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  • View in gallery

    Difference in implied surface flux [ from Eq. (3)] as evaluated from ERA-I data (March 2000–February 2007 average) using Kelvin or Celsius scale for atmospheric temperatures. The color legend starts at −30 W m−2 and ends at 30 W m−2 in increments of 2 W m−2.

  • View in gallery

    Schematic showing temperature and latent heat transports associated with atmospheric vapor Fa and oceanic water transports Fo across a circle of latitude φ. Temperature is assumed to be uniform across both domains. Atmospheric mass transport M must be balanced by oceanic mass transport −M for continuity reasons. Also, TK and TC represent temperature measured in Kelvin or Celsius, respectively. (a) Temperature and latent heat transport using traditional conventions in atmospheric and oceanic diagnostic studies, respectively. The sum of the transports in (a) is Fa + Fo = M × Lυ(T00) + M(cgTKclTC). (b) Temperature and latent heat transport as computed using the here-presented definitions. The sum of the transports in (b) is Fa + Fo = M × Lυ(TK), as expected from the assumption of uniform temperature.

  • View in gallery

    Illustration of the atmospheric and oceanic total energy budget following the (a) full budget equation given in Eq. (19), (b) full budget equation with frozen and liquid water as well as h0 removed [Eq. (20)], and (c) commonly used budget Eq. (21). Blue boxes indicate terms describing vapor or liquid water enthalpy transports. Red boxes indicate inconsistencies in the traditional formulation. Note that and include h0 (as opposed to and defined in the main text) and that T00 = 273.16 K and h0 = 0 J kg−1 in ocean products. Also note that both Figs. 3b and 3c exclusively use Lυ(T00) to facilitate comparison.

  • View in gallery

    Implied net surface energy flux (W m−2) based on CERES and ERA-I data using various approaches and simplifications: (a) traditional, (b) new_no_vapor, (c) new_Tskin, and (d) new_exact.

  • View in gallery

    Difference of implied surface energy flux (W m−2) as estimated from traditionally estimated divergence of atmospheric energy transport based on Eq. (21) and (a) divergence of atmospheric energy transports consistently considering enthalpy of moisture based on Eq. (20) and (b) divergence of atmospheric energy transports with consistently removed effects of enthalpy of moisture based on Eq. (24). Results based on ERA-I data.

  • View in gallery

    Time-averaged ΔFp = cl(TpTskin) × P (W m−2), where Tp is approximated via 2-m wet bulb temperature using temperature data from ERA-I and precipitation data from GPCP. Note that ΔFp can be interpreted as cooling of the surface due to precipitation being cooler than the skin temperature Daily mean fields have been used for evaluation of the product.

  • View in gallery

    Divergence of three-dimensional vapor enthalpy fluxes (W m−2) equal to , as estimated from the difference of surface enthalpy fluxes associated with P and E and lateral divergence of atmospheric vapor enthalpy transports. Units are W m−2.

  • View in gallery

    Contribution of latent heat flux associated with snowfall to the net surface energy flux (W m−2). Snowfall data are based on ERA-I.

  • View in gallery

    Difference of indirectly computed energy divergence and directly computed energy divergence using the (a) traditional and (b) new_no_vapor formulations of the divergence term.

  • View in gallery

    Difference of implied net surface energy flux (W m−2) based on RadTOA from CERES and (a) ERA-I-based traditionally computed energy divergence, (b) ERA-I-based energy divergence with the effect of water vapor removed, (c) JRA-55-based traditionally computed energy divergence, and (d) JRA-55-based energy divergence with the effect of water vapor removed, and net surface energy flux based on RadS from CERES and OAFlux turbulent fluxes.

  • View in gallery

    As in Fig. 10, but with net surface energy flux from ORAP5.

  • View in gallery

    As in Fig. 10, but with net surface energy flux from GECCO2.

  • View in gallery

    As in Fig. 10, but with net surface energy flux from CERES and SEAFLUX turbulent fluxes.

  • View in gallery

    Implied net surface energy flux over land (W m−2) based on CERES and ERA-I data using various approaches and simplifications: (a) traditional, (b) new_no_vapor, (c) new_Tskin, and (d) new_exact.

  • View in gallery

    As in Fig. 14, but using JRA-55 data instead of ERA-I.

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Toward Consistent Diagnostics of the Coupled Atmosphere and Ocean Energy Budgets

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  • 1 Department of Meteorology and Geophysics, University of Vienna, Vienna, Austria
  • | 2 Met Office Hadley Centre, Exeter, United Kingdom
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Abstract

The widely used diagnostic vertically integrated total energy budget equations of atmosphere and ocean contain inconsistencies that should no longer be disregarded. The neglect of enthalpy fluxes associated with precipitation and evaporation leads to a spurious dependence on reference temperature. This seemingly small inconsistency is amplified because enthalpy of water vapor implicitly included in lateral atmospheric energy transports usually is computed on the Kelvin scale, leading to inconsistencies that, although zero when globally averaged, attain values on the order of 20 W m−2 in the tropics. A more consistent energy budget framework is presented, which is independent of reference temperature and which takes full account of enthalpy fluxes associated with mass transfer through the surface. The latter include effects of snowfall and additional nonlatent contributions, which have a net cooling effect on the earth’s surface (−1.3 W m−2). In addition to these diagnostic issues, comparatively small inconsistencies in the energetic formulations of current weather and climate models are highlighted. Using the energy budget formulation presented here, instead of that commonly used, yields enhanced self-consistency of diagnosed atmospheric energy budgets and substantially improved spatial agreement between fields of net surface energy flux inferred from the divergence of lateral atmospheric energy transports in conjunction with satellite-based radiative fluxes and independent surface flux products. Results imply that previous estimates of radiative plus turbulent surface fluxes over the ocean, balancing the observed ocean warming, are biased low by ~1.3 W m−2. Moreover, previous studies seriously underestimated cross-equatorial atmospheric and oceanic energy transports. Overall, the presented framework allows for unambiguous coupled energy budget diagnostics and yields more reliable benchmark values for validation purposes.

Denotes content that is immediately available upon publication as open access.

This article is licensed under a Creative Commons Attribution 4.0 license (http://creativecommons.org/licenses/by/4.0/).

© 2017 American Meteorological Society.

Corresponding author: Michael Mayer, michael.mayer@univie.ac.at

Abstract

The widely used diagnostic vertically integrated total energy budget equations of atmosphere and ocean contain inconsistencies that should no longer be disregarded. The neglect of enthalpy fluxes associated with precipitation and evaporation leads to a spurious dependence on reference temperature. This seemingly small inconsistency is amplified because enthalpy of water vapor implicitly included in lateral atmospheric energy transports usually is computed on the Kelvin scale, leading to inconsistencies that, although zero when globally averaged, attain values on the order of 20 W m−2 in the tropics. A more consistent energy budget framework is presented, which is independent of reference temperature and which takes full account of enthalpy fluxes associated with mass transfer through the surface. The latter include effects of snowfall and additional nonlatent contributions, which have a net cooling effect on the earth’s surface (−1.3 W m−2). In addition to these diagnostic issues, comparatively small inconsistencies in the energetic formulations of current weather and climate models are highlighted. Using the energy budget formulation presented here, instead of that commonly used, yields enhanced self-consistency of diagnosed atmospheric energy budgets and substantially improved spatial agreement between fields of net surface energy flux inferred from the divergence of lateral atmospheric energy transports in conjunction with satellite-based radiative fluxes and independent surface flux products. Results imply that previous estimates of radiative plus turbulent surface fluxes over the ocean, balancing the observed ocean warming, are biased low by ~1.3 W m−2. Moreover, previous studies seriously underestimated cross-equatorial atmospheric and oceanic energy transports. Overall, the presented framework allows for unambiguous coupled energy budget diagnostics and yields more reliable benchmark values for validation purposes.

Denotes content that is immediately available upon publication as open access.

