## 1. Introduction

Several inner products, based on “energy” squared norms, have been used in four-dimensional variational assimilation tools to minimize cost functions (Talagrand 1981; Courtier 1987; Thépaut and Courtier 1991). It was supposed that the energy corresponding to observational errors could be distributed equally among these different basic prognostic fields. Inner products based on these energy squared norms are used to define dry semi-implicit operators and dry normal modes of GCMs or NWP models, as long as they are invariant by the linear set of primitive equations (Thépaut and Courtier 1991).

Here, the term energy means that the sum of quadratic terms is considered for perturbations of the wind components (*u*′)^{2} + (*υ*′)^{2}, temperature (*T*′)^{2}, and surface pressure

The same inner products and norms are currently used for computing dry or moist singular vectors and for determining forecast errors or sensitivity to observations based on tangent linear and adjoint models (Buizza and Palmer 1995; Palmer et al. 1998; Mahfouf and Bilodeau 2007; Janisková and Cardinali 2017).

However, all these norms suffer from a lack of consistency with physical relationships in thermodynamics, because (i) these energy squared norms are not based on the standard definition of energy as expressed in general thermodynamics; (ii) the use of the squared norm for water including the quadratic term

Ideally, all these quadratic terms should be derived from some general laws of physics. This is true for the average of the kinetic energy *u*′ and *υ*′ are the unbiased departures between analyses and short-range forecasts.

In contrast, the usual temperature component of the squared norm

To derive quadratic squared norms in both wind components and temperature, a relevant method might be based on the study of the sum of the kinetic energy and “a form of the Available Potential Energy” (APE) of Lorenz (1955). This method is chosen in Talagrand (1981), the old ARPEGE-IFS documentation (1989, unpublished), Joly and Thorpe (1991), Joly (1995), Ehrendorfer and Errico (1995), Errico and Ehrendorfer (1995), E99, Ehrendorfer (2000), Errico (2000), and Descamps et al. (2007).

*T*

_{r}or an equivalent. This leads to

*T*

_{r}is used in Courtier (1987), Thépaut and Courtier (1991), Buizza et al. (1993), Ehrendorfer and Errico (1995), Buizza et al. (1996), Mahfouf and Buizza (1996), E99, Errico (2000), Barkmeijer et al. (2001), Zadra et al. (2004), Errico et al. (2004), Mahfouf and Bilodeau (2007), Rivière et al. (2009), Holdaway et al. (2014), Janisková and Cardinali (2017), among others.

However, it is worth noting that the use of a constant value *T*_{r} for *T* according to Lorenz (1955). No other definition is allowed, and the use of a constant temperature *T*_{r} makes the theory incompatible with that of Lorenz and weakens the theoretical basis for present formulations of the norm for temperature.

All temperature, pressure and water vapor components of existing squared norms correspond to the quadratic terms *Although it is called a measure of the energy, it has not been demonstrated that it is indeed such in the contexts to which it has been applied. The fact that it has units of energy per unit mass does not by itself qualify it as a measure of energy*”). Moreover, the moist-air generalization of the APE by Lorenz (1978, 1979) does not lead to any easy-to-use analytical formulation that could replace

Therefore, other ideas had to be tested in order to solve the problems described so far. Since the temperature component (2) is presently derived from an approximate version of the APE of Lorenz, which was improved by Pearce (1978) and Marquet (1991) for the dry air, and then by Marquet (1993, hereafter M93) for the moist air, this article examines the possibility of deriving the quadratic terms in temperature, pressure and water content from a general principle based on the concept of “moist available enthalpy” defined in M93.

The available enthalpy is one form of what is known as “exergy” in general thermodynamics. This new exergy norm is used in Borderies et al. (2019) to measure the relative impact of the assimilation of observations on the analysis and short-term forecasts for the French AROME model, with a large impact of the new water-content quadratic term. Indeed, the weighting factors of the exergy norm are significantly different from those used up to now in dry and moist squared norms, in particular by several orders of magnitude for the water content.

To achieve some numerical validation of the theoretical formulations for the exergy norm, the same comparisons of the squared norms with inverse analysis increment estimates are made as in MB07.

The motivations for these comparisons can be found in Errico et al. (2004), where a moist norm was used with weights “proportional to estimates of the variances of analysis uncertainty”. It was also explained in Barkmeijer et al. (2001) that “in the case of forecast-error covariance prediction, a norm at initial time based on the analysis-error covariance matrix is the more appropriate” (Ehrendorfer and Tribbia 1997; Palmer et al. 1998; Barkmeijer et al. 1998). At that time, “the analysis-error covariance metric became the reciprocal of the total-energy metric currently used at ECMWF to compute singular vectors for the EPS” (Barkmeijer et al. 1998). And “a specific-humidity norm based on error variances” was experimented by Derber and Bouttier (1999) at ECMWF, leading to a specific-humidity norm defined in Barkmeijer et al. (2001) from the ECMWF “averaged error variances for *q*_{υ},” with a strong decrease of this norm above 500 hPa, a property that has remained unexplained until now.

This paper is organized as follows. Existing moist-air squared norms are recalled in section 2a. Section 2b presents some theoretical motivations for the use of exergy functions based on the concepts of relative entropy and Kullback–Leibler divergence. The derivations of the moist-air available-enthalpy are conducted in appendixes B–G and the corresponding quadratic approximate squared norm components are shown in section 2c for temperature, pressure, and water. The datasets from the Canadian Meteorological Centre (CMC), the NASA Goddard Earth Observing System (GEOS), and the French ARPEGE models are described in section 3. These datasets are used to compare the norm components for water and temperature with the root-mean-square (RMS) of analysis increments, with cross sections and vertical profiles shown in sections 4a–c for the three models, leading to an explanation of the decrease with height of the water vapor exergy terms described in section 4d. Forecast observation impacts are described in section 4e for the GEOS model. Conclusions are drawn in section 5.

## 2. Theoretical considerations

### a. Existing moist-air energy norms

*u*′,

*υ*′,

*T*′,

*dm*=

*ρdτ*is equal to

*dpd*Σ/

*g*, where Σ is the horizontal surface area. The volume integrals over

*dm*/Σ and the surface integral over

*d*Σ/Σ represent energies per unit of horizontal area, all expressed in units of J m

^{−2}. The pressure component is expressed in E99 as a volume integral of

The two formalisms using the surface pressure or its logarithm are nearly equivalent, providing that

The justification for the last integral of (3) depending on the variance of water vapor content can be found in Ehrendorfer et al. (1995), Buizza et al. (1996), Mahfouf et al. (1996) and E99. The water contribution of the squared norm is derived from the temperature contribution *c*_{pd}(*T*′)^{2}/(2*T*_{r}) with the additional hypothesis that changes of temperature and moisture are related by *H*_{r}) and water content (*Q*_{r}) were defined, leading to the equivalent formulation

The question addressed in E99 is the relevance of that special formulation for the water contribution. Due to the uncertainty in the assumption *w*_{q}(*z*) (also denoted by *w*^{2} or *ϵ*, depending on papers) is added in the last integral of (3). The effects of making this relative weight larger or smaller than the standard value 1 are discussed in E99 and Barkmeijer et al. (2001), where *w*_{q}(*z*) may increase with height in the upper troposphere and in the stratosphere.

*q*

_{υ}/

*q*

_{sw}. This assumption is expected to be realistic in cloudy areas where relative humidity reaches 100%, however, it may not be realistic in frequently undersaturated moist areas. The constraint of zero departure (at constant pressure) in the quantity

*q*

_{υ}/

*q*

_{sw}/(

*T*,

*p*) corresponds to

*w*

_{q}(

*z*) with altitude considered in Zadra et al. (2004) in moist singular vector computations. The aim was to suppress the impact of humidity perturbations in the stratosphere according to the results of Buizza et al. (1996) and E99, who showed that for increasing

*w*

_{q}the contribution of the dry fields dominates initially, whereas the contribution of moisture dominates at the final time (and vice versa when

*w*

_{q}is smaller).

