Semi-organized structures and turbulence in the atmospheric convection

The atmospheric convective boundary layer (CBL) consists of three basic parts: (i) the surface layer unstably stratified and dominated by small-scale turbulence of very complex nature; (ii) the CBL core dominated by the energy-, momentum- and mass-transport of semi-organized structures (large-scale circulations), with a small contribution from small-scale turbulence produced by local structural shears; and (iii) turbulent entrainment layer at the upper boundary, characterized by essentially stable stratification with negative (downward) turbulent flux of potential temperature. The energy- and flux budget (EFB) theory developed previously for atmospheric stably-stratified turbulence and the surface layer in atmospheric convective turbulence is extended to the CBL core using budget equations for turbulent energies and turbulent fluxes of buoyancy and momentum. For the CBL core, we determine global turbulent characteristics (averaged over the entire volume of the semi-organized structure) as well as kinetic and thermal energies of the semi-organized structures as the functions of the aspect ratio of the semi-organized structure, the scale separation parameter between the vertical size of the structures and the integral scale of turbulence and the degree of thermal anisotropy characterized the form of plumes. The obtained theoretical relationships are potentially useful in modeling applications in the atmospheric convective boundary-layer, and analysis of laboratory and field experiments, direct numerical simulations and large-eddy simulations of convective turbulence with large-scale semi-organized structures.


I. INTRODUCTION
Conventional theory of turbulence and methods of calculation of turbulent transport coefficients are based on the following classical paradigm [see, e.g., Refs.[1][2][3][4][5].Turbulent flow represents a superposition of the two types of motion: fully organized mean flow and fully chaotic turbulence produced, e.g., by the meanflow velocity shears.Turbulence comprises an ensemble of chaotic motions (turbulent eddies) of different scales characterized by the forward energy cascade from larger to smaller eddies.The spectrum of turbulence has an inertial range characterized by the energy flux towards smaller eddies with constant energy dissipation rate.The energy flux is balanced by the viscous dissipation at the smallest eddies at the viscous range of scales.
Accordingly, the mean flow is treated deterministically, while turbulence is described statistically.The local characteristics of turbulence (in particular, turbulent fluxes that appear in the Reynolds-averaged equations) are controlled by local features of the mean flow.The turbulent flux of any transporting property is proportional to the mean gradient of the property multiplied by appropriate turbulent-exchange coefficient.This concept of down-gradient turbulent transport reduces the turbulence-closure problem to determining the above exchange coefficients: eddy viscosity K M , eddy diffusivity K D and turbulent heat conductivity K H that, in turn, are assumed proportional to the product of turbulent ki-netic energy E K and turbulent timescale t T .The above turbulence paradigm and the concept of down-gradient transport have been formulated for the shear-generated turbulence in neutrally stratified flows [see, e.g., Refs.[1][2][3][4], and have proven applicable to a wide range of neutrally and weakly stably or unstably stratified flows.
However, there is increasing evidence of their poor applicability to both strongly stable stratification and strongly unstable stratification [see, e.g., Refs.[6][7][8][9][10][11][12][13][14][15].The present paper is devoted to atmospheric convective boundary layer (CBL), which involves besides the mean flow and Kolmogorov's turbulence, the two additional types of motion disregarded in the conventional theory: • Small-scale buoyancy-driven vertical plumes, which exhibit inverse energy transfer, namely, merge to form larger and larger plumes instead of breaking down and feeding kinetic energy of horizontal velocity fluctuations, as it should be in the case of the forward cascade [16]; • Large-scale semi-organized convective structures (energetically supplied by merging plums), which embrace the entire CBL, and perform non-local transports irrespective of mean gradients of transporting properties [see, e.g., Refs.17 -20].
We recall that the CBL in the atmosphere develops against strongly stable stratification in the free flow.This leads to the formation of comparatively thin, stably stratified turbulent entrainment layer at the CBL upper boundary.The turbulent entrainment layer separates CBL from the free atmospheric flow and acts similarly to the upper lid in laboratory experiments, causing the development of semi-organized structures: large-scale convective cells (the cloud cells) in the shear-free CBL (analogous to large-scale circulations in laboratory experiments) and large-scale convective rolls (the cloud streets) in the sheared CBL [see, e.g., Refs.19,20].The convective semi-organized structures disturb the CBL-free flow interface, which leads to exciting internal gravity waves in the free atmospheric flow and pumping the energy out of CBL [21,22].Taking into account the above processes, it is convenient to divide CBL into three basic parts (see Fig. 1): • Shallow surface layer strongly unstably stratified and dominated by vertical transport due to the small-scale three-dimensional turbulence produced by both mean-wind shears and structural convective-wind shears of semi-organized structures, and strongly anisotropic buoyancy-driven merging-plum turbulence; • Deep CBL core with preferable vertical transport due to the semi-organized structures and small contribution from three-dimensional turbulence produced by local shears of the semi-organized structures, and very small vertical gradient of the mean potential temperature; • Shallow turbulent entrainment layer at the CBL upper boundary with strong stable stratification dominated by turbulent transport and the downward turbulent flux of potential temperature.
Observations in the atmospheric CBL, laboratory experiments and large-eddy simulation (LES) confirm that convective structures principally differ from turbulent eddies.The characteristic scales of the semi-organized structures are much larger than the integral turbulence scale and their life-times are much longer than the largest turbulent time scales [see, e.g., 19,20].In the shearfree atmospheric CBL, the semi-organized structures (the cloud cells) are similar to Bernard cells.They consist of narrow uprising flows surrounded by wide downdraughts, embrace the entire CBL (up to 1-3 km height), and include pronounced convergence flows toward the cell axes in the near-surface layer, as well as divergence flows at the CBL upper boundary.In the sheared CBL, the semiorganized structures (the cloud streets) have the form of rolls stretched along the mean wind.Various features in the atmospheric convective turbulence have been studied theoretically and numerically [see, e.g., Refs.[23][24][25][26][27][28][29][30][31], and in the field experiments [see, e.g., 17,32,33], see reviews [34,35], and references therein.
Deterministic treatment of semi-organized convective structures as distinct from turbulence treated statistically has been employed to derive non-local convective heat/mass-transfer law for the shear-free CBL [17].Interesting features of convective turbulence now attributed to merging-plume mechanism implying inverse energy transfer from smaller to larger plumes were found long ago [16].The ideas that the shear-produced turbulence interacts with convective semi-organized structures in the same way as with usual mean flow have been used in a number of studies [18,19].
In the present paper we focus on physical processes in the CBL core and extend the energy-and flux budget (EFB) turbulence closure theory developed previously for atmospheric stably-stratified turbulence [12,21,22,[36][37][38], turbulent transport of passive scalar [39] and the surface layers in atmospheric convective turbulence [40], to convective turbulence in the CBL core in shear-free convection with very weak mean wind.
The EFB theory for the stably stratified turbulence explains the existence of strong turbulence produced by large-scale shear for any stratification [12,21,22,[36][37][38].The physics related to self-maintaining of a stably stratified turbulence for any stratification is caused by the following.The increase in the buoyancy due to an enhancement of the vertical gradient of the mean potential temperature results in a conversion of turbulent kinetic energy (TKE) into turbulent potential energy (TPE).This decreases the negative down-gradient vertical turbulent flux of potential temperature by a positive non-gradient turbulent flux of potential temperature originated from enhanced TPE.This mechanism of the self-control feedback decreases the buoyancy and maintains stably stratified turbulence for any stratification [5,39,40] in agreement with wide experimental evidence [6-8, 10, 11, 13, 15, 41].
The EFB theory for the surface layers in atmospheric convective turbulence [40] describes a smooth transition between a stably-stratified turbulence and a convective turbulence, providing analytical expressions for the vertical profiles for all turbulent characteristics in the entire surface layer including TKE, the intensity of turbulent potential temperature fluctuations, the vertical turbulent fluxes of momentum and buoyancy (proportional to potential temperature), the integral turbulence scale, the turbulence anisotropy, the turbulent Prandtl number and the flux Richardson number.The obtained analytical vertical profiles describe also the transition range between the lower and upper parts of the surface layer.
The EFB theory for the CBL core developed in the present study, is based on the threefold decomposition: mean flow, semi-organized strictures and small-scale turbulence, in combination with analytical description of convective semi-organized structures in the shear-free CBL.We find the global turbulent characteristics (averaged over the entire volume of the semi-organized structure) and kinetic and thermal energies of the semiorganized structures, which depend on the aspect ratio of the semi-organized structure and scale separation parameter between the vertical size of the structures and the integral scale of turbulence.The obtained theoretical relationships are potentially useful in modeling applications in the atmospheric convective boundary-layer.This paper is organized as follows.In Sec.II we discuss turbulent flux of potential temperature and its effect on the formation of semi-organized structures.In this section we derive expressions for the velocity and potential temperature of convective semi-organized structures.In Sec.III we study turbulence in the CBL core, starting with budget equations for turbulent energies and turbulence fluxes of potential temperature and momentum (Sec.III A), and formulate main assumptions for the energy and flux budget turbulence closure theory for convective turbulence (Sec.III B).In the framework of this theory, we derive expressions for the global characteristics of convective turbulence and semi-organized structures (Sec.III C).In Sec.IV we consider a transition from the convective surface layer to the CBL core, where we perform a matching between the solutions obtained for the convective surface layer and the CBL core.Finally, in Sec.V we discuss the obtained results and draw conclusions.

