1. Introduction

Isentropic analysis is a technique that was originally introduced in the late 1930s to study synoptic-scale motions (Rossby 1937; Namias 1939; Namias and Stone 1940). It relies on analyzing atmospheric motions on surfaces of constant potential temperature. While not exactly conserved for atmospheric flows, potential temperature varies slowly when compared to its spatial location or to other thermodynamic variables such as pressure or temperature. This makes it possible to track air parcels in situations where reconstructing their trajectories is impossible because of a lack of observations. The original framework has been greatly extended over the years to further account for the role of rotation and vorticity (Hoskins et al. 1985), to investigate the global circulation (Johnson 1989; Held and Schneider 1999; Pauluis et al. 2008, 2010), and more recently to study moist convection (Pauluis and Mrowiec 2013; Mrowiec et al. 2016).

The name isentropic analysis itself is somewhat of a misnomer. Indeed, while potential temperature is an adiabatic invariant, it is not a good measure of entropy. More specifically, there is not a one-to-one correspondence between surfaces of constant potential temperature and surfaces of constant entropy. Even though isoplethic analysis may be a more accurate terminology, the usage in our field has been that isentropic analysis refers to the study of atmospheric motions on isosurfaces of an adiabatic invariant. Different choices for the adiabatic invariant can lead to strikingly different outcomes. For example, Pauluis et al. (2008) show that the meridional mass transport on surfaces of constant equivalent potential temperature is twice as large as the mass transport on surfaces of constant potential temperature. Any study using isentropic analysis thus requires a clear specification of the variable used to define isopleths.

Adiabatic invariants are thermodynamic quantities that are conserved for reversible adiabatic transformations. A peculiar problem in atmospheric sciences lies in that, over time, many adiabatic invariants have been introduced, most often as a potential temperature of some sort, such as equivalent, virtual, liquid, and entropy potential temperatures. Worse, different definitions have been used over time under the same terminology. For instance, the entropy temperature of Hauf and Holler (1987) is the same as the equivalent potential temperature of Emanuel (1994) except for its treatment of the ice phase, but it is quite different from the entropy temperature of Marquet (2011). While such an overabundance of adiabatic invariants may be daunting, only two adiabatic invariants are necessary to determine the evolution of the flow.

Marquet (2017, hereafter M17) draws attention to some of the problems that arise from the many potential temperatures. While this is a fair warning, M17 goes further and argues that the isentropic analysis should be limited to the use of the entropy potential temperature defined in Marquet (2011). This is contrary to the original idea of Rossby (1937) and to the longstanding practice in our field. It is also very limiting; much insight can be gained from studying a circulation in different coordinate systems in the same way as looking as a sculpture under different angles reveals more about the artist’s intention. Several methods have been introduced to explain how the mean circulation depends on the choice of coordinates, such as the use of joint distributions (Pauluis et al. 2010), reconstruction techniques that relate the mean circulation in different coordinate systems (Juckes 2001; Laliberte et al. 2012; Pauluis et al. 2011), and a methodology to extract thermodynamic cycles from the isentropic analysis (Pauluis 2016). While M17 draws attention to the impacts of the coordinate system on the resulting circulation, there is no mention of any of these previous discussions. This paper aims to correct this omission and to explain how the choice of a coordinate system impacts the mean circulation.

A second issue raised by M17 is related to the estimate of the work done by an atmospheric flow. In several papers, such as Emanuel (1994) and Pauluis (2011), the work performed by an atmospheric flow is computed on the basis of the “work per unit mass of dry air” transported. M17 takes issue with the choice and argues that the work should be computed on a “per unit mass of moist air” basis. The difference between the two frameworks is substantial but can be easily explained from the fact that air parcels moving in the atmosphere lose water mass through rainfall. In particular, computations on the basis of per unit mass of moist air as in M17 severely underestimate the actual work performed by atmospheric motions.

