Clim Dyn DOI 10.1007/s00382-009-0718-1

Climate entropy budget of the HadCM3 atmosphere–ocean general circulation model and of FAMOUS, its low-resolution version

Salvatore Pascale • Jonathan M. Gregory • Maarten Ambaum • Re´mi Tailleux

Received: 6 August 2009 / Accepted: 27 November 2009 Ó Springer-Verlag 2009

Abstract The entropy budget is calculated of the coupled an important influence on the climate entropy budget. atmosphere–ocean general circulation model HadCM3. Since this is the first diagnosis of the entropy budget in a Estimates of the different entropy sources and sinks of the climate model of the type and complexity used for pro- climate system are obtained directly from the diabatic jection of twenty-first century climate change, it would be heating terms, and an approximate estimate of the plane- valuable if similar analyses were carried out for other such tary entropy production is also provided. The rate of models. material entropy production of the climate system is found to be *50 mW m-2 K-1, a value intermediate in the Keywords Entropy budget General circulation models -2 -1 Á Á range 30–70 mW m K previously reported from dif- Material entropy production ferent models. The largest part of this is due to sensible and latent heat transport (*38 mW m-2 K-1). Another 13 mW m-2 K-1 is due to dissipation of kinetic energy in 1 Introduction the atmosphere by friction and Reynolds stresses. Numerical entropy production in the atmosphere dynamical core is The climate is an important example of a non-equilibrium found to be about 0.7 mW m-2 K-1. The material entropy steady state system. The non-equilibrium nature of the production within the ocean due to turbulent mixing is climate system means that a variety of irreversible pro- *1 mW m-2 K-1, a very small contribution to the mate- cesses must be taking place continuously within it, whereas rial entropy production of the climate system. The rate of the steady characteristic implies that a time-independent change of entropy of the model climate system is about balance must exist for some relevant physical quantities, 1 mW m-2 K-1 or less, which is comparable with the for example energy, angular momentum and water. typical size of the fluctuations of the entropy sources due to The physical quantity which is able to describe both the interannual variability, and a more accurate closure of the irreversibility (non-equilibrium) and the steadiness of the budget than achieved by previous analyses. Results climate system at once (and of any non-equilibrium steady are similar for FAMOUS, which has a lower spatial state system) is the entropy production rate of the system resolution but similar formulation to HadCM3, while more (Kondepudi and Prigogine 1998; DeGroot and Mazur substantial differences are found with respect to other 1984). models, suggesting that the formulation of the model has In contrast to energy fluxes, at the top of the atmosphere, in a steady state, the outgoing entropy flux exceeds the incoming: entropy is produced. The rate of entropy pro- S. Pascale (&) J. M. Gregory M. Ambaum R. Tailleux duction of the system, due to irreversible processes, must Department of Meteorology,Á UniversityÁ of Reading,Á be positive in order to satisfy the second law of thermo- Earley Gate, Reading RG6 6BB, UK dynamics. However, the rate of change of entropy must be e-mail: [email protected] zero if the system is in a steady state. J. M. Gregory General circulation models (GCMs) are generally not Met Office Hadley Centre, Exeter EX1 3PB, UK designed to diagnose entropy, but at the moment there is

123 S. Pascale et al.: Climate entropy budget increasing interest in entropy production as a diagnostic Sect. 6, while in Sect. 7 the numerical entropy source of tool for GCMs. This is motivated in part by the intriguing the HadCM3 dynamical core is analysed. Section 8 deals Maximum Entropy Production (MEP) conjecture (Paltridge with the ocean component of the model and provides direct 1975), for a general review see Martyushev and Seleznev estimates of the rate of entropy production in the body of the (2006) or Kleidon and Lorenz (2005), and Kleidon (2009) ocean by turbulent mixing. Section 9 discusses the uncertainty for a view of all the variety of applications to the Earth in the material entropy production and compares different system) and its possible applications for improving GCMs model values. The paper ends with a summary of the key (Kunz et al. 2008). Moreover, entropy seems to have a role results in Sect. 10. in the basic thermodynamics of the climate: for example Pauluis and Held (2002) showed that the efficiency of convection is reduced by the entropy production of non- 2 Models and experiments viscous processes (such as water vapour diffusion). Fur- thermore, Johnson (1997) argued that spurious numerical The AOGCM used in this paper is HadCM3 (Gordon et al. entropy sources (of either sign—a negative source is a 2000; Pope et al. 2000) and its low resolution version, sink) can affect the simulated general circulation of the FAMOUS (Jones et al. 2005), of which we are using ver- atmosphere; his example was the cold bias at high latitude sion FAMOUS-xdbua (Smith et al. 2008). HadCM3 has and altitude found in many GCMs. Recently, Lucarini et al. been widely used to simulate present day and future cli- (2009) used entropy production concepts and other ther- mate and compares well with other current AOGCMs of modynamic quantities to study a solar constant variation the Coupled Model Intercomparison Project (Reichler and hysteresis experiment. Kim 2008). Some previous entropy budgets have been published HadCM3 and FAMOUS are configurations of the Met based on observational analyses, theoretical calculations Office (UK) unified forecast and climate model. Apart from and model simulations. Fraedrich and Lunkeit (2008) used some minor differences the physics and the dynamics are a 25-year integration of the Planet Simulator, a model of the same. By virtue of the reduced horizontal and vertical intermediate complexity (Fraedrich et al. 2005). Goody resolution and increased timestep FAMOUS runs about ten (2000) diagnosed entropy terms from a six-year control run times faster than HadCM3. This characteristic is particu- using the atmospheric component of Goddard Institute of larly useful for long runs or large ensembles of integrations, Space Studies (GISS) model with sea-surface temperatures for which HadCM3 is too expensive in terms of time and (SSTs) maintained at climatological temperatures and also resources. However, the reduced resolution means inevita- gave some theoretical values for the most important bly that the climate in FAMOUS is somewhat less realistic, material entropy production terms. Peixoto et al. (1991) though still satisfactory for our investigative purposes. also made some order-of-magnitude estimates on the basis The runs subjected to entropy budget analysis are of energy flux values of Peixoto and Oort (1984). 50-year runs with a pre-industrial CO2 concentration. As noted in Fraedrich and Lunkeit (2008) entropy budgets of complex GCMs have not attracted much attention in the 2.1 The atmosphere scientific literature. In this paper we present the first analysis of the entropy budget of a complex coupled atmosphere– The atmosphere model of HadCM3 has a horizontal grid ocean general circulation model (AOGCM). This is the type spacing of 2.5° 9 3.75° and 19 vertical levels. The time- of model used for projection of anthropogenic climate step is 30 min. The horizontal atmospheric grid-spacing of change during the twenty-first century. It would be desirable FAMOUS is 5° 9 7.5°, twice that used in HadCM3, and if similar entropy budget diagnostics were calculated in other the vertical resolution is reduced to 11 levels. A timestep of AOGCMs in order to enable comparisons. 1 h is used. The paper is organised in the following way. Section 2 The dynamic core is a split-explicit finite difference describes the setup of runs being analysed. The entropy scheme (Cullen and Davies 1991). Each timestep consists budget equations are outlined in Sect. 3 and the method of an adjustment step followed by a Heun advection step. used to diagnose the associated entropy sources is descri- The prognostic variables are the zonal and meridional wind bed. In Sect. 4, we consider the radiation and show an component, v : (u, v), the surface pressure p*, the specific approximate method to calculate the irreversible entropy humidity q and the potential air temperature h. At the end production and the net entropy flux at the top of the of every integration step a Fourier filter is applied to avoid atmosphere. Section 5 discusses the atmospheric diabatic the numerical need for a very short timestep. The advection processes in HadCM3 due to the hydrological cycle and should be a fully adiabatic process but the numerical sensible heat. The meaning of the energy correction term in scheme can introduce spurious numerically generated HadCM3 and the kinetic energy dissipation is outlined in entropy sources/sinks (Egger 1999; Johnson 1997).

