# Neutrality Versus Materiality: A Thermodynamic Theory of Neutral Surfaces

## Abstract

**:**

## 1. Introduction

## 2. A Critical Assessment of McDougall et al. (2014)

#### 2.1. Objectives of This Section

#### 2.2. Summary of McDougall et al. (2014)’s Arguments

- the definition of the buoyancy force acting on a single fluid parcel entering the dynamical interpretation of the neutral tangent plane Equation (1),
- the assumption that it is physically meaningful to parameterise isopycnal and diapycnal dispersion in terms of second-rank diffusion tensor as proposed by [29],
- the observation that lateral dispersion is about 7 orders of magnitude larger than quasi-vertical dispersion,
- the observed smallness of viscous dissipation in the ocean,
- the assumption that it is legitimate to regard the displacement $\delta \mathbf{x}$ entering the neutral tangent plane Equation (1) as an actual fluid parcel displacement.

- (i)
- involve no small-scale turbulent mixing, in which case the combined (two-step) process is adiabatic and isohaline and so is equivalent to epineutral dispersion (meaning along a neutral tangent plane), or
- (ii)
- the sinking and rising parcels would mix and entrain in a plumelike fashion with the ocean environment, and therefore experience irreversible mixing.

#### 2.3. A Critical Discussion of McDougall et al. (2014)

#### 2.3.1. Expanding on Point (i) of McDougall et al. (2014)

- First, a non-neutral stirring event associated with the displacement $\delta {\mathbf{x}}_{1}$, which as it takes the parcel away from its equilibrium position, must entail a non-zero energy cost and some finite buoyancy force, implying ${b}_{1}=-\mathbf{d}\xb7\delta {\mathbf{x}}_{1}\ne 0$;
- second, a re-laminarisation process during which the fluid parcel seeks to find its closest level of neutral buoyancy, which is also associated with a displacement $\delta {\mathbf{x}}_{2}$ experiencing a nonzero buoyancy force ${b}_{2}=-\mathbf{d}\xb7\delta {\mathbf{x}}_{2}$ and finite energy cost.

#### 2.3.2. Are Neutral Trajectories Really Neutral?

#### 2.3.3. Are Neutral Trajectories Really Adiabatic and Isohaline?

#### 2.3.4. Do Neutral Rotated Diffusion Tensors Really Minimise Spurious Diapycnal Mixing?

^{−4}, the effective diapycnal diffusivity ${K}_{d}^{Veronis}$ exceeds 10

^{−5}m

^{2}/s and becomes dominated by horizontal mixing. Details of the derivations needed to arrive at Equation (17) are given in Appendix A.

## 3. Neutrality and the Energetics Cost of Adiabatic and Isohaline Stirring on Isopycnal Surfaces

#### 3.1. Objectives of This Section

#### 3.2. Link between the Energy Cost of Parcel Exchanges and Lateral Dispersion

#### 3.3. Theory of the Energetics of Two-Parcel Exchanges on Material Isopycnal Surfaces

^{−1}(dbar)

^{−1}, using $\Delta p=10\phantom{\rule{0.166667em}{0ex}}\text{dbars}$, $\Delta \theta =1$ °C, and $({p}_{r}-\overline{p})=1000\phantom{\rule{0.166667em}{0ex}}\text{dbar}={10}^{7}$ Pa yields

