Article Semicompressible Ocean Thermoynamics an Boussinesq Energy Conservation William K. Dewar 1, *, Joseph Schoonover 1, Trevor McDougall 2 an Rupert Klein 3 1 Department of Earth, Ocean, an Atmospheric Science, Floria State University, Tallahassee, FL 32309, USA; schoonover.numerics@gmail.com 2 School of Mathematics an Statistics, UNSW, Syney, NSW 2052, Australia; trevor.mcougall@unsw.eu.au 3 Institut für Mathematik, Freie Universität, Berlin 14195, Germany; rupert.klein@math.fu-berlin.e * Corresponence: wewar@fsu.eu; Tel.: +1-850-644-4099 Acaemic Eitor: Pavel S. Berloff Receive: 29 Januray 2016 ; Accepte: 24 March 2016 ; Publishe: 8 April 2016 Abstract: Equations more accurate than the Boussinesq set that still filter out soun were recently introuce. While these equations were shown to have a consistent potential energy, their thermoynamical behavior an associate implications were not fully analyze. These shortcomings are remeie in the present note that argues both sets are fully consistent from a thermoynamic perspective. It is further argue that both sets remain computationally competitive with the Boussinesq set. Keywors: thermoynamics; compressibility; equations of motion 1. Introuction Dewar et al. [1] recently evelope two sets of ocean ynamical equations that inclue some egree of compressibility an exclue soun waves. They were argue to be simultaneously more accurate than the Boussinesq equations an computationally competitive with them. These were referre to as the Type I an Type II Semicompressible equations. In subsequent communications with Prof. R. Klein, he correctly pointe out that the thermoynamic consistency of these equations ha not yet been emonstrate. Thermoynamic consistency in this context means that total energy is conserve an that an entropy variable exists for the system that obeys the secon thermoynamic law. This remark was mae in comparison to pseuoincompressible equations erive in [2] for a general equation of state in which a much more thorough thermoynamic analysis ha been unertaken. The purpose of this short communication is to analyze similar characteristics of both types of the semicompressible equations. 2. The Semicompressible Equations As iscusse in [1], the Type I equations are ρ ( t u + u u + 2Ω u) = p [ρ ρ r (z) + pρ r c 2 s ρ ]gk + ρ ν visc u t ρ + (ρ u) = 0 ρ = ρ(s A, Θ, P ) (1a) t Θ = Θo t S A = S o A where ensity, ρ, is evaluate using Conservative Temperature, Θ, Absolute Salinity, S A an the hyrostatic pressure, P, etermine from the reference ensity profile, ρ r (at most a function of z) accoring to (1b) Fluis 2016, 1, 9; oi:10.3390/fluis1020009 www.mpi.com/journal/fluis
Fluis 2016, 1, 9 2 of 8 z P = ρ r g (2) The quantity ν visc is viscosity an ν = 1/ρ. De Szoeke an Samelson [3] have shown that the hyrostatic Boussinesq equations are equivalent to the compressible hyrostatic Navier-Stokes equations written in pressure coorinates. Equation (1a) improves upon the classic hyrostatic balance in the hyrostatic limit through the presence of the soun spee. The ensity approximation it represents is a more accurate approximation of true ensity than ρ, ensity evaluate at the hyrostatic pressure. Equation (1) is also more flexible in that they can be use in non-hyrostatic settings. The notation X o in Equation (1) enotes the non-avective contributions to X. Absolute Salinity, being a purely conserve quantity (ignoring chemistry, see [4]), the form is clear S o A = ν F S (3) where F S is the iffusive flux of Absolute Salinity. We purposely leave the form for Θ o unspecifie, as part of the present exercise is to etermine it. Other notation is stanar. The equations in Equation (1) satisfy the energy equation t (ρ (K + h + I )) + (u(ρ (K + h + I )) + pu) = pr + ρ ν visc K ρ ɛ (4) with potential energy