The energy-and ﬂux budget theory for surface layers in atmospheric convective and stably stratiﬁed turbulence

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I. INTRODUCTION
In spite of turbulent transport have been studied theoretically, in laboratory and field experiments and numerical simulations during a century [1][2][3][4][5][6][7][8], some crucial questions remain.This is particularly true in applications such as geophysics and astrophysics, where the governing parameter values are too large to be modelled either experimentally or numerically.The classical Kolmogorov's theory has been formulated for a neutrally stratified homogeneous and isotropic turbulence [9][10][11][12].This turbulence is different from convective and stably stratified turbulence.
Modern understanding of atmospheric convective turbulence is based on the following [13]: • buoyancy produces chaotic vertical plumes, which are different from small-scale turbulent eddies; • the small-scale turbulent eddies which are produced by the mean-flow shear and the shear of self-organised coherent structures, are unstable and break down in smaller unstable eddies, thus causing the direct cascade of the turbulent kinetic energy; • merging of small plumes into larger plumes results in an inverse energy cascade towards their conversion into the self-organized large-scale coherent structures; • Surface layer strongly unstably stratified and dominated by small-scale turbulence of very complex nature including usual 3-D turbulence, generated by mean-flow surface shear and structural shears (the lower part of the surface layer), and unusual strongly anisotropic buoyancy-driven turbulence (the upper part of the surface layer); • CBL core dominated by the structural energy-, momentum-and mass-transport, with only minor contribution from usual 3-D turbulence generated by local structural shears on the background of almost zero vertical gradient of potential temperature (or buoyancy); • turbulent entrainment layer at the CBL upper boundary, characterised by essentially stable stratification with negative (downward) turbulent flux of potential temperature (or buoyancy).
The goal of this paper is to develop the energy and flux budget (EFB) turbulence closure theory for the surface layer in convective and stably stratified turbulence using budget equations for turbulent energies and fluxes.The EFB theory of turbulence closure has been previously developed for stably stratified dry atmospheric flows [28][29][30][31][32][33] and for passive scalar transport in stratified turbulence [34].The EFB turbulence closure theory is based on the budget equations for the densities of turbulent kinetic and potential energies, and turbulent fluxes of momentum and heat.
In agreement with wide experimental evidence [35][36][37][38][39][40][41][42][43], the EFB theory for the stably stratified turbulence [28][29][30][31][32][33] demonstrates that strong turbulence is maintained by large-scale shear for any stratification, and the "critical Richardson number", treated many years as a threshold between the turbulent and laminar regimes, actually separates two turbulent regimes: the strong turbulence typical of atmospheric boundary layers and the weak threedimensional turbulence typical of the free atmosphere, and characterized by strong decrease in heat transfer in comparison to momentum transfer.
The physical mechanism of self-existence of a stably stratified turbulence is as follows [32,34].The increase of the vertical gradient of the mean potential temperature (i.e., the increase of the buoyancy) causes a conversion of turbulent kinetic energy into turbulent potential energy.On the other hand, the negative down-gradient vertical turbulent heat flux is decreased by the counteracting positive non-gradient heat flux that is increased with the increase of the turbulent potential energy.The latter is the mechanism of the self-control feedback resulting in a decrease of the buoyancy.Due to this feedback, the stably stratified turbulence is maintained up to strongly supercritical stratifications.The EFB theory have been verified against scarce data from the atmospheric experiments, direct numerical simulations (DNS), large-eddy simulations (LES) and laboratory experiments relevant to the steady state turbulence regime.
In the present study we develop the EFB theory for the surface layers in convective and stably stratified turbulence which allows us to determine the vertical profiles for all turbulent characteristics.This paper is organized as follows.In Section II we formulate governing equations for the energy and flux budget turbulence-closure theory for convective and stably stratified turbulence.In this section we also discuss assumptions used in the EFB theory.In Section III we develop the EFB theory for surface layers in stratified turbulence considering the steadystate and homogeneous regime of turbulence.In Section IV we discuss the EFB theory for the atmospheric stably stratified boundary-layer turbulence.In Section V we apply the EFB theory to surface layers in turbulent convection.Finally, conclusions are drawn in Section VI.