*σ*

_{k}is the thickness of the layer

*k*in the

*σ*vertical coordinate and

*ω*

_{ij}is the fractional coverage of the model grid box defined by the zonal (

*i*) and meridional (

*j*) indices.

The weighting factors *V*_{u}, *V*_{υ}, (*V*_{T1})_{jk}, (*V*_{p1})_{j}, and (*V*_{q1})_{jk} will hereafter be referred to as “*V* terms.” They are interpreted as variances of analysis errors in Errico et al. (2004) and MB07. The indices *j* and *k* mean that temperature, surface pressure and water variances can a priori depend on latitude (*j*) and/or altitude (*k*).

*V*terms in E99 can be written as

*V*

_{u},

*V*

_{υ},

*V*

_{T1}, and

*V*

_{p1}are all constant, whereas (

*V*

_{q1})

_{k}may depend on altitude for water, via the arbitrary weight

*w*

_{q}(

*z*).

All terms in parentheses in (6) are dimensionless in Errico et al. (2004) and MB07, where the dimensions of the square root of (*V*_{T1})_{jk}, (*V*_{p1})_{j}, and (*V*_{q1})_{jk} are K, hPa, and kg kg^{−1}, respectively. The square root of these *V* terms will be called “*SqV* terms” hereafter. The dimensionless characteristic of (6) can be explained by first multiplying all terms of (3) by the dimensionless value 2, and then by dividing all terms by the same energy term *V*_{0} = 2 J kg^{−1}. Therefore, the dimensions of *c*_{pd}*T*_{r}, *R*_{d}*T*_{r}, and *L*_{υ}*Q*_{r} are same as the one of *V*_{u} = *V*_{υ} = *V*_{0}, namely in units of m^{2} s^{−2} or J kg^{−1}. The value of the dummy specific content *Q*_{r} has no impact in (8); it is introduced to highlight the relevant dimension of kg^{2} kg^{−2} for (*V*_{q1})_{jk}.

*V*

_{q2})

_{jk}is expressed in kg

^{2}kg

^{−2}, because

*V*

_{0}. This means that the dimension of the square root of (

*V*

_{q2})

_{jk}is the same as the specific content

^{−1}and, from (10), varies with altitude via the ratio of the average terms

### b. Relative entropy, exergy, and available enthalpy

Due to the uncertainty and plurality in *V*_{T1}, *V*_{q1}, or *V*_{q2} defined in E99 or MB07, and due to the arbitrary values for *w*_{q}(*z*), it is necessary to find a more general and comprehensive “measure,” “norm” or “distance” between a perturbed thermodynamic state defined by (*T*_{2}, *q*_{υ2}, *p*_{s2}) and a reference one defined by (*T*_{1}, *q*_{υ1}, *p*_{s1}).

*x*

_{j}s represent a real state (

*x*) and the

*y*

_{j}s represent a reference state (

*y*) of the system (see Cover and Thomas 1991).

^{1}

This Kullback–Leibler divergence *K* is usually interpreted as being a nonsymmetric measure of how much the *x*_{j}s deviate from the *y*_{j}s. It also represents the “gain in information” of the state characterized by the distribution (*x*_{j}) with respect to the equilibrium distribution (*y*_{j}). Therefore, it is unclear whether *K* corresponds to the measure or the distance between the two thermodynamic states (*T*_{2}, *q*_{υ2}, *p*_{s2}) and (*T*_{1}, *q*_{υ1}, *p*_{s1}).

*x*

_{j}s and the

*y*

_{j}s that correspond to these two thermodynamic states. Moreover, the relative entropy

*K*is clearly different from the entropy

*y*

_{j}included in (11). However, it is possible to show that the macroscopic value of

*K*roughly corresponds to the free energy function

*e*

_{i}−

*T*

_{r}

*s*, which is different from the entropy

*s*because it depends on the internal energy

*e*

_{i}and a reference temperature

*T*

_{r}. More precisely, it is shown for instance in Procaccia and Levine (1976), Eriksson and Lindgren (1987), and Karlsson (1990) that the exergy of moist air can be computed by the “available energy” function

*a*

_{e}=

*k*

_{B}

*T*

_{r}

*K*, with

*K*(

*x*||

*y*) given by (11). This function

*a*

_{e}can be written in terms of the local atmospheric variables (

*p*,

*T*,

*q*

_{n}), leading to

*r*” denotes a reference state and where the sum over “

*n*” represents the dry air, water vapor, liquid water and ice species. The specific volume is

*α*= 1/

*ρ*and the specific contents

*q*

_{n}are multiplied by the reference Gibbs functions

*μ*

_{rn}=

*h*

_{rn}−

*T*

_{r}

*s*

_{rn}. The quantity

*a*

_{e}given by (12) is called “

*maximum available work from a nonflow system*” by Bejan (2016, Eq. 5.12) for system at rest reaching a pressure equilibrium with the environment (the laboratory). The last sum over

*n*in (12) is called “

*chemical exergy*” by Bejan, while the other terms form the “

*nonflow exergy*.”

*e*

_{i}−

*e*

_{ir}) and −

*p*

_{r}(

*α*−

*α*

_{r}) in (12) must be replaced by the difference in specific enthalpy (

*h*−

*h*

_{r}) to form the “

*thermomechanical and chemical flow exergy*” defined in Bejan (2016, Eq. 5.25). It is the same available enthalpy function as that studied in Marquet (1991) and M93 and corresponding to (B1), with all other terms of (12) remaining the same, leading to

*h*to replace the internal energy is motivated by the natural application of

*h*to the flowing moist-air atmosphere. No hypothesis is made from this point of view, since the use of enthalpy does not impose movements that would be made “at constant pressure.” The change in the variable

*h*=

*e*

_{i}+

*p*/

*ρ*is simply mathematical, with no underlying physical assumptions. One of the interests of the introduction of the enthalpy

*h*is the existence of the Bernoulli function

*h*+

*gz*+ (

*u*

^{2}+

*υ*

^{2})/2, which is constant during stationary, adiabatic and frictionless motions, with a similar Bernoulli’s law derived in M93 for

*a*

_{m}+

*gz*+ (

*u*

^{2}+

*υ*

^{2})/2.

The flow exergy *a*_{m} given by (13) ensures the definition of the aforementioned general distance between a perturbed atmospheric state and a reference one. Indeed, since the available enthalpy is the maximum work (or energy) that a system can deliver when passing from a reference state to the real state, this work is produced by transformations from different forms of energy to other forms of energy.

In particular, it is shown in M93 that a Bernoulli equation exists and that the sum *a*_{m}(*T*, *p*, *q*_{υ}, *q*_{l}, *q*_{i}) + (*u*^{2} + *υ*^{2})/2 + *ϕ* is conserved along any streamline of an adiabatic frictionless and reversible steady flow of a closed parcel of moist air. This means that the conversions between the potential energy, the kinetic energy and the temperature, pressure and water components of *a*_{m}(*T*, *p*, *q*_{υ}, *q*_{l}, *q*_{i}) given by (13) can be evaluated with the weighting factors *V*_{T}, *V*_{p}, and *V*_{q}, ensuring relevant thermodynamic transformations of energy from one form to another.

