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Observational and Energetics Constraints on the Non-Conservation of Potential/Conservative Temperature and Implications for Ocean Modelling

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Observational and Energetics Constraints on the Non-Conservation of Potential/Conservative Temperature and Implications for Ocean Modelling Observational and energetics constraints on the non-conservation of potential/conservative temperature and implications for ocean modelling R´emi Tailleux Department of Meteorology, University of Reading, Earley Gate, PO Box 243, Reading, RG6 6BB, United Kingdom. Email: [email protected], Telephone: +44(0)1183788328. Fax: +44(0)1183788316. Abstract This paper elucidates the nature of heat non conservation by showing that the nonconservative production/destruction of po- tential temperature and entropy is related to the production/destruction by surface buoyancy fluxes, while the non conservation of potential enthalpy/conservative temperature is a direct measure of the compressible work of expansion/contraction. These results are used to show how to estimate the nonconservative production of potential temperature, entropy and conservative temperature from observations and from numerical ocean model estimates of the ocean energy cycle. The results suggest that the non conser- vation of potential temperature could be easily accounted for in numerical ocean models with a simple modification of existing codes, providing a potential alternative option to TEOS10 recommendation of switching from potential temperature to conservative temperature. Keywords: ocean modelling, conservation equations, heat non-conservation, energy conservation, potential temperature, conservative temperature 1. Introduction θ in coarse-grained primitive hydrostatic Boussinesq models. Specifically, the method works as follows. Starting from the The issue of whether potential temperature θ is the most ap- hydrostatic Boussinesq primitive equations formulated in terms propriate heat variable to be used in numerical ocean general of either potential temperature, conservative temperatureor en- circulation models (OGCMs) has recently come under scrutiny tropy, one writes down the evolution equation for the total en- following McDougall (2003) and IOC (2010) suggestion that ergy assuming that the evolution equation for any of the heat θ should be replaced by Conservative Temperature (CT or Θ). variable is the sum of a conservative part (expressed as the di- The main argument, originally made by McDougall (2003), is vergence of some flux) plus an a priori unknown nonconserva- that the current practice of treating θ as a conservative quantity tive part. Both conservative and nonconservative terms appear is significantly inaccurate, and to the extent that one should in- in the equation for total energy thus obtained. As discussed by sist in treating heat as conservative in OGCMs, it appears to be Tailleux (2010), each nonconservative term is associated with significantly more accurate to do so by using CT instead. Since some inconsistency in the model formulation, such as using the CT is a relatively new quantity, its formal properties have yet total hydrostatic pressure instead of the Boussinesq pressure in to be fully understood, so that the full implications of switch- the equation of state for instance. One of the nonconservative ing from θ to CT in OGCMs are not all entirely clear. The term in the total energy equation is directly related to the heat alternative option — to accept that potential temperature is fun- nonconservation term. Imposing such a term to be zero, as re- damentally nonconservative and to modify its model formula- quired by the principle of energy conservation, provides an ex- tion accordingly — has not received attention so far, but needs plicit expression for the heat nonconservative term in terms of to be understood to inform the debate about whether to switch the heat and salt fluxes, which depends on the assumed form of or not. The main purpose of this paper is to achieve a deeper the turbulent ocean mixing processes. Graham and McDougall understanding of the nature of heat non conservation that can (2013) uses a similar approach to quantify heat non conserva- help the oceanography community better understand the pros tion, but which relies on the existence of a different conservative and cons of each modelling choice. quantity (a locally referenced potential enthalpy) than total en- From a fundamental viewpoint, the non conservation of heat ergy. Their approach yields a different expression for the non- as measured by potential temperature, conservative temperature conservative terms than that of Tailleux (2010), which among or entropy is now well understood to be a natural consequence other things, lacks pressure gradient terms. of energy conservation, in the sense that were heat to be conser- vative, total energy would be nonconservative and conversely, The main objective of this paper is to clarify the nature of as shown in Tailleux (2010). Tailleux (2010) proposed to the non-conservation of potential/conservative temperature