## 1. Introduction

Lorenz (1955), building on the work of Margules (1910), formulated the concept of available potential energy (APE) as that portion of the total potential energy, that is, the sum of the gravitational and internal energy in hydrostatic balance, that is available for conversion into kinetic energy. Since then the concept has been a cornerstone of large-scale dynamic meteorology and the theory of the general circulation (e.g., Lorenz 1967; Peixoto and Oort 1991). The Lorenz formulation is a Lagrangian one that determines the APE for an isentropic rearrangement of air parcels. For an atmosphere that is statically stable, the entropy increases with height. Thus, an isentropic vertical rearrangement of the parcels is excluded and the theory only assesses the available energy associated with the leveling of isentropic layers toward geopotential surfaces. This feature is an ideal means to assess the energy available for conversion to kinetic energy by baroclinic instability processes. However, this feature precludes the theory from evaluating the energy available for a purely vertical (e.g., convective) rearrangement. The theory requires the specification of a reference state. For example, if the APE is defined on isobaric coordinates, a reference (potential) temperature must be assigned. The specification of the reference temperature remains an unsolved problem in the theory (e.g., Dutton and Johnson 1967; Lorenz 1979; Pauluis 2007). [Randall and Wang (1992) advance a numerical, parcel-moving algorithm to find the temperature structure for their generalized convective available potential energy (GCAPE) that is applied to a single atmospheric sounding.]

Eulerian formulations of the available energy have appeared in several formulations: entropic energy (Dutton 1973; Livezey and Dutton 1976), static exergy (Karlsson 1990), extended exergy (Kucharski 1997), and available energy (Bannon 2005). These formulations differ slightly in their base state and their treatment of water vapor and hydrometeors. They all require the specification of a reference temperature. Dutton (1973) uses a reference temperature defined from the total potential energy. Karlsson (1990) uses a reference temperature that minimizes the entropy difference between the atmosphere and its reference atmosphere. Kucharski (1997) uses a reference temperature profile based on the horizontally averaged density field.

The present work reexamines and refines the formulation of the Eulerian atmospheric available energy (AE) of Bannon (2005) that defines the available energy as a generalized Gibbs function between the atmosphere and an isothermal reference atmosphere at the temperature *T _{r}*. The reference atmosphere is in thermal and hydrostatic equilibrium (Fig. 1) and, hence, is dynamically and convectively “dead.” It has the same mass per unit area of dry air and water as the atmosphere. The reference temperature is determined uniquely by minimizing this function. Section 2 presents a thermodynamic proof that the maximum kinetic energy that can be extracted from the total energy is specified by the minimum of the generalized Gibbs function. Minimization is defined to occur at the isothermal reference temperature

*T*

_{0}. The outline of the proof follows but generalizes that in Reif (1965, section 8.3). Independently Karlsson (1990) suggested this minimization of his exergy formulation but did not pursue it. Section 3 describes the construct of the equilibrium atmosphere and the need for the careful treatment of the hydrometeors. Section 4 derives the governing equation for the evolution of the available energetics of a flow. The sources and sinks of AE are quantified as well as the partitioning of the AE into available potential and available elastic components. The positive definite nature of the AE is demonstrated along with its relation to the exergy approach of engineering thermodynamics (Bejan 1997). Section 5 presents some applications of AE to the standard atmosphere, idealized baroclinic zones, and observed moist soundings. It is demonstrated that the available energy shares properties with the Lorenz APE as well as convective available potential energy (CAPE). The available energy is partitioned into available baroclinic energy (ABE) and available convective energy (ACE) components. Section 6 estimates the available energetics of the general circulation.

## 2. Thermodynamic derivation of the atmospheric available energy

*A*and a large multicomponent reservoir, denoted with the subscripts

*A*and res, respectively. The system is the atmosphere in thermal, mechanical, and diffusive contact with the reservoir that is a motionless reference atmosphere (see Fig. 1). State variables of the reference atmosphere are denoted for brevity with a subscript

*r*. By the second law of thermodynamics, the entropy

*S*of this universe tends to increase:where Δ denotes a finite change between a final and initial state. For example,

*u*is the specific internal energy,

*T*is the temperature,

*s*is the specific entropy,

*p*is the pressure, and

*α*is the specific volume. Here

*μ*is the specific chemical potential of the

_{j}*j*th component of mass

*m*and

_{j}*j*th component (i.e., the mass of the

*j*th component per total mass). The summation convention is assumed for repeated indices. A finite temporal change in the internal energy of the whole reservoir iswhere

