methods for inferring effective eddy isopycnic diffusivity from point release tracers. The first method is based on the growth rate of the area occupied by the tracers (the equivalent variance). During the period when tracer dispersion is dominated by stirring, the equivalent variance is found to increase at a rate between the second power law (for a pure shearing flow regime) and the exponential law (for a pure stretching flow regime). The second method is based on the length of the tracer contours. In the framework of equivalent radius, the two methods of inferring eddy diffusivity can be understood as two different averagings over the tracer patch. Over a shorter period of tracer dispersion the two methods give different eddy diffusivities, and only over a longer time when tracer dispersion approaches the final stage of diffusion do they give a similar value of diffusivity. A new diagnostic quantity called stirring efficiency is introduced to indicate different flow regimes by measuring the efficiency of stirring against mixing. The new diagnostic quantity has the advantage that it can be calculated directly from the gradients of tracer distribution without needing to estimate strain rate or background diffusivity.

Corresponding author address: Mei-Man Lee, National Oceanography Centre, Southampton, Empress Dock, European Way, Southampton SO14 3ZH, United Kingdom. Email: mmlee@noc.soton.ac.uk

1. Introduction

It is well established that a scalar tracer released in a turbulent flow will be subject to stirring by the shear and mixing by the molecular diffusion. The stirring process is adiabatic with the tracer contours stretched and gradients sharpened. This creates small-scale features that molecular diffusion eventually acts upon, leading to irreversible mixing and smoother tracer gradients. Thus, the action of diffusion is enhanced by the shears of advective flow, and one seeks an effective diffusivity that incorporates stirring-enhanced mixing. See Garrett (2006) for a review of stirring and mixing by turbulence in the ocean.

In the last decade or so, oceanographers have tried to infer eddy diffusivity by releasing inert tracers such as SF6 into the ocean. Most tracer release experiments focus on estimating diapycnal mixing in a particular environment such as in the quiet main thermocline of the subtropical gyre (Ledwell et al. 1998), within the convection region in the Greenland Sea (Watson et al. 1999), over rough topography in the abyssal Brazil Basin (Ledwell et al. 2000), or associated with salt fingers in the main thermocline of the tropical Atlantic (Schmitt et al. 2005). Estimates of diapycnal diffusivity vary from 0.1 cm2 s−1 in the main thermocline to 10 cm2 s−1 or more in the presence of rough topography. Eddy diffusivity along isopycnals is rarely estimated from these experiments because it is almost impossible to recover all the tracer after the release. However, in the North Atlantic Tracer Release Experiment (NATRE) in 1992, the eddy isopycnal diffusivity was estimated to be about 1000 m2 s−1 (Ledwell et al. 1998). But, it is not clear how robust such an estimate is based on a few observations with long time intervals between them. The purpose of this study is to examine in more detail the method of determining effective isopycnal eddy diffusivity from a point release tracer.

The dispersion of the tracer has three distinct stages (Garrett 1983). Initially, the tracer diffuses from a point into a Gaussian distribution. The area increases linearly with time at a rate proportional to small-scale (background) diffusivity. The second stage begins when the tracer patch reaches a length scale large enough for the stirring by shear flow to dominate the initial diffusion. Finally, when the length scale of the tracer patch is larger than that of the eddies, streaks are wrapped around and merged together, resulting in a more homogenized tracer patch. Thus, at a longer time period, the ensemble-averaged tracer diffuse in a Fickian manner with an eddy effective diffusivity. The three-stage tracer dispersion has been illustrated in a two-layer quasigeostrophic vorticity model calibrated with the NATRE floats data (Sundermeyer and Price 1998).

Some characteristic of flows may be inferred from each stage of the dispersion: the background diffusivity from the initial stage, strain rate from the second stage, and the eddy effective diffusivity over a longer time period. However, the question is how can we identify which stage the tracer dispersion has reached? The most common measure is the rms distance from the center of mass of tracer distribution. We will compare this with the other two measures: the area occupied by tracers (Joseph and Sender 1958) and the length of tracer contours (Nakamura 1996). These two measures are explained in detail in section 2. Briefly here, the “area” method measures the rate of increase of area enclosed by tracer contours. The “contour length” method (called equivalent length) measures the length of interface available for background diffusion. Both are distinctly different from the rms distance to the center of mass method in their insensitivity to the shape of tracer distribution.

The equivalent length method is an elegant way of diagnosing stirring-enhanced mixing (Nakamura 1995). It is based on the fact that nondivergent flows are area (or volume) preserving and so advection alone cannot change the area (or volume) enclosed by tracer contours. However, the shear flow can deform tracer contours and the available interface for diffusion is elongated. Thus, the more complex the geometry of tracer distribution is, the more effective diffusion will be. Diagnosis of effective diffusivity using this method has been applied to quasi-steady chemical tracers in atmospheric models (Nakamura and Ma 1997; Allen and Nakamura 2001) and passive tracers in an idealized Southern Ocean model (Marshall et al. 2006). In this study, we apply it to point release tracers and introduce two new diagnostic tools (the mean effective diffusivity and stirring efficiency) so the evolution of tracers can be quantified.

We use a global eddy-resolving model in two horizontal resolutions at 1/12° and 1/4° with the tracer released at a location as close to the NATRE as possible. The details of the model experiments are given in section 3 and the evolutions of tracers are in section 4. In sections 5 and 6, diagnoses using each method are illustrated and compared. The possible impact of diapycnal processes on the methods is discussed in section 7. There is a summary in section 8.

2. Background

a. The equivalent radius variance

Consider a simple two-dimensional case where the evolution of tracer is controlled by diffusion:
i1520-0485-39-4-894-e1
where C(x, y, t) is the tracer concentration and kx,y are the constant diffusivities in the x and y directions.
The solution for a tracer initially released at a point (x, y) = (0, 0) is a Gaussian distribution (e.g., Sanderson and Okubo 1986):
i1520-0485-39-4-894-e2
where Q is the total tracer load and σx,y2 = 2kx, yt are the variances in the x and y directions. The quantitiy σxσy is called the mean variance. The peak tracer concentration, Cmax = Q/(2πσxσy), decreases inversely with time and the contours of constant tracer concentration form a set of ellipses. The variance in the direction of the principal axes of the ellipses, σx,y2, grows linearly in time at a rate twice the respective diffusivity.

In the ocean, there are spatially varying flows and so a point-released tracer will no longer evolve into a simple Gaussian distribution with elliptical tracer contours. Instead, the action of shearing and stretching by differential advection causes tracer contours to be deformed into irregular shapes with steep gradients and fine filaments. In such situation, it may be preferrable to use some kind of variance without needing to specify any particular direction such as the directions of principle axes. One way to do this is to use the area enclosed by tracer contours (Joseph and Sender 1958; Okubo 1971), which we describe below.

For a given a tracer concentration value c, define an equivalent radius γe(c) that satisfies πγe2 = Ac, where Ac is the area enclosed by the tracer contour C = c. Thus, for any tracer distribution C there is a corresponding radially symmetrical function Ĉ such that Ĉ[γe(c)] = c. From this function, one obtains a tracer-weighted average of equivalent radius squared:
i1520-0485-39-4-894-e3
where the area element is da = 2πγee. We will call σe2 the equivalent (radius) variance and σe the equivalent deviation. A more intuitive interpretation of the equivalent variance is to observe that πσe2 is simply the tracer-weighted average of the area enclosed by tracer contours.

In the case of a Gaussian distribution as in Eq. (2), it is straightforward to show that the equivalent radius variance is twice the mean variance, σe2 = 2σxσy, and πσe2 is equal to the amount of area enclosed by the contour of tracer concentration c1 = Cmaxe−1. It is worth noting that the contour C = c1 encompasses a tracer load of Q(1 − e−1), which is about 63% of the total amount of tracer. This may be compared to the contour of tracer concentration Cmaxe−3, which encloses an area of 3πσe2 that encompasses about 95% of total tracer load. We will use these facts later when comparing different methods of inferring diffusivity.

