It is well known that there is an infinite number of ways of constructing a
globally defined density variable for the ocean, with each possible density
variable having, a priori, its own distinct diapycnal diffusivity. Because no
globally defined density variable can be exactly neutral, numerical ocean
models tend to use rotated diffusion tensors mixing separately in the
directions parallel and perpendicular to the local neutral vector at rates
defined by the isoneutral and dianeutral mixing coefficients respectively. To
constrain these mixing coefficients from observations, one widely used tool
is inverse methods based on Walin-type water mass analyses. Such methods,
however, can only constrain the diapycnal diffusivity of the globally defined
density variable

Tracers in the oceans are stirred and mixed preferentially along
isopycnal surfaces

We assume fixed composition, thus allowing one to treat
practical (conductivity) salinity and Absolute Salinity as equivalent, since
the two are then linked to each other by a fixed conversion factor. Note that
all the arguments could be reformulated using the more recent conservative
temperature

A conceptual difficulty with neutral rotated diffusion tensors, however, is
that it is not possible to construct a well-defined and materially conserved
density variable

This implies that the “effective cross-isopycnal mixing” experienced by a
material density variable

In other words, the diapycnal mixing seen by any isopycnal surface, including
the neutral surfaces

Although the issue was raised before

A quantification of this effect in terms of diapycnal diffusion is the first
aim of this work. To do so, we develop a mathematical framework to estimate
the implied diapycnal mixing due to isoneutral mixing on any density surface.
Using the observed ocean climatology, we then quantify the contamination of
diapycnal diffusion by isoneutral mixing for a series of commonly used
density surfaces. We will consider the following:

Our results provide the first estimate of the uncertainties associated with
diagnosing diapycnal mixing in the presence of isoneutral mixing. They
further suggest that their effect might, in fact, be more important than
usually assumed, therefore warranting more attention than it has received.
Another motivation stems from a recent justification for the well-known
one-dimensional advection–diffusion model for heat uptake in the ocean, e.g.

The present work also raises questions about how to measure and interpret the measurement of diapycnal mixing. Indeed measured diapycnal mixing is not easily separated from isoneutral mixing and depends on the choice of density surface used for the diagnostic. Related to that, the mathematical framework we use below clearly reveals that for a given turbulent flux, an infinite number of projections and thus of iso-diapycnal diffusion coefficients, each associated to a choice of density surface, are possible.

Section 2 presents the theoretical framework used for defining effective diffusivities for each variable. We also discuss how our framework relates to similar concepts and approaches previously published. Section 3 presents estimates of the diapycnal diffusion contamination due to isoneutral mixing for the aforementioned five density variables. The sensitivity of the results to the choice of isoneutral mixing and location is also discussed. In Sect. 4, we discuss the wider implications of our findings and the related issue of defining, measuring and comparing mixing coefficients. Section 5 summarizes and discusses the results.

Thermodynamic properties in numerical ocean models are commonly formulated in
terms of

The evolution equation of any material density variable

The diffusive flux of

Schematic showing the neutral plane and neutral direction

We define the

Equation (

To construct an effective turbulent diffusivity

Let

It should be noted that

In

In this section we estimate the effective diffusivity (

Reference density for

As expected, the range of values taken by the reference density of the three
potential density variables increases with the reference pressure.

Figure

Histogram of the decimal logarithm of the squared sine between the
gradient of

Figure

.

The first case (Fig.

To investigate the importance of the localised large departure from
neutrality in the construction of

This calculation shows that the isoneutral contribution to effective diapycnal mixing is very localised spatially with 5 % of
each surface accounting for most of the effective diffusivity for all the density variables under consideration here. However,
even without this top 5 %,

Decimal logarithm of the sine between the neutral vector, and the
gradient of

The largest angles between the neutral vector and the gradient of the density variable are found mostly in the
ACC at all depths for

Mixing of heat and salt in numerical ocean models is commonly parameterised
by means of a neutral rotated diffusion tensor using the dianeutral and
isoneutral mixing coefficients

In this paper, we have presented a new framework for assessing the
contribution of isoneutral diffusion to the effective diapycnal mixing
coefficient

Our results thus suggest that the potential contamination due to isoneutral
mixing should always be assessed for any inference of diapycnal mixing based
on the use of any density variable

Overall, the

Our results show that the evaluation of effective diapycnal mixing using a
sorting algorithm of density

This work advocates for the construction of a density function

The World Ocean Atlas dataset used in this study is available on the NOAA website:

The following steps describe the calculation of the effective
diffusivity coefficient for a given

The reference depth

The neutral vector is calculated as the gradient of the locally referenced potential density.

The sinus of the angle between

The product

The evolution equation for

The authors declare that they have no conflict of interest.

This work was supported by the grant NE/K016083/1 “Improving simple climate models through a traceable and process-based analysis of ocean heat uptake (INSPECT)” and its follow-up NE/R010536/1 “New prOcess-based UndersTanding of ocean heat Uptake with an application to improved Climate pRojections for pOlicy and Planning” (OUTCROP) of the UK Natural Environment Research Council (NERC). Modeling results presented in this study are available upon request to the corresponding author. Edited by: Eric J. M. Delhez Reviewed by: Sjoerd Groeskamp and two anonymous referees