### c. The new moist-air available-enthalpy norm

*V*terms corresponding to (7)–(10) for temperature, pressure, and water content can be written as

*V*

_{q})

_{jk}is independent of

*r*

_{r}. The last formulation in (18) is obtained with

*R*

_{υ}=

*R*

_{d}/

*r*

_{0}and

*r*

_{0}=

*r*

_{r}(

*p*

_{r}−

*e*

_{r})/

*e*

_{r}≈ 622 g kg

^{−1}, where

*r*

_{0}is proportional to the reference mixing ratio

*r*

_{r}. This shows that the dimensions of (

*V*

_{q})

_{jk}and of

^{2}kg

^{−2}, since

*V*

_{0}= 2 m

^{2}s

^{−2}and

*R*

_{d}

*T*

_{r}have the same dimension. Therefore, the square root of (

*V*

_{q})

_{jk}has the dimension of a mixing ratio, as expected.

From (8) and (17) the pressure *V* terms *V*_{p1} and (*V*_{p})_{j} may be close to each other if *V*_{p})_{j} only depending on *p*_{r}.

Differently, the temperature and water *V* terms can differ significantly because *V*_{q})_{jk} since

*w*

_{q}(

*z*) in E99, leading to

*c*

_{pd}

*T*

_{r}and

*R*

_{d}

*T*

_{r}with

*L*

_{υ}

*r*

_{r}, also of

*e*

_{r}with

*p*

_{r}−

*e*

_{r}and of

*r*

_{r}with

*r*

_{υ}.

*w*

_{q}(

*z*), which increases with height for decreasing values of

*V*term in (10) with the constant MSE

*V*term in (8), leading to

*w*

_{q2}≈ (

*w*

_{q})

^{2}because

*w*

_{q2}(

*z*) with height in MB07.

Although the reference value of water content has no impact on the water term (*V*_{q})_{jk} given by (18), it is possible to compute, for the sake of internal consistency and realism, both *e*_{r} and *r*_{r} for several of the values of *T*_{r} and *p*_{r} that, from Table 1, are typically used in atmospheric research (semi-implicit algorithms, computation of singular vectors and studies of sensitivity to observations or forecast errors). The result is shown in Table 2 for saturating pressures *e*_{r} = *e*_{sw}(*T*_{r}) or *e*_{si}(*T*_{r}) with respect to the more stable state (liquid water or ice), depending on the temperature *T*_{r}. The zero Celsius and 280 K temperatures are added to show the rapid increase of both *e*_{r} and *r*_{r} with *T*_{r} for an increase of a few degrees between 270 and 280 K. The higher temperature *T*_{r} = 350 K leads to unrealistically large values of *r*_{r}, which are even undefined (negative) for 367.8 hPa. The explanation for these impossible values for some couple (*T*_{r}, *p*_{r}) comes from the fact that *e*_{r} is defined as the saturation pressure at the temperature *T*_{r}. We therefore assume that *p*_{r} > *e*_{r}, which is not verified for example for *T*_{r} = 350 K for which *e*_{r} = 411 hPa is greater than 367.8 hPa in Table 2. But this assumption *p*_{r} > *e*_{r} does not limit the validity of the theory, in the same way that the assumption *p* > *e* for humid air does not limit the two state equations for dry air and water vapor. Therefore the available enthalpy function and the exergy norm are well-defined for values *T*_{r} < 300 K for which the ratios |*X*_{υ}/*Y*_{υ}| are greater than 10 in Table E1, regardless of the pressure *p*_{r}.

The reference temperatures *T*_{r} (K) and pressures *p _{r}* (hPa) used (from the left to the right) in Pearce (1978) and M93, Buizza et al. (1996) and Mahfouf and Buizza (1996), E99 and Holdaway et al. (2014), Errico et al. (2004) and MB07, and Janisková and Cardinali (2017).

The reference mixing ratio *r*_{r}(*T*_{r}, *p*_{r}) defined as *r*_{0} *e*_{r}(*T*_{r})/[*p*_{r} − *e*_{r}(*T*_{r})] in g kg^{−1} and the saturated pressure *e*_{r}(*T*_{r}) in hPa computed for several reference temperatures *T*_{r} in K and pressures *p*_{r} in hPa.

## 3. The datasets

The RMS of analysis increments *S*_{q} and the *SqV* terms are computed for three systems using 3DVAR or 4DVAR algorithms. The periods correspond to either individual days, month or seasonal periods. The aim is to show that the temperature and water components of the exergy norm lead to robust results (i.e., that are valid for a wide range of durations and for different systems).

ARPEGE is the NWP model used at the French weather service at Météo-France (Courtier et al. 1991). The horizontal Gauss grid is based on a Schmidt projection with a spectral truncation T1198 and a stretching factor of 2.2 (i.e., with a varying resolution from 7 km over France to 33 km over the South Pacific). The vertical grid has 105 hybrid levels extending from 10 m to 0.1 hPa. The data assimilation is based on a 6-hourly incremental 4DVAR (Courtier et al. 1994), with increments computed at the truncations T149c1 (135 km) and T399c1 (50 km).

The Global Environment Multiscale (GEM) model (Côté et al. 1998a,b) studied in MB07 is used at the Canadian Meteorological Centre (CMC). The global horizontal grid has a uniform resolution of 1.5° in longitude and latitude. The resolution is variable in the vertical, with 28 *σ* levels extending from the surface up to 10 hPa. The analysis increments are diagnosed by the CMC 3DVAR system (Gauthier et al. 2007).

The Goddard Earth Observing System version 5 (GEOS-5) is an atmospheric global circulation model developed by the National Aeronautics and Space Administration’s (NASA) Global Modeling and Assimilation Office (GMAO). The model is based on the finite volume cubed-sphere (FV3) dynamical core (Putman 2007). The Modern-Era Retrospective Analysis for Research and Applications (MERRA-2), version 2 (Gelaro et al. 2017), is a global reanalysis produced by GMAO using the GEOS forecast model and gridpoint statistical analysis data assimilation system (Wu et al. 2002; Kleist et al. 2009). The 3D-Var system MERRA-2 produces an analysis every 6 h from 1980 to the present day. The horizontal resolution of the data assimilation and model is around 50 km, or 0.5°. In the vertical, 72 hybrid sigma-pressure levels are used, reaching from the surface to 0.01 hPa. The linearized version of GEOS includes the FV3 dynamical core and a linearization of the relaxed Arakawa–Schubert convection scheme (Holdaway et al. 2014, hereafter H14), single moment cloud scheme (Holdaway et al. 2015) and a simplified boundary layer scheme.

## 4. The results

### a. Seasonal means of ARPEGE: The water norms

The ARPEGE seasonal averages of RMS of analysis increments *S*_{q} and exergy *SqV* term

The general patterns for *S*_{q} and ^{−1}) and seasonal latitude oscillations following the regions of maximum surface temperatures (from −15° in DJF to +15° in JJA).

Values close to the ground are of the same order of magnitude for the analysis increments (≈0.7 g kg^{−1}), the exergy term (≈0.4 g kg^{−1}) and the E99 term (^{−1} or 0.57 g kg^{−1} computed with *T*_{r} = 300 K and *w*_{q} = 1.0 or *w*_{q} = 0.3).

The JJA and DJF seasonal means of the “constant RH” value *S*_{q} and *w*_{q2} ≈ (*w*_{q})^{2} derived from (20)–(21) leading to values of ^{−1} in the stratosphere (purple color). These values of

As in Figs. 1b and 1d, but for the DJF and JJA seasonal average of the MB07 water term

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

As in Figs. 1b and 1d, but for the DJF and JJA seasonal average of the MB07 water term

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

As in Figs. 1b and 1d, but for the DJF and JJA seasonal average of the MB07 water term

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

Vertical profiles are plotted in Fig. 3 for the horizontal means of the RMS of analysis increment *S*_{q} and for the *V* terms

Vertical profiles of horizontal mean of seasonal averages computed from ARPEGE outputs every 6 h and for three latitude domains: (a),(d) southern extratropical midlatitudes from −60° to −30°; (b),(e) tropical latitudes from −30° to +30°; (c),(f) northern extratropical midlatitudes from +30° to +60°. The vertical profiles of the DJF means are plotted in (a)–(c); those for the JJA means in (d)–(f). The E99 water terms *w*_{q} = 1.0 and *w*_{q} = 0.3. The exergy water term *S*_{q} (blue solid lines).

