- freely available
Fluids 2017, 2(2), 20; https://doi.org/10.3390/fluids2020020
2. Buoyancy, Locally-Referenced Potential Density, Lagrangian Energetics, and Transience
3. What Does the Ocean Care about? Human Versus Nature, Local Versus Global
3.1. Rebuttal (in Boldface) of Representative Comments about the Human Versus Nature Controversy
- “Tailleux (2016) asserts that `adiabatic and isohaline parcel exchanges can only be meaningfully defined on material surfaces of the form ’. We disagree with this statement.” The fact remains that stirring surfaces are by definition the ones that are invariant upon adiabatic and isohaline parcel exchanges; this constrains them to being defined in terms of material functions of S and alone, whether MGG agree or not. MGG fail to realise this because their construction of LRPD is mathematically incorrect. Mathematically, the only correct way to define LRPD is as the material function , maximising neutrality locally in some sense. This can be done either by following a local version of ’s approach or by following ’s suggestion to minimise the Jacobian term in the vicinity of the point considered. Whether to regard as local or global depends solely on the investigator’s intention to use it as a local or global descriptor of stirring. In neutral density theory, this distinction would be between LRPD and ’s globally-defined empirical neutral density . How these distinctions enter the problem is further clarified below as one goes from Stages 1–3 to Stage 4. “There is no fundamental reason that the ocean should oblige in this regard. Rather, we oceanographers should examine ocean mixing in terms of known physical processes that occur at the in-situ pressure of the mixing.” These are rhetorical arguments, not scientific ones. On the other hand, I think that one should first define the particular problem one is trying to solve, then formulate a mathematically well-posed problem to study it, and finally use the tools that one deems appropriate to solve it. Ideally, one should attempt to formulate testable hypotheses in a way that others can understand and scrutinise. This is the basis of the scientific method. In contrast, MGG seem to be asserting that they can make definitive statements about ocean stirring without considering any particular model of stirring, or that it is somehow possible to claim that there is one and only one possible kind of stirring surface endorsed by Nature based only on the equation of state from seawater, the consideration of a single one-dimensional fluid parcel trajectory, and observations that viscous dissipation rates are small in some unspecified sense. How can this possibly be enough information to say anything meaningful about turbulent ocean stirring?
- “…then Tailleux (2016) incorrectly asserts that the globally defined variable must have some significance to the issue of local mixing processes and that we should attempt to minimize the extra unintended mixing across the iso-surfaces of . There is no reason for the ocean to pay special attention to an arbitrary variable of our construction.” The relevance of my to the local and global descriptions of stirring is justified by the fact that is constructed so that the energy cost of all adiabatic and isohaline parcel exchanges taking place on iso- surfaces satisfies , which characterises lateral stirring in the ocean according to the simple model of stirring considered by . As mentioned previously, solving the NANSE does not require the introduction of any density variable, so MGG are obviously right than there is no reason for the ocean to pay special attention to an arbitrary variable of our construction. I find it therefore baffling that MGG should think that their own construction— which is no more than a particular subcase of the more general class of variables considered by —is the only non-arbitrary density variable somehow “endorsed” by Nature—a belief that is obviously more mystical than rational.
- “Rather than attempting to impose human-conceived, globally-defined variable as a new anthropogenic constraint on ocean mixing dynamics, we instead look to ocean physics for inspiration regarding ocean mixing. The relevant function to be considered when local mixing processes are considered is the locally-defined potential density . The reference pressure of this potential density variable is the local in-situ pressure of this potential density variable at which the mixing is occurring and is clearly not a globally-defined variable.” What MGG think is the right approach is well known; asserting it once more does not help us understand why they think they can crack the ocean turbulence problem from just musing about the equation of state for seawater, a single one-dimensional fluid parcel trajectory, and observations of viscous dissipation rates, or why they think that LRPD is less arbitrary than any other material or non-material density variable.