This article is licensed under a Creative Commons Attribution 4.0 license (http://creativecommons.org/licenses/by/4.0/).

© 2017 American Meteorological Society.

Corresponding author: Michael Mayer, michael.mayer@univie.ac.at

1. Introduction

Energy budget diagnostics are a standard method to quantify the mean state, variability, and trends of the earth’s climate system (e.g., Peixoto and Oort 1992; Hantel 2005; Trenberth et al. 2016). The results are important to understand the global climate system (e.g., Mayer et al. 2014; Donohoe et al. 2014) as well as for climate model validation (e.g., Williams et al. 2015; Mayer et al. 2016).

The energy exchange between atmosphere and ocean is of particular interest, since it determines patterns of ocean heat transport and storage. However, quantification of net surface energy flux FS is difficult, since in situ observations are sparse and model-based surface fluxes from reanalyses as well as other parameterization-based flux products can have large systematic errors (e.g., Valdivieso et al. 2017; Berrisford et al. 2011).

Another approach to estimate FS is combining satellite-based net radiation at the top of the atmosphere RadTOA and the divergence of vertically integrated horizontal atmospheric energy transports. Given that global mean RadTOA is unbiased by anchoring to estimates of global mean ocean heat storage as done in the Clouds and the Earth’s Radiant Energy System–Energy Balanced and Filled product (CERES-EBAF; Wielicki et al. 1996; Loeb et al. 2009), the obvious advantage of this method is that global mean constraints are satisfied because the global mean of the divergence of energy transports is zero.

Atmospheric reanalyses tremendously facilitated this approach compared to earlier works using rawinsonde data (e.g., Vonder Haar and Oort 1973), but several authors pointed out that reanalyzed winds needed an adjustment for conservation of dry mass to obtain meaningful results for atmospheric energy transports and implied net surface energy flux (Hantel and Haase 1983; Trenberth 1991, 1997; Graversen et al. 2007). This issue also persists when evaluating state-of-the-art reanalysis products (Mayer and Haimberger 2012).

Divergences of atmospheric energy transports based on analyzed fields are also less prone to temporal inhomogeneities than parameterized fluxes (e.g., Chiodo and Haimberger 2010). Hence, FS estimates based on the atmospheric energy budget are believed to be superior to parameterization-based products (see, e.g., von Schuckmann et al. 2016a,b; Trenberth and Fasullo 2017). However, realistic spatial patterns of employed fields of divergence of energy transports are crucial for the quality of the derived surface energy fluxes, and there still is room for improvement. For example, Liu et al. (2017) have shown that residuals of the implied FS over land can largely be attributed to spurious patterns in the fields of the divergence of energy transports. Moreover, considerable spatial differences exist between budget-derived surface fluxes and independent flux products (Liu et al. 2017).

Here we revisit the formulation of the vertically integrated energy budget equations in atmosphere and ocean as currently used in diagnostic studies. Thereby, we identify inconsistencies in the traditional diagnostic evaluation practices of these equations. First, we show that traditional formulations of the atmospheric energy budget treat three-dimensional moisture fluxes inconsistently by neglecting vertical enthalpy fluxes at the surface associated with precipitation and evaporation, which erroneously introduces a dependency on reference temperature. Second, we scrutinize the choice of reference state. Since absolute energies are not measurable, thermodynamic quantities can be defined only relative to some reference state. In meteorology and oceanography, the choice of the reference state is rarely discussed and is often implicit in the choice of units. In practice, this leads to an inconsistency arising from the fact that dynamical meteorologists usually view liquid water at 0°C as the reference state of water (by using latent heat of vaporization at this temperature) but employ heat capacity of vapor in conjunction with temperature referenced to 0 K. The latter choice is convenient when considering moist air in the ideal gas equation or adjusting the heat capacity of air for its moisture content (Curry and Webster 1998). However, we will show that this convention is inconsistent with oceanographic practices of viewing liquid as the reference state but employing the Celsius temperature scale (Griffies et al. 2016).

We present a consistent energy budget framework as well as simplifications thereof, which introduce only small approximation errors. It is shown that the agreement of implied surface energy fluxes with various independent surface flux products improves substantially when employing the revised budget framework described here.

The paper is structured as follows. Sections 2 and 3 describe inconsistencies in present coupled energy budget evaluations. In section 4, we derive the flux form of the total atmospheric energy budget including consistent treatment of water (vapor) enthalpy fluxes. Section 5 presents two simplifications substantially facilitating evaluation of the here-proposed equations. Section 6 describes the employed data used to obtain the results presented in section 7. A summary and implications follow in section 8. Appendix A provides a detailed derivation of the continuity equation employed herein and a list of acronyms is given in appendix B.

2. Ambiguity associated with the choice of atmospheric reference temperature

We motivate an improved energy budget formulation by first identifying an ambiguity in the currently widely used formulation (denoted by the tilde). The common total atmospheric energy budget equation in flux form in a steady state reads as follows (see, e.g., Oort and Vonder Haar 1976; Boer and Sargent 1985; Trenberth and Solomon 1994; Mayer and Haimberger 2012; Liu et al. 2015; Kato et al. 2016):
e1
Here, is moist static energy, k is kinetic energy, v is the horizontal wind vector, and is the net surface energy flux, representing the sum of net surface radiation RadS and sensible heat flux and latent heat flux associated with evaporation (SH and LH, respectively). Atmospheric pressure p at the surface is denoted by ps and gravitational acceleration by g. Moist static energy usually is defined as = caT + ϕ + Lυ(T00)qg, where ca is specific heat of dry air at constant pressure, T is temperature (usually given in kelvin), ϕ is geopotential, Lυ(T00) is the latent heat of vaporization at T = T00 = 273.16 K, and qg is the specific vaporous moisture content. Variation of the heat capacity of moist air with moisture content is usually neglected, as well as temperature dependence of Lυ.
The divergence term in Eq. (1) can be decomposed as follows:
e2
The advective part of the divergence term (first term on the right) is independent of reference temperature since it involves temperature differences. In contrast, the term involving the mass divergence (second term on the right) as well as the surface advection term (third term on the right) depend on reference temperature, because they involve temperature itself.
The two terms involving absolute temperature yield different results when using different reference temperatures. The indirectly estimated from Eq. (1) thus includes an ambiguous part Δ as follows:
e3
Thereby, we have introduced subscripts K and C to denote the use of either the Kelvin or Celsius scale, respectively. Moreover, we have used the fact that it is usually necessary to adjust atmospheric winds from reanalyses such that dry mass is conserved (see Trenberth 1991). Consequently, under assumption of a steady state, vertically integrated divergence of lateral atmospheric mass transports equals the lateral moisture flux divergence [balancing the sum of total precipitation P and evaporation E; both defined positive downward; units are kg m−2 s−1; compare also Eq. (15) below]. Assuming “no-slip” boundary conditions, the last term in Eq. (3) will vanish over land and will be very small over oceans. Hence, the mass divergence term in Eq. (3) dominates .

The field of as evaluated from European Centre for Medium-Range Weather Forecasts (ECMWF) interim reanalysis (ERA-Interim, hereinafter ERA-I; Dee et al. 2011) data for the period March 2000–February 2007 (see section 6 for a rationale for the choice of this period) is presented in Fig. 1. The plot shows that there is a strong dependence on reference temperature in the divergence of energy transports and consequently implied surface energy flux estimates using the conventional Eq. (1). Highest values are reached in areas of strong moisture divergence and convergence, like in the intertropical convergence zone (ITCZ). Values are on the order of 20 W m−2 over large regions in the tropics, with generally smaller values in the extratropics. The magnitude of this ambiguity essentially scales with the strength of the hydrological cycle in the employed dataset [see Eq. (3)]. The estimated using Japan Meteorological Agency Japanese 55-year Reanalysis (JRA-55; Kobayashi et al. 2015) is qualitatively similar but even more pronounced (RMS = 7.6 W m−2; not shown) than the field based on ERA-I (RMS = 6.5 W m−2; see Fig. 1), indicating a more vigorous water cycle in JRA-55 compared to ERA-I.