*A*. Here

*j*th component transferred out of the reservoir into the system

*A*. The first law applied to the reference atmosphere iswhere

*A*, including that due to mass exchange. The two relations (2.3) and (2.4) combine to express the temporal entropy change of the reservoir asThen the total temporal entropy change isSolving this relation for the thermal energy transfer

*A*iswhere

*A*on the reservoir with pressure

*p*and

_{r}*A*. In particular the additional work can represent the energy transfer into the kinetic energy of

*A*. Eliminating the thermal energy transfer

*G*of the atmosphere

_{A}*A*at the reference temperature, pressure, and chemical potential of the reservoir. Then the total entropy change isThis inequality implies that any additional work

*W** done by

*A*requires a decrease in its Gibbs function with time:If

*W** = 0, then

*A*can obtain its reference state and define its Gibbs function in that state as

*δ*defines a finite difference between

*A*and its reference state. In contrast,

*A*is not necessarily in the equilibrium state,

*A*will be in equilibrium with

*T*that minimizes

_{r}*T*=

_{r}*T*

_{0}, the change in the Gibbs function is a minimum

*A*, then the inequality (2.10) becomesThe maximum possible increase in kinetic energy is that for the reversible case

*A*,It proves convenient to rewrite this expression in terms of the specific enthalpy

**x**= (

*x*,

*y*,

*z*) and time

*t*but the equilibrium entropy, pressure, and chemical potentials are only functions of height

*z*. Again, the symbol

*δ*defines a finite departure of the atmosphere from its equilibrium state. For example,

## 3. Specification of the equilibrium state

*T*

_{0}. Then, the dry air and vapor pressures are, in a Cartesian geometry,where the asterisks denote a surface (

*z*= 0) value and the scale heights are

*H*

_{s}*= R*

_{d}T_{0}/

*g*and

*H*

_{sυ}*= R*

_{υ}T_{0}/

*g*(details of the thermodynamic formulations and the values of physical constants are summarized in appendix A). In order that the equilibrium state has the same mass per unit area of dry air

*M*and water vapor

_{d}*M*as atmosphere

_{υ}*A*, the surface pressures arewhere the exponential factor accounts for the finite height

*z*

_{top}of the atmosphere (Fig. 1). If topography is present, (3.2) is modified in a straightforward manner to conserve mass between

*A*and its equilibrium atmosphere. If the surface vapor pressure

*T*

_{0}, then

*P*of water per unit area has precipitated into the water reservoir (Fig. 1),Then the amount of water vapor in the equilibrium state becomes

*j*= 1 and 2 refer to the dry air and water vapor, respectively, and the subscripts

*j*= 3 and 4 refer to liquid water and ice. For a gas we havewhere the quantities with subscript

*c*are arbitrary constants that do not affect the computations. Because of the exponential decay of the gas pressures with height (3.1), the chemical potentials of dry air and water vapor decrease linearly with height

This asymmetry is physically appropriate for the equilibrium atmosphere. The criterion (Gibbs 1873, 1874, p. 146) for equilibrium of a system in a gravitational field is that the total chemical potential (i.e., the sum of the intrinsic chemical potential *μ* and the geopotential *gz*) is constant. The expression (3.7) for the dry air and water vapor satisfies this criterion. In contrast, the total potentials for the hydrometeors are not constant and these components are not in equilibrium. This disequilibrium is physically correct because all hydrometeors have nonzero terminal fall speeds and will eventually settle out.

*z >*0 produces no superfluous accounting changes in the entropy and potential energy budgets. The derivation of the next section uses (3.8). Appendix B shows that the consequences of omitting the geopotential in (3.8) lead to an equivalent result.

## 4. Available energetics

This section uses a fluid dynamical approach to generalize the thermodynamic derivation of section 2 to include diabatic and frictional processes for an open atmosphere. It also quantifies the sources and sinks of the available energy and resolves issues related to the asymmetry in the chemical potentials discussed in section 3.

