5.2. TKE Closures for the Martian PBL
 Many models of Martian circulation employ ReSMs for parameterization of the turbulent boundary layer. As explained earlier, Martian and terrestrial boundary layers are similar in many aspects and so they can be described by similar models. Indeed, the Reynolds stress modeling approach used in terrestrial circulation models has been adopted in several leading Martian atmospheric models, e.g., by Forget et al.  and Rafkin et al. . Despite the portability of terrestrial ReSMs to the Martian environment, there has been a notable absence of a comprehensive source of information on the ReSM technique for the Martian community. One of the purposes of this paper is to mitigate this omission and provide the community of Mars researchers with a state-of-the-art review on those aspects of the Reynolds stress modeling approach, as applied to the terrestrial atmosphere and ocean, that can be applied to Mars.
 The Reynolds stress models rely upon the Reynolds averaging of the governing equations and a set of closure assumptions relating unknown correlations to the known ones as well as to the mean fields. ReSM is a powerful tool for modeling turbulent flows that has been used extensively in engineering, environmental, and geophysical sciences, and its foundations have been outlined in numerous review articles and books [e.g., Tennekes and Lumley, 1972; Mellor and Herring, 1973; Monin and Yaglom, 1975; Mellor and Yamada, 1982; Pope, 2005; Alfonsi, 2009]. All ReSM models are based upon the Reynolds decomposition of fluctuating flow characteristics, such as the instantaneous velocity, i = Ui + ui, where = Ui is either the ensemble- or time- or space-averaged velocity and ui is its fluctuating counterpart. The subtleties of the averaging have been extensively analyzed, e.g., by Monin and Yaglom  or Pope , and will not be discussed further. We shall refer to either of these averagings as Reynolds averaging and assume that they are used consistently throughout the derivations.
 The application of Reynolds averaging to the governing equations in the Boussinesq approximation and utilizing Einstein's summation rule yield [Galperin et al., 1989]
where Θ and θ are the mean and fluctuating potential temperatures, respectively, ρ is the density, ρ0 is a constant reference density, P is the mean pressure, fk = 2Ω(0, cosϕ, sinϕ) is the Coriolis vector, Ω is the angular velocity of the planet's rotation, and gi = (0, 0, −g) is the acceleration due to gravity.
 Reynolds averaging of the governing equations for turbulent momentum and heat fluxes, and , respectively, encounter the classical problem of turbulence closure since those equations involve many unknown correlations that cannot be “closed” at any level of correlation, i.e., a closed expression for the unknown correlations cannot be derived without introducing additional assumptions. The most problematic correlations are those involving the pressure because it is a nonlocal variable [e.g., Mellor and Herring, 1973]. Two approaches have been developed to model the pressure-velocity correlation terms, one by Launder et al.  and Zeman and Lumley  (we shall refer to these as LRR models), and the other by Mellor and Herring  and Mellor and Yamada  (hereafter referred to as MY models). Models of the LRR family are more comprehensive and more complicated than their MY counterparts. The LRR approach was initially designed for engineering flows with weak or no shear, while the MY models were more commonly applied to geophysical and environmental flows with strong shear. Later, the LRR models were also adapted to geophysical flows [e.g., Canuto et al., 2001; Cheng et al., 2002]. Models of intermediate complexity have also been proposed [e.g., Kurbatskiy and Kurbatskaya, 2006, 2009]. Note that the closure constants in both families of closures have been assumed invariant [e.g., Lewellen, 1977], although for some applications the constants were allowed to be flow dependent [e.g., Ristorcelli, 1997; Girimaji, 2000]. The closures used in current models of the Martian circulation are almost exclusively of the MY family [Forget et al., 1999; Rafkin et al., 2001; Moudden and McConnell, 2005], and so we shall only consider models of this class hereafter.