playe by the quantity h + I P ρ ρ P r = P = ρρ r b I gρ r P (5) The quantity ɛ = ν visc u 2 is the viscous issipation of kinetic energy, b I is the Type I buoyancy, the quantity R R = Θ ν t Θ ν S A t S A (6) enotes iffusive terms an Equation (1b) emonstrates the presence of compressibility in these equations. The Type II equations are where ρ II ( t u + u u + 2Ω u) = p (ρ ρ II )gk p c 2 sii t ρ II + (ρ II u) = (ρ II u) = 0 ρ = ρ(s A, Θ, P ) t Θ = Θo t S A = S o A ρ II (z) = ρ II (0)e z o g c 2 z sii gk + ρ II ν visc u (7a) (7b) (8) is either a chosen reference ensity profile yieling the soun spee profile c 2 s II or a profile constructe from a chosen soun spee profile, c 2 s II. In either case, ensity, ρ II, is a function of epth only an the hyrostatic pressure is compute from it z P = ρ II g (9)
Fluis 2016, 1, 9 3 of 8 Compressibility appears in these equations in Equation (7b) where the time inepenence of ρ II has been recognize. The Type II equations satisfy the energy constraint t (ρ II(K + h + II )) + (uρ II(K + h + II ) + pu) = ρii ν visc K ρ II ɛ (10) where the role of potential energy is now playe by h + II = P ρ ρ P II ρ 2 P = II b II gρ II P (11) The quantity b II is the Type II buoyancy. The avantages an isavantages of these equations are iscusse in [1]. Extracting Equations (1) an (7) from the full Navier Stokes equations is akin to substituting the Boussinesq ensity for the full ensity in most places, an retaining a reference ensity profile where neee for potential energy consistency. 3. Semicompressible Thermoynamics We now consier how to augment (1) an (7) with a consistent thermoynamic behavior. The First Law of Thermoynamics is t h ν t P = T t η + µ t S A = 1 ρ F Q + ɛ (12) where T is temperature, h(η, S A, P) is specific enthalpy, η is specific entropy, specific volume is enote by ν = 1/ρ, P is pressure an µ is the relative chemical potential of salt in seawater. The quantity ɛ is the heat of viscous energy issipation, an F Q is a generalize heat flux. Enthalpy is efine by h = e + Pν (13) where e is specific internal energy. Enthalpy from Equation (12) is seen to epen naturally on the variables η, S A, an P, an in this form plays the role of a thermoynamic potential from which all thermoynamic variables can be obtaine by ifferentiation. For example, erivatives of enthalpy with respect to its natural variables are T, µ, an ν, respectively. The erivation of Equation (12) is iscusse extensively in the TEOS-10 manual ([4]; see also [5,6]). Equation (12) is exact. The semicompressible equations consier moifications to the thermoynamic variables cause mainly by the ifference between static an full pressure, an we analyze Equation (12) from that perspective. In what follows, we will aopt entropy, salinity, an pressure as state variables, an will often cast pressure epenency in terms of static pressure. For example, h = h (S A, η, P ) + h P (S A, η, P )p + O(p 2 ) (14) where P is static pressure an p the ynamic pressure, given by p = P P. The superscript ( ) will enote quantities evaluate at the static pressure. Specific volume, ensity, etc. will be written in a manner similar to Equation (14). Note, accoring to Equation (12), Equation (14) is equivalent to h = h (S A, η, P ) + ν p (15)