123 S. Pascale et al.: Climate entropy budget

Hyper-diffusion (sixth-order in HadCM3, eighth in the oceans, the solid Earth and the cryosphere), and radiation, FAMOUS) is applied after the advection to the model which extends beyond the Earth system but which penetrates prognostics. The temperature hyper-diffusion leads to a it and interacts with the matter. The major effect of this small local heating and the momentum hyper-diffusion to a interaction is an energy gain or loss: the climate gains q_rad dissipation of kinetic energy (Boville and Bretherton 2003). and the radiation loses q_rad; where q_rad is the radiative heating The physical parametrisation schemes (see Sects. 5–8) rate. The balanced entropy budget of the steady climate include: (a) the radiation scheme (Edwards and Slingo and the radiation is described by the two following equations 1996), (b) the convection scheme (Gregory and Rowntree (Weiss 1996; Goody 2000), respectively: 1990), (c) the large-scale cloud scheme (Smith 1990), q_rad (d) the large-scale precipitation scheme (Gregory 1995), s_mat qdV 0; climate 1 T þ ¼ ð Þ (e) the boundary layer scheme (Smith 1990). At the end of ZV  every timestep an energy and mass correction is applied to q_rad irr impose conservation. The atmosphere model includes the s_pl ndA À s_ qdV; radiation 2 Á ¼ T þ rad ð Þ surface model, over both land and ocean, which simulates TOAZ ZV  surface fluxes of latent and sensible heat and momentum, where the volume integrals are over the whole volume V of i.e. windstress. the climate system and the surface integral in (2) over the top of the atmosphere (TOA), n is the normal unit vector 2.2 The ocean defined on TOA; s_pl the (upward) entropy flux at TOA. For convenience, we list the most important symbols we use in The ocean component in HadCM3 is a 20 level version of the Table 1, and abbreviations in Table 2. Note that lower-case Cox (1984) model on a 1.25° 9 1.25° latitude–longitude s_ and q_ are intensive quantities, while upper-case S_ and Q_ grid. In FAMOUS this resolution is reduced to 2.5° 9 3.75° are extensive. with the 20 vertical levels unchanged. A 1-h time step is used Equation 1 is the entropy budget equation for the matter to integrate the model in HadCM3, 12 h in FAMOUS. The and it states that in a steady state climate the radiative Boussinesq approximation is adopted, in which density dif- rev entropy source S_ q_rad=T qdV; where T is the tem- ferences are neglected except in the buoyancy term. The rad perature at which the¼ energyð isÞ gained or lost, must be hydrostatic assumption is made in which local acceleration R balanced by the rate of material entropy production S_ and other terms of equal order are eliminated from the mat due to material irreversible processes: viscous and thermal equation of vertical motion. In order to suppress rapidly dissipation, latent and sensible heating, etc. travelling external waves, the model has a flat rigid surface, In a steady state the entropy of the climate system is not which exerts a downward pressure in lieu of the weight of a changing, so the net radiative entropy flux at the top of the variable sea level. The prognostic variables are sea-water atmosphere S_pl s_pl ndA must equal the planetary potential temperature h, salinity and horizontal velocity. ¼ TOA Á entropy production. Equation 2 describes the entropy In the ocean shortwave radiation is partly absorbed at the R budget for the radiation interacting with the climate sys- surface (red component) and partly (blue component) selec- tem. It says that the planetary entropy production in a tively with depth using a double exponential decay. Within steady state is equal to the sum of the radiative entropy sink the ocean various parametrisation schemes are employed to rev of the matter S_ q_rad=T qdV; which Goody represent small-scale turbulent mixing (see Sect. 8). rad (2000) calls reversibleÀ ¼(sinceðÀ the heatÞ is transferred from The atmospheric and ocean model are coupled once per R the radiation to the matter through a series of local equi- day. The atmosphere model is run with fixed SSTs through librium states which are by definition locally reversible), the day and the various fluxes are accumulated after each and the irreversible entropy production S_irr dVs_irr due atmospheric time step. At the end of the day these fluxes rad ¼ rad to thermalization of the absorbed and emitted photons are passed to the ocean model which is then integrated R (Goody and Abdou 1996). The processes which contribute forwards in time and supplies SSTs and sea-ice conditions to S_irr are discussed in Ozawa et al. (2003) and Fraedrich back to the atmosphere. rad and Lunkeit (2008) and Sect. 4. By adding (1) and (2) we obtain:

3 The climate entropy budget irr s_pl ndA s_mat s_ qdV 3 Á ¼ þ rad ð Þ Z Z 3.1 Planetary entropy production TOA V ÀÁ which now gives us a joint description of matter and radiation The spatial domain of the climate system encompasses in the Earth system. Equation 3 states that the total entropy matter (i.e. the fluid which constitutes the atmosphere and carried out of the Earth system by electromagnetic radiation

123 S. Pascale et al.: Climate entropy budget

Table 1 List of symbols Symbol Description Units

-1 q_k Specific heating rate due to process k W kg

Q_ k Global integral of heating due to process k W -1 -1 s_k Specific entropy source due to process k W kg K -1 S_k Global integral entropy source due to process k WK S_ Global integral rate of change of entropy of the climate system W K-1 -1 S_mat Global integral rate of material entropy production W K -2 -1 s_pl Outgoing entropy flux at TOA W m K irr -1 -1 s_rad Specific rate of radiative irreversible entropy production W kg K -2 sx,y Zonal and meridional component of the turbulent stresses N m H Sensible heat flux at the surface, positive upwards W m-2 Latent heat flux at the surface, positive upwards W m-2 L e Specific dissipation rate of kinetic energy W kg-1 D Global integral dissipation rate of kinetic energy W

hL Liquid-frozen potential temperature K q Specific humidity g kg-1 -1 qL/F Liquid/frozen cloud water g kg -2 Lz Longwave radiative flux at level z Wm -2 ez Longwave emission at level z Wm

z Transmittivity at level z – T

Table 2 List of abbreviations used in the text equals the sum of the material entropy production S_mat and the irreversible source of radiative entropy production S_irr Abbreviations Description rad taking place inside the Earth system ‘‘adv’’ Advection _ _ _irr _rev _irr Spl Smat Srad Srad Srad: 4 ‘‘at’’ Atmosphere ¼ þ ¼À þ ð Þ ‘‘bl’’ Boundary layer 3.2 Method for diagnosis of entropy sources ‘‘conv’’ Convection ‘‘diff’’ Temperature diffusion In non-equilibrium thermodynamics (Kondepudi and ‘‘diss’’ Dissipation Prigogine 1998) the specific material entropy production, ‘‘encor’’ Energy correction i.e. not including radiation, has the form: ‘‘hor’’ Horizontal diffusion s_mat =T F 1=T Ji li=T ; 5 ‘‘ice’’ Ice physics ¼ þ Árð ÞÀ Árð Þ ð Þ ‘‘irr’’ Irreversible where F is the material heat flux, Ji the diffusive mass flux ‘‘lh’’ Latent heat of the ith chemical species, with chemical potential li, and ‘‘solid’’ Solid Earth and cryosphere e the specific dissipation rate of kinetic energy due to ‘‘ls’’ Large scale precipitation viscous forces. We omit the contribution from chemical reactions. The first term can be diagnosed directly in a ‘‘lw’’ Longwave GCM (Sect. 6). The second term is hard to deal with ‘‘mat’’ Matter because in a GCM F is not represented by processes ‘‘ml’’ Ocean mixed-layer mixing transporting heat down a temperature gradient; many ‘‘oc’’ Ocean parametrised processes are typically involved, which do ‘‘pl’’ Planetary not involve fluxes and gradients explicitly. On the other ‘‘rad’’ Radiation hand, local heating rates are quite easily diagnosed for ‘‘sh’’ Sensible heat these processes. A heating rate can be written in terms of ‘‘sur’’ Surface the convergence of F and, from elementary calculus, ‘‘sw’’ Shortwave ‘‘TOA’’ Top of the atmosphere F =TdV F 1=T dV F=T ndA ‘‘tur’’ Turbulent ðÀrÁ Þ ¼ Árð Þ À ð ÞÁ ZV ZV ZA ‘‘ver’’ Vertical diffusion 6 ð Þ

123 S. Pascale et al.: Climate entropy budget over a volume V with boundary A. If we integrate over the vapour, etc., by direct calculation involving chemical whole climate system, by construction there is no boundary potentials. The global integral of this term can be term, because heat fluxes through the boundary are purely evaluated instead from the rate at which heat is added radiative, not material. Hence, when the whole climate (removed) to (from) the hydrological cycle, that is from system is taken into account, the entropy production rate the latent heating associated with evaporation at the due to F (first on the right) equals the volume integral of surface and phase changes within the atmosphere and from the entropy source (on the left). the temperatures at which this occurs. This gives the net In the present work, following Fraedrich and Lunkeit entropy production by the hydrological cycle indirectly (2008), the climate entropy sources/sinks are computed (Kleidon 2009). ‘‘directly’’ based on temperature and temperature tenden- cies. In this method the entropy source associated with the 3.3 Material entropy production by the climate system physical process k is calculated from the local diabatic heating q_k according to: In terms of material sources (7–9) the global integral rate of material entropy production is written: q_k 1 1 s_k ; WKÀ kgÀ ; 7 c ¼ T ð Þ ð Þ q_k 1 S_mat qdV ; WKÀ 10 ¼ T ð ÞðÞ i.e. in terms of the temperature increments: k c Z X X Vc 1 oTk 1 DTk s_k cp cp ; 8 where the sum over c refers to the subsystems of the cli- ¼ T ot ’ T Dt ð Þ mate system, i.e. the atmosphere, the ocean, and the solid where t is time, cp the specific heat capacity at constant Earth and cryosphere, which we denote together as ‘‘solid’’. pressure, DTk the temperature increment after the pro- T is the local temperature in each subsystem where the heat c cess k and Dt the physics timestep. In the model time- q_k is released, k are the non-radiative diabatic processes, step the temperature is incremented sequentially after and the integrals are extended over the volume of the cli- each process. In the present study T is chosen to be the mate subsystems Vc. temperature after the process. In the limit of small Let us write (10) in terms of the quantities available from timesteps the processes are simultaneous and T is the model in order to identify them. The heating rates con- uniquely defined. Since the entropy source is due to a sidered are those due to viscous dissipation and material heat local convergence or divergence of heat, it can be transport, including latent heating. In the model, viscous positive or negative, unlike entropy production, which is dissipation in the atmosphere  q_at ; but viscous dissipa- ¼ diss due to processes that transport heat but cause zero net tion is neglected in the ocean, and is absent in the solid Earth. heating, and which is positive definite according to the Atmosphere model processes cause sensible and latent at at second law of thermodynamics. heating q_sh; q_lh within the body of the atmosphere and oc; solid oc; solid To express the heating rates and the entropy sources in q_sh ; q_lh in the surface layer of the ocean and solid Wm-2 and W m-2K-1 we have to multiply (8) by the Earth. Within the ocean model, there are heating rates due to mass per unit area between the upper and lower vertical turbulent mixing processes (diffusion and mixed layer oc boundaries of the grid-box. Because the model makes the physics) q_tur: We neglect the entropy production due to the hydrostatic approximation, the required factor is (-Dp/g), heat conduction within the ground which has been estimated where Dp is the pressure difference between the upper and by Weiss (1996) and Kleidon (2009) to be about 1 mW lower vertical boundaries of the grid-box. m-2 K-1 In the ocean we have available from the model output the Hence (10) can be expanded as: potential temperature h and the potential temperature at at at q_diss q_lh q_sh increments due to the different parametrisation schemes, S_mat qdV þ qdV ¼ T þ T dhk/dt for process k. By using the general relation between VZat VZat potential temperature h and entropy s (Vallis 2006), q_solid q_solid q_oc q_oc q_oc lh sh dV lh sh dV tur dV ds = c (p , h)dh/h, and noting that in HadCM3 cp pR; h þ q þ q q ; p R ð Þ¼ þ T þ T þ T c~ is constant, we can diagnose the entropy source using the VsolidZ VZoc VZoc following equation: 11 ð Þ 1 Dhk s_k c~ : 9 where Vat, Voc, Vsolid are the volumes of the atmosphere, ’ h Dt ð Þ solid solid oc oc ocean and solid Earth. Because q_sh ;q_lh ;q_sh ;q_lh exist The third term on the right of Eq. 5 accounts for only in the surface layer of the solid Earth and ocean, we changes of phase of water, molecular diffusion of water can rewrite them:

123 S. Pascale et al.: Climate entropy budget

q_solid q_solid Hsolid solid sh lh dV dA 12 Now (20) can be easily identified as the integrand function þ q þL ; rev T ¼À Tsur ð Þ in (1), which states that S_ S_ S_ 0; i.e. the rate of Z Z rad mat Vsolid Aland change of entropy of the climate¼ þ system¼ is zero in a steady q_oc q_oc Hoc oc state. This holds also for any subsystem of the climate sh þ lh qdV dA þL 13 T ¼À Tsur ð Þ separately, e.g. the atmosphere. VZoc AZoc where T is the surface temperature, H and are the sur L sensible and latent heat flux respectively (positive upwards, 4 Radiative entropy production within the climate so these fluxes cool the surface), and Aoc and Aland are the system ocean and land surfaces. The latent and sensible heat fluxes are available from the model, so these terms can be readily 4.1 The radiation scheme evaluated. Noting that Aoc Aland A; i.e. the whole [ ¼ interface between the atmosphere and the land–ocean, The atmosphere comprises N emitting-absorbing levels (11) can be further rewritten: (N = 19 in HadCM3 and N = 11 in FAMOUS). The

at at radiation scheme consists of two parts, solar (shortwave) q_diss q_sh H S_mat qdV qdV dA and thermal (longwave) (Edwards and Slingo 1996), using ¼ T þ 0 T À Tsur1 VZat VZat ZA six shortwave bands and eight longwave bands. It includes @ A the effects of greenhouse gases: CO2,H2O, O3,O2,N2O, q_at q_oc lhqdV dA L dV turqdV: CH4, CFC11 and CFC12. The calculations of the radiative þ 0 T À Tsur1 þ T fluxes within the atmosphere allow an estimate of the VZat ZA VZoc @ A shortwave heating rate q_sw and the longwave heating 14 rev rate q_lw: From these last we obtain s_ q_sw=T q_lw=T ð Þ rad ¼ þ By defining (Eqs. 1, 2). The radiation generates a large entropy sink -2 -1 at (*-390 mW m K ) in the body of the atmosphere q_diss S_diss qdV; 15 because it is dominated by longwave cooling, the atmo- T  ð Þ sphere being heated mainly non-radiatively from the sur- VZa face by latent and sensible heat. _ _at _sur Ssh lh Ssh lh Ssh lh 16 þ  þ þ þ ð Þ qat qat H 4.2 The irreversible radiative entropy production _at _lh _sh _ sur Ssh lh þ qdV; Ssh lh dA þL; þ  T þ À Tsur VZa AZsur As pointed out by Weiss (1996) and Goody (2000) the _irr 17 irreversible radiative entropy production Srad is a large term ð Þ _ oc compared with the material entropy production Smat and oc q_tur S_ qdV; 18 thus it dominates the planetary entropy production S_pl (see tur T  ð Þ Table 3). The radiative processes accounted for by S_irr are VZoc rad _irr the thermalization of absorbed shortwave radiation Ssw and we summarise Eq. 14 thus: _irr the absorption and reemission of longwave radiation Slw _ _ _ _oc within the climate system. Thermalization is the process in Smat Sdiss Ssh lh Stur: 19 ¼ þ þ þ ð Þ which the molecules after the absorption of photons rear- To this we may add two unphysical sources of entropy from range their energy levels towards lower energy modes. This _ the atmosphere model: Sdiff due to temperature hyperdiffu- is a small-scale process whereby local thermodynamic _ sion, which is included for numerical stability, and Sadv due to equilibrium is maintained; when considering the climate inaccuracy in the advection by the dynamical core. entropy budget, we are concerned with larger scales, and In Table 4 the vertically integrated global averages are local thermodynamic equilibrium is assumed always to shown of the entropy sources and diabatic heatings apply. The large irreversible radiative entropy production is described in the following sections. not relevant to the operation of the material climate system, When radiative heating rates are also included in a as can be understood by a thought experiment, in which we _ formula like (10) we obtain the rate of change of entropy S imagine there is no radiation, but instead we provide the of the climate system: climate system at every point with heat sources and sinks q_c q_c equal to shortwave absorption and net longwave emission. S_ qdV k rad : 20 ¼ T þ T ð Þ This substitution of a different mechanism for external c Z k ! X Vc X energy exchange would not affect the evolution of the

123 S. Pascale et al.: Climate entropy budget

Table 3 The global integral entropy budget obtained from HadCM3 and FAMOUS is compared with other GCM simulations and observations Process HadCM3 FAMOUS Fraedrich and Lunkeit (2008) Goody (GISS) Goody Peixoto

Radiative entropy terms

S_pl 911.3 897.8 882 – – 892 _irr Ssw 811.8 790.9 812 802 – 819 _sur;at Slw 10.6 10.2 6 – – 24 _at;at Slw 38.6 42.7 28 – – – _irr Srad 861.0 843.7 846 – – 843 _rev Srad -51.0 -54.1 – -72.8 – – Material entropy terms

S_sh lh 37.8 37.9 29 58.7 21.2 25 þ _bl Ssh 2.2 2.3 1 3.4 2.4 2.1 a a b S_diss 12.5 13.6 6 11.5 11.3 7

S_diff 0.8 1.1 – – – –

S_adv -0.1 -0.4 – – – – _oc Stur 0.8 1.0 – – – – S_mat 51.8 53.3 35 70.2 32.5 34.1 Rate of change of entropy S_ 0.8 -0.8 -7 -2.6 – -17 In the table here both material and radiative terms discussed in Sects. 3–8 are summarized. All the terms are in mW m-2 K-1 a The term denoted by ‘‘diss’’ is from the energy correction in HadCM3 and FAMOUS b The estimate in Fraedrich and Lunkeit (2008) is deduced from the total entropy imbalance _irr Some values from Peixoto et al. (1991) have been updated after Kleidon and Lorenz (2005). The value of Ssw in Goody has been calculated from _oc values given in Table 6 in Goody (2000) assuming TSun = 5,777 K. Values of Stur are here averaged over the whole surface and therefore different from the ones shown in Table 6 climate system, since the fluid equations are concerned _irr 1 1 sur 1 1 Ssw dA dzq_sw dAFsw : only with the amount of heat gained or lost. Nonetheless, ¼ T À TSun þ Tsur À TSun we calculate S_irr following Ozawa et al. (2003), Peixoto Z Z Z  rad; 21 et al. (1991) and Fraedrich and Lunkeit (2008). We also ð Þ calculate S_pl from the incoming shortwave and outgoing The first term is associated with the absorption of light longwave radiation; the entropy flux carried by a radiation in the atmosphere and the second accounts for absorption at beam is obtained by dividing the flux of radiative energy by the surface. Equation 21 is formally identical to Eqs. 16a, the emission temperature. 16b in Ozawa et al. (2003) but allows for a differential absorption throughout the atmosphere. 4.3 The shortwave radiation 4.4 The longwave radiation Let us first consider the solar shortwave radiation. In the irr radiation scheme the solar radiation is represented as a Longwave irreversible entropy production s_lw is calculated vertically orientated downward beam which is partly as the sum of two components: emission–absorption at;at absorbed at every level and finally at the surface. We will longwave interaction within the atmosphere, s_lw ; and not consider the entropy associated with scattering and emission–absorption longwave interaction between the sur;at reflection. If we assume the brightness temperature of the surface and the atmosphere, s_lw : at;at sur;at incoming radiation to be that of the surface of sun, Exact calculation of the s_lw and s_lw would require the _pl TSun = 5.777 K, the incoming entropy flux Ssw knowledge of the transmittivity and emissivity at every TOA _ ¼ Fsw =TSun (noting that the net Spl is defined as positive frequency m. Here we have decided to provide a first TOA _irr outwards), where Fsw is the net downward shortwave approximation for Slw based on spectrally integrated radi- radiative flux at the TOA. This flux is absorbed within the ative transfer. In the radiation scheme it is easy to obtain TOA climate system, such that Fsw dA qdzq_sw the upward and downward longwave fluxes at every model sur sur ¼ þ dAF ; where F is the net downward flux at the sur- half level (interfaces between model layers), say Lz?1/2(:) sw sw R R face. Then from Eq. 4 we have and Lz?1/2(;). By neglecting the spectral dependence for R