Fig. 1.
Fig. 1.

Difference in implied surface flux [ from Eq. (3)] as evaluated from ERA-I data (March 2000–February 2007 average) using Kelvin or Celsius scale for atmospheric temperatures. The color legend starts at −30 W m−2 and ends at 30 W m−2 in increments of 2 W m−2.

Citation: Journal of Climate 30, 22; 10.1175/JCLI-D-17-0137.1

It might be surprising that there is a dependence on reference temperature although state-of-the-art energy budget diagnostics adjust the winds such that the three-dimensional atmospheric mass budget, including P and E, is closed (see, e.g., Trenberth and Solomon 1994; Mayer and Haimberger 2012), which is a necessary precondition for unambiguous energy budget diagnostics (Schauer and Beszczynska-Möller 2009). The reason for the revealed ambiguity is that Eq. (1) does not acknowledge the fact that energy fluxes associated with moisture fluxes are actually three-dimensional. Specifically, this means that there are also enthalpy fluxes associated with P and E, but only lateral enthalpy transports by moisture are considered in Eq. (1). It is not a particularly new finding that P and E carry enthalpy (see, e.g., Businger 1982; Griffies et al. 2016), but these enthalpy fluxes have been neglected in diagnostic studies so far. We will show in section 4 that these vertical enthalpy fluxes are naturally included in the total atmospheric energy budget equation when correctly implementing the flux form and that implementation of these fluxes makes the atmospheric energy budget equation independent of reference temperature.

3. Specific enthalpy of water in the climate system

Water in the climate system is present in all three states and is exchanged continuously across the boundaries of the subsystems (atmosphere, ocean, and cryosphere). To track energy flows within the climate system consistently, enthalpy transports associated with water, vapor, and ice transports need to be treated consistently; that is, careful “bookkeeping” is necessary (Businger 1982).

In the following, cs, cl, and cg denote specific heats of water in its various states [solid, liquid, and vaporous (gas), respectively] at constant pressure. Latent heat of fusion is denoted by Lf and Lυ denotes latent heat of vaporization. We start from a reference state of ice at 0 K with a specific enthalpy of 0 J kg−1 and assume ice melts at T00 = 273.16 K. Hence, the specific enthalpy of water at 0°C is h0 = cs × 273.16 + Lf (T00). More sophisticated choices for the reference enthalpy h0 are possible (see, e.g., Marquet 2015), but we will show that this is irrelevant for the steady state.

After these preliminary notes, we describe specific enthalpy changes associated with the process of water evaporating from the sea surface and subsequently changing its temperature in the atmosphere. We start with seawater at the sea surface with temperature TSST, the specific enthalpy of which is
e4
The second term on the right in Eq. (4) describes the additional enthalpy required to warm water from T00 to TSST.
To evaporate seawater, the latent heat of evaporation at temperature TSST, that is, Lυ(TSST), has to be added. Right after evaporation and without changing temperature, the enthalpy of the vapor is then
e5
After evaporation, enthalpy required for further temperature changes of water vapor within the atmosphere is proportional to cg. The enthalpy of water vapor at atmospheric temperature Ta is then
e6
The temperature dependency of Lυ is due to the difference in specific heats of liquid water and vapor, respectively (Heintz 2011; Weinhold 2009). Thus, we can rewrite the latent heat of vaporization at atmospheric temperature Ta using Kirchhoff’s equation:
e7
This can be rearranged to
e8
Inserting Eq. (8) into Eq. (6) yields
e9
The enthalpy of water vapor in the atmosphere is thus independent of the temperature at which evaporation occurred, which is a result of Kirchhoff’s equation. It is evident that if the water cools down to TSST again, condenses (thereby releasing its latent heat of vaporization), and then falls down to the ocean again, it arrives at the ocean surface with the same amount of enthalpy as stated in Eq. (4).
The equations found so far are also valid for sublimating ice (subscript i). The enthalpy of frozen water at Ti < T00 is
e10
where Lf is latent heat of fusion. After sublimation and without temperature change, the enthalpy of the sublimated water mass is
e11
where we used the definition of Lf(Ti) = Lf(T00) + (clcs)(TiT00) and which is consistent with Eq. (5). Thus, the use of cl in Eq. (9) consistently includes sublimation.
Based on the results of this section, a more accurate definition of moist static atmospheric energy than (as defined in section 2), additionally including specific liquid water content ql and ice content qs, reads as follows:
e12
This more thorough definition of m, especially the introduction of a reference enthalpy and state, is a prerequisite for unambiguous energy budget diagnostics. Note that there is also a reference enthalpy for the dry air part, but this can be neglected as long as the dry mass budget is closed. Moreover, it is equivalent to use Lυ(T00) instead of Lυ(Ta) in Eq. (12), but then vapor enthalpy scales with cg instead of cl [cf. also Eq. (6)]. Here it is more convenient to use cl in all water (vapor) enthalpy terms.

The important point of the equations derived so far is that water in the ocean and water vapor in the atmosphere carry the same reference enthalpy h0; that is, the amount of transported reference enthalpy is the same regardless of whether the water is transported in the ocean or the atmosphere. We illustrate the importance of this point by considering a simple example of oceanic and atmospheric water (vapor) transports across an arbitrary circle of latitude φ, assuming uniform temperature. From mass continuity, the sum of oceanic and atmospheric mass transports must balance each other. Thus, there is exclusively latent heat transport across φ and temperature transports in atmosphere and ocean cancel each other. This balance can only be achieved if we make consistent choices regarding reference temperature and reference state and consequently reference enthalpy h0 of water in atmosphere and ocean. This usually is not done in diagnostic studies because of conflicting traditions, as discussed in section 1: Meteorologists tend to use vapor at 0 K as the reference state of water, while oceanographers naturally view liquid water at 0°C as the reference state. Please see Fig. 2 for an illustration of this issue.

Fig. 2.
Fig. 2.

Schematic showing temperature and latent heat transports associated with atmospheric vapor Fa and oceanic water transports Fo across a circle of latitude φ. Temperature is assumed to be uniform across both domains. Atmospheric mass transport M must be balanced by oceanic mass transport −M for continuity reasons. Also, TK and TC represent temperature measured in Kelvin or Celsius, respectively. (a) Temperature and latent heat transport using traditional conventions in atmospheric and oceanic diagnostic studies, respectively. The sum of the transports in (a) is Fa + Fo = M × Lυ(T00) + M(cgTKclTC). (b) Temperature and latent heat transport as computed using the here-presented definitions. The sum of the transports in (b) is Fa + Fo = M × Lυ(TK), as expected from the assumption of uniform temperature.

Citation: Journal of Climate 30, 22; 10.1175/JCLI-D-17-0137.1

Hence, in addition to the ambiguity associated with the choice of reference temperature as discussed in section 2, the different traditions concerning the reference state of water introduce another inconsistency to current coupled budget evaluations. We will develop a more consistent budget framework circumventing these caveats in the following section.