The equivalent variance, σe2, is different from the conventional distance variance,
i1520-0485-39-4-894-e4
where r = x2 + y2 is the distance to the center of mass and the area element is da = dxdy. In the case of the simple diffusion example in Eq. (2), the conventional distance variance is σr2 = σx2 + σy2. In this case, it is clear that σe2σr2. It can be shown that this inequality is always true for any distribution.
The common practice is to infer diffusivities from the growth rate of variance. In theory any variance can be used, but the inferred diffusivity will depend on the choice of variance for any distribution other than a symmetrical Gaussian distribution. All the variances mentioned so far except the equivalent variance strongly depend on the shape of the tracer distribution. For this reason, we would emphasize the use of equivalent variance and define an apparent diffusivity, as in Okubo (1971):
i1520-0485-39-4-894-e5
Thus, apparent diffusivity is a measure (up to a scale 4π) of how fast the tracer-weighted average of the area enclosed by tracer contours spreads, regardless of the geometrical shape of the tracer distribution. The factor of 1/4 is such that ka gives the same diffusivity for a symmetric Gaussian distribution from a simple diffusion problem.

b. The time evolution of variances

How does a point-release tracer evolve in a turbulent flow? Garrett (1983) described the dispersion of point-release tracers as a three-stage process. Assume the ensemble-averaged tracer distribution over many realizations is a Gaussian distribution and define the tracer area Γ to be the area enclosed by the tracer contour with concentration value c = peak value × e−1. Note that this way of defining Γ uses the property of Gaussian distribution and so Γ is a special case of the equivalent variance σe2.

Initially, tracer diffuses from the point of release to a near-symmetrical Gaussian distribution by small-scale diffusion with Γ = 4πkt, where k is the background diffusivity. The diffusive process dominates until the length scale of tracer LC is comparable to the length scale of the flow LU and then stirring by shear flows begins to take effect. This is supposed to take place at the time when the advective time scale LU/U (where U is the velocity scale) is shorter than the diffusive time scale LC2/k (where LC ∼ 2kt, the radius of the circular area 4πkt). This implies a time scale Ta = (4λ)−1 and a tracer length scale LCk/λ, where λ = U/LU is the scale of the strain rate. During the stirring-dominated stage, the distorted tracer patch is thought of as a deformed Gaussian distribution with Γ(t) = Γ0eαλ(tTa), where α is an O(1) constant and Γ0 = Γ|t=Ta. This indicates that the tracer area Γ increases exponentially in time at a rate proportional to the strain rate λ. Finally, when the length scale of the tracer is much larger than that of the flow, streaks of the tracer are wrapped around and eventually merged together by diffusion, resulting in a more homogenised tracer field. The tracer distribution at this final stage is nearly Gaussian with Γ = 4πkht, where kh is called the effective eddy diffusivity.

For some simple cases where the flow is steady, the tracer advection–diffusion equation can be solved explicitly (Okubo 1966). The simplest case is when there is no stretching, no shearing, and only rotation, and so tracer is diffused by background diffusion. In the case of u = u0 + λy, υ = 0 (pure shearing), the mean variance σxσykλt2 for large t (Novikov 1958). In the case of u = ηx, υ = −ηy (pure stretching), the mean variance σxσyk/ηe2ηt for large t (Townsend 1951).

The scalars λ and η are the constant shear rate and constant stretching rate, respectively, see appendix B for definition. So, the mean variance increases in time as a power of 2 for pure shearing flows and exponentially for pure stretching flows. Therefore, Garrett’s prediction of exponential growth of Γ is at least consistent with the case for pure stretching flows. One might expect that in the ocean the variance growth rate is somewhere between the two flow regimes. In any case, during the second stage of tracer dispersion, the increase of variance will be faster than linear, so the apparent diffusivity will be time dependent.

c. The transformed tracer equation

The concept of the equivalent radius is very useful in that apart from inferring the apparent diffusivity it can be used as a coordinate to simplify the tracer advection–diffusion equation. The equivalent radius coordinate has been applied to quasi-steady tracers for studying stirring and mixing in the ocean and atmosphere (Nakamura 1995, 1996; Shuckburgh and Haynes 2003; Marshall et al. 2006). Here, we apply it to a point-released tracer and compare to the apparent diffusivity in the previous section.

Consider the advection–diffusion equation of tracer:
i1520-0485-39-4-894-e6
where u = (u, υ) is the divergence-free velocity and k is the constant background diffusivity. The derivation for rewriting (6) in equivalent radius coordinate can be found in the literature. All our diagnosis uses the isopycnic layer thickness formulation (see appendix A), but for the convenience of discussion the following equations omit the layer thickness.
In equivalent radius coordinates, the tracer equation in Eq. (6) is transformed into a diffusion-only equation:
i1520-0485-39-4-894-e7
where Ĉ(γe, t) is the radially symmetric function as before, κe(γe, t) is the effective diffusivity,
i1520-0485-39-4-894-e8
Le(γe, t) is the equivalent length,
i1520-0485-39-4-894-e9
and L0(γe, t) = 2πγe is the minimal length that a tracer contour can have for enclosing the same amount of area.

There is no explicit advective term in (7) since divergence-free flows are area preserving and so Ĉ (γe) can only be changed by diffusion. That is to say, if there is no background diffusion (k = 0) then the value of tracer concentration enclosing a given amount of area will remain unchanged at all time and so the lhs of (7) will be zero. The crucial point of the transformed tracer equation is that if there are shears then the diffusion of tracer is much more effective than that given by the background diffusivity. The shear-enhanced diffusion is manifested in the effective diffusivity κe, which is the background diffusivity multiplied by a factor of Le2/L02. The Cauchy–Schwartz inequality tells us that Le is always greater or equal to the actual length of the contour, L = ∮C=c dl (Shuckburgh and Haynes 2003). Clearly, LL0, the length of a tracer contour is always greater than the minimal length that a contour can have for the same enclosed area. Thus, the ratio Le/L0 (≥ L/L0) measures how long a tracer contour is relative to the minimal length that a contour can have for the same enclosed area. The idea is that as a tracer contour is deformed by shear flow, the ratio Le/L0 increases, resulting in a longer interface for diffusion to operate and, therefore, a shear-enhanced diffusion.

The effective diffusivity κe in (8) is an averaged diffusivity for each tracer contour (i.e., it can vary across tracer contours but not along the contours). The formulation of (8) implies that κe can be calculated at any instant without prior knowledge of the history of a tracer. However, the disadvantage is that it is necessary to know the value of background diffusivity before one can obtain a realistic value of effective diffusivity.

To get around this problem, we propose an alternative way of calculating effective diffusivity that does not require the knowledge of background diffusivity or the gradients of tracer. Integrating (A5) (from appendix A) with respect to c to give
i1520-0485-39-4-894-e10
where A(c) is the area enclosed by tracer concentration c. The first term is the total diffusive flux of tracer across a fixed contour C = c and so (10) is a flux–gradient relationship. The effective diffusivity can be calculated from equating the total diffusive flux (the first term) to the third term in (10). Interestingly, the first equality in (10) implies that background diffusivity can also be calculated in a similar way.

At this point, it is worth comparing the effective diffusivity κe to the apparent diffusivity κa discussed in section 2a. First of all, the effective diffusivity κe is an averaged value for each tracer contour and so it can vary from contour to contour. This means that κe has one degree of freedom in the horizontal space. In contrast, the apparent diffusivity κa is an averaged quantity for the entire tracer distribution. Second, the effective diffusivity κe in (8) gives an impression of its dependence on the length. However, if the effective diffusivity is expressed using the flux–gradient relationship in (10), then κe can be reformulated in the terms of the change of area enclosed by tracer contours. This bears some similarities to the apparent diffusivity κa, which is related to the rate of change of the mean area enclosed by tracer contours. Before we compare them further, we need to have some kind of averaged effective diffusivity for the entire tracer and not just for each tracer contour.

