## 1. Introduction

Interactions between thermal and kinetic energy changes have been historically investigated in the literature under available potential energy, for example, by Margules (1910) and Lorenz (1955). The focus of available potential energy is mainly on the hydrostatically balanced atmosphere so far. Zilitinkevich et al. (2007) proposed the idea of total turbulent energy conservation by considering interactions between turbulent kinetic energy (TKE) and turbulent potential energy (TPE) changes, in which TPE is defined as the temperature variance normalized by the Brunt–Väisälä frequency. Because turbulent mixing is a process that results from the nonhydrostatic pressure balance, essentially Zilitinkevich et al. (2007) have extended the investigation of interactions between kinetic and thermal energy changes to the nonhydrostatic atmosphere. Note that total turbulence energy (TTE) defined by Zilitinkevich et al. (2007) as the sum of TKE and TPE is different from total energy conservation, which is described in this study, and TTE is not constrained by an independent conservation law.

By analyzing observed turbulent mixing in the atmospheric boundary layer, Sun et al. (2016) found that interactions between atmospheric kinetic and thermal energy changes are crucial in determining turbulence intensity, and turbulent mixing plays an important role in heat transfer for temperature redistribution and the atmospheric stratification. In addition, they demonstrated that the most energetic turbulence eddies are large and nonlocal. Their study stresses the important role of energy conservation in determining air motion and atmospheric thermodynamic structures. However, using the traditional thermal energy conservation equation, we are unable to explain observed diurnal variations of air temperature even for a simple sunny day over a flat homogeneous terrain (see details in section 4). The unsatisfactory theoretical explanation of the observation leads us to examine thermal energy conservation for the atmosphere in this study.

In the literature, names of conservation laws have been used differently; we first clarify those names. Note that all the energy conservation laws address the balance between energy changes or the rate of energy, and environmental forcings, not energy itself, are conserved unless the net forcing is zero. Kinetic energy conservation describes the balance between the net change of kinetic energy or the rate of kinetic energy within a system of a finite volume and the net change of work done at the boundary of the system. Following classical thermodynamics, we refer to the first law of thermodynamics as the one for describing internal energy conservation when flow is at rest. Batchelor (1967, p. 21) describes the first law of thermodynamics as “Work and heat are regarded as equivalent forms of energy, and the change in the internal energy of a mass of fluid *at rest* consequent on a change of state is defined, by the first law of thermodynamics, as being such as to satisfy conservation of energy when account is taken of both heat given to the fluid and work done on the fluid.” That is, kinetic energy is not included in the first law of thermodynamics as the equilibrium state is considered; the system only includes molecular motion. The familiar thermal energy conservation equation for the atmosphere describes internal energy changes of a system within a finite volume as a result of net heating/cooling or thermal forcing, and is derived based on the first law of thermodynamics (e.g., Feynman et al. 1963; Batchelor 1967; Fleagle and Businger 1972; Gill 1982).

In a fluid system that is characterized with flow stratification and motion under the influence of gravity, conservation of energy studies the total energy including kinetic, internal (can be expressed as thermal), and potential energy (e.g., Fleagle and Businger 1972). To distinguish it from other conservation laws, we refer to conservation of energy for this type of flow as total energy conservation, which is the same as the total energy equation [Eq. (4.7.5) in Gill (1982)]. Total energy conservation describes the balance between the total energy change of the system and the rate of the net mechanical work done to the system and thermal forcing added to the system. Therefore, total energy conservation provides a constraint for changes of the sum of the thermal, kinetic, and potential energy conservation in the atmosphere. Total energy conservation has also been called the first law of thermodynamics in the literature (e.g., Bennett and Myers 1962; Balmer 2010) or simply conservation of energy (e.g., Fleagle and Businger 1972; Kuo 2005).

