Convection occurs within most stars. Convection causes the transfer of energy and momentum, and the mixing of matter in stellar interiors. Therefore, it strongly influences the internal structure, evolution, and pulsational stability of stars. Stellar convection is always turbulent due to its large scale. It is impossible to expect to have a perfect convection theory until a better understanding of turbulence has been achieved. Up to now the most popularly applied convection theory is still the MLT developed by Bohm-Vitense (1958), and its many versions. The most obvious advantages of the MLT are its straightforwardness in physical picture and simplicity of use. However, the MLT is not a dynamic theory following the hydrodynamic equations and turbulence theory, but is a phenomenological theory based on a simple analogy of turbulence with the kinetic theory of gas molecules. In fact, turbulence is more complex than the kinetic theory of gas molecules. The fundamental shortcoming of the MLT is that it cannot give a correct description for the dynamic behaviors of turbulent convection. When dealing with the dynamic problems of turbulent convection, such as time-dependent and non-local convection, this shortcoming becomes prominent. A possible approach to turbulent convection is direct hydrodynamic numerical simulation. Taking into account the fact that both the time and space scales of turbulence are more than ten orders of magnitude less than the stellar ones, a direct hydrodynamic numerical simulation of stellar convection would be impossible for the calculation of stellar structure and evolution in the foreseeable future. Therefore, it is a reasonable choice to develop a theoretical approach exact enough and simple enough for treating turbulent convection in the calculation of stellar evolution and oscillations. Helio- and astero-seismology have made significant progress in the past two decades thanks to the GONG, SOHO, OGLE, MACHO, 2MASS, CoRoT, and Kepler ground-based and space projects. They offer the best opportunity to test stellar evolution and convection theories. In the present paper we pay close attention to convection theory and the relevant problems in stellar structure, evolution, and pulsational stability. Based on the reflections above, we decided to abandon the phenomenological MLT, and to develop a non-local and time-dependent theory of convection based on hydrodynamic equations and turbulence theory (Xiong, 1978; Xiong, 1980; Xiong, 1981; Xiong, 1989; Xiong et al., 1997; Deng et al., 2006). In contrast to the MLT, it is a dynamic theory of correlation functions of turbulent velocity and temperature (and, in addition, of the element abundance for chemically inhomogeneous stars) following the hydrodynamic equations and turbulence theory. So, it can be expected that our theory has a more solid hydrodynamic foundation and can give a more exact description of the hydrodynamic behavior of turbulent convection than the MLT does. Our purpose is to improve the treatment of overshooting mixing in calculations of stellar evolution and the treatment of dynamic and thermodynamic coupling between convection and oscillations in calculations of stellar oscillations. Canuto (1993, 1997, 1999) and Li of the Yunnan group of China (Zhang, 2012a; Zhang and Li, 2012a; Zhang, 2012b; Zhang and Li, 2012b; Zhang, 2013) proposed a similar theory. In the present paper, we do not attempt to make a comprehensive review and comparison for convection theories and relevant problems, which would be a large and difficult amount of work. In A Non-Local and Time-Dependent Theory of Convection section, a brief description of our non-local and time-dependent convection theory is given. The following sections review the progress of its applications in theoretical calculations of the structure of the solar convection zone (Structure of the Solar Convection Zone section), of stellar evolution and lithium depletion in the atmosphere of the Sun and late-type dwarfs (Overshooting Mixing and Stellar Evolution section), and of stellar oscillations (in PartⅡ). These applications are not only the primary motivations of our research on the theory of stellar convection, but also an important means for testing convection theory. A summary and discussion follow in section 7 of PartⅡ, where we try to evaluate the successes and failures of our theory, analyze their reasons and identify direction for improvement. It is known that this paper will contain strong personal bias. Our viewpoint on the excitation mechanism for red giants may not gain widespread approval. Nowadays, our understanding of many problems is still inconclusive, and will need more time to verify. Disputes over different academic viewpoints will contribute to the development of science. Our theory is far from perfect, however it has shown obvious improvements in comparison with MLT.

This article is divided into two parts. Convection theory and its applications in theoretical calculations of the structure of the solar convection zone and for stellar evolution are contained in PartⅠ; the applications of the theory in calculations of stellar oscillations, and summary and discussions are contained in PartⅡ.