### a. General derivation of the governing equation

**v**iswhere the material derivative isand

*m*and

_{j}**v**

_{j}are the mass and velocity of the

*j*th component. Then the continuity equation iswhere

*ρ*is the total density. For the hydrostatic equilibrium state, the force balance iswhere Φ is the geopotential. The Gibbs relation in the form

*dh = Tds + αdp + μ*

_{j}*dχ*for the equilibrium state implies that the term in square brackets in (4.2) vanishes. Then, using (4.3) and (4.4), (4.2) becomesThe last term vanishes because the material derivatives of the chemical potentials only contain a gravity contribution and the sum of the differences in concentrations vanish:This result is a consequence of (3.8) that includes a gravity component to the chemical potentials of the hydrometeors. This assumption is addressed further in appendix B. The specific kinetic energy, ke, and enthalpy equations arewhere

_{j}*τ*is the viscous stress tensor. Then (4.5), (4.6), and (4.7) sum to give

*ρ*

_{j}is the density of the

*j*th component, whose velocity relative to the mass-weighted average velocity

**v**is

*h*relative to the mean velocity). Here

_{j}*e*is the rate of strain tensor. Then, the available energetics equation becomesThe internal sources of available energy are given by the last term in (4.11), whereor, using the definition of the chemical potentials,For example, in the case of a phase change between water vapor and liquid water, we havewhere

_{ij}*H*is the relative humidity. Thus, inclusion of the geopotential term in the equilibrium chemical potential for the hydrometeors (3.8) correctly handles entropy production during phase changes.

*j*th component is

### b. Partitioning and linearization of the available energy

The available energy (4.1) can be partitioned into the sum of the available energies of each component. Using relations of the form

*j*= 1, 2) may be partitioned (e.g., Bannon 2005) into available potential and available elastic contributions:whereandThe specific entropy is

*j*= 3 or 4) iswhere the geopotential

*κ*= 0.

The preceding formulas emphasize the enthalpic nature of the available energy. Their definitions in terms of normalized departures are computationally advantageous to ensure positive definite calculations as well as providing easy derivations of their quadratic nature.

## 5. Applications

The available energy formalism is applied to several atmospheres to determine their equilibrium temperature *T*_{0} and associated available energy.

### a. Standard atmosphere

The case of the standard atmosphere in Fig. 2 is an example of the minimization of the atmospheric Gibbs function *δG _{A}*. This hydrostatic atmosphere extends to a height of 25 km with a surface temperature and pressure of 288.15 K and 1013.25 hPa. The lapse rates are 6.5, 0, −1.0, and −2.8 K km

^{−1}for

*z*< 11, 20, 32, and 50 km, respectively. The potential and elastic contributions are defined by (4.17) and (4.18). The temperature profile is plotted in Fig. 3a, where it is compared to its equilibrium temperature

*T*

_{0}= 251.95 K. The total energy (TE) is slightly greater than that of its equilibrium atmosphere for both the 25- and 50-km-deep cases (Table 1). The internal energy (IE =

*U*) differences between the atmosphere and its equilibrium are small and change sign between the two cases. For both, the potential energy (PE) is less for the equilibrium atmosphere. This reduction in PE is consistent with a lower center of gravity associated with the lower temperature of the equilibrium atmosphere in the lower troposphere (Fig. 3a). Because of the finite vertical extent of each atmosphere, the ratio of the potential to internal energy is less than that of

*R/c*= 40%. Thus, the total energy is only approximately the enthalpy (cf. Lorenz 1955).

_{υa}Traditional energetics of two standard atmospheres of different depths. Values for the equilibrium atmosphere are in parentheses. The total energy is the sum of the internal and potential energies

The available energy (AE) for both atmospheres resides primarily (Table 2) in the available potential energy (APE) with about 10% in the available elastic energy (AEE). For the 25 km deep atmosphere, the available energy of 11.45 MJ m^{−2} is much less than its total energy (Table 1) of 2.4733 GJ m^{−2}. Thus, only 0.46% is available for conversion into kinetic energy; 0.48% for the 50 km deep atmosphere. In contrast the available energy agrees well with the difference in total energy

Available energetics of the two standard atmospheres.

The vertical distribution of the available energy (Fig. 3b) is bimodal with a maximum at the surface, a secondary maximum at the tropopause, and a midtropospheric minimum near where the equilibrium temperature and the sounding (Fig. 3a) intersect. The secondary peak lies at the height of the change in the lapse rate. The elastic energy is significant in the stratosphere and its contribution dominates that of the potential energy above 20 km. This behavior holds for all of the cases analyzed and reflects that the elastic energy (4.18) is inversely proportional to the square of the pressure field.

### b. Idealized baroclinic zones

The idealized baroclinic zones are based on a generalized compressible Eady base state (appendix C). The available energetics is summarized in Table 3 as a function of the total meridional temperature gradient *T*_{0}) decreases slightly. For each case, the difference in total energy between the atmosphere and its equilibrium state is the available energy.