Theoretically, the thermal energy conservation equation can be derived as the residual balance between total energy conservation and kinetic energy conservation as demonstrated in, for example, Kuo (2005). We extend the traditional derivation of thermal energy conservation in engineering textbooks, and derive the thermal energy balance based on the physics principles of total energy conservation and momentum conservation for the atmosphere, ideal gas law, and mass conservation (section 2). Application of mass conservation explains density fluxes as a result of air thermal expansion/compression for the incompressible atmosphere, as incompressibility is associated with the air volume change with pressure under a constant temperature (American Meteorological Society 2018). Because vertical air density changes strongly impact potential energy changes, while potential energy changes are associated with kinetic energy changes, thermal energy conservation derived with the constraint of total energy conservation has to include this potential energy change as well. For this kind of nonequilibrium system, the first law of thermodynamics valid for an equilibrium system could not be applied. The resulting interaction between kinetic and thermal energy changes is different from the energy conversion between thermal and kinetic energy at a point as a result of high velocities and viscous energy dissipation such as shock waves and deceleration of a fluid approaching a subsonic stagnation point (e.g., Kays 1966). We then use observations to confirm the role of potential energy changes in explaining the diurnal variations of air temperature (section 4) based on the derived thermal energy conservation equation for the turbulent atmosphere (section 3). A summary is presented in section 5.

## 2. Conservation equations of total, kinetic, and thermal energy

Mathematical derivatives are commonly used to describe changes of physical variables within an infinitesimally small volume. However, following the concept of mathematical limits, such as when spatial intervals

### a. Total energy conservation

*ρ*can be expressed as

*t*in the dominator is the time,

*V*and

*w*are the horizontal and the vertical wind speeds, respectively),

*Q*represents the rate of net heat added to the system or diabatic thermal forcing (for changing energy) such as molecular thermal conduction, and

*g*and

*z*are the gravity acceleration constant and the height above the surface); that is,

^{−2}) expressed in Eq. (1) reflects the balance between total energy changes or the rate of total energy inside the system on the left-hand side (lhs) and the net heating or diabatic thermal forcing and the rate of mechanical work done to the system on the right-hand side (rhs).

*p*is the external or environmental pressure on the boundary of the system) and by the viscous stress related to the rate of angular deformation done to the system

*i*and

*j*directions,

*μ*is the dynamic viscosity, and

*Q*in the atmosphere can come from vertical divergence of the atmospheric radiation and latent heat from water condensation, as well as molecular diffusion. Near the heated/cooled ground surface, molecular diffusion between the solid surface and the air above leads to air temperature changes. Based on the ideal gas law,

*T*is the air temperature and

*R*is the gas constant for dry air, for which the influence of water vapor on

*R*can be expressed in terms of the virtual temperature), air temperature changes can lead to air density changes as

Over the heated ground surface, the air density of the heated air through molecular diffusion would be less than that of the unheated air above, resulting in positive buoyancy and upward air movement, or negative vertical density fluxes. The energy for the generation of the negative vertical density flux is obtained from the air thermal expansion as a result of the thermal forcing *Q*. When the surface is cooled through longwave radiation, high-density air is generated from air thermal compression as the air is cooled through molecular diffusion near the surface. As a result, the air density decreases with height, and the air is stably stratified. When air motions are generated by

In both surface heating and cooling situations described above, the vertical air density flux is associated with the external forcings of the system—*Q* alone for the surface heating case, and both *Q* and

*Q*need to be included in the total energy conservation equation [Eq. (1)]. The potential energy change associated with the vertical density fluxes as a result of thermal expansion/compression can be expressed as

In the atmosphere, molecular motions are responsible for heat transfer in the surface viscous sublayer adjacent to the ground, where the dramatic air thermal expansion/compression occurs. Above it, turbulent mixing is much more effective in transferring heat than molecular diffusions. Because of the effective turbulent mixing above the surface viscous sublayer and its contribution to vertical density fluxes, the consequence of the thermal expansion/compression as a result of thermal energy transfer at the surface can impact air flows above the viscous sublayer where local thermal expansion/compression is relatively small.