d. The transformed tracer variance equation

To do this, we make use of the transformed equation for tracer variance.1 Multiply (7) by Ĉ(γe, t) and integrate over the tracer domain,
i1520-0485-39-4-894-e11
where da = L0e = 2πγee is the area element. From (11), the mean effective diffusivity κe can be defined as
i1520-0485-39-4-894-e12
Thus, the mean effective diffusivity κe is the ratio between the rate of change of total tracer variance ∫ 1/2Ĉ2 da and the total tracer gradients squared in equivalent radius coordinate. Alternatively, equate (11) and (12) to obtain
i1520-0485-39-4-894-e13
Thus, the mean effective diffusivity κe is the average of effective diffusivity κe[γe(c)] weighted by the squared tracer gradient. The weighted average means that more weight is placed toward the center of a tracer patch where the tracer contours are more compact. This interpretation of κe will help us to understand the comparison with the apparent diffusivity.
It is also worth comparing the transformed tracer variance equation in Eq. (11) with the conventional tracer variance equation:
i1520-0485-39-4-894-e14
where da = dx dy. By comparing (14) with (12), it can be seen that the effect of shear on tracers is embedded in the tracer gradients when using (x, y) coordinates, whereas it is automatically included in the mean effective diffusivity when using equivalent radius coordinates.
Since the total variance is the same regardless of the coordinates, the lhs of (11) and (14) are identical. Take the ratio to define the “stirring efficiency” as
i1520-0485-39-4-894-e15
By definition, the stirring efficiency measures how much steeper tracer gradients are relative to the gradients that the tracer would have if it had not been distorted by shear flows. Thus, it is an indicator of how efficient differential advection is sharpening the tracer gradients against the background diffusion that acts to smooth the tracer gradient. It is not difficult to see that in fact ζ = κe/k and so ζ also tells us how much efficient diffusion has become as a result of stirring. However, the real advantage of defining the stirring efficiency as we did in (15) is that it simply uses the degree of tracer distortion to tell us something about stirring versus mixing without needing to calculate strain rate and small-scale diffusivity separately. This way of using tracers to obtain flow information is particularly useful for the real ocean where strain rate and background diffusivities are difficult to measure.

e. Comparing effective and apparent diffusivities

We are now ready to compare the effective and apparent diffusivities. Using the transformed tracer equation in Eq. (7), we can express the apparent diffusivity in the following way. First, we rewrite the time derivative of the numerator of the equivalent variance in (3) as
i1520-0485-39-4-894-e16
The last equality uses integration by parts twice. Use A = πγe2 (the area enclosed by the contour C = c) as a variable to rewrite the rhs of (16):
i1520-0485-39-4-894-e17
Since the total amount of tracer is constant, the apparent diffusivity from (5) can be written as
i1520-0485-39-4-894-e18
In this way, the apparent diffusivity is interpreted as the tracer-weighted average of ∂/∂A(κeA). The unusual outcome is that the quantity ∂/∂A(κeA) takes into account the across-contour variation of effective diffusivity κe.

So, (13) and (18) together tells us that the mean effective diffusivity κe and the apparent diffusivity κa can be interpreted as two different ways of averaging κe. There is no particular reason to believe why one should be better than the other. The diffusivities κa and κe can be calculated anytime during the tracer dispersion. During the initial background diffusion-dominant stage, the two diffusivities should give similar values since there is not much tracer distortion. During the stirring-dominant stage, diffusivities κa and κe will deviate from each other since each averaging procedure will bias a different aspect of the tracer distribution. At much longer times when tracer contours are merged and the tracer distribution is close to Gaussian, κa and κe should converge a similar value again since now the ratio Le/L0 (hence, κe) is nearly constant. So, over the longer time period, it is expected all three eddy diffusivities discussed so far will be similar, khκaκe.

3. Model

The model we use is a global ocean model, the Ocean Circulation and Climate Advanced Model (OCCAM; Coward and de Cuevas 2005). Here we only describe the part of model setup that is relevant to the passive tracer experiment. Other model details may be found in Lee et al. (2007). The OCCAM model has a suite of runs with three different horizontal resolutions at 1°, 1/4°, and 1/12°. All runs have the same 66 vertical levels and the same 6-hourly atmospheric fields [from the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR)] for calculating surface fluxes. Note that the applied surface fluxes depend on the surface ocean temperature and so may be different in different runs. For our study, we use the runs at 1/4° and 1/12° resolutions. The horizontal advection of tracers uses the modified split QUICK (MSQ) scheme. This involves a fourth-order accurate advection scheme together with a velocity-dependent biharmonic diffusion. There is no explicit horizontal diffusion. The vertical advection uses simple second-order center differences with an explicit diffusivity of 0.1 cm2 s−1. In the 1/4° run, there is also Gent–McWilliams parameterization and isopycenic diffusion with both thickness diffusivity and isopycnal diffusivity set to be 50 m2 s−1.

a. Tracer releases

To better understand how the diffusivity metrics described in section 2 relate to the real ocean, we perform a series of numerical tracer experiments designed to be as similar as possible to NATRE. Although we cannot replicate the exact conditions of NATRE, the general conditions of an open-ocean pycnocline tracer release is a useful setting for better understanding the different measures of diffusivity. The NATRE release site is located in the southeastern part of the subtropical gyre in the North Atlantic at a depth of about 300 m over a region about 25 km by 25 km in the horizontal. The tracer in the model is injected into nine grid cells in the 1/12° run with 3, 3, and 1 in the x, y, and z directions, respectively, and into one grid cell in the 1/4° run. The center of each tracer patch (the triangle marker in Fig. 1) is as close to that in NATRE as model grids allow. Despite this, there are still considerable differences between the model and observational releases. At 300 m, the model vertical grid spacing is about 34 m, much greater than the 2-m vertical spread in the NATRE. The tracer in the models is set to be uniformly 1 throughout the initialization grid cells. In NATRE, the tracer is released along cruise tracks and so they are streaky and discontinuous.

When the tracer varies over a length scale, LC = δΔx, the biharmonic diffusion associated with the horizontal advection scheme implies a diffusivity k′ = 1/16Ux)3/(LC2) (where δ is a scalar, U is the velocity scale, and Δx is the grid spacing; Webb et al. 1998; Lee et al. 2002). The biharmonic diffusion becomes larger than the advection when k′/Lc2U/LU, which implies δ ≤ 1/2(LUx)1/4. Let U ∼ 4 cm s−1 and the strain rate λ ∼ 10−6 s−1 (these are the averaged values over the tracer patch in the 1/12° model run), then the length scale of the flow is LU = U/λ ∼ 40 km and the scale of tracer is LC ∼ 1/2 Δx. So, the scale-selected biharmonic diffusion ensures that tracer filaments are marginally resolved at the model grid scale.

The models start from 1985 (see Coward and de Cuevas 2005 for model startup). Tracers are integrated online, but because of the limitation of disk space and computational resources we can only run one tracer at one time. The model tracers are not released at the same year as in NATRE, but they were released at the same month whenever possible. For the 1/4° run, a test run of the tracer was started in May 1993 and run for 360 days (called EXPl/4test). The tracer was reinitialized in May 1994 and run until 2002 (called EXP1/4). For the 1/12° run, a test run was started in August 1994 and run for 260 days (called EXPl/12test). The tracer was reinitialized in May 1995 and run until 2000 (called EXP1/12). The two test runs allow us to assess the sensitivity to the release time. Our main results will be focused on the first 3 yr of two main runs: EXP1/4 and EXP1/12. For convenience, we use elapsed days since the release without referring to specific months and years.

b. Method of binning

The z-level data is binned into 10 density layers defined by potential density σ3 referenced to 300 m. The potential density classes are 27.1, 27.3, 27.5, 27.7, 27.9, 28.1, 28.3, 28.5, 28.7, and 28.9 (in kg m−3). We use the same binning procedure as in Lee et al. (2007), so ensuring properties (i.e., mass and tracer substance) are conserved. On the density layer, the tracer concentration is defined to be the layer-thickness-weighted mean τ̂(x, y, t) = /h, where the overbar is the vertical integral over the target density layer. Note that any binning procedure will inevitably create averaged tracer values, so a tracer on a density layer might appear to be more mixed than on a z level. However, we tested many different choices of density bins and the results do not change qualitatively. For the rest of paper, we will use the notation τ rather than τ̂ for convenience. The target density surface in NATRE is 28.05. In the model, the density layer that contains most of the initial tracer substance is the density layer 27.9, which will be our model target density layer.

Once properties are binned into isopycnic layers, we perform a second binning procedure by binning properties according to the tracer value on the target density layer. The tracer-based binnings allow us to calculate quantities such as the area enclosed by tracer contours on the target density layer. As the tracer concentration is diluted in time, we need to have tracer bins that also vary with time, unlike the density bins, which are time independent. We choose the values for tracer bins according to an exponential function {Cmax (t)exp[3(n/N − 1)], n = 1, N}, where Cmax (t) is the maximum value of C(x, y, t) in the target density layer. The number of tracer bins is N = 20 and N = 40 for the 1/4° and 1/12° runs, respectively. The smaller number of bins for the 1/4° run is to avoid too much noise caused by the tracer occupying very few grid cells initially. Two additional bins, −1 and 0, were added to account for negative tracer values. The negative value tracer typically takes up about 0.2% of total tracer substance.

