The present work re-examines and generalizes the vertical thermal energy transport by the CCE concept in the TR zone. The heat and mass fluxes in the boundary layer (BL) over the ground surface is governed the “laws of the wall,” dealt with extensively in boundary layer theory [2][3]. Atmospheric scientists, following Molin and Obukhov [4]-[6], adopted the eddy diffusivity and turbulent dissipation due to large-eddy mixing and fluctuations in the lowest layers of TR with the vertical, turbulent heat flux, ω ′ T ′ , and mass flux, u ′ ω ′ , expressed as covariances of the velocities and the temperature components, ω ′ , T ′ and u ′ . In Obukhov’s concept, these components were assumed to be constant with elevation in BL. Weather science does not use the surface heat and mass transport coefficient concepts for laminar or turbulent, stable, or unstable BC layers over the ground, missing the convenience of an intermittent, measurable, transport model component, described and used in [1]. The BC layer is a very small, but important part of TR, assumed to be below the elevation of the monitored point for the air temperature, T a . Two other areas above the BL have active, convective vertical transport: 1) the zone of TR between the elevation of the measured point of T a , and below the height of the lower edge of the tropopause; and 2) the zone above it, between the elevation of the lower edge (umbrella layer) of ST and below the height of the upper edge of the tropopause layer at some 20 km elevation.

The zone of 1) in TR is often handled with the same model as of the BL, expanding Obukhov’s concept, and increasing the applicable height of a BL model, high into the atmosphere [7]. Convective transport (driven by gradients in a strict definition) is broadly defined in atmospheric science interchangeably with advection (that is driven, strictly, by advective velocity in the transport literature, especially in hydrology). Vertical transport of thermal energy and mass by convection uses the Convective Available Potential Energy (CAPE) parameter in mathematical models, as well as in observational explanations of cloud formations and storms, briefly reviewed in [1]. CAPE is used to explain the formation of cumulus clouds and thunderstorm clouds, as if the driving energy to lift a lump of mass in TR were along an adiabatic transport process in TR. Assuming mechanical energy balance along an adiabatic, instead of a constant compression index trajectory, n , ignores the combined effects of changing pressure, elevation, temperature and the loss of energy due to friction loss, explained in [1]. Using CAPE as the driving force for vertical lift of air disagrees with the conservation of the mechanical energy along a movement trajectory of constant n for a lump of mass in TR as it was shown in [1]. Empirical adjustments are needed to alleviate the conceptual weakness of using CAPE as the maximum energy available to an ascending parcel in dry or moist-air convection. Multiple types of CAPE are described in the literature, such as one for downdraft (DCAPE), surface based (SBCAPE), mixed layer or mean layer CAPE (MLCAPE), most unstable or maximum usable (MUCAPE), or normalized (NCAPE), serving the needs for adjustments.

Hypothesis 1 (H1): Under a quiet fair-weather condition, zero horizontal gradients for energy, momentum and mass flux define a useful subclass of weather processes in the TR. The H1 is an assumption that such a subclass is not an empty set in the ST and there exists tangible time periods (hours, days, or weeks) during which no significant, horizontal change in air temperature, barometric pressure, horizontal air velocity, or other properties are experienced within the representative element volume (REV) defined as 10 km × 10 km × 10 km [1]. H1 is intuitive as more than one monitoring station is needed to experimentally prove it. However, the spatially sparse set of monitoring stations for meaningful and useful weather reporting assume that each data point has a long, horizontal range of validity, with negligibly changing gradients, supporting H1.

Hypothesis 2, (H2). The single column solver (SCS) model for its REV volume to represent the STR must have horizontally zero net air mass flux exchange with its surrounding volumes in the atmosphere. H2 is a simplifying stipulation for the applicability of the SCS model, at least for time periods, confirmed by H1.

Theorem 1 (T1): Assuming the validity of H1 and H2, the total thermodynamic (TT) and the total mechanical (TM) energy must be both conserved within the REV volume of the STR with zero horizontal gain or loss components.

The proof of T1 is trivial, following the conservation laws for energy and the teaching from the first law thermodynamics. The REV volume is assumed to be closed by the insulating boundaries amid horizontally zero energy gradients on the vertical planes.

Theorem 2 (T2): Assuming the validity of H1 and H2, the total air mass within the REV volume of the STR is not affected by mass fluxes of the horizontal wind or on the ground wall.

The proof of T2 comes from the mass conservation law for a closed volume due to zero gradient-(that is, insulating) boundaries in the vertical planes and zero vertical air flow across the ground wall. The only open boundary for air mass to flow across is at the TR upper horizontal layer, which allows for vertical, in-and-out breathing or diffusion-dispersion mass transport with the tropopause layer in the ST.