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

Vertical profiles of horizontal mean of seasonal averages computed from ARPEGE outputs every 6 h and for three latitude domains: (a),(d) southern extratropical midlatitudes from −60° to −30°; (b),(e) tropical latitudes from −30° to +30°; (c),(f) northern extratropical midlatitudes from +30° to +60°. The vertical profiles of the DJF means are plotted in (a)–(c); those for the JJA means in (d)–(f). The E99 water terms *w*_{q} = 1.0 and *w*_{q} = 0.3. The exergy water term *S*_{q} (blue solid lines).

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

Vertical profiles of horizontal mean of seasonal averages computed from ARPEGE outputs every 6 h and for three latitude domains: (a),(d) southern extratropical midlatitudes from −60° to −30°; (b),(e) tropical latitudes from −30° to +30°; (c),(f) northern extratropical midlatitudes from +30° to +60°. The vertical profiles of the DJF means are plotted in (a)–(c); those for the JJA means in (d)–(f). The E99 water terms *w*_{q} = 1.0 and *w*_{q} = 0.3. The exergy water term *S*_{q} (blue solid lines).

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

Almost the same features are observed for the two seasons and for the three latitude domains. The large decrease with height by at least three orders of magnitude for the analysis increments *S*_{q} cannot be represented by the E99 constant values ^{−1} with *w*_{q} = 1.0 or *w*_{q} = 0.3, nor for any other constant value for *w*_{q}.

The differences between the vertical profiles of the RMS of analysis increments, those for the exergy terms and those for the MB07 term remain small from the surface up to about 200 hPa (less than one order of magnitude). The exergy term

For these reasons, the RMS of the analysis increments, the exergy norm and the MB07 norm are thus similar to each other, while the values for E99 are more different from the other three. The aim was not to perfectly simulate the RMS of the analysis increments, but to approach them qualitatively, both for their vertical variation and for their order of magnitude.

The lack of a contribution from condensed water species to the moist-air exergy norm, together with the absence of any latent heat terms *L*_{υ} or *L*_{s}, may seem surprising. However, the condensed water contents *q*_{l} and *q*_{i} do exist in (B1) for the moist-air exergy function *a*_{m}, which forms the starting point for deriving the moist exergy squared norm.

It is this theory that ultimately allows *q*_{l} and *q*_{i} to be neglected in the squared norm components *N*_{T}, *N*_{p} and *N*_{υ}, as small correction terms. Moreover, the seasonal averages plotted in Fig. 1 for ARPEGE confirm that there is no need to add independent norms related to the condensates *q*_{l} or *q*_{i}, because the comparisons between the latitude-section of *S*_{q} and *q*_{l} and *q*_{i} are large (tropical cumulus and extratropical frontal regions).

### b. Seasonal means of ARPEGE: The temperature norms

The exergy norm seemed able to induce new results, especially for the moisture term

For this purpose, ARPEGE winter averages of the RMS of analysis increments for temperature *S*_{T} and of the temperature exergy term

As in Figs. 1a and 1b, but for temperature (K).

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

As in Figs. 1a and 1b, but for temperature (K).

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

As in Figs. 1a and 1b, but for temperature (K).

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

The same ARPEGE seasonal mean (DJF) as in Fig. 3a but for temperature (K) and for the RMS of analysis increments *S*_{T} (solid blue), the E99 term

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

The same ARPEGE seasonal mean (DJF) as in Fig. 3a but for temperature (K) and for the RMS of analysis increments *S*_{T} (solid blue), the E99 term

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

The same ARPEGE seasonal mean (DJF) as in Fig. 3a but for temperature (K) and for the RMS of analysis increments *S*_{T} (solid blue), the E99 term

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

Although the comparisons of norms for each latitude and pressure are less relevant for the temperature components than for the water components (especially within the tropics), the general appearance for *S*_{T} and

The variations with height of *S*_{T}, while the constant value deduced from the E99 temperature component of the norm (*T*_{r} = 300 K) is further from the *S*_{T} profile.

Therefore, although variations with height of *S*_{T} and *S*_{q} and *S*_{T} and

### c. A specific day for CMC and GEOS systems

The results presented in the previous sections regarding ARPEGE seasonal averages are encouraging, but the need for daily applications of the exergy norm would require similar variations with height and latitude for a given situation for both the analysis increments and the norms. In addition, the encouraging results obtained with the 4D-Var incremental assimilation of the ARPEGE variable mesh model must be confirmed with different models and/or assimilation schemes.

To do this, the results obtained for the humidity variable are shown in Fig. 6 for one single analysis (0000 UTC 26 December 2002). Outputs from the GEM-CMC system are on the left in Figs. 6a, 6c, and 6e and those from the GEOS-MERRA-2 system are on the right in Figs. 6b, 6d, and 6f. The latitude–pressure sections for the RMS of analysis increments *S*_{q} in Figs. 6a and 6b are similar to those in Figs. 1a and 1b. The vertical profiles of the exergy term *T*_{r} = 300 K and *w*_{q} = 1.0 or *w*_{q} = 0.3 are similar to those in Fig. 3a.

Latitude–pressure sections and vertical profiles of horizontal averages for the water term for 26 Dec 2002: (a),(c),(e) GEM-CMC; (b),(d),(f) GEOS-MERRA-2. (a),(b) Sections of the RMS of analysis increments *S*_{q} (g kg^{−1}). (c),(d) Sections of exergy norms ^{−1}). (e),(f) Vertical profiles of horizontal averages of E99 (dotted purple), MB07 (dashed black), and exergy (dashed red) norms and the analysis increments *S*_{q} (solid blue).

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

Latitude–pressure sections and vertical profiles of horizontal averages for the water term for 26 Dec 2002: (a),(c),(e) GEM-CMC; (b),(d),(f) GEOS-MERRA-2. (a),(b) Sections of the RMS of analysis increments *S*_{q} (g kg^{−1}). (c),(d) Sections of exergy norms ^{−1}). (e),(f) Vertical profiles of horizontal averages of E99 (dotted purple), MB07 (dashed black), and exergy (dashed red) norms and the analysis increments *S*_{q} (solid blue).

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

Latitude–pressure sections and vertical profiles of horizontal averages for the water term for 26 Dec 2002: (a),(c),(e) GEM-CMC; (b),(d),(f) GEOS-MERRA-2. (a),(b) Sections of the RMS of analysis increments *S*_{q} (g kg^{−1}). (c),(d) Sections of exergy norms ^{−1}). (e),(f) Vertical profiles of horizontal averages of E99 (dotted purple), MB07 (dashed black), and exergy (dashed red) norms and the analysis increments *S*_{q} (solid blue).

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

While the RMS of analysis increments are noisier for those GEM-CMC and GEOS-MERRA-2 daily outputs than for the ARPEGE seasonal averages, the same decay with height and relative maxima in the lower layers in the tropics is observed for this particular day. The differences between the three ARPEGE, GEM-CMC, and GEOS-MERRA-2 systems are more pronounced above 200 hPa in the upper troposphere and in the stratosphere, where GEM-CMC exhibits larger analysis increments than ARPEGE, while those for GEOS-MERRA-2 are smaller than ARPEGE.

The latitude–pressure sections plotted for the water component of the exergy norm in Figs. 6c and 6d for GEM-CMC and GEOS-MERRA-2 are similar to those for ARPEGE in Figs. 1a and 1b.

The water exergy *SqV* term *S*_{q}.

The results presented in this section for a specific day and for two different systems are therefore broadly comparable to those shown for the ARPEGE seasonal averages. We can therefore be confident that the results derived in this paper from the exergy norm will be robust for other systems with similar patterns of analysis fields.