- “Tailleux (2016) prefers to imagine that the ocean mixes along a surface, but why this would be the case in doing so the ocean would automatically delete a real physical process, thermobaricity, that occurs in the ocean?” As said above, the irreversible mixing of heat and salt in the NANSE is represented in terms of (isotropic) Laplacian operators, not rotated ones. I therefore consider that the idea that the ocean mixes heat and salt separately along some Nature-endorsed isopycnal and diapycnal directions has no physical basis. On the other hand, we may always decide—if we find it useful as seems to be the case in the ocean—to decompose mixing into isopycnal and diapycnal components in terms of some density-like material function of our choice. I  have discussed the physics that should inform this choice, which is different from MGG’s view. Contrary to what MGG or  assert, it is perfectly possible for a variable to represent thermobaricity, as this is the case of ’s thermodynamic neutral density , owing to its use of a reference pressure that is itself a material function. Moreover, was found to be the most accurate material approximation of and of its gradient ever constructed, much better than the previous attempt by , which suggests that it can reproduce all of what does (apart from non-material behaviour).
- “This dispersion mechanism appears to be a new mixing mechanism, but it relies on the unphysical demand that the ocean mix along an anthropogenically chosen surface.” No, it does not. The dispersion mechanism exists independently of any anthropogenically chosen surface. As explained below, the -chosen surface is constructed after the fact. The way it works is as follows: two fluid parcels swap position in the ocean. Then, anthropos invents a material surface that fits the data and declares that the two parcels have exchanged position on such a surface. Fluid parcels are never aware of how anthropos chooses to explain their behaviour. This process is schematically illustrated in Figure 1.
3.2. The Five Stages of a Rigorous Study of Stirring and Mixing, from Observations to Parameterisations
- The first stage consists of introducing a model of stirring. Among the models available, the most comprehensive and detailed is probably that first discussed by . For the present purposes, it is sufficient to use the simpler model used by  that regards stirring as a general permutation of a discretised version of the ocean conserving and S, of which the building blocks are pairwise permutations of fluid parcels. This model aims to formalise what one means by “stirring” in the context of the problem considered and facilitate the communication of one’s ideas to other investigators in the field by defining clearly the particular hypotheses made.
- The second stage consists of introducing a notional observing system capable of recording the thermodynamic properties and geographical locations of each fluid parcel making up the pairwise permutations defining “stirring events”. At the end of the observing phase, a preliminary classification of the various observed permutations into “vertical stirring” and “lateral stirring” categories could be attempted based on the ratio for instance, or any other criterion deemed appropriate.
- The third stage consists of introducing stirring surfaces by fitting the data, as illustrated in the right panel of Figure 1. The simplest fit assumes a linear function of S and ,
- In the fourth stage, we now seek a globally defined material density variable , as previously undertaken by [10,12,16] for instance. The aim is to go from a representation of water masses to a density/spiciness representation, where is meant to parameterise spiciness. We note that if in-situ density is regarded as a function of , then the neutral vector is given byAs noted by , the sign and amplitude of the partial derivative of with respect to spiciness control both the degree of non-neutrality of and the energy cost of adiabatic and isohaline parcel exchanges. All globally-defined variables capable of enforcing the constraint for all possible locally-defined adiabatic and isohaline parcel exchanges can claim relevance to the description of lateral stirring in the ocean. Note that such an approach does not assume that mixing occurs along iso- surfaces, because—as stated earlier—irreversible mixing of heat and salt is described by isotropic Laplacian diffusion, not rotated diffusion. Rather, the goal is to find a density variable that can be useful to work in an isopycnal coordinate system. A priori, the number of such variables is infinite, which implies that the concepts of “isopycnal” and “diapycnal” are fundamentally non-unique and different for each variable , raising the question of which particular density variable measurements of “diapycnal” and “isopycnal” mixing actually pertain to.