This paper is organized as follows. Section 2 reviews adiabatic invariants for moist air and their connection to entropy. Section 3 discusses the impacts of the coordinate system on isentropic analysis in hurricane simulations. Section 4 explains the differences between computing the work in a thermodynamic cycle on the basis of per unit mass of dry air or on the basis of per unit mass of moist air and argues that the former provides the correct estimate for the work done in a precipitating atmosphere.

2. Entropy, potential temperatures, and adiabatic invariants in a moist atmosphere

An adiabatic invariant is a thermodynamic state variable that is conserved for reversible, adiabatic, and closed transformations. It should be emphasized here that the concept of an adiabatic invariant only applies to closed transformations—that is, transformations in which the system does not exchange mass with the outside world. By virtue of the second law of thermodynamics, entropy is an adiabatic invariant. Assuming that moist air behaves as a perfect mixture, its specific entropy can be written as the sum of its components:

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where , , , and are the specific entropies of dry air, water vapor, liquid water, and ice and , , , and are the specific humidities for water vapor, liquid water, ice, and total water. These specific entropies are computed by using the Gibbs relationship and by taking advantage of Nernst’s law that states that the entropy of a crystalline structure converges to 0 as the absolute temperature goes to 0 K (Hauf and Holler 1987; Marquet 2011). This will be referred hereto as the absolute entropy.

In atmospheric science, it has however been a common practice to define entropy by using different reference states. We refer to such alternative definitions of entropy as relative entropy, as it depends on the choice of a reference state. Moist entropy is the relative entropy of moist air in which the reference states are liquid water at the freezing point and dry air at the same temperature and pressure of 10⁵ Pa. The difference between the moist entropy and the absolute entropy is given as

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Here, is the specific entropy of dry air in the reference state (pressure of 10⁵ Pa and temperature of 273.15 K), and is the specific entropy of liquid water at 273.15 K. The mathematical expressions of the second law of thermodynamics and the Gibbs relationship can be written either in terms of the absolute entropy or in terms of any relative entropy. It also ensues that any relative entropy such as moist entropy is an adiabatic invariant for moist air.

There are in fact an infinite number of adiabatic invariants for moist air. For closed, reversible, and adiabatic transformations, both entropy and total water content are conserved. It follows that any arbitrary function of entropy and total water is also conserved. Dalton’s law implies that any thermodynamic properties of moist air can be written as a function of entropy, total water content, and pressure. Thus, any thermodynamic invariant of moist air can also be written as a function of entropy and total water content alone. For example, the equivalent potential temperature from Emanuel (1994) is given by

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while the entropy temperature of Marquet (2011) can be obtained as

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where and are the specific heat capacity of dry air and liquid water at constant pressure and T₀ = 273.15 K is the temperature of the reference state used for and .

Different adiabatic invariants are affected differently by the addition or removal of water from an air parcel, by the diabatic heating, and by irreversible processes. One author thus may prefer to use equivalent potential temperature as it is only weakly affected by the loss of liquid water during precipitation, while another may choose a virtual potential temperature that better captures the variation of density of unsaturated air. This overabundance of invariants can be unnecessarily confusing as only two independent invariants are necessary to determine all the others. The choice of the two invariants may be a matter of personal preference. Hauf and Holler (1987) propose to express any potential temperature definition to the entropy and total water content. Much difficulty could indeed be avoided if this suggestion were more broadly followed.

3. Isentropic circulations and the choice of the coordinate system

The isentropic analysis introduced by Rossby (1937) focused on studying horizontal motions on surfaces of constant potential temperature. More recently, Pauluis and Mrowiec (2013) suggested sorting vertical motion in terms of the equivalent potential temperature. To this effect, an isentropic streamfunction is defined as

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where is the mass of dry air per unit volume, w is the vertical velocity, and H is the Heaviside function defined as for and otherwise. Quantitatively, this is equal to the net upward mass flux of all air parcels with an equivalent potential temperature less than or equal to . The mathematical definition of the streamfunction can be applied to any variables. In practice, preference has been given to adiabatic invariants such as the equivalent potential temperature or some of its variations.

