1. Introduction
The atmospheric boundary layer (ABL) is the lowermost part of the atmosphere that is directly influenced by Earth’s surface and across which turbulent transports exchange momentum, energy, and matter between the surface and the free atmosphere. Under conditions of clear skies, in the evening net surface longwave emission exceeds shortwave absorption, and the ground surface becomes colder than the overlying atmosphere. Turbulence in the resulting stable boundary layer (SBL) can then only be maintained by mechanical generation of turbulence kinetic energy (TKE). The evolution of stratification with time is determined by how efficiently turbulence can transport energy toward the surface relative to how rapidly it is lost by radiative fluxes. Such stable boundary layers can also form during fair skies and weak pressure gradients in basins, in local depressions, and in valleys with weak down-valley slopes (Mahrt 2014). The longest-enduring stable boundary layer is found during the polar night, which can last over 80 days (Grachev et al. 2005). Near-calm conditions can be associated with low surface temperatures, an accumulation of pollutants near the ground, and the formation of dense fog (Mahrt 2011).
Understanding and modeling the SBL is essential for both regional and global atmospheric models. Regional models are used to model, for example, pollutant dispersal (Nappo 1991; Arya 1999; Salmond and McKendry 2005), wind speed power and distribution for wind power assessments (Petersen et al. 1998), and the variability and extremes of near-surface winds (He et al. 2010; Monahan et al. 2011; He et al. 2012). For larger-scale models, such as numerical weather prediction or global climate models, an accurate representation of turbulent mixing is necessary to model the near-surface wind, temperature, and humidity. In particular, changes in turbulent mixing may impact the representation of surface temperature extremes, fog, and clouds (Holtslag et al. 2013). All of these models require turbulence parameterizations, as physically relevant dynamics occur on scales of motion that are smaller than the grid scales can capture.
Both modeling and observational studies indicate that the interplay between the suppression of turbulence due to static stability and the mechanical generation of turbulence due to wind shear gives rise to multiple stability regimes within the stable boundary layer. The simplest classification scheme distinguishes the weakly stable boundary layer (WSBL) from the very stable boundary layer (VSBL) (Webb 1970; Walters et al. 2007; Mahrt 1998a, 2014). Weakly stable conditions occur in the presence of moderate to strong pressure gradients or cloudy skies and are characterized by the presence of continuous (if weak) turbulent mixing. Classic Monin–Obukhov similarity theory (MOST) generally holds in the WSBL (Mahrt 1998a; Grachev et al. 2005; Pahlow et al. 2001; Mahrt 2014). In contrast, a VSBL typically occurs in the presence of weak pressure gradients or clear skies (with strong surface radiative cooling), and turbulence can weaken to the point of collapse. Although turbulence may never completely cease in the VSBL, the scales of motion become small enough that the flow near the surface can decouple from the overlying atmosphere and MOST fails (Derbyshire 1999; Banta et al. 2007; Williams et al. 2013; Mahrt 2011; Optis et al. 2015).
As local stability parameters, such as the Richardson number (Ri), tend to be lower in the WSBL and higher in the VSBL, attempts have been made to define a transitional Richardson number that separates the two regimes (Mahrt 2014). Specifically, a so-called critical Richardson number (Ric) above which turbulence collapses has been sought out. The existence of such a value is debated (Holtslag and De Bruin 1988; Beljaars and Holtslag 1991), and if it exists, its value is highly uncertain (Fernando and Weil 2010). Other authors have emphasized the need for additional metrics to separate the regimes (Van de Wiel et al. 2002; Grachev et al. 2005; Monahan et al. 2015). More recent studies have argued that the collapse of turbulence is best interpreted in terms of the maximum sustainable downward heat flux (MSHF) (Van de Wiel et al. 2007, 2012b; Van Hooijdonk et al. 2015).
The existence of a maximum sustainable heat flux can be understood by considering the two factors that determine the heat flux: the strength of the temperature gradient and the intensity of turbulence (De Bruin 1994; Malhi 1995; Van Hooijdonk et al. 2015). In near-neutral conditions, the heat flux is weak because of the small temperature gradient, while in strongly stable conditions the heat flux is weak because turbulent transports are suppressed. For a given wind shear, there is an optimal combination of the turbulent intensity and the temperature gradient that gives rise to a maximum possible sensible heat flux.