In order to overcome the defects of the MLT, we developed a non-local and time-dependent theory of convection based on the hydrodynamic equations and turbulence theory (Xiong, 1978; Xiong, 1980; Xiong, 1981; Xiong, 1989; Xiong et al., 1997; Deng et al., 2006). In our theory, the classical Reynolds decomposition is used. Each of the physical variables X is expressed as the sum of its average value X ¯ and (Eulerian) turbulent fluctuation X ′ , X = X ¯ + X ′ , (1) substituting Eq. 1 into the hydrodynamic equations and the radiation transfer equation, and making a Taylor expansion for X ′ , retaining only its first-order terms, ignoring all the second and higher-order terms, then averaging for each of the equations, we can obtain the following average hydrodynamic equations and the radiation transfer equation,   D ρ ¯ D t + ρ ¯ ∇ k u ¯ k = 0 ,   (2)     D u ¯ i D t + 1 ρ ¯ ∇ k ( g i k P ¯ + ρ ¯ u ′ i u ′ k ¯ ) + g i k ∇ k Φ ¯ = 1 ρ ¯ ∇ k σ ( u ¯ ) i k ,   (3) ρ ¯ C p ¯ D T ¯ D t − α D P ¯ D t + ∇ k ( C p ¯ ρ u ′ k T ′ ¯ ) + ∇ k F r k ¯ = ρ ε N ¯ + σ ( u ′ ) i k ∇ k u ′ i ¯   , (4) F r i = − 4 a c T ¯ 3 3 ρ ¯ κ ¯ g i k ∇ k T ¯ ,   (5) where     D D t = ∂ ∂ t + u ¯ k ∇ k   . (6)

is the comoving differential, and T, P, ρ are, respectively, the temperature, pressure, and density of gas, C P is the specific heat at constant pressure, α   =   − ( ∂ ln ⁡ ρ / ∂ ln ⁡ T ) P the expansion coefficient, ε N the effective nuclear energy-generation rate per gram, σ i k the viscosity tensor, Φ   the gravitational potential, κ the radiative opacity, and F r i the radiation flux vector. It can be seen that, when convection sets in, an extra term ρ u ′i u ′k   ¯ in the average equation of momentum conservation and another extra term C p ¯ ρ u ′i T ′   ¯ in the average equation of energy conservation will emerge. They are, respectively, the well-known Reynolds stress and convective enthalpy flux. In the present paper, all the equations are expressed in the tensor format. g i k is the metric tensor for the current framework. The implicit summation rule of tensors is used, i.e., a summation should be performed for an index k running from 1 to 3 if both the subscript and superscript k appear in a term. Subtracting the average equations, Eqs. 3, 4, from the corresponding original equations of momentum and energy conservation, after a long derivation, it is not difficult to obtain the following dynamic equations of turbulent velocity and temperature:       D w ′ i D t + w ′k ∇ k u i   ¯ − 1 ρ ¯ ∇ k ( g i k P ′ + ρ u ′ i u ′ k − ρ u ′ i u ′ k   ¯ ) − α T ′ T ¯ ( g i k ∇ k Φ ¯ +     D u i   ¯ D t ) = 1 ρ ¯ ∇ k σ ( u ′ ) i k ,   (7)       D D t ( T ′ T ¯ ) + T ′ T ¯ { [ 1 − α + C P , T ] D ⁡ ln T ¯ D t + [ C P , P + ( 1 − α ) ∇ a d ] D ⁡ ln P ¯ D t } + w ′k ( ∇ k ⁡ ln T ¯ − ∇ a d ∇ k ⁡ ln P ¯ ) +     1 ρ ¯ C P T ¯ ∇ k [ ρ ¯ C P T ¯ ( w ′ k T ′ T ¯ − w ′ k T ′ T ¯ ¯ ) ] = 1 ρ ¯ C P T ¯ [ ( ρ ε N ) ′ + σ ( u ′ ) i k ∇ k u ′ i − σ ( u ′ ) i k ∇ k u ′ i ¯ − ∇ k F ′ r k ] ,   (8) where w ′i = ρ u ′i / ρ ¯ is the density-weighted turbulent velocity, ∇ a d = α P / ρ C p T =   ( Γ 2 − 1 ) / Γ 2 is the adiabatic temperature gradient, C P , T = ( ∂ ln ⁡ C p / ∂ ln ⁡ T ) p and C P , P = ( ∂ ln ⁡ C p / ∂ ln ⁡ P ) T are, respectively, the partial derivative of Cp with respect to T and P, and Γ 2   ( along   with   Γ 1   a n d   Γ 3   below ) are the adiabatic exponents introduced by Chandrasekhar (1939). Starting from Eqs. 7, 8, it is not difficult to constitute the dynamic equations of auto- and cross-correlations for turbulent velocity and temperature. For example, we have the equation of velocity correlations as,     D D t w ′ i w ′ j ¯ + w ′ i w ′ k ¯ ∇ k u j   ¯ + w ′ j w ′ k ¯ ∇ k u i   ¯ + 1 ρ ¯ ∇ k ( ρ ¯ u ′ k u ′i u ′j ¯ ) − w ′ i T ′ T ¯ ¯ ( D u j   ¯ D t + g j k ∇ k Φ ¯ ) − w ′ j T ′ T ¯ ¯ ( D u i   ¯ D t + g i k ∇ k Φ ¯ ) + 1 ρ ¯ ∇ k [ g i k w ′j P ′ ¯ + g j k ′ i P ′ ¯ + w ′ j σ ( u ' ) i k ¯ + w ′ i σ ( u ′ ) j k ¯ ] − 1 ρ ¯ ( g i k P ′ ∇ k w ′ j ¯ + g j k P ′ ∇ k w ′ i ¯ ) =   − 1 ρ ¯ [ σ ( u ′ ) i k ∇ k w ′ j + σ ( u ′ ) j k ∇ k w ′i ] . (9)