Two CCE process components are needed to form a Carnot work-heat process in a vertically stratified TR volume. The two overlapping Carnot processes are shown together Figure 1, one in blue and one in red, in a p − ν diagram, where ν = 1 / ϱ . A unit mass particle activates the Carnot engine by moving along an n = n i constant line 1-2 (in blue) for the upper-side expansion route 1-1H-2 and a lower-side compression route 2-2L-1; and the other is an overall compression along 2-1 (in red) for the upper-side compression route 2-1H-1 and a lower-side expansion route 1-2L-2. When overlapped in REV, both CE processes have an upper and a lower side and may transport thermal energy between points 1 and 2 without net air mass exchange and without the need for turbulent dispersion.

Figure 1. Two Carnot cycle compression-expansion processes shown in red and blue colors between points 1 and 2 along an atmospheric, polytropic trend curve with n = n i compression index. The thermal energy flow directions are also shown for the counterclockwise (red) and clockwise (blue) circulation directions (after [1]).

The overall compression (in blue) has in series isothermal and adiabatic compression elements on the upper (H) side, and a series adiabatic and of isothermal expansion elements in the lower (L) side. The Q H (in blue) is the incoming thermal energy to a unit mass lump during 1-2 expansion on the upper side. The Q L (in blue) is the exiting thermal energy from a unit mass lump during 2-1 compression on the lower side.

In a complete, overall expansion (blue) Carnot cycle, the upper side must gain Q H thermal energy during 1-2 isothermal expansion and the lower side must lose Q L thermal energy (both in blue) during 2-1 isothermal compression to a unit mass lump. The area in the 1-1H-2-1 triangle is proportional to Q H (in blue). The area in the 2-2L-1-2 triangle is proportional to Q L (in blue). The Q H and Q L components are not expected to be equal.

The complete, overall expansion (red) Carnot cycle between 2-1 is the mirror image of that of the previous, overall compression (blue) process; only the signs of the energy terms need to change. The “blue” and “red” processes may not be equally probable in the TR.

The mathematical model of the Q H and the Q L thermal energy components for the overall compression process for a discretized vertical column uses the notations d Q u and d Q d [J/kg] thermal energy terms for the Q H and the Q L components, respectively.

The d Q u and d Q d [J/kg] thermal energy terms are calculated from the isothermal processes using (3b) and the P 1 H and P 2 L pressures at the mid-points of the upper and lower paths, These pressures are expressed with the help of the adiabatic sections with zero net thermal energy exchange.

The isothermal processes are modeled in the upper and lower sections between point 1 at ( i , j ) and point 2 at ( i , j + 1 ) , using the isothermal streamline Equation in (3b). The thermal energy is expressed first from the logarithm of pressures ratio along the isothermal paths, and then the P 1 , P 1 H and P 2 , P 2 L pressures are expressed with temperatures along the n i streamline, employing (3) and algebra. The result is obtained in an interesting form, using temperature ratios only on the n i = c o n s t process curve as follows [1]:

where

The n I i coefficient may be regarded as the difference of n i n i − 1 , and κ κ − 1 , where κ = c p / c v , (the adiabatic compression index), as indeed, n I i = n i n i − 1 − κ κ − 1 = n i n i − 1 − c p R .

Therefore, if the atmospheric energy transport were adiabatic, the d Q u and d Q d terms would be correctly both zero from (9a) and (9b).

The thermal energy of d Q u and d Q d are in [J/kg] in units, comparable to each other for a constant unit mass of air to satisfy zero net vertical mass transport. Vertical positions change by switching places with no net vertical mass change is conceivable, for example, by crisscrossing vertical positions along the overall, horizontal, average trajectory of air by, e.g., flow over surface waviness [1].

To interpret the CCE terms in (9a) and (9b) for energy conservation in a constant volume, each term at z ( j ) location must be multiplied by the air density at the same elevation. To still maintain zero vertical mass balance, the average density must be used as a multiplier, ρ ¯ ( i , j ) = ( ρ ( i , j ) + ρ ( i , j + 1 ) ) / 2 :

The differences between d Q u and d Q d thermal energy terms in absolute values are differentially small for fine d z division. The integral values of d Q u and d Q d terms are calculated to obtain the total values of CCE energy densities in TR over D z elevation in [J]. The Rieman summations for the integrals Q C E u ( i ) and Q C E d ( i ) are used by taking the scalar product of ρ ¯ ( i , j ) and the d Q u ( i , j ) and d Q d ( i , j ) as vectors, d z = z j + 1 − z j = 1 , d x = d y = 1 , for j ∈ [ 1 , D z − 1 ] :

The Q C E u and Q C E d energy functions in the single column are depicted from the SCS model for the selected hours in Figure 2(a) and Figure 2(b) during a fair-weather day, out of a 90-day (2208 hrs) time in a previous study [1]. The CCE components may add or remove thermal energy to or from the single column of volume 1 m × 1 m × Dz m. It is assumed that the upward and downward air flow exchange cycles actuates the CCE process over the entire D z height, separated spatially into cells by the horizontal wind. Note that the exchanges are not necessarily simultaneous, and not in the same vertical, unit-area (1 m × 1 m) column in the large REV volume over a 1 km × 1 km ground wall surface area within 1 hour.