Although we use the air heating/cooling at the surface as the examples for providing diabatic thermal forcing *Q* into the system, physically, Eq. (11) captures the potential energy change as a result of density fluxes generated by air thermal expansion/compression in general based on the ideal gas law and mass conservation. In the atmosphere, water phase changes and radiation of clouds can also provide similar diabatic heating/cooling sources; similar physical processes can occur. That is, vertical density fluxes can travel to a distance where local *Q* is negligibly small such as buoyancy fluxes under convective conditions.

### b. Kinetic energy conservation

The kinetic energy conservation equation [Eq. (12)] reveals that kinetic energy changes are derived through the work done by mechanical forcings of the pressure gradients

*Q*in Eq. (14) is used to emphasize that Eq. (14) represents the kinetic energy conservation equation with consideration of the impacts of thermal forcing

*Q*as well as mechanical forcing

*Q*. That is,

*Q*, the pressure

*p*inside the system is not the ambient pressure

*p*but

In the turbulent atmosphere, vertical density variations are considered in decomposing air density into mean and perturbed components, and

### c. Thermal energy conservation

^{−1}s

^{−1}is relatively small for air temperature of 300 K.

*h*and

^{1}

Fundamentally, the thermal energy conservation equation [Eq. (23)] is derived as the residual energy balance between the total energy conservation equation [Eq. (18)] and the kinetic energy conservation equation [Eq. (14)]. Because total energy conservation includes the impacts of thermal expansion/compression on energy changes, essentially the derived thermal energy balance [Eq. (23)] also includes the impacts of thermal expansion/compression on internal energy changes. Because the additional mechanic forcing *Q*, and is not an external forcing for total energy changes, its appearance in the kinetic energy conservation equation has to balance its appearance in the thermal energy conservation equation. Thus, a positive

The derived thermal energy conservation equation clearly demonstrates different balance equations between temperature and other scalars such as atmospheric compositions. Because heat transfer represents energy transfer, it is capable of changing kinetic energy while concentration changes of atmospheric compositions cannot unless the concentration change can lead to potential energy changes. Dissimilarities between temperature and water vapor have indeed been observed in the literature (e.g., Cava et al. 2008; Van de Boer et al. 2014; Guo et al. 2016).

### d. Physical differences between the traditional and the derived energy conservation equations

*W*is the rate of work done to the system [Eq. (24) is the same as Eq. (1.5.2) in Batchelor (1967), except here

*Q*and

*W*have units of the rate of energy changes instead of the amount of energy per unit mass as in Batchelor (1967)] even through Eq. (24) requires equilibrium conditions where kinetic energy is not considered. This traditional practice is different from the approach in engineering textbooks (e.g., Kuo 2005), where conservation of total energy is explicitly considered as a basic physics principle for general fluids, instead of the first law of thermodynamics. Therefore, fundamentally, the difference is whether we consider the first law of thermodynamics, which is only valid for equilibrium conditions, or total energy conservation, which is valid for nonequilibrium conditions, as a basic principle of physics for atmospheric flows.

*Q*does not impact the kinetic energy conservation equation in his derivation. If we ignore the impacts of thermal expansion/compression on vertical density fluxes, or potential energy changes in the atmosphere, which is equivalent to setting

Without considering vertical density transfer as a consequence of thermal expansion/compression on potential energy changes, total energy also appears to be conserved; however, the important physical connections between thermal expansion/compression and potential energy changes and interactions between kinetic and thermal energy changes through potential energy changes are not accounted for. Because of the familiarity of the Boussinesq approximation in the atmosphere community, impacts of the work through vertical density fluxes on kinetic energy changes can be accepted relatively easily. However, it is difficult to understand the extra term

Schematic illustration of the differences between the traditional and the new concepts of kinetic *p*, external pressure; *ε*, the viscous deformation stress to the system). For the atmosphere, kinetic energy changes are traditionally considered to result from the mechanical forcing *Q* is used for thermal expansion/compression, leading to vertical density fluxes and potential energy changes, *Q*, and

Citation: Journal of Applied Meteorology and Climatology 58, 2; 10.1175/JAMC-D-17-0350.1