The terms in square brackets on the left-hand side of Eq. 9 are the energy fluxes of pressure and viscous stress. They are negligible in comparison with the next two terms when it is assumed that the size scale of turbulent elements is far smaller than the characteristic length of the mean fluid fields. This implies that the turbulence is near quasi-isotropic. The terms on the right-hand side of Eq. 9 are the turbulent dissipation due to molecular viscosity. Based on isotropic turbulence theory, this dissipation can be expressed as (Hinze, 1975)   1 ρ ¯ [ σ ( u ′ ) i k ∇ k w ′ j ¯ + σ ( u ′ ) j k ∇ k w ′ i ¯ ] = 2 3 η e x w ′ i w ′ j ¯ l e   ,   (10) where η _e is the Heisenberg eddy coupling constant, l e is the characteristic length of energy-containing eddies of turbulence, and x is the rms turbulent velocity, x = g i j w ′ i w ′ j ¯ 3   . (11)

In the dynamic equations for second-order correlations, the third-order correlations must appear due to nonlinearity of the hydrodynamic equations. The fourth-order correlations will appear if we constitute the dynamic equation for the third-order correlations. The equations of the correlation functions are never closed, as is well-known in turbulence theory (Hinze, 1975). In order to close the equation of second-order correlations, we use the gradient-type diffusion approximation for treatment of the third-order correlations (Xiong, 1980; Xiong, 1981; Xiong, 1989) u ′ k w ′ i w ′ j ¯ = − w ′ k w ′ d τ c ∇ α w ′ i w ′ j ¯   ,   (12) where τ c =   ∧ x   . (13) is the lifetime of turbulence and ∧ is the diffusion length of turbulence. We then further assume that both   l e and ∧ are proportional to the local pressure scale height H p , l e   = C 1 H p = C 1 r 2 P ¯ G M r ρ ¯   ,   (14) ∧   =   3 4 C 2 H p = C 2 3 r 2 P ¯ 4 G M r ρ ¯   . (15)