For context to other atmospheric variables, the monitored, hourly air temperatures and the modeled ground wall temperatures are shown in Figure 3(a) for the selected hours. The variations of the modeled compression index, n i , and it’s modified counterpart from (10), n I i in TR for the same hours are shown in Figure 3(b). The corresponding compression index for the coldest hour at 7 am ( i = 368 hr) is n i = 1.2452 , ( n I i = 1.5765 ), and for the hottest hour at 4 pm ( i = 377 hr) is n i = 1.3774 . ( n I i = 0.1482 ).

For the effective heat conductivity and thermal diffusivity functions, the heat flux density function over a unit (1 m²) surface area is calculated from one component, e.g., the Q C E u ( i ) function, as | Q C E u ( i ) | − | Q C E d ( i ) | is zero within rounding limit:

(a)

(b)

Figure 2. (a) The Q C E u energy function; (b) The Q C E d energy function.

(a)

(b)

Figure 3. (a) Air and ground wall temperatures; (b) Compression index, n i ; and n I i , [Equation (10)].

For comparison only, the other component:

The heat flux density variation with time due to the upward and downward paths in all CCE cells in TR are shown for the selected time period in Figure 4(a), and Figure 4(b), respectively.

The effective thermal conductivity of k e f f ( i ) in the TR is evaluated from the defining equation of q W ( i ) = k e f f ( i ) ⋅ d T a ( i ) / d z = k e f f ( i ) ⋅ a l r ( i ) :

The dispersity coefficient D ( i ) is calculated from the defining equation of:

(a)

(b)

Figure 4. (a), q W u and q W d components together from CCE; (b), | q W u | to represent CCE in TR.

where a l r ( i ) , C p ( i ) , and ρ ¯ ( i ) are the lapse rate, specific heat, and average density, respectively, at time index i in the TR.

The effective heat conductivity and the thermal dispersity coefficient are shown in Figure 5(a), and Figure 5(b), respectively for a single, but simultaneous CCE event in each hour along the single column in TR.

It is helpful to recall the thermodynamic energy balance for closed work cycles, such as in the upper and lower closed triangles in Figure 1, explained in the foregoing. It is stated in the art that the total thermal energy terms must be equal to the mechanical energy terms in closed cycles [11]. Without re-proving with algebra, the total thermodynamic energy terms are written for the TR, adding the extra thermal terms to those in the mechanical components in (1):

(a)

(b)

Figure 5. (a) Heat conductivity from one CCE/hr event; (b) Thermal Dispersity from one CCE/hr.

Recollect that the W s − W τ term is included as a positive turbulent energy input of the wind by initial estimation and correction. The W s − W τ is removed to the left side in the hydrostatic law in (5) for combining it with the gravitational constant as a “hydrostatic defect” for resolving P ( z ) with elevation in TR, and eliminating this mechanical term by dissolving it in the compression index via ( P 1 , T 1 ), and ( P 2 , T 2 ) where 1 is surface and 2 is upper, umbrella edge location. The Q h and Q τ are the added (or removed) thermal energy, and friction heat from W , respectively. The TR is an energy-balanced system, defined by H1, H2, T1 and T2. If the energy system absorbs W s − W τ in TR via n , and W τ = Q τ , then Q τ must be added to the thermal energy balance in the total TD equation. For a well-balanced weather system with no trade wind, W τ may be estimated to be W τ = W s / 2 .

Combining (2) with (13), incorporating W s into n , and bringing the previously discarded thermal CCE term as heat from the mechanical energy balance to the right side, and assuming the practical case of n ≠ 1 gives the full thermodynamic conservation equation:

The left side of (17) is considered to be equal to the sum of the d Q u ( i , j ) and d Q d ( i , j ) terms defined by (9a) and (9b) for [J/kg] units, as v = 0 for the vertical advection velocity component in TR.

For the specific energy in unit volume in [J/m³], (17) must be multiplied with mean value of the air densities at their entering points for mass conservation between the upper and the lower triangles in CCE. For this case, the d Q u ( i , j ) and d Q d ( i , j ) terms are defined by (11a) and (11b) for [J/m³] units.