Schematic illustration of the differences between the traditional and the new concepts of kinetic *p*, external pressure; *ε*, the viscous deformation stress to the system). For the atmosphere, kinetic energy changes are traditionally considered to result from the mechanical forcing *Q* is used for thermal expansion/compression, leading to vertical density fluxes and potential energy changes, *Q*, and

Citation: Journal of Applied Meteorology and Climatology 58, 2; 10.1175/JAMC-D-17-0350.1

Schematic illustration of the differences between the traditional and the new concepts of kinetic *p*, external pressure; *ε*, the viscous deformation stress to the system). For the atmosphere, kinetic energy changes are traditionally considered to result from the mechanical forcing *Q* is used for thermal expansion/compression, leading to vertical density fluxes and potential energy changes, *Q*, and

Citation: Journal of Applied Meteorology and Climatology 58, 2; 10.1175/JAMC-D-17-0350.1

The magnitude of the vertical density fluxes on potential energy changes or the magnitude of the pressure perturbation

## 3. Energy conservation equations in the turbulent atmosphere

In this section, we derive the kinetic energy conservation equation [Eq. (12)], where density variations are explicitly included through turbulence perturbations, and the derived thermal energy conservation equation [Eq. (23)] in the turbulent atmosphere. We use the traditional practice of decomposing each variable *ϕ* into the Reynolds-averaged mean value

*e*. We keep both MKE and TKE in the kinetic energy conservation equation because of interactions between turbulent and mean flows [the second term on the rhs of Eq. (26)]. The mechanical energy dissipation rate

We now discuss interactions between the kinetic and thermal energy changes in the turbulent atmosphere. Turbulent motion in the atmosphere can be generated either by wind shear or positive buoyancy. Turbulence intensity measured by TKE depends not only on the energy exchange between TKE and MKE [the second term on the rhs of Eq. (26)] but also on the rate of the work associated with the vertical density fluxes *Q* on thermal energy changes. As a result, the traditional thermal energy balance does not satisfactorily explain the observed air temperature changes in the atmosphere (more in section 4).

## 4. Observational evidence of potential energy changes in the derived thermal energy conservation equation

Using the field dataset from the 1999 Cooperative Atmosphere–Surface Exchange Study (CASES-99), which is described in Cuxart et al. (2002) and Sun et al. (2013, 2015, 2016) for detailed observations and data processes, we examine the role of ^{−1} from 0.2 to 5.9 m above the surface; 5-min block-averaged ^{−1}; ^{−1} at 1.5 and 30 m; and net radiation

We first describe the observed vertical variations of turbulent heat flux

(a) Diurnal variations of the standard deviation of potential temperature *z* = 1.5 and 0.5 m, marked in black and red, respectively. (d) The bin-averaged daytime sensible heat flux,

Citation: Journal of Applied Meteorology and Climatology 58, 2; 10.1175/JAMC-D-17-0350.1

(a) Diurnal variations of the standard deviation of potential temperature *z* = 1.5 and 0.5 m, marked in black and red, respectively. (d) The bin-averaged daytime sensible heat flux,

Citation: Journal of Applied Meteorology and Climatology 58, 2; 10.1175/JAMC-D-17-0350.1

(a) Diurnal variations of the standard deviation of potential temperature *z* = 1.5 and 0.5 m, marked in black and red, respectively. (d) The bin-averaged daytime sensible heat flux,

Citation: Journal of Applied Meteorology and Climatology 58, 2; 10.1175/JAMC-D-17-0350.1

Diurnal variations of (a) the net radiation *w* to TKE, (b) TKE *e* at 1.5 m, (c) the 5-min standard deviation of the thermocouple temperature

Citation: Journal of Applied Meteorology and Climatology 58, 2; 10.1175/JAMC-D-17-0350.1

Diurnal variations of (a) the net radiation *w* to TKE, (b) TKE *e* at 1.5 m, (c) the 5-min standard deviation of the thermocouple temperature

Citation: Journal of Applied Meteorology and Climatology 58, 2; 10.1175/JAMC-D-17-0350.1