The third-order correlations represent the non-local transport of turbulent convection. ρ ¯ u ′k w ′i w ′j ¯   is the energy flux of turbulent stress. The gradient-type diffusion approximation means that turbulence diffuses from one location where it is strong to another location where it is weak. This is a very simple but reasonable physical assumption. Another closure scheme is to construct dynamic equations of third-order correlations (Canuto, 1993; Xiong et al., 1997). The fourth-order correlations will appear; they are expressed as three products of two second-order correlations of turbulent fluctuations in terms of the fourth-order correlation (Orszag, 1977; Lesieur, 1987). This standard normal approximation seems to have a more solid basis in stochastic theory. However, the physically positive quantities, such as the auto-correlations of turbulent velocity and temperature (namely x 2 and Z in the present paper), sometimes become negative. The numerical calculations will eventually fail to converge. Grossman (1996) compares in detail these two closure schemes (namely, our gradient-type diffusion approximation for third-order correlations and the quasi-normal approximations for fourth-order correlations). He concluded “. . . the Xiong solution predicts second moments better, with third-moment agreement not as good. The Full solution [namely with the quasi-normal approximation] predicts third moments better, but the second moments [namely x 2 , Z, and V in present paper] show inferior agreement. . . . Since the second moments are most important for constructing stellar models, we conclude that the Xiong closures perform impressively well.” Therefore, we abandoned the closure schemes of the quasi-normal approximations, and adopted the gradient-type diffusion approximation for the closure of the dynamic equations of turbulent correlations, because it is a simpler and more practical scheme than the quasi-normal approximation for calculations of stellar evolution and oscillations (Xiong et al., 1997).

Convection results from the thermal instability of a fluid medium. In a gravitationally stratified fluid medium, a perturbed fluid element will be accelerated by buoyant forces along the direction of gravity, when the local temperature gradient exceeds the adiabatic one. The original direction of convective motion is along with gravity, i.e., along the radial direction of the star. Therefore, convection is highly anisotropic in the low wave number range of the turbulent spectrum. Due to the continuity and nonlinearity of hydrodynamics, a part of the kinetic energy of convective elements will be converted into horizontal motion. Turbulence becomes more and more isotropic in the high wave-number range. Rotta (1951) pointed out that the correlation of the pressure and velocity gradients tends to make turbulence isotropic. Therefore, we assume (Deng et al., 2006) 1 ρ P ′ ( g i k ∇ k w ′i + g j k ∇ k w ′i − 2 3 g i j ∇ k ′ k ) ¯   = − C 3 4 3 η e G M r ρ ¯ x 3 C 1 r 2 P ¯ χ i j ,   (16) where x ² and χ i j are, respectively, the isotropic and anisotropic component of the velocity correlation. w ′ i w ′ j ¯   = g i j x 2 + χ i j . (17)

Substituting Eq. 17 into Eq. 9 and contracting with respect to indices i and j and noting Eqs. 10–16, we have 3 2 D x 2 D t − x 2 D ⁡ ln ρ ¯ D t + χ i j ∇ i u j   ¯ − 3 2 ρ ¯ ∇ k ( Q α k ∇ α x 2 ) − α V k ( D u k ¯ D t + ∇ k Φ ¯ ) = − 2 3 η e G M r C 1 r 2 P ¯ x 3 . (18)

Subtracting the product of g i k and Eq. 18 from Eq. 9 and noting Eq. 16, we can obtain the dynamic equation of the anisotropic component of the velocity correlation χ i j , D χ i j D t + x 2 ( g i k ∇ k u j   ¯ + g j k ∇ k u i   ¯ − 2 3 g i j ∇ k u k ¯ ) + χ i k ∇ k u j   ¯ + χ j k ∇ k u i   ¯ − 2 3 g i j x α β ∇ α u β ¯ − α ( g i k V j + g j k V i − 2 3 g i j V k ) ( D u k ¯ D t + ∇ k Φ ¯ ) − 1 ρ ¯ ∇ k ( Q k α ∇ α χ i j ) = − 4 3 η e ( 1 + C 3 ) G M r 3 C 1 r 2 P ¯ χ i j . (19)