The ground wall surface temperature varies by seconds from periodical, diurnal solar irradiation heating, and nocturnal infrared radiation cooling. The air temperature field follows course in TR, measurable by weather stations close to the ground wall and resolved from the SCS model giving a time- and elevation-dependent temperature and pressure fields. Due to thermal expansion and contraction, two processes are in play: 1) the air density changes with elevation and time; and 2) the elevations of the center of gravity of the expanding mass elements in each pixel change with time, relative to the fixed vertical coordinates of z in TR.

A straightforward way to model the elevation change for each pixel and integrate them over elevation is to calculate the change in the elevation of the center of gravity of the TR, z c g T R , due to air density change in a changed temperature and pressure field, varying in time.

The z c g T R ( i ) is numerically calculated in SCS as the first moment of the density function ρ ( i , j ) over elevation z ( j ) ;

The vertical velocity of z c g T R within each hour at time index i is written as a finite differential:

The q C E v c g ( i ) energy transported by the CCE process at every hour in the TR is calculated from the column-averaged values of the q C E u ( i ) and q C E d ( i ) terms of (11a) and (11b), expressed from (12a) and (12b) as follows:

The rate of energy density is calculated from the column-averaged q C E u ¯ ( i ) and q C E d ¯ ( i ) values by selectively multiplying each by the v c g T R ( i ) velocity component, according to the vertical flow direction:

Effective heat conductivity and thermal dispersity functions are calculated as before.

The v c g T R velocity and q C E v c g rate of energy density (power) in 1 m × 1 m × Dz [m] single column are shown in Figure 6(a), and Figure 6(b), respectively. The effective heat conductivity, driven by the variation of air density, and the corresponding, effective thermal dispersity are shown in Figure 7(a) and Figure 7(b), respectively.

(a)

(b)

Figure 6. (a). Average, vertical uplift velocity in TR; (b) The up or down CCE power in TR.

(a)

(b)

Figure 7. (a) Effective heat conductivity from air density variation; (b) Effective thermal dispersity.

The sum of the upper and lower components, q C E u ( i ) + q C E d ( i ) may be close to zero at every time index, but their individual paths work intensively as shown in Figure 4 and Figure 5 for a single CCE cycle incident in one hour in each 1 m vertical section in the TR volume. With a horizontal, 2 m/s average air velocity in the example over a 3-month period, millions of CCE cycle incidents may happen. The potential may be estimated by an accompanying function from the RMS of the large-eddy variations of the vertical velocity component of the wind velocity, of which only the horizontal component is reported routinely.

As recalled from [1], a horizontal, stochastically or chaotically varying wind may cross the REV horizontally over a level ground but may change elevation of the streamlines along the travel over surface perturbations while statistically satisfying mass conservation at every elevation in a finite, e.g. 5 min or 1 hour, time interval. Individual Carnot cycles may be driven by the elevation change, along the crisscrossing horizontal wind patterns and creating virtual, irrotational vortices. If horizontal layers travel not parallel as [ 2 → 2 1 → 1 ] but stochastically or chaotically crisscross as [ 2 ↘ 1 1 ↗ 2 ] in each dz = 1 [m] pixels as a mixing instant, the result is mixing the vertical, (2, 1) order into (1, 2) order without rotating the pixel. Shifting the position of the 1 × 1 [m] column by dx = vdt horizontal distance within the REV’s surface boundaries that may be 10 km × 10 km is still within the averag domain of the SCS model. The lack of experimental results in the literature is surprising about direct measurements of low-frequency variations of vertical wind velocity components.

The ground wall surface temperature varies by seconds from periodical, diurnal solar irradiation heating, and nocturnal infrared radiation cooling. The air temperature field follows course in TR, measurable by weather stations close to the ground wall and resolved from the SCS model giving a time- and elevation-dependent temperature field. Due to thermal expansion and contraction, two processes are in play: 1) the air density changes with elevation and with time [(already modeled in the d Q u and d Q d terms in (11a) and (11b)]; and 2) the elevations of the center of gravity of the expanding mass elements in each pixel change with time, relative to the fixed vertical coordinates of z in TR.

A straightforward way to model the elevation change for each pixel and integrate them over elevation is to calculate the change in the elevation of the center of gravity of the TR due to air density change in a changed temperature and pressure field, varying with time.

As shown in Figure 6(a), and Figure 6(b), the vertical air velocity and the rate of energy density (power) are low-frequency functions with time. The thermal energy transport measured in effective heat conductivity is higher that of the wind-enhanced CCE value, for a single CCE cycle in one hour in a full, 1 m × 1 m × Dz [m] air column. The extremally high effective heat conductivity means that the atmospheric air will vigorously seek to comply with the linear temperature distribution between points 1 and 2 in two different elevations, that is, in the single column example, between surface and the top of TR.