Normally, stability analyses in fluid motion study the transition from laminar to turbulent flow. The primary focus of our analysis will be the more general dynamical systems concept of linear stability of equilibria, assessing the response of the model under consideration to small perturbations around equilibrium states corresponding to turbulent flows. We will refer to stability in the first, more restricted, sense as hydrodynamic stability and the more general sense simply as stability.
We will investigate the stability properties of the equilibrium states of Couette flow for different parameterizations of the turbulent fluxes. As first noted by Taylor (1971), the existence of an MSHF implies the existence of two equilibrium surface stresses for a specified surface heat flux (below the MSHF). Taylor’s hypothesis that one of the solutions was stable and the other unstable was confirmed analytically for Couette flow with turbulent fluxes parameterized using a Businger–Dyer-type similarity function by Van de Wiel et al. (2007). The equilibrium state at which the stability properties of the linearized dynamics changed from stable to unstable was shown to correspond to the MSHF state. Van de Wiel et al. (2007) used numerical simulations of the time-dependent problem to show that one branch of the equilibrium curve was stable while the other was unstable.
Businger–Dyer-type parameterizations are known to be decreasingly accurate as stratification increases, and other flux parameterizations exist (e.g., Webb 1970; Hicks 1976; Clarke 1970; Holtslag and De Bruin 1988; Chenge and Brutsaert 2005; Brown et al. 2008; Delage 1997). As these parameterizations are obtained as empirical fits to datasets with large scatter, none of these parameterizations is expected to be absolutely correct, and qualitative features of the MSHF mechanism for turbulent collapse should be robust to changes in parameterization for this mechanism to be relevant to the atmospheric SBL. In this study, we will explore several of the available parameterizations to answer the question of whether the MSHF framework is sensitive to the choice of model.
Using data from the Royal Netherlands Meteorological Institute (KNMI) Cabauw observatory, Monahan et al. (2015) provided evidence of two distinct states of the SBL, corresponding to the stable and unstable branches of the Couette model equilibrium curves found by Van de Wiel et al. (2007). Roughly half of the observed states fell along the unstable branch. The population of the unstable branch motivates a more thorough analysis of the stability properties of the equilibrium curve of the Couette flow model. In this study, we extend the results of Van de Wiel et al. (2007) to directly analyze the linear stability of equilibria for a range of turbulent flux parameterizations by numerically computing the eigenstructure of the linearized operator.
This paper is organized as follows. In section 2 we describe the Couette flow model, including the Richardson number–dependent parameterizations of turbulent viscosity and diffusivity. The framework for the equilibrium and linear stability analyses is presented in section 3. Results of numerically computed equilibrium curves and linear stability analysis are presented in section 4. Conclusions and discussion appear in section 5.
2. The Couette flow model
Following Van de Wiel et al. (2007), we consider a horizontally homogeneous, one-dimensional Couette flow model in which momentum and temperature tendencies result only from turbulent flux convergence. The Coriolis effect, pressure gradient force, radiative flux divergence, and molecular viscosity/diffusivity are neglected for simplicity. In particular, we neglect the existence of a molecular boundary layer that becomes relevant when the surface is so smooth that the viscous sublayer is actually deeper than the roughness elements. A schematic diagram for our model of wind speed U and temperature T in the SBL is shown in Fig. 1. The height of the upper boundary is fixed at
Schematic diagram of the 1D Couette flow model. The lower boundary is fixed, and the upper boundary is constantly moving. The temperature profile is shaded in grayscale.
Citation: Journal of the Atmospheric Sciences 73, 9; 10.1175/JAS-D-16-0057.1
The velocity upper boundary condition was motivated in Van de Wiel et al. (2012b) by the observation that near-surface winds tend to weaken in the evening, while the flow above the nocturnal boundary layer tends to accelerate as a result of inertial effects. At an intermediate altitude, decameters above the ground, the magnitude of the wind remains approximately constant. This point was denoted the velocity crossing point (CP) by Van de Wiel et al. (2012b). At Cabauw, the altitude of the CP is observed to be about 40 m (Van de Wiel et al. 2012a). Thus, there is reason to believe that the Couette flow model can provide a useful first approximation even though it represents a dramatic simplification of the dynamics of the SBL.
