 Fully prognostic equations for all Reynolds stresses, , and turbulent heat fluxes, in the MY approach can be classified in terms of the departure from isotropy quantified by the expressions
where q2 = = 2EKT, EKT is TKE and aij and bi are tables of nondimensional coefficients. Let us introduce the following notations:
where ∥.∥ denotes a matrix norm [Mellor and Yamada, 1974; Galperin et al., 1988]. Assuming a and b are small but of the same order of magnitude, i.e., O(a) = O(b), and systematically neglecting terms of progressively higher order in a and b, one establishes the MY hierarchy of turbulence closure models [Mellor and Yamada, 1974]. By neglecting all terms up to O(a2) and O(b2), one arrives at the quasi-equilibrium turbulence energy model [Galperin et al., 1988], which could be classified as the level 2 model in the MY hierarchy. For this model, the Reynolds stresses and turbulent heat fluxes are related by algebraic equations, first for the Reynolds stresses themselves,
for the heat flux,
and for the temperature variance,
The TKE is given via a prognostic equation for q2,
where β is the thermal expansion coefficient, β = (∂ρ/∂θ)p/ρ0, Sq is the vertical nondimensional exchange coefficient for q2 (usually taken to be equal to 0.2 [Mellor and Yamada, 1982]), and various turbulence length scales are related to the master macroscale, ℓ, following Mellor and Yamada ,
It can be shown [e.g., Mellor, 1975] that the remaining constant, C1, is not independent but is related to the other constants by
In the MY model, the dissipation rate, ε, is given by
 The Coriolis terms in equations (32) and (33) significantly complicate the algebra. These terms were investigated by Galperin et al.  for the case of stable stratification and by Hassid and Galperin  for the cases of neutral and unstable stratification. It was found for the stable case that the contribution of the Coriolis terms does not exceed about 10% of the total stress, while for neutral and unstable stratification, the magnitude of the Coriolis terms can be large. These conclusions could be important for the daytime Martian atmospheric boundary layer, which is often strongly unstable and very deep. Despite these findings, the explicit Coriolis terms in the Reynolds stress and heat flux equations are usually neglected. The effect of the horizontal component of the Coriolis parameter on the mean flow has been considered in a relatively small number of studies, among them Kasahara , Dellar and Salmon , and Gerkema et al. , where it was shown that it could be significant though again usually neglected in most implementations.
 In the boundary layer approximation, using equations (32)–(35) with the explicit Coriolis terms neglected, the vertical turbulent fluxes of momentum and temperature can be represented in the following format:
where U and V are the components of the mean horizontal velocity, z is the vertical coordinate, KM is the vertical eddy viscosity, and KH is the vertical eddy thermal diffusivity. KM and KH are generally given by mixing length parameterizations of the form KM = qℓSM and KH = qℓSH, where SM and SH are nondimensional stability functions. Galperin et al.  showed that, in the quasi-equilibrium turbulence energy model, SM and SH are given by
with N being the Brunt-Väisälä frequency. These equations are assumed valid for both stable and unstable stratification. Using somewhat modified closure assumptions but staying within the MY hierarchy, Kantha and Clayson  obtained slightly different equations for SM and SH.
 Calculation of KM and Kh relies upon the precise specification of the turbulence macroscale, ℓ, which presents a formidable problem for turbulence modeling because the macroscale does not obey any known conservation law. As a result, all existing models for ℓ, either prognostic or diagnostic, are empirical. For neutrally stratified atmospheric boundary layers, Blackadar's algebraic equation has been commonly used [Blackadar, 1962],
where ℓ0 is a reference length scale. This equation ensures a smooth transition of ℓ from κz in the logarithmic velocity profile near the solid ground to a constant value at the top of the boundary layer. For the Martian atmosphere, Forget et al.  fix ℓ0 at 160 m while Moudden and McConnell  set ℓ0 to 200 m. Rafkin et al.  use a different definition of ℓB,
where, following Mellor and Yamada ,
and where h is the top of the boundary layer. A similar definition of ℓ0 was used by Haberle et al. [1993a], but they employed a coefficient of 0.2 instead of 0.1.
 In stably or neutrally stratified boundary layers and in flows that combine layers with different senses of stratification, Blackadar's formulation fails and more a sophisticated formulation may be required. In stably stratified boundary layers, the size of the overturning eddies is conditioned by their kinetic energy exceeding the potential energy of the background. This leads to the length scale limitation first introduced by Deardorff  and then used in turbulence models by André et al. , Hassid and Galperin , Galperin et al. , and many others,
The turbulence macroscale in stably stratified boundary layers can either be clipped to ℓs [Galperin et al., 1988; Rafkin et al., 2001] or determined from the equation
which ensures a smooth transition from the Blackadar formulation to ℓs in regions dominated by stable stratification. Some models take into account the dependence of ℓ on the vertical Coriolis parameter, f, by adding another term to equation (51),
where Cf is a constant [see, e.g., Zilitinkevich et al., 2007, and references therein]. Combined with equation (46), inequality (50) imposes a limitation on GH,
 In the original MY models, the stability functions, SM and SH, depended not only on GH but also on the nondimensional mean shear parameter, GM, given by
where S is the mean shear magnitude. In practice, however, these formulations led to spurious oscillations and instabilities [Mellor and Yamada, 1982; Hassid and Galperin, 1983]. The quasi-equilibrium turbulence energy model by Galperin et al.  was designed to alleviate this problem. Various aspects of these instabilities were discussed by Deleersnijder and Luyten , Mellor , Deleersnijder and Burchard , and Umlauf and Burchard . An insightful study by Deleersnijder et al.  attributes these oscillations to the ways SM and SH depend on the mean gradients of the velocity and temperature, i.e., GM and GH. To highlight the source of the instability, they consider a simple scalar diffusion equation in the variable ψ,
where λ is the diffusivity. It is assumed that λ = λ(ψz), ψz ≡ , and λ is positive definite. For ψz, a corresponding diffusion equation can also be derived,
where is the “effective diffusivity;” = λ + ψz. Unlike λ, it is possible for to become negative, giving rise to an unbounded growth of ψz, even though ψ itself remains bounded in time. As a result, ψ may exhibit small-scale, finite-amplitude oscillations that appear as an instability of the large-scale flow. Deleersnijder et al.  designed a criterion which allows one to determine whether or not a particular dependence, [SH(GM, GH), SM(GM, GH)], would lead to oscillations of this kind. This is a useful criterion as it makes it possible to screen various stability functions and select those that will not cause unphysical oscillations of the solution. Among other interesting results, this criterion demonstrates that the stability functions of the quasi-equilibrium turbulence energy model by Galperin et al.  can never cause spurious oscillations of this kind.
