The Coupling State of an Idealized Stable Boundary Layer [PDF]

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Boundary-Layer Meteorol (2012) 145:211–228 DOI 10.1007/s10546-011-9676-3 ARTICLE

The Coupling State of an Idealized Stable Boundary Layer Otávio C. Acevedo · Felipe D. Costa · Gervásio A. Degrazia

Received: 5 May 2011 / Accepted: 14 November 2011 / Published online: 7 January 2012 © Springer Science+Business Media B.V. 2012

Abstract The coupling state between the surface and the top of the stable boundary layer (SBL) is investigated using four different schemes to represent the turbulent exchange. An idealized SBL is assumed, with fixed wind speed and temperature at its top. At the surface, two cases are considered, first a constant temperature, 20 K lower than the SBL top, and later a constant 2 K h−1 cooling rate is assumed for 10 h after a neutral initial condition. The idealized conditions have been chosen to isolate the influence of the turbulence formulations on the coupling state, and the intense stratification has the purpose of enhancing such a response. The formulations compared are those that solve a prognostic equation for turbulent kinetic energy (TKE) and those that directly prescribe turbulence intensity as a function of atmospheric stability. Two TKE formulations are considered, with and without a dependence of the exchange coefficients on stability, while short and long tail stability functions (SFs) are also compared. In each case, the dependence on the wind speed at the SBL top is considered and it is shown that, for all formulations, the SBL experiences a transition from a decoupled state to a coupled state at an intermediate value of mechanical forcing. The vertical profiles of potential temperature, wind speed and turbulence intensity are shown as a function of the wind speed at the SBL top, both for the decoupled and coupled states. The formulation influence on the coupling state is analyzed and it is concluded that, in general, the simple TKE formulation has a better response, although it also tends to overestimate turbulent mixing. The consequences are discussed. Keywords

Coupling · Stable boundary layer · Turbulence formulations

1 Introduction It is common in recent years to classify the stable boundary layer (SBL) as weakly or very stable (Holtslag and Nieuwstadt 1986; Mahrt et al. 1998; Mahrt 1999; Mahrt and Vickers

O. C. Acevedo (B) · F. D. Costa · G. A. Degrazia Departamento de Física, Universidade Federal de Santa Maria, Santa Maria, RS 97105-900, Brazil e-mail: [email protected]

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2006). Typically, the weakly SBL shows continuous turbulence, which follows similarity relationships, while in the very stable regime turbulent mixing is drastically reduced and may become intermittent. A more objective classification criterion uses the sensible heat flux dependency on stability, and is such that in weakly stable conditions the heat flux increases in magnitude with stability, while under very stable conditions the thermal stratification is strong enough to dampen the fluxes, which decrease with increasing stability. In a recent study, Sun et al. (2011) provided further observational evidence that supports such a classification, and showed that the weak and strong turbulence states are separated by an abrupt transition, characterized by a vertically dependent mean wind threshold. An alternative, but not different, classification of the SBL regards the coupling state between the air next to the surface and the upper levels. Under turbulent conditions, different locations within a given region remain connected to each other and to the upper SBL, so that there is little spatial variability of properties such as temperature or scalar concentration. In this situation, surface variables such as temperature, humidity and scalar concentrations are usually controlled by processes that operate on larger scales than the SBL thickness. In this case, which usually coincides with the weakly stable regime, the surface may be regarded as coupled to the upper SBL. On the other hand, if there is insufficient mechanical turbulence production by the larger scale flow, the surface tends to decouple from upper levels. Local processes control the evolution of scalar variables near the surface and large spatial variability arises, something that has been known to occur for temperature for a long time (Geiger 1965). The decoupled state may, therefore, be associated with the very stable boundary-layer regime. Acevedo and Fitzjarrald (2003) presented observational evidence of decoupling using a regional network of stations. Their results showed that, within a region, some stations, usually at higher locations, tend to remain turbulent through the entire night, while others experience rapid early evening cooling and a wind speed decrease on colder nights. Coupling may occur intermittently or in an organized form at the different locations and, when it occurs, the surface air temperature may increase by as much as 10 K, exactly matching the surface air temperature registered at higher locations that remained turbulent for the entire period. Such sharp transitions between the two states suggest a bi-stable condition, which has been reproduced by simple numerical representations of the SBL. McNider et al. (1995) showed that a simple turbulence formulation based on a critical Richardson number, which is a threshold above which the turbulent mixing is totally suppressed, not only solves the coupling/decoupling state of the boundary layer but also presents a very sharp transition between the two states, highly dependent on the initial conditions. Derbyshire (1999) presented a detailed analysis of the decoupling dependence on the turbulence representation used, analytically showing that decoupling is a natural response of the SBL system, as long as the turbulence intensity is somehow sensible to the atmospheric stability expressed by a parameter such as the Richardson number. Besides, his analysis showed that decoupling does not strictly depend on assuming a critical stability threshold beyond which no turbulence remains. Given Derbyshire’s results, that decoupling arises in a variety of turbulence representations, the purpose of the present study is to investigate in further detail how typically used turbulence formulations resolve the state of coupling between the surface and the upper levels. The analysis will focus on aspects such as when (at which value of mechanical forcing) the transition between the two states occurs, how sharp it is, and what are the equilibrium vertical profiles of variables such as temperature, wind speed and turbulence intensity in each boundary-layer state for each of the formulations considered. To proceed, an idealized SBL is imposed, with fixed prescribed upper and lower boundary conditions. Such an idealization is chosen with the purpose of comparing solely the SBL response to the schemes used to

