Mixing Length Formulation Research Articles

Designing a Fully‐Tunable and Versatile TKE‐l Turbulence Parameterization for the Simulation of Stable Boundary Layers

AbstractThis study presents the development of a so‐called Turbulent Kinetic Energy (TKE)‐l, or TKE‐l, parameterization of the diffusion coefficients for the representation of turbulent diffusion in neutral and stable conditions in large‐scale atmospheric models. The parameterization has been carefully designed to be completely tunable in the sense that all adjustable parameters have been clearly identified and the number of parameters has been minimized as much as possible to help the calibration and to thoroughly assess the parametric sensitivity. We choose a mixing length formulation that depends on both static stability and wind shear to cover the different regimes of stable boundary layers. We follow a heuristic approach for expressing the stability functions and turbulent Prandlt number in order to guarantee the versatility of the scheme and its applicability for planetary atmospheres composed of an ideal and perfect gas such as that of Earth and Mars. Particular attention has been paid to the numerical stability and convergence of the TKE equation at large time steps, an essential prerequisite for capturing stable boundary layers in General Circulation Models (GCMs). Tests, parametric sensitivity assessments and preliminary tuning are performed on single‐column idealized simulations of the weakly stable boundary layer. The robustness and versatility of the scheme are assessed through its implementation in the Laboratoire de Météorologie Dynamique Zoom GCM and the Mars Planetary Climate Model and by running simulations of the Antarctic and Martian nocturnal boundary layers.

Improving the Intensity Forecast of Tropical Cyclones in the Hurricane Analysis and Forecast System

Abstract A modification to the mixing length formulation in a planetary boundary layer (PBL) scheme is introduced to improve the intensity forecast of tropical cyclones (TCs) in a basin-scale Hurricane Analysis and Forecast System (HAFS) for the real-time experiment in 2021. The 2020 basin-scale HAFS with the physics suite of the NCEP operational Global Forecast System performs well in terms of the reduced root-mean-square (RMS) errors in track and intensity except for the mean intensity bias, compared with NCEP operational hurricane models. To address the large intensity bias issue, the vertical mixing length near the surface used in the PBL scheme is increased to follow the similarity theory, consistent with that used in the surface layer scheme. Test results show that the RMS error and bias in intensity are further reduced without the degradation of the track forecast. An idealized one-dimensional TC PBL model is used to understand the model response to the modification, indicating that the radial wind is strengthened to dynamically balance the enhanced downward momentum mixing. This is also exhibited in the case study of a three-dimensional HAFS simulation, with the improved vertical distribution of the simulated wind speed in the eyewall area. Given the improvement, the modification has been implemented in one of the configurations of the first version of the operational HAFS at NCEP. Finally, the adjustment of the parameterization of diffusion and mixing in TC simulations is discussed. Significance Statement A modification to the mixing length formulation in a PBL scheme is described, which improves the intensity forecast of tropical cyclones simulated in the Hurricane Analysis and Forecast System (HAFS). Retrospective tests indicate that the modification can reduce the root-mean-square error and bias of the simulated TC intensity by 5%–10% and 50%, respectively. This modification has been implemented in one of the operational configurations of HAFS, version 1, at NCEP, improving the hurricane model guidance.

A Single-Column Model Evaluation of Mixing Length Formulations and Constraints for the sa-TKE-EDMF Planetary Boundary Layer Parameterization

Abstract Parameterizing boundary layer turbulence is a critical component of numerical weather prediction and the representation of turbulent mixing of momentum, heat, and other tracers. The components that make up a boundary layer scheme can vary considerably, with each scheme having a combination of processes that are physically represented along with tuning parameters that optimize performance. Isolating a component of a PBL scheme to examine its impact is essential for understanding the evolution of boundary layer profiles and their impact on the mean structure. In this study we conduct three experiments with the scale-aware TKE eddy-diffusivity mass-flux (sa-TKE-EDMF) scheme: 1) releasing the upper limit constraints placed on mixing lengths, 2) incrementally adjusting the tuning coefficient related to wind shear in the modified Bougeault and Lacarrere (BouLac) mixing length formulation, and 3) replacing the current mixing length formulations with those used in the MYNN scheme. A diagnostic approach is adopted to characterize the bulk representation of turbulence within the residual layer and boundary layer in order to understand the importance of different terms in the TKE budget as well as to assess how the balance of terms changes between mixing length formulations. Although our study does not seek to determine the best formulation, it was found that strong imbalances led to considerably different profile structures both in terms of the resolved and subgrid fields. Experiments where this balance was preserved showed a minor impact on the mean structure regardless of the turbulence generated. Overall, it was found that changes to mixing length formulations and/or constraints had stronger impacts during the day while remaining partially insensitive during the evening.