In section 2, we have assumed no diapycnal flow. However, models do have diapycnic flows and so the total tracer substance in the density layer will not be exactly conserved. Between 5-day means, tracer substance in the target density layer changes by no more than 0.3% in both 1/4° and 1/12° runs. By the end of 3 yr about 60% and 70% of the total tracer substance remains in the target density layer for the 1/12° and 1/4° run, respectively. We will discuss how this might affect diagnostic methods in section 7.

4. Evolution of tracer

As in NATRE, the tracers in the model were released in a region where eddy kinetic energy is relatively low compared to that in the vicinity of the Gulf Stream. Figure 1 (shading) shows annual mean eddy kinetic energy at 314 m (model level 25) for the year 1995 from the 1/12° run. The superimposed contours show the year 1995 annual mean pressure field at 314 m, indicating the large-scale circulation of the subtropical gyre. However, flows are by no means quiescent. At the time of release, the pressure at 314 m in the 1/12° run is filled with small-scale turbulence (Figs. 2a,b, the relative size of box is drawn in Fig. 1). The eddy-rich structure has a great impact on the initial spreading of tracer. In EXPl/12test, the release site was near the southwestern edge of a cyclonic eddy (Fig. 2a). At day 180, the tracer becomes elongated in the east–west direction and the center of mass is to the southwest direction following the tail end of the cyclonic eddy (Fig. 2c). In contrast, the release site in EXP1/12 was near the southwestern edge of an anticyclonic eddy (Fig. 2b). As a result of following the anticyclonic eddy, the center of mass at day 180 is to the northeast direction (Fig. 2d), opposite to that in the test run EXPl/12test.

On the other hand, the tracers in the 1/4° run are not as sensitive to the time of release as in the 1/12° run. This is because there is no small-scale eddy field in the 1/4° run, as shown by the pressure fields (Figs. 3a,b). For both EXP1/4 and EXPl/4test, the center of mass at day 180 is to the southwest and the spatial pattern remains near-circular with only slight deformation (Figs. 3c,d).

In NATRE, the center of mass moves to the southwest after 6 months, similar to EXPl/12test. However, as we have demonstrated this clearly depends on the flow field at the time of release. Since the model does not reproduce exactly the same flow in real time, we do not expect the model tracer to have the same distribution as in NATRE. Our focus is on the characteristics of the spatial pattern. The tracer in the 1/12° run is similar to NATRE in that after 180 days it has a rich spatial structure with filaments and pinched tracer contours, which are completely absent from the 1/4° run.

The continuous straining by the flow in the 1/12° run eventually breaks up the tracer into four or five patches (Fig. 4, top panels). After 365 days, it spreads over 10° in the east–west direction and 7° in the north–south direction. By the end of 2 yr, the tracer has spread over a considerably wider region (the top-right panel shows part of whole region covered by tracer). In comparison, tracer in the 1/4° run is slow to develop (Fig. 5, top panels). It begins to stretch in the east–west direction after 180 days and becomes more distorted at the end of 2 yr. However, it remains as a self-contained entity without fine filaments or breakups. Such differences between the two model tracers are due to the different strain rate of the flows, which will be discussed further in later sections. But first, we present the diagnosis of effective eddy diffusivity.

5. The apparent diffusivity κa

a. The equivalent radius variance

Figure 6 shows the time series of the equivalent variance, σe2, and the conventional distance variance, σr2. To take a closer examination of σe2, we plot the time series in log–log scale (Fig. 7, left panels). The initial linear growth period of σe2 seems to be reproduced well by σe2(0) + 4kt (thin solid lines), where σe2(0) is the variance at the time of release, and the background diffusivity k is set to be 15 and 60 m2 s−1 for the 1/12° and 1/4° run, respectively. For the 1/12° run, there is no explicit horizontal diffusion and so the background diffusivity arises from the implicit biharmonic diffusion in the horizontal advection scheme. For the 1/4° run, the larger background diffusivity is due to the additional explicit diffusivity of 50 m2 s−1 from the isopycnal diffusion. So the mean numerical diffusivity is estimated to be of order 10–15 m2 s−1. These values are large compared to the estimate of NATRE, where the diffusivity is about 2 m2 s−1 for the scale between 1 and 10 km.

During the stirring-dominant stage, we compare the growth of σe2 with two curves: one is λkt2, corresponding to the pure-shearing flow regime, and the other is σe2|t=Teαη(tT)., corresponding to the pure-stretching flow regime. We set T = 70 and 360 days for EXP1/12 and EXP1/4, respectively, and the strain rate λ = uy + υx and stretching rate η = uxυy to be the yearly-averaged values for the corresponding period (first year for EXP1/12 and second year for EXP1/4; see appendix B). Although the parameter α is predicted to be of the order of 1 (Garrett 1983), here the value 0.2 gives us a closer fit to σe2 for both 1/12° and 1/4° runs. We see that in the 1/12° run the rate of increase lies between the t2 law (for a pure-shearing flow regime) and the exponential law (for a pure-stretching regime). Using the observed dye data Okubo (1971) suggested a t2.3 law, while from the NATRE tracer Ledwell et al. (1998) diagnosed a law close to t2. It is less clear for the 1/4° run because the stirring-dominant stage is not long enough yet to separate t2 from exponential growth.

If we assume that variance is controlled by the pure-shearing flow regime, σe2λkt2, then σeλkt. The quantity ∂σe/∂tλk is called diffusion velocity (Okubo 1968). This means that the rate of increase of equivalent deviation σe is controlled by the combined effects of shearing and background diffusion. To verify this scaling using the two model runs, we need to choose the period from each model run when the rate of variance increase is closest to the power of 2 law (i.e., the first year for EXP1/12 and the second year for EXP1/4). From the diagnosed λ and k over these periods, we estimate the diffusion velocity λk to be about 0.42 and 0.27 cm s−1 for EXP1/12 and EXP1/4, respectively. These diffusion velocities are comparable to the estimate of 1 cm s−1 by Joseph and Sender (1958). The time series of equivalent deviation σe (Fig. 8) shows that the increase of σe is about twice in EXP1/12 than in EXP1/4, which is close to the ratio of diffusion velocity 0.42/0.27 ∼1.6.

Garrett (1983) estimated that the time scale for tracer dispersion to reach the final stage of diffusion is about 1 yr. His estimate is based on the exponential growth rate during the stirring-dominant stage and kh = 1000 m2 s−1 for eddy diffusivity, k = 10−2 m2 s−1 for background diffusivity, 10−6 s−1 for strain rate, and α = 0.5. For our model, if we take background diffusivity k = 10 m2 s−1 and α = 0.2, then we also obtain an estimated time scale about 365 days. If we instead use the second power law, then 4kht = λkt2 implies t ∼ 4kh/(λk) ∼ 12 yr. However, the growth rate of variance in the model is slower than exponential but faster than the second power, and so it is likely that the time scale for approaching final diffusion is order of 5–6 yr. In any case, it can be seen from Fig. 7 that σe2 has not reached the linearly increasing stage after 3 yr.

So, we cannot estimate the eddy diffusivity kh from the growth of equivalent variance σe2. However, we can still calculate the apparent diffusivity, κa = 1/4(∂σe2/∂t), between two 5-day means (solid line in Fig. 9). Since the rate of growth of σe2 after the initial period is faster than linear, the apparent diffusivity must be time dependent. During the first 500 days of EXP1/12, κa increases to 1500 m2 s−1 and then fluctuates between 1000 and 1800 m2 s−1. Since the tracer in 1/12° and 1/4° models evolves at different time scales, one way to compare the corresponding apparent diffusivity is to see how they vary as functions of equivalent deviation, σe. If we assume the pure-shearing flow regime (σe2λkt2), then κaλktλkσe− the diffusion velocity appears again. Figure 10 shows that when σe = 250 km (corresponding to 450 days in EXP1/12 and 900 days in EXP1/4), the κa is about 350 and 180 m2 s−1 for each run. The ratio of the diffusivities is again closer to that given by the diffusion velocity. This suggests it may not be unreasonable to scale κa with a length scale, say, σe, and the diffusion velocity λk.

b. Comparison with σr2 and Γ

Figure 6 shows that the equivalent variance evolves differently from the conventional variance. During the first 300 days, σr2 increases more rapidly than σe2 and afterward more slowly than σe2. A seemingly robust feature is σr2σe2.