and use the same idea to evaluate the non conservation of CT and of entropy. Building upon the results by Tailleux (2012), we Preprint submitted to Ocean Modelling May 7, 2014 argue that the nonconservative production of conservative tem- temperature θ and conservative temperature Θ. As these vari- perature or potential enthalpy fundamentally measures the ther- ables are all conservative, they can all a priori be assumed to modynamic work of expansion/contraction. In other words, in satisfy evolution equations of the form the same way that McDougall (2003) argues that potential en- Dη thalpy is the most appropriate variable to measure “heat” into ρ = −∇ · (ρF ) + ρη˙ , (4) Dt η irr the oceans, we argue that the non-conservation of potential en- thalpy measures the “work” done by compressible effects. We Dθ ρ = −∇ · (ρF ) + ρθ˙ , (5) also argue that the nonconservativeproduction of potential tem- Dt θ irr perature is of a fundamentally different nature, and actually DΘ measures the production of potential temperature by surface ρ = −∇ · (ρFΘ) + ρΘ˙ irr, (6) enthalpy fluxes, in the same way that irreversible entropy pro- Dt duction reflects the entropy production by surface fluxes in a where Fη, FΘ, and Fθ are the molecular diffusive fluxes of each steady-state system. Our results suggest that it might be rela- quantity, whileη ˙irr, θ˙irr and Θ˙ irr are the non-conservation terms tively straightforward to modify existing codes to take into ac- for each quantity that are the main focus of the present paper. count the leading order non conservation of potential tempera- ture that has been neglected so far. 2.2. Passage relations for “heat” variables This paper is organised as follows. Section 2 recalls the gen- How entropy, potential temperature and conservative temper- eral construction of the nonconservative production terms and ature are related to each other has been previously discussed in their link to energy conservation initiated in Tailleux (2010) Tailleux (2010) building upon previous work by Bacon and Fo- and Tailleux (2012), and further shows how to link the non fonoff (1996) and McDougall (2003) (see also IOC (2010)). conservation of potential temperature and entropy to ocean sur- All the necessary relations are usually obtained from the total face properties; such a link is well known for entropy, but not differential of the specific enthalpy h, viz., for potential temperature. Section 3 discusses a priori estimates for the non conservation terms, as well as some of their theo- dh = Tdη + µdS + υdP, (7) retical properties. Section 4 uses observations to illustrate and which can alternatively be written in terms of temperature, quantify empirically the results of Section 3. Section 5 offers a salinity and pressure as follows: summary and discussion of the implications of our results for ocean modelling, in particular whether to switch from θ to CT. ∂µ dh = cpdT + µ − T dS + υ(1 − αT)dP. (8) ∂T ! 2. Non-conservation of “heat” variables for the fully com- The passage from (7) to (8) follows from the fact that the total pressible Navier-Stokes equations for seawater differential of specific entropy in terms of temperature, salinity and pressure is given by: 2.1. Compressible Navier-Stokes equations and “heat” vari- cp ∂µ α dη = dT − dS − dP. (9) ables T ∂T ρ The non-conservativeness of different measures of heat is How the term Tdη + µdS in the enthalpy differential (7) trans- discussed in the context of the full compressible Navier-Stokes forms in the (θ, S ) and (Θ, S ) representations is given by Eqs. equations (NCS) in a rotating frame, viz., Eqs. (B.2) and (B.3) of Tailleux (2010), specifically R Dv ∂µ Tcp ρ + 2Ω × v + ∇P = −ρ∇Φ+ ρ∇· S (1) µdS + Tdη = µ − T R dS + dθ, (10) Dt ∂θ ! θ Dρ 0 + ρ∇· v = 0 (2) TµR Tcp Dt µdS + Tdη = µ − dS + dΘ. (11) θ θ DS Also useful are relations allowing to pass from the (θ, S ) to ρ = −∇ · (ρF ) (3) Dt S (Θ, S ) representation, which can be obtained by eliminating η between (10) and (11), which yields: where v = (u, v, w) is the three-dimensional velocity field, P is R the pressure, ρ is the density, S is (absolute) salinity, Φ= g Z is 1 ∂µ cp 0 Θ= µ − θ R + θ, Ω d 0 R dS 0 d (12) the geopotential, g0 is the acceleration of gravity, is Earth’s cp ∂θ ! cp rotation vector, S is the stress tensor, Φ = g0Z is the geopo- 0 tential formulated in terms of a constant gravitational potential cp 1 ∂µ dθ = dΘ − µ − θ R dS. (13) acceleration g0 and geopotential height Z = z, with z the regular R R R cp cp ∂θ ! height increasing upwards, FS is the diffusive salinity flux. As in Graham and McDougall (2013), we focus on the three These two equations correspond to Eqs. (A.14) and (A.15) in classical measures of heat based on specific entropy η, potential Tailleux (2010). 2 2.3. Passage relations for diffusive fluxes Tailleux (2010) derived the following expressions for ηirr, θirr Θ In order to link the nonconservative production/destruction and irr, of η, θ and Θ to surface fluxes of heat and salt, we need to un- ε − F · ∇µ − F · ∇T µ 1 ε K S η − ·∇ ·∇ K derstand the links between the different diffusive fluxes.
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