II. TURBULENT FLUX OF POTENTIAL TEMPERATURE AND SEMI-ORGANIZED STRUCTURES
The convective boundary layer involves three principally different types of motion: • regular plain-parallel mean flow homogeneous in the horizontal plain (coordinates x, y), but heterogeneous in the vertical (coordinate z); • vertically and horizontally heterogeneous long-lived CBL-scale semi-organized convective structures; and • small-scale turbulence.
We use capital letters with superscript (m) to denote the mean-flow fields of wind: , pressure P (m) and potential temperature Θ (m) ; capital letters with superscript (s) to denote the semi-organized structure fields, U (s) = U (s) x , U y , U , and P (s) , Θ (s) ; lower-case letters to denote turbulent fields: u, p and θ; and just capital letters to denote actual (total) fields, e.g., the total velocity is U = U (m) + U (s) + u, the total pressure is P = P (m) + P (s) + p and the total potential temperature is Θ = Θ (m) + Θ (s) + θ.In the present study, we consider a shear-free convection with negligible mean flow field, U (m) = 0, but with a non-zero vertical gradient of the mean potential temperature ∇ z Θ (m) = 0.
Turbulent fluxes of potential temperature and momentum are defined as F = u θ and τ ij = u i u j , respectively, where angle brackets denote ensemble averaging.For the sake of definiteness, we restrict our consideration to dry atmosphere, so that buoyancy is proportional to the potential temperature Θ = T (P * /P ) 1−γ −1 , where T is the fluid temperature with the reference value T * , P is the fluid pressure with the reference value P * and γ = c p /c v is the specific heat ratio.The familiar downgradient approximation of the turbulent fluxes of potential temperature and momentum reads: , K H and K M are turbulent heat conductivity and turbulent viscosity traditionally treated as scalars [1].
On the other hand, there are long-lived CBL-scale semi-organized convective structures and the velocity field inside large-scale convective structures is strongly nonuniform.These nonuniform motions can produce anisotropic velocity fluctuations which can contribute to the turbulent flux of potential temperature.In particular, the classical turbulent flux of potential temperature, F = −K H ∇Θ, does not take into account the contribution from anisotropic velocity fluctuations.