II. ENERGY-AND FLUX-BUDGET EQUATIONS AND BASIC ASSUMPTIONS
We consider plain-parallel, unstably and stably stratified dry-air flow and employ the budget equations underlying turbulence-closure theory in the Boussinesq approximation.We assume that vertical component of the mean-wind velocity is negligibly small compared to horizontal component, and horizontal gradients of all properties of the mean flow (the mean velocity and the mean potential temperature) are negligibly small compared to vertical gradients.
In this section we outline the energy and flux budget (EFB) closure theory based on the budget equations for the density of turbulent kinetic energy, the intensity of potential temperature fluctuations and turbulent fluxes of momentum and heat.In our analysis, we use budget equations for the one-point second moments to develop a mean-field theory.We do not study small-scale structure of turbulence like intermittency described by high-order moments for turbulent quantities.We are interested by large-scale long-term dynamics and consider effects in the spatial scales which are much larger than the integral scale of turbulence and in timescales which are much longer than the turbulent timescales.
We start with the basic equations of the EFB theory.The budget equation for the density of turbulent kinetic energy (TKE), E K = u 2 /2, reads where the first term, −τ iz ∇ z U i , in the right-hand side of Eq. ( 1) is the rate of production of TKE by the vertical gradient of horizontal mean velocity U (z), D/Dt = ∂/∂t + U •∇ is the convective derivative, τ iz = u i u z with i = x, y are the off-diagonal components of the Reynolds stress describing the vertical turbulent flux of momentum, and the angular brackets imply ensemble averaging.The second term β F z in Eq. ( 1) describes buoyancy, β = g/T * is the buoyancy parameter, g is the gravity acceleration, F z = u z θ is the vertical component of the turbulent flux of potential temperature, Θ = T (P * /P ) 1−γ −1 is the potential temperature, 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 potential temperature Θ = Θ + θ is characterized by the mean potential temperature Θ(z) and fluctuations θ, the fluid velocity U + u is characterized by the mean fluid velocity [which generally includes the meanwind velocity U (w) (z) = (U x , U y , 0) and the local threedimensional mean velocity U (s) related to the large-scale semi-organised coherent structures in a convective turbulence, and small-scale fluctuations u = (u x , u y , u z ).
The last term, ε K = ν (∇ j u i ) 2 , in the right-hand side of Eq. ( 1) is the dissipation rate of the density of the turbulent kinetic energy, where ν is the kinematic viscosity of fluid.The term Φ K = ρ −1 0 u z p + ( u z u 2 − ν ∇ z u 2 )/2 determines the flux of E K , where the fluid pressure P = P + p is characterized by the mean pressure P and fluctuations p, and ρ 0 is the fluid density.
The budget equation for the intensity of potential temperature fluctuations where Φ θ = u z θ 2 − χ ∇ z θ 2 /2 describes the flux of E θ and ε θ = χ (∇θ) 2 is the dissipation rate of the intensity of potential temperature fluctuations E θ , and χ is the molecular temperature diffusivity.
The budget equation for the turbulent flux F i = u i θ of potential temperature is given by where δ ij is the Kronecker unit tensor, Φ is the dissipation rate of the turbulent heat flux.The first term, −τ iz ∇ z Θ, in the right-hand side of Eq. ( 3) contributes to the traditional turbulent flux of potential temperature which describes the classical gradient mechanism of the turbulent heat transfer.The second and third terms in the right-hand side of Eq. (3) describe a non-gradient contribution to the turbulent flux of potential temperature.The budget equation for the vertical turbulent flux F z = u z θ of potential temperature is given by where E z = u 2 z /2 is the density of the vertical turbulent kinetic energy.
The budget equation for the off-diagonal components of the Reynolds stress τ iz = u i u z with i = x, y reads where Φ The budget equations for the horizontal and vertical turbulent kinetic energies E α = u 2 α /2 can be written as follows: where α = x, y, z, the term ε α = ν (∇ j u α ) 2 is the dissipation rate of E α and Φ α determines the flux of x,y − ν ∇ z u 2 x,y )/2.The terms Q αα = 2ρ −1 0 ( p∇ α u α are the diagonal terms of the tensor Q ij .In Eq. ( 6) we do not apply the summation convention for the double Greek indices.Different aspects related to budget equations (1)-( 6) have been discussed in a number of publications [28-34, 50, 58-60].
The energy and flux budget turbulence closure theory assumes the following.The characteristic times of variations of the densities of the turbulent kinetic energies E K and E α , the intensity of potential temperature fluctuations E θ , the turbulent flux F i of potential temperature and the turbulent flux τ iz of momentum (i.e., the off-diagonal components of the Reynolds stress) are much larger than the turbulent timescale.This allows us to obtain steady-state solutions of the budget equations (1)- (6).
Dissipation rates of the turbulent kinetic energies E K and E α , the intensity of potential temperature fluctuations E θ and F i are expressed using the Kolmogorov hypothesis, i.e., 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 [1,2,8,9,11].Note also that the dissipation rate of the TKE components E α (where α = x, y, z) is ε α = E K /3t T .This is because the main contribution to the rate of dissipation of the TKE components is from the Kolmogorov viscous scale where turbulence is nearly isotropic, so that 5) is the effective dissipation rate of the off-diagonal components of the Reynolds stress τ iz [28,32,34], where ε (τ ) iz is the molecular-viscosity dissipation rate of τ iz , that is small because the smallest eddies associated with viscous dissipation are nearly isotropic [61].In the framework of EFB theory, the role of the dissipation rate of τ iz is assumed to be played by the combination of terms −β F i − Q iz , and it is assumed that ε , where C τ is the effective-dissipation time-scale empirical constant for stably stratified turbulence [28,32,34], while for a convective turbulence C τ is a function of the flux Richardson number (see Sect.V).
The effective dissipation rate assumption has been justified by Large Eddy Simulations (see Fig. 1 in [32]), where LES data by [62,63] have been 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 rate assumption was based on our prior analysis of the Reynolds stress equation in the k-space using the τ -approximation (see [20,21]).Remarkably, the effective dissipation as-sumption directly yields the familiar down-gradient formulation of the vertical turbulent flux of momentum [see Eq. ( 7) below], that is well-known result which is valid for any turbulence with a non-uniform mean velocity field.
Note that the diagonal and off-diagonal components of the Reynolds stress have different physical meaning.The diagonal components of the Reynolds stress describe turbulent kinetic energy components.They have the Kolmogorov spectrum ∝ k −5/3 , that is related to the direct energy cascade.The latter is the main reason for turbulent viscosity and turbulent diffusivity.The off-diagonal components of the Reynolds stress are related to the tangling mechanism of generation of anisotropic velocity fluctuations.They have different spectrum ∝ k −7/3 [64][65][66][67].The off-diagonal components of the Reynolds stress are determined by spatial derivatives of the mean velocity field.The diagonal components of the Reynolds stress are much larger than the off-diagonal components.
We assume that the term ρ −1 0 θ ∇ z p in Eq. ( 4) for the vertical turbulent flux of potential temperature is parameterised so that β θ 2 −ρ −1 0 θ ∇ z p = 2C θ β E θ , with the positive dimensionless empirical constant C θ which is less than 1.This assumption has been justified by Large Eddy Simulations (see Fig. 2 in [32]), where LES data by [62,63] have been used for the two types of atmospheric boundary layer: "nocturnal stable" and "conventionally neutral".In addition, this assumption has been justified analytically (see Appendix A in [28]).