4. A mass-consistent formulation of the atmospheric total energy budget

We start our derivation from the advective form of the total atmospheric energy budget in a steady state including the nonmaterial source term N:
e13
Please note that from here we use the more exact definition of m as defined in Eq. (12). To obtain the flux form of Eq. (13), we use the equation of continuity for moist air including liquid water:
e14
Note that the velocities in Eq. (14) are barycentric (i.e., they include vapor and liquid water fluxes). Thus, ∂ω/∂p includes contributions from vertical liquid water and vapor fluxes. Inclusion of liquid water removes the conversion term (evaporation and condensation) from the continuity equation, which represents a difference from Trenberth (1991), who considers only dry air plus vapor [see appendix A for a derivation of Eq. (14)]. Assuming a steady state, vertical integration of Eq. (14) yields
e15
This is the balance required when adjusting divergent winds for diagnostics studies (e.g., Trenberth and Solomon 1994; Mayer and Haimberger 2012).
Multiplication of Eq. (14) by (m + k) and addition to Eq. (13) yields the flux form of the total atmospheric energy budget:
e16
Vertical integration of Eq. (16) with Eq. (15) in mind yields
e17
It is the last term on the rhs of Eq. (17) that has been neglected in derivations of the total energy budget equation so far. At this point it is necessary to specify this term, which represents the enthalpy flux associated with mass transfer through the surface. There is no flux of dry air through the surface, and we assume negligible kinetic energy as well as negligible geopotential flux associated with P and E. Based on these assumptions, the last term on the rhs of Eq. (17) can be written as the sum of enthalpies carried by rain Prain, snowfall Psnow, and evaporation (including sublimation) Thus, we can rewrite that term using the results of section 3 as follows:
e18
The snowfall term in Eq. (18) states that the fraction of water mass falling as snow leaves the column with an enthalpy reduced by Lf(Tp), and the atmosphere is warmed with this additional energy. Following Eq. (11), the evaporation term in Eq. (18) remains unaffected by sublimation. Note that, for generality, we replaced TSST with skin temperature Ts (= TSST, over ocean) and we set temperature of evaporating water to Ts. The last term in Eq. (18) represents the latent heat flux, which we combine with net surface radiation and sensible heat flux to the commonly used net surface energy flux FS = RadS + LH + SH. Furthermore, the energy source at the top of the atmosphere is represented by RadTOA. Thus, the more complete total atmospheric energy budget reads as follows:
e19
A schematic depicting Eq. (19) is presented in Fig. 3a.
Fig. 3.
Fig. 3.

Illustration of the atmospheric and oceanic total energy budget following the (a) full budget equation given in Eq. (19), (b) full budget equation with frozen and liquid water as well as h0 removed [Eq. (20)], and (c) commonly used budget Eq. (21). Blue boxes indicate terms describing vapor or liquid water enthalpy transports. Red boxes indicate inconsistencies in the traditional formulation. Note that and include h0 (as opposed to and defined in the main text) and that T00 = 273.16 K and h0 = 0 J kg−1 in ocean products. Also note that both Figs. 3b and 3c exclusively use Lυ(T00) to facilitate comparison.

Citation: Journal of Climate 30, 22; 10.1175/JCLI-D-17-0137.1

We neglect liquid and frozen water in horizontal atmospheric transports (i.e., freezing and condensation occurs locally) because this effect is small and these fields strongly depend on the model physics. Moreover, we assume a closed moisture budget (i.e., terms involving h0 drop out). Then Eq. (19) reduces to
e20
Here, we introduced expressions for the precipitation enthalpy flux (occurring at temperature Tp referenced to T00) Fp = cl(TpT00) × P and evaporative enthalpy flux Fe = cl(TsT00) × E. For comparison, we rewrite Eq. (1) using the terminology of this paper:
e21
Please note that we do not include h0 of the dry air part of Eq. (21) since it drops out as long as dry mass is conserved (see also section 3). The formulation of the total atmospheric energy budget proposed here, as given in Eq. (20), differs from the traditional formulation given in Eq. (21) in several aspects.

First and most importantly, Eq. (21) neglects enthalpy fluxes associated with precipitation and evaporation. Neglect of these fluxes introduces a spurious dependence on the reference temperature and hence reference enthalpy h0, because the h0 term in the divergence expression is not cancelled out from Eq. (21). As shown in section 2, the spurious inclusion of this term leads to errors of up to 30 W m−2 in the inferred surface energy fluxes. Another subtlety is the fact that by setting T00 = 0 K as usually done in diagnostic studies of the atmosphere, h0 in Eq. (21) is implicitly set to h0 = ca × 273.16 rather than cs × 273.16 + Lf(T00) (as defined in section 3). Compared to h0 as defined here, this reduces the effect of the inconsistent treatment of three-dimensional enthalpy fluxes associated with moisture by about 70%, but it is nevertheless sizeable (as already shown in Fig. 1). Second, energetic effects of snowfall are not taken into account by Eq. (21). It is important to stress that these differences are a shortcoming of current diagnostic practices and not an inconsistency of (climate) models.

Third and fourth, Eq. (21) neglects the differences in the specific heats of dry air and water vapor as well as temperature dependency of Lυ. These two simplifications are also made in the IFS (ECMWF 2015) and other weather and climate models (e.g., the Met Office Unified Model; Walters et al. 2017). These inconsistencies in model formulations, which are small compared to the diagnostic inconsistencies pointed out above, might nevertheless be partly responsible for model drift as seen from coupled model runs (Hobbs et al. 2016).

The energy budget formulation proposed here [Eq. (20)] and the traditional formulation [Eq. (21)] are illustrated in Figs. 3b and 3c, respectively, summarizing their differences.

To illustrate the relevance of the outlined inconsistencies also for the ocean heat budget, we include the divergence of ocean heat transport in Fig. 3. Ocean models usually use 0°C as reference temperature and hence the h0 term in the ocean will be zero. Nevertheless, there will be a remaining inconsistency in the ocean energy budget if Fp and Fe are neglected as indicated in Fig. 3b. Thus, a consistent formulation of the vertically integrated ocean energy budget assuming a steady state reads as follows (cf. also Griffies et al. 2016):
e22
Here, ρ0 represents seawater density θ, ocean potential temperature, and c the horizontal ocean currents. Please note the consistent usage of heat capacities in Eqs. (20) and (22). We also note that the bias introduced by neglecting Fp and Fe as usually done in oceanic energy budget diagnostics (e.g., Trenberth and Fasullo 2008) is comparatively small because h0 is set to zero implicitly by the common usage of Celsius scale in ocean datasets.

5. Simplifications for practical evaluation of the proposed equations

Evaluation of Eq. (20) is more difficult than that of the traditional formulation [Eq. (1)] since it requires specification of P, E, Tp, and Ts. Precipitation and evaporation are problematic terms since in reanalyses they can only be obtained from model forecasts, which increases undesired model dependency of the results. We thus present two possible simplifications of Eq. (20), which should alleviate this problem.

The first simplification is to assume Tp = Ts in the Fp term, which is common practice in coupled climate models (Griffies et al. 2016). For this approach, it is sufficient to know P + E, which can be estimated from the divergence of analyzed atmospheric moisture transports [Trenberth and Caron 2001; see also Eq. (15)]. Equation (20) in this case simplifies to
e23
A further simplification of Eq. (23) is to assume negligible temperature change of water (vapor) on its passage through the atmospheric column. This means that the divergence of lateral enthalpy transports associated with vapor transports balances vertical enthalpy fluxes associated with P and E; that is,
eq1
and these terms consequently cancel out. For energy budget diagnostics confined to the tropics one can also neglect the effect of snowfall. Equation (23) then simplifies to
e24
Equation (24) states that the net surface energy flux FS is balanced by the divergence of dry-air enthalpy, latent heat, and geopotential and kinetic energy transports and neglects enthalpy transports associated with water (vapor) fluxes. The important difference of Eq. (24) compared to Eqs. (1) and (21) is the consistent removal of vapor enthalpy in Eq. (24). We will quantify the introduced errors associated with simplified Eqs. (23) and (24) in the results section.

Inspection of Eqs. (20)(24) reveals that it is only the simplified Eq. (24) that allows for evaluation of reference-temperature-independent purely horizontal atmospheric energy transports based only on horizontal winds and atmospheric state quantities. While the divergence term in the traditional formulation [Eq. (21)] is dependent on reference temperature due to its inconsistent treatment of three-dimensional vapor fluxes, Eqs. (20) and (23) require evaluation and subtraction of Fp, Fe, and Lf(Tp)Psnow in order to obtain meaningful (i.e., reference temperature independent) lateral energy transports.