A. Turbulent flux of potential temperature
The contribution to the turbulent flux of potential temperature from anisotropic velocity fluctuations plays a crucial role in the formation of large-scale semi-organized structures in turbulent convection.Indeed, the turbulent flux of potential temperature F which takes into account anisotropic velocity fluctuations reads [18,19]: where m) is the classical background turbulent flux of potential temperature in the absence of nonuniform large-scale flows, α is the degree of thermal anisotropy that characterizes the form of plumes and is defined by Eq. ( 18), t T is the characteristic turbulent time at the integral turbulent scale, is the mean vorticity characterized the semi-organized structure, the mean velocity z that is decomposed into the horizontal U  [18,19,23,42] that these new contributions cause the excitation of large-scale convectivewind instability and the formation of large-scale semiorganized structures.
The mechanism of the large-scale convective-wind instability is related to the second term h in Eq. ( 1) for the turbulent flux of potential temperature, which causes the redistribution of the vertical background turbulent flux of potential temperature F * z by the perturbations of the convergent (or divergent) horizontal large-scale flows U (s) h (see Fig. 2) during the life-time of turbulent eddies.Therefore, this effect increases the vertical turbulent flux of potential temperature by the converging horizontal motions, which enhances both, the upward (positive) turbulent flux of potential temperature and buoyancy.The latter forms the upward flow and strengthens the horizontal convergent flow, resulting in the large-scale convective-wind instability.
On the other hand, the last term ∝ [(2α + 3)/10] t T (W (s) ×F * z ) in Eq. ( 1) produces the horizontal turbulent flux of potential temperature by a "rotation" of the vertical background turbulent flux F * z with the perturbations of the horizontal mean vorticity W h , decreasing the local potential temperature in rising motions.The latter weakens the buoyancy acceleration, and reduces perturbations of the vertical large-scale velocity and vorticity, contributing to the damping of large-scale convectivewind instability [18,19,23].

B. Analytical solution for semi-organized structures
Let us determine the large-scale velocity U (s) and potential temperature Θ (s) of the semi-organized structures formed in small-scale convective turbulence.Note that the timescale of the growth of the CBL height is much larger than the characteristic time scale of evolution of the semi-organized structures.During the formation of these coherent structures there is a two-way nonlinear coupling: • the effect of small-scale turbulent convection on the formed semi-organized coherent structures; and • a back-reaction of the formed semi-organized coherent structures on small-scale turbulent convection.
The velocity and potential temperature inside the semiorganized structures are strongly non-uniform, causing an anisotropy of convective turbulence.In particular, the convective plumes are extended in vertical direction in regions with strong buoyancy.To describe such complicated process, we have to solve nonlinear equations for turbulence and mean-field equations for semi-organized structures simultaneously.This cannot be done analytically for large Reynolds and Rayleigh numbers.To solve this problem, we apply another approach.We use a linearized mean-field equations for the velocity and potential temperature of the semi-organized coherent structures with the parameterized turbulent flux of potential temperature determined by Eq. ( 1).This allows us to describe direct coupling of small-scale convective turbulence with the formed semi-organized coherent structures.However, to take into account back-reaction of the formed semi-organized coherent structures on small-scale convective turbulence, we introduce a phenomenological parameter α that determines a thermal anisotropy and describes an anisotropic form of plumes [see Eq. ( 18) in Sec.III C].In such phenomenological approach we take into account the back-reaction of the coherent structures on small-scale convective turbulence.
The equations for evolution of the vorticity W (s) and potential temperature Θ (s) of semi-organized structures in a fluid flow in the Boussinesq approximation are given by where the turbulent flux of potential temperature F is given by Eq. ( 1).Here e is the vertical unit vector, β = g/T * is the buoyancy parameter, g is the gravity acceleration and D/Dt = ∂/∂t + U (s) • ∇.We restrict our consideration to the quasi-stationary regime and search for axisymmetric (ϑ-independent) solution to equations for the vorticity W (s) and potential temperature Θ (s) , in cylindrical coordinates (r, ϑ, z), where the velocity and vorticity W (s) are expressed in terms of the stream function Ψ as Taking the eddy viscosity K M and eddy conductivity K H independent of r and z, linearized equations ( 2) and ( 3), we obtain the following steady-state solution: where and J m (x) is the Bessel function of the first kind possessing the properties: Here λ = 3.83 is the first root of equation J 1 (x) = 0, so that J 1 (λ) = 0, and A * = π R/(λ L z ) is aspect ratio of the semi-organized structures, where R and L z are the radius and height of the semi-organized structures, respectively; U z0 , Θ 0 and Ψ 0 = U z0 L z are amplitudes of the velocity, temperature and streamfunction, respectively.Substituting Eqs. ( 6)-(10) into Eqs.( 2)-( 5), we obtain relationships for the amplitudes and the aspect ratio of the semi-organized structures as and Ψ 0 = U z0 R/λ, where Pr T = K M /K H is the turbulent Prandtl number, σ = 4 (8α − 3)/45 and µ = (2α + 3)/(8α−3), and α is the degree of thermal anisotropy defined by Eq. ( 18) in Sec.III C. The above solution mimic comparatively narrow uprising flow surrounded by wider and weaker downdraught in reasonable agreement with large-eddy simulations [20].

III. TURBULENCE IN CBL CORE
To describe turbulence in the CBL core, we use the budget equations for the density of turbulent kinetic energy, the intensity of potential temperature fluctuations and turbulent fluxes of potential temperature and momentum in the Boussinesq approximation (see, e.g., Ref. [40]).We consider a shear-free turbulent convection with negligibly small mean velocity U (m) in comparison with the velocity U (s) of the semi-organized structures.