123 S. Pascale et al.: Climate entropy budget

transmittivity m and assuming the emissivity to be where Ksur is the longwave radiation originated at the Tð Þ 1 ; we can express the fluxes at each half-level as ground reaching the top of the atmosphere, which in our ÀT approximation is: Lz 1=2 ez zLz 1=2 ; þ ð"Þ ¼ þT À ð"Þ 22 Ksur esur 1 2; ...; N 27 Lz 1=2 ez zLz 1=2 ð Þ À ð#Þ ¼ þT þ ð#Þ ¼ T T T ð Þ and Kz the longwave flux emitted at the atmospheric model from which we can obtain the following estimates for z T level z and still reaching the top of the atmosphere: and the Plank sources ez at every model level z: Kz ez z 1 z 2; ...; N : 28 Lz 1=2 Lz 1=2 ¼ T þ T þ T ð Þ z þ ð"ÞÀ À ð#Þ; ez Lz 1=2 zLz 1=2 : T ¼ Lz 1=2 Lz 1=2 ¼ þ ð"ÞÀT À ð"Þ Emissions ez and transmittivities z are calculated as À ð"ÞÀ þ ð#Þ T 23 shown in (23). ð Þ The approximation (22) is quite good. This can be at;at TOA s_lw is determined by the longwave interaction of any two checked comparing the value of Flw obtained from the atmospheric layers i and j at temperature Ti and Tj. The functions (27), (28) with the one available in the model layer i emits an energy amount ei via longwave radiation, diagnostics in which the radiative transfers are properly which crosses the intermediate layers through which it is calculated. The two fields are very similar indeed. Despite partially transmitted arriving at layer j, where a fraction the crudeness of the approximation (22) it is surprising to -2 1 j is absorbed. The same happens for the emission ej see that typical differences at TOA are 0.4 W m or less ð ÀT Þ _at;at partially absorbed at the level i. Hence Slw can be written in FAMOUS (i.e. a relative error of about 0.2%) and 0.02 as: in HadCM3 (*0.01%). More significant differences are on N the polar regions and on the South Pole particularly though, at;at 1 1 S_ ij 24 where our approximation leads to underestimates of up to lw T T ¼ i 1 j [ i C j À i ð Þ X¼ X  *2%. Thus the planetary entropy production is: where the coefficients ij are given by: C N TOA j 1 Ksur Kz F ei 1 j ej 1 i lÀi 1 l j i [ 1 S_ S_lw S_sw sw 29 ij ð ÀT ÞÀ ð ÀT Þ ¼ þ T À pl pl pl ; C ¼ ei 1 j ej 1 i j i 1: ¼ À ¼ Tsur þ z 1 Tz À TSun ð Þ  ÂÃð ÀT ÞÀ ð ÀT Þ Q À ¼ X¼ _sur;at which is formally identical to Eq. 12 in Ozawa et al. The estimate of Slw (Eq. 16c) in Ozawa et al.(2003) requires knowledge of the upwelling longwave radiation (2003). emitted by the surface (say e ) and partially absorbed at We estimate the error in the longwave entropy flux as sur _ the level i and conversely of the radiation e emitted at the DSpl DL=T; where DL is a typical mean value for the flux i  -2 level i and absorbed at the surface. This term can be written error and T a typical temperature. Taking DL*0.2 W m as: from the model output and T *270 K we have 2 1 DS_pl 0:7mW mÀ KÀ ; which is quite small if compared to N 1 1  _ 2 1 _sur;at the typical values of Spl 900 mW mÀ KÀ (see Table 3). Slw i 25  ¼ G T À T ð Þ This estimate of the error is of the same order of magnitude of i 1 i sur X¼ the residual in the balance Eq. 2 we found in the model where the coefficients i are: -2 -1 -2 -1 G (1.2 W m K in HadCM3 and 0.2 W m K in esur 1 1 ei i 1 FAMOUS), as can be seen in Table 3. i ð ÀT ÞÀ i 1 ¼ G ¼ esur 1 i ei lÀ1 l i [ 1:  ½Šð ÀT ÞÀ ¼ T 4.5 The planetary entropyQ production 5 Sensible and latent heating The planetary entropy production is the difference between _pl the outgoing planetary longwave entropy flux Slw and the 5.1 Convection _pl TOA incoming solar shortwave entropy flux Ssw Fsw =TSun at the top of the atmosphere. In order to estimate¼ the long- The convection incorporates several irreversible processes wave entropy flux the longwave energy flux at the top of which lead to diabatic heating q_conv and an entropy source TOA the atmosphere Flw must be decomposed into its emission s_conv: Water vapour condensation of the saturated rising level components: parcels releases a considerable amount of heat in the sur- N rounding cloud environment. At the same time the rising TOA parcel interacts with the ‘‘external’’ environment through Flw Ksur Kz; 26 ¼ þ z 1 ð Þ entrainment and detrainment at the edges of the cloud. X¼

123 S. Pascale et al.: Climate entropy budget

In the convection scheme all these processes are imple- This air temperature change is due to latent heat cp dT mented in the convective cloud model, and lead to an associated with condensation/evaporation forced by the increment of the cloud potential temperature. After dynamics and all the physical processes which ‘‘unbal- updating the cloud properties, the scheme finds out the ance’’ the cloud, i.e. which make it inconsistent with the impact on the environment, i.e. on the model primary required diagnostic relations. variables h and q. In fact, as usual in all mass flux schemes, The large scale precipitation scheme (Gregory 1995) the plume interacts with its environment through en/ deals with micro-physical processes associated with large detrainment of heat, moisture and cloud liquid water and scale precipitation: evaporation of rain and sublimation of subsidence induced in the cloud environment which com- frozen precipitation in sub-saturated regions, freezing and pensates for the parcel’s upward mass flux. The expres- melting of the cloud water if this is in an inappropriate state sions for the net change of environmental potential before it is converted to liquid or frozen precipitation. The temperature and specific humidity are quite involved temperature increments contribute to the total large scale and can be found in Gregory and Rowntree (1990), cloud diabatic heating q_ls and to the entropy source s_ls; Eqs. (21a, b). which peaks in the mid-latitude regions (see Fig. 1c). By looking at Table 4 and Figs. 1b, 2b we note that the entropy source associated with the convection scheme is 5.3 Boundary layer the largest non-radiative term, and is concentrated mainly in the upper tropical atmosphere. The boundary layer scheme (Smith 1990) determines the transport processes of momentum, latent heat and sensible 5.2 The large-scale cloud and precipitation scheme heat in the lowest atmospheric levels and at the surface (the boundary layer levels are five in HadCM3 and three in The large scale cloud scheme (Smith 1990) operates FAMOUS). The boundary layer heating term q_bl and diagnostically using the values of the so-called ‘‘cloud- entropy source s_bl are the sum of two contributions: one conserved variables’’ to calculate the new cloud amount in due to the divergence of sensible heat (enthalpy) fluxes and every grid box and the new amounts of water vapour, liquid the other due to condensation of some moisture forced by and frozen cloud water. These variables are the total water the vertical turbulent motions leading to the formation of content qt = q ? qL ? qF (where q is the water vapour low clouds. The scheme is implemented by using the cloud and qL and qF the liquid and frozen water contents, conserved temperature TL and hence ‘‘TL fluxes’’ respectively) and the liquid-frozen water potential tem- ’ FT_L = q w0 TL are used on the boundary layer levels to perature hL = h - (LC/cp)qL - {(LC ? LF)/cp}qF (where h i work out TL increments (here we have used the usual for- LC and LF are the latent heat of condensation and fusion malism of fluctuations and averaging of the statistical respectively and cp the specific heat capacity at constant theory of the turbulence). The relation between the tem- pressure). hL and qt are not affected by changes of phase, perature increment dT and dT is, from the T definition: and nor is the corresponding liquid-frozen temperature L L LC LC LF TL = T - (LC/cp)qL - {(LC ? LF)/cp }qF), while qL and dT dTL dqL þ dqF: 31 ¼ þ cp þ cp ð Þ qF are incremented by amounts dqL, dqF. As a consequence a net temperature increment dT is implied: From dT we calculate the boundary layer entropy source LC LC LF (Figs. 1d, 2d), which is quite intense but of course confined dT dqL þ dqF: 30 ¼ cp þ cp ð Þ to the lower troposphere.

Table 4 List of the area-averaged vertically integrated entropy sources/sinks and diabatic heatings for the atmosphere and the surface for a 50 year control run in FAMOUS and HadCM3 Atmosphere Ocean and solid

_rev ______sur _rev Srad Sconv Sls Sbl Sdiff Sencor Sadv Total Ssh lh Srad Total þ HadCM3 -392.1 235.3 69.9 73.1 0.8 12.5 -0.1 -0.7 -340.5 341.1 0.6 FAMOUS -397.6 265.4 44.4 72.2 1.1 13.6 -0.4 -1.2 -344.1 343.8 -0.3

Q_ rad Q_ conv Q_ ls Q_ bl Q_ diff Q_ encor – Total H Q_ rad Total þL HadCM3 -101.9 63.3 17.5 21.1 0.1 3.1 – 3.2 -101.2 101.7 0.5 FAMOUS -103.1 71.6 10.5 20.9 0.1 3.4 – 3.5 -103.1 102.5 0.6 The different terms are associated to the different parametrisation schemes discussed in Sect. 4–7. There is no heating associated with the numerical entropy sinks. The entropy sources are expressed in mW m-2K-1 and the heating rates in W m-2

123 S. Pascale et al.: Climate entropy budget

Fig. 1 Fifty year zonal means a -1 -1 b -1 -1 of the specific entropy sources RADIATION [mW kg K ] CONVECTION [mW kg K ] s_rad; s_conv; s_ls; s_bl; s_diff and 1 1 0.2 0.2 s_encor mW kgÀ KÀ ) within the ð atmosphere from FAMOUS 0.4 0.4

0.6 0.6

0.8 0.8

90N 45N 0 45S 90S 90N 45N 0 45S 90S Eta (hybrid sigma-pressure) Eta (hybrid sigma-pressure)

-0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 -0.1 -0.05 0 0.05 0.1

c LARGE SCALE [mW kg-1 K-1] d BOUNDARY LAYER [mW kg-1 K-1]

0.2 0.8 0.4

0.6 0.9 0.8

90N 45N 0 45S 90S 90N 45N 0 45S 90S Eta (hybrid sigma-pressure) Eta (hybrid sigma-pressure)

-0.06 -0.04 -0.02 0 0.02 0.04 0.06 -0.1 0 0.1 0.2 0.3 e TEMPERATURE DIFFUSION [mW kg-1 K-1] f ENERGY CORRECTION [mW kg-1 K-1]

0.2 0.2

0.4 0.4

0.6 0.6

0.8 0.8

90N 45N 0 45S 90S 90N 45N 0 45S 90S Eta (hybrid sigma-pressure) Eta (hybrid sigma-pressure)