III. THE EFB THEORY FOR SURFACE LAYERS IN STRATIFIED TURBULENCE
In this section we develop the EFB theory for surface layers in convective and stably stratified turbulence.We use the down-gradient formulation of the vertical turbulent flux of momentum which follows from Eq. ( 5), i.e., the turbulent fluxes of the momentum are where is the turbulent dissipation timescale, ℓ z is the vertical integral scale and E z is the vertical turbulent kinetic energy.The production rate, Π K = −τ iz ∇ z U i of the turbulent kinetic energy by the vertical gradient of horizontal mean velocity [see Eq. ( 1)] can be rewritten by means of Eq. ( 7) as 2 is the squared mean velocity shear caused by the horizontal mean wind velocity.
The steady-state version of the budget equations for the density of turbulent kinetic energy where the dissipation rate ε K of the turbulent kinetic energy is expressed using the Kolmogorov hypothesis, We stress that all results obtained in the present study are mainly valid for temperature stratified turbulence (convective turbulence or stably stratified turbulence), where fluctuations of the vertical velocity u z depend on the buoyancy, β F z .Since for temperature stratified turbulence, ρ −1 0 u z p and u z u 2 do depend on the buoyancy, the third-order moments Φ K should depend on buoyancy.We assume that the vertical gradient ∇ z Φ K of the flux of E K is determined by the buoyancy, i.e., ∇ z Φ K = −C Φ β F z , where C Φ is the dimensionless empirical constant.The justification of this assumption for a convective turbulence has been performed in Ref. [13], where experimental data obtained from meteorological observations at the Eureka station (located in the Canadian territory of Nunavut) in conditions of the long-lived convective boundary layer typical of the Arctic summer have been used for validation of the assumption ∇ z Φ K = −C Φ β F z (see the right panel in Fig. 1 in Ref. [13]).Turbulent fluxes were calculated directly from the measured velocity and temperature fluctuations.In these meteorological observations warming of the convective layer from the surface is balanced by pumping of colder air into the layer via the general-circulation mechanisms.Note also that no principal contradictions have been found between the available data from observations at mid-or low latitudes and the data from Eureka [68].
Using the expression, Eq. ( 9) is reduced by simple algebraic calculations to a nonlinear equation for the vertical profile of the normalized TKE, ẼK ( Z) = E K ( Z)/E K0 as where the normalised height is the vertical share of TKE, u * is the local (zdependent) friction velocity, and L is the local Obukhov length defined as and F z is the local vertical turbulent flux of potential temperature.For stably stratified turbulence, the vertical turbulent flux F z of potential temperature is negative, and the local Obukhov length L is positive.For stably stratified turbulence ( Z > 0), Eq. ( 11) has two asymptotic solutions: (i) for a lower part ( Z ≪ 1) of the surface layer, Eq. ( 11) yields (ii) for an upper part ( Z ≫ 1) of the surface layer, it is In the framework of the EFB theory of surface layers, we use the same definition (12) for L in convective turbulence as well, where the vertical turbulent flux F z of potential temperature is positive, and L is negative.Equation (11) for the surface layer in convective turbulence ( Z < 0) reads and it has two asymptotic solutions: (i) for a lower part (| Z| ≪ 1) of the surface convective layer, Eq. ( 15) yields (ii) for an upper part (| Z| ≫ 1) of the surface convective layer, the balance of the first and the second terms in Eq. ( 11) yields ẼK = Z2/3 , i.e., Note that as follows from the definition of Z = ℓ z /(C * L), the ratio z/L for convective turbulence is where and κ 0 = 0.4 is the von Karman constant.Note that generally, Eq. ( 18) can be valid also for arbitrary z, but in this case C ℓ should be a function of height (see Section V).
In Fig. 1 we show a numerical solution of Eq. ( 11).In particular, in Fig. 1 we plot the normalized turbulent kinetic energy ẼK = E K /E K0 versus Z for convective ( Z < 0) and stably stratified ( Z > 0) turbulence.This numerical solution is in a good agreement with the above asymptotic solutions for convective and stably stratified turbulence.
Now we define the flux Richardson number as so that for stably stratified turbulence, Ri f is positive and varies from 0 to the limiting value R ∞ = 0.2 at very large gradient Richardson number Ri ≫ 1.Here the gradient Richardson number, Ri, is defined as where N 2 = β ∇ z Θ and N is the Brunt-Väisälä frequency.In the framework of the EFB theory of surface layers, we use the same definition (19) for the flux Richardson number in turbulent convection, so that Ri f is negative in turbulent convection, and its absolute value is not limited and can be large.Equations ( 12) and ( 19) allow us to relate the turbulent viscosity K M with the flux Richardson number Ri f as [34] where τ is given by Eq. ( 10).Using Eqs. ( 8) and ( 21), we rewrite the flux Richardson number as Equations ( 10) and ( 21) allow us to relate the large-scale shear S with the flux Richardson number as Using Eqs. ( 8), we rewrite Eq. ( 9) as the dimensionless ratio In addition, by means of Eqs. ( 8), ( 21) and ( 24), we obtain the normalized vertical integral scale ℓ z as the function of the flux Richardson number: where 1 For stably stratified turbulence R ∞ = 0.2, so that C Φ < 4. Thus, the normalised height Z = ℓ z /(C * L) as the function of the flux Richardson number reads Note also that using Eq. ( 26) we can rewrite Eq. ( 25) as Since convective turbulence is essentially different from stably stratified turbulence, the behaviour of the flux Richardson number Ri f ∝ Z Ẽ1/2 K is also different for these two kinds of turbulence.In particular, in convection both, the buoyancy and large-scale shear produce convective turbulence, so that the flux Richardson number can be enough large.Contrary, in stably stratified turbulence the large-scale shear produces turbulence, while the buoyancy decreases TKE, so that the flux Richardson number is limited by some value, R ∞ = 0.2.However, in the presence of internal gravity waves the maximum value of the flux Richardson number can be larger in several times in comparison with the case without waves [30,33].