A New Horizontal Mixing-Length Formulation for Numerical Simulations of Tropical Cyclones

AbstractA new physically based horizontal mixing-length formulation is introduced and evaluated in the Hurricane Weather and Research Forecasting (HWRF) Model. Recent studies have shown that the structure and intensity of tropical cyclones (TCs) simulated by numerical models are sensitive to horizontal mixing length in the parameterization of horizontal diffusion. Currently, many numerical models including the operational HWRF Model formulate the horizontal mixing length as a fixed fraction of grid spacing or a constant value, which is not realistic. To improve the representation of the horizontal diffusion process, the new formulation relates the horizontal mixing length to local wind and its horizontal gradients. The resulting horizontal mixing length and diffusivity are much closer to those derived from field measurements. To understand the impact of different mixing-length formulations, we analyze the evolutions of an idealized TC simulated by the HWRF Model with the new formulation and with the current formulation (i.e., constant values) of horizontal mixing length. In two real-case tests, the HWRF Model with the new formulation produces the intensity and track forecasts of Hurricanes Harvey (2017) and Lane (2018) that are much closer to observations. Retrospective runs of hundreds of forecast cycles of multiple hurricanes show that the mean errors in intensity and track simulated by HWRF with the new formulation can be reduced approximately by 10%.

Large eddy simulations of the Martian convective boundary layer: Towards developing a new planetary boundary layer scheme

The grid resolutions typically employed in atmospheric global circulation and mesoscale models are not sufficient to explicitly resolve the turbulence processes within the planetary boundary layer (PBL). turbulent fluxes are, therefore, fully parameterized in those models, based on empirical relationships for the mixing length scale using PBL schemes. However, microscale models use the large-eddy simulation (LES) technique to resolve turbulence processes. Here, we perform LES computations for the daytime Martian planetary boundary layer, ranging from weakly to strong convective conditions, using the Mars implementation of planetWRF, the MarsWRF model. In this study, our main focus is to investigate the turbulence statistics and turbulent spectrum utilizing our LES results. Then, using the computed turbulence kinetic energy and its dissipation rate, a generic formulation for the mixing length scale variation in the Martian convective boundary layer is proposed. This mixing length formulation is used to derive a Mars-specific PBL scheme and its performance is compared to the PBL scheme currently in use by the MarsWRF model, the MRF scheme. The proposed scheme is tested both in global and mesoscale simulations, which are used to evaluate the convective boundary layer height and near-surface meteorology conditions at the InSight landing site. The presently proposed PBL scheme results in an improved prediction of convective boundary layer height that agree better with the observational estimations acquired by radio occultations of Mars Express in comparison to the MRF scheme. Also, the prediction of near-surface winds at the InSight landing site is slightly improved.

A Generalized Mixing Length Closure for Eddy‐Diffusivity Mass‐Flux Schemes of Turbulence and Convection

AbstractBecause of their limited spatial resolution, numerical weather prediction and climate models have to rely on parameterizations to represent atmospheric turbulence and convection. Historically, largely independent approaches have been used to represent boundary layer turbulence and convection, neglecting important interactions at the subgrid scale. Here we build on an eddy‐diffusivity mass‐flux (EDMF) scheme that represents all subgrid‐scale mixing in a unified manner, partitioning subgrid‐scale fluctuations into contributions from local diffusive mixing and coherent advective structures and allowing them to interact within a single framework. The EDMF scheme requires closures for the interaction between the turbulent environment and the plumes and for local mixing. A second‐order equation for turbulence kinetic energy (TKE) provides one ingredient for the diffusive local mixing closure, leaving a mixing length to be parameterized. Here, we propose a new mixing length formulation, based on constraints derived from the TKE balance. It expresses local mixing in terms of the same physical processes in all regimes of boundary layer flow. The formulation is tested at a range of resolutions and across a wide range of boundary layer regimes, including a stably stratified boundary layer, a stratocumulus‐topped marine boundary layer, and dry convection. Comparison with large eddy simulations (LES) shows that the EDMF scheme with this diffusive mixing parameterization accurately captures the structure of the boundary layer and clouds in all cases considered.