The time series in log–log scale shows that the conventional variance σr2 in EXP1/12 (Fig. 7, top-right panel) seems to approach a linear growth after about 1 yr (note that in the 1/4° run, σr2 has not reached the final diffusion stage yet). As predicted by Garrett (1983), σr2 (in the 1/12° run) indeed approaches the final diffusion stage earlier than σe2 (or his Γ). This means that while the distance to the center of mass is not growing as fast as during the stirring-dominant stage, the area covered by the tracer continues the faster than linear growth, implying that the tracer contours are merged to “fill in” the gaps between contours. This “filling in” can also be seen from the time series of areas inside the tracer contours containing the 65% and 95% tracer load (Fig. 6). We see that πσr2 is close to the 95% curve near the end of 3 yr, but much greater than 95% over the first year because of the much more irregular tracer distribution during the stirring-dominant stage.

From the linear growth of σr2, we estimate the eddy diffusivity according to πσr2 = 12πkht. Note that here we use 12πkht rather than 4πkht because πσr2 is closer to the area with the 95% tracer load, and (see section 2a) for a Gaussian distribution the area containing the 95% tracer load is ∼ 12πkht. This implies kh ∼ 1000 m2 s−1. On the other hand, the equivalent variance σe2 is very close to the 65% curve. Note that neither the 65% nor the 95% curve approaches the final diffusion stage (Fig. 6). Assuming σr2 reached the final diffusion at year 1, then the radius of the tracer patch would be 23kht ∼ 660 km. This suggests σe needs to be about 660 km before tracer dispersion enters the final diffusion stage. If σe follows the second power law, then this would take about 5 or 6 yr.

For completeness, we also calculate Γ (Fig. 6) using the simple relationship ∫C2 da = Q2/2Γ (Garrett 1983). It shows that the time evolution of Γ is very similar to those of the equivalent radius variance, πσe2, and also to the area enclosing the 65% tracer load. We mentioned in section 2a that if the tracer distribution is Gaussian, then πσe2 is equal to the area containing the 65% tracer load and Γ is equivalent to πσe2. Although these two integral properties seem to hold for our tracer, it does not necessarily imply that the tracer distribution is Gaussian for a single realization. In fact, we found cA/∂c is not spatially constant as it would be for a Gaussian distribution (not shown).

6. The effective diffusivity κe

a. Examples

Here, we show examples of κe as a function of tracer contours at a given 5-day mean. The effective diffusivity κe can be evaluated from (8), but we need to first estimate background diffusivity. We could have used the background diffusivity k obtained from the initial linear growth of the equivalent variance σe2 in the previous section, but for an independent estimate we will use the tracer variance equation in Eq. (14). The 3-yr time series of the background diffusivity is shown in Fig. 11 (the thick dashed lines). The large spikes at the beginning of the 1/4° run are due to the steep gradients resulting from the tracer taking up very few grid points initially. The 3-yr average value is about 19.6 ± 4.8 m2 s−1 and 90 ± 22 m2 s−1 for EXP1/12 and EXP1/4, respectively (where ± means one standard deviation). Thus, the background diffusivity estimated previously from the initial growth rate of σe2 lies near the lower bound of the present estimate. As we expected, the background diffusivities for the two test runs (the thin dashed lines) do not differ significantly from those in the main runs.

We also calculate how the background diffusivity varies as a function of tracer contours using the first equality in (10). We choose one example from EXP1/12 at day 360 and one example from EXP1/4 at day 730 and plot them as functions of the equivalent radius (the dashed lines in Fig. 12). The background diffusivity does not seem to vary much across tracer contours.

We now present the equivalent length, Le, the minimum length, L0, and the actual length, L, as functions of tracer contours from EXP1/12 at day 360 (Fig. 13). Note that in calculating lengths all the line integrals are replaced by area integrals using the identity in (A6) to avoid integrating along curves. The equivalent length Le is longer than the actual length L as expected from the Cauchy–Schwartz inequality. The equivalent length Le is closer to the minimum length L0 near the center of the patch and increases to at least 10 times longer toward the edge of the tracer patch. This is due to the fact that the tracer contours at the outer edge of the tracer patch cover wider areas and so allow more stirring by the shear flow. This may be explained by the following scaling. The equivalent length may be scaled as LeAλ/k, where A is the size of the mixing region with strain rate λ and k/λ is the width scale of filaments (Shuckburgh and Haynes 2003). Let Aγe2, then Leγe2λ/k, suggesting Le varies as the square of the equivalent radius γe. So, Le/L0γeλ/k. This means that a larger area enclosed by contours (or, equivalently, a longer tracer contour) corresponds to a higher ratio of equivalent length to minimal length. The example in EXP1/12 shows that Le indeed increases faster than γe but slightly slower than γe2 (Fig. 13). Similarly, Le/L0 increases with γe.

In Fig. 12 (the dotted lines), the effective diffusivity κe = kLe2/L02 is shown as a function of equivalent radius. For the background diffusivity, we should use the spatial-mean value at the specified day rather than the time-mean value. That is, k = 15.7 and 62 m2 s−1 for the 1/12° run at day 360 and for the 1/4° at day 730, respectively. The larger κe toward the edge of the patch reflects the fact that there is a larger ratio Le/L0 as explained in Fig. 13. From the scaling Le2/L02γe2λ/k, we can scale κeλγe2. So, at a given γe, the effective diffusivity κe is scaled by the straining rate. If we compare κe in the 1/12° and 1/4° at the same equivalent radius γe, say, at 200 km, then the ratio of κe is about 3 although the ratio of strain rate gives about 6–10 (Fig. 12).

The alternative way of calculating κe using the flux–gradient relationship in (10) is shown by the solid lines in Fig. 12. They show that κe from (10) is similar to that from (8). Since this method does not depend on the background diffusivity, the similarity between the two reassures us that the spatial-mean choice of background diffusivity for (8) is fairly good. However, such similarity will not hold if we were to use the time-mean background diffusivity for (8), especially for the 1/4° run where the background diffusivity varies considerably with time.

The effective diffusivity can be mapped onto the horizontal plane using the value of tracer at each point in space (bottom panels in Figs. 4 and 5). The mapping is purely for visualization: recall that κe does not vary along the contour of constant tracer concentration. Within each time frame, it can be seen that more deformed tracer contours correspond to larger effective diffusivity. Looking at the tracer patch as a whole, the overall effective diffusivity increases in time, reflecting more distortion of the tracer as time increases. It is worth noting that the effective diffusivity in the 1/4° run at day 730 has a similar order of magnitude to the 1/12° run at day 180. Since the background diffusivity in the 1/4° run is about 4 times larger than that in 1/12° run, this means that the Le2/L02 is much smaller in the 1/4° run than that in the 1/12° run. One can see that the tracer in the 1/12° run at day 180 has steeper gradients with fine filaments. Thus, the same value of effective diffusivity does not necessarily imply the same effect of eddy stirring on the tracer. Different parts of the ocean may have different background diffusivities at different times, thus one needs to be cautious about how to quantify the effect of eddies on the dispersion of tracer. In this sense, the stirring efficiency ζ in (15) may turn out to be a useful quantity. We discuss this after we consider the time series of mean effective diffusivity.

b. The mean effective diffusivity κe

The time series of the mean effective diffusivity κe using (13) is shown in Fig. 11 (the thick solid lines). The effective eddy diffusivity increases in time as the tracer evolves into a more complex distribution. The diffusivity in EXP1/12 starts at about 20 m2 s−1 initially and increases to about 254 m2 s−1 at day 180. After this, the large fluctuations continue and reach 500 m2 s−1 at day 500 and 850 m2 s−1 at day 700 or 800. The increase carries on until the end of 3 yr. EXP1/4 shows a sharp transition at the first 40 days, which is because of steep gradients caused by the initial tracer occupying only one grid cell. After the initial period, there is a gradual increase from 50 m2 s−1 to about 250 m2 s−1 at day 600. The time series of κe from the two test runs EXPl/12test and EXPl/4test are also shown (the thin solid lines). They exhibit slightly different patterns from those in the main runs. In particular, κe in EXPl/4test is consistently lower than in EXP1/4. This can be explained by the different amount of stretching of tracer as shown in the Figs. 3c,d at day 180.