Available energetics of the idealized baroclinic zone as a function of horizontal temperature gradient

The spatial variation of the available energy (Fig. 4a) exhibits a bimodal variation with larger values where the difference between the temperature and the equilibrium temperature is larger. The dominance of the temperature variation, rather than pressure variation, reflects the dominance of the available potential energy contribution (4.17) to the total available energy. The efficiency factor

The effect of a surface pressure gradient associated with a mean geostrophic wind is small (Table 4) with a slight increase in available energy as the wind transitions from westerly to easterly. Henceforth, the surface wind is set to zero.

Available energetics of the idealized baroclinic zone with ^{−1}.

The effect of sloping topography on the available energy is readily included in this Eulerian formulation. Table 5 provides a comparison for topography sloping linearly upward toward the pole (henceforth poleward) or toward the equator (henceforth equatorward) to reach a maximum height of 3 km (appendix C). The poleward case contains more available energy. This result is consistent with the linear Eady model analysis of Mechoso (1980) that indicates greater growth rates for unstable baroclinic waves for terrain sloping with the isentropes.

Available energetics of the idealized baroclinic zones over topography sloping upward toward the pole or toward the equator. The horizontal temperature gradient is

The effect of the inclusion of a moist boundary layer (appendix C) on the available energy is summarized in Table 6. The vapor pressure has a surface relative humidity of 70% and decays vertically with a scale height of 3 km. Inclusion of moisture has increased the equilibrium temperature by ~5 K to 265.12 K and increased all available energy components of the dry air. The vapor contributes to the available energy primarily through its elastic component. The distribution of the available energy and the efficiency factor (not shown) is similar to that of Fig. 4 for the dry case. The major effect of the moisture is through the increased equilibrium temperature that moves the zero efficiency isopleth downward. Figure 5 illustrates the effects of water vapor on the minimization of the Gibbs function. For reference temperatures below about 270 K, the reference atmosphere contains an ice reservoir of tens of kilograms per square meter of water. The enthalpy change associated with this phase transformation increases the Gibbs function curve at those temperatures and moves the temperature minimum toward warmer temperatures. The equilibrium atmosphere is saturated with an ice reservoir that is 1.29 cm thick due to precipitation (3.3). The ice contributes the major amount of the difference in total energy between the atmosphere and its equilibrium by its enthalpy of deposition (Table 6). The vertical distributions of the available energies (not shown) are similar to those presented in section 5d, which analyzes observed moist atmospheric soundings.

Available energetics of the moist idealized baroclinic zone with *T*_{0} = 265.12 K with a saturated surface vapor pressure of ^{−2} to the total energy difference.

### c. Comparison with the available potential energy of Lorenz

*vertically*to its state of lowest total energy. A subsequent adjustment of the system then occurs

*laterally*among the columns to the equilibrium temperature

Available baroclinic energetics of the idealized baroclinic zones with horizontal temperature gradient

### d. Individual moist soundings

A real data case is afforded by the sounding from the Second Verification of the Origins of Rotation in Tornadoes Experiment [VORTEX2; 2155 UTC 5 June 2009, National Severe Storms Laboratory (NSSL1)] presented in Fig. 6. The sounding exhibits a large potential for deep convection for parcels ascending above about 700 hPa along the 70°C saturated adiabat. The convective available potential energy (CAPE) evaluated from 700 to 170 hPa is 2370 J kg^{−1}. Assuming hydrostatic balance, the available energy may be analyzed in pressure coordinates. The minimization of the atmospheric Gibbs function (not shown) is similar to that depicted in Fig. 5. Table 8 and Fig. 7 present the results of an AE analysis of the profile. The dry air AE (Fig. 7a) exhibits a bimodal distribution with the potential dominating the elastic contribution. In contrast, the water vapor AE (Table 8) is dominated by the elastic contribution. This result implies that the vapor pressure perturbations dominate the potential temperature perturbations (section 4b). The water vapor AE (Fig. 7b) is also bimodal but with a thick 200-hPa layer of near-zero AE. This minimum reflects the impact of the low dewpoints between 700 and 500 hPa in the sounding (Fig. 6). The AE may be compared to the convective available potential energy by dividing by the total mass of the sounding *M*_{0} = 6.845 × 10^{4} kg m^{−2}, yielding a CAPE of 1541 J kg^{−1}.