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represent it. Therefore, this is, in some cases, accomplished at the expense of representation realism. Among the turbulence schemes compared are those that solve a prognostic equation for turbulent kinetic energy (TKE) and those that merely prescribe turbulence as a function of stability and in each case the simplest possible formulations were chosen. A specific purpose of the study is to identify what aspects of the solutions are forced by the choice of SF. To do that, besides comparing the outcomes of the different formulations, a special case where the only dependence of the turbulence intensity on stability is that directly imposed by the TKE budget, is also considered. Although quantitatively unrealistic, this particular exercise provides important insights into the nature of the SBL coupling state and how it qualitatively affects various relevant quantities, such as turbulence intensity and fluxes.

2 Models 2.1 Model Equations The idealized SBL considered in the present study has a constant thickness h, and prescribed conditions both at h and at the ground. The SBL is divided in n equally spaced levels (Fig. 1, solid lines). Potential temperature (θ ) and wind speed (u) are determined at the model levels from the turbulence flux divergence: ∂θ ∂w  θ  =− , ∂t ∂z ∂u ∂  w u  =P− . ∂t ∂z

(1) (2)

In Eq. 2, P represents a forcing term, such as a pressure gradient or an advective acceleration, and is prescribed. The choice of an imposed accelerating term in such a form is necessary because the horizontal wind is considered to be one-dimensional, so that a geostrophic balance and its associated flow acceleration cannot be assumed. A similar approach was taken by van de Wiel et al. (2002). The turbulent fluxes are determined at intermediate levels (Fig. 1, dashed lines), located at the middle distance between adjacent levels. They are parametrized by a simple first-order formulation as w  θ  = −K H ∂θ/∂z and w  u  = −K M ∂u/∂z, where K H and K M are respectively the heat and momentum turbulent exchange coefficients. Their values depend on the turbulence formulation used, as described next. 2.1.1 TKE Model TKE models generally prescribe the momentum exchange coefficient as a function of TKE and momentum mixing length: √ (3) K M = c elM where c is a proportionality constant, e is the TKE and lM is the momentum mixing length. Such representation for the exchange coefficients has been adopted by Bélair et al. (1999), Cuxart et al. (2000), Xue et al. (2000), Lenderink and Holtslag (2004), Costa et al. (2011), among others. In the TKE model, u ∗ is directly related to the TKE (e) by u 2∗ = e/5.5. This approach was taken by Duynkerke (1988), who remarked that the constant 5.5 is the average from a number of estimates given by Panofsky et al. (1984) for the relationship between u ∗

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Fig. 1 Level distribution for the schemes with one main level (left) and n main levels (right). Intermediate levels are shown as dashed lines. Variables determined at each level are indicated

and e. If the Prandtl number is considered unity, both coefficients for momentum and heat are equal, and Eq. 3 can be written as a direct function of u ∗ (Costa et al. 2011). Following McNider et al. (1995), the mixing length is taken as l = κz, where κ is the von Karman constant, and Eq. 3 becomes: K H = K M = κu ∗ z (4) √ where the constant c is assumed to be 1/ 5.5 for simplicity, a choice within the range of values typically used (Cuxart et al. 2006). TKE is determined prognostically from its budget equation:    w ∂e ∂u p ∂ g w e + = −w  u  + w θ  − − ε, (5) ∂t ∂z  ∂z ρ0 where the right-hand-side terms are respectively the shear production, buoyant destruction, turbulent vertical transport and dissipation terms. The transport term is parametrized as (w  e + p  w  /ρ0 ) = −K ∂e/∂z, while the dissipation rate is represented, from dimensional analysis as ε = u 3∗ / (κz). 2.1.2 TKE/SF Model In the second turbulence formulation considered, similarly to the one previously described, a prognostic equation for TKE is considered. The difference lies in the fact that the exchange coefficients do not depend only on the response to the TKE budget, but also on an imposed SF, which is usually obtained from similarity relationships. Therefore, such a formulation is named “TKE/SF”, as it shares the characteristics of both a TKE and a SF model. A large variety of stability dependences has been suggested for the exchange coefficients (Louis 1979; Beljaars and Holtslag 1991; McNider et al. 1995; Delage 1997). Here, it is considered that K H = K M = κu ∗ z/ [φ (Ri)]2 , where φ(Ri) is the SF, responsible for dampening the exchange coefficients as the Richardson number increases. The SF considered here

