## 1. Introduction

The nonequilibrium thermodynamics corresponding to climatological atmospheric entropy production is a fundamental problem that needs to be extensively explored.

In the last several decades, entropy, related to nonequilibrium thermodynamics, has been applied to various problems in the atmospheric sciences, for example, the general circulation of the atmosphere (Peixoto and Oort 1992; Johnson 1997; Goody 2000; Takamitsu and Kleidon 2005; Fraedrich and Lunkeit 2008; Pascale and Gregory 2009; Lucarini et al. 2011), moist convection (Pauluis and Held 2002a,b), and synoptic-scale severe atmospheric systems (Liu and Liu 2004). Additionally, Nicolis and Nicolis (1980) and Pujol and Llebot 1999b have applied the second differential of the entropy to instability in low-dimensional climate models. A significant effort has also been to study the maximum entropy production principle using simple climate models (Paltridge 1975; O’Brien and Stephens 1995; Pujol and Llebot 1999a; Lorenz et al. 2001; Ou 2001; Ozawa et al. 2003; Goody 2007; Shimokawa and Ozawa 2007; Li 2009; Lucarini 2009; Lucarini et al. 2010; and others).

*s*is the entropy per unit mass,

*ρ*is density,

*Q*is the thermal heating rate, and

*T*is the temperature. Since

*Q*~ −

**∇**·

**J**, where

**J**is the thermal flow, we haveThe first and the second terms are called the external and internal entropy production, respectively. The external entropy production deals with the local convergence (divergence) of the entropy flow

**J**/

*T*. By Gaussian theorem, the volume integral of

**∇**· (

**J**/

*T*) can be transformed into a surface integral representing the flow across the boundary. Therefore, the external entropy provides the boundary entropy supply to a system. The internal entropy production describes the physics of dissipation, as flow

**J**moving downgradient of temperature is an irreversible process, which reduces the orderliness of the system and is accompanied by positive internal entropy production. According to Prigogine (1980), dissipation is the most fundamental process in nature.

In the atmosphere, the corresponding entropy balance equation is much more complicated than the simple case discussed above. However, the basic physics should be similar. The atmosphere is a thermodynamic system containing many different physical processes. Most of them are irreversible, such as rainfall, convection, and transport of large-scale eddy flow, etc. These irreversible processes are associated with positive internal entropy production. Therefore, the orderliness of the atmosphere will decrease if the boundary does not provide enough negative entropy or if there is not a sufficient sink of entropy (due to radiation) inside the system. The main purpose of this work is to understand the boundary entropy supply and the convergence of entropy flow and to understand the extent of atmospheric internal entropy production and its impact on the climate’s orderly appearance. To achieve this, the basic atmospheric entropy equation will be derived through the mean and transient thermal and moisture equations, in that all of the physical processes are included from the large scale to the small scale. Then, the basic entropy equation will be applied to the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis data (Kistler et al. 2001) to evaluate the atmospheric external and internal entropy production.

## 2. Theoretical background for atmospheric entropy

*θ*is the potential temperature;

*T*is the temperature;

*q*is the specific humidity for water vapor;

_{υ}*q*is the water mass mixing ratio;

_{w}*c*

_{pd}and

*c*

_{pv}are the specific heat at constant pressure for dry air and water vapor, respectively;

*c*is the specific heat for water;

_{w}*R*is the gas constant of water vapor;

_{υ}*λ*is the latent heat of evaporation of water; and

*q*are less than 10

_{υ}^{−2}and

*q*<

_{w}*q*. Therefore, we can neglect

_{υ}*q*+

_{υ}*q*in the first term and the whole second term in the right-hand side of (1). Hence, we have (Emanuel 1994) the following:From (2) we obtain the following entropy balance equation (details in the appendix):where the overbar and prime are the time mean and transient, respectively;

_{w}*J*

_{θi}and

*J*

_{υi}are the large-scale eddy thermal flow and moistening heat flow, respectively;

*σ*is the internal entropy production shown in the appendix. The term

In this paper, the subscript *i* (or *j*) takes values 1, 2, and 3, representing two horizontal and one vertical directions. For example, *x*_{1}, *x*_{2}, *x*_{3} = *x*, *y*, *z* for coordinates and *u*_{1}, *u*_{2}, *u*_{3} = *u*, *υ*, *w* for velocities in the *x*, *y*, and *z* directions, etc. The subscript *α* takes the values 1 and 2, representing two horizontal directions. The convention of summing the repeated index *i* (or *j*) from 1 to 3 and the repeated index *α* from 1 to 2 is adopted, for example, *a _{i}b_{i}* =

*a*

_{1}

*b*

_{1}+

*a*

_{2}

*b*

_{2}+

*a*

_{3}

*b*

_{3}and

*a*=

_{α}b_{α}*a*

_{1}

*b*

_{1}+

*a*

_{2}

*b*

_{2}.