The stirring efficiency ζ in (15) (Fig. 14) is shown as a function of equivalent deviation, σe. In theory, ζ should be at least 1 since effective diffusivity cannot be smaller than the background diffusivity. In our models, this is only true after a certain number of days since initially there is no substantial straining/stretching of the tracer contours and so the tracer patch is small with large gradients at the grid scale. For the 1/12° run ζ is consistently greater than 1 after 70 days and for the 1/4° run after 285 days. Therefore, we can set the time when ζ ≥ 1 to be the time when straining by shear flows begins to operate. This time scale also corresponds well with the time scales when the growth rate of equivalent variance switches from linear to faster than linear.

In the following, we attempt to explain why the stirring efficiency is 10 times larger in the 1/12° run than in the 1/4° run (Fig. 14). If we scale κeκaλkσe (from section 5a), then ζκe/kλ/kσe. This implies that at a given equivalent deviation, σe, the stirring efficiency is scaled as λ/k, representing the competition between stirring and mixing. In our experiments, the ratio of λ/k between the 1/12° and the 1/4° runs is about 6.3.

As stated in section 2d, the true advantage of ζ is that it can be calculated directly from the tracer fields without needing to estimate strain rate or background diffusivity. This may be useful as a way to separate different flow regimes in the ocean by applying it to tracers in the different locations. To get some idea of the order of magnitude of ζ in the ocean, we can in theory estimate ζ from kh/k. Assume kh ∼ 103 m2 s−1, then the stirring efficiency in the real ocean could vary from ∼104 to 500, depending what background diffusivity is considered [according to Ledwell et al. (1998), k = 0.07 m2 s−1 for the scale 0.1–1 km and k = 2 m2 s−1 for the scale 1–10 km].

c. Comparison with the apparent diffusivity

The apparent and mean effective diffusivities plotted together in Fig. 9 shows that κa is twice as large as κe. As explained in section 2e, these two diffusivities are two different ways of averaging effective diffusivity κe. Here, we illustrate this difference using an example from EXP1/12 at day 730. The diffusivity κe is an average weighted by the squared tracer gradient in equivalent radius coordinate. It puts more weight toward the center of the tracer patch where the tracer contours are bunched together and so the tracer gradients are larger (dashed line in the top panel of Fig. 15a). Near the center of the patch the effective diffusivity is relatively smaller (solid line in Fig. 15 a) and so the result of the tracer gradient weighting is a nearly uniform diffusivity across the tracer contours (dotted line in Fig. 15a). In this example, while κe increases from 0 near the center of the tracer patch to 2500 m2 s−1 toward the outer edge, the mean diffusivity κe is only about 450 m2 s−1, which is about the value of diffusivity at the tracer contour containing 10% of the tracer load.

This may be compared to a simpler average where the weighting is just tracer concentration rather than the tracer gradients, κs = ∫ κeĈ da/∫Ĉ da. This average puts more weight away from the center of the tracer patch (dashed line in Fig. 15b). As a result, the weighted diffusivity maintains a similar shape as the unweighted one and the averaged value is about 1400 m2 s−1, which is about the value of diffusivity at the contour containing 50% of the tracer load.

For the apparent diffusivity, the quantity to be averaged is ∂(κeA)/∂A. Although it varies the same way as κe, it has a much larger value toward the outer edge of tracer patch (cf. solid lines in Figs. 15b,c). This is because κe is not constant—it increases toward the outer edge. The weighting for κa also puts more weight near the outer edge of the tracer and so the combination gives κa a value of about 1900 m2 s−1. This is about the value of diffusivity at the contour containing 75% of the tracer load.

Using the tracer contour C65, enclosing the 65% tracer load as a guide, the mean effective diffusivity κe puts 93% of the weight inside the contour C65, whereas the apparent diffusivity κa by definition has exactly 65% of the weight inside the 65% contour. The problem with κa is that the quantity to be averaged is twice of the original κe, which leads to a much larger average value. Intuitively, the simple average κs seems to give a fairly representative average, although it is not clear how to link κs with some equation to give it a physical meaning.

7. The possible effect of tracer loss on estimating diffusivity

The tracer in the model is not conserved on the target density layer. The transfer of the tracer across density layers can be due to both vertical diffusivity (explicit and implicit) and the horizontal advection scheme. To separate the effect of each process, the details of the tracer budget at each grid cell would need to be calculated, which we are not able to do for this study. However, there are clues that may help us to estimate the likely impact of diapycnal processes on our results.

To give an estimate, we consider a simple approximation where the loss of tracer substance is proportional to the tracer concentration (e.g., assume the tracer concentration on either side of the target density layer is negligible and so the higher tracer concentration corresponds to larger vertical tracer gradient and larger diffusive flux divergence):
i1520-0485-39-4-894-e19
where Ċ is diapycnal flux of tracer across a density layer and 1/μ is an spatially averaged time scale. In reality, μ could vary from point to point. From (19), we estimate the additional tracer loss term in the variance equation in Eq. (11) as
i1520-0485-39-4-894-e20
where C* = (∫C2 da)/(∫C da) is the mean tracer concentration in the density layer. Using the rhs of (20), the tracer loss term S can be calculated using the two integral quantities, the mean tracer concentration, C*, and the total loss of tracer substance in the density layer, ∫Ċ da.

Thus, an adjusted effective diffusivity including the effect of the tracer loss term can be defined as κeadjust = κe + κeS, where κeS = S/∫|∂Ċ(γe, t)/∂γe|2 da. Similarly, the adjusted mean background diffusivity is κadjust = κ + κS, where κS = S/∫|∇C|2da.

The adjusted effective diffusivity κeadjust is shown by the dotted lines in Fig. 11. For the 1/12° run, the tracer loss term is about 16% of the rate of change of the tracer variance. The adjustment for the effective diffusivity κeS is −82 ± 88 m2 s−1. For mean background diffusivity, the adjustment is about κS = −3.6 ± 2.9 m2 s−1 (not shown). For the 1/4° run, the tracer loss term is about 14% of the rate of change of the variance. This corresponds to the adjustment of κeS = −16.9 ± 17 m2 s−1 to the effective diffusivity and κS = −6.9 ± 5 m2 s−1to the background diffusivity.

How does the diapycnal diffusion affect the equivalent variance? Take the solution for a point release tracer in a three-dimensional uniform-shear flow with u = u0 + λyy + λzz, υ = w = 0 (where λy,z are the constant shears in the y and z direction; Okubo 1968). For a time long enough that the stirring dominates the background diffusion, the mean variance on a horizontal slice can be approximated as σxσykyλyt2 1 + 1/4(λz/λy)2kz/ky (where ky,z are the constant diffusivities in the y and z direction). This implies that the variance can be affected by the combination of vertical shear of horizontal flow and vertical diffusivity (i.e., the tracer is diffused vertically to different levels and advected horizontally before being diffused vertically back to the original level). Let λz/λyLy/Lz ∼ 3 × 102 (where Ly ∼100 km and Lz ∼ 300 m are the horizontal and vertical length scale of eddies) and kz/ky ∼ 10−6 (where kz ∼ 0.1 cm2 s−1 and ky ∼ 10 m2 s−1 are the estimates of model vertical and horizontal numerical diffusivities). Using these estimates, the additional term would change the mean variance by 1%. This implies that in our model the influence of vertical shear/diffusivity is small. It would not be unreasonable to expect the equivalent variance to have a similar behavior. For the real ocean, if we take ky = 2 m2 s−1 for the scale 1–10 km, then the vertical shear term can alter the variance by 5%. If we take ky = 0.07 m2 s−1 for the scale 0.1–1 km then the ratio kz/ky becomes much larger. However, at this scale eddies becomes smaller so the ratio λz/λy becomes smaller. Thus, it is not clear that at the scale of 0.1–1 km what would be the impact of the vertical shear/diffusivity on the variance over a horizontal slice. Futher investigation is needed in order to properly quantify the effect of vertical processes on the variance. However, this is beyond the scope of our study.

8. Summary

Lagrangian observations such as tracers and floats are frequently used to estimate eddy diffusivity. Because of the sparse spatial–temporal coverage of the data, these estimates are inevitably uncertain. In addition, the relationship between different methods of inferring diffusivity is often not clear, which makes it difficult to interpret the meaning of eddy diffusivity. This study examined two diagnostic methods that are applied to point release tracers.