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is that proposed by Louis (1979), based on observations that even for stabilities beyond the critical Richardson number the surface cooling rate is not really as fast, since it results from models that impose a critical Richardson number. In models that assume a critical Richardson number, often unrealistic low nighttime ground temperatures are produced when the critical value is exceeded, due to a rapid surface cooling that arises when the ground becomes energetically disconnected from the atmosphere (Louis 1979). Therefore, to avoid this kind of behaviour, for stable conditions Louis (1979) suggested a SF for which turbulence never totally disappears: φ (Ri) = 1 + 4.7Ri.

(6)

2.1.3 Short-Tail Stability Function Model In the SF formulations, the turbulence intensity is not determined by a prognostic equation but, rather, by an arbitrarily imposed dependence on the atmospheric stability. A large variety of such formulations has been suggested, and they can be generally classified as short-tail and long-tail, depending on their high-stability limit. The third formulation considered in the present study is a short-tail SF, used by McNider et al. (1995): u ∗ = κz

du  f (Ri) dz

(7)

where the stability dependence is given through the function f(Ri) as:  f (Ri) ≡

(1 − Ri/Ri cr )2 for Ri < Ri cr 0 for Ri ≥ Ri cr .

(8)

Short-tail SFs follow classical turbulence theory in considering that turbulence is totally suppressed at a given stability threshold. For Richardson-number based SFs, such a threshold is usually referred to as the critical Richardson number (Ri cr ).

2.1.4 Long-Tail Stability Function Model In long-tail functions, as discussed in Sect. 2.1.2, there is no upper stability limit for the existence of turbulence. Although this idea may contradict the local TKE budget in very stable conditions, surface scalar forecasts using long-tail SFs tend to be better on the average than those obtained using short-tail functions (Louis 1979; Delage 1997). A reason for this has been suggested by Mahrt (1987) and by Delage (1997), and relies on the fact that within a given region there may be localized patches of turbulence even when the mean regional stability is beyond critical for the existence of turbulence. The fourth formulation considered is a long-tail SF, similar to that suggested by Louis (1979): u∗ =

κz du φ 2 (Ri) dz

(9)

where the function φ (Ri) is given by Eq. 6.

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2.2 Model Discretization The implementation of the turbulence formulations described above on a discrete vertical structure such as those shown in Fig. 1 leads to the following prognostic equations for variables θ and u at level i:  κ  ∂θi = θi−1 u ∗i−1/2 z i−1/2 +θi+1 u ∗i+1/2 z i+1/2 −θi u ∗i−1/2 z i−1/2 −θi u ∗i+1/2 z i+1/2 , ∂t ( z)2 (10) κ  ∂u i = P+ u i−1 u ∗i−1/2 z i−1/2 + u i+1 u ∗i+1/2 z i+1/2 − u i u ∗i−1/2 z i−1/2 ∂t ( z)2  −u i u ∗i+1/2 z i+1/2 . (11) In the TKE and TKE/SF models, prognostic equations are solved for TKE at the intermediate heights between each of the main levels: ∂ei−1/2 κz i−1/2 u ∗i−1/2 (u i − u i−1 ) gz i−1/2 κu ∗i−1/2 (θi − θi−1 )     + = ∂t  z φ 2 Ri i−1/2 φ 2 Ri i−1/2 ( z)2 + where

u3 Ti − Ti−1 − ∗, z κz

(12)

   κz i u ∗i+1/2 + u ∗i−1/2 ei+1/2 − ei−1/2      Ti = z φ 2 Ri i+1/2 + φ 2 Ri i−1/2

and

Ri i−1/2 =

g z 



θi − θi−1 (u i − u i−1 )2

.