Fluis 2016, 1, 9 4 of 8 3.1. Type I Thermoynamics We first evelop what is a consistent statement of Type I thermoynamics. consiering the form of Equation (12) to O(p 2 ); i.e., We begin by where t (h + ν p) = (ν + p P ν) t P + ν t p + T+ t η + µ+ t S A + O(p 2 ) (16) T + = T + p P T (17) an similarly for µ +. Evaluating the erivative of the static pressure, Equation (16) becomes t (h + ν p) = (ν + p P ν)( wgρ r) + ν t p + T+ t η + µ+ t S A (18) While Equation (18) is literally correct to O(p 2 ), it will turn out that a less accurate approximation, to O(p), is require on the right han sie of Equation (12) T + t η + µ+ t S A = ν F Q + ɛ (19) The reason for this will become clear later, here we stress only that it assures thermoynamic consistency of the Type I equations. 3.2. Type I Mechanical Energy Equation The mechanical energy equation erive from Equation (1a) is ρ t K = u p wg(ρ ρ r + ρ r p ρ c 2 ) + ρ ν visc K ρ ɛ (20) s where K = (u u)/2 is the kinetic energy ensity. Introucing the gravitational potential Φ = gz an using Equation (18) ρ ( t K + Φ + h + pν ) = u p + t p + ρ ν visc K ρ ɛ + ρ (T + t η + µ+ t S A) (21) so where Using Equation (12) P h = ν (22) P P h 1 P = h o + νp = h o + P + ρ r b P (23) ρ r g b = g (ρ ρ r ) ρ (24) The quantity h o is recognize as the potential enthalpy efine by [6], an the last integral in Equation (23) is the so-calle ynamic enthalpy of [7]. The quantity is a constant reference surface pressure. The mile integral in Equation (23) reuces upon inspection to Φ. Thus, Equation (21) becomes ρ t (K + h o + h + I ) = pu + ρ ν visc K ρ ɛ + ρ (T + t η + µ+ t S A) (25)
Fluis 2016, 1, 9 5 of 8 Lastly, we exploit Equation (19) to obtain ρ t (K + h o + h + I ) = pu + ρ ν visc K F Q (26) which is in conservative form. This implies the Type I equations have a well-forme energy principle, provie they are augmente by Equations (18) an (19) as their thermoynamic equations. Note that the entropy in Equation (19) can be rewritten as ρ t η = FQ µ + F S T + + (F Q µ + F S ) ( 1 T + ) F S ( µ+ T + ) + ρ ɛ T + (27) which is entropy non-conservation in its familiar form. Hence, the Type I equations also have a consistent entropy variable. At this point, a practical ifficulty is that entropy is not a typical ocean moelling variable. Instea, McDougall s Conservative Temperature Θ = h o c o P (28) McDougall ([6]) is preferable an so it is necessary to work out a Conservative Temperature equation to complement (27). From Equation (18), one fins t h ν t P = 1 ρ F Q p P T t η p P µ t S A + ɛ (29) By efinition h o = h(η, S A, ) (30) so t h o = θ t η + µ o t S A (31) where θ is potential temperature an µ o = µ(η, S A, ). Potential enthalpy is thus relate to entropy via t η = t h o µ o θ t S A (32) From Equation (23) t h o = c o p t Θ = 1 ρ F Q (T + θ) t η (µ+ µ o ) t S A + ɛ (33) so using Equation (32) t Θ = ν F Q + ɛ c o P (1 + (T+ θ) ) θ (( T+ θ )µ o µ + ) t S A c o P (1 + (T+ θ) θ ) (34) The unerline terms in Equation (34) are quite small, accoring to [8]. Neglecting them leas to the statement that Conservative Temperature is very accurately portraye as a conserve quantity; however, all of those terms must be inclue in orer to ensure full energy conservation. It is seen, therefore, that the Type I equations yiel energy conservation, but the cost is the insertion of a large quantity (c o PΘ) into the energy equation. A similar result was labelle in [7] as a crippling isavantage of the Boussinesq-Bernoulli equation. It might thus be more useful to work with a partial equation when consiering energy; i.e., one that oesn t explicitly inclue Conservative Temperature, but in any case, the full potential enthalpy Equation (34) is require. We have also foun the Type I Semicompressible equations have a proper entropy variable an conserve total energy, an so are