#### 3.4. Energy-Based Definition of Global Neutral Surfaces

## 4. Neutrality and Energetics of Parcels Exchanges Using a Non-Material Density Variable

#### 4.1. Objectives of This Section

#### 4.2. Description of Adiabatic and Isohaline Stirring Using Orthobaric Density

^{−1}, $\Delta {\theta}^{*}=10$ °C, and $\Delta p=10$ dbar. This yields $\Delta {\rho}^{LR}\approx {10}^{3}\times {2.10}^{-8}\times 10\times 10={2.10}^{-3}\phantom{\rule{0.166667em}{0ex}}\text{kg}\xb7{\mathrm{m}}^{-3}$. This is significantly smaller than the estimate for potential density away from its reference pressure. This clearly establishes the superior neutrality properties of orthobaric density over potential density variables when assessed in terms of the present energy-based neutrality criterion. This is in contrast to [36], who has claimed that the neutrality of orthobaric density is not superior to that of ${\sigma}_{2}$; McDougall and Jackett’s [36] conclusion, however, derives from the use of a somewhat idiosyncratic definition of neutrality that tends to favour ${\gamma}^{n}$ over other density variables. Specifically, McDougall and Jackett’s [36] approach is based on evaluating the fraction of the ocean over which what they call “the spurious diapycnal mixing” associated with a given variable is greater than ${10}^{-5}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{2}/\mathrm{s}$. They find that this fraction is not smaller for orthobaric density compared to ${\sigma}_{2}$. Whether this is a valid or fair way to assess neutrality is unclear, however, as other criteria exist according to which orthobaric density is clearly more neutral than ${\sigma}_{2}$.

## 5. A posteriori Thermodynamic Justification for Computing $\gamma (S,\theta )$ from the Minimisation of $\left|{J}_{n}\right|$

#### 5.1. Objectives of This Section

#### 5.2. Density/Spiciness Representation of Water Masses

## 6. Summary and Conclusions

- Elementary but rigorous physical considerations clearly indicate that the physical processes for lateral dispersion in the ocean must in general have a non-zero buoyancy, which is confirmed by a survey of the literature on the topic. The concept of epineutral dispersion, therefore, only makes sense if viewed as the aggregate result of many individual non-neutral (i.e., having non-zero buoyancy) stirring events, so that it is only the net displacement $\delta \mathbf{x}={\sum}_{i=1}^{N}\delta {\mathbf{x}}_{i}$ aggregated over N individual stirring events that is approximately neutral and solution of the neutral tangent plane equation $\mathbf{d}\xb7\delta \mathbf{x}\approx 0$;
- It is not true that neutral trajectories obtained as solutions of the neutral tangent plane Equation (1) can describe actual trajectories, contrary to what is usually assumed, because such trajectories implicitly require the existence of non-material sources of heat and salt. It is also not true that neutral tangent planes represent surfaces along which fluid parcels can be exchanged without experiencing (restoring or otherwise) buoyancy forces. Indeed, irreducible thermobaric forces always accompany adiabatic and isohaline parcels exchanges, and will force any pair of fluid parcels out of their original neutral tangent plane as soon as the parcel exchange takes place. If parcel exchanges on neutral tangent planes truly occurred without experiencing buoyancy forces, they would not experience thermobaric dianeutral dispersion;
- The widespread idea that the neutral tangent plane Equation (1) can be justified in terms of momentum considerations appears to be invalid since the quantity $b=-\mathbf{d}\xb7\delta \mathbf{x}$ cannot represent the full vertical force ${F}^{\left(z\right)}$ acting on a fluid parcel experiencing an adiabatic and isohaline lateral displacement owing to its lack of dependence on the horizontal pressure gradient. The new concept of ‘dynamic neutrality’ is introduced to describe lateral displacements satisfying ${F}^{\left(z\right)}=0$ to distinguish it from Mcdougall buoyancy-based ‘statistic neutrality’. In contrast to McDougall’s neutral displacements, ‘dynamic’ neutral displacements occur in the wedge of instability, have a non-zero buoyancy, a negative energy cost (they release available potential energy) and are necessarily transient;
- Since the stirring events making up epineutral/isopycnal/lateral dispersion are usually individually non-neutral, it is argued that the neutral tangent plane Equation (1) can only be a valid model for epineutral dispersion if interpreted in an averaged sense; however, because traditional Eulerian averages give rise to eddy-correlation terms—absent from (1)—it is postulated that (1) can only be justified from first principles as a Lagrangian or quasi-Lagrangian average of the density Equation (5). This requires that the sought-for Lagrangian or quasi-Lagrangian density variable γ whose identification is the ultimate goal of the neutral density theory should be identified prior to the computation of the mean neutral vector appearing in (1); this suggests that γ can only be meaningfully constructed in thermodynamic space, and cast doubt on the possibility to provide a rigorous justification for density variables also varying geographically;
- We established, using both energetics and thermodynamics arguments, that the criterion measuring the degree of neutrality of a material density variable $\gamma (S,\theta )$ is the smallness of the absolute value of the Jacobian term$${J}_{n}=\frac{\partial (\nu ,\gamma )}{\partial (S,\theta )}$$
- The present theory naturally explains why most material density variables fail to be uniformly neutral in the ocean, because they are such that ${J}_{1}\ne 0$ in the following Taylor series expansion for ${J}_{n}$:$${J}_{n}={J}_{0}+{J}_{1}(p-{p}_{r})+O({(p-{p}_{r})}^{2})$$
- It was demonstrated that the neutral and non-neutral stirring events contributing to epineutral dispersion could be characterised in terms of their energy signature, and suggested that the events with negative energy cost $\Delta E<0$—that is, releasing available potential energy—are associated with enhanced lateral dispersion.
- A new mechanism for enhanced lateral dispersion was identified whose source of energy stems from the coupling between thermobaricity and density-compensated temperature/salinity anomalies. Such a mechanism does not exist in a salt-less ocean, and is speculated to act as a physical process for the removal of density-compensated $\theta /S$ anomalies and neutral helicity in the ocean;
- It was established that the use of a neutral rotated diffusion tensor, as is the current practice in numerical ocean modelling, implies that the effective diapycnal diffusivity of all conceivable material density variables is potentially much larger than the value of dianeutral diffusivity used in such tensors, raising the issue of whether the use of such tensors avoids or causes spurious diapycnal diffusion;
- It was established that orthobaric density appears to significantly more neutral based on the present energy-based definition of neutrality than suggested by McDougall and Jackett’s [36] evaluation.

