Spiciness Theory Revisited, with New Views on Neutral Density, Orthogonality, and Passiveness

Ocean Sci., 17, 203–219, 2021 https://doi.org/10.5194/os-17-203-2021 © Author(s) 2021. This work is distributed under the Creative Commons Attribution 4.0 License. Spiciness theory revisited, with new views on neutral density, orthogonality, and passiveness Rémi Tailleux Dept. of Meteorology, University of Reading, Earley Gate, RG6 6ET Reading, United Kingdom Correspondence: Rémi Tailleux ([email protected]) Received: 26 April 2020 – Discussion started: 20 May 2020 Revised: 8 December 2020 – Accepted: 9 December 2020 – Published: 28 January 2021 Abstract. This paper clarifies the theoretical basis for con- 1 Introduction structing spiciness variables optimal for characterising ocean water masses. Three essential ingredients are identified: (1) a As is well known, three independent variables are needed to material density variable γ that is as neutral as feasible, (2) a fully characterise the thermodynamic state of a fluid parcel in material state function ξ independent of γ but otherwise ar- the standard approximation of seawater as a binary fluid. The bitrary, and (3) an empirically determined reference function standard description usually relies on the use of a tempera- ξr(γ / of γ representing the imagined behaviour of ξ in a ture variable (such as potential temperature θ, in situ temper- notional spiceless ocean. Ingredient (1) is required because ature T , or Conservative Temperature 2), a salinity variable contrary to what is often assumed, it is not the properties im- (such as reference composition salinity S or Absolute Salin- posed on ξ (such as orthogonality) that determine its dynam- ity SA), and pressure p. In contrast, theoretical descriptions ical inertness but the degree of neutrality of γ . The first key of oceanic motions only require the use of two “active” vari- 0 result is that it is the anomaly ξ D ξ − ξr(γ /, rather than ξ, ables, namely in situ density ρ and pressure. The implication that is the variable most suited for characterising ocean wa- is that S and θ can be regarded as being made of an active ter masses, as originally proposed by McDougall and Giles part contributing to density and a passive part associated with (1987). The second key result is that oceanic sections of nor- density-compensated variations in θ and S – usually termed malised ξ 0 appear to be relatively insensitive to the choice of “spiciness” anomalies – which behaves as a passive tracer. ξ, as first suggested by Jackett and McDougall(1985), based Physically, such an idea is empirically supported by numer- on the comparison of very different choices of ξ. It is also ical simulation results showing that the turbulence spectra argued that the orthogonality of rξ 0 to rγ in physical space of density-compensated thermohaline variance is generally is more germane to spiciness theory than orthogonality in significantly different from that contributing to the density thermohaline space, although how to use it to constrain the (Smith and Ferrari, 2009). choices of ξ and ξr(γ / remains to be fully elucidated. The re- Although behaving predominantly as passive tracers, sults are important for they unify the various ways in which density-compensated anomalies may occasionally “activate” spiciness has been defined and used in the literature. They and couple with density and ocean dynamics. This may also provide a rigorous theoretical basis justifying the pursuit happen, for instance, when isopycnal mixing of θ and S of a globally defined material density variable maximising leads to cabbeling and densification, which may create avail- neutrality. To illustrate the latter point, this paper proposes a able potential energy (Butler et al., 2013); when density- new implementation of the author’s recently developed ther- compensated temperature anomalies propagate over long modynamic neutral density and explains how to adapt exist- distances to de-compensate upon reaching the ocean sur- ing definitions of spiciness and spicity to work with it. face, thus modulating air–sea interactions (Lazar et al., 2001); when density-compensated salinity anomalies propagate from the equatorial regions to the regions of deepwa- ter formation, thus possibly modulating the strength of the thermohaline circulation (Laurian et al., 2006, 2009); and/or Published by Copernicus Publications on behalf of the European Geosciences Union. 204 R. Tailleux: Spiciness theory revisited when isopycnal stirring of density-compensated θ=S anoma- following. “Secondly, it is readily apparent that the per- lies releases available potential energy associated with ther- pendicular property itself has no inherent physical mean- mobaric instability (Ingersoll, 2005; Tailleux, 2016a). For ing since a simple rescaling of either the potential temper- these reasons, the mechanisms responsible for the formation, ature θ or the salinity axis S destroys the perpendicular prop- propagation, and decay of spiciness anomalies have received erty”. The Jackett and McDougall(1985) remarks are impor- much attention, with a key research aim being to understand tant for at least two reasons: (1) first, for suggesting that it 0 their impacts on the climate system; e.g. Schneider(2000), is really the isopycnal anomaly ξ D ξ − ξr(γ / defined rela- Yeager and Large(2004), Luo et al.