## 3. External entropy production—Surface entropy supply and convergence of eddy entropy flow

*z*= 0 is the surface,

*z*=

*z*is the top of atmosphere (TOA), and c.c. implies the convergence contribution from large-scale eddies, which is defined asIt will show later that c.c. ≈ 0 for the global mean. Over a long time average, the global mean of the Lagrangian derivative of entropy should be close to zero as well. Therefore, in the sense of global mean,where 〈·〉

_{t}*represents a global mean. Locally, (6) does not hold.*

_{g}*g*is the gravitation constant,

*F*is the net diabatic heat flux at height

*z*(or pressure

*p)*,

*Q*is the corresponding diabatic heating rate, and

*F*

_{TOA}is the net flux at the top of the atmosphere. Here,

*F*

_{TOA}is zero for sensible, latent, and convective fluxes.

It is seen in Fig. 1 that in the tropics and lower-latitude regions, the boundary supply of entropy is generally positive; only in the northern and southern polar regions is the boundary supply of entropy in weakly negative value. The distribution of the entropy supply is very nonhomogeneous. Over large areas of the Atlantic Ocean and the Indian Ocean, the surface entropy supply is in the range of 0.4–0.6 W m^{−2} K^{−1}. Over the tropical ocean near South America, there is an elongated region with a relatively low entropy supply, with a value as low as 0.1 W m^{−2} K^{−1}. In such regions there is strong upwelling of cold water from the deep ocean, which leads the weak upwelling sensible/latent heat flux and surface entropy supply.

There are two regions with very high boundary entropy supply: one is in the northwestern Pacific and the other is in the northwestern Atlantic near the eastern coast of North America. In these two regions, the surface entropy fluxes are greater than 0.8 W m^{−2} K^{−1}. These are regions where warm ocean western boundary currents (the Kuroshio and the Gulf Stream) tend to warm the atmosphere.

Figure 2 shows the time evolution of the global mean boundary supply of entropy. Additionally, the regional mean results are shown for the tropics (18°S–18°N), the Northern Hemisphere (18°–90°N), and the Southern Hemisphere (18°–90°S). It is found that the global mean boundary entropy flux is close to 0.35 W m^{−2} K^{−1} with a weakly seasonal oscillation. The time-averaged value is listed in Table 1. Since the boundary supply of entropy is dominated by the latent heat and sensible heat, the results of boundary entropy fluxes by the latent heat and sensible heat are also shown in Fig. 2.

Global mean and regional mean of the net boundary entropy supply for individual physical processes (W m^{−2} K^{−1}).

In the tropics the boundary supply entropy flux is large, but the seasonal oscillation is weak. Outside of the tropics, in both the Northern and Southern Hemispheres, the boundary supplies of the entropy flux are smaller than those of the tropics; however, the seasonal oscillations become larger, especially in the Southern Hemisphere. By decomposing the entropy flux into the latent heat and sensible heat components, strong seasonal oscillations are found for both parts in the Northern and Southern Hemispheres, but their phases are different. In the winter season of the Northern Hemisphere, the surface latent heat flux is relatively high and the surface temperature is relatively low. Both factors contribute a higher surface entropy flux. However, the relatively low surface temperature can cause the low surface sensible flux and its related entropy flux. Therefore, the phases of the surface entropy flux are different for the latent heat and sensible heat parts. This difference in phase results in a weak seasonal oscillation in the total. The same is true for the summer season. In the Southern Hemisphere, the seasonal oscillation of the surface entropy flux is strong, since the total result is mainly determined by the latent heat part.