A. Budget equations
We start with the basic equations of the energy and flux budget (EFB) closure theory.The budget equation for the density of turbulent kinetic energy (TKE), E K = u 2 /2, the intensity of potential temperature fluctuations E θ = θ 2 /2, the turbulent flux F i = u i θ of potential temperature, and the Reynolds stress τ ij = u i u j are given by where tive and δ ij is the Kronecker unit tensor.The first term, i , on the RHS of Eq. ( 13) is the rate of production of TKE by the gradients of the velocity U (s) of the semi-organized structures.In particular, turbulence in the CBL-core is produced largely by local shears in semiorganized structures.We will show that the turbulent kinetic energy is very low compared to kinetic energy of motions in semi-organized structures.These conclusions are fully confirmed by LES and various laboratory experiments (see, e.g., Ref. [20], and references therein).The second production term β F z in Eq. ( 13) describes buoyancy.The first two terms, on the RHS of Eq. ( 14) are the rates of production of potential temperature fluctuations, where the first contribution due to semi-organized structures is dominant one.
The first term, −τ iz ∇ z Θ (m) , on the RHS of Eq. ( 15) contributes to the turbulent flux of potential temperature due to the small vertical gradient of the mean potential temperature, while the second term, −τ ij ∇ j Θ (s) , contributes to the turbulent flux caused by the semiorganized structures.Both terms correspond to the classical gradient mechanism of the turbulent heat transfer.The third term 2β E θ δ i3 on the RHS of Eq. ( 15) describes a non-gradient contribution to the turbulent flux of potential temperature.The term, ε K = ν (∇ j u i ) 2 , in the RHS of Eq. ( 13) is the dissipation rate of the density of the turbulent kinetic energy, where ν is the kinematic viscosity of fluid.The term, ε θ = χ (∇θ) 2 in the RHS of Eq. ( 14) is the dissipation rate of the intensity of potential temperature fluctuations E θ , and χ is the molecular temperature diffusivity.The term, ε on the RHS of Eq. ( 15) is the dissipation rate of the turbulent flux of potential temperature.The term, ε 16) is the molecular-viscosity dissipation rate, and the tensor Q ij = ρ −1 0 ( p∇ i u j + p∇ j u i ) describes correlations of pressure fluctuations and turbulent velocity gradients.

B. Basic assumptions
In the framework of the energy and flux budget turbulence closure theory, we assume the following.The characteristic times of variations of the densities of the TKE, the potential temperature fluctuations intensity, the turbulent flux of potential temperature and the Reynolds stress are substantially longer than the turbulent timescales.This assumption yields the steady-state solutions of the budget equations ( 13)-( 16).
This allows one to express the dissipation rates of E K , E θ and F i applying the Kolmogorov hypothesis.This implies that ε K = E K /t T , ε θ = E θ /(C p t T ), and ε is the turbulent dissipation timescale, ℓ z is the vertical integral scale, and C p and C F are dimensionless empirical constants.In addition, the dissipation rates ε α of the TKE components [44]), since the dominant contribution to E α is from the Kolmogorov viscous scale where turbulence is nearly isotropic.Here the summation convention for the double Greek indices is not applied.
The term ε 16) is the effective dissipation rate of the off-diagonal components of the Reynolds stress τ iz [12,36,39].The dissipation rate of τ iz is assumed to be due to the combination ε Here C τ is the effective-dissipation timescale empirical constant [12,36,39,40].
The effective dissipation assumption was justified by Large Eddy Simulations (see Fig. 1 in Ref. [12]), where the data from Ref. [45,46] were used for the two types of atmospheric boundary layer: "nocturnal stable" (with essentially negative buoyancy flux at the surface and neutral stratification in the free flow) and "conventionally neutral" (with a negligible buoyancy flux at the surface and essentially stably stratified turbulence in the free flow).The effective dissipation assumption directly yields the well-known down-gradient formulation of the vertical turbulent flux of momentum u i u z = −K M ∇ z U i , where i = x, y, and The latter result is valid for a shear-produced turbulence or a convective turbulence with a non-uniform large-scale velocity field.We point out that the diagonal components of the Reynolds stress are much larger than the offdiagonal components.The diagonal components of the Reynolds stress determines the TKE components which obey the Kolmogorov spectrum ∝ k −5/3 , while the offdiagonal components of the Reynolds stress are produced by the tangling mechanism of generation of anisotropic velocity fluctuations, and they obey the ∝ k −7/3 spectrum [47].
The final assumption is related to the term ρ −1 0 θ ∇ z p in the budget equation for F z which is parameterized as β θ 2 − ρ −1 0 θ ∇ z p = 2C θ β E θ , where C θ < 1 is the positive dimensionless empirical constant.The latter assumption has been justified analytically (see Appendix A in [36]) and by Large Eddy Simulations, where the data from Refs.[45,46] have been used for the two types of at-mospheric boundary layer: "nocturnal stable" and "conventionally neutral" (see Fig. 2 in [12]).
For convective turbulence, we choose the following values of the non-dimensional empirical constants which have been used in the EFB theory for stably stratified turbulence [39,40]: C p = 0.417, C θ = 0.744, C τ = 0.1 and C F = 0.125.This corresponds to the turbulent Prandtl number for a non-stratified turbulence Pr (0) T = 0.8.