M17 notes that the isentropic streamfunction computed by using the entropy potential temperature, (5), differs substantially from the isentropic streamfunction for the equivalent potential temperature, (3). This should not be too surprising given the definition of the streamfunction itself. Figure 1 shows the streamfunction, (6), obtained for the equivalent potential temperature (left panel) and for the entropy potential temperature (right panel). As in M17, the computation for and excludes all condensed water. The data used here are from a numerical simulation of a hurricane with the Weather Research and Forecasting Model (Skamarock et al. 2008) previously analyzed by Pauluis and Zhang (2017). The streamfunction is computed for day 5 of the simulation. The streamfunctions differ primarily in their tilt: to the right in θ_s–z and to the left in θ_e–z.

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Comparison between the isentropic streamfunction, (6), obtained when the adiabatic invariant is (left) equivalent potential temperature and (right) entropy potential temperature . Condensed water here has been omitted from the computation of both and .

Citation: Journal of the Atmospheric Sciences 75, 10; 10.1175/JAS-D-18-0059.1

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In a steady state, the streamlines correspond to the mean flow in z–θ coordinates. In Fig. 1, negative values of the streamlines correspond to a counterclockwise rotation. The streamfunction in θ_e–z (Fig. 1a) indicates that air rises at high values of and descends at lower values of , consistent with a direct overturning and upward energy transport. Most of the streamlines are tilted to the left, indicating that the equivalent potential temperature of air parcels decreases as they rise, which can be interpreted as the result of entrainment of dry air into the updrafts. As noted in Mrowiec et al. (2016), the streamline farthest to the right corresponds to air parcels ascending in the eyewall at very high values of . That this streamline is almost vertical in Fig. 1a implies that air parcels rising within the eyewall do so without much loss of water vapor from entrainment.

In contrast, the streamlines in show little tilt in the lower troposphere but a substantial tilt to the right in the upper troposphere. This implies that increases substantially as air parcels rise. This can be explained by the fact that, under atmospheric conditions, the absolute entropy of liquid water (or ice) is less than the absolute entropy of dry air. When an air parcel loses liquid water through rainfall, its absolute entropy increases and so does its entropy potential temperature. (This is similar to the increase in liquid water potential temperature during precipitation.) In the lower troposphere, where the streamlines are almost vertical, changes of are due to the combination of entrainment, which reduces , and precipitation, which increases it. The isentropic analysis indicates that there is a close compensation between the two processes in this simulation, although there is no obvious reason as to why this should be the case under different conditions. In fact, the outermost streamline exhibits a clear tilt to the right even in the lower troposphere. This streamline corresponds to air parcels rising in the eyewall, which, as noted earlier, experience little entrainment. In the upper troposphere, the right tilt of the streamlines indicates that the entropy potential temperature of rising air parcels increases, which can be attributed to the loss of condensed water. The difference in tilt between the streamline obtained in θ_e–z and θ_s–z can be explained directly by the fact that adding or removing water in various phases makes different contributions to and . In particular, the removal of liquid water through rainfall has a very small impact on but significantly increases .

While the two streamfunctions may appear to differ at first glance, they are strongly related to each other. Indeed, one can reconstruct the streamfunction in θ_s–z directly from the streamfunction in θ_e–z. To do this, we first compute the mass-weighted averaged value of the entropy potential temperature as a function of both and z as defined in Pauluis and Mrowiec (2013). The isentropic streamfunction in θ_s–z can then be estimated by taking the value of the streamfunction in θ_e–z estimated at the value of the equivalent potential temperature is such that , that is,

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This procedure is illustrated in Fig. 2. Figure 2a shows . Comparing the definitions of both potential temperatures, (3) and (5), indicates that the entropy potential temperature is systematically lower than the equivalent potential temperature by an amount proportional to the total water content of the parcel. The distribution of shows that the entropy potential temperature is lower than the potential temperature by about 20 K in the lower troposphere, but the difference decreases with height and vanishes above 12 km. The reconstructed streamfunction is shown in Fig. 2b and is a very close match to the direct computation in Fig. 1b. From a mathematical point of view, the transformation (7) takes the streamlines in θ_e–z (as in Fig. 1a) and displaces them horizontally to the location corresponding to (i.e., shifting them to the left) without affecting the total mass transport.