- In the fifth stage, we seek to parameterise the turbulent heat and salt fluxes and in terms of rotated diffusion tensors:
Conflicts of Interest
- Tailleux, R. Neutrality versus materiality: A thermodynamic theory of neutral surfaces. Fluids 2016, 1, 32. [Google Scholar] [CrossRef]
- Eckart, C. An analysis of stirring and mixing processes in incompressible fluids. J. Mar. Res. 1948, 7, 265–275. [Google Scholar]
- McDougall, T.J.; Groeskmap, S.; Griffies, S.M. On geometrical aspects of interior ocean mixing. J. Phys. Oceanogr. 2014, 44, 2164–2175. [Google Scholar] [CrossRef]
- Redi, M.H. Oceanic isopycnal mixing by coordinate rotation. J. Phys. Oceanogr. 1982, 12, 1154–1158. [Google Scholar] [CrossRef]
- McDougall, T.J.; Church, J.A. Pitfalls with the numerical representation of isopycnal diapycnal mixing. J. Phys. Oceanogr. 1986, 16, 196–199. [Google Scholar] [CrossRef]
- Veronis, G. The role of models in tracer studies. In Numerical Models of Ocean Circulation; National Academy of Science: Washington, DC, USA, 1975; pp. 133–146. [Google Scholar]
- Boning, C.; Holland, W.R.; Bryan, F.O.; Danabasoglu, G.; McWilliams, J.C. An overlooked problem in model simulations of the thermohaline circulation and heat transport in the Atlantic ocean. J. Clim. 1995, 8, 515–523. [Google Scholar] [CrossRef]
- Lazar, A.; Madec, G.; Delecluse, P. The deep interior downwelling, the Veronis effect, and mesoscale tracer transport parameterizations in an OGCM. J. Phys. Oceanogr. 1999, 29, 2945–2961. [Google Scholar] [CrossRef]
- Gent, P.R.; Willebrand, J.; McWilliams, T.J.; McWilliams, J.C. Parameterizing eddy-induced tracer transports in ocean circulation models. J. Phys. Oceanogr. 1995, 25, 463–474. [Google Scholar] [CrossRef]
- Eden, C.; Willebrand, J. Neutral density revisited. Deep Sea Res. Part II 1999, 46, 33–54. [Google Scholar] [CrossRef]
- Nycander, J. Energy conversion, mixing energy, and neutral surfaces with a nonlinear equation of state. J. Phys. Oceanogr. 2011, 41, 28–41. [Google Scholar] [CrossRef]
- Jackett, D.R.; McDougall, T.J. A neutral density variable for the world’s oceans. J. Phys. Oceanogr. 1997, 27, 237–263. [Google Scholar] [CrossRef]
- Iudicone, D.; Madec, G.; McDougall, T.J. Water-mass transformations in a neutral density framework and the key role of light penetration. J. Phys. Oceanogr. 2008, 38, 1357–1376. [Google Scholar] [CrossRef]
- Tailleux, R. Generalized patched potential density and thermodynamic neutral density: Two new physically based quasi-neutral density variables for ocean water masses analyses and circulation studies. J. Phys. Oceanogr. 2016, 46, 3571–3584. [Google Scholar] [CrossRef]
- McDougall, T.J.; Jackett, D.R. The material derivative of neutral density. J. Mar. Res. 2005, 63, 159–185. [Google Scholar] [CrossRef]
- De Szoeke, R.A.; Springer, S.R. Orthobaric density: A thermodynamic variable for ocean circulation studies. J. Phys. Oceanogr. 2000, 30, 2830–2852. [Google Scholar] [CrossRef]
- Griffies, S.M. The Gent-McWilliams skew flux. J. Phys. Oceanogr. 1998, 28, 831–841. [Google Scholar] [CrossRef]
- Ferrerira, D.; Marshall, J.; Heimbach, P. Estimating eddy stresses by fitting dynamics to observations using a residual-mean ocean circulation model and its adjoint. J. Phys. Oceanogr. 2005, 35, 1891–1910. [Google Scholar] [CrossRef]
- McDougall, T.J.; Jackett, D.R. An assessment of orthobaric density in the global ocean. J. Phys. Oceanogr. 2005, 35, 2054–2075. [Google Scholar] [CrossRef]
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