C. Global characteristics of convective turbulence with semi-organized structures
In this section we determine global characteristics of convective turbulence by averaging over the entire volume of the semi-organized structure.As follows from laboratory experiments [48,49], direct numerical simulations [50][51][52][53], and mean-field numerical simulations [42], the vertical gradient of the mean potential temperature can be positive and negative inside the large-scale circulations in a convective turbulence.In particular, the vertical gradient of the mean potential temperature can be positive when the vertical turbulent flux of potential temperature is negative.
To describe this effect, we introduce the degree of thermal anisotropy α in convective turbulence that depends on the form of plumes.In particular, the plumes can be characterized by the two-point instantaneous correlation function θ(t, x) u z (t, x + r) , where ℓ (pl) h and ℓ (pl) z are the horizontal and vertical scales in which the two-point instantaneous correlation functions θ(t, x) u z (t, x + r) tend to 0 in the horizontal and vertical directions, respectively.The degree of thermal anisotropy α is defined as [18] For the isotropic case, ℓ when the plumes have the form of ball and the degree of thermal anisotropy α = 1.For α < 1, the plumes are extended in the vertical direction having the form of columns, ℓ z , the parameter α can be estimated as For α > 1, the plumes have the form of "pancake", ℓ z , the parameter α can be estimated as Using Eqs. ( 12) and ( 17), we determine the normalized vertical turbulent flux of potential temperature averaged over the entire volume of the semi-organized structure as where ... V denote averaging over the entire volume of the semi-organized structure, ℓ is the integral scale of the turbulence, F = β F z ℓ/E 3/2 K , function f F (A * ) is given by Eq. (A1) in Appendix A. We remind that the aspect ratio of the semi-organized structures is A * = π R/(λ L z ) (see Sec. II).Here we assume that the turbulent dissipation timescale is K and the vertical anisotropy parameter is As follows from Eqs. (21) and Eq.(A1) in Appendix A, the vertical flux of potential temperature F z V is negative when f F (A * ) < 0, i.e., 2α 4A 2 * − 1 < 3 1 + A 2 * .This implies that the vertical flux of potential temperature F z V is negative when where we use Eq.(19).Note that the large-scale circulations are formed when A * ≥ 1 [18,19].Applying Eqs. ( 18) and ( 22), we obtain that the vertical flux of potential temperature F z V is negative when In Fig. 3, we plot the normalized vertical turbulent flux of the potential temperature F versus the aspect ratio 2R/L z of the semi-organized structures for different values of the degree of thermal anisotropy α.Here we take into account that semiorganized structures are formed when the scale separation parameter is L z /ℓ z > 5 and the aspect ratio for the semi-organized structures is 2R/L z ≥ 2 [18,19].It is seen in Fig. 3 that when plumes are extended in the vertical direction, i.e., ℓ In Fig. 4, we also show the dependence of the normalized vertical turbulent flux of the potential temperature F V on the aspect ratio 2R/L z of the semi-organized structures for α = −0.55(ℓ (pl) h /ℓ (pl) z = 0.5) and different values of the scale separation parameter L z /ℓ z .The absolute value of the normalized vertical turbulent flux of potential temperature decreases with increase of scale separation between vertical size of the semi-organized structures L z and the vertical integral scale ℓ z .
As follows from Eq. ( 21) and Figs.2-3, that the normalized vertical turbulent flux of potential temperature averaged over the entire volume of the semi-organized structure is small ( F V ≪ 1) because the vertical integral scale ℓ z is much smaller than the vertical size L z of the semi-organized structure.Note also the coefficient C τ is small.This means that the volume averaged TKE dissipation rate is much larger than the turbulence production rate, β F z V , caused by buoyancy.This effect is due to the fact that for large Rayleigh numbers, the convective turbulence is mainly produced by the local shear of the semi-organized structures rather than the buoyancy (see below).
The production rate, Π K , of the turbulent kinetic energy by local large-scale shear ∇ j U (s) i of the semiorganized structures is given by Π see the first term on the RHS of Eq. ( 13)], where S is the large-scale shear.Using the steady-state version of the budget equation ( 13), we obtain that turbulent kinetic energy is To find the production rate of the turbulent kinetic energy averaged over the entire volume of the semi-organized structure, we use the analytical solution ( 6)-( 7) for the velocity U (s) of the semi-organized structure.This yields the averaged squared large-scale shear S 2 V given by Eq. (B1) in Appendix B. Therefore, the turbulent kinetic energy density averaged over the entire volume of the semi-organized structure, is given by where the function f S (A * ) is given by Eq. (A3) in Appendix A. Equation (24) implies that the turbulent kinetic energy density E K V is much less than the squared velocity , transported by the semi-organized structures, is determined using Eqs.( 11) and ( 24), and Eq.(B6) in Appendix B: where function f Fs (A * ) is given by Eq. (A4) in Appendix A. By means of Eq. ( 25), we find a characteristic convective velocity U D defined as Now we define a characteristic convective temperature Θ D from a condition Using the definition (26) of the convective velocity U D , we obtain a relation between the convective velocity U D and the convective temperature Θ D as . By means of Eq. ( 26) and Eq.(B6) in Appendix B, we find the convective temperature Θ D as The velocity U D and temperature Θ D characterize the large-scale properties of convection.Let us determine the global energetic characteristics of semi-organized structures.Equations ( 26) and (B7) in Appendix B yield an expression for the kinetic energy density of semi-organized structures as where function f U (A * ) is given by Eq. (A5) in Appendix A. In Fig. 5, we show the normalized velocity Ũ = 2 E U V /U D versus the aspect ratio 2R/L z of the semiorganized structures for different values of the scale separation parameter L z /ℓ z .The kinetic energy density E U V of the semi-organized structures increases with increase of the scale separation parameter L z /ℓ z [see Eq. ( 29)].Note that the kinetic energy density E U V is nearly independent of the parameter α (and the ratio ℓ z ) which characterizes the thermal anisotropy of convective turbulence.
Equation (28) and Eq.(B8) in Appendix B yield the thermal energy density of the semi-organized structures as where function f Θ (A * ) is given by Eq. (A6) in Appendix A. In Fig. 6, we plot the normalized potential temperature Θ = 2 E Θ V /Θ D versus the aspect ratio 2R/L z of the semi-organized structures for different values of the scale separation parameter L z /ℓ z .Equation (30) and Fig. 6 demonstrate that the thermal energy density of the semi-organized structure E Θ V increases with the aspect ratio 2R/L z of the semi-organized structures approaching to the value which is of the order of Θ 2 D .Equations ( 29) and ( 30) imply that the flux Ũ Θ of the potential temperature transported by the semiorganized structures is independent of the scale separation parameter L z /ℓ z .
Using Eqs. ( 24) and ( 26), we express the turbulent kinetic energy density E K V in terms of the squared convective velocity U 2 D as where function f u (A * ) is given by Eq. (A7) in Appendix A. Equation (31) implies that the turbulent kinetic energy density is much smaller than the squared velocity U 2 D .Equations ( 29) and ( 31) allow to determine the ratio of the turbulent kinetic energy density E K V to the kinetic energy density of semi-organized structures In Fig. 7, we plot the ratio of the turbulent kinetic energy density to the kinetic energy density of semi-organized structures, E K V / E U V versus the aspect ratio 2R/L z of the semi-organized structures for different values of the scale separation parameter L z /ℓ z .As follows from Eq. ( 32) and Fig. 7, the turbulent kinetic energy density E K V is much smaller than the kinetic energy density of semi-organized structures E U V .This is because the vertical integral scale ℓ z is much smaller than the vertical size L z of the semi-organized structure.Indeed, intensity of velocity fluctuations can be estimated as , where we take into account that the production rate of the turbulent kinetic energy density is due to the local large-scale shear of the  21), ( 24) and ( 25) allow us to determine the ratio of the vertical turbulent flux of potential temperature F z V to the vertical flux of potential temperature