It is important to stress that mass-consistent wind fields are an essential prerequisite for practical evaluation of Eqs. (20), (21), (23), and (24), as described, for example, by Mayer and Haimberger (2012). Otherwise, spurious divergence patterns in the original reanalyzed wind fields will contaminate the results. We also note that in reality the steady state is not satisfied even for multiyear averages due to trends in global mean atmospheric water vapor, but these effects are very small (global mean latent heat trend <0.01 W m−2 and local latent heat trends <0.5 W m−2 everywhere during March 2000–February 2007 period using ERA-I data).

6. Data

Divergence of atmospheric energy transports using mass-consistent winds (see Trenberth and Caron 2001; Mayer and Haimberger 2012) are computed using 6-hourly analyzed model level data from ERA-I and JRA-55. Data were obtained on the respective native grids. For evaluation of the surface enthalpy fluxes, we also obtained 2-m temperature and 2-m dewpoint temperature from JRA-55 and ERA-I, respectively. While skin temperature is available from ERA-I, we had to assemble JRA-55 skin temperature from Centennial In Situ Observation-Based Estimates (COBE) of SST (Ishii et al. 2005) and JRA-55 ground temperature over land. However, it is impossible to obtain JRA-55 skin temperature in sea ice– and snow-covered regions, since this variable was not archived. Snowfall data are taken from 12-hourly accumulated fields from ERA-I and JRA-55. For RadTOA, we employ satellite-based CERES-EBAF version 2.8 data, and we use daily Global Precipitation Climatology Project (GPCP) 1° daily (1DD) version 1.2 (Adler et al. 2003) precipitation data. We additionally use ERA-I precipitation data at 6-hourly resolution to check the impact of time sampling. To assess the self-consistency of the atmospheric energy budget derived from ERA-I, we employ an indirect estimate of the energy divergence term in Eq. (24) computed as a residual from parameterized RadTOA and FS (i.e., averages of 12-hourly model forecasts) and the analysis increments from ERA-I, as described in Chiodo and Haimberger (2010) and Mayer and Haimberger (2012).

For comparison with our energy budget-based surface energy flux estimates, we employ four different surface flux products which represent a choice of state-of-the-art datasets of their respective types. We use parameterized fluxes from the flux-corrected Ocean Reanalysis Pilot 5 (ORAP5; Zuo et al. 2015) and the energy-balanced German Contribution to Estimation of the Circulation and Climate of the Ocean version 2 (GECCO2; Köhl 2015) ocean reanalyses. Furthermore, we combine net surface radiation data from CERES-EBAF-Surface version 2.8 (Kato et al. 2013) with turbulent fluxes from OAFlux version 3 (Yu et al. 2008) and SEAFLUX version 1.0 (Curry et al. 2004). The combined CERES-EBAF-Surface/OAFlux and CERES-EBAF-Surface/SEAFLUX surface flux estimates exhibit large global imbalances (+22.2 and +16.2 W m−2, respectively). Since here we are mainly interested in the spatial structure of the fields from these products, we remove their yearly global mean values. The time period for the evaluations presented here is limited to March 2000–February 2007, as given by the availability of CERES and SEAFLUX data.

7. Results

a. Improved surface energy flux estimates and quantification of the impact of simplifications

We present time-averaged implied net surface energy fluxes based on Eq. (21) (the “traditional” method), the here-proposed improved Eq. (20) (“new_exact” method), and the two suggested simplifications in Eq. (23) (“new_Tskin”) and Eq. (24) (“new_no_vapor”) based on ERA-I data in Fig. 4, respectively. The overall structures agree very well in all plots, but differences exist that will be discussed below. Global mean FS is 0.8 W m−2 in the traditional and new_no_vapor estimates, consistent with the global mean RadTOA value from CERES. Global mean FS from new_Tskin and new_exact are higher, which is related to the global mean cooling effect of precipitation. Results are similar when using JRA-55 data (not shown).

Fig. 4.
Fig. 4.

Implied net surface energy flux (W m−2) based on CERES and ERA-I data using various approaches and simplifications: (a) traditional, (b) new_no_vapor, (c) new_Tskin, and (d) new_exact.

Citation: Journal of Climate 30, 22; 10.1175/JCLI-D-17-0137.1

The rather subtle differences between the various proposed methods can be better appreciated from difference plots. Both differences between the traditional and new_exact estimates (Fig. 5a) and traditional and new_no_vapor estimates (Fig. 5b) show a distinct P + E pattern, indicating that the consistent treatment of the three-dimensional water flux has the strongest impact of all methodological improvements. The similarity between Fig. 5 and Fig. 1 indicates that the h0 term hidden in Eq. (1) introduces the largest inconsistency in the conventional budget formulation. The largest discrepancy between Figs. 5a and 5b seems to occur over regions with climatologically strong snowfall (northern and southern extratropical storm tracks as well as northern Eurasia and North America).

Fig. 5.
Fig. 5.

Difference of implied surface energy flux (W m−2) as estimated from traditionally estimated divergence of atmospheric energy transport based on Eq. (21) and (a) divergence of atmospheric energy transports consistently considering enthalpy of moisture based on Eq. (20) and (b) divergence of atmospheric energy transports with consistently removed effects of enthalpy of moisture based on Eq. (24). Results based on ERA-I data.

Citation: Journal of Climate 30, 22; 10.1175/JCLI-D-17-0137.1

In the following, we assess the impact of the simplifications introduced in Eqs. (23) and (24), the net effect of which we showed in Figs. 5a and 5b, in more detail. Please note that we choose the sign such that the fields show the warming and cooling of the surface by the respective terms, but the sign of their contribution to implied FS is actually reversed [see Eq. (20)].

The error introduced with the assumption of Tp = Ts made in Eq. (23) can be quantified by calculating Fp using a more realistic estimate of Tp, which however is not a standard output from reanalyses or climate models. A good approximation for Tp is wet bulb temperature near the surface (Gosnell et al. 1995). Thus, the introduced error is ΔFp = cl(TpTs) × P. Figure 6 shows ΔFp for March 2000–February 2007 using skin and wet bulb temperature from ERA-I and daily precipitation data from GPCP. As expected, ΔFp is largest in regions of strong precipitation. Since these are humid regions, wet bulb temperature is close to skin temperature and hence ΔFp is generally small. However, we note the global mean surface cooling effect of ΔFp, which is −0.4 W m−2. This value is almost identical when using ERA-I precipitation data (−0.4 W m−2). Moreover, tests with ERA-I showed that the impact of using 6-hourly instead of daily precipitation data is marginal (<3%).

Fig. 6.
Fig. 6.

Time-averaged ΔFp = cl(TpTskin) × P (W m−2), where Tp is approximated via 2-m wet bulb temperature using temperature data from ERA-I and precipitation data from GPCP. Note that ΔFp can be interpreted as cooling of the surface due to precipitation being cooler than the skin temperature Daily mean fields have been used for evaluation of the product.

Citation: Journal of Climate 30, 22; 10.1175/JCLI-D-17-0137.1

The assumption behind complete removal of the enthalpy transport associated with the three-dimensional moisture flux, as proposed in Eq. (24), is that water (vapor) does not change its temperature on its passage through the atmospheric column. We can estimate the size of the introduced error by computing the sum of
eq2
and Fp + Fe, where we use the approximation introduced in Eq. (23):
eq3
Results based on ERA-I and JRA-55 are presented in Figs. 7a and 7b, respectively. Positive values can be interpreted as regions where water vapor is warmed (cooled) on its passage through the column, thereby cooling (warming) the column. Values are generally small (RMS = 1.0 W m−2) with maximum values on the order of 5 W m−2 in the ITCZ and a few locations over land, where some noisy patterns of the moisture flux divergence field seem to be reflected. Values are generally positive in the ITCZ regions, where precipitation dominates over evaporation and positive values indicate that laterally converging atmospheric moisture is generally cooler than precipitation. An exception is the negative values of ERA-I in the eastern Pacific ITCZ region, which could be a result of defective low-level moisture analyses of ERA-I in that region as documented by Josey et al. (2014). The global mean of
eq4
is zero (Gauss’s theorem) and hence the global mean of Figs. 7a and 7b represents the global mean of Fp + Fe, which is −0.3 W m−2 based on both ERA-I and JRA-55. This value is negative because precipitation tends to occur over cooler SST regions than evaporation and is in very good agreement with the model-based estimates provided by Griffies et al. (2016). We note that results are very similar when using fields with 6-hourly time frequency (not shown).
Fig. 7.
Fig. 7.