Table 1 lists the 10-yr mean values of entropy supply corresponding to latent heat, sensible heat, irreversible surface evaporation, and surface stress. The term

To comprehend the seasonal pattern, Fig. 3 shows the net surface entropy flux for the December–February (DJF) Northern Hemisphere (or called boreal winter) and the June–August (JJA) Southern Hemisphere (or called austral winter). The corresponding zonal mean values are plotted in the bottom panels. Compared to the JJA Southern Hemisphere, the DJF Northern Hemisphere shows more structure. In the DJF Northern Hemisphere, there are two strong positive centers: one is in the northwestern Pacific and the other is in the western Atlantic near the coast of North America corresponding to the Kuroshio and the Gulf Stream. In the two contour centers, the entropy fluxes are greater than 1.6 W m^{−2} K^{−1}, while in the other regions the surface entropy supply is more homogeneous and relatively small. In the JJA Southern Hemisphere, the main structure of the net surface entropy flux is a large circle of entropy flux over 0.2 W m^{−2} K^{−1} around 30°–50°S. There is a negative surface entropy flux in the Antarctic polar region, which is due to the negative (downward) surface sensible heat flux over regions covered with sea ice and snow.

*d*Σ

*is the surface integral element in the*

_{i}*i*direction and Ω is the integral area covering the atmosphere, that is, the boundary of the surface and the boundary of the TOA. The integral of (7) is zero under the condition that

*J*

_{θi}=

*J*

_{υi}= 0 at the surface and the TOA. This condition is usually assumed in large-scale dynamics (James 1994).

Figure 4 shows the distribution of c.c. for the DJF Northern Hemisphere and the JJA Southern Hemisphere (top panels). The calculation is based on (5). Since the NCEP data are only up to 10 hPa, the upper limit of the integral is set to *z _{m}*, where

*z*is the height at 10 hPa. Generally the values are negative in the lower-latitude regions and become positive in the higher-latitude regions. This indicates that the large-scale eddy entropy flow is divergent in the lower-latitude regions and convergent in the higher-latitude regions.

_{m}In the top-left panel of Fig. 4, it is shown that the large divergences occur in the northwestern Pacific and western Atlantic near the coast of North America. They are close to the locations with large positive values of the surface entropy flux displayed in Fig. 3. It is the strong local surface entropy supply that causes the strong local divergence of the entropy flow in these regions, since the system tends to reach a balance in entropy production. It is interesting to note that the two strong convergent centers are near to the two strong divergent centers, as the poleward entropy flow gradually converges in the transport process. Similar to the surface entropy supply case, the JJA Southern Hemisphere shows less structure for the entropy convergence.

The corresponding zonal mean results are presented in the bottom panels of Fig. 4. It is clearly shown that the local maxima of the divergences of entropy occur at about 30° in the winter hemisphere and at about 40° in the summer hemisphere for both DJF and JJA. The divergence of the large-scale eddy entropy flow in the midlatitudes and the convergence of it in the higher latitudes works as the meridional transport of the entropy flow, since the polar regions have large entropy sink because of the longwave cooling as the term of

## 4. Internal entropy production—Climate dissipation

Internal entropy production illustrates the relationship between the gradient of mean fields (temperature, potential temperature, and moisture as shown in ln

*k*is the diffusivity. Hence, we obtain

Equation (10) shows that the flow moves along the temperature gradient from the higher-temperature region to the lower-temperature region. During this process, the flow tends to smooth out the gradient of the temperature field by extracting energy from the high-temperature regions. This results in the redistribution of the temperature profile and the reduction of the existing slope of temperature. Therefore, the positive internal entropy production is referred to as dissipation (de Groot and Mazur 1984).

In the atmosphere, the structure in the temperature field is always present because of the nonhomogeneous distribution of radiative heating. A region with large internal entropy production corresponds to a large dissipative process. This indicates that in such a region, there is a highly organized temperature structure that can be dissipated. Otherwise, the dissipation process cannot last long. From this point of view, a region of positive internal entropy production must correlate with the dissipative temperature structure.

In the DJF Northern Hemisphere, there are well-defined patterns of internal entropy production with two maximum centers in the northwestern Pacific and the northwestern Atlantic. Based on the above discussions, there must be orderly atmospheric thermal structures in such regions that are being continuously dissipated. In the JJA Southern Hemisphere, there are large internal entropy productions in the polar region as well as in the subtropical high pressure region. In the bottom panels of Fig. 5, the corresponding zonal mean results of the vertically integrated internal entropy production are shown. It is seen that the maximum regions are in the midlatitudes for both the Northern and Southern Hemispheres.