semi-organized structures
where function f FT (A * ) is given by Eq. (A8) in Appendix A. In Fig. 8 we show the normalized vertical turbulent flux of the potential temperature F z V /(Θ D U D ) versus the aspect ratio 2R/L z of the semi-organized structures for α = −0.55(i.e., for ℓ (pl) z = 0.5) and different values of the scale separation parameter L z /ℓ z .It follows from Eq. ( 33) and Fig. 8 that the vertical flux of potential temperature, Θ , transported by the semi-organized structures is much larger than the vertical turbulent flux, F z V , of potential temperature, i.e., This is because the vertical integral scale ℓ z is much smaller than the vertical size L z of the semi-organized structure.Indeed, the vertical turbulent flux can be estimated as , where we take into account that u 2 z ∼ t T K M S 2 , the shear is estimated as S ∼ U D /L z , the gradient of the mean potential temperature is estimated as ∇ z Θ (s) ∼ Θ D /L z and t T S ≤ 1.Note that the analysis of large-scale instability in shear-free convection [18,19] shows that the semiorganized structures are formed when the scale separation parameter L z /ℓ z > 5.
Using Eqs.(B10)-(B14) in Appendix B, we obtain expression for the turbulent thermal energy density E θ V  as where function f θ (A * ) is given by Eq. (A9) in Appendix A. Thus, the ratio of the turbulent thermal energy density E θ V to the thermal energy density of the semiorganized structures E Θ V is given by where we use Eq.(30).In Fig. 9 we show the ratio of the turbulent thermal energy density to the thermal energy density of the semi-organized structures, E θ V / E Θ V , versus the aspect ratio 2R/L z of the semi- organized structures for different values of the scale separation parameter L z /ℓ z .Equation ( 35) and Fig. 9 demonstrate that the turbulent thermal energy E θ V is much smaller than the thermal energy of the semiorganized structures E Θ V .This is because the vertical integral scale ℓ z is much smaller than the vertical size L z of the semi-organized structure.Indeed, the turbulent thermal energy density can be estimated as , where we take into account that the production rate of the potential temperature fluctuations is estimated as , the vertical turbulent flux of the potential temperature is F z ∼ −K H ∇ z Θ (s) and the gradient of the mean potential temperature is estimated as Using Eqs. ( 33) and (B16) in Appendix B, we obtain the expression for the vertical gradient of the mean potential temperature, where function f ∇ (A * ) is given by Eq. (A11) in Appendix A. In Fig. 10 we plot the normalized vertical gradient of the mean potential temperature, m) , versus the aspect ratio 2R/L z of the semi-organized structures for different values of the scale separation parameter L z /ℓ z .As follows from Eq. (36) and Fig. 10 that the normalized vertical gradient of the mean potential temperature ∇ z Θ is small and positive, because for the considered parameter range [see Eq. ( 22)] the vertical turbulent flux of the potential temperature F z V is negative as well as the function f ∇ (A * ) is negative.
Therefore, the global characteristics of a convective turbulence depend on the aspect ratio of the semi-organized structures, the scale separation parameter between the vertical size L z of the structures and the integral scale of turbulence ℓ z , and the degree of thermal anisotropy (i.e., the form of plumes).In the limit of large aspect ratio of the semi-organized structures, the global turbulence characteristics reaches their asymptotic values which depend on the degree of thermal anisotropy and the scale separation parameter (see Figs. [2][3][6][7][8][9].

IV. TRANSITION FROM CBL CORE TO CONVECTIVE SURFACE LAYER
In this Section we discuss a matching of solutions obtained for the CBL core and the convective surface layer.We start with the solutions obtained for the convective surface layer.

A. Convective surface layer
First, we outline the results of the EFB theory for the atmospheric convective surface layer [40].The vertical profiles of various turbulent characteristics are determined by the following equations: where Ri f = −| Z| for | Z| ≪ 1 and Ri f = − Z4/3 for | Z| ≫ 1; • the turbulent viscosity, where • the turbulent Prandtl number, where Pr T = Pr (0) where • the large-scale shear, • the vertical gradient of the mean potential temperature, where Here Ri where E K0 = u 2 * /(2C τ A z ) 1/2 , and ẼK = 1 + | Z|/2 for | Z| ≪ 1 and ẼK = Z2/3 for | Z| ≫ 1.Using Eq. ( 41), we obtain the mean horizontal velocity at the surface layer as For illustration, the vertical profile (44) of the normalized mean horizontal velocity U h (z)/u * at the surface layer is shown in Fig. 11, where we use the numerical solution of nonlinear equation (43) (see Ref. [40]).