Available energetics of the VORTEX2 sounding. The equilibrium temperature is *T*_{0} = 262.1 K, and a surface vapor pressure ^{−5} m thick. The total AE divided by the mass of the sounding yields a CAPE of 1541 J kg^{−1}.

A second case is phase III of the Global Atmospheric Research Program Atlantic Tropical Experiment (GATE) sounding analyzed by Randall and Wang (1992; see their Table 1 for the “given sounding”). This tropical sounding (Fig. 8) is relatively moist with the temperature profile close to the saturated adiabat of 70°C. This structure suggests a small value of CAPE. In contrast, the AE analysis of Table 9 and Fig. 9 indicates that the sounding is a potentially large source of kinetic energy. The general features of the AE profiles are consistent with those for the extratropical VORTEX2 sounding. These include a bimodal vertical distribution, a dominant potential contribution by the dry air, and a dominant elastic contribution by the water vapor. With a total mass of *M*_{0} = 9.135 × 10^{4} kg m^{−2}, the bulk CAPE is 2212 J kg^{−1}. In contrast, Randall and Wang estimate their GCAPE to be 11 J kg^{−1}.

Available energetics of the GATE Phase III sounding. The equilibrium temperature is *T*_{0} = 271.4 K with ^{−5} m thick. The total AE divided by the mass of the sounding yields a CAPE of 2212 J kg^{−1}.

## 6. Global available energy and the general circulation

**n**is the unit outward normal,where the rate of conversion of available energy into kinetic isand the integral is over the volume of the atmosphere. This conversion term indicates that flows down the pressure and geopotential gradients are kinetic energy producing. The rate of dissipation of kinetic energy isand includes both interior and surface contributions. Here

*A*is the surface area of the earth. In a steady state, the conversion to KE balances the dissipation of KE.

_{e}Quantification requires the determination of the equilibrium temperature *T*_{0} for the atmosphere. Analysis of the 25-km-deep standard atmosphere with a 70% surface relative humidity and 3 km water vapor scale height yields an equilibrium temperature of 256 K with an available energy of 14 MJ m^{−2}. A more definitive determination requires analysis of a global dataset. Such a calculation is beyond the scope of the current investigation. For definiteness in the following discussion, we take *T*_{0} = 255 K, the planetary/effective temperature.

### a. Dissipation of kinetic energy

The dissipation term (6.3) includes three distinct processes: viscous dissipation of kinetic energy within the atmosphere and the surface flux of kinetic energy out of the atmosphere by the surface wind stresses and by the precipitation of hydrometeors.

#### 1) Internal dissipation

^{−2}globally. For the tropics Pauluis et al. (2000) estimate the turbulent cascade by convection to be 1 W m

^{−2}and the hydrometeor dissipation to be 2–4 W m

^{−2}. Using satellite data, Pauluis and Dias (2012) estimate the dissipation to be 1.8 W m

^{−2}. In the extratropics there is a reduction in precipitation and a lowering of the mean height of hydrometeor formation. Globally we take the hydrometeor dissipation to be 1–3 W m

^{−2}.

#### 2) Kinetic energy loss due to precipitation

*P*is the precipitation rate,

*h*denotes a hydrometeor. The overbar denotes a representative mean. For a fall speed of 1 m s

^{−1}and a precipitation rate of 1 m yr

^{−1}(

#### 3) Kinetic energy loss due to wind stress

With surface wind speeds exceeding that of the underlying surface, there is a downward transport of kinetic energy out of the atmosphere given by ^{−1} and bulk aerodynamic drag formulations with a drag coefficient of 10^{−3}, the downward transport is

Thus, the total dissipation rate is about 5–7 W m^{−2}. Table 10 summarizes the discussion.

Dissipation of kinetic energy. In a steady state the total dissipation rate equals the rate of conversion of available energy to kinetic energy.

### b. Generation of available energy

The generation rate (6.5) includes surface fluxes of exergy into the atmosphere and interior diabatic processes. The interior terms are weighted by the efficiency factor *T*_{0}. A contour plot (not shown) of the efficiency factor would have a zero contour along the *T*_{0} = 255 K isotherm that runs from the middle tropical troposphere downward toward the poles (with additional zero isopleths in the upper stratosphere and lower mesosphere). For the same magnitude temperature difference

#### 1) Radiation

*H*= 8 km and

_{s}^{−1}. This mean efficiency varies from

The effect of clouds on the radiative generation could be significant. Low-level cloud bases, where the efficiency is positive generally, absorb longwave radiation emitted from the surface, leading to a positive AE generation. Emission of longwave radiation to space from high-level cloud tops, where the efficiency is negative, will also generate positive AE.