In the same way, we can derive the dynamic equations of the temperature auto-correlation and cross-correlation of velocity and temperature from Eqs. 7, 8: D Z D t + Z { [ 1 − α + C P , T ] D ⁡ ln T ¯ D t + [ C P , P + ( 1 − α ) ∇ a d ] D ⁡ ln P ¯ D t } + 2 ∨ k ( ∇ k ⁡ ln T ¯ − ∇ a d ∇ k ⁡ ln P ¯ ) − 1 ρ ¯ 2 C p 2 ∇ k ( ρ ¯ C p 2 Q α k ∇ α Z ) = − 2 3 η e G M r ρ ¯ C 1 r 2 P ¯ ( x + x c ) Z   , (20) D V i D t + V k + V i { [ 1 − α + C P , T ] D ⁡ ln T ¯ D t + [ C P , P + ( 1 − α ) ∇ a d ] D ⁡ ln P ¯ D t } − α Z ( D u i ¯ D t + g i k ∇ k Φ ¯ ) + ( g i k x 2 + χ i k ) ( ∇ k ⁡ ln T ¯ − ∇ a d ∇ k ⁡ ln P ¯ ) − 1 ρ ¯ C p ∇ k ( C p Q α k ∇ α V i ) = − 3 η e G M r ρ ¯ C 1 r 2 P ¯ ( 3 x + x c ) V i ,   (21) where Z = ( T ′ T ¯ ) 2 ¯   ,   (22) V i = u ′ i T ′ T ¯   ¯   ,   (23) Q i j = 3 C 2 r 2 P ¯ 4 G M r ρ ¯ ( g i j x + χ i j x ) ,   (24) x c = 3 a c G M r T ¯ 3 C 1 ρ ¯ C P ¯ κ ¯ r 2 P ¯ .   (25) P e = x / x c   is the effective Peclet number of turbulent convection. Eqs. 3, 4 can be rewritten as D u i   ¯ D t + 1 ρ ¯ ∇ k [ g i k ( P ¯ + ρ ¯ x 2 ) + ρ ¯ χ i k ] + g i k ∇ k Φ ¯ = 0 , ( 3 ′ ) D ⁡ ln T ¯ D t + ∇ a d D ⁡ ln P ¯ D t + 1 ρ ¯ C p ¯ T ¯ [ ρ ¯ x 2 ( 3 D ⁡ ln ⁡ x D t − D ⁡ ln ρ ¯ D t ) + ρ ¯ χ i k ∇ k u i ¯ + ∇ k ( F r k ¯ + F c k + F t k ) ] = 0 ,       ( 4 ′ ) where F r k ¯ , F c k , F t k are, respectively, the radiative, convective enthalpy, and turbulent kinetic energy fluxes, F c k =   ρ ¯ C P T ¯ V k ,   (26) F t k =   − 3 ρ ¯ x Q k α ∇ α x .   (27)

The contribution of pressure fluctuations to the enthalpy flux Eq. 26 has been neglected, because it is much less than the contribution of the temperature fluctuation. The combination of the average hydrodynamic equations Eqs. 2, 3′, 4′, 5, and the dynamic equations of auto- and cross-correlations for turbulent velocity and temperature Eqs. 18–21 form a complete and closed system of radiation-hydrodynamic equations for the calculation of stellar structure and oscillations.

The details of the derivation and main simplifications can be found in our original works (Xiong, 1978; Xiong, 1980; Xiong, 1981; Xiong, 1989; Xiong et al., 1997; Deng et al., 2006). The main simplifications and assumptions can be summarized as follows:

1) Turbulence is near quasi-isotropic; the characteristic linear size of turbulent elements is far less than the characteristic length of the average fields.