Modelling and assessing the near-wake representation and turbulence behaviour of control-oriented wake models

Due to the interaction between the wake of an upstream turbine on a downstream turbine, power losses and increased fatigue loads occur. By yawing the upstream turbine with regard to the wind direction, one can potentially reduce the power losses of the downstream turbine and reduce the fatigue loads. The evolution of the wake depends on the pressure gradient within the near-wake region and the turbulent kinetic energy and must be incorporated in existing wake steering algorithms to provide an accurate estimation of the wake flow. This paper will show a first approach to implement a near-wake model and a turbulence model in the curled wake model within the controls-oriented framework FLORIS. The near-wake model is based on an analytical expression of the velocity profile to model the pressure gradient. Furthermore, two turbulence models are incorporated within the curled wake model based on a Gaussian-distribution and a mixing length formulation. The adapted curled wake model is then assessed with the measurement data acquired in the wind tunnel at ForWind – University of Oldenburg. The evaluation of the models show good agreement for the velocity deficit and representation of the near-wake region. Furthermore, the turbulent kinetic energy behaved as expected in comparison to other work, showing a ring of high turbulent kinetic energy at non-yawed condition which is deflected to a curled shape at large yaw angles with the turbulence model based on a mixing length formulation.

Effects of a Fence on Pollutant Dispersion in a Boundary Layer Exposed to a Rural-to-Urban Transition

Simultaneous particle-image velocimetry and laser-induced fluorescence combined with large-eddy simulations are used to investigate the flow and pollutant dispersion behaviour in a rural-to-urban roughness transition. The urban roughness is characterized by an array of cubical obstacles in an aligned arrangement. A plane fence is added one obstacle height h upstream of the urban roughness elements, with three different fence heights considered. A smooth-wall turbulent boundary layer with a depth of 10h is used as the approaching flow, and a passive tracer is released from a uniform line source 1h upstream of the fence. A shear layer is formed at the top of the fence, which increases in strength for the higher fence cases, resulting in a deeper internal boundary layer (IBL). It is found that the mean flow for the rural-to-urban transition can be described by means of a mixing-length model provided that the transitional effects are accounted for. The mixing-length formulation for sparse urban canopies, as found in the literature, is extended to take into account the blockage effect in dense canopies. Additionally, the average mean concentration field is found to scale with the IBL depth and the bulk velocity in the IBL.

A Scale-Adaptive Turbulent Kinetic Energy Closure for the Dry Convective Boundary Layer

Abstract A pragmatic scale-adaptive turbulent kinetic energy (TKE) closure is proposed to simulate the dry convective boundary layer for a variety of horizontal grid resolutions: from 50 m, typical of large-eddy simulation models that use three-dimensional turbulence parameterizations/closures, up to 100 km, typical of climate models that use one-dimensional turbulence and convection parameterizations/closures. Since parameterizations/closures using the TKE approach have been frequently used in these two asymptotic limits, a simple method is proposed to merge them with a mixing-length-scale formulation for intermediate resolutions. This new scale-adaptive mixing length naturally increases with increasing grid length until it saturates as the grid length reaches mesoscale-model resolution. The results obtained using this new approach for dry convective boundary layers are promising. The mean vertical profiles of potential temperature and heat flux remain in good agreement for different resolutions. A continuous transition (in terms of resolution) across the gray zone is illustrated through the partitioning between the model-resolved and the subgrid-scale transports as well as by documenting the transition of the subgrid-scale TKE source/sink terms. In summary, a natural and continuous transition across resolutions (from 50 m to 100 km) is obtained, for dry convection, using exactly the same atmospheric model for all resolutions with a simple scale-adaptive mixing-length formulation.