While MOST predicts that the similarity functions should be universal, it does not predict their functional form. Many different forms of
Stability functions for momentum
To assess the robustness of the MSHF framework for predicting the collapse of turbulence to the parameterizations of the turbulent fluxes, we examine four sets of stability functions. Their functional forms are given explicitly in Table 1 and are plotted in Fig. 2 alongside the similarity functions. The various stability function formulations have been proposed based on empirical fits to available data (e.g.,Webb 1970; Businger et al. 1971; Dyer 1974; Hicks 1976; Clarke 1970; Holtslag and De Bruin 1988; Chenge and Brutsaert 2005; Brown et al. 2008). However, flux observations in the very stable regime are subject to large variability and problems with sampling (Nieuwstadt 1984; Mahrt 1985), so it is difficult to ascertain empirically which function is most appropriate. Some of the relations imply the existence of an Ric (Businger et al. 1971; Brown et al. 2008; Holtslag and De Bruin 1988) above which
(a),(c) Similarity functions are shown for Businger et al. (1971) (Bus), Holtslag and De Bruin (1988) (HD), Beljaars and Holtslag (1991) (BH) and Brown et al. (2008) (Brown). (b),(d) The stability functions of (a) and (c), respectively. Corresponding formulas are given in Table 1.
Citation: Journal of the Atmospheric Sciences 73, 9; 10.1175/JAS-D-16-0057.1
Andreas (2002) reviewed various stability functions used to model the stably stratified surface layer over snow and ice. They examined the characteristics of seven different sets of stability functions, including three of the functions presented in Table 1 (Businger et al. 1971; Holtslag and De Bruin 1988; Beljaars and Holtslag 1991). Assuming the existence of a critical Richardson number and that the turbulent Prandtl number
Finally, we also considered the stability function introduced by Brown et al. (2008), characterized by a unit Prandtl number. Although there is no Ric for this function, Ri decreases rapidly with increasing stability. This parameterization has been tuned to improve predictions by the Met Office (UKMO) Unified Model [e.g., cloud coverage and thickness (Price et al. 2015)].
Part of the rationale for the consideration of these parameterizations is their use in current operational models. The Businger–Dyer form with
3. Equilibrium and linear stability





The definition of



The dynamical stability calculation finds the eigenmodes of the discretized matrix operator
For each of the stability functions presented in Table 1 and a range of
Numerical simulations of the fully nonlinear Couette flow equations were performed using finite differences in space and time on a logarithmically spaced grid ranging from
4. Results


For the Businger–Dyer-type function (Table 1), (a) the equilibrium diagrams for
Citation: Journal of the Atmospheric Sciences 73, 9; 10.1175/JAS-D-16-0057.1
As in Fig. 3, but for the Holtslag and De Bruin (1988) function (Table 1).
Citation: Journal of the Atmospheric Sciences 73, 9; 10.1175/JAS-D-16-0057.1
As in Fig. 3, but for the Beljaars and Holtslag (1991) function (Table 1).
Citation: Journal of the Atmospheric Sciences 73, 9; 10.1175/JAS-D-16-0057.1
As in Fig. 3, but for the Brown et al. (2008) function (Table 1).