*p*is the surface pressure. Equation (12) is equivalent to a mass-weighted mean. In addition, the distributions of the vertical mean large-scale eddy thermal flow

_{s}The top panels of Fig. 6 show the distribution of the vertical mean latitudinal temperature gradient. In the DJF Northern Hemisphere, there are three centers located in the regions close to the centers of internal entropy production shown in Fig. 5. In addition, the pattern of the JJA Southern Hemisphere is similar to that of the internal entropy production. The bottom two rows of Fig. 6 show the distribution of the vertical mean eddy flows in the latitudinal direction. In the DJF Northern Hemisphere, there are strong correlations between the temperature gradient and the eddy flows, especially in the western Atlantic region. Therefore, the linear proportion of the eddy flows of

In fact, the relationship of (10) has been studied as the macroturbulence diffusion in Held (1999). Readers are referred to it for a detailed physical discussion. In Fig. 2 of Held (1999), a result at 850 hPa was shown for

*θ*/∂

*y*~ ∂

*u*/∂

*p*. Then, the first term on the right-hand side of (13) issince

*p*=

*p*and 0. Generally, in the midlatitude region above the boundary layer, the poleward thermal flow,

_{s}Therefore, (13) describes the typical dissipation process in which the eddy flow acts to redistribute the temperature and dynamical structures. This process tends to irreversibly change the system from order to disorder and thus corresponds to an increase of internal entropy production. As mentioned above, the temperature gradient is mostly created by the nonhomogeneous distribution of radiation energy. Therefore, it is the radiation that maintains the temperature and dynamical structures and the eddy flow tends to dissipate them. Regionally, the air–sea interactions at the western boundary currents also help maintain the atmospheric temperature structure due to strong ocean fronts there. As it is shown above, the strong dissipation structures occur in the northwestern Pacific near Japan and near the eastern coast of North America, which indicates the maintained orderly temperature structures in these western boundary regions.

The vertical distribution of the zonal mean internal entropy production due to large-scale eddy flows is shown in the top panels of Fig. 7 for DJF and JJA. In DJF, below 250 hPa there are two centers of large internal entropy production. These occur at about 40°N and 45°S, coincident with the regions of large-scale eddy kinetic energy (James 1994). Large-scale eddy meridional heat flows are found in these regions of large meridional temperature gradient, where the eddies act against the temperature gradient to make it smaller than it would otherwise be.

In JJA, there is also a large internal entropy production within the boundary layer around altitude 60°–75°S. The detailed analysis shows that the large internal entropy production is mostly produced by the dissipation in the meridional direction, as the term of

It is very interesting to find that there is negative internal entropy production in the lower stratosphere, though the value is very small. The negative internal entropy production in the lower stratosphere is mainly produced by the large-scale eddy thermal flow in the latitudinal direction as the large-scale eddy thermal flow moves along the countergradient of temperature. The large-scale eddy flow comes from the temporal variation. Therefore, the large-scale eddy flows in the lower stratosphere are not directly driven by the temperature gradient as shown in (10), which is necessary for a positive internal entropy production as discussed above.

The countergradient large-scale eddy thermal flow has been discussed previously (Wallace 1978). Understanding this phenomenon from an atmospheric entropy point of view has not been addressed. The negative entropy production results indicate that large-scale eddies have very weak correlation to the temperature gradient in the lower stratosphere. Thus, the relation of (10) does not hold true as that shown in Fig. 6.

The eddy moisture flow diffusion term *J*_{υi}/*λ* moving along the gradient of *J _{i}*∂

*T*/∂

*x*in (9), which shows the internal entropy production is due to the thermal flow

_{i}*J*moving along the gradient of

_{i}*T*. The vertical distribution of the zonal mean internal entropy production due to eddy moisture flow diffusion is shown in the bottom panels of Fig. 7. The result is only up to 200 hPa, since the NCEP data of

*ψ*such as water vapor, temperature, etc., the balance equation for its mean iswhere

*O*is the unresolved small-scale source. In the global mean,provided

*ψ*. This is very different from the entropy case. Even for the global mean, large-scale eddies contain the contribution to the entropy production as shown above. Entropy is a measure of the orderliness of a thermodynamic system. Any irreversible motion will lead to an increase of entropy, since the irreversible motion dissipates the orderliness of the system.

The time evolution of the vertically integrated large-scale eddy internal entropy production is presented in Fig. 8, with the global mean and three regional means. In the tropics, the curve is very flat and its value is very small. For both the Northern and Southern Hemispheres, the result is always larger in the winter hemisphere, since large-scale eddies are generally stronger in the winter season. The time-averaged global mean and the regional mean values are also shown in Fig. 8.