Diurnal variations of (a) the net radiation *w* to TKE, (b) TKE *e* at 1.5 m, (c) the 5-min standard deviation of the thermocouple temperature

Citation: Journal of Applied Meteorology and Climatology 58, 2; 10.1175/JAMC-D-17-0350.1

We then examine the phase relationship between the temporal variation of air temperature and the vertical convergence of turbulent heat fluxes on 10 October, which should be in phase if the traditional thermal energy conservation equation [Eq. (31) with zero rhs] is valid. The close correlations between the variations of *w*,

We then examine the contribution of vertical density fluxes

We now investigate the impacts of *μ* and the weak wind speed associated with the stable conditions.

(a) The wind speed

Citation: Journal of Applied Meteorology and Climatology 58, 2; 10.1175/JAMC-D-17-0350.1

(a) The wind speed

Citation: Journal of Applied Meteorology and Climatology 58, 2; 10.1175/JAMC-D-17-0350.1

(a) The wind speed

Citation: Journal of Applied Meteorology and Climatology 58, 2; 10.1175/JAMC-D-17-0350.1

Examples of both the daytime and nighttime observations suggest that the traditional thermal energy conservation balance cannot explain the observed diurnal variation of air temperature. Including

## 5. Summary

Applying total and kinetic energy conservation, we revisit thermal energy conservation (related to internal energy) in the atmosphere. Kinetic energy conservation is impacted not only by environment mechanical work *Q* (diabatic thermal forcing), *Q* and dissipation heat

The presence of

The traditional thermal energy conservation equation for the atmosphere is guided by the first law of thermodynamics even though it is only valid for an equilibrium state for which air motions are not included. Because the first law of thermodynamics does not explicitly include air motions, except for molecular motions, the impacts of environmental mechanical and thermal forcing on a system are somewhat isolated: mechanical forcing is used for kinetic energy changes and thermal forcing is used for thermal energy changes only. Applying the first law of thermodynamics to a nonequilibrium system not only violates the required equilibrium condition, but also misses the contribution of thermal forcing to air motions. Vertical convergence/divergence of heat fluxes in the traditional thermal energy balance represents internal energy changes, not potential energy changes associated with thermal expansion/compression. Therefore, in the traditional thermal energy balance,

The derived thermal energy conservation equation also demonstrates that temperature and atmospheric compositions have different balance equations. Unlike regular scalars such as atmospheric compositions, temperature transfer is associated with energy transfer, and impacts not only thermal energy changes but also kinetic energy changes. Differences between the derived thermal energy balance and the conservation equation of atmospheric compositions may shed light on observed dissimilarities between temperature and water vapor in the literature.

The derived thermal energy conservation equation could also potentially improve our understanding of the observed atmospheric thermodynamic structures, for example, in explaining the well-observed surface energy imbalance problem, which is based on the traditional thermal energy balance. Impacts of the derived thermal energy balance on atmospheric thermodynamics could be important not only in the atmospheric boundary layer, as demonstrated in this study, but also in large-scale and mesoscale atmospheric and oceanic thermodynamics whenever vertical density fluxes prevail in space. The concept of vertical density fluxes in connecting kinetic and thermal energy changes with consideration of total energy conservation could also contribute to a better understanding of the available potential energy in the literature (e.g., Tailleux 2013). Further field and laboratory investigations into energy conservation, especially total and thermal energy conservation, are needed to quantitively verify the derived thermal energy balance. Exploration of interactions between kinetic and thermal energy changes by numerical models would require consideration of thermal expansion/compression in generating vertical density fluxes in both kinetic and thermal conservation equations because all current numerical models including high-resolution ones are based on the traditional thermal energy balance, in which thermal expansion/compression is not explicitly included to relate to air motions and atmospheric thermodynamic structures.