Divergence of three-dimensional vapor enthalpy fluxes (W m−2) equal to , as estimated from the difference of surface enthalpy fluxes associated with P and E and lateral divergence of atmospheric vapor enthalpy transports. Units are W m−2.

Citation: Journal of Climate 30, 22; 10.1175/JCLI-D-17-0137.1

The time-averaged effect of snowfall cooling the surface is plotted in Fig. 8. As already noted from the comparison of Figs. 5a and 5b, the latent heat flux associated with snowfall can be considerable, going up to −5 W m−2 over the Southern Ocean. The result is very similar when using JRA-55 snowfall data (not shown). Using a range of different independent products, Behrangi et al. (2016) provide a total precipitation uncertainty of about 50 mm yr−1 for both northern and southern high latitudes. Assuming a snowfall fraction of approximately 70% (based on model results), this estimate converts to regional average uncertainties of the Lf(Tp)Psnow term of about 0.3 W m−2, providing confidence in the pattern shown in Fig. 8. The snowfall effect based on GPCP seems unrealistically strong in some regions, such as in northern Eurasia [not shown; see discussion in Behrangi et al. (2016)], and hence is not considered any further.

Fig. 8.
Fig. 8.

Contribution of latent heat flux associated with snowfall to the net surface energy flux (W m−2). Snowfall data are based on ERA-I.

Citation: Journal of Climate 30, 22; 10.1175/JCLI-D-17-0137.1

To summarize, it is evident from Figs. 5a and 5b that (i) the traditional surface flux estimate exhibits a spurious P + E pattern due to the inconsistent treatment of the three-dimensional moisture flux and (ii) the effect of the simplifications introduced in Eqs. (23) and (24) is comparatively small (cf. RMS values of fields shown in Figs. 5a,b, 6, 7, and 8, respectively). A relatively large contribution stems, however, from latent heat flux associated with snowfall, which should be taken into account when investigating high-latitude energy budgets. For example, the snowfall effect north of 70°N is −1.8 W m−2 (−2.3 W m−2) based on ERA-I (JRA-55). However, given the uncertainties associated with the separate estimation of Fp and Fe, it is justifiable to use Eq. (24) for estimating the implied net surface energy flux, especially when focusing on the tropics.

Another result from this section is that there is a global mean cooling of the ocean due to different temperatures of P and E. This global mean effect is about −0.7 W m−2, as estimated from the sum of the global means of fields shown in Figs. 6 and 7 (remember that we set Tp = Ts for the results shown in Fig. 7). Additionally, there is a net surface cooling effect by snowfall of about −0.6 W m−2 (−0.7 W m−2) based on ERA-I (JRA-55). This implies that maintaining global ocean warming at the observed rate (on the order of 1 W m−2; see Trenberth et al. 2016) actually requires a global mean net surface energy flux (FS = RadS + SH + LH) of about 2.3 W m−2. This value is well within uncertainty bounds of current estimates of the surface flux components (see, e.g., Wild et al. 2013; Trenberth et al. 2009).

b. Self-consistency of the ERA-I energy budget

Before comparing our results for inferred FS to other products, we assess the self-consistency of the atmospheric energy budget based on ERA-I. Assuming that the statistics of the atmospheric circulation during 12-hourly forecasts from ERA-I are very similar to those of the analyses, the direct and indirect estimates of the energy divergence term should not differ much. We compare difference maps between indirectly and directly estimated energy divergence based on (i) the common budget formulation [Eq. (21)] in Fig. 9a and (ii) the improved budget formulation [Eq. (24)] in Fig. 9b. It is evident that the P + E pattern present in Fig. 9a is largely removed in Fig. 9b. The RMS difference drops by 21% for the global difference fields and even 28% when considering only ocean points. The improved agreement of the direct and indirect estimate of the energy divergence is an unambiguous indication that Eq. (24) allows for more consistent energy budget diagnostics compared to the traditional formulation.

Fig. 9.
Fig. 9.

Difference of indirectly computed energy divergence and directly computed energy divergence using the (a) traditional and (b) new_no_vapor formulations of the divergence term.

Citation: Journal of Climate 30, 22; 10.1175/JCLI-D-17-0137.1

c. Comparison to independent surface flux products

We now compare implied net surface energy fluxes based on RadTOA from CERES and fields of divergence of energy transports based on ERA-I or JRA-55 to FS fields based on CERES-EBAF/OAFlux (Fig. 10), ORAP5 (Fig. 11), GECCO2 (Fig. 12), and CERES-EBAF/SEAFLUX (Fig. 13). Plots are only shown for implied surface fluxes based on Eq. (21) (traditional) and Eq. (24) (new_no_vapor), since differences between new_no_vapor, new_Tskin, and new_exact are small in the tropics (see Fig. 4 and discussion in section 7a).

Results presented in Figs. 1013 show that all difference fields based on the traditional FS estimate exhibit a spurious P + E pattern, which is removed in the new_no_vapor estimate. Root-mean-square differences between implied FS and the independent FS products are reduced by up to 40%, yielding an RMS difference of 10.3 W m−2 between the JRA-55-based and the CERES-EBAF/OAFlux-based FS estimate (cf. Figs. 10c and 10d), which is quite small given the almost full independence of the two estimates and the generally large uncertainty associated with this quantity. Reductions of the RMS differences are generally larger for the JRA-55 estimates, since this dataset exhibits a more vigorous water cycle than ERA-I does, leading to a stronger impact of the consistent treatment of water vapor enthalpy (see also note in section 2).

Fig. 10.
Fig. 10.

Difference of implied net surface energy flux (W m−2) based on RadTOA from CERES and (a) ERA-I-based traditionally computed energy divergence, (b) ERA-I-based energy divergence with the effect of water vapor removed, (c) JRA-55-based traditionally computed energy divergence, and (d) JRA-55-based energy divergence with the effect of water vapor removed, and net surface energy flux based on RadS from CERES and OAFlux turbulent fluxes.

Citation: Journal of Climate 30, 22; 10.1175/JCLI-D-17-0137.1

Fig. 11.
Fig. 11.

As in Fig. 10, but with net surface energy flux from ORAP5.

Citation: Journal of Climate 30, 22; 10.1175/JCLI-D-17-0137.1

Fig. 12.
Fig. 12.

As in Fig. 10, but with net surface energy flux from GECCO2.

Citation: Journal of Climate 30, 22; 10.1175/JCLI-D-17-0137.1

Fig. 13.
Fig. 13.

As in Fig. 10, but with net surface energy flux from CERES and SEAFLUX turbulent fluxes.

Citation: Journal of Climate 30, 22; 10.1175/JCLI-D-17-0137.1

In terms of RMS differences, both the ERA-I-based and JRA-55-based surface energy flux estimate agrees better with GECCO2 and the CERES-EBAF/OAFlux than with the other two surface flux products. Patterns of differences between the products are generally dissimilar. However, there are also common features in Figs. 10b13b (Figs. 10d13d), possibly indicating shortcomings in the ERA-I (JRA-55) data. For example, the ERA-I-based surface energy flux appears to be too strong (positive into the ocean) in the central Pacific ITCZ, which might also be related to the defective moisture analyses of ERA-I in that region, as already pointed out in section 7a. In contrast, the JRA-55-based estimate seems to be too low in the far eastern Pacific ITCZ and the eastern Atlantic ITCZ as well as in the Bay of Bengal. This comparison thus reveals shortcomings in the reanalysis data, which were masked by the large biases of traditional evaluation methods.

d. Improvement of surface flux imbalances over land

Long-term averages of net surface energy flux over land should be close to zero locally. This constraint can be used as a sanity check of surface energy flux data. Regions of high snow amounts represent an exception since there FS should balance the latent heat flux of snowfall (see also Fig. 8). Based on ERA-I snowfall data, we assume 1.5 W m−2 to be a reasonable value for the global land average net surface energy flux. Fasullo and Trenberth (2008), using the commonly used budget formulation and older-generation reanalyses than those employed here, obtained an implied FS over land in the range from −2.5 to 3 W m−2.