Equations ( 11) and ( 22) yield the normalized turbulent kinetic energy ẼK = E K /E K0 as a function of the flux Richardson number as This implies that the normalized turbulent kinetic energy ẼK in stably stratified turbulence decreases up to the minimum value As follows from Eq. ( 22), the function Z Ẽ1/2 Since in convective turbulence, the flux Richardson number is negative, the normalized turbulent kinetic energy, , increases with the flux Richardson number.Now we derive expression for the turbulent Prandtl number, Pr T = K M /K H .To this end, we use the steadystate versions of Eqs. ( 2) and ( 4): Equations ( 31)- (32) and the expression for the turbulent heat flux, F z = −K H ∇ z Θ, yield the turbulent heat conductivity K H as By means of Eq. ( 9) for TKE, and Eq. ( 19) for Ri f , we derive the identity for the dimensionless ratio as Thus, Eqs. ( 8), ( 33) and ( 35) yield the turbulent Prandtl number, Pr T = K M /K H as where Pr (0) T = C τ /C F is the turbulent Prandtl number for a non-stratified turbulence when the mean potential temperature gradient vanishes.The gradient Richardson number Ri and the flux Richardson number Ri f are related as Ri = Ri f Pr T .
Using Eqs. ( 31)- (32), we determine the level of temperature fluctuations characterised by the dimensional ratio where 23), and expressions for the friction velocity, u 2 * = K M S, and the turbulent heat flux, F z = −K H ∇ z Θ, yield the vertical gradient of the mean potential temperature as The steady-state version of Eq. ( 3) for homogeneous turbulence yields the horizontal turbulent flux F i of potential temperature: Since in convective turbulence the vertical turbulent flux F z is positive, the horizontal turbulent flux F i of potential temperature is directed opposite to the wind velocity U i , i.e., Eq. ( 39) describes the counter-wind horizontal turbulent flux in convective turbulence.Contrary, in a stably stratified turbulence the vertical turbulent flux F z is negative, so that Eq. ( 39) determines the co-wind horizontal turbulent flux.
The physics of the counter-wind turbulent flux in a convective turbulence is as follows [34].In horizontally homogeneous, convective turbulence with a large-scale shear velocity (e.g., directed along the x-axis), the mean shear velocity U x increases with increasing height, while the mean potential temperature Θ decreases with height.Uprising fluid particles produce positive fluctuations of potential temperature (θ > 0) since ∂θ/∂t ∝ −(u • ∇)Θ, and negative fluctuations of horizontal velocity (u x < 0) since ∂u x /∂t ∝ −(u • ∇)U x .It results in negative horizontal temperature flux, u x θ < 0. Similarly, sinking fluid particles cause negative fluctuations of potential temperature (θ < 0), and positive fluctuations of horizontal velocity (u x > 0), that implies negative horizontal temperature flux u x θ < 0. Therefore, the net horizontal turbulent flux is negative ( u x θ < 0) even for a zero horizontal mean temperature gradient.This is the counterwind turbulent flux of potential temperature that results in modification of the potential-temperature flux by the non-uniform velocity field.
Let us find dependence of the horizontal turbulent flux F i of potential temperature on the flux Richardson number.To this end we use the identity, that is derived by means of Eqs. ( 8), ( 10) and (24).Therefore, the ratio of the horizontal and vertical turbulent fluxes of potential temperature, F x /F z , is given by Most of the results obtained in this section depend on the vertical share of TKE, A z ≡ E z /E K , which is determined below.The mean shear velocity U x (z) produces the energy of longitudinal velocity fluctuations E x , which in turns feeds the transverse E y and the vertical E z components of turbulent kinetic energy.The intercomponent energy exchange term Q αα in the right-hand side of Eq. ( 6) is traditionally parameterized through the "return-to-isotropy" hypothesis [69].On the other hand, the temperature stratified turbulence is usually anisotropic, and the inter-component energy exchange term Q αα should depend on the flux Richardson number Ri f .
Here we adopt the following model for the intercomponent energy exchange term Q αα which generalizes the "return-to-isotropy" hypothesis to the case of the convective and stably stratified turbulence.We use the normalised flux Richardson number Ri f /R ∞ that is varying from 0 for a non-stratified turbulence to 1 for a strongly stratified turbulence, where the limiting value of the flux Richardson number, R ∞ ≡ Ri f | Ri→∞ , is defined for very strong stratifications when the gradient Richardson number Ri → ∞.The model for the inter-component energy exchange term Q αα is described by where and C r is the dimensionless empirical constant.When Ri f = 0, Eqs. ( 42)-( 45) describe the "return-to-isotropy" hypothesis [69].To derive equation for the vertical share of TKE in a stratified turbulence, we use the steady-state version of Eq. ( 6) for vertical TKE E z as We assume that the vertical gradient ∇ z Φ z of the flux of E z is determined by the buoyancy, i.e., ∇ z Φ z = −C z β F z , where C z is the dimensionless empirical constant.The justification of this assumption for a convective turbulence has been performed in Ref. [13], where experimental data obtained from meteorological observations at the Eureka station have been used for validation of this assumption (see the left panel in Fig. 1 in Ref. [13]).Thus, by means of Eqs. ( 35) and ( 44)-( 46), we determine the vertical share of TKE A z ≡ E z /E K as a function of the flux Richardson number: According to Eq. ( 47), the vertical share A z of TKE for a non-stratified turbulence is This implies that the vertical share of TKE in a convective turbulence is given by In convective turbulence for large |Ri f | ≫ 1, the vertical share of TKE A z → 1 [13].This condition yields Substituting Eq. ( 49) into Eq.( 48), we obtain that the vertical share of TKE in a stably stratified turbulence is while in a convective turbulence (where |Ri f | ≪ |R ∞ | and Ri f < 0), the vertical share of TKE is Note that Eqs. ( 42)-( 45) describes a simple generalization of the "return-to-isotropy" hypothesis [69].These equations affect only Eq. ( 50) for the dependence of the vertical share of TKE on the flux Richardson number, A z (Ri f ).This function is the most unknown in observations.
When turbulence is isotropic in the horizontal plane, the horizontal shares of TKE are A x = A y = 1 − A z .This yields the horizontal components of TKE as where A x = E x /E K and A y = E y /E K .When turbulence is anisotropic in the horizontal plane, the intercomponent energy exchange term Q αα and the horizontal shares of TKE are given in Appendix.
Let us consider stably stratified turbulence.Neglecting the term ∇ z Φ K in Eq. ( 9), we rewrite this equation as Ri where we use the definition (19) for the flux Richardson number.By means of this equation and the expressions for the squared Brunt-Väisälä frequency, N 2 = β ∇ z Θ, and the turbulent heat flux, F z = −K H ∇ z Θ, we obtain equation for the turbulent heat conductivity K H as In very strong stable stratification, the gradient Richardson number admits a limit Ri → ∞ and the flux Richardson number Ri f → 0.2.This implies that the turbulent heat conductivity for a very strong stable stratification K H ≈ 0.25 ε K /N 2 .This is a well-known Cox-Osborn equation [70,71] that plays an important role in Physical Oceanography.