-0.02 -0.01 0 0.01 0.001 0.0012 0.0013 0.0015 0.0016 0.0018

It is not known how much of dqL and dqF is due to qL = qF = 0, i.e. there is no surface flux of cloud water, condensation and how much simply to convergence of the only water vapour. Furthermore, we know that FT_L = qL and qF fluxes. In order to know that we would need to FT = 0 above the boundary layer levels. Therefore the know those fluxes, but the scheme just uses the fluxes of qt. convergence of the vertical turbulent fluxes of sensible Hence it is not possible to separate uniquely the two dif- heat must take place completely within the boundary layer. ferent contributions to s_bl from sensible heat and conden- A very simple approximation is: sation. The cloud scheme case is different in this respect 1 1 because no flux divergences are present and thus changes S_bl H dA; 32 sh ~ in qL and qF are entirely due to local water vapour con- ’ T À Tsur ð Þ surZ  densation into cloud water. In other words the HadCM3 boundary layer scheme is intrinsically ignorant about the where T~ is a temperature representative of the boundary partitioning of sensible and latent heating. This is an layer convergence of the sensible heat flux. We have example of the shortcomings of models that are not gen- arbitrarily chosen the temperature on the second model erally constructed to deal with an accurate treatment of half-level (the first half level is the surface) corresponding entropy. to a pressure level of approximately 950 hPa since the An approximate estimate of the entropy produced by the convergence of FT_L takes place mostly between the sur- vertical turbulent fluxes of sensible heat within the face and the first model levels. Within this approximation, boundary layer can be obtained when we recall that at we can see from Table 3 that the entropy production due to the surface (which coincides with the first boundary layer the hydrological cycle is *35.6 mW m-2K-1, that is * _ _bl level in the model), FT_L = H, because at the surface Ssh lh Ssh: þ À

123 S. Pascale et al.: Climate entropy budget

Fig. 2 Fifty year mean of the a -2 -1 b -2 -1 mass weighted vertical integrals RADIATION [W m K ] CONVECTION [W m K ] 90N 90N of the entropy sources s_rad; s_conv; s_ls; s_bl; s_diff and s_encor 45N 45N (W m-2K-1) from FAMOUS 0 0

45S 45S

90S 90S 0 90E 180 90W 0 0 90E 180 90W 0

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 c LARGE SCALE PRECIPITATION [W m-2 K-1] d BOUNDARY LAYER [W m-2 K-1] 90N 90N

45N 45N

0 0

45S 45S

90S 90S 0 90E 180 90W 0 0 90E 180 90W 0

-0.05 0 0.05 0.1 0.15 0.2 0.25 -0.1 0 0.1 0.2 0.3

e TEMPERATURE DIFFUSION [W m-2 K-1] f ENERGY CORRECTION [W m-2 K-1] 90N 90N

45N 45N

0 0

45S 45S

90S 90S 0 90E 180 90W 0 0 90E 180 90W 0

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.009 0.01 0.011 0.012 0.013 0.014 0.015

5.4 Temperature hyper-diffusion change in the atmosphere total kinetic and moist static energy over each timestep with the convergence of surface The temperature hyper-diffusion terms q_diff and s_diff have a and TOA heat fluxes. The imbalance is ascribed to the characteristic vertically striped pattern due to the numerics neglect by the atmosphere model of the heating D due to of the diffusion operator. This is quite unphysical as it can kinetic energy dissipation by the internal stresses, i.e. be observed in Figs. 1e, 2e. The hyper-diffusion term has almost no effect on the overall heating ( 0.09 W m-2) but D dVs : v 33 * E¼ ¼ r ð Þ although no net heat is added to the atmosphere, the heat is ZV moved down a temperature gradient which generates where s is the stress tensor. is converted into a temperature entropy of around 1 mW m-2K-1. E increment dT = cpM ; M being the mass of the atmo- ¼E ð Þ sphere, which is applied uniformly to the temperature field in the model. The cross section of the associated specific entropy 6 The energy correction and the dissipation of kinetic source is shown in Fig. 1f, where it can be noted that it is energy inversely proportional to the temperature field (since when 1 1 expressed in mW kgÀ KÀ ; s_encor dT=T)whereasinFig. 2f / 6.1 The meaning of the energy correction in HadCM3 the mass weighted vertical integral of s_encor shows the pro- portionality to the mass of the atmospheric vertical column. The atmosphere energy correction (Gregory 1998) is We can check that the energy correction effec- E introduced in the model to account for the energy loss after tively corresponds to the total amount of dissipated every timestep. This loss is diagnosed by comparing the kinetic energy by calculating the change in the total

123 S. Pascale et al.: Climate entropy budget

(thermal ? gravitational) potential energy P = $(cv T ? the curves are substantially the same. The bias might be gz)qdV during the advection step in the model. In fact from attributed to numerical inaccuracy or the small imbalance the energy cycle formalism (Peixoto and Oort 1992) the in the diabatic heating (0.1 W m-2, see Table 4). links between dissipation D = $qe dV, kinetic energy K = $qdV (v v)/2, the rate of conversion of potential Á 6.2 The dissipative processes in the model energy into kinetic energy C(P, K) =-$dV v p, the Ár total diabatic heating Q_ and P are: In the model there are four explicit processes causing the K_ D C P; K 34 dissipation of the specific kinetic energy (v v)/2: (a) the ¼À þ ð ÞðÞ turbulent eddy stresses in the boundary layerÁ scheme P_ C P; K Q_ 35 ¼À ð Þþ ð Þ associated with the vertical diffusion of horizontal which says that in a steady state (K_ P_ 0) the dissipa- momentum; (b) the hyper-diffusion term in the momentum ¼ ¼ tion must be equal to the rate at which potential energy is equation for (u, v); (c) the gravity wave drag and (d) the converted into kinetic energy (D = C) and that Q_ D: impact on the large scale flow due to the convective eddy- ¼ Now let us divide the atmospheric timestep of the model flux stresses. Of course in reality the mechanical dissipa- into an advective step, during which the equations are tion of kinetic energy takes place only at the very small integrated and no diabatic heating is added, and a following scales of the molecular viscosity (millimetres or less) physics step, when the diabatic heating is added up by all towards which it is drained down by the turbulent energy the physics schemes. This division is unphysical but it cascade. But in a GCM we do not resolve such small scales reflects the way the model works. Hence all the quantities and therefore we have to parametrize it by using phe- involved can written as the sum of an ‘‘advective’’ part and nomenological expressions for the eddy-stresses. a ‘‘physics’’ part, e.g. P_ P_adv P_phy; where P_adv is the The total amount of viscous dissipation D = $V_at qe dV ¼ þ rate of change of P during the advective step and P_phy due to the eddy-stresses is given by the mass integral over the during the physics step. During the advection in the model entire atmospheric volume of the specific kinetic energy ten- 2 there is no diabatic heating, Q_ 0; hence P_adv Cadv: dency qt(v /2) = v qtv caused by them. This can be readily Á We note that C(P, K) = 0 during¼ the physicsÀ step¼ since seen if we write the hydrostatic kinetic energy equation: there is no conversion of potential energy into kinetic v2=2 1 (during the physics step kinetic energy, and thus the hori- d ð Þ v hp FK ; 36 dt ¼Àqð Ár þrÁ ÞÀ ð Þ zontal wind (u, v), is altered only by the viscous drag and this term is accounted in D; apart from this the wind is where d/dt = qt ? v h, h the horizontal gradient, s the Ár r -1 unaffected). It follows that P_ P_ Q_ D: viscous stress tensor , FK = v s, and e = q hv s the adv phy Á r Á Therefore in a steady state modelÀ climate¼ the dissipation¼ ¼ local viscous dissipation. All advective and horizontal rate must be equal to minus the variation of total potential gradient terms are treated in the the dynamical core, so 2 energy during the advective step. We estimate P_adv using within the parametrizations (a–d) (36) is just qt(v /2) = -1 the Lorenz formula P = $cp T qdV, which is valid for an q FK - e. If we integrate over the whole atmospheric rÁ atmosphere over a flat surface. This is not the case for volume and neglect the flux of kinetic energy into the HadCM3 but here we are not interested in a very precise ocean (*0.007 W m-2, Peixoto and Oort 1992), we obtain 2 estimate of P but just in corroborating the interpretation of D = $V_at qedV = qt$V_at qdV (v /2). In the following, we the energy correction as dissipation. describe how HadCM3 deals with the kinematic impact of In Fig. 3a, b the time series of and P_adv are plotted the boundary layer, convection, gravity wave schemes and showing that despite a constant biasE of lessÀ than 0.1 W m-2 horizontal hyperdiffusion.