## Acknowledgments

The author would like to thank Drs. Ming Cai, Larry Mahrt, Henk de Bruin, Eric DeWeaver, Andy Kowalski, Eugene Takle, Dan Rajewsk, Richard Rotunno, Qi Li, and anonymous reviewers for their helpful comments. The University Corporation for Atmospheric Research manages the National Center for Atmospheric Research (NCAR) under sponsorship by the National Science Foundation (NSF). This material is based upon work supported by NSF while the author worked at NCAR and serves at NSF. Any opinions, findings and conclusions, or recommendations expressed in this publication are those of the author and do not necessarily reflect the views of the NSF.

## APPENDIX

### Derivation of Energy Conservation Equations

#### a. Kinetic energy conservation

*V*and

*w*in the directions of

*x*and

*z*), the horizontal and the vertical momentum balance equations for a small-volume system can be expressed as (e.g., Kuo 2005)

*μ*is the dynamic air viscosity and

*V*and Eq. (A2) by

*w*as

*u*,

*υ*, and

*w*in the orthogonal directions of

*x*,

*y*, and

*z*would be

*u*, Eq. (A8) multiplied by

*υ*, and Eq. (A9) multiplied by

*w*; thus, the change in total kinetic energy resulting from the Coriolis force would be

#### b. Work associated with viscosity

#### c. Turbulent atmosphere

##### 1) Conservation equations of kinetic and thermal energy in the turbulent atmosphere

Here, we derive the conservation equations of kinetic and thermal energy for the turbulent atmosphere. Because of the effectiveness of turbulence in heat transfer in comparison with molecular thermal transfer, the air expansion/compression associated with molecular motions can be assumed to be approximately negligible (i.e., *ϕ* in the balance equation as

*ρ*. Reynolds averaging the decomposed kinetic energy balance indicates that

##### 2) Balance equations for temperature-related variables

## REFERENCES

American Meteorological Society, 2018: Incompressibility.

*Glossary of Meteorology*, Amer. Meteor. Soc., http://glossary/ametsoc.org/wiki/incompressibility.Balmer, R. T., 2010:

*Modern Engineering Thermodynamics*. Academic Press, 904 pp.Bannon, P. R., 1996: On the anelastic approximation for a compressible atmosphere.

,*J. Atmos. Sci.***53**, 3618–3628, https://doi.org/10.1175/1520-0469(1996)053<3618:OTAAFA>2.0.CO;2.Batchelor, G. K., 1967:

*An Introduction to Fluid Dynamics*. Cambridge University Press, 615 pp.Bennett, C. O., and J. E. Myers, 1962:

*Momentum, Heat, and Mass Transfer*. McGraw-Hill, 697 pp.Cava, D., G. G. Katul, A. M. Sempreviva, U. Giostra, and A. Scrimieri, 2008: On the anomalous behaviour of scalar flux–variance similarity functions within the canopy sub-layer of a dense alpine forest.

,*Bound.-Layer Meteor.***128**, 33–57, https://doi.org/10.1007/s10546-008-9276-z.Cuxart, J., G. Morales, E. Terradellas, and C. Yagüe, 2002: Study of coherent structures and estimation of the pressure transport terms for the nocturnal stable boundary layer.

,*Bound.-Layer Meteor.***105**, 305–328, https://doi.org/10.1023/A:1019974021434.Feynman, R. P., R. B. Leighton, and M. Sands, 1963:

*The Feynman Lectures on Physics*. Vol. 1. Addison-Wesley, 562 pp.Fleagle, R., and J. Businger, 1972:

*An Introduction to Atmospheric Physics*. Academic Press, 346 pp.Garratt, J. R., 1992:

*The Atmospheric Boundary Layer*. Cambridge Atmospheric and Space Science Series, Cambridge University Press, 316 pp.Gill, A. E., 1982:

*Atmosphere–Ocean Dynamics*. Academic Press, 662 pp.Green, D. W., and R. H. Perry, 2007:

*Perry’s Chemical Engineers’ Handbook*. 8th ed. McGraw-Hill, 2640 pp.Guo, X., Y. Sun, and S. Miao, 2016: Characterizing urban turbulence under haze pollution: Insights into temperature–humidity dissimilarity.