The tool used in this study is based on the concept of equivalent radius, γe. It allows us to bypass x and y coordinates and to concentrate on the intrinsic nature of tracer dispersion. The variance of equivalent radius, σe2, for example, is related to the average of the area enclosed by tracer contours. The apparent diffusivity, κa, is thus defined as the growth rate of equivalent variance. The initial tracer dispersion is dominated by small-scale background diffusion with σe2 increasing linearly. From this, we diagnose a mean numerical diffusivity associated with the model’s advection scheme to be about 10–15 m2 s−1. After the initial diffusion stage, the tracer dispersion is dominated by the stirring due to shear flows. In our 1/12° run the growth rate of equivalent variance during this stage is between the pure shearing flow regime (power of 2 growth) and the pure stretching flow regime (exponential growth). In the ocean, earlier studies suggested that the variance increases with time between second and third power law (Okubo 1971) and more recent result from NATRE also found a close link to second power law (Ledwell et al. 1998).

Another way of using the equivalent radius is to use it as a coordinate to transform the advection–diffusion tracer equation (Nakamura 1996). The result is a simple diffusion equation with an effective diffusivity κe = κLeq2/L02. As a tracer contour is distorted by shear flows, the available interface for small-scale diffusion increases and so gives a higher value of effective diffusivity. In this context, effective diffusivity is a function of tracer contours and it reflects the geometrical complexity of tracer distribution regardless of the history of the tracer dispersion.

We take a step further to propose a new way of evaluating κe by relating the diffusive flux of the tracer across the tracer contours to the gradients of the tracer w.r.t. the area enclosed by the tracer contours. In this way, the effective diffusivity has a physical meaning (the flux–gradient relationship) in addition to the geometric one. The new way of calculating κe also has the advantage that it does not require prior knowledge of background diffusivity, which is difficult to obtain. On the other hand, it does require tracer fields between short time intervals, which may be difficult to acquire for tracers released in the ocean.

Previous studies (e.g., Allen and Nakamura 2001; Marshall et al. 2006) applied effective diffusivity diagnosis to quasi-steady tracers. For such tracers, the tracer contours can be associated with geographical locations and so the spatial pattern of effective diffusivity can be linked to the spatial characteristics of the flow such as the region of strong mixing. A point release tracer is always evolving in time and so it is not meaningful to follow a given tracer contour. Thus, we introduce a new quantity called the mean effective diffusivity κe, which assigns an effective diffusivity at any instant time for the whole tracer patch rather than for individual tracer contours. It also represents an average of κe weighted by the tracer gradients [cf. Eq. (13)].

The mean effective diffusivity κe may be compared to the apparent diffusivity κa. First, we interpret the apparent diffusivity as the tracer-weighted average of a quantity that takes into account variations of κe across the tracer contours [cf. Eq. (18)]. So, the mean effective diffusivity κe (based on the tracer variance, ∫C2) and the apparent diffusivity κa (based on the equivalent variance, σe2) are in fact two different ways of averaging κe. It is not clear which averaging is more meaningful except that κe represents an average that puts more weight over the area inside a tracer contour containing 65% of the total tracer load. The two diffusivities have a similar value only when the tracer dispersion reaches the final diffusion stage.

A more traditional approach is to infer an overall eddy diffusivity from the evolution of variance at a later time. The idea is that during the final stage of tracer dispersion the variance converges to a linear growth rate (Garrett 1983). In our models, after 3 yr the traditional distance (to the center of mass) variance σe2 seems to approach a linear growth stage that would give an overall eddy diffusivity kh = 1000 m2 s−1. However, the equivalent variance σe2 has not reached the final diffusion stage. We estimate it would take about 5 or 6 yr for σe2 to converge to linear growth. The point is that distance-based variance (e.g., σr2), always reaches the final linear growth stage sooner than area-based variance (e.g., σe2). So, when inferring diffusivity one needs to be cautious and note that the time scale associated the final stage of tracer evolution is different for different methods.

In summary, the apparent diffusivity, κa, and the effective diffusivity, κe, represent different averagings over the tracer patch. Over a longer time when tracer contours are merged and the tracer patch is nearly Gaussian, all three diffusivities kh, κa, and κe are expected to converge to a similar value. In our experiment, this value could be about 1000 m2 s−1.

The tracer dispersion is ultimately an interplay between the stirring due to shear and mixing due to background diffusion. In our opinion, the fundamental quantity is the equivalent radius (which gives rise to the equivalent variance σe2 and the equivalent length Le). The effective diffusivity is a secondary quantity as it is not uniquely defined (e.g., κe, κa, and kh) and during the stirring-dominating stage the diffusivity will be time dependent. If the radius of a tracer patch is scaled as σe, then the rate of increase of the radius of tracer is scaled ∂σe/∂tλk, indicating that stirring and mixing work together to increase the size of tracer patch.

We have introduced a new diagnostic quantity called the stirring efficiency, ζ. It simply measures the degree of deformation (in terms of gradients) of the tracer patch relative to what it would otherwise be without shear flows. Thus, it is an indicator of how efficient the stirring is against mixing. Like the Peclet number, stirring efficiency characterizes the flows in terms of advection versus diffusion. However, the advantage of ζ is that it can be calculated from the tracer fields without needing to estimate the length scale of flow or the background diffusivity. The stirring efficiency may be scaled as ζλ/kσe, which is consistent with the intuition that the efficiency of stirring is a result of the competitions between stirring and mixing.

The stirring efficiency may be useful as a way of assessing models’ simulations of the tracer evolution. For example, the comparison between the stirring efficiency for the tracers in the 1/12° and 1/4° runs suggests that the 1/12° model is about 10 times more efficient than the 1/4° model in terms of stirring the tracers. If the stirring efficiency in the real ocean is estimated to be about 102–104, depending on the length scale considered for mixing, then much lower values of stirring efficiency in the models indicates the deficiency of the models. Such a deficiency may be due to the model’s large numerical diffusion (e.g., the 1/12° run), or due to the combination of the weak strain rate and large explicit diffusion (e.g., the 1/4° run).

This leads to the question of what impact the 50 m2 s−1 explicit isopycnic diffusivity in the 1/4° model has on the tracer simulation. From the perspective of equivalent variance σe2λkt2, the explicit isopycnic diffusivity compensates for the smaller strain rate. For example, the diffusion velocity λk in the 1/4° model is half of that in the 1/12° model rather than 10 times smaller as it would be implied by the strain rate alone. Thus, one can argue larger isopycnic diffusivity is necessary in order to rectify the smaller strain rate. On the other hand, the stirring efficiency is greatly reduced with the presence of explicit diffusion (as explained in the previous paragraph). Thus, one might argue that smaller explicit isopycnic diffusivity for the 1/4° model is preferrable. On top of all this, it is not clear to what degree the explicit isopycnic diffusivity affects the strain rate in the 1/4° model. Since the time scale for the onset of stirring domination is determined by the strain rate, it would always take longer for the 1/4° tracer to catch up to the same effective diffusivity in the 1/12° model. Even when tracers in the two model resolutions give a similar value of effective diffusivity, they will be achieved through different mechanisms. For example, we see in the Figs. 4 and 5 that the tracer in the 1/12° model (day 365) has fine filaments with pinch-off whereas the tracer in the 1/4° model (day 735) remains as a coherent structure. It remains a difficult issue to decide what values of isopycnic diffusivity should be used in the lower-resolution model. The answers depend on the criteria for assessing model performance.

Acknowledgments

We thank Alberto Naveira Garabato for the motivations and discussions of this work. We are especially grateful to the referees whose constructive comments on the earlier drafts helped to improve the manuscript.