Implied FS values over land based on ERA-I and JRA-55 budgets are presented in Figs. 14 and 15, respectively. The improved methods yield somewhat better results in tropical regions; for example, the spurious surface energy fluxes over Indonesia in ERA-I and over the Amazon in JRA-55 are strongly reduced compared to the traditional method. Moreover, the traditional method yields higher RMS values for both datasets, also indicating some improvement of the data consistency in the improved methods. However, regional values of implied FS are still far from zero in many locations, and the patterns do not resemble the spatial structure of snowfall-related latent heat fluxes (cf. Fig. 8).

Fig. 14.
Fig. 14.

Implied net surface energy flux over land (W m−2) based on CERES and ERA-I data using various approaches and simplifications: (a) traditional, (b) new_no_vapor, (c) new_Tskin, and (d) new_exact.

Citation: Journal of Climate 30, 22; 10.1175/JCLI-D-17-0137.1

Fig. 15.
Fig. 15.

As in Fig. 14, but using JRA-55 data instead of ERA-I.

Citation: Journal of Climate 30, 22; 10.1175/JCLI-D-17-0137.1

Negative (positive) global land averages of the ERA-I (JRA-55) indicate that ocean-to-land energy transports are too weak (strong) in ERA-I (JRA-55). Both products are several watts per meter squared off our best estimate (1.5 W m−2) for global land average net surface energy flux, and especially JRA-55 obviously produces unrealistic global ocean-to-land energy transports. The meridional structure of implied surface energy fluxes shows that ocean-to-land energy transports are too strong in the tropics and too weak in the extratropics in both products. Thus, the consistent treatment of water vapor in the energy budget equations yields only modest improvements in this aspect, indicating that other shortcomings of the reanalysis products are largely responsible for the spurious nonzero values over land. Hence, adjustment methods for implied surface flux products as proposed in Liu et al. (2015) or Liu et al. (2017) still seem a useful postprocessing step.

e. Implications for cross-equatorial heat transport in ocean and atmosphere

On average, there is a net northward atmospheric moisture transport across the equator associated with the mean position of the ITCZ. Consequently, the traditional formulation of the divergence of atmospheric energy transports [Eq. (21)] yields a positively biased atmospheric cross-equatorial energy flux (CEF) estimate. The March 2000–February 2007 average value from ERA-I (JRA-55) based on Eq. (21) is −0.21 PW (−0.15 PW) (see also Table 1). Removing the effect of vapor enthalpy transport following Eq. (24) yields an atmospheric CEF of −0.40 PW (−0.48 PW) for ERA-I (JRA-55).

Table 1.

Southern and Northern Hemisphere net surface energy flux inferred from RadTOA based on CERES data and atmospheric CEF computed from ERA-I and JRA-55. Also given is the implied oceanic CEF based on the hemispheric net surface energy fluxes and hemispheric ocean heat content changes [0.36 PW in the Southern Hemisphere (SH) and 0.20 PW in the Northern Hemisphere (NH)] based on ORAP5. CERES data have been adjusted to match global mean ocean heat content changes (0.56 PW), yielding an SH (NH) TOA imbalance of 0.49 PW (0.07 PW). Additionally, three recent estimates from the literature are provided. Units are PW.

Table 1.

The stronger southward atmospheric CEF has also implications for indirect estimates of oceanic CEF. To estimate this, we use a spatially uniform adjustment to tune the global mean CERES-based RadTOA to the March 2000–February 2007 average global mean ocean heat content changes based on ORAP5 (0.56 PW). After this adjustment, the Southern (Northern) Hemisphere average energy input at TOA is 0.49 PW (0.07 PW). Taking into account the Southern (Northern) Hemisphere average ocean heat content changes of 0.36 PW (0.20 PW), the hemispheric TOA imbalance requires a total (atmosphere plus ocean) northward energy transport of 0.13 PW. The oceanic CEF can then be inferred as a residual using our estimate of atmospheric CEF and is estimated to be 0.53 PW (0.61 PW) based on the ERA-I (JRA-55) new_no_vapor method, which is considerably higher than the values based on the traditional method (0.34 and 0.28 PW, respectively). It is not surprising that recent estimates of Liu et al. (2017) or Loeb et al. (2016), who give an oceanic CEF estimate of 0.32 ± 0.16 and 0.44 ± 0.07 PW, respectively, are closer to our traditional estimate, since the inconsistent equations were used in those studies. Stephens et al. (2016), who provide an estimate of oceanic CEF based on surface energy fluxes and ocean heat content changes from ocean reanalyses, give a slightly higher value (0.45 ± 0.60 PW) than the two former studies, but their uncertainty bounds are very large. This large spread is likely related to difficulties of ocean reanalyses to maintain a dynamical balance at the equator, which can adversely affect computed oceanic CEF (Valdivieso et al. 2017). This is also the reason why verification of oceanic CEF with direct computation from ocean reanalyses data is problematic.

Although our estimates of CEF provided here are based on more consistent evaluation methods than other recent estimates, these values should not be viewed as definite. It is possible that the here-employed reanalysis data exhibit shortcomings affecting these estimates and that results based on future reanalysis products are more accurate. For example, Figs. 14 and 15 reveal that ERA-I and JRA-55 have too strong atmospheric energy transports into equatorial land regions. If these spurious transports have a meridional component (which is impossible to check), this would also affect our CEF estimates.

8. Summary and implications

We reexamined the common formulation of the vertically integrated atmospheric total energy budget as used in a large number of diagnostic studies in the past. We identified a spurious dependency of the divergence term on the employed temperature scale, although this should not be the case as long as the mass budget is closed. This introduces an ambiguity when, for example, inferring net surface energy flux from the atmospheric energy budget. The found shortcoming arises from the inconsistent treatment of the three-dimensional vapor and liquid water enthalpy fluxes in the atmosphere: Lateral enthalpy transports by moisture are taken into account, while vertical enthalpy fluxes at the surface associated with P and E are usually neglected (except for latent heat), which results from the incorrect implementation of the flux form to the atmospheric energy budget equation. Another more subtle issue arises from different traditions concerning the reference state of water in different communities. Dynamical meteorologists implicitly assume water vapor at 0 K as the reference state and somewhat inconsistently add latent heat of vaporization at 0°C to the moist static energy, while oceanographers usually view liquid water at 0°C as the reference state. Both communities have good reasons for their choices, but they conflict when it comes to energy budget diagnostics of the coupled atmosphere–ocean–ice system. In addition to these major shortcomings of common diagnostic practices, we also pointed out minor inconsistencies stemming from assumptions generally made in weather and climate models, such as constant latent heats.

We resolved the revealed issues by a consistent choice for the reference state of water and correct implementation of the flux form in the atmospheric total energy budget equation. Based on this, we presented consistent total energy budget equations for atmosphere and ocean, which both include enthalpy fluxes associated with precipitation and evaporation and are independent of the employed temperature scale. Furthermore, we presented simplification of the improved atmospheric budget equation facilitating its evaluation with modest approximation error. Results unambiguously show that the self-consistency of the diagnosed atmospheric energy budget is improved when using the proposed instead of the traditional energy budget equation.