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(left) Mass-weighted averaged entropy potential temperature as a function of and height and (right) reconstructed isentropic streamfunction for the entropy potential temperature based on (7).

Citation: Journal of the Atmospheric Sciences 75, 10; 10.1175/JAS-D-18-0059.1

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The assumption behind the reconstruction (7) is that the streamlines in θ_e–z can be recast in any thermodynamic coordinates by computing the mass-weighted averaged value along the streamline. This is similar to the procedure used in the Mean Airflow as Lagrangian Dynamics Approximation (MAFALDA) to reconstruct the thermodynamic cycles associated with deep convection and hurricanes (Pauluis 2016; Pauluis and Zhang 2017). The transformation (7) should be viewed here as a null hypothesis, that is, a default reconstruction method that, if successful, indicates that the two circulations computed in different coordinate systems are equivalent. This gives us confidence that the overturning flow in the hurricane simulation is accurately captured by the isentropic analysis. When the reconstruction fails, the differences in streamfunction can be used to assess in which ways the circulation departs for this simple overturning scenario.

To illustrate this point, the streamfunctions in θ_e–z and θ_s–z coordinates are computed again, but this time, all condensed water—including precipitation—is included in the computation of and . The streamfunctions are shown in Figs. 3a and 3b. In addition to the difference in tilt, the streamfunction in entropy potential temperature is substantially weaker than the streamfunction in equivalent potential temperature, and it exhibits a region of positive value in the upper troposphere. The transformation (7) conserves the maximum and minimum of the streamfunction and, as such, cannot explain the differences between Fig. 3a and Fig. 3b.

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Comparison between the isentropic streamfunction, (6), obtained when the adiabatic invariant is (left) equivalent potential temperature and (right) entropy potential temperature . Computations of both and include condensed water.

Citation: Journal of the Atmospheric Sciences 75, 10; 10.1175/JAS-D-18-0059.1

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To explain these, following a similar method used in Pauluis et al. (2008, 2010), we compute the upward mass transport as a function of both and defined by

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This amounts to computing the mass transport along adiabatic filaments. These adiabatic filaments have the advantage that any adiabatic invariant is constant along such adiabatic filaments, which implies that the adiabatic filaments are independent of the choice of the adiabatic invariants used to define them. Furthermore, the upward mass transport along any isopleth of any adiabatic invariants can be computed directly from M. The vertical mass transport on isentropic filaments in Fig. 4a shows this mass transport evaluated at a height of 7 km, a level where the difference in the two streamfunctions is very pronounced.

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(left) Joint distribution of the upward mass transport as a function of and at a height of 7 km. (right) As in (left), but for the total condensed water content.

Citation: Journal of the Atmospheric Sciences 75, 10; 10.1175/JAS-D-18-0059.1

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The mass transport on adiabatic filaments shows a broad ascent region (in θ_e–θ_s) at high values of and a more narrow region of descent near the main diagonal with , corresponding to air parcels with little to no water content. The separation between ascending and descending air is less pronounced in terms of : for a given value of , one often find a mix of ascending and descending air parcels at different value of . The streamfunction for either θ_e–z or θ_s–z can be computed directly from the mass transport on adiabatic filaments:

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Thus, the streamfunction for is obtained by integrating M on a horizontal half plane in Fig. 4a, while the streamfunction for is obtained by integrating M on a vertical half plane. The structure of M in Fig. 4a shows that there is be a substantial amount of cancellation in the computation of the streamfunction in but much less on surfaces of constant . In other words, the equivalent potential temperature better separates between updrafts and downdrafts than and better captures the overturning circulation when condensed water is included.