Fig. 3 Energy correction FAMOUS [W m -2] HadCM3 [W m-2] (continuous line) and E 3.8 3.8 consumption of total potential energy P_adv (dashed line) by atmosphericÀ dynamics for 3.6 3.6 FAMOUS and HadCM3 3.4 3.4

3.2 3.2

3.0 3.0 6600 6610 6620 6630 6640 2290 2300 2310 2320 2330

123 S. Pascale et al.: Climate entropy budget

(a) Turbulent stresses in the boundary layer: vertical Table 5 Area-averaged values for the vertically integrated dissipa- momentum diffusion. The vertical turbulent mixing of tion rate of kinetic energy due to various processes in HadCM3 and momentum affects the large scale circulation through FAMOUS bl bl bl the action of the eddy-stress s s i s j (Smith Dbl Dhor Dconv Dgw Residual ¼ x þ y E 1993). The model works out their impact on the HadCM3 3.1 1.5 0.6 0.6 0.3 0.1 horizontal flow according to: FAMOUS 3.4 1.4 0.8 0.8 0.2 0.2 bl ov 1 os -2 : 37 All the values are W m . The kinetic energy lost in the filtering is ot ¼Àq oz ð Þ -2 bl very small (*0.03 W m ) and hence not shown in the table above (b) Horizontal momentum diffusion. A hyper-diffusion operator is applied to horizontal velocity v yielding: boundary layer stresses. The energy correction term in D FAMOUS is slightly larger than in HadCM3 and the other ov v ; 38 terms are of comparable magnitude. Let us note though that ot ¼Dð Þ ð Þ hor the kinetic energy drain from the large scale flow (Dbl) is being a biharmonic operator of sixth order in slightly higher in HadCM3 than in FAMOUS. The dissi- D HadCM3 and eight in FAMOUS (details in Cullen pation of kinetic energy due to the turbulent down-scale et al. 1993). energy cascade generates an entropy source of *13 mW -2 -1 (c) Gravity wave drag. The drag of the mountains on the m K . For more discussion see Sect. 9. atmosphere is manifested in the production of gravity waves. The gravity wave drag scheme (Gregory et al. gw 7 Numerical entropy production in the dynamical core 1997b) determines the surface stress ssur and the distribution of this stress through the atmospheric column sgw. Once the stresses are determined, the In numerical models there is a small entropy source of a impact on the horizontal momentum is: numerical nature (Johnson 1997; Egger 1999). Theoreti- cally the advection is a perfectly adiabatic process. How- ov 1 osgw : 39 ever in the numerical integration phenomena of pure ot ¼Àq oz ð Þ gw numerical nature such as the numerical dispersion, Gibbs oscillation and numerical diffusion and filtering yield an (d) Convective momentum transport. The mass-flux entropy source. Moreover, the hyper-diffusion (Sect 5.4), convection scheme implemented in the Unified Model required for the numerical stability and, de facto, part of (Gregory and Rowntree 1990) also accounts for the the dynamic core, can be conceptually considered as part of vertical mixing ofhorizontal momentum associated with the numerical entropy (Woollings and Thuburn 2006) convective updraughts and downdraughts. The effects of although it is associated with an explicit heating. However the subgrid-scale circulation upon the large-scale wind this term, which is an unphysical one, may also be con- can be written as: sidered to represent the mixing of the small-scale eddies. ov ov0x0 40 The numerical entropy produced by the advection scheme ot ¼À op ð Þ conv is not associated with a heating term. As a consequence we cannot apply (8) but it must be estimated as in Woollings and where x = Dp/Dt. The vertical eddy-flux of horizon- Thuburn (2006) using the more general relation between tal velocity can be expressed in terms of large-scale entropy s and potential temperature h, ds = cp (dh/h) for a dry and cloud variables as (Gregory et al. 1997a) v0x0 ¼ atmosphere. In our case if ha is the potential temperature after Mu vu v Md vd v ; where M and v are the ð À Þþ ð À Þ u u the advection timestep t and just before, the numerical values of the cloud mass-flux and velocities in the D hb entropy production during the timestep Dt is: updraught and Md and vd in the downdraught. From (37–40) the kinetic energy tendencies and the S_adv qdVcp ln ha qdVcp ln hb =Dt: 41 dissipation terms are estimated and compared with the ¼ 0 À 1 ð Þ Z Z energy correction which is about 3.1 W m-2 in HadCM3 Vat Vat @ A -2 2 1 and 3.4 W m in FAMOUS. The several dissipative We found in the control runs S_adv 0:1 mW mÀ KÀ -2 -À1 contributions for the two models are shown in Table 5. (HadCM3) or *- 0.4 mW m K (FAMOUS), which Their sum is 3.0 W m-2 in HadCM3 and 3.2 W m-2 in is the combined effect of adjustment, advection and Fourier FAMOUS and so a small residual still remains which has filtering (Cullen and Davies 1991). Woollings and Thuburn to be ascribed probably to filtering and the discretization (2006) found a term of similar magnitude but positive error. It is found that about half of the dissipation is due to (*0.5 mW m-2K-1) using the Reading IGCM1 spectral

123 S. Pascale et al.: Climate entropy budget model (Hoskins and Simmons 1975). They considered the 8.2 Diffusion, mixing, convection, dynamics dynamical core as a whole, thus including also the effect of the hyper-diffusion in the computation of numerical Several irreversible processes are parametrised in the ocean entropy production. When we likewise sum S_diff S_adv; we model in order to reproduce the effects of the turbulence due þ-2 -1 find numerical entropy sources of 0.7 mW m K in to the sub-grid scale mixing: vertical diffusion, mixed layer HadCM3 and 0.6 mW m-2 K-1 in FAMOUS, similar to physics, convective adjustment and isopycnal diffusion. IGCM1. From Table 6 we can see that these processes do not con- tribute to an overall heating of the ocean although locally they may be non-zero. That means that they just move heat 8 Entropy production in the ocean from one point to another one inside the ocean without a flux contribution at the surface (a boundary flux). Conversely, The material entropy production in the ocean is due to they have a non-zero contribution to the ocean entropy three different processes: heat transport, viscous dissipation budget since they move heat down a temperature gradient. and molecular diffusion of salt ions. The last two are The dominant entropy source among the diabatic pro- negligible and estimated to be one and two orders of cesses inside the ocean is S_ver caused by the vertical diffu- magnitude smaller than that due to heat transport (Gregg sion of the potential temperature. This is the sum of two 1984; Shimokawa and Ozawa 2001). Furthermore in contributions: first, an explicit vertical diffusion, second HadCM3 ocean the dissipated kinetic energy is not the vertical component of the isopycnal diffusion. The accounted for at all in the energy equation. As a conse- explicit vertical diffusion is implemented as qz(j(z)qzh) quence we will consider only the entropy associated with where the vertical diffusion coefficient j is the sum of a diabatic heating in the model. Numerical entropy produc- depth-dependent constant background term and a term tion is negligible in the ocean model. dependent on the stability of the ocean and hence on the Richardson number as in Pacanowsky and Philander (1981). 8.1 Processes at the surface Mixing in the ocean occurs predominantly along iso- pycnal surface, i.e. surfaces of equal density, which is why Table 6 shows the area-averaged estimates of the vertically water masses tend to preserve their temperature and integrated sources and sinks of entropy in the ocean as salinity characteristics over very large distances. In order to simulated by HadCM3 and FAMOUS. It can be seen that represent such a process a diffusion scheme which orients the major part of the entropy is produced or destroyed at the mixing tensor along isopycnal surface is needed (Redi the top layers of the ocean model. The model divides these 1982). The isopycnal surfaces are mostly horizontal, par- into three: the blue component of shortwave radiation, ticularly at the low-latitude regions, but in the high-latitude fluxes due to latent heating from sea-ice melting and regions they can be quite steep. Therefore both a horizontal freezing and snowfall into the ocean, and the net effect of and a vertical component are present. all other atmosphere fluxes. AfurtherschemeimplementedintheHadCM3oceanmodel is Visbeck et al. (1997)versionoftheGentandMcWilliams (1990)schemefortheadiabaticthicknessdiffusion.Thisisa Table 6 List of ocean entropy sources and diabatic heatings for scheme introduced in order to parametrise the effects of HadCM3 and FAMOUS mesoscale eddies on large scale density and tracer structures. ______Snpen Spen Sice Sver Shor Sml Total This is achieved by mixing of isopycnal layer thickness along HadCM3 -290.2 294.6 -6.0 0.6 0.3 0.3 -0.4 isopycnal surfaces and advecting tracers by the eddy-induced FAMOUS -291.1 296.5 -6.5 0.8 0.3 0.3 0.3 transport velocity derived from the isopycnal thickness mixing. As expected, since the scheme is intrinsically formulated to be Q_ npen Q_ pen Q_ ice Q_ ver Q_ hor Q_ ml Total adiabatic, its entropy contribution is insignificant. HadCM3 -85.3 86.8 -1.6 0 0 0 -0.1 The mixed layer scheme (Kraus and Turner 1967) works FAMOUS -85.8 87.8 -1.8 0 0 0 0.2 out a balance between the energy required for mixing the All the terms are averages over the ocean area of the vertically water column and the energy available as turbulent kinetic integrated specific entropies and heatings, units are mW m-2 K-2 and -2 energy from the wind and from the introduction of buoy- Wm . The ‘‘npen’’ terms refer to the sum of longwave cooling, latent and sensible heat fluxes and shortwave radiation except the blue ancy at the ocean surface, leading to convective instability. component, which is counted in the ‘‘pen’’ terms. The ‘‘ice’’ terms are Water is mixed down from the surface to a level at which for latent heating associated with sea-ice processes and snowfall into no more energy is available for mixing, producing a ver- the ocean. The ‘‘ver’’ terms refer to vertical diffusion, ‘‘hor’’ to tically homogeneous layer. The spatial pattern of the ver- horizontal diffusion, ‘‘ml’’ to mixed-layer mixing and convection. Terms not shown in the table give heatings of less than 10-6 Wm-2 tically integrated ocean entropy terms discussed in this and entropy sources of order 10-6 Wm-2K-1 or less section is shown in Fig. 4(a–f).

123 S. Pascale et al.: Climate entropy budget

Fig. 4 Thirty years mean of the a -2 -1 b -2 -1 vertically integrated HORIZONTAL DIFFUSION [W m K ] VERTICAL DIFFUSION [W m K ] (W m-2K-1) entropy sources 90N 90N within the ocean: 45N 45N s_hor; s_ver; s_sfc; s_pen; s_ice; s_ml from FAMOUS 0 0

45S 45S

90S 90S 0 90E 180 90W 0 0 90E 180 90W 0

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 00.0010.0020.0030.0040.005

c SURFACE FLUXES [W m-2 K-1] d PENETRATIVE SW RADIATION [W m-2 K-1] 90N 90N

45N 45N

0 0

45S 45S

90S 90S 0 90E 180 90W 0 0 90E 180 90W 0

-0.8 -0.6 -0.4 -0.2 0 0.05 0.15 0.25 0.35 0.45

e ICE PHYSICS [W m-2 K-1] f MIXED LAYER PHYSICS [W m-2 K-1] 90N 90N

45N 45N

0 0

45S 45S

90S 90S 0 90E 180 90W 0 0 90E 180 90W 0

-0.1 0 0.1 0.2 4e-4 8e-4 0.0012 0.0016 0.002

The sum of the heat transport terms described in this (see Sect. 4.2), which dominates their estimated entropy oc _oc _ -2 -1 _irr section gives q_tur q_ver q_hor q_ml and thus Stur Sver production of 555.7 mW m K ; typical values for Srad ¼ þ þ ¼ þ -2 -1 S_hor S_ml (see Eq. 18), the material entropy production are *550 mW m K (Kleidon and Lorenz 2005). Yan þ _oc purely internal to the ocean. We obtain Stur 1.2 and 1.4 et al. (2004) do not provide any order-of-magnitude esti- -2 -1 ¼ mW m K in HadCM3 and FAMOUS, averaged over mates for the material entropy production, which is hard to the ocean surface only (0.8 and 1.0 mW m-2K-1 respec- estimate from observations since it depends on the local tively when averaged over the whole Earth surface). gradients of temperature.