### d. The decrease with height of w_{q}

The advantage of the exergy approach is that it provides an analytic formulation for the weighting factor *w*_{q} given by (19). As an example, values of *w*_{q}(*q*_{υ}) are plotted in Fig. 7 for 0.1 < *q*_{υ} < 25 g kg^{−1}.

The dimensionless exergy weighting factor *w*_{q}(*q*_{υ}) given by (19) plotted with *q*_{υ} in ordinates.

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

The dimensionless exergy weighting factor *w*_{q}(*q*_{υ}) given by (19) plotted with *q*_{υ} in ordinates.

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

The dimensionless exergy weighting factor *w*_{q}(*q*_{υ}) given by (19) plotted with *q*_{υ} in ordinates.

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

The weighting factor *w*_{q}(*r*_{υ}) is smaller than unity for moist low levels where *q*_{υ} > 6.7 g kg^{−1} for *T*_{r} = 300 K, and it is equal to 0.33 for *q*_{υ} ≈ 20 g kg^{−1}. Conversely, it is much larger than unity for small values of *q*_{υ}, reaching *w*_{q} ≈ 67 for *q*_{υ} ≈ 0.1 g kg^{−1} in the upper troposphere and *w*_{q} ≈ 6700 for *q*_{υ} ≈ 0.001 g kg^{−1} in the stratosphere.

It is also possible to plot the vertical profiles of *w*_{q} in terms of the horizontal mean value *w*_{q} with height, with a factor varying nonlinearly from 1 to 40 for the pressure varying from 1000 to 300 hPa, is similar to the one proposed empirically in previous studies; for instance, a weight of *w*_{q}(*r*_{υ}) ≈ 5 was evaluated for the lower part of the atmosphere in Barkmeijer et al. (2001) from the ECMWF averaged error variances for *q*_{υ}, with *w*_{q}(*r*_{υ}) strongly increasing above 500 hPa. This description is consistent with the exergy weight displayed in Fig. 8.

The dimensionless exergy weighting factor *w*_{q}(*z*) given by (19) for the vertical profile of average values

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

The dimensionless exergy weighting factor *w*_{q}(*z*) given by (19) for the vertical profile of average values

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

The dimensionless exergy weighting factor *w*_{q}(*z*) given by (19) for the vertical profile of average values

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

The same relation used to plot these diagrams “*w*_{q} in terms of *q*_{υ}” is used to plot the exergy norm in the pressure (*p*) and latitude (*φ*) sections shown in Figs. 1b, 1d, 6c, and 6d, where the zonal averages

### e. FSOI

The forecast sensitivity to observation impact (FSOI) method can be used to assess and compare the capacity of various observing systems to reduce a given short-range forecast error produced by a NWP model (e.g., Baker and Daley 2000; Langland and Baker 2004; Cardinali 2009; Gelaro et al. 2010). Typically, fields from a 24 h forecast are compared against a verifying analysis, in terms of *u*, *υ*, *T*, *p*_{s}, and *q*_{υ} using an inner product based on the E99 energy norm with different values of *w*_{q} in the moist term. The adjoint of the forecast model is used to propagate a sensitivity backward from verifying time (24 h) to obtain a sensitivity at analysis time (0 h). The adjoint model can include both dry physical processes (turbulent diffusion, radiation, gravity wave drag) and moist processes (large-scale condensation, moist convection).

Impacts shown in the present paper are examined in averages per observation system and for the global domain with the E99 norms (7)–(8) where *T*_{r} = 270 K, *p*_{r} = 1000 hPa, and *w*_{q} = 0.3. The value of 0.3 is chosen empirically in H14 to produce approximately equal weighting between the temperature and specific humidity components of the norm.

The metrics monitored at GMAO are the following: impact per analysis, impact per observation, fraction of beneficial observations, and observation count per analysis. The observation impacts are computed as reductions in the final 24 h forecast errors due to any given extra set of observations included in the initial analysis. The adjoint model can be used to propagate the final energy norm gradient backward 24 h in order to obtain sensitivities of these forecast errors at the initial time (Trémolet 2008). These sensitivities are then passed through the adjoint of the data assimilation system to convert them into observation space and to provide the impacts.

Figure 9 compares the 24 h forecast error reductions produced by various observing systems included in the MERRA-2 data assimilation system with three different inner products for the estimation of the global forecast error: the E99 “dry energy squared norm” with *w*_{q} = 0.0, the E99 “moist energy squared norm” with *w*_{q} = 0.3, and the “exergy squared norm” *N*_{T} + *N*_{p} + *N*_{υ} introduced in Eqs. (14)–(16) of section 2c.

The 24-h forecast observation impacts per analysis for each observation system. Comparisons of (i) the dry norm (white); (ii) the moist norm E99 with *w*_{q} = 0.3 (gray), namely the same as Fig. 9 in H14; and (iii) the moist exergy norm (dark).

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

The 24-h forecast observation impacts per analysis for each observation system. Comparisons of (i) the dry norm (white); (ii) the moist norm E99 with *w*_{q} = 0.3 (gray), namely the same as Fig. 9 in H14; and (iii) the moist exergy norm (dark).

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

The 24-h forecast observation impacts per analysis for each observation system. Comparisons of (i) the dry norm (white); (ii) the moist norm E99 with *w*_{q} = 0.3 (gray), namely the same as Fig. 9 in H14; and (iii) the moist exergy norm (dark).

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

The impacts of the dry energy E99 squared norms are those computed and studied in Fig. 9 of H14 for the month 17 March–17 April 2012. The impacts for the two moist squared norms (E99 with *w*_{q} = 0.3 and exergy formulations) are computed for another month (1–30 September 2015). For convenience, the impacts of the three dry and moist squared norms are compared on the same plot despite having been computed over those two distinct periods. In all experiments, the adjoint model includes a comprehensive set of physical processes with moist processes as described in H14.

As expected from the definition of the moist energy norm, impacts are larger when they include the moist term, as already shown in H14. It is interesting to note that the increase in observation impacts not only holds for observations sensitive to atmospheric water vapor, such as radiosoundings, but also for observation systems where only a small subset of the observations directly measure moisture, such as the Infrared Atmosphere Sounding Interferometer (IASI) radiances, Advanced Microwave Sounding Unit (AMSU-A) radiances that are sensitive to atmospheric temperature, and atmospheric motion vectors (AMVs) that are directly sensitive to horizontal wind components. These results show that a reduction of forecast error in the moisture field is possible through observations of temperature and wind. This could occur through dynamical balance, for example.

The ranking, in terms of contributions of the various observing systems to the forecast error reduction, is unchanged when moving from E99/*w*_{q} = 0.0 to E99/*w*_{q} = 0.3. Similarly, when examining the impact with the exergy norm instead, it is clear that the overall observation impact is larger, but that the ranking of the observation systems relative to each other is almost the same. Larger values come from the difference in the weighting factor *w*_{q} applied to the moisture at upper levels, which does not depend on height for the E99 norm and increases with height for the exergy norm according to Fig. 8.

The most striking feature, when using the exergy norm, is the very large increase by a factor of three (or > +200%, see Table 3) of the only observing system highly sensitive to atmospheric water vapor: the Microwave Humidity Sounder (MHS). According to Fig. 9, radiosonde observations (RAOBs) ranks first for the exergy norm, which may have important implications given that the operational radiosonde observing network is expensive to operate. These results suggest that radiosonde humidity sensors play an important role in the 24 h forecast accuracy, even more than MHS.

## 5. Conclusions

The main objective of this paper is to provide a general and more satisfactory method for combining thermodynamic variables of the atmosphere into a norm. There are several formulations for these norms currently in use for a wide variety of important applications, yet until now all have been derived using heuristic methods and approximations.