B. Matching of solutions for CBL core and convective surface layer
The matching between the solutions obtained for the CBL core and the convective surface layer are performed as follows., and we take into and (44), we obtain the ratio of the friction velocity u * to the convective velocity U D as where The ratio of the friction velocity u * to the maximum value U r is given by u * (r) where U r (r = r max , z → 0) and r max is the radius at which the function J 1 (Y ) reaches the maximum value.In Fig. 12 we show the radial profiles of the ratios L O /L z and u * /U (max) r , which demonstrate that these ratios are small.Therefore, this analysis allows us to connect the global turbulent characteristics in the atmospheric CBL-core with the basic characteristics of the convective surface layer.

V. CONCLUSIONS
In the present study we investigate essential features of turbulence and semi-organized structures in the core of the atmospheric convective boundary-layer (the CBL core) by means of the energy-and flux budget (EFB) theory.
• Using the analytical solution ( 6)- (10) for the semiorganized structures and budget equations ( 13)-( 16) for the basic second moments in convective turbulence, we find the global characteristics (averaged over the entire volume of the semiorganized structure) including turbulent kinetic energy density and intensity of potential temperature fluctuations, turbulent flux of potential temperature, as well as the kinetic and thermal energy densities of the semi-organized structures.
• Applying the analytical description for plumes based on the two-point instantaneous correlation functions θ(t, x) u z (t, x + r) , we connect the global characteristics of convective turbulence and semi-organized structures with degree of the thermal anisotropy α defined by Eq. ( 18) and characterized the form of plumes.
• We demonstrate that when plumes are extended in the vertical direction (ℓ ), the vertical turbulent flux of potential temperature aver-aged over the entire volume of the semi-organized structure F z V is negative (see Fig. 3), where ℓ (pl) h and ℓ (pl) z are the horizontal and vertical scales in which the two-point instantaneous correlation functions θ(t, x) u z (t, x+r) characterized the plumes, vanish in the horizontal and vertical directions, respectively.
• When ℓ • The turbulent kinetic energy density E K V is much smaller than the kinetic energy density of semi-organized structures E U V [see Eq. ( 32) and Fig. 7].Increase of the scale separation between the vertical size L z of the semi-organized structures and the vertical integral scale of turbulence ℓ z , increases the kinetic energy density E U V of the semiorganized structures and the ratio E U V / E K V .
• The turbulent thermal energy density E θ V is much smaller than the thermal energy density of the semi-organized structures E Θ V [see Eq. (35) and Fig. 9].
• The global turbulence characteristics depend on the aspect ratio of the semi-organized structure, the scale separation parameter between the vertical size L z of the structures and the integral scale of turbulence ℓ z , and the degree of thermal anisotropy (i.e., the form of plumes).In the limit of large aspect ratio of the semi-organized structures, these global turbulence characteristics reaches their asymptotic values which depend on the degree of thermal anisotropy and the scale separation parameter.
• We connect the global turbulent characteristics in the atmospheric CBL-core with the basic characteristics of the convective surface layer.This analysis yields the ratio of the local Obukhov length scale, L O = −u 3 * /(β Fz ), for the convective surface layer to the vertical size of the semi-organized structure L z as well as the ratio of the friction velocity u * to the maximum value of the horizontal velocity of the semi-organized structure U (s) r [see Eqs.(45) and (47), as well as Fig. 12].
The obtained theoretical results are important for modeling applications in the atmospheric convective boundarylayer.These results are also very useful for analysis of laboratory and field experiments, direct numerical simulations as well as large-eddy simulations of convective turbulence with large-scale semi-organized structures.

DEDICATION
This paper was dedicated to Prof. Sergej Zilitinkevich (1936Zilitinkevich ( -2021) ) who initiated this work and discussed some of the obtained results.Data sharing is not applicable to this article as no new data were created or analyzed in this study.= (ν + χ) (∇ j u i ) (∇ j θ) dissipation rate of the turbulent heat flux ε K = ν (∇ j u i ) 2 dissipation rate of E K ε α = ν (∇ j u α ) 2 dissipation rate of horizontal and vertical turbulent kinetic energy density components E α ε θ = χ (∇θ) 2 dissipation rate of E θ

FIG. 1 .
FIG. 1. Structure of the atmospheric convective boundary layer (CBL), where U (s) and W (s) = ∇×U (s) are the velocity and vorticity characterising the semi-organized structures in CBL.

h
and vertical U (s) z components.The new contributions to the turbulent flux of potential temperature are caused by anisotropic velocity fluctuations and depend on the mean velocity gradients of the nonuniform large-scale flow.It has been demonstrated in Refs.
other words, the contribution to the turbulent flux of potential temperature ∝ [(2α + 3)/10] t T (W (s) ×F * z ) creates the horizontal turbulent flux of potential temperature via rotation of the vertical turbulent flux F * z by the large-scale horizontal vorticity, W

FIG. 2 .
FIG. 2. The mechanism of the large-scale convective-wind instability associated with the new contribution of the turbulent flux of potential temperature Fnew = −tT α F * z div U (s) h , which increases (or decreases) the vertical turbulent flux of potential temperature shown by the red arrow in b (or by the blue arrow in d) via redistribution of the uniform vertical turbulent flux F * z by convergent (or divergent) horizontal mean flows U (s) h (shown by the green arrows in a and c).The vertical turbulent flux Fnew enhances the upward (positive) turbulent flux of potential temperature, increasing the local mean potential temperature and producing the upward large-scale flow.Likewise, the vertical turbulent flux Fnew decreases the vertical turbulent flux of potential temperature by the divergent horizontal motions, decreasing the local mean potential temperature and producing the downward large-scale flow.