Improving Wind-Ramp Forecasts in the Stable Boundary Layer

The viability of wind-energy generation is dependent on highly accurate numerical wind forecasts, which are impeded by inaccuracies in model representation of boundary-layer processes. This study revisits the basic theory of the Mellor, Yamada, Nakanishi, and Niino (MYNN) planetary boundary-layer parametrization scheme, focusing on the onset of wind-ramp events related to nocturnal low-level jets. Modifications to the MYNN scheme include: (1) calculation of new closure parameters that determine the relative effects of turbulent energy production, dissipation, and redistribution; (2) enhanced mixing in the stable boundary layer when the mean wind speed exceeds a specified threshold; (3) explicit accounting of turbulent potential energy in the energy budget. A mesoscale model is used to generate short-term (24 h) wind forecasts for a set of 15 cases from both the U.S.A. and Germany. Results show that the new set of closure parameters provides a marked forecast improvement only when used in conjunction with the new mixing length formulation and only for cases that are originally under- or over-forecast (10 of the 15 cases). For these cases, the mean absolute error (MAE) of wind forecasts at turbine-hub height is reduced on average by 17%. A reduction in MAE values on average by 26% is realized for these same cases when accounting for the turbulent potential energy together with the new mixing length. This last method results in an average reduction by at least 13% in MAE values across all 15 cases.

On the Coherence in the Boundary Layer: Development of a Canopy Interface Model

A 1D Canopy Interface Model (CIM) is developed to act as an interface between a meso-scale and a micro-scale atmospheric model and to better resolve the surface turbulent fluxes in the urban canopy layer. A new discretisation is proposed to solve the TKE equation finding solutions that remain fully concordant with the surface layer theories developed for neutral flows over flat surfaces. A correction is added in the buoyancy term of the TKE equation to improve consistency with the Monin-Obukhov surface layer theory. Obstacles of varying heights and dimensions are taken into account by introducing specific terms in the equations and by modifying the mixing length formulation in the canopy layer. The results produced by CIM are then compared with wind and TKE profiles simulated with a LES experiment and results obtained during the BUBBLE meteorological intensive observation campaign. It is shown that the CIM computations are in good agreement with the results simulated by the LES as well as the measurements from BUBBLE. The applicability of the correction term in an urban canopy layer and to further validate CIM in multiple stability conditions and various urban configurations is discussed.

On the surface physics affecting solar oscillation frequencies

Abstract Adiabatic oscillation frequencies of stellar models, computed with the standard mixing-length formulation for convection, increasingly deviate with radial order from observations in solar-like stars. Standard solar models overestimate adiabatic frequencies by as much as ∼ 20 μHz. In this Letter, we address the physical processes of turbulent convection that are predominantly responsible for the frequency differences between standard models and observations, also called ‘surface effects’. We compare measured solar frequencies from the Michelson Doppler Imager instrument on the SOlar and Heliospheric Observatory spacecraft with frequency calculations that include 3D hydrodynamical simulation results in the equilibrium model, non-adiabatic effects, and a consistent treatment of the turbulent pressure in both the equilibrium and stability computations. With the consistent inclusion of the above physics in our model computation, we are able to reproduce the observed solar frequencies to ≲3 μHz without the need of any additional ad hoc functional corrections.

Parameterization of eddy fluxes based on a mesoscale energy budget

It has recently been proposed to formulate eddy diffusivities in ocean models based on a mesoscale eddy kinetic energy (EKE) budget. Given an appropriate length scale, the mesoscale EKE can be used to estimate an eddy diffusivity based on mixing length theory. This paper discusses some of the open questions associated with the formulation of an EKE budget and mixing length, and proposes an improved energy budget-based parameterization for the mesoscale eddy diffusivity. A series of numerical simulations is performed, using an idealized flat-bottomed β-plane channel configuration with quadratic bottom drag. The results stress the importance of the mixing length formulation, as well as the formulation for the bottom signature of the mesoscale EKE, which is important in determining the rate of EKE dissipation. In the limit of vanishing planetary vorticity gradient, the mixing length is ultimately controlled by bottom drag, though the frictional arrest scale predicted by barotropic turbulence theory needs to be modified to account for the effects of baroclinicity. Any significant planetary vorticity gradient, β, is shown to suppress mixing, and limit the effective mixing length to the Rhines scale. While the EKE remains moderated by bottom friction, the bottom signature of EKE is shown to decrease as the appropriately non-dimensionalized friction increases, which considerably weakens the impact of changes in the bottom friction compared to barotropic turbulence. For moderate changes in the bottom-friction, eddy fluxes are thus reasonably well approximated by the scaling relation proposed by Held and Larichev (1996), which ignores the effect of bottom friction.