It is argued in this paper that such approximations can be avoided by instead considering the principles of fundamental physics more carefully. Specifically, the approach is to start with some general exergy functions, which are constructed by combining the first (enthalpy) and second (entropy) law of thermodynamics, leading to the available enthalpy function *a*_{m} derived in M93. This kind of exergy function is also based on the concept of relative entropy or Kullback distance, two equivalent concepts that are already used in many papers dealing with assimilation techniques.

The choice of the exergy (available enthalpy) squared norms provides not only the quadratic terms (*T*′)^{2}, *T* and *q*_{υ} vary with height in the same way as the RMS of analysis increments. This ensures an even weighting of all variables and all levels when computing the global norm. Such results are valid for both seasonal average periods and for a particular day.

The fact that the weights for the exergy norm for *T* and *q*_{υ} are close to the RMS of analysis increments is not straightforward. Indeed, if the observation system is radically changed, the increments could be very different, while the exergy-norm weights would not be modified. To better understand the complex links that can exist between fields as different as thermodynamics, information theory and data assimilation, it is possible to refer to papers cited in section 3 of Marquet and Dauhut (2018).

Inspired by previous studies by Kleeman (2002) and Majda et al. (2002), the paper of Xu (2007) examined the use of the relative entropy or Kullback-Leibler distance *K*(*x*||*y*) given by (11) “*to measure the information content of the pdf produced by an optimal analysis of observations (or compressed super-observations) with respect to a prior background pdf used by the analysis (…) where the background pdf can be always considered as an approximation of the analysis pdf.*” Xu showed that the integral form of the relative entropy *K*(*x*||*y*) “*is a quadratic form of the analysis increment vector weighted by* *yields an explicit formulation in which the signal part is given by the inner-product of the analysis increment vector weighted by the inverse of the background covariance matrix*” (

Since Xu (2007) demonstrated a close relationship between *K*(*x*||*y*) and the weighting factors *V*_{u}, *V*_{υ}, *V*_{υ}, *V*_{p}, *V*_{T}, and *V*_{q}, the next step is to use the close relationship shown by Procaccia and Levine (1976), Eriksson and Lindgren (1987), Eriksson et al. (1987), Karlsson (1990) and Honerkamp (1998) between *K*(*x*||*y*) and exergy functions, to foresee a direct link between the moist-air exergy defined in thermodynamics and the weighting factors used in data assimilation.

The new exergy (available enthalpy) squared norm may solve the main disadvantage of using the constant E99 moist *V* term stated in Rivière et al. (2009), namely that the weight for water is no longer proportional to the weight for temperature with the exergy formulation, leading to new results with the use of the *V*_{q} term.

A first usage of the exergy norm in the context of FSOI experiments has shown that it increases observation impact in a way similar to what has previously been described when going from a dry energy norm to a moist energy norm (e.g., H14). However, the enhancement of the impact is larger, since the exergy norm accounts more evenly for moisture forecast errors between the various atmospheric layers, whereas the moist energy norm penalizes the upper-tropospheric levels. The results are very similar among the various observing systems, however, with a noticeable difference for the MHS and RAOBs, for which the contributions are particularly enhanced with the exergy norm. This is in agreement with the known impact of microwave humidity sounders from direct observing system experiments (Karbou et al. 2010; Chambon et al. 2015). In consequence, it is expected that the various observing systems would be more fairly ranked through more balanced contributions between wind, temperature and water vapor forecast errors through the use of the exergy norm in FSOI experiments.

Another usage of the exergy norm has been shown by Borderies et al. (2019) to demonstrate the impact of airborne cloud radar reflectivity data assimilation.

The important point is that the analytical formulation of the exergy norm is not complicated. It is comparable in complexity to existing formulations (E99; MB07) and can be easily coded and used in operational systems, for moist singular vector and FSOI calculations as well as forecast verifications. The only new aspect is the need to take into account horizontal averages, or averages on each latitude circle, for the mean temperature and vapor content variables *N*_{T}, *N*_{υ}, *V*_{T}, and *V*_{q}.

## Acknowledgments

The definitions of the squared norm components *N*_{T}, *N*_{p}, and *N*_{υ} were obtained during the Pan-GCSS meeting in Athens, Greece, in May 2005. The results presented in this paper are thanks to Philippe Courtier’s initial encouragements, with numerous preliminary tests carried out between 2005 and 2018. The authors wish to thank the editor and the three reviewers for their comments, which helped to improve the manuscript.

## APPENDIX A

### List of Symbols and Acronyms

B_{p} | A dummy notation for a pressure norm |

APE | The global available potential energy (Lorenz) |

α | The specific mass of moist air (the density 1/ |

a_{e} | The moist specific available energy |

a_{h}, a_{m} | The dry and moist specific available enthalpies |

a_{T}, a_{p} | Temperature and pressure components of |

a_{υ} | The water component of |

c_{pd} | Specific heat of dry air (1004.7 J K |

c_{pv} | Specific heat of water vapor (1846.1 J K |

c_{l} | Specific heat of liquid water (4218 J K |

c_{i} | Specific heat of ice (2106 J K |

c_{p} | The specific heat at constant pressure for moist air, = |

δ | = |

e | The water vapor partial pressure |

e_{i} | The specific internal energy |

e_{r} | The water vapor reference partial pressure, with |

F, H | Dimensionless functions of |

g | Magnitude of Earth’s gravity (9.8065 m s |

The Lorenz stability parameter | |

A weight in the water component of MB07 norm | |

GCM | General circulation model |

h, H | Specific and global enthalpies |

H_{r} | A dummy-scale height (C87) |

k_{B} | The Boltzmann constant |

K | Kullback function, contrast, relative entropy |

L_{f} | = |

L_{υ} | = |

L_{s} | = |

L_{f}(T_{r}) | = 0.334 × 10 |

L_{υ}(T_{r}) | = 2.501 × 10 |

L_{s}(T_{r}) | = 2.835 × 10 |

m | A mass of moist air |

dm | The element of mass (= |

N | The global available enthalpy squared norms |

NWP | Numerical weather prediction |

ω_{ij} | The fractional coverage of the model grid box |

p | The pressure ( |

p_{s} | The surface pressure |

q | The specific content (e.g., |

Q_{r} | A dummy specific water content (C87) |

r | The mixing ratio (e.g., |

r_{0} | = |

ρ | Specific mass of moist air (= |

R_{d} | Dry-air gas constant (287.06 J K |

R_{υ} | Water vapor gas constant (461.52 J K |

R | Gas constant for moist air (= |

s | The specific entropy |

σ | The vertical coordinate of the model grid box |

Σ, dΣ | Global and element of horizontal surface of Earth |

T | The absolute temperature |

T_{r} | The reference zero Celsius temperature (273.15 K) |

U | The horizontal wind and its components ( |

U | The horizontal wind speed |

μ | The specific Gibbs’ function ( |

ϕ | The gravitational potential energy ( |

V | The variances of analysis errors |

V_{0} | A special variance of 2 J kg |

w_{q} | A relative weight in water components of norms |

x_{j}, y_{j} | The micro states that define the function |

Z | A dimensionless water vapor variable |

#### a. Lower indices (for h, s, p, μ, ρ, q, r, V, X, Y, Z):

r | Reference value (entropy, available enthalpy) |

d, υ | Dry-air and water vapor gases phases |

l, i | Liquid water and ice condensed phases |

sw, si | Saturating value (over liquid or ice) |

t | Total water value (vapor plus liquid plus ice) |

T, p, υ | Temperature, pressure, and water components |

T_{1}, p1 | Notations for pressure components ( |

q, q2 | Notations for water components ( |

1, 2 | Notations in separating laws |

i, j, k | Indices for longitude, latitude and altitude |

#### b. Upper indices/operator:

Departure terms from average values | |

Average values |

## APPENDIX B

### The Specific Moist-Air Available Enthalpy

*a*

_{m}is an exergy function defined in M93 [see Eq. (17), p. 574] as a sum of four partial moist available enthalpies for dry air (

*a*

_{m})

_{d}, water vapor (

*a*

_{m})

_{υ}, liquid water (

*a*

_{m})

_{l}, and ice (

*a*

_{m})

_{i}, leading to

*T*

_{r}is a constant reference pressure.