- They connect for the first time epineutral dispersion with the actual stirring events that cause it; moreover, the realisation that epineutral dispersion is actually made up of non-neutral stirring events resolves some longstanding apparent paradoxes and inconsistencies between “neutral thinking”, “turbulence thinking” and “baroclinic instability thinking” that have caused much confusion and controversy in the field;
- They clearly establish the relevance of energetics for categorising the different possible dispersion regimes in the ocean, with epineutral dispersion being associated with energy neutral and unstable processes, whereas diapycnal dispersion is associated with positive energy consumption;
- They dispel the widespread misconception, e.g., [1], that the buoyancy forces involved in parcel exchanges in potential density surfaces are necessarily restoring;
- They clearly indicate that the systematic study of density-compensated (spiciness) salinity/ temperature anomalies, e.g., [48], will be essential to progress our understanding of epineutral dispersion, of the observed smallness of helicity, and of the ocean ”thinness” in $(S,\theta ,p)$ space established by [39];
- They provide a potential unifying framework for discussing a number of widely disparate results all connected to the thermobaric instability identified here, such as the thermobaric instability sustaining solitary Rossby waves discussed by [50], the existence of a spiciness mode [46], the thermobaric production of potential vorticity [47], thermobaric numerical instabilities in isopycnic numerical ocean circulation models [51], the possibility for thermobaricity to cause a form of conditional instability akin to that at the origin of thunderstorms in the atmosphere [52], that [53] speculated might be responsible for past climate change.
- They provide for the first time concrete ways to test the validity of neutral diffusion tensors; for instance if we could establish on the basis of direct numerical simulations or observations that the diapycnal diffusivity of ${\sigma}_{0}$, ${\sigma}_{2}$, ${\sigma}_{4}$ and Lorenz reference density was comparable to observed values of diapycnal mixing, it would unambiguously invalidate the idea that neutral rotated diffusion tensors are necessarily the best possible practice. Until now, to paraphrase Stommel [54], our ideas about neutral density and neutral rotated diffusion have had so far a peculiarly dreamlike quality, and it has been unclear whether any of the premises on which neutral density thinking relies are actually testable or falsifiable. The present results suggest that they might be.