Improvements to stellar structure models, based on a grid of 3D convection simulations – II. Calibrating the mixing-length formulation

We perform a calibration of the mixing length of convection in stellar structure models against realistic 3D radiation-coupled hydrodynamics (RHD) simulations of convection in stellar surface layers, determining the adiabat deep in convective stellar envelopes. The mixing-length parameter $\alpha$ is calibrated by matching averages of the 3D simulations to 1D stellar envelope models, ensuring identical atomic physics in the two cases. This is done for a previously published grid of solar-metallicity convection simulations, covering from 4200 K to 6900 K on the main sequence, and 4300-5000 K for giants with logg=2.2. Our calibration results in an $\alpha$ varying from 1.6 for the warmest dwarf, which is just cool enough to admit a convective envelope, and up to 2.05 for the coolest dwarfs in our grid. In between these is a triangular plateau of $\alpha$ ~ 1.76. The Sun is located on this plateau and has seen little change during its evolution so far. When stars ascend the giant branch, they largely do so along tracks of constant $\alpha$, with $\alpha$ decreasing with increasing mass.

THE ROLE OF SUBSURFACE FLOWS IN SOLAR SURFACE CONVECTION: MODELING THE SPECTRUM OF SUPERGRANULAR AND LARGER SCALE FLOWS

We model the solar horizontal velocity power spectrum at scales larger than granulation using a two-component approximation to the mass continuity equation. The model takes four times the density scale height as the integral (driving) scale of the vertical motions at each depth. Scales larger than this decay with height from the deeper layers. Those smaller are assumed to follow a Kolomogorov turbulent cascade, with the total power in the vertical convective motions matching that required to transport the solar luminosity in a mixing length formulation. These model components are validated using large scale radiative hydrodynamic simulations. We reach two primary conclusions: 1. The model predicts significantly more power at low wavenumbers than is observed in the solar photospheric horizontal velocity spectrum. 2. Ionization plays a minor role in shaping the observed solar velocity spectrum by reducing convective amplitudes in the regions of partial helium ionization. The excess low wavenumber power is also seen in the fully nonlinear three-dimensional radiative hydrodynamic simulations employing a realistic equation of state. This adds to other recent evidence suggesting that the amplitudes of large scale convective motions in the Sun are significantly lower than expected. Employing the same feature tracking algorithm used with observational data on the simulation output, we show that the observed low wavenumber power can be reproduced in hydrodynamic models if the amplitudes of large scale modes in the deep layers are artificially reduced. Since the large scale modes have reduced amplitudes, modes on the scale of supergranulation and smaller remain important to convective heat flux even in the deep layers, suggesting that small scale convective correlations are maintained through the bulk of the solar convection zone.

Global Patterns of Mesoscale Eddy Properties and Diffusivities

Abstract Mesoscale eddies play a major role in the transport of tracers in the ocean. Focusing on a sector in the east Pacific, the authors present estimates of eddy diffusivities derived from kinematic tracer simulations using satellite-observed velocity fields. Meridional diffusivities are diagnosed, and how they are related to eddy properties through the mixing length formulation of Ferrari and Nikurashin, which accounts for the suppression of diffusivity due to eddy propagation relative to the mean flow, is shown. The uniqueness of this study is that, through systematically varying the zonal-mean flow, a hypothetical “unsuppressed” diffusivity is diagnosed. At a given latitude, the unsuppressed diffusivity occurs when the zonal-mean flow equals the eddy phase speed. This provides an independent estimate of eddy phase propagation, which agrees well with theoretical arguments. It is also shown that the unsuppressed diffusivity is predicted very well by classical mixing length theory, that is, that it is proportional to the rms eddy velocity times the observed eddy size, with a spatially constant mixing efficiency of 0.35. Then, the suppression factor is estimated and it is shown that it too can be understood quantitatively in terms of easily observed mean flow properties. The authors then extrapolate from these sector experiments to the global scale, making predictions for the global surface eddy diffusivity. Together with a prognostic equation for eddy kinetic energy and a theory explaining observed eddy sizes, these concepts could potentially be used in a closure for eddy diffusivities in coarse-resolution ocean climate models.