*c*

_{pd},

*c*

_{pv},

*c*

_{l},

*c*

_{i}) and gas constants (

*R*

_{d},

*R*

_{υ}) are all constant for the atmospheric range of temperature (from 180 to 320 K), leading to

*e*

_{r}is equal to the ice-vapor value

*e*

_{si}(

*T*

_{r}) for

*T*

_{r}< 0°C or to the liquid-vapor value

*e*

_{sw}(

*T*

_{r}) for

*T*

_{r}> zero Celsius. The moist available enthalpy (B1) is computed by including (B6)–(B10) in (B2)–(B5), yielding

*q*

_{l}and

*q*

_{i}are not neglected, but appear in the moist values of

*c*

_{p}and

*q*

_{d}= 1 −

*q*

_{υ}−

*q*

_{l}−

*q*

_{i}, anywhere else.

## APPENDIX C

### The Temperature Component of *a*_{m}

*a*

*a*

_{T}of the available enthalpy defined in Marquet (1991, hereafter M91) and M93 in terms of the function

*F*(

*X*) according to

*c*

_{p}is equal to

*q*

_{d}

*c*

_{pd}+

*q*

_{υ}

*c*

_{pv}+

*q*

_{l}

*c*

_{l}+

*q*

_{i}

*c*

_{i}and is not a constant, since it depends on varying specific contents of dry air and water species.

*F*(*X*) is positive and asymmetric with respect to *X* = 0, see Fig. C1. It is a quadratic-like function because *F*(*X*) ≈ *X*^{2}/2 for |*X*| < 0.3. This terminology “quadratic-like” corresponds to functions with Taylor series of the form: *X*^{2}/2 + *aX*^{3} + *bX*^{4} + …, where the quadratic term *X*^{2}/2 is the first-order approximation and where the other higher-order terms can be discarded. This approximation is typically valid for 210 K < *T* < 390 K if *T*_{r} = 300 K. *F*(*X*) = 0 only for *X* = 0, namely for *T* = *T*_{r}.

The two functions *F*(*X*) = *X* − ln(1 + *X*) and *X*^{2}/2 plotted for −1 > *X* > +2.5.

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

The two functions *F*(*X*) = *X* − ln(1 + *X*) and *X*^{2}/2 plotted for −1 > *X* > +2.5.

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

The two functions *F*(*X*) = *X* − ln(1 + *X*) and *X*^{2}/2 plotted for −1 > *X* > +2.5.

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

## APPENDIX D

### The Pressure Components of *a*_{m}

*a*

Terms in the second line of (B11) can be rearranged in order to compute the separate quadratic contributions due to total pressure *p* = *p*_{d} + *e* on the one hand, and to water species contents (*q*_{υ}, *q*_{l}, or *q*_{i}) on the other hand.

*p*=

*ρRT*,

*p*

_{d}=

*q*

_{d}

*ρR*

_{d}

*T*and

*e*=

*q*

_{υ}

*ρR*

_{υ}

*T*, respectively, leading to

*R*=

*q*

_{d}

*R*

_{d}+

*q*

_{υ}

*R*

_{υ}is not a constant since it varies with

*q*

_{d}and

*q*

_{υ}.

*q*

_{d}

*R*

_{d}and

*q*

_{υ}

*R*

_{υ}given by (D1) and (D2) can be inserted into (B11), yielding

*a*

_{p}defined by (D4), leading to the separation of

*a*

_{m}into

*a*

_{υ}.

*p*/

*p*

_{r}), since it is negative for

*p*<

*p*

_{r}. This apparent drawback was already mentioned in M91 and M93. However, it is possible to integrate by parts

*a*

_{p}in (D4) with respect to

*p*, leading to

*H*(

*X*) can be introduced by choosing the constant of integration

*C*= 1, yielding

*X*

_{p}is the dimensionless pressure control variable.

It is shown in Fig. D1 that *H*(*X*) is positive and asymmetric with respect to *X* = 0. It is a quadratic-like function because *H*(*X*) ≈ *X*^{2}/2 up to higher-order terms. *H*(*X*) = 0 only for *X* = 0, namely for *p* = *p*_{r}.

The two functions *H*(*X*) = (1 + *X*) ln(1 + *X*) − *X* and *X*^{2}/2 plotted for −1 < *X* < +2.5.

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

The two functions *H*(*X*) = (1 + *X*) ln(1 + *X*) − *X* and *X*^{2}/2 plotted for −1 < *X* < +2.5.

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

The two functions *H*(*X*) = (1 + *X*) ln(1 + *X*) − *X* and *X*^{2}/2 plotted for −1 < *X* < +2.5.

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

The constant reference pressure *p*_{r} can enter the derivative in (D7) and the term *p*_{r} *H*(*X*_{p}) is equal to the function *p*_{r} − *p* + *p* ln(*p*/*p*_{r}) ≈ (*p* − *p*_{r})^{2}/(2*p*_{r}) called “store of work for any layer under isothermal conditions” in Margules (1910) and studied in Eq. (Ia)′ page 505, the bottom of page 506 and the top of page 507 of this old paper.

## APPENDIX E

### The Water Components of *a*_{m}

*a*

The aim of this section is to show that *a*_{υ} given by (D5), which depends on the six pressures *p*, *p*_{r}, *p*_{d}, (*p*_{d})_{r}, *e*, and *e*_{r}, can be expressed in terms of the sole water mixing ratios *r*_{υ} = *q*_{υ}/*q*_{d} and *r*_{r}. In this way, *a*_{υ} will be interpreted as the water vapor component of *a*_{m}.

*e*/

*p*

_{d}=

*r*

_{υ}/

*r*

_{0}, where

*r*

_{0}≡

*R*

_{d}/

*R*

_{υ}= 0.622. The same result is valid for the reference state, leading to

*e*

_{r}/(

*p*

_{d})

_{r}=

*r*

_{r}/

*r*

_{0}and to a reference value for the mixing ratio given by

*T*

_{r}and

*p*

_{r}are known, because

*p*

_{r}= (

*p*

_{d})

_{r}+

*e*

_{r}and

*e*

_{r}(

*T*

_{r}) is the saturation pressure of water at

*T*

_{r}and (

*p*

_{d})

_{r}=

*p*

_{r}−

*e*

_{r}(

*T*

_{r}) can then be computed.

*e*/

*p*

_{d}=

*r*

_{υ}/

*r*

_{0}and

*e*

_{r}/(

*p*

_{d})

_{r}=

*r*

_{r}/

*r*

_{0}, leading to

*a*

_{υ}given by (D5) can then be written as

*a*

_{υ}has already been derived in the exergetic analysis of moist-air processes described (in German) in Szargut and Styrylska [1969, Eq. (10)] and recalled in Bejan [2016, Eq. (5.48), p. 207], though with different notations.

The bracketed terms in (E4) only depend on *r*_{υ} and on the two known reference values *T*_{r} and *p*_{r}, since the reference mixing ratio is *r*_{r} = *r*_{0} *e*_{r}(*T*_{r})/[*p*_{r} − *e*_{r}(*T*_{r})]. Therefore, *a*_{υ} will be called the water vapor component of *a*_{m}.

The impacts of *q*_{l} and *q*_{i} are not neglected up to this point, because the condensed water contents impact the gas constant *R*, which depends on *q*_{υ} and *q*_{d} = 1 − *q*_{υ} − *q*_{l} − *q*_{i}. Conversely, the bracketed terms in (E4), which generates the quadratic-like part of *a*_{υ}, do not depend on *q*_{l} or *q*_{i}. These results could not be expected and are just imposed by the exact computations.