IV. THE ATMOSPHERIC STABLY STRATIFIED BOUNDARY-LAYER TURBULENCE
In view of the applications of the obtained results to the atmospheric stably stratified boundary-layer turbulence, we outline below the useful in modelling theoretical relationships [32,34].It is known that the wind shear in stably stratified turbulence has two asymptotic results: (i) S = τ 1/2 /(κ 0 z) at ς ≪ 1, which describes the logprofile for the mean velocity, and (ii) S = τ 1/2 /(R ∞ L) when ς ≫ 1.The latter result follows from Eq. ( 23), where ς = z 0 dz ′ /L(z ′ ) is the dimensionless height based on the local Obukhov length scale L(z), and κ 0 = 0.4 is the von Karman constant.For surface layer in stably stratified turbulence (defined as the lower layer which is 10 % of the turbulent boundary layer), the Obukhov length scale L is independent of z and the dimensionless height ς = z/L.Interpolating these two asymptotic results, we obtain that the wind shear S(ς) can be written as The latter allows us to get the vertical profile of the turbulent viscosity K M (ς) = τ /S as Using Eqs. ( 21) and (55), we arrive at the expression for the vertical profile of the flux Richardson number Ri f (ς) as Equation ( 56) yields the expression for ς as In this case, the vertical share of TKE A z (ς) ≡ E z /E K reads and the vertical profile of the turbulent Prandtl number Pr T (ς) is given by: Note that the gradient Richardson number Ri and the flux Richardson number Ri f are related as Ri(ς) = Ri f (ς) Pr T (ς).Equations ( 55)-( 59) are in agreement with Monin-Obukhov-Nieuwstadt similarity theories [72,73].In the Monin-Obukhov similarity theory [72], the turbulent fluxes of momentum τ , heat F z , the Obukhov length scale L and other scalars are approximated by their surface values, while the similarity theory by Nieuwstadt [73] is extended to the entire stably stratified boundary layer employing local z-dependent values of the turbulent fluxes τ (z) and F z (z), and the length L(z) instead of their surface values.
Using Eqs. ( 22) and ( 57), we can relate ς and Z for stably stratified turbulence as For the surface layer ( Z ≪ 1) of the stably stratified turbulence, the dimensionless height is ς = z/L, and the normalised TKE is ẼK ≈ 1 [see Eq. ( 16)].Therefore, Eq. ( 60) in this case is reduced to This equation coincides with Eq. ( 18) derived for the low part (| Z| ≪ 1) of the surface layer in convective turbulence.