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Appendix B

**Figure B1.**Schematics illustrating the key directions controlling the energy cost of adiabatic stirring. The red arrows define the optimal direction for stirring and is located in the so-called wedge of baroclinic instability whose origin can be traced back to [31]. The neutral direction associated with zero energy cost is indicated by the big blue dot and is perpendicular to the page.

**Figure B2.**Schematics illustrating approximate neutral surfaces and approximate optimal stirring surface obtained by making a continuous surface to be tangent at all points to the neutral tangent plane and to the optimal stirring direction.

## Appendix C

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**Figure 1.**Schematics adapted and modified from Figure 1 of [6], intended to be a: “Sketch of a central seawater parcel being moved adiabatically and without change in its salinity to either the right or the left of its original position in a direction that is not neutral. When the parcel is then released it feels a vertical buoyant force and begins to move vertically (upward on the left and downward on the right) toward its original “isopycnal”". The sentence (direction of stirring) was added to the original figure.

**Figure 2.**Relative positions of isobaric surface (blue), McDougall neutral surface (red), and "dynamic" neutral surface (green).

**Figure 3.**Fluid parcel trajectories, depicted as the red arrows, must lie at the intersection of surfaces of constant potential temperature and salinity for adiabatic and isohaline displacements caused by stirring. Due to the turbulent character of the ocean, fluid parcel trajectories are expected to undergo large lateral displacements responsible for isopycnal mixing being much larger than diapycnal mixing.

**Figure 4.**Schematics of the three main physical situations characterising the two-parcels exchange studied in this paper. (Top panel) Spontaneous exchange taking place on a thermodynamic surface going through the “wedge of instability” thus releasing available potential energy. Following the exchange, the fluid parcels become statically unstable and attracted back to their original neutral surfaces; as they do so, they move further apart from each other, causing enhanced lateral dispersion, while also possibly undergoing some irreversible diffusive mixing through entraining some of the surrounding fluid in the process (not considered in this paper). (Middle panel) Energy neutral parcels exchange associated with regular lateral dispersion. (Bottom panel) Forced exchange on a non-neutral thermodynamic surface not going through the wedge of instability thus requiring an external energy input. Following the exchange, parcels are attracted back to their original surfaces, with a possible reduction of the distance separating them, thus with no or little lateral dispersion, while also possibly undergoing some irreversible diffusive mixing as in the unstable case (not considered in this paper).

**Figure 5.**Schematics illustrating the adiabatic vertical dispersion associated with the exchange of two fluid parcels being initially on the same non-material density surface (such as orthobaric density or neutral density for instance, whose iso-surfaces are depicted by the dotted lines). Before the parcel exchange, the non-material density surface coincides with a material density surface $\gamma (S,\theta )=\text{constant}$, indicated by the purple solid line, on which the adiabatic and isohaline parcel exchange actually takes place. As the result of the parcels exchange, the density of each parcel changes by $\pm \Delta \gamma $, resulting in a net mass loss for the original orthobatic or neutral density density class, and a mass gain for the two orthobaric or neutral density surfaces below and above. Since in physical space, the two fluid parcels are supposed to exchange their position, adiabatic and isohaline stirring on orthobaric or neutral density surfaces must result in the latter moving relative to the material iso-γ surface.

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Tailleux, R.
Neutrality Versus Materiality: A Thermodynamic Theory of Neutral Surfaces. *Fluids* **2016**, *1*, 32.
https://doi.org/10.3390/fluids1040032

**AMA Style**

Tailleux R.
Neutrality Versus Materiality: A Thermodynamic Theory of Neutral Surfaces. *Fluids*. 2016; 1(4):32.
https://doi.org/10.3390/fluids1040032

**Chicago/Turabian Style**

Tailleux, Rémi.
2016. "Neutrality Versus Materiality: A Thermodynamic Theory of Neutral Surfaces" *Fluids* 1, no. 4: 32.
https://doi.org/10.3390/fluids1040032