(2005), Tailleux et al. tive to some reference function of density ξr(γ / that is dy- (2005), and Zika et al.(2020). namically passive and therefore the quantity truly measur- From a dynamical viewpoint, in situ density ρ is the most ing spiciness. This was further supported by McDougall and relevant density variable for defining density-compensated Giles(1987) subsequently arguing that it is such an anomaly θ=S anomalies, but its strong pressure dependence makes that represents the most appropriate approach for character- the associated isopycnal surfaces strongly time-dependent ising water mass intrusions. Note here that if one defines and therefore impractical to use. This is why in practice reference salinity and temperature profiles Sr(γ / and θr(γ / oceanographers prefer to work with isopycnal surfaces de- such that γ .Sr(γ0/;θr(γ0// D γ0, using a Taylor series ex- 0 0 fined by means of a purely material density-like variable pansion shows that S D S − Sr(γ / and θ D θ − θr(γ / are γ - 0 0 γ D γ .S;θ/ unaffected by pressure variations. Since density- compensated at leading order, i.e. they satisfy γSS C γθ θ ≈ compensated θ=S anomalies are truly passive only if defined 0, and hence approximately passive. (2) The second reason in terms of in situ density, γ needs to be able to mimic the is for establishing that the imposition of any form of orthog- dynamical properties of in situ density as much as feasible. onality between ξ and γ is even less meaningful than previ- As discussed by Eden and Willebrand(1999), this amounts ously realised, since even if ξ is constructed to be orthogonal to imposing that γ be constructed to be as neutral as feasible. to γ in some sense, this orthogonality is lost by the anomaly 0 Because of the thermobaric non-linearity of the equation of ξ D ξ − ξr(γ /. state, it is well known that exact neutrality cannot be achieved If one accepts that it is the spiciness anomaly ξ 0 D ξ − by any material variable. As a result, investigators have re- ξr(γ / rather than the spiciness as a state function (ξ) that is sorted to using either neutral surfaces (McDougall and Giles, the most appropriate measure of water mass contrasts, as ar- 1987) or potential density referenced to a pressure close to gued by Jackett and McDougall(1985) and McDougall and the range of pressures of interest (Jackett and McDougall, Giles(1987), the central questions that the theory of spici- 1985; Huang, 2011; McDougall and Krzysik, 2015; Huang ness needs to address become the following. et al., 2018). In this paper, I propose instead using a new implementation of the Tailleux(2016b) thermodynamic neutral 1. Are there any special benefits associated with one par- density variable γ T, which is currently the most neutral ma- ticular choice of spiciness as a state function (ξ) over terial density-like variable available. The resulting new vari- another, and if so, what are the relevant physical argu- T ments that should be invoked to establish the superiority able is referred to as γanalytic in the following, and details of its construction and implementation are given in Sect.2. of any given particular choice of ξ? Once a choice for γ has been made, a second material vari- 2. How should γ and the reference function ξ (γ / be con- able ξ D ξ.S;θ/ is required to fully characterise the thermo- r structed and justified? dynamic properties of a fluid parcel. From a mathematical viewpoint, the only real constraint on ξ is that the transfor- 3. While any ξ independent of γ can be used to meaning- mation .S;θ/ ! (γ;ξ/ defines a continuously differentiable fully compare the spiciness of two different water sam- one-to-one mapping (that is, an isomorphism) so that .S;θ/ ples lying on the same isopycnal surface γ D constant, properties can be recovered from the knowledge of (γ;ξ/. it is generally assumed that it is not possible to mean- For this, it is sufficient that the Jacobian J D @(ξ;γ /[email protected];θ/ ingfully compare the spiciness of two water samples be- differs from zero everywhere in .S;θ/ space where invert- longing to two different isopycnal surfaces γ1 and γ2 ibility is required. Historically, however, spiciness theory ap- (Timmermans and Jayne, 2016). Is this belief justified? pears to have been developed on the predicate that for ξ Is the transformation of ξ into an anomaly ξ 0 sufficient to be dynamically inert, it should be constructed to be “or- to address the issue? thogonal” to γ in .S;θ/ space, as originally put forward by Veronis(1972) (whose variable is denoted by τ ν in the fol- Regarding the first question, Jackett and McDougall(1985) lowing).

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