8.3 Comparison of estimates of ocean entropy production 9 Climate entropy budget and the uncertainty in the material entropy production Shimokawa and Ozawa (2001) used an idealised Atlantic Ocean with rectangular domain of 72° longitude by 140° The important issue we are dealing with in this paper is the latitude to diagnose the entropy productions due to heat entropy balance of the climate system. Obviously the cli- transport, finding a value of about 2.5 mW m-2K-1, rather mate, strictly, is never in a steady state but over a clima- _oc similar to our result for Stur: tological period (decades) we can consider it to be, Yan et al. (2004) studied the entropy budget of the statistically speaking, in a steady state since we do not ocean using the energy fluxes and the temperature at the expect its properties to change substantially. General cir- _irr ocean surface. However they included Srad due to ther- culation models are usually built to be energetically bal- malization of shortwave radiation absorbed by the ocean anced, e.g. to assure a null net radiative flux at the top of

123 S. Pascale et al.: Climate entropy budget the atmosphere, but not to be entropy balanced. Therefore the treatment of processes associated with water phase it is an important question to check how well HadCM3 is transitions. Phase changes of convective and large scale entropy balanced and compare it with other models for precipitation are not considered in the Planet Simulator which entropy diagnostics are available. while in HadCM3 these processes are dealt with; likewise The complete climate entropy budget for HadCM3 and in the Planet Simulator no storage of water in clouds is FAMOUS is shown in Table 3, where it is compared with implemented and condensation occurs if the air is super- estimates obtained from other models and theoretical saturated, whereas HadCM3 has prognostic cloud water computations (Fraedrich and Lunkeit 2008; Goody 2000; variables. S_sh lh is still larger than the order-of-magnitude Peixoto et al. 1991). estimate by Peixotoþ et al. (1991) (25 mWm-2 K-1) and The overall imbalance of the HadCM3 climate, i.e. S_ Goody (2000) (21.2 mW m-2 K-1), but smaller than the (see Sect.3), is shown in Table 3. In terms of the quantities one in Goody (2000) from the GISS model (59 mW _ _rev _ -2 -1 listed in that table S Srad Smat: HadCM3 has m K ) where this term is dominated by moist and dry 2 1 ¼ þ S_ 0:8 mW mÀ KÀ ; which is quite small if compared convection. Differences among GCM diagnostics can be with the typical fluctuations due to the interannual vari- understood in terms of differences in the ways the energy ability of most of the entropy terms which is of order fluxes are represented in the models, e.g. the GISS model 1 mW m-2 K-1. FAMOUS likewise has an entropy has a very large value for the sensible and latent heat -2 -1 2 imbalance *-0.8 mW m K for the whole climate fluxes, H 120 WmÀ : The approximate estimate of _bl þL system, which is within the range of the interannual vari- Ssh described in Sect. 5.3 is quite close to the one in ability. The other two GCMs show a much higher imbal- Peixoto et al. (1991) and lies between those from the other ance as can be seen in Table 3. GCMs. This highlights that the maximum uncertainity is There is greater planetary entropy production S_pl in associated with the entropy produced by the moist pro- HadCM3 and FAMOUS than in the Planet Simulator of cesses leading to latent heating, being the range *(20–36) Fraedrich and Lunkeit (2008), who gave the only other value mW m-2 K-1. obtained from a GCM. However, at *900 mW m-2 K-1, The dissipation term in the atmosphere is quite important both are close to the order-of-magnitude estimate by Peixoto and deserves some discussion. In Fraedrich and Lunkeit et al. (1991). The next three rows are the irreversible entropy (2008) this term is not worked out directly because in the production terms of the radiative field as in Sect. 4.2. The Planet Simulator the energy loss by friction is not trans- shortwave absorption part is similar in HadCM3 and the ferred back by heating. This contribution is assumed to Planet Simulator, while for FAMOUS we find a lower value. equal the entropy imbalance and estimated to be *6 mW This difference is due mainly to differences in the shortwave m-2 K-1. However, other explanations are possible for an fluxes in the two models, i.e. HadCM3 has a higher value of entropy imbalance; we would argue that it cannot be con- the shortwave energy radiation absorbed in the atmosphere fidently attributed solely to this particular missing term. than FAMOUS (283 vs. 266 W m-2). Also the difference in Goody (2000) provides two estimates for the dissipative S_pl is linked to different longwave flux at the top of the entropy production of about the same size. The first atmosphere (240 vs. 235 W m-2). The atmosphere–atmo- (11.5 mW m-2 K-1) is diagnosed in the GISS model and sphere longwave and surface–atmosphere longwave inter- includes dissipation due to dry convection and surface drag action is generally greater in the UM models (*50 vs. *34 (8.1 mW m-2 K-1), moist convection (2.7 mW m-2 K-1) mW m-2 K-1), which also has to be attributed to the dif- and a stratospheric drag (0.7 mW m-2 K-1). In the dry ferences in the model radiative properties. convection term the contribution from the numerical drag In the remaining rows material entropy production terms term associated with the use of a binomial filter is included can be seen. Unlike the radiative terms, the material terms as well. The second estimate of 11.3 mW m-2K-1 comes are little affected by the differences in resolution between from a theoretical order-of-magnitude calculation in which FAMOUS and HadCM3. Both UM models produce the typical values of the surface drag and velocities are taken same amount of entropy in the interaction of the solid Earth into account. Goody estimates the entropy produced by and ocean with the atmosphere through sensible and latent stresses in the frictional boundary layer to be *6 mW 2 1 -2 -1 heating, S_sh lh 38 mW mÀ KÀ (since both models have m K , due to dry convection in the free atmosphere to þ  2 -2 -1 almost the same mean values of H 102 WmÀ ). be around 4 mW m K and due to moist convection þL -2 -1 More important differences can be noted though when a 1.3 mW m K . The order-of-magnitude value provided comparison is made with the other models. The HadCM3 by Peixoto et al. (1991) is 7 mW m-2 K-1, estimated value of S_sh lh , mainly due to the hydrological cycle, is from a diabatic frictional heating in the boundary layer of larger thanþ the value found by Fraedrich and Lunkeit 1.9 W m-2 at a temperature of 280 K. (2008) (29 mW m-2 K-1). This is not surprising since HadCM3 and FAMOUS have an entropy production due HadCM3 and the Planet Simulator differ substantially in to the dissipation of kinetic energy which is higher than the

123 S. Pascale et al.: Climate entropy budget other model and observational estimates. The reason for HadCM3 and FAMOUS are rather similar as far as that is the higher value of the dissipation rate D *3.1 W entropy terms are concerned although the resolution is m-2 discussed in Sect. 6 (see Table 5), which agrees with different. On the other hand, even though the results con- Goody (2000) but exceeds estimates from observations, firm to a certain extent the order-of-magnitude numbers e.g. *1.9 W m-2 (Peixoto and Oort 1992), *2.4W m-2 obtained from (Goody 2000; Fraedrich and Lunkeit 2008), (Arpe et al. 1986) and other GCMs, e.g. *2.0 W m-2 in differences between HadCM3 and the other two models are NCAR community atmosphere model (Boville and much more substantial and this shows that the differences Bretherton 2003). It is quite surprising how poor still is our in the model formulation are very influential on the entropy knowledge of this fundamental thermodynamic quantity of budget. the atmosphere and how different still are the estimates The material entropy production is found to be around 2 1 from most of the GCMs. S_mat 50 mW mÀ KÀ , and it is dominated by the latent _ _ _ _ _oc  The sum of Ssh lh; Sdiss; Sdiff ; Sadv and Stur is the material heat transfer from the surface (i.e. ocean and land) to the þ -2 -1 entropy production of the climate system, S_mat. In the atmosphere and within the atmosphere, *36 mW m K , 2 1 HadCM3 simulation we found S_mat 51:8 mW mÀ KÀ that is to say about 70% of the material entropy production. 2 1 ¼ and S_mat 53:3 mW mÀ KÀ in FAMOUS. These are The next most important process is the atmospheric viscous ¼ -2 -1 2 1 intermediate between *35 mW m K in Fraedrich and dissipation S_diss 13 mW mÀ KÀ , which accounts for -2 -1  Lunkeit (2008) and *70 mW m K in Goody (2000). nearly 25% of climate material entropy production. The _bl As we have seen above, the part which is most model remaining terms account for around the 3.5 % (Ssh) and 1 % _ _oc _ dependent is Ssh lh , i.e. the hydrological cycle, and it is the (Stur; Sdiff). The study of the ocean reveals that whereas its þ one causing the wide spread. The uncertainity in the vis- contribution to S_sh lh is significant since most of the latent cous dissipation term is less important, say [6-13] mW heat flux is concentratedþ over the oceans, the entropy due to m-2K-1. The material entropy production has been argued the turbulent mixing within it is a very small contribution oc 2 1 to be an important diagnostic tool in the analysis and (S_ 1mW mÀ KÀ ) to the overall material entropy. tur  comparison of climate models (Kunz et al. 2008; Lucarini The numerical entropy produced by the atmosphere model 2009; Lucarini et al. 2009), therefore a better knowledge of dynamical core is even smaller, *0.7 mW m-2K-1 in both it from both observations and modelling would be valuable. HadCM3 and FAMOUS, and it is the combined effect of the diffusion (positive) and the filtering-adjustment-advection (negative). This confirms the findings in Woollings and 10 Conclusions Thuburn (2006) who computed very similar values for a spectral dynamical core in isolation (0.5 mW m-2K-1). A detailed and comprehensive entropy budget analysis of a Comparison with other estimates reveals substantial complex atmosphere–ocean general circulation model, disagreement in the most important terms. By comparing HadCM3, and its low resolution version, FAMOUS, has S_mat with the other estimates we see that HadCM3 estimate been presented in this paper. This study represents the first is intermediate within a range (30, 70) mW m-2 K-1 case for an AOGCM, since previous ones were about a constrained by Goody (2000) and Fraedrich and Lunkeit slab-model (Fraedrich and Lunkeit 2008) and an AGCM (2008). The uncertainty is of almost 100%. Also a funda- (Goody 2000). mental quantity, the total dissipation D, is quite poorly The irreversible physics embedded in the model known from models and observations (range 1.9–3.6 W schemes yielding climate entropy sources has been accu- m-2). All this suggests the need for future work to improve rately discussed and connected with the different kinds of our knowledge of the basic thermodynamics of the climate entropy produced within the climate system, as well as the system. diagnostic methods to calculate the radiative entropy pro- duction and the material entropy production terms. Acknowledgments Salvatore Pascale would like to thank Robert A perfectly entropy balanced model should have S_ 0 Plant for the help provided in the diagnostic coding. Jonathan Gregory ¼ was partly supported by the Joint DECC, Defra and MoD Integrated in ideal steady state conditions. We find that in Climate Programme, DECC/Defra (GA01101), MoD (CBC/2B/ 2 1 HadCM3 S_ 0:8 mW mÀ KÀ and in FAMOUS S_ 0:8 0417_Annex C5). The authors also thank the two anonymous 2 1 À reviewers for their comments, which led to significant improvements mW mÀ KÀ . The typical magnitude of entropy variations due to interannual variability is 1mW m-2 K-1, hence on the original version of the manuscript. HadCM3 and FAMOUS can be considered close to a steady state and quite well entropy balanced. In this respect either References the model or the diagnostic techniques are better than the 2 1 Planet Simulator (S_ 7 mW mÀ KÀ ) and the GISS Arpe K, Brankovic C, Oriol E, Speth P (1986) Variability in time and À2 1 model (S_ 3 mW mÀ KÀ ). space of energetics from a long series of atmospheric data À 123 S. Pascale et al.: Climate entropy budget