,*Bound.-Layer Meteor.***158**, 501–510, https://doi.org/10.1007/s10546-015-0104-y.Högström, U., 1990: Analysis of turbulence structure in the surface layer with a modified similarity formulation for near neutral conditions.

,*J. Atmos. Sci.***47**, 1949–1972, https://doi.org/10.1175/1520-0469(1990)047<1949:AOTSIT>2.0.CO;2.Kays, W. M., 1966:

*Convective Heat and Mass Transfer*. McGraw-Hill, 387 pp.Kuo, K. K., 2005:

*Principles of Combustion.*2nd ed. John Wiley and Sons, 732 pp.Lorenz, E., 1955: Available potential energy and the maintenance of the general circulation.

,*Tellus***7**, 157–167, https://doi.org/10.3402/tellusa.v7i2.8796.Mahrt, L., 1986: On the shallow motion approximations.

,*J. Atmos. Sci.***43**, 1036–1044, https://doi.org/10.1175/1520-0469(1986)043<1036:OTSMA>2.0.CO;2.Margules, M., 1910: On the energy of storms.

*The Mechanics of the Earth’s Atmosphere*, C. Abbe, Ed., Smithsonian Miscellaneous Collection, Vol. 51, Smithsonian Institution, 533–595.Stull, R. B., 1988:

*An Introduction to Boundary Layer Meteorology*. Kluwer Academic, 666 pp.Sun, J., S. K. Esbensen, and L. Mahrt, 1995: Estimation of surface heat flux.

,*J. Atmos. Sci.***52**, 3162–3171, https://doi.org/10.1175/1520-0469(1995)052<3162:EOSHF>2.0.CO;2.Sun, J., L. Mahrt, R. M. Banta, and Y. L. Pichugina, 2012: Turbulence regimes and turbulence intermittency in the stable boundary layer during CASES-99.

,*J. Atmos. Sci.***69**, 338–351, https://doi.org/10.1175/JAS-D-11-082.1.Sun, J., D. H. Lenschow, L. Mahrt, and C. Nappo, 2013: The relationships among wind, horizontal pressure gradient, and turbulent momentum transport during CASES-99.

,*J. Atmos. Sci.***70**, 3397–3414, https://doi.org/10.1175/JAS-D-12-0233.1.Sun, J., L. Mahrt, C. Nappo, and D. H. Lenschow, 2015: Wind and temperature oscillations generated by wave–turbulence interactions in the stably stratified boundary layer.

,*J. Atmos. Sci.***72**, 1484–1503, https://doi.org/10.1175/JAS-D-14-0129.1.Sun, J., D. H. Lenschow, M. A. LeMone, and L. Mahrt, 2016: The role of large-coherent-eddy transport in the atmospheric surface layer based on CASES-99 observations.

,*Bound.-Layer Meteor.***160**, 83–111, https://doi.org/10.1007/s10546-016-0134-0.Tailleux, R., 2013: Available potential energy and exergy in stratified fluids.

,*Annu. Rev. Fluid Mech.***45**, 35–58, https://doi.org/10.1146/annurev-fluid-011212-140620.Van de Boer, A., A. Moene, A. Graf, D. Schüttemeyer, and C. Simmer, 2014: Detection of entrainment influences on surface-layer measurements and extension of Monin–Obukhov similarity theory.

,*Bound.-Layer Meteor.***152**, 19–44, https://doi.org/10.1007/s10546-014-9920-8.Vickers, D., and L. Mahrt, 2006: Contrasting mean vertical motion from tilt correction methods and mass continuity.

,*Agric. For. Meteor.***138**, 93–103, https://doi.org/10.1016/j.agrformet.2006.04.001.Zilitinkevich, S. S., T. Elperin, N. Kleeorin, and I. Rogachevskii, 2007: Energy- and flux-budget (EFB) turbulence closure model for stably stratified flows. Part I: Steady-state, homogeneous regimes.

,*Bound.-Layer Meteor.***125**, 167–191, https://doi.org/10.1007/s10546-007-9189-2.

^{1}

With significant water vapor changes, virtual potential temperature can be used (Stull 1988).