Fluis 2016, 1, 9 6 of 8 fully thermoynamically consistent. These equations are very similar to the pseuo-incompressible equation set for general equations of state in [2]. 3.3. Type II Equations: Potential, Kinetic, an Thermal Energies The Type II equations are quite similar to the Type I equations except for the appearance of the reference ensity profile, rather than the Boussinesq ensity, in the mass conservation equation an in front of the momentum acceleration. Most of the analysis carries over in a manner similar to the Type I analysis, with the one important istinction that the velocity ivergence iagnose from mass conservation has no iabatic contributions u II = gw c 2 sii Therefore, the mechanical energy equation for the system analogous to Equation (21) is wg ρ II K = pu + p t c 2 wg(ρ ρ II + sii p c 2 sii (35) ) + ρ II ν visc K ρ II ɛ (36) with the unerline terms cancelling. Taking the material erivative of Equation (11) an substituting in Equation (36) yiels P [ ] ρ II t K = pu + ρ bii II P S A gρ r t S A (37) P [ ] bii + ρ II P η gρ r t η ρ II t h+ II + ρii ν visc K ρ II ɛ To put Equation (37) in conservative form requires the potential enthalpy equation t h = t h o + ν P t P + which is equivalent to Equation (33). This leas to ν P P S A t S ν A + η P t η (38) ρ II t (K + h+ II + co P Θ) = pu F Q + ρ II ν visc K (39) The Type II Conservative Temperature equation becomes t Θ = ν II F Q + ɛ (( T+ θ )µ o µ + ) t S A c o P T+ θ (40) The most significant ifference relative to the Type I set is in Equation (35), where the iabatic contributions to expansion are absent. Like for Type I, it is still necessary to retain all terms for Conservative Temperature, rather than the approximate form recommene in [6]. We conclue that the Type II equations also have a consistent mechanical an thermal energy structure. Our Type II equations are very similar to the thermoynamically consistent anelastic set erive in [9]. 3.4. A Comparison to the Boussinesq Set To ensure full energetic consistency for the semicompressible equations, it has been necessary to carry along some aitional non-conservative effects in the calculations. We now analyze the full energetics behavior of the Boussinesq set to gauge any aitional computational buren of the semicompressible set relative to them. If we aopt Conservative Temperature, salinity, an pressure as
Fluis 2016, 1, 9 7 of 8 the thermoynamic coorinates an follow the proceure outline in [7], we arrive at the Boussinesq version of the Bernoulli equation t (K + p + h B) = t p + P [ ] b P P [ S A gρ o t S A + Θ b gρ o ] P t Θ ɛ (41) (see Equation (48) in [7]) where b = g(ρ ρ o )/ρ is the Boussinesq buoyancy an ρ o is the Boussinesq reference ensity. Dynamic enthalpy as efine in [7] is h B = 1 P bp (42) gρ o Returning to a form of the potential enthalpy consistent with the Boussinesq approximation h (S A, Θ, P ) = h o + 1 P (1 + b )P (43) ρ o g where h o is potential enthalphy, an taking its material erivative yiels t h ν t P = c o P [ P t Θ + b S A gρ o P [ b + Θ gρ o ] P t S A (44) ] P t Θ = 1 ρ o F Q + ɛ Using Equation (44) to eliminate the pressure integrals in Equation (41) turns it into the conservative Boussinesq energy equation t (K + p + h B + c o P Θ) = t p + ρii ν visc K 1 ρ o F Q (45) The Boussinesq Conservative Temperature equation is given by t Θ = ν o F Q + ɛ (( T θ )µ o µ + ) t S A c o P T θ (46) The point of this section is that to obtain a conservative energy equation out of the Boussinesq set, it is necessary to carry along the same aitional iffusive terms as for the semi-compressible equations. A similar result has recently been erive in [10,11]. This is not one in any Boussinesq ocean circulation moel of which we are aware. 4. Summary The Type I an II semicompressible equations iscusse in [1], which are analogues of the pseuo-incompressible an anelastic equations known in meteorology, have unergone an examination of their thermoynamic structure. To summarize, the Types I an II equations are fully consistent from an energetic an thermoynamic perspective. To obtain this, it is necessary to inclue a more complete equation for Conservative Temperature as a part of the equation sets than is normally one. Interestingly, it turns out the Boussinesq equations must also carry analogous quantities to be in purely conservative energetic form. We thus see the semicompressible equations remain computationally competitive with the Boussinesq equations while amitting some egree of compressibility. Other results here inclue the associate efinitions of the ynamic enthalpy variables belonging to the sets. From a simple computer throughput perspective, this analysis recommens Type II relative to Type I because they are slightly less computationally intensive. However, the Type II equations