Now we investigate the total internal entropy production corresponding to both large-scale eddy flows and small-scale diabatic heat fluxes based on (8). Takamitsu and Kleidon (2005) have addressed the effect of small-scale diabatic heat fluxes. In Fig. 9 the vertically integrated total internal entropy production is shown for the DJF Northern Hemisphere and the JJA Southern Hemisphere. Three maxima were found in the DJF Northern Hemisphere. These centers are found in the western Pacific, the western Atlantic, and the polar region (in the Atlantic sector), similar to the result for large-scale eddies only.

For the zonal mean result shown in the bottom panels of Fig. 9, the internal entropy production has large values in the tropical region, which is different from that of the internal entropy production by the large-scale eddy only, as shown in Fig. 5.

In the JJA Southern Hemisphere, the positive internal entropy production shows large centers around Antarctica. The pattern is similar to that of the boundary entropy supply shown in the top panels of Fig. 3. The corresponding zonal mean result in the bottom panel also shows a peak value in the tropical region.

The vertical distribution of the zonal mean total internal entropy production is shown in the top panels of Fig. 10. It is found that the internal entropy production is primarily positive below the tropopause. Large positive values occur in the tropics and midlatitude regions (especially in the boundary layer) for both DJF and JJA. In such regions, the orderly temperature structures are undergoing strong dissipation by large-scale eddy flows and small-scale diabatic heat fluxes. In the stratosphere, though there are regions of negative internal entropy production similar to that of Fig. 7, the total result of the negative internal entropy production is negligible in comparison with that of the total positive result. The atmosphere as a whole does obey the second law of thermodynamics.

In the bottom panels of Fig. 10, it is found that the internal entropy production due to moisture diffusion and phase change appears mostly in the tropics and subtropical high regions. The negative result appears in the tropical region due to the sink of convective moisture there. However, as shown in the bottom panels of Fig. 10, the total result of internal entropy production is positive in the tropical region.

The time evolution of the vertically integrated total internal entropy production by (8) is presented as dashed lines in Fig. 8. Compared to the large-scale eddy results (solid lines), the total internal entropy production is much larger, especially in the tropics. In the Northern Hemisphere, the dashed and solid lines are of opposite phases. The entropy production due to small-scale diabatic heating is larger in the summer season, while the entropy production due to large-scale eddies is larger in the winter season. In the Southern Hemisphere, the shift in phase becomes less. This is because the contribution due to large-scale eddies becomes smaller in the Southern Hemisphere, thus the diabatic heat contribution dominates in the total internal entropy production.

In Fig. 8, it is shown that the 10-yr averaged global mean vertically integrated value is 0.037 49 W m^{−2} K^{−1} for the total internal entropy production and 0.008 40 W m^{−2} K^{−1} for the large-scale eddy part. Pascale and Gregory (2009) obtained 0.0518 W m^{−2} K^{−1} for the total internal entropy production based on the calculation in the third climate configuration of the Met Office Unified Model (HadCM3). Also, Lucarini et al. (2011) obtained the results from 0.052 to 0.058 W m^{−2} K^{−1} for 13 climate models. The difference could be due to the different datasets from the NCEP reanalysis and from the climate model simulations. In addition, the approximate method of equivalent temperature was used in Pascale and Gregory (2009) and Lucarini et al. (2011). It is interesting to find that the internal entropy production due to a large-scale eddy is comparable to the horizontal part of the internal entropy production obtained in Lucarini et al. (2011), since the horizontal components dominate the large-scale eddy.

*z*is the height of 10 hPa. Using the correlated-

_{m}*k*distribution radiation algorithm by Li and Barker (2005) with nonlocal thermodynamic equilibrium (non-LTE) correction in the mesosphere (Fomichev 2009), we obtained the global mean ofThis positive result is mostly from the ozone heating centered near the stratopause. Therefore, the global mean vertically integrated entropy production by radiation is −0.3982 W m

^{−2}K

^{−1}. In Table 1, it is shown that the total surface diabatic entropy flux is 0.3422 W m

^{−2}K

^{−1}. Figure 8 shows the vertically integrated internal entropy production is 0.037 49 W m

^{−2}K

^{−1}; also, the contribution above

*z*is missing, but the value should be very small. The sum of the three values is very close to zero, which indicates the time-averaged global mean result of (6) holds in the atmosphere.