The streamfunction in θ_s–z also presents a dipole structure, with a region of positive values in the upper troposphere. The pattern shown in Fig. 3b indicates that air rises at high and low values of and subsides at intermediate values. This is consistent with Fig. 4a, which shows descending air at intermediate values of and rising air at both low and high . The presence of rising air parcels at high but low can be explained by the respective impact of condensed water on the two potential temperatures. Comparing the definitions for and shows that the difference between the two is approximately given by

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which corresponds to a difference of about 1.1 K for a change of 1 g kg⁻¹ in total water content. At a constant value of , variations in are due to changes in water content. In particular, an increase in the amount of condensed water decreases . This is confirmed in Fig. 4b, which shows the distribution of condensed water as a function of and at 7 km. Ascending parcels with high values of but low values of exhibit high amount of condensed water and explain why there is a dipole structure in the streamfunction in θ_s–z but not in θ_e–z. When condensed water is taken into consideration, the entropy potential temperature does not separate well between the ascending and descending air parcels and severely underestimates the mass transport in the upper troposphere when compared to the isentropic analysis based on equivalent potential temperature.

The differences in the streamfunctions when computed with and without condensed water are very pronounced for but are also substantial for . These differences reveal the high amount of ice present, as much as 10 g kg⁻¹, with most of it in the form of graupel and snow. High ice content reduces and in the updrafts. When enough ice is present, or in the updrafts may be similar to or even lower than that of surrounding dry air so that it may not be possible to separate updrafts and downdrafts on the basis of or alone.

It is desirable to have a robust way to compute the isentropic analysis, that is, one that is not overly sensitive to the amount of condensed water. Two methods have been proposed. First, Pauluis and Mrowiec (2013) exclude falling precipitation from the computation of and but keep the cloud water and ice. While the amount of condensed water that qualifies as “cloud” water depends on the microphysical assumptions, it is usually small, on the order of 1 g kg⁻¹. As a result, streamfunctions computed with cloud water and ice (not shown) are nearly identical to the streamfunctions without any liquid water or ice. Alternatively, Pauluis (2016) uses a different adiabatic invariant, the frozen equivalent potential temperature defined as

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Here, is the specific entropy of ice at , and is the specific heat of ice. The definition of is similar to that of the equivalent potential temperature, (3), except for the fact that the reference values and for liquid water have been replaced by those for ice and . From physical point of view, the frozen equivalent potential temperature corresponds to the temperature that an air parcel would have after first expanding adiabatically so that all water vapor condenses as ice then being compressed back to the reference pressure without allowing for any phase transition, that is, with all water remaining in the solid phase. As a result, it is slightly larger than the equivalent potential temperature. The primary advantage of for the isentropic analysis lies in that it is only slightly affected by the addition or removal of ice. The isentropic streamfunction for computed without condensate and with condensate are shown in Fig. 5 and exhibit only minor differences between each other.

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The isentropic streamfunction, (6), for the frozen equivalent potential temperature (left) without and (right) with condensed water included in the definition of .

Citation: Journal of the Atmospheric Sciences 75, 10; 10.1175/JAS-D-18-0059.1

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4. Work by a thermodynamic cycle

A second issue raised by M17 lies in the quantification of the work done by atmospheric motions. This issue is unrelated to the broader questions of the adiabatic invariants, but the misconceptions highlighted by M17 need to be corrected. Pauluis (2011, 2016) uses the Gibbs relationship to provide a framework to relate the work generated by atmospheric motions to the entropy transport. M17 criticizes the fact that Pauluis (2011, 2016) normalizes the cycle by the mass of dry air being transported and suggests that the work should be normalized by the mass of moist air instead. Under the per-unit-mass-of-moist-air framework, the work produced by a cycle is

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where α is the specific volume of moist air and P is the total pressure. Under the per-unit-mass-of-dry-air framework, the mass of dry air is constant through a thermodynamic cycle, and the work is

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where is the specific volume of dry air. For a given parcel trajectory, the computation of and will differ significantly.

Adopting either the framework of per unit mass of moist air or per unit mass of dry air implies very different assumptions on the exchange of mass. Under the framework of per unit mass of dry air, water can be added or removed, but the amount of dry air remains unchanged. In contrast, under the framework of per unit mass of moist air, any addition or removal of water from the system must be balanced by an equal removal or addition of the same mass of dry air. In the steam cycle proposed by Pauluis (2011), water vapor is added at warm temperature and removed at cold temperature, with no change in the amount of dry air. As the mass of dry air through the cycle is constant, the per-unit-mass-of-dry-air framework should be used to correctly assess the thermodynamic transformations, and the work done by the steam cycle is given by .