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APPENDIX A

The Transformed Tracer Equation

Here, we give a layer formulation and all our diagnostics are done in the layer format accordingly. The evolution equation for a passive tracer in an isopycnal layer between isopycnals σ and σ + δσ is
i1520-0485-39-4-894-ea1
where τ is the tracer concentration, h is the isopycnal layer thickness per unit of density, ∇ is the isopycnal gradient, u is the isopycnal velocity, and τ̇/dt and σ̇/dt are the material derivatives of τ and σ, respectively, representing diffusion and source/sink.
Following Nakamura (1995), a control volume bounded by a tracer contour τ and lying between isopycnals σ and σ + δσ can only be changed through the nonadvective nature of tracer flux across the tracer contour and through the nonadvective nature of density flux across isopycnals. This is expressed as
i1520-0485-39-4-894-ea2
where da is the area element and Aτ is the area inside the tracer contour τ.
For simplicity, we assume that cabelling and diffusive density flux may be ignored (σ̇ ∼ 0). Thus, the volume can only be changed through nonadvective process acting on the tracer:
i1520-0485-39-4-894-ea3
Since there is no forcing of the tracer, τ̇ can be only due to the diffusive flux of tracer. This diffusive flux consists of the horizontal divergence of the horizontal diffusive flux and the vertical divergence of the diapycnal component of diffusive flux:
i1520-0485-39-4-894-ea4
where D = −κ3τ is the three-dimensional diffusive flux in the x, y, z coordinate; D2 is the horizontal component of D; κ is the background diffusivity, which can be either molecular diffusivity (in the real ocean) or numerical diffusivity (in models); and n is the unit vector normal to the isopycnal.
For simplicity, we further assume the diapycnal component, D · n, is small compared to the horizontal component. Thus, from (A3) and (A4), we have
i1520-0485-39-4-894-ea5
where dl is the line element and so da = dl dτ/|∇τ|. The second equality uses Gauss’s theorem. The third equality uses the identity
i1520-0485-39-4-894-ea6
Note that the sign convention is such that τ increases toward the center of the tracer patch, as it would be for a point-released tracer.

To really appreciate the essence of tracer-based coordinates, we need to use the volume enclosed by the contours of tracer as a coordinate. Define Vτ = ∫Aτ h da and τ(V, t) such that τ(Vτ*, t) = τ*(t).

Following Nakamura (1996) and using the equality ∂τ(V, t)/∂t = −[∂τ(V, t)/∂V](∂V/∂t), (A5) can be transformed to
i1520-0485-39-4-894-ea7
where
i1520-0485-39-4-894-ea8
In (A7), κLe2 does not yet have units of diffusivity. To make it more transparent, we use the “equivalent radius” instead of the volume as a coordinate. For simplicity, we assume the region of interest is on a flat plane rather than on a sphere. For a given volume V, we define the equivalent radius γe to be such that V = πγe2h, where h is the mean isopycnal layer thickness for V.

APPENDIX B

The Strain Rate and Stretch Rate

The strain rate in the model was calculated using two different expressions: the rms of strain rate λ = ∂u/∂y + ∂υ/∂x, representing the shearing deformation, and the rms of stretching rate η = ∂u/∂x − ∂υ/∂y, representing the extension (or contraction) deformation. The rms is the average over the region occupied by the tracer.

The three annual mean strain rates in units of 10−6 s−1 are 1.18, 0.77, and 0.88 for the 1/12° run and 0.08, 0.12, and 0.17 for the 1/4° run. The annual mean stretching rates in the same units are 0.95, 0.75, and 0.85 for the 1/12° run and 0.08, 0.12, and 0.16 for the 1/4° run.

Fig. 1.
Fig. 1.

The shading gives the mean eddy kinetic energy (cm2 s−1) at 314-m depth (model level 25) for year 1995 from the 1/12° run. The highest eddy kinetic energy (indicated by darker shading) is in the region of the Gulf Stream. The contour (in units of 104 Pa) gives the mean pressure field at the same depth for the same year. The triangle indicates the site of tracer release in the model and the rectangular box corresponds to the plots shown in Figs. 2 –5.

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3902.1

Fig. 2.
Fig. 2.

Five-day mean fields from the 1/12° run. (a), (b) The pressure fields at 314 m. The color scale is on the left with the units of 104 Pa and contour interval is 80 Pa. (a) The time of tracer release for EXPl/12test. (b) The time of tracer release for EXP1/12. (c),(d) The tracer concentration on the target density layer after the 180 days [both (c) and (d) correspond to the release in (a) and (b), respectively]. The tracer concentration is normalized to have the maximum value of 1. The color scale is on the right with the magenta color showing the negative tracer value. The release site is marked by the triangle. For comparison, the domain of plots is also drawn in Fig. 1.

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3902.1

Fig. 3.
Fig. 3.

(a)–(d) As in Fig. 2, but for the 1/4° run. The tracer fields in (c),(d) are 360 days after releases.

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3902.1

Fig. 4.
Fig. 4.

For EXP1/12. (top) The normalized tracer concentration on the target density layer (color scale on the left). (bottom) The effective eddy diffusivity κe (m2 s−1; color scale is on the right). (left) 180, (middle) 365, and (right) 730 days.

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3902.1

Fig. 5.
Fig. 5.

As in Fig. 4, but for EXP1/4.

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3902.1

Fig. 6.
Fig. 6.

The 3-yr time series of areas from EXP1/12: from the equivalent variance, πσe2 (thick solid line), from the conventional distance variance, πσr2 (dashed line), the tracer area containing 65% tracer load (dotted line), the tracer area containing the 95% tracer load (dot–dashed line), and the tracer area, Γ (thin solid line).

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3902.1

Fig. 7.
Fig. 7.

(a),(b) The time series of the equivalent variance, σe2 (thick solid lines), in log–log scale for the run of (a) 1/12° and (b) 1/4°. The three triangles mark 180 days for 1 and 2 yr. The thin solid lines show σe2(0) + 4kt, where k = 15 and 60 m2 s−1 in the 1/12° and 1/4° run, respectively (thin solid lines). The σe2(0) is the variance at the time of release. Note that the thin and solid lines almost overlap. Also plotted are curves of the second power law, λkt2 (dashed lines), and the exponential law, σe2(T)e0.2η(tT) (dotted lines), relevant to the stirring-dominated stage. σe2(T) is the value of σe2 at the time T (70 and 360 days for the 1/12° and 1/4° runs, respectively). See text for the values of straining rate λ and stretching rate η. (c),(d) The conventional distance variance, σr2 (thick solid line). The superimposed line in EXP1/12 is 12kht, where kh = 1000 m2 s−1 (thin solid line).

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3902.1

Fig. 8.
Fig. 8.

The time series of the equivalent deviation σe from EXP1/12 (solid line) and EXP1/4 (dashed line).

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3902.1

Fig. 9.
Fig. 9.

The time series of the apparent diffusivity κa (solid line, smoothed with a 30-day filter) and the mean effective diffusivity κe (dashed line) from EXP1/12.

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3902.1

Fig. 10.
Fig. 10.

The time series of the apparent diffusivity κa (m2 s−1) plotted as a function of the equivalent deviation σe for EXP1/12 (solid line) and EXP1/4 (dashed line).

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3902.1

Fig. 11.
Fig. 11.

The diffusivity (m2 s−1) over 3 yr: (left) the 1/12° run and (right) the 1/4° run. The background diffusivity k is given by the dashed lines and the mean effective diffusivity κe is given by the solid lines. The thick lines are from the main runs and the thin lines are from the test runs. The dotted lines are κeadjust = κe + κeS.

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3902.1

Fig. 12.
Fig. 12.

The diffusivity (m2 s−1) as a function of equivalent radius γe (km). The dashed lines are the background diffusivities. The dotted and the solid lines are the effective diffusivities calculated from (8) and (10), respectively. (left) EXP1/12 at day 365 and (right) EXP1/4 at day 730.

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3902.1

Fig. 13.
Fig. 13.

Length for the contours of tracer from EXP1/12 at day 365 as a function of equivalent radius γe plotted in log–log scale: the equivalent length Le (thick solid line), the actual length L (dashed line), and the minimal length L0 = 2πγe (thick dotted line). The superimposed lines are γe2 (thin solid line) and Le/L0 (thin dotted line).

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3902.1

Fig. 14.
Fig. 14.

The eddy efficiency ζ plotted against the equivalent deviation σe. The solid line is from EXP1/12 and the dashed line is from EXP1/4.

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3902.1

Fig. 15.
Fig. 15.

Quantities contribute to mean diffusivities at day 730 of EXPl/12: (a) Mean effective diffusivity κe, (b) κs = ∫ κeC da/∫C da, and (c) apparent diffusivity κa. The x axis is the normalized tracer load encompassed by the tracer contours with 1 corresponding to the total tracer load enclosed by the lowest tracer concentration. In (a), κe (solid line), weighting |∂Ĉ/∂γe|2da/∫ |∂Ĉ/∂γe|2da (dashed line), and the multiplication of the two (dotted line). In (b), κe (solid line), weighting C da/∫C da (dashed line), and the multiplication of the two (dotted line). In (c), ∂/∂A(κeA) (solid line), weighting Cda/∫C da (dashed line), and the multiplication of the two (dotted line).

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3902.1

1

The term variance is not to be confused with the variance w.r.t. distribution (e.g., σr2).

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