Citation: Journal of the Atmospheric Sciences 73, 9; 10.1175/JAS-D-16-0057.1
In broad terms, the equilibrium curves corresponding to the different stability functions exhibit qualitatively similar structures. In all cases, there is a maximum in heat flux for intermediate values of
The dependence of the nondimensional MSHF
(a) The MSHF and (b) corresponding
Citation: Journal of the Atmospheric Sciences 73, 9; 10.1175/JAS-D-16-0057.1
The value of
To assess the linear stability of the equilibrium states, for each
The value of the unstable eigenmode
Linearized stability analysis is a useful tool for understanding the stability properties of a dynamical system, but it is not always sufficient. For example, it fails to predict the onset of turbulence in some classical flows (including Couette and Poiseuille flows). This failure can be attributed to the nonnormality of the governing operator (Gustavsson 1991; Farrell and Ioannou 1993; Reddy and Henningson 1993; Baggett et al. 1995; Trefethen et al. 1993; Farrell and Ioannou 1993). For the discretized Couette flow system,
Nonnormal growth rates for the Businger–Dyer-type function with
The maximum real eigenvalue of
Citation: Journal of the Atmospheric Sciences 73, 9; 10.1175/JAS-D-16-0057.1
To better understand the physical relevance of the instabilities along the unstable branch, we compute the dimensional e-folding times
Dimensional versions of the results shown in Fig. 3 obtained using h = 25 m.
Citation: Journal of the Atmospheric Sciences 73, 9; 10.1175/JAS-D-16-0057.1
For a better understanding of how well the nonlinear dynamical system is represented by our linear stability analysis, we simulated Eqs. (1) and (2) numerically using the Businger–Dyer
The time it took for turbulence to collapse was less than an hour for some perturbations, while for others it was on the order of tens of hours. We verified that the initial growth rates were consistent with the predictions of our linear stability analysis
5. Physical interpretation and conclusions
Although the Couette flow model is an oversimplification of atmospheric boundary layers, it serves as a useful heuristic tool for examining turbulent collapse in the SBL. Using specific closure assumptions, Van de Wiel et al. (2007) demonstrated that the collapse of turbulence could be explained as a linear instability of the system associated with the existence of an MSHF. The clustering of observed turbulence data around the equilibrium curve predicted by the Van de Wiel et al. (2007) model observed in Monahan et al. (2015), including around the unstable branch, motivated a more detailed analysis of the stability characteristics of the model and the sensitivity of its equilibrium structure to turbulence closure assumptions. We extended this analysis by testing a range of turbulence parameterizations (Table 1) and performed a full linear stability analysis to investigate the details of the equilibrium solutions.
The MSHF framework for predicting turbulent collapse, as proposed by Van de Wiel et al. (2007), provides a physical explanation of the transition from the WSBL to the VSBL. Our analysis demonstrated that this framework (Van de Wiel et al. 2012b, 2007; Van Hooijdonk et al. 2015) is robust to substantial changes in the representation of turbulence within the model. However, both the MSHF and the shape of the equilibrium curve varied among the different turbulent flux parameterizations. In particular, for some schemes the local extrema of the equilibrium curves were not unique. With the exception of the Businger–Dyer-type formulations, both qualitative and quantitative features of the equilibrium curves varied with
Using a Businger–Dyer-type parameterization of turbulent fluxes (Businger et al. 1971; Dyer 1974), Van de Wiel et al. (2007, 2012b) derived an analytical expression for the maximum equilibrium sustainable heat flux and showed that this framework predicted turbulent collapse for both Couette and pressure-driven flows. Equivalently, the MSHF framework can be expressed in terms of the speed at a given altitude. A specified surface heat flux determines a characteristic velocity scale
Linear stability analysis showed that, for all of the stability functions considered here, the stability properties depend on
For each
While the Couette model provides a heuristic description of the transition from the WSBL to the VSBL, it cannot account for the reverse transition. A complete understanding of transitions from the VSBL to the WSBL requires a more comprehensive model. Recent work has focused on the role of the pressure gradient force (Donda et al. 2015), but more work is needed to capture the full complexity of this dynamical system (Sun et al. 2002, 2004).
Acknowledgments
The authors acknowledge support by the Natural Sciences and Engineering Research Council of Canada (NSERC). This research was enabled in part by support provided by WestGrid (www.westgrid.ca) and Compute Canada/Calcul Canada (www.computecanada.ca). We also acknowledge helpful comments from Carsten Abraham, Bas van de Wiel, and two anonymous referees.
APPENDIX
Equilibrium Analysis and Linearized Dynamics



















The equilibrium profile given by Eq. (10) depends on only the momentum similarity function
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