Under the per-unit-mass-of-moist-air framework, there is no net addition of mass. Whenever water is either added to or removed from the system, an equal mass of dry air must be removed or added. For a convective cycle in which water vapor is added at warm temperature and removed at colder temperature, an equal mass of dry air must be added at cold temperature and removed at warm temperature. This mass of dry air is also added at low pressure and removed at high pressure and must be compressed, which reduces the amount of work produced by the cycle.

The specific volume of dry air and moist air are related by , where is the total mixing ratio. Thus, the difference of work between the two frameworks is

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The difference between and correspond to the expansion of an additional mass of air circulating through the cycle. It is positive in the steam cycle, as the mass of water present in the parcel—and thus the mass of the parcel itself—is larger during expansion than during compression. If, in addition, one assumes the pressure to be hydrostatic, that is, , with g the gravitational acceleration and Z the geopotential height, the difference between the two framework can be written as

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This is the geopotential energy imparted to the water as it is being lifted, and it is lost through dissipation as water precipitates (Pauluis et al. 2000).

From a physical point of view, the per-unit-mass-of-dry-air framework is the correct one to assess the atmospheric overturning of precipitating convection. A cycle based on this framework assumes that there is not net upward mass transport of dry air. This is consistent with the fact that mass conservation ensures that, on average, there is no net upward mass flux of dry air. In contrast, the per-unit-mass-of-moist-air framework implies that the upward transport of water vapor is balanced by a net downward transport of dry air, which is simply not the case in Earth’s atmosphere. More specifically, the per-unit-mass-of-moist-air framework underestimates the amount of work produced by an amount approximately equal to the geopotential energy gained by the water substance.

5. Concluding remarks

Isentropic analysis was originally introduced by Rossby (1937) to study synoptic motions and has recently been adapted to investigate the overturning in moist convection and hurricanes (Pauluis and Mrowiec 2013; Mrowiec et al. 2016). It relies on assessing the atmospheric flows on the isopleths of some adiabatic invariants, typically potential temperature or the equivalent potential temperature. The resulting circulation depends on the choice of the adiabatic invariant. Recently, M17 suggested limiting the isentropic analysis to the surface of constant entropy potential temperature. This recommendation is ill-advised. By comparing the isentropic analysis obtained for adiabatic invariants, one can often gain new insights on the atmospheric circulation.

Here, the isentropic streamfunctions obtained by using the equivalent potential temperature and the entropy potential temperature are compared. When condensed water is excluded from the computation, one streamfunction can be reconstructed from the other through a coordinate transformation. This gives us confidence that the isentropic analysis captures the atmospheric overturning in the simulation. The largest impact on the isentropic streamfunction in our simulations occurs when one take into account the precipitating water and ice in the computation of and . This leads to a reduction of the mass transport in both cases, albeit the impacts are much more pronounced for . This issue arises from the large amount of precipitating water present in the updrafts, which can reduce by 3 K and by up to 10 K. As a result, updrafts and downdrafts cannot be as easily separated in terms of either and , leading to an underestimate of the atmospheric overturning. This problem can be avoided altogether by limiting the condensed water to cloud water and ice or by using a frozen equivalent potential temperature originally defined in Pauluis (2016).

It has also been shown here that, in order to compute the work done by atmospheric motions, the use of the framework of “per unit mass of dry air” is preferable to the framework of “per unit mass of moist air.” The latter framework underestimates the total work done by atmospheric motions by omitting the work performed to lift water that is then lost when precipitation falls back to Earth’s surface. As precipitation-induced dissipation accounts for a large fraction of the total frictional dissipation in Earth’s atmosphere (Pauluis et al. 2000; Pauluis and Dias 2012), the use of the per-unit-mass-of-moist-air framework can result in a large error in the estimate of the work done by the atmospheric circulation.