V. SURFACE LAYERS IN CONVECTIVE TURBULENCE
In this section we apply results obtained in Section III to convective turbulence.In this case, the nonlinear equation for the vertical profile of the normalized TKE, ẼK ( Z) = E K ( Z)/E K0 is given by Eq. ( 15).In Fig. 2 we show the normalized turbulent kinetic energy E * K = E K /u 2 * versus z/L obtained in the EFB theory.This dependence has been compared with the data [13] obtained from meteorological observations at the Eureka station * versus z/L obtained in the EFB theory (solid line) which is compared with the data obtained from meteorological observations at the Eureka station located in the Canadian territory of Nunavut [68] in the conditions of long-lived convective boundary layer typical of the arctic summer.located in the Canadian territory of Nunavut [68] in the conditions of long-lived convective boundary layer typical of the arctic summer.The better agreement between theoretical predictions and the observation data is achieved when C τ is the following function of z/L (see Fig. 3): where X(z) = Ri f (z)/R ∞ .We remind that C τ is related to the effective dissipation time scale of the Reynolds stress.Asymptotic solutions of Eq. ( 15) for the normalized TKE, ẼK ( Z) are given by Eq. ( 16) for a lower part (| Z| ≪ 1) of the surface convective layer, and by Eq. ( 17) for an upper part (| Z| ≫ 1) of the surface convective layer.Below we present asymptotic formulas for various turbulent characteristics based on Eqs. ( 21), ( 22)-( 26), ( 36)-( 38), ( 41) and ( 51)- (52) for the lower and upper parts of the surface layer in convective turbulence.In particular, the turbulence characteristics for a lower part (| Z| ≪ 1) of the surface convective layer are given by • the large-scale shear, • the turbulent viscosity, • the vertical share of TKE, • the turbulent Prandtl number, • the level of temperature fluctuations, • the vertical gradient of the mean potential temperature, • the ratio of the horizontal and vertical turbulent fluxes of potential temperature, • the horizontal components of TKE, where In Eq. ( 64) we take into account that for the surface layer in convective turbulence, the vertical integral turbulent scale, ℓ z = C ℓ z and in Eq. ( 70) we consider the case when the mean velocity U i is directed along the x-axis.
Let us discuss the choice of the dimensionless empirical constants [32,34].There are two well-known universal constants: the limiting value of the flux Richardson number R ∞ = 0.2 for an extremely strongly stratified     turbulence (i.e., for Ri → ∞) and the turbulent Prandtl number Pr (0) T = 0.8 for a nonstratified turbulence (i.e., for Ri → 0) [78][79][80].The constant C p describes the deviation of the dissipation timescale of E θ = θ 2 /2 from the dissipation timescale of TKE.The constant C θ is given by  Absolute values of the gradient Richardson number Ri in convective turbulence are much larger than in stably stratified turbulence.The reason is that the large-scale shear in convective turbulence is much smaller than in stably stratified turbulence (see Fig. 10).This is because TKE in convective turbulence is much larger than in stably stratified turbulence (see Fig. 12), because in convection, both, the buoyancy and large-scale shear produce turbulence.Contrary, in stably stratified turbulence, the large-scale shear produces TKE, while the buoyancy decreases TKE and produces the temperature fluctuations.
On the other hand, the normalized intensity of potential temperature fluctuations Ẽθ = E θ /θ 2 * (see Fig. 10) in convective turbulence is much weaker than in stably stratified turbulence.The latter is caused by a weak gradient of the mean potential temperature in convective turbulence in comparison with that of stably stratified turbulence (see Fig. 8).The vertical share A z of turbulent kinetic energy in stably stratified turbulence is changed stronger than in the surface layers of convective turbulence (see Fig. 11).Indeed, turbulence tends to be two-dimensional one for very large gradient Richardson number in stably stratified turbulence, i.e., A z becomes very small.Contrary, in convection the buoyancy is dominated in the energy production in the upper part of the surface layer, resulting in a strong increase of the vertical TKE, i.e., the vertical share A z → 1.
Since the normalized turbulent kinetic energy ẼK0 = E K0 /u 2 * is inversely proportional to the vertical share A z , it changes significantly in stably stratified turbulence in comparison with convective turbulence (see Fig. 12).In Fig. 13 we show the normalized vertical integral scale ℓ z /L versus z/L for convective and stably stratified turbulence.In stably stratified turbulence, the vertical integral scale reaches the Obukhov length scale at high gradient Richarson numbers.Contrary, in convective turbulence the ratio ℓ z /|L| is strongly increases with height.