 The gradient Richardson number, Rig (compare equation (16)), is an important characteristic parameter for stratified turbulence that determines the strength of stratification with respect to vertical shear. Under strongly stable conditions, a critical value of Ri (denoted as Ricr) may be attained at which turbulent mixing is often assumed to be fully suppressed. In MY models, Ricr is under 0.2, leading to underpredicted mixing in some situations [Martin, 1985; Simpson et al., 1996; Rippeth, 2005]. Galperin et al.  discussed the general notion of a Ricr, based upon recent observational and theoretical studies that considered the effects of nonstationarity, internal waves, and flow anisotropization. They concluded that all these factors preclude the full laminarization of turbulence and thus make the concept of a Ricr devoid of its conventional meaning. Consequently, they suggested that the use of Ricr as a criterion of turbulence extinction should be avoided.
 Turbulence intensity in stably stratified flows can also be judged by another parameter, the buoyancy Reynolds number, Reb = ε/ν0N2, with ν0 being the molecular viscosity [see, e.g., Galperin and Sukoriansky, 2010, and references therein]. For Reb = O(1), vertical turbulent mixing may become laminarized. However, in horizontal planes, the mixing can still be much larger than in laminar flows. Conclusions similar to those of Galperin et al.  on the absence of a meaningful Ricr were also reached by Zilitinkevich et al.  based upon the total energy approach, with the total energy being the sum of kinetic and potential energies of turbulent fluctuations. A number of ReSMs with no Ricr were subsequently developed [e.g., Canuto et al., 2008; Violeau, 2009; Alexakis, 2009; Kantha and Carniel, 2009; Kitamura, 2010].
 In the case of unstable stratification, a superequilibrium balance equation for q2 (with all tendency terms dropped), equation (35), yields
Galperin et al.  showed that equation (57) yields a limitation on GH for the case of unstable stratification,
which can be used as a clipping condition in simulations. Even though models of the MY family can be, and have been, applied to flows with unstable stratification and strong convection, Forget et al.  express some doubts about the validity of the model in such conditions. Canuto et al.  explored the limitations of models of the MY family in depth and concluded that the condition (58) essentially imposes a limitation on the size of the eddies that can take part in convective transport. This limitation forces the model to include local interactions only and filters out the nonlocality which would otherwise be allowed by the third-order and higher-order correlations. To overcome this shortcoming, Canuto et al.  suggest that the third-order and possibly the higher-order correlations should be included in MY-type models.
 The effects of the nonlocality in convective flows are partially captured by the length scale specification, which must respond to ascending and descending motions of air parcels typical of convection. A widely accepted formulation of ℓ for turbulent convection goes back to works by André et al. , André and Lacarrére , Bougeault and André  and Bougeault and Lacarrére . It is postulated that for each level, z, a parcel with a EKT corresponding to that level (equal to EKT(z)) can travel upward and downward before being damped by buoyancy forces. One defines ℓup and ℓdown according to
If a parcel encounters on its way a layer with stable stratification, then equations (59a) and (59b) readily provide the limitation (50). Keeping in mind a possible large disparity between ℓup and ℓdown and the need to keep the bias toward smaller values, ℓ is related to ℓup and ℓdown via ℓc = min(ℓup, ℓdown). In realistic situations, boundary layers combine regions with both stable and unstable stratifications and to reflect this, the length scale equation (52) should be supplemented by the term ℓc−1. Such an equation has been widely used in atmospheric models and numerical weather prediction in the framework of the so-called CBR model [Cuxart et al., 2000].