20 FIG. 7 .
FIG. 7.The ratio of the turbulent kinetic energy density to the kinetic energy density of semi-organized structures, EK V / EU V versus the aspect ratio 2R/Lz of the semiorganized structures for α = −0.55 and different values of the scale separation parameter Lz/ℓz = 7 (solid line); 8 (dashed line) and 10 (dashed-dotted line).

2 FIG. 9 .
FIG. 9.The ratio of the turbulent thermal energy density to the thermal energy density of the semi-organized structures, E θ V / EΘ V , versus the aspect ratio 2R/Lz of the semiorganized structures for α = −0.55 and different values of the scale separation parameter Lz/ℓz = 7 (solid line); 8 (dashed line) and 10 (dashed-dotted line).
where S( Z) = u * /(κ 0 z) for | Z| ≪ 1 and S( Z) = (u * /|L O |) Z−4/3 for | Z| ≫ 1; is the flux Richardson number, Fz is the vertical turbulent flux of the potential temperature at the surface layer, u 2 * = K M S with u * being the local (z-dependent) friction velocity, S = dU h /dz is the large-scale shear at the surface layer, U h (z) is the mean horizontal velocity at the surface layer, θ * = | Fz |/u * = u 2 * /β |L O | with L O being the local Obukhov length defined as L O = −u 3 * /(β Fz ), the normalized height Z = κ 0 z/L O with κ 0 = 0.4 being the von Karman constant, and Pr (0) T = C τ /C F is the turbulent Prandtl number for a non-stratified turbulence at Ri f = 0. Note that the flux Richardson number Ri f is negative in the convective turbulence in surface layer, and its absolute value is not limited and can be large.The local Obukhov length L O is negative in the convective turbulence as well, but the product L O Ri f is positive.The vertical profile of the normalized turbulent kinetic energy density, ẼK ( Z) = E K ( Z)/E K0 , is determined by the following nonlinear equation:

20 FIG. 11 .
FIG.11.The vertical profile of the normalized mean horizontal velocity U h (z)/u * at the surface layer.
of the horizontal velocity of the semi-organized structure U (s)

< 1 ,
Fig. 10].The gradient ∇ z Θ (m) increases with decrease of the scale-separation parameter L z /ℓ z .•We demonstrate that the vertical flux of potential temperature Θ (s) U (s) z

NOMENCLATUREAV
x,y = E x,y /E K horizontal anisotropy parameters A z = E z /E K vertical anisotropy parameter A * = π R/(λ L z ) aspect ratio of the semi-organized structures E K = u 2 /2 density of turbulent kinetic energy (TKE) ẼK = E K /E K0 normalized density of turbulent kinetic energy E K0 = u 2 * /(2C τ A z ) 1/2 density of turbulent kinetic energy at the surface E z = u 2 z /2 density of the vertical turbulent kinetic energy E α = u 2 α /2 horizontal and vertical turbulent kinetic energies (α = x, y, z) E U V kinetic energy density of the semi-organized structureE θ = θ 2 /2 intensity of potential temperature fluctuations E Θ V thermal energy density of semi-organized structure F i = u i θ turbulent flux of potential temperature F x,y horizontal turbulent flux of potential temperature F z = u z θ vertical turbulent flux of potential temperature F secondary vertical turbulent flux F V ≡ β F z V ℓ/ E 3/2 KV normalized averaged vertical turbulent flux of the potential temperature g gravity acceleration J m (x) Bessel function of the first kind K H turbulent heat conductivityK M turbulent (eddy) viscosity L O = −τ 3/2 /(β F z ) local Obukhov length L z height of the semi-organized structure ℓ z vertical integral scale ℓ (pl) h characteristic horizontal scale of plumes ℓ (pl) z characteristic vertical scale of plumes N = (β |∇ z Θ|) 1/2 Brunt-Väisälä frequency Q ij = ρ −1 0 ( p∇ i u j + p∇ j u i ) inter-component energy exchange term Q αα = 2ρ −1 0 ( p∇ α u α diagonalterms of the tensor Q ij p fluctuations of the fluid pressure P = P (m) +P (s) +p total fluid pressure with the reference value P * P (m) mean fluid pressure P (s) mean fluid pressure related to the large-scale semiorganized structures Pr T = K M /K H turbulent Prandtl number Pr (0) T = C τ /C F turbulent Prandtl number for a nonstratified turbulence R radius of the semi-organized structure Ri = N 2 /S 2 gradient Richardson number Ri f = −β F z /(K M S 2 ) flux Richardson number R ∞ = Ri f (Ri → ∞) flux Richardson number at very large gradient Richardson number S mean velocity shear caused by semi-organized structure S mean velocity shear in the surface layer t T = ℓ z /E 1/2 z turbulent dissipation timescale T fluid temperature with the reference value T * u = (u x , u y , u z ) fluctuations of the fluid velocity u * local (z-dependent) friction velocity U = U (m) + U (s) + u total velocity U D characteristic convective velocity U = U (m) + U (s) total mean velocity U (s) mean velocity related to the semi-organized structure U (m) (z) = (U x , U y , 0) mean-wind velocity U (s) h horizontal component of mean velocity related to the semi-organized structure U (s) z vertical component of mean velocity related to the semi-organized structures W (s) = ∇×U (s) mean vorticity characterized the semiorganized structure Z = κ 0 z/L O normalized height α degree of thermal anisotropy β = g/T * buoyancy parameter γ = c p /c v specific heat ratio δ ij Kronecker unit tensor ε (F) i