#### 2) Dissipation

#### 3) Precipitation exergy flux

*j*= 3 or 4) isBecause exergy is a positive definite quantity, precipitation always carries exergy out of the atmosphere and thus acts as a sink of AE. Taking

#### 4) Evaporation exergy flux

*H*is the relative humidity. This input is composed of three contributions. The first cancels the precipitation sink (to first order) but without the geopotential term:The second is due to the enthalpy of evaporation:The third isfor

*H*= 50% and

*H*

_{0}= 100%. Then the total evaporative generation is

#### 5) Dry air exergy flux

^{−2}(Trenberth et al. 2009), the mean air – surface temperature difference is

*C*= 10

_{H}^{−3}. By analogy, the exergy flux is defined by

*C*

_{ex}= 10

^{−3},

### c. Irreversible entropy production

The last term in (6.4) represents irreversible entropy production processes (e.g., the subcloud evaporation of raindrops). It is positive definite and hence is a sink of available energy. The term ^{−2} that includes a surface contribution due to irreversible evaporation. The latter process is contained here in the third contribution of the surface evaporation (6.17). Thus, we take

Table 11 summarizes the discussion on the available energy generation budget.

Generation of available energy. In a steady state, the total rate of generation of available energy less the lost work equals the rate of conversion of available energy to kinetic energy.

## 7. Conclusions

Minimization of an atmospheric Gibbs function has been utilized to determine the temperature

The analysis of the available energy cycle of the general circulation in section 6 is summarized in Fig. 10 and quantified in Table 12. A measure of the efficiency

Estimates of the available energy cycle of the global atmosphere.

The author benefited from suggestions from colleagues John A. Dutton, Dennis Lamb, Sukyoung Lee, Paul M. Markowski, and Raymond G. Najjar. Paul M. Markowski provided the VORTEX2 sounding. The comments of Olivier Pauluis and an anonymous reviewer also helped improve the manuscript.

# APPENDIX A

## Thermodynamics

The thermodynamic formulations are standard (e.g., Bohren and Albrecht 1998). The dry air and water vapor are treated as ideal gases with constant specific heats. The ice and liquid water are taken to be incompressible with constant specific heats. The enthalpies of phase change vary linearly with temperature following Kirchhoff’s law. This dependence is included in the Clausius–Clapeyron equations for the equilibrium vapor pressures. Table A1 summarizes the notation and values of various constants.

Physical constants.

# APPENDIX B

## Asymmetric Chemical Potentials

*–gz*in (3.8). Then (4.8) retains the last term in (4.5) to becomeThe last term in (B.1) arises from the asymmetry in the potentials. It iswhere the chemical potentials for liquid and solid water (

*j*= 3, 4) are now constants, while those for dry air and water vapor (

*j*= 1, 2) are each a constant −

*gz*. In addition the concentrations for the atmosphere and for the equilibrium atmosphere independently sum to unity with

*H*is the relative humidity. Thus, inclusion of the geopotential term in the equilibrium chemical potentials (3.8) handles entropy production during phase changes appropriately.

# APPENDIX C

## Idealized Baroclinic Zones

Idealized dry baroclinic zones are constructed with inspiration from the model of Eady (1949) but for a compressible atmosphere. The geometry is Cartesian with (*y, z*) indicating meridional and vertical coordinates. The central sounding is that for the standard atmosphere as are the lapse rates. The surface temperature varies linearly over the domain with a total variation of *U*_{0} with constant Coriolis parameter of 10^{−4} s^{−1}. The pressure and density are determined from the hydrostatic relation and the ideal gas law. This construction is applied over a 6 × 10^{3} km × 12 km grid *y*_{0} = 3 × 10^{3} km and *z*_{top} = 12 km. The resolution is 100 m in the vertical and 100 km in the horizontal direction.

The topography is *z*_{0} over the 2*y*_{0} domain. The plus sign indicates topography that is a maximum at the high latitude *y = y*_{0} (henceforth, the poleward case) and the minus sign one that is maximum at the low latitude *y = −y*_{0} (henceforth, the equatorward case).

The effects of moisture on the baroclinic isolated jet are assessed by including a water vapor field that has a relative humidity of 70% at the surface with a scale height of 3 km. Then, the warmer lower latitudes will have greater moisture content than those at higher latitudes.

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