produced by ECMWF. Beitraege zur Physik der Atmosphaere Kleidon A, Lorenz R (2005) Non-equilibrium thermodynamics and 59:321–355 the production of entropy. Understanding complex systems. Boville BA, Bretherton CS (2003) Heating and kinetic energy Springer, Berlin dissipation in the NCAR community atmosphere model. J Clim Kleidon A, Schymansky S (2008) Thermodynamics and optimality of 16:3877–3887 the water budget on land: a review. Geophys Res Lett 35:L20404 Cox MD (1984) A primitive equation, three dimensional model of the Kondepudi D, Prigogine I (1998) Modern thermodynamics: from heat ocean. GFDL ocean group technnical report no. 1, Princeton NJ, engines to dissipative structure. Wiley, Hoboken USA, 143 pp Kraus E, Turner J (1967) A one dimensional model of the seasonal Cullen M, Davies T (1991) A conservative split-explicit integration thermocline. Part II. Tellus 19:98–105 scheme with fourth order horizontal advection. Q J R Meteorol Kunz T, Fraedrich K, Kirk E (2008) Optimisation of simplified GCMs Soc 117:993–1002 using circulation indices and maximum entropy production. Cullen M, Davies T, Mawson M (1993) Unified model documentation Clim Dyn 30:803–813 paper no. 10. Tech. rep., Meteorological Office, UK Lucarini V (2009) Thermodynamic efficiency and entropy production DeGroot S, Mazur P (1984) Non-equilibrium thermodynamics. in the climate system. Phys Rev E 80:021118 Kluwar, Dover Lucarini V, Fraedrich K, Lunkeit F (2009) Thermodynamic analysis Edwards J, Slingo A (1996) Studies with a flexible new radiation of snowball earth hysteresis experiment: efficiency, entropy code. Part I: Choosing a configuration for a large-scale model. Q production and irreversibility. QJR Meterol Soc (in press) J R Meteorol Soc 122:689–719 Martyushev L, Seleznev V (2006) Maximum entropy production Egger J (1999) Numerical generation of entropies. Mon Weather Rev principle in physics, chemistry and biology. Phys Rep 426:1–45 127:2211–2216 Ozawa H, Ohmura A, Lorenz RD, Pujol T (2003) The second law of Fraedrich K, Jansen H, Kirk E, Lunkeit F (2005) The planet thermodynamics and the global climate system: a review of the simulator: towards a user friendly model. Meteorologische maximum entropy production principle. Rev Geophys 41(4):1018. Zeitschrift 14:305–314 doi:10.1029/2002RG000113 Fraedrich K, Lunkeit F (2008) Diagnosing the entropy budget of a Pacanowsky R, Philander G (1981) Parametrisation of vertical mixing climate model. Tellus A 60:921–931 in numerical models of the tropical oceans. J Phys Oceanogr Gent P, McWilliams J (1990) Isopycnal mixing in ocean circulations 11:1443–1451 models. J Phys Oceanogr 20:150–155 Paltridge GW (1975) Global dynamics and climate—a system of Goody R (2000) Sources and sinks of climate entropy. Q J R Meteorol minimum entropy exchange. Q J R Meteorol Soc 101:475–484 Soc 126:1953–1970 Pauluis O, Held M (2002) Entropy budget of an atmosphere in Goody R, Abdou W (1996) Reversible and irreversible sources of radiative–convective equilibrium. Part I: Maximum work and radiation entropy. Q J R Meteorol Soc 122:483–496 frictional dissipation. J Atmos Sci 59:125–139 Gordon C, Cooper C, Senior C, Banks H, Gregory J, Johns T, Peixoto J, Oort A, de Almeida M, Tome´ A (1991) Entropy budget of Mitchell J, Wood RA (2000) The simulation of sst, sea ice the atmosphere. J Geophys Res 96:10981–10988 extents and ocean heat transports in a version of the Hadley Peixoto JP, Oort AH (1984) Physics of climate. Rev Mod Phys Centre coupled model without flux adjustments. Clim Dyn 56:365–429 16:147–168 Peixoto JP, Oort A (1992) Physics of the climate. Springer, New York Gregg MC (1984) Entropy generation in the ocean by small-scale Pope VD, Gallani ML, Rowntree PR, Stratton RA (2000) The impact mixing. J Phys Oceanogr 14:688–711 of new physical parametrizations in the Hadley Centre climate Gregory D (1995) A consistent treatment of the evaporation of rain model—HadAM3. Clim Dyn 16:123–146 and snow for use in large-scale models. Mon Weather Rev Redi M (1982) Oceanic isopycnal mixing by coordinate rotation. 123:2716–2732 J Phys Oceanogr 12:1154–1158 Gregory D (1998) Unified model documentation paper no. 28. Tech. Reichler T, Kim J (2008) How well do coupled models simulate rep., Meteorological Office, UK today’s climate? Bull Am Meteorol Soc 89:303–311 Gregory D, Rowntree P (1990) A mass flux convection scheme with Shimokawa S, Ozawa H (2001) On the thermodynamics of the representation of cloud ensemble characteristics and stability- oceanic general circulation: entropy increase rate of an open dependent closure. Mon Weather Rev 118:1483–1506 dissipative system and its surroundings. Tellus 53A:266–277 Gregory D, Kershaw R, Innes P (1997a) Parametrization of Smith R (1990) A scheme for predicting layer clouds and their water momentum transport by convection. Part II: Tests in single- content in a general circulation model. Q J R Meteorol Soc column and general circulation models. Q J R Meteorol Soc 116:435–460 123:1153–1183 Smith R (1993) Unified model documentation paper no. 24. Tech. Gregory D, Shutts GJ, Mitchell JR (1997b) A new gravity-wave-drag rep., Meteorological Office, UK scheme incorporating anisotropic orography and low-level wave Smith RS, Gregory JM, Osprey A (2008) A description of the breaking: impact upon the climate of the UK Meteorological FAMOUS (version xdbua) climate model and control run. Office Unified Model. Q J R Meteorol Soc 124:463–493 Geosci Mod Dev 1:147–185 Hoskins BJ, Simmons AJ (1975) A multi-layer spectral model and the Vallis GK (2006) Atmospheric and oceanic fluid dynamics. Cam- semi-implicit method. Q J R Meteorol Soc 101:637–655 bridge University Press, Cambridge Johnson D (1997) ‘‘General coldness of climate’’ and the second Visbeck M, Marshall J, Haine T, Spall M (1997) On the specification law: implications for modelling the earth system. J Clim of eddy transfer coefficients in coarse resolution ocean circula- 10:2826–2846 tion models. J Phys Oceanogr 27:381–402 Jones C, Gregory J, Thorpe R, Cox P, Murphy J, Sexton D, Valdes P Weiss W (1996) The balance of entropy on earth. Contin Mech (2005) Systematic optimisation and climate simulation of Thermodyn 8:37–51 FAMOUS, a fast version of Had CM3. Clim Dyn 25:189–204 Woollings T, Thuburn J (2006) Entropy sources in a dynamical core Kleidon A (2009) Nonequilibrium thermodynamics and maximum atmosphere model. Q J R Meteorol Soc 132:43–59 entropy production in the earth system. Naturwissenschaften Yan Y, Gan Z, Qi Y (2004) Entropy budget of the ocean system. 96:653–677 Geophys Res Lett 30:L14311

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