An Integrated TKE-Based Eddy Diffusivity/Mass Flux Boundary Layer Closure for the Dry Convective Boundary Layer

Abstract This study presents a new approach to the eddy diffusivity/mass flux (EDMF) framework for the modeling of convective boundary layers. At the root of EDMF lies a decomposition of turbulent transport mechanisms into strong ascending updrafts and smaller-scale turbulent motions. The turbulent fluxes can be therefore described using two conventional approaches: mass flux (MF) for the organized thermals and eddy diffusivity (ED) for the remaining turbulent field. Since the intensities of both MF and ED transports depend on the kinetic energy of the turbulent motions, it seems reasonable to formulate an EDMF framework based on turbulent kinetic energy (TKE). Such an approach allows for more physical and less arbitrary formulations of parameters in the model. In this study the EDMF–TKE coupling is achieved through the use of (i) a new parameterization for the lateral entrainment coefficient ɛ and (ii) the MF contribution to the buoyancy source of TKE. Some other important features of the EDMF parameterization presented here include a revised mixing length formulation and Monin–Obukhov stability scaling for the surface layer. The scheme is implemented in a one-dimensional (1D) model. Several cases of dry convective boundary layers (CBL) with different surface sensible heat fluxes in the free-convection limit are investigated. Results are compared to large-eddy simulation (LES). Good agreement between LES and the 1D model is achieved with respect to mean profiles, boundary layer evolution, and updraft characteristics. Some disagreements between the models are found to most likely relate to deficiencies in the TKE simulation in the 1D model. Comparison with other previously established ɛ parameterizations shows that the new TKE-based formulation leads to equally accurate, and in many respects better, simulation of the CBL. The encouraging results obtained with the proposed EDMF framework indicate that full integration of EDMF with higher-order closures is possible and can further improve boundary layer simulations.

Interior structure models of solid exoplanets using material laws in the infinite pressure limit

The detection of low-mass extrasolar planets has initiated growing interest in massive rocky bodies (super-Earths) for which no Solar System analogue does exist. Here, we present a new model approach to investigate their interior structure and thermal state. We improve and extend previous interior models mainly in two areas: the first improvement is due to the consequent application of equations of state (EoS) that are compliant with the thermodynamics of the high-pressure limit and facilitate reinvestigating mass–radius relations for terrestrial-type exoplanets. To quantify the uncertainty due to extrapolation, we compare a generalized Rydberg and a Keane EoS, which are both consistent with the high-pressure limit. Furthermore, we consider a reciprocal K′ EoS that fits the seismologically obtained Preliminary Reference Earth Model (PREM), thereby accounting for the mineralogical composition of the Earth. As a result, the predicted planetary radii of terrestrial-type exoplanets of up to 10 Earth masses would differ by less than 2% between all three EoS, well within current observational limits. The second extension arises from the adoption of a mixing length formulation instead of the commonly used, more simplified parameterized approach to model convective heat transport in planetary mantles. In comparison to parameterized convection models, our results indicate generally hotter interiors with increasing planetary mass and a cumulative tendency to extended regimes of sluggish convection in the lowermost mantle. The latter is attributed to less efficient convective heat transport with increasing mantle pressures. An improved knowledge of the present thermal state is prerequisite to gain a better understanding of the pathways of internal evolution of terrestrial-type exoplanets.

THE MASS MIXING LENGTH IN CONVECTIVE STELLAR ENVELOPES

The scale length over which convection mixes mass in a star can be calculated as the inverse of the vertical derivative of the unidirectional (up or down) mass flux. This is related to the mixing length in the mixing length theory of stellar convection. We give the ratio of mass mixing length to pressure scale height for a grid of 3D surface convection simulations, covering from 4300\,K to 6900\,K on the main-sequence, and up to giants at $\log g = 2.2$, all for solar composition. These simulations also confirm what is already known from solar simulations, that convection doesn't proceed by discrete convective elements, but rather as a continuous, slow, smooth, warm upflow and turbulent, entropy deficient, fast down drafts. This convective topology also results in mixing on a scale as that of the classic mixing length formulation, and is simply a consequence of mass conservation on flows in a stratified atmosphere.

Convection in stellar models

Convection has two main effects of interest to stellar modelers: It changes the stratification compared to a radiative or conductive region, and it mixes the matter efficiently. This was recognized early on, and a rudimentary description of convection was soon worked out—this is the familiar mixing length formulation. This paper will compare and contrast this much used formulation with results from realistic 3D simulations of interacting convection and radiation in the surface layers of stars. Due to space limitations, this review will be concerned with convective envelopes, only.