The turbulent dissipation, diffusion, and anisotropy are carefully taken into account. Based on the turbulence theory, they are expressed, respectively, by Eqs. 10, 12, 16, 19. There are three convective parameters (C₁, C₂, C₃) in our convection theory. They are associated, respectively, with turbulent dissipation, diffusion, and anisotropy. They can be calibrated by using the comparison between the observed and theoretical depth and T-P structure of the solar convection zone, the turbulence velocity-temperature fields in the solar atmosphere, the lithium depletion in the atmosphere of the Sun and late-type dwarfs, and 3D simulations (Deng et al., 2006). The calibration of the convective parameters (C₁, C₂, C₃) is somewhat complex. First, the convective parameters (C₁, C₂, C₃) are not a set of constants, but they vary slowly as a function of stellar parameters such as mass M, luminosity L, effective temperature T e (Ludwig et al., 1999); even in the interior of the same star they vary with radius r. Fortunately, in the deep interior of stars, apart from the surface layer, convective transport of energy is very effective ( P e =   x / x c ≫ 1 ) . Therefore, the temperature gradient is very close to the adiabatic one, independent of the choice of convective parameters. The depth of the convection zone is almost defined by the structure of the surface super-adiabatic convection zone, where convective energy transport is inefficient ( P e = x / x c ≤ 1 ) . Our research shows that, if only the pulsational stability is of concern, by using a fixed set of (C₁, C₂, C₃) calibrated to the Sun, all of the pulsational instability strips in the H-R diagram, such as the RR Lyrae (Xiong et al., 1998b), δ   S c u t i − γ   D o r a d u s   (Xiong et al., 2016), Mira (Xiong et al., 1998a), and LPV (Xiong et al., 2018) instability strips, can be defined. The effects of (C₁, C₂, C₃) on convection are not completely independent and the available observations for calibration are too rare, so it is difficult to calibrate (C₁, C₂, C₃) independently. After theoretical considerations, these entanglements for calibration of (C₁, C₂, C₃) can be removed, at least partially. C₃, unlike C₁ and C₂, is a rather independent convective parameter. It is related to the anisotropy of turbulent convection. In the deep interior of a convectively unstable zone, the ratio of the squared radial component of turbulent velocity to the horizontal one w r 2 ¯ /     w h 2 ¯   ≈   ( 3 + C 3 ) /   2   C 3   . Anisotropy increases and convective overshooting decreases with decreasing C_3. C₃ = 3 is a reasonable value, which agrees well with the observations of turbulent velocity in the solar atmosphere, the lithium depletion of the Sun and late-type main-sequence stars, and hydrodynamic simulations (Deng et al., 2006; Deng and Xiong, 2008). C₁ and C₂ are two convective parameters related to turbulent dissipation and diffusion. It can be seen from Eqs. 14, 15 that l e and ∧ are, respectively, the dissipation and diffusion length of turbulence. So, one can expect that C₂/C₁ is approximately a constant independent of the stellar parameters M, L, and T e . C₂/C₁ = 1/2 is a good choice. Once C₂/C₁ and C₃ are identified, C₁ becomes the only adjustable convective parameter. Our research shows that a non-local and anisotropic convective model of the Sun with (C₁, C₂, C₃) = (0.64, 0.50, 3.0) can reproduce almost all of the observed characteristics of the solar convection zone very well (Deng et al., 2006; Xiong and Deng, 2001).

Astronomy is a science based on the observations of astronomical objects. Stellar convection theory originated from the requirement for a treatment of convective transport of energy and momentum, and of convective mixing of materials in the stellar interior. In the previous section, we described a non-local and time-dependent theory of convection. In this section we will present its application in studies of the solar convection zone.

The Sun is the closest star to us, and is the only star that can provide high spatial resolution observations. Therefore, the Sun is an ideal natural laboratory for testing convection theory. By using our non-local and anisotropic convection theory described in the previous section, we calculated a model of the solar convective envelope (Unno, et al., 1985; Xiong and Cheng, 1992; Xiong and Deng, 2001). The MHD equation of state (Däppen et al., 1988; Hummer and Mihalas, 1988; Mihalas et al., 1988) and OPAL opacity (Rogers and lglesias, 1992) supplemented by low-temperature opacities (Alexander and Ferguson, 1994) were used. The chemical abundance is X = 0.70 and Z = 0.02. The surface boundary was located at optical depth τ = 10⁻³. By choosing a first trial set of T and r at the surface boundary T = T₀, r = R₀, and P =   a T 0 4 , by using a relaxation procedure, it is not difficult to obtain that T = T e , r = R ⊙ , and L =   4 π R ⊙ 2 T e 4 , at τ = 2/3. The original fundamental equations are our radiation-hydrodynamic equations for the calculation of stellar structure and oscillations, Eqs. 2, 3′, 4′, 5, 18–21. The Henyey method (Henyey et al., 1964) for the integration of differential equations was used. The details of the working equations and boundary conditions can be found by referring to the original works of the author cited above.

We know from the theory of stellar structure and evolution that the stellar structure should be defined once the mass and element abundances of a star are assigned. Therefore, stellar evolution in the H-R diagram is, in fact, a reflection of nuclear evolution in the stellar interior. Convection (including overshooting) is the most important means of element mixing in the stellar interior. Our non-local convection theory has been successfully used to treat non-local convective mixing of elements in the evolution of massive stars (Xiong, 1986) and to calculate lithium depletion in the atmosphere of the Sun (Xiong and Deng, 2002) and late-type main sequence stars (Xiong and Deng, 2009).