(13)

(14)

Using the relationship u 2∗ = e/5.5 (Duynkerke 1988) and considering n vertical levels, Eqs.10–12 comprise a system of (3n + 1) equations for (3n + 1) variables (n levels of θ, n levels of u and (n+1) levels of u ∗ ). For the TKE model, the φ functions equal unity, while for the TKE/SF model they are given by (6). In the SF models, only Eqs. 10 and 11 are solved, while u ∗ is given by Eqs. 7 or 9.   At the first intermediate level, u ∗1/2 = κ f Ri 1/2 u 1 / ln (z/z 0 ) for the short-tail model and    u ∗1/2 = κu 1 / φ 2 Ri 1/2 ln (z/z 0 ) for the long-tail case. At the remaining levels, the fric   tion velocity is u ∗i−1/2 = κ f Ri i−1/2 (u i − u i−1 ) / ln (z i /z i−1 ) for the short-tail scheme,    and u ∗i−1/2 = κ (u i − u i−1 ) / φ 2 Ri i−1/2 ln (z i /z i−1 ) for the long-tail one. 2.3 Initial and Boundary Conditions In the simulations analyzed here, the SBL thickness (h) is always assumed to be 50 m. At this height, it is assumed that the potential temperature is constant and that the friction velocity vanishes (McNider et al. 1995; van de Wiel et al. 2002; Costa et al. 2011). The use of a constant-height SBL is necessary for model simplicity, using a limited number of vertical levels. It affects the model results by restraining all flux convergences to a shallower layer than would occur if the SBL thickness evolved along the night. As a consequence, the simulated turbulence and flux magnitudes under the connected state lose realism, especially in

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cases with intense winds at the SBL top. On the other hand, the disconnected state and the transition between states tend to be unaffected by such a choice. Temperature and wind speed are prescribed at the top and bottom levels, so that θn+1 = 300 K, u n+1 = u h , θ0 = 300−(∂θ/∂t) t where (∂θ/∂t) is a prescribed cooling rate and u 0 = 0. The wind speed at the SBL top is the main external forcing of the system and is constant for any given run. The acceleration term P in Eq. 10 is P = 10−4 u h , so that it has a similar magnitude to mid-latitude geostrophic forcing. At the first intermediate level, Eq. 12 requires evaluation of the surface turbulent transfer term T0 , which is assumed to be zero. Linear initial vertical profiles are assumed for both θ and u at the n levels. The initial ground temperature is either 280 K for simulations with no cooling rate or 300 K when a cooling rate is assumed. Initial TKE is (0.003/z) m2 s−2 and this is also the minimum value allowed for this variable at each level in both the TKE and TKE/SF models. The use of a residual turbulence intensity is necessary in the schemes that solve a TKE prognostic equation to avoid them having an equilibrium state at e¯ = 0. Although this is an artificial imposition, it is supported by observations that a non-zero level of turbulent mixing is always observed, even in the most stable cases (Mahrt and Vickers 2006). The value chosen for the residual TKE was taken from observations made in very stable conditions at a site where such a situation is typical at night (Sakai et al. 2004). Sensitivity tests indicate that using smaller values for the residual TKE would not affect the results significantly. 2.4 Model Integration The model is integrated for 10 h, with a timestep of 0.1 s. For the different simulations, the wind speed at the domain top (u h ) varies from 1 to 20 m s−1 and the number of vertical levels varies from 1 to 9. The remaining constants considered are κ = 0.4, g = 9.8 m s−2 ,  = 300 K and, when necessary, we assume z 0 = 0.1 m. Sensitivity tests indicate that different roughness lengths may change the model equilibrium temperatures, but without affecting the coupling state of the simulated SBL.

3 Results 3.1 Constant Lower Boundary Condition We first analyze the results from an SBL whose upper and lower boundaries are fixed. Precisely, a 300 K potential temperature is assumed at h, while the surface temperature is kept at 280 K. The contrast between the coupled and uncoupled states for each scheme considered is exemplified by the difference in the solutions with 2 and 12 m s−1 winds at the SBL top (Fig. 2), for a 1-level scheme. In all cases, 12 m s−1 winds produce coupling, characterized by the fact that the single level considered reaches a warm equilibrium at a temperature close to that existing at the SBL top. As such winds are sufficient to maintain the SBL turbulent in all cases, the solutions do not depend on the turbulence scheme chosen. On the other hand, for the 2 m s−1 case, some remarkable differences can be perceived. Although in most cases the single level remains cold, disconnected from the SBL, in both the TKE/SF and long-tail schemes the equilibrium is achieved after a long evolution, as a consequence of the small turbulent intensities at the intermediate level between the surface and the main level considered. In the short-tail scheme, no changes of the main level temperature occur for more than 2 h. This is the time necessary for the local wind to increase sufficiently to reduce the

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Fig. 2 Solutions of the different 1-layer models with u h = 2 m s−1 (black lines) and u h = 12 m s−1 (blue lines). In each case, solid lines shows temperature (scale at the left), while dotted lines are the friction velocity (scales at the right)

stability, forcing the Richardson number at the first intermediate level to fall below its critical value. Once this happens, the air tends to the equilibrium at the cold surface temperature. The most different solution is that of the TKE model, for which the mechanical forcing is never enough to cause the friction velocity to vary from its residual value, very close to zero. The consequence is that the potential temperature at the main level finds