The issues outlined in the present paper have a number of implications. Results show that enthalpy fluxes associated with precipitation (including snowfall) and additional nonlatent contributions from evaporation have a global mean cooling effect on the surface of −1.3 W m−2, which is a result of different spatial distributions of these two quantities. This implies that global mean estimates of the common net surface energy flux FS should be increased to about 2.3 W m−2 to balance the additional surface enthalpy fluxes quantified here and to maintain the observed ocean warming (on the order of 1 W m−2; see Trenberth et al. 2016). This adjustment is easily within uncertainty bounds of FS as provided, for example, by Wild et al. (2013) or Trenberth et al. (2009).

Regionally, the additional surface enthalpy fluxes associated with P and E are much larger (on the order of 20 W m−2 in the tropics). These fluxes must be taken into account when aiming for unambiguous estimates of inferred FS from the atmospheric energy budget. Comparison of FS inferred from satellite-based radiation and reanalysis data to independent surface flux products reveals substantially better agreement when using the improved budget formulation, with a reduction of the RMS differences of the time-averaged fields by up to 40% compared to the traditional formulation.

It is a common approach to combine satellite-based radiation at TOA and atmospheric energy transports from reanalyses to infer ocean heat transports. Hence, because of the hemispheric asymmetry of P and E, the ambiguities found in the atmospheric energy budget will also project onto the inferred partition between atmospheric and oceanic energy transports. One aspect of this is that previous studies seriously underestimated cross-equatorial energy transport by atmosphere and ocean. The straightforward approach to resolve this issue is to use the simplified version of the here-presented atmospheric energy budget equation with the enthalpy of moisture removed. This allows for evaluation of lateral energy transports independent of reference temperature and without quantifying Fp and Fe. Results from that approach should be consistent with transports obtained from climate models, which usually are inferred as a residual of RadTOA and FS. Another consistent but somewhat artificial solution would be to add lateral atmospheric energy transports associated with moisture to the lateral ocean heat transport, the sum of which could be viewed as energy transports of the “hydrosphere.” The advantage of that approach would be that Fp and Fe do not have to be quantified.

We emphasize that the shown ambiguities in diagnosed energy budgets are exclusively due to the shortcomings of the commonly used equations and are not related to the employed datasets. Hence, energy budget evaluations based on other existing and future reanalysis products will benefit from the here-presented diagnostic framework in a similar manner. It would certainly be desirable to extend the here-presented diagnostic equations to the nonsteady state. If local mass variations are nonnegligible, reference enthalpies do not cancel from the equations. In that case it is unavoidable that the choice of temperature scale influences the results. Performing diagnostics in Celsius scale instead of Kelvin scale will considerably reduce this effect, but a further reduction is certainly possible by optimizing the chosen reference temperature.

The discussion here has been concerned with analyzing the diagnostic formulation of the atmospheric energy budget, but it is natural to ask to what extent these issues affect the underlying formulation of atmospheric models. In fact, models are typically so formulated that they avoid the major inconsistencies identified here. Models do not employ an explicit representation of the energy budget but rather have a dynamical core that represents the adiabatic evolution of the temperature and of physical parameterizations that represent diabatic processes. In effect, the dynamical core yields changes in enthalpy and so is not affected by the choice of the reference state. The parameterizations consistently account for latent heating, although the variation of the latent heat with temperature is normally neglected, with the values at 0°C being used, while the specific heats of water are approximated. Consequently, models are affected only by the smaller inconsistencies associated with approximation of the specific heats.

The (coupled) climate model community has put considerable efforts into consistent formulation of coupled energy budgets (see Griffies et al. 2016). A comparable effort has been lacking in the observation and reanalysis community so far, mainly due to the heterogeneity of the community and the fact that coupled atmosphere–ocean–ice reanalyses have become available only recently (e.g., Laloyaux et al. 2016). The present study thus represents an important step toward more consistent diagnostics of the coupled atmosphere–ocean–ice system, which will lead to more reliable benchmark values for validation purposes.

Acknowledgments

The first author thanks A. Donohoe, S. M. Griffies, and M. Hantel for discussions initially motivating this research. The authors thank K. E. Trenberth and C. Liu for helpful comments on an earlier version of this article. The authors also thank three anonymous reviewers for their insightful comments and useful suggestions. This work was financially supported by Austrian Science Fund project P28818 and ERA-CLIM2 (EU FP7 Grant 607029). Patrick Hyder was supported by the Joint UK BEIS/Defra Met Office Hadley Centre Climate Programme (GA01101).

APPENDIX A

Equation of Continuity Including Liquid Water

Following Trenberth (1991), the continuity equation for dry air plus water vapor, explicitly taking liquid water into account and assuming a steady state, reads as follows:
ea1
Here e and c represent evaporation and condensation as a function of pressure, respectively.
The continuity equation of liquid water in the atmosphere is analogous to Eq. (A1):
ea2
Here, q1 represents specific liquid water content, and ωql schematically denotes vertical liquid water flux. Note that e(p) and c(p) appear with different signs in Eqs. (A1) and (A2), respectively.
Adding Eqs. (A1) and (A2) yields the continuity equation for the moist atmosphere including liquid water [see also Eq. (14) in the main text]:
ea3
Equation (A3) appears identical to the textbook form of the continuity equation, but the inclusion of liquid water and vapor has important implications for the vertical integral of Eq. (A3):
ea4
This is identical to Eq. (15) in the main text. In the first step in Eq. (A4) we have identified E + P as the vertical flux mass flux through the surface. There is no vertical dry air flux at the surface. In the second step in Eq. (A4) we acknowledged the fact that dry mass divergence is zero (conservation of dry mass) in the steady state and we have neglected horizontal liquid water transports. Equation (A4) states that the vertically integrated divergence of atmospheric moisture transports balances P and E. We note that inclusion of ice to the presented derivation is straightforward, starting from an equation analogous to Eq. (A2).

The convenient consequence of inclusion of liquid water to the continuity equation is the fact that e(p) and c(p) vanish, which considerably simplifies implementation of the flux form to the total energy budget equation of the atmosphere. Otherwise, vertical integrals of the product of moist static energy and e(p) and c(p) would have to be taken into account. Equation (15) in the main text is identical to the vertically integrated continuity equation provided by Trenberth (1991), but we think its derivation is more transparent when explicitly taking into account liquid water in the atmosphere.

APPENDIX B

List of Terms and Acronyms

C

Celsius

ca

Specific heat capacity of dry air

cg

Specific heat capacity of vapor

cl

Specific heat capacity of liquid water

Horizontal ocean current vector

c(p)

Local condensation rate (as a function of pressure)

E

Evaporation rate at the surface (positive downward)

e(p)

Local evaporation rate (as a function of pressure)

Fa

Atmospheric energy transport

Fe

Surface enthalpy flux associated with evaporation

Fo

Oceanic energy transport

Fp

Surface enthalpy flux associated with precipitation

FS

Net surface energy flux (latent plus sensible heat flux plus net radiation)

g

Gravitational acceleration

h0

Reference enthalpy of water

K

Kelvin

k

Atmospheric kinetic energy

LH

Latent heat flux

Lf

Latent heat of fusion

Lυ

Latent heat of vaporization

M(ϕ)

Zonal integral of mass flux across latitude ϕ

Moist static energy as usually defined

m

Moist static energy as defined in this paper

N

Non-material energy source

ω

Vertical velocity

P

Total precipitation (sum of rain and snow; positive downward)

Prain

Rain rate

Psnow

Snowfall rate

p

Atmospheric pressure

qg

Specific vapor content

ql

Specific liquid water content

qs

Specific ice content

RadS

Net radiation at the surface

RadTOA

Net radiation at the top-of-the-atmosphere

SH

Sensible heat flux

ρ0

Seawater density

SST

Sea surface temperature

T

Temperature

Ta

Atmospheric temperature

Ti

Ice temperature

Tp

Temperature of precipitation

TS

Skin temperature

T00

Reference temperature

Θ(z)

Seawater potential temperature (as a function of depth z)

Φ

Geopotential

φ

Latitude

v

Horizontal wind vector

vs

Horizontal wind vector at the surface

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