VI. CONCLUSIONS
We develop the energy and flux budget theory for the atmospheric surface layers in turbulent convection and stably stratified turbulence.This theory uses the budget equations for turbulent energies and fluxes.In the framework of this theory we determine the vertical profiles for all turbulent characteristics and for the mean velocity and mean potential temperature.In particular, we find the vertical profiles of turbulent kinetic energy, the intensity of turbulent potential temperature fluctuations, the vertical turbulent fluxes of momentum and buoyancy (proportional to potential temperature), the integral turbulence scale, the turbulent anisotropy, the turbulent Prandtl number and the flux Richardson number.
Since the large-scale shear in convective turbulence is much than in stably stratified turbulence, the absolute values of the gradient Richardson number in convective turbulence are much larger than in stably stratified turbulence.This is natural result, since turbulent kinetic energy (produced by both, the buoyancy and largescale shear) in convective turbulence is much stronger than in stably stratified turbulence.On the other hand, the large-scale shear produces turbulent kinetic energy in stably stratified turbulence, and the buoyancy decreases TKE and produces the temperature fluctuations.In convective turbulence, the gradient of the mean potential temperature is usually small in comparison with stably stratified turbulence.Therefore, potential temperature fluctuations are much smaller than in stably stratified turbulence.The vertical integral scale in stably stratified turbulence can only reach the Obukhov length scale at high gradient Richarson numbers.On the other hand, the vertical integral scale in convective turbulence can be much larger than the absolute value of the Obukhov length scale.In this Appendix, we discuss the model for the intercomponent energy exchange term Q αα for anisotropic turbulence in the horizontal plane.In particular, the inter-component energy exchange terms Q αα are given