_{m}In this work, the entropy advection,

## 5. Conclusions

The primary objective of this study was to investigate atmospheric entropy and its association with climate dissipation. The balance equation for entropy is derived using the mean and transient thermal and moisture equations. The external entropy production due to small-scale diabatic heat corresponds to the boundary entropy supply, which is determined by the unresolved small-scale diabatic heat fluxes. In the tropics, the net boundary supply of entropy is much larger than that in the extratropics. In the DJF Northern Hemisphere, there are two strong positive centers of boundary supply of entropy: one is in the northwestern Pacific and the other is in the northwestern Atlantic.

The external entropy production due to the large-scale eddy is the convergence of eddy entropy flow. Generally, the divergence of large-scale eddy entropy occurs in the lower latitudes and the convergence of large-scale eddy entropy occurs in the higher latitudes, which indicates the polarward transport of entropy flow.

The internal entropy production represents the physical process of dissipation. In the atmosphere, only in regions with strong thermal structure can strong dissipation occur. On the basis of this argument, we can determine the atmospheric temperature orderliness by the calculation of the distribution of internal entropy production. For the internal entropy production due to large-scale eddies only, there are informative patterns for the DJF Northern Hemisphere, with three contour centers in the western Pacific, the western Atlantic, and the north polar regions. This indicates that strong dissipation to the thermal and dynamic structures occurs in such regions. An interesting finding is that large-scale eddy internal entropy production is negative in the lower stratosphere, which corresponds to a process as the eddy flow moves counter to the gradient of temperature in this region. For the total internal entropy production due to the large-scale eddy flow and the small-scale diabatic heat flux, it is found that the internal entropy production is much larger in the tropical region than that in the extratropics, with a pattern similar to the surface entropy flux.

The authors thank Drs. C.-D. Johnson, S. Lambert, V. Lucarini, W. Merryfield, and anonymous reviewers for their constructive comments, and Dr. M. Shrinivas (at NCEP) for his help in interpreting NCEP data.

# APPENDIX

## Entropy Balance Equation

*ρ*is the air density. For flow in planetary atmospheres, the variation of density in the vertical direction is much larger than any horizontal fluctuations, and scale analysis shows that

*s*

_{0}is a constant of integration. Given (A2) the mean entropy isIn the derivation of (A3), terms proportional to

*n*≥ 2) are neglected, since they are at least one order smaller than that of terms

*ψ*is defined (James 1994) aswhere

*τ*is the time length for the mean and the transient is

*ρ*′ becomes important only in a detailed process related to buoyancy. Given (A3)where

*Q*,

_{r}*Q*, and

_{h}*Q*are diabatic heating rates corresponding to radiation, sensible heating, and frictional heating, respectively;

_{f}*Q*

_{1}is the heating rate for deep–shallow convection; and

*C*and

*E*are the condensation and evaporation rates, respectively. In NCEP data, the sensible heating rate corresponds to the turbulent vertical diffusion process. In (A6)

*Q*

_{1}and

*λ*/

*c*

_{pd}(

*C*−

*E*) are in the same category, but one is for convective scale and the other is for large scale, separately treated in NCEP data. Written in flux form, (A6) becomeswhere

*F*is the sensible heat flux and

_{h}*τ*

_{i,j}is the diffusion acceleration tensor (viscous stress). In (A7), we only keep the vertical components of the fluxes, since the vertical contribution dominates in diabatic heating. Similarly, the balance equation

*q*from (Emanuel 1994)where

_{υ}*F*is the latent heat flux corresponding to the vertical turbulent moistening diffusion process, and

_{l}*Q*

_{2}/

*λ*is the moistening rate for deep/shallow convection. Also −

*c*

_{pd}

*Q*

_{2}/

*λ*and

*E*−

*C*are in the same category but for different scales. The vertical diffusion moistening heating rate (latent heat) is

*Q*= −1/(

_{l}*c*

_{pd}

*ρ*)∂

*F*/∂

_{l}*z*.

*Q*

_{1}and

*Q*

_{2}cannot be cancelled out because deep cumulus cloud can carry the released latent heat of condensation to the upper troposphere. Following Emanuel (1994),where [·] and * represent the spatial mean and transient, respectively, and

*Q*

_{1}and

*Q*

_{2}. We restrict

*Q*

_{1}and

*Q*

_{2}to being only related to convection, the same as Emanuel (1994).

*J*

_{υi}/

*λ*the large-scale eddy moisture flow and

*F*/

_{i}*λ*the small-scale moisture flux.

*Q*

_{2}, with

*Q*

_{2}= 1/(

*c*

_{pd}ρ)(∂

*F*

_{m}/∂

*z*)].

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