Fluis 2016, 1, 9 8 of 8 represent the oceanic ensity structure by a single vertical profile. Any such profile cannot perform well globally, so the cost of the simplicity is a less accurate ensity representation. This promises to be the most troubling when computing sea level changes via the continuity equation. The aitional accuracy provie by the Boussinesq ensity in the Type I set might well compensate for the (slight) aitional complexity. Any further exploration with these equation sets shoul bear in min the above points. Acknowlegments: This research was make possible by grants from the National Science Founation Grants OCE-1537304 an OCE-100743 an the Gulf of Mexico Research Initiative through consortium funing to the University of Miami for the Consortium for Avance Research on Transport of Hyrocarbon in the Environment (CARTHE). William Young is thanke for many interesting conversations that ae significantly to the manuscript. Author Contributions: William K. Dewar an Rupert Klein conceive the research. Trevor McDougall an Joseph Schoonover contribute substantially to the thermoynamical analysis. William K. Dewar wrote the paper. Conflicts of Interest: The authors eclare no conflict of interest. References 1. Dewar, W.; Schoonover, J.; McDougall, T.; Young, W. Semicompressible ocean ynamics. J. Phys. Oceanogr. 2015, 45, 149 156. 2. Klein, R.; Pauluis, O. Thermoynamic consistency of a psueoincompressible approximation for general equations of state. J. Atmos. Sci. 2012, 69, 961 968. 3. De Szoeke, R.; Samelson, R. The uality between the Boussinesq an non-boussinesq hyrostatic equations of motion. J. Phys. Oceanogr. 2002, 32, 2194 2203. 4. Intergovernmental Oceanographic Commission (IOC); Scientific Committee on Oceanic Research (SCOR); International Association for the Physical Sciences of the Oceans (IAPSO). The International Thermoynamic Equation of Seawater 2010: Calculation an Use of Thermoynamic Properties; 2010 Manuals an Guies No. 56, UNESCO (English); Intergovernmental Oceanographic Commission: West Perth, Australia, 2010; p. 196. 5. Lanau, L.M.; Lifshitz, E.M. Flui Mechanics, 1st e.; Pergamon Press: Oxfor, UK, 1959. 6. McDougall, T. Potential enthalpy: A conservative oceanic variable for evaluating heat content an heat fluxes. J. Phys. Oceanogr. 2003, 33, 945 963. 7. Young, W. Dynamic enthalpy, conservative temperature an the seawater Boussinesq approximation. J. Phys. Oceanogr. 2010, 40, 394 400. 8. Graham, F.S.; McDougall, T.J. Quantifying the Nonconservative Prouction of Conservative Temperature, Potential Temperature, an Entropy. J. Phys. Oceanogr. 2013, 43, 838 862. 9. Pauluis, O. Thermoynamic consistency of the anelastic approximatio for a moist atmosphere. J. Atmos. Sci. 2008, 65, 2719 2729. 10. Tailleux, R. On the energetics of stratifie turbulent mixing, irreversible thermoynamics, Boussinesq moels an the ocean heat engine controversy. J. Flui Mech. 2009, 638, 339 382. 11. Tailleux, R. Irreversible compressible work an available potential energy issipation in turbulent stratifie fluis. Phys. Scr. 2013, oi:10.1088/0031-8949/2013/t155/014033. c 2016 by the authors; licensee MDPI, Basel, Switzerlan. This article is an open access article istribute uner the terms an conitions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).