 Specification of the turbulence length scale in flows with multiple regimes has been a persistently difficult problem for modeling. Aside from the algebraic formulations of the kind described earlier, another widely used approach employs prognostic equations for quantities related to ℓ, such as the dissipation rate, ε. Models utilizing prognostic equations for EKT and ε are often known as K − ε models [e.g., Rodi, 1987; Pope, 2005]. Mellor and Yamada  consider a prognosic equation for the quantity q2ℓ, which can be constructed by analogy with the equation for q2,
where Ps = KMS2 and Pb = KHN2 represent the mechanical production and the buoyant destruction of the turbulence energy, respectively, ε is the dissipation rate given by equation (39), and the constants E1, E2, and E3 are 1.8, 1.33, and 1.8, respectively. This equation is quite popular in oceanographic modeling and has been employed, for instance, in the Princeton Ocean Model (POM; http://www.aos.princeton.edu/WWWPUBLIC/htdocs.pom/). The values of these constants were reevaluated by Burchard  using simulations of three oceanic flows, and their recommendation was to increase E3 to about 5. Umlauf and Burchard  formulated a generic transport equation for a quantity ψ = (cμ0)p EKTmℓn that, with a proper choice of parameters p, m, and n, reverts to equations for either ε or q2ℓ or a quantity ω ∝ ε/EKT discussed, e.g., by Wilcox  and Umlauf et al. . Note that, even though great progress has been achieved in Reynolds stress modeling using prognostic length scale equations, this approach is mostly local and thus may have difficulties in convective boundary layers where ℓ becomes nonlocal, as illustrated by equations (59a) and (59b).
 As mentioned earlier, in convective flows proper accounting for the effects of buoyancy-driven convection cells and the ensuing nonlocality requires modification, not only of the turbulence length scale equation but also of the equations for turbulence correlations [Moeng and Wyngaard, 1989; Holtslag and Boville, 1993]. Deardorff , Holtslag and Moeng , Wyngaard and Weil  and Canuto et al.  introduced ad hoc nonlocal terms that improved their models' performance. However, a comprehensive approach to overcome the shortcomings of local models entails consideration of the balance equations for the third and fourth moments, as was done, e.g., by Canuto , Canuto et al. , Cheng and Canuto , Cheng et al. , Zilitinkevich et al. , Gryanik et al. , and Ferrero and Colonna . A recent study by Ferrero and Racca  demonstrated that accounting for nonlocal effects via higher-order correlations can improve simulations of the boundary layer height, even in the case of neutral stratification.
 The use of turbulence schemes in atmospheric models requires their consistency with the imposed boundary conditions. A conventional method to derive such boundary conditions near the underlying surface is to use a constant-flux layer approximation between the surface and the first grid point and to employ Monin-Obukhov similarity functions (Φq, where q = m, h etc.; see section 2) to calculate the required values of the mean profiles. Mellor  showed how Monin-Obukhov similarity functions can be derived directly from the turbulence model. However, in many cases these functions are taken from observations and so the mismatch between the observed functions and those obtained from the turbulence model may introduce spurious fluxes of momentum, heat, and other quantities.
 A new family of ReSMs was developed recently, based upon a spectral approach which is an alternative to the Reynolds stress modeling [Sukoriansky et al., 2005b]. This approach has been coined a quasi-normal scale elimination, or QNSE. Within this theory, internal waves and turbulence are treated as one entity rather than as an ad hoc dichotomy. QNSE provides a rigorous procedure of successive coarsening of the system's domain of definition that produces the effective, scale-dependent, vertical viscosity (KM) and thermal diffusivity (KH) as well as their horizontal counterparts. Dependent upon the range of eliminated scales, QNSE provides either subgrid-scale parameterization for LESs or an equivalent of an ReSM [Sukoriansky et al., 2005b, 2006]. In the latter case, the eddy viscosities and eddy diffusivities become functions of either the local gradient Richardson number, Ri, given by (9), or the Froude number, Fr = ε/NEKT. One of the important advantages of the QNSE model is the absence of the critical Richardson number, Ricr [Sukoriansky et al., 2005b; Galperin et al., 2007]. The QNSE-based expressions for KM and KH have been successfully tested in both K − ε and K − ℓ applications [Sukoriansky et al., 2005a, 2006; Sukoriansky and Galperin, 2008]. Along with the QNSE-based Monin-Obukhov similarity functions for the near-surface layer, these expressions were implemented in recent releases of the Weather Research and Forecasting (WRF) model for Earth applications. We note, however, that a version of the WRF model known as PlanetWRF has been in use recently for global simulations of the Martian atmosphere [Richardson et al., 2007], and other versions are now being used for mesoscale and LES modeling for Mars [Spiga and Forget, 2009; Spiga et al., 2010].