8 FIG. 2 .
FIG.2.The normalized turbulent kinetic energy E * K = EK/u 2 * versus z/L obtained in the EFB theory (solid line) which is compared with the data obtained from meteorological observations at the Eureka station located in the Canadian territory of Nunavut[68] in the conditions of long-lived convective boundary layer typical of the arctic summer.

FIG. 4 .
FIG. 4. The flux Richardson number Ri f versus z/L for convective and stably stratified turbulence.

5 FIG. 5 .
FIG. 5.The turbulent Prandtl number Pr T versus z/L for convective and stably stratified turbulence.

FIG. 6 .
FIG.6.The gradient Richardson number Ri versus z/L for convective and stably stratified turbulence.

60 FIG. 8 .FIG. 9 .
FIG. 8.The normalised mean temperature difference Θ = (T − T b )/θ * versus z/L for convective and stably stratified turbulence, where T b is the mean temperature at the lower boundary.

8 FIG. 11 .
FIG. 11.The normalized vertical share Az of turbulent kinetic energy versus z/L for convective and stably stratified turbulence.

FIG. 13 .
FIG.13.The normalized vertical integral scale ℓz/L versus z/L for convective and stably stratified turbulence.

ACKNOWLEDGMENTS
This paper is dedicated to Prof. SergejZilitinkevich  (1936Zilitinkevich  ( -2021) )  who initiated this work and discussed the obtained results.This research was supported in part by the PAZY Foundation of the Israel Atomic Energy Commission (grant No. 122-2020), and the Israel Ministry of Science and Technology (grant No. 3-16516).

APPENDIX 1 :
THE MODEL FOR THE INTER-COMPONENT ENERGY EXCHANGE TERMS Qαα FOR ANISOTROPIC TURBULENCE IN THE HORIZONTAL PLANE