*a*

_{υ}given by (E4) can be transformed into the sum of the two terms depending on the function

*H*, leading to

*F*or

*H*of the variable (

*r*

_{υ}−

*r*

_{r})/

*r*

_{r}.

The ratio |*X*_{υ}/*Y*_{υ}| = *r*_{0}/*r*_{r} = [*p*_{r} − *e*_{r}(*T*_{r})]/*e*_{r}(*T*_{r}) shown in Table E1 is computed for the set of reference values *T*_{r} and *p*_{r} used in Tables 1 and 2. The ratio is larger than 20 for *T*_{r} ≤ 300 K and *p*_{r} = 800 or 1000 hPa. This result justifies the name “large” and “small” given to *X*_{υ} and *Y*_{υ}, respectively.

The ratio |*X*_{υ}/*Y*_{υ}| = [*p*_{r} − *e*_{r}(*T*_{r})]/*e*_{r}(*T*_{r}) computed for several reference temperatures *T*_{r} and pressures *p*_{r}. See the Table 2 for values of *e*_{r}(*T*_{r}).

The higher temperature *T*_{r} = 350 K leads to small values of |*X*_{υ}/*Y*_{υ}|, which are close to unity, with an undefined (negative) ratio for 367.8 hPa. Values of *T*_{r} > 300 K are thus beyond the scope of the next definition for the water component of the exergy norm, where both *r*_{υ} and *r*_{r} are much lower than *r*_{0} ≈ 622 g kg^{−1} only for *T*_{r} ≤ 300 K, leading to *X*_{υ} ≈ (*r*_{υ} − *r*_{r})/*r*_{r} and *Y*_{υ} ≈ −(*r*_{υ} − *r*_{r})/*r*_{0}. The best candidate for a water dimensionless variable similar to *X*_{T} = (*T* − *T*_{r})/*T*_{r} is thus the large component *X*_{υ}.

## APPENDIX F

### Separating Properties of *F* and *H*

Previous results cannot be used as such by replacing the terms (*T* − *T*_{r})^{2}, (*p*_{s} − *p*_{r})^{2}, and (*r*_{υ} − *r*_{r})^{2} by the departure terms (*T*′)^{2}, *T*′, *T* − *T*_{r}, *p*_{s} − *p*_{r}, and *r*_{υ} − *r*_{r} cannot cancel for all vertical levels and for constant values of *T*_{r}, *p*_{r}, and *r*_{r}.

It is thus important to introduce the mean values *T*, *p*_{s}, and *r*_{υ} computed for a given circle of latitudes, or for a given pressure level, or for any other kind of average like those considered in Fig. F1. The eddy departure terms will then be defined in the usual way by

The separation of the flow into an uneven basic state (*x* term stands for the meteorological variables *T*, *p*, *Z*_{υ}, or *r*_{υ}, also *u* and *υ*.

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

The separation of the flow into an uneven basic state (*x* term stands for the meteorological variables *T*, *p*, *Z*_{υ}, or *r*_{υ}, also *u* and *υ*.

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

The separation of the flow into an uneven basic state (*x* term stands for the meteorological variables *T*, *p*, *Z*_{υ}, or *r*_{υ}, also *u* and *υ*.

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

Therefore, the aim is to express the available-enthalpy functions *a*_{T}, *a*_{p} and *a*_{υ} depending on *T* − *T*_{r}, *p*_{s} − *p*_{r}, and *r*_{υ} − *r*_{r} in terms of the “energies of the mean state,” which depend on

*X*is separated into a mean part

*X*

_{1}for which

*X*

_{2}for which

*F*(

*X*) in Marquet (1991, 2003), and the one valid for

*H*(

*X*) is shown in this appendix. For any variable written as

*X*=

*X*

_{1}+

*X*

_{2}+

*X*

_{1}

*X*

_{2,}the two properties

*X*

_{1}> −1 and

*X*

_{2}> −1, which means

*X*

_{1}+

*X*

_{2}+

*X*

_{1}

*X*

_{2}= (1 +

*X*

_{1})(1 +

*X*

_{2}) − 1 > −1. The flow

*X*is then separated into the same mean and eddy parts used to derived (F2) and with

The physical consequence of (F2), (F5), and (F6) is the appearance of exact self-similarity properties verified by the total, mean and eddy parts of the flow: quadratic *F* or *H* functions generate quadratic *F* or *H* functions for the mean and the eddy parts of the flow. More precisely, the quadratic approximation of (F5) will allow computations of (*T* − *T*_{r})^{2}/2 in terms of *T*′)^{2}/2, with similar results derived from (F6) and valid for surface pressure and water vapor mixing ratio.

## APPENDIX G

### Mean and Eddy Components of *a*_{T}, *a*_{p}, *a*_{υ}

*a*

*a*

*a*

*a*

_{T}given by (C1) can be computed by replacing

*X*

_{T}=

*T*/

*T*

_{r}− 1 in (C2) by

*X*

_{1}and

*X*

_{2}in (F5), respectively. It is then assumed that

*c*

_{p}≈

*c*

_{pd}and

*F*(

*X*) ≈

*X*

^{2}/2, leading to

*T*

_{r}. The integral of the second quadratic term represents the “available enthalpy” of the perturbations

*T*′ of the actual state

*T*with respect to the mean state

*a*

_{p}given by (D7) is computed by assuming that

*R*≈

*R*

_{d}, leading to

*X*

_{p}= −1 for

*p*= 0 and

*H*(−1) = 1, leading to a constant value

*R*

_{d}

*T*

_{r}

*p*

_{r}/

*g*that will not enter the definition of the squared norm component for pressure. The aim is thus to compute mean and eddy components of

*X*

_{1}and

*X*

_{2}in (F6), respectively. The separating property (F6) can then be applied to (G7), leading to

*H*(

*X*) ≈

*X*

^{2}/2.

*B*

_{p}in the rhs of (G10) represents the unavailable enthalpy of the mean state

*p*

_{r}. The second quadratic term represents the available enthalpy of the perturbations

*p*

_{s}with respect to the mean state

*X*

_{υ}| ≫ |

*Y*

_{υ}|. This result is used together with the assumptions

*R*≈

*R*

_{d},

*r*

_{0}≫

*r*

_{υ}, and

*r*

_{0}≫

*r*

_{r}, leading to

*Z*

_{υ}≈

*r*

_{υ}/

*r*

_{0},

*Z*

_{r}≈

*r*

_{r}/

*r*

_{0}, and

*X*

_{υ}≈

*r*

_{υ}/

*r*

_{r}− 1, to approximate the (isobaric, horizontal or uneven) surface mean value of

*a*

_{υ}by

*R*

_{υ}=

*R*

_{d}/

*r*

_{0}has been used.

*X*

_{1}and

*X*

_{2}in (F6), respectively, with the property

*H*(

*X*) ≈

*X*

^{2}/2, leading to

*r*

_{r}. The integral of the second quadratic term represents the available enthalpy of the perturbations

*r*

_{υ}with respect to the mean state

If the exact moist value *R* = (1 − *q*_{t}) *R*_{d} + *q*_{υ} *R*_{υ} was not approximated by *R*_{d} in (E6), leading to *R*_{d}/*r*_{0} = *R*_{υ} in (G14)–(G17), then a factor *R*_{υ} in (G18), but leading to small terms in comparison with the definition (G18) for *N*_{υ}.

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^{1}

The original notations of Shannon and Kullback using *p*(*p*_{j}) and *q*(*q*_{j}) are replaced here to avoid confusion with the pressure *p* and the specific water content quantities *q*_{t}, *q*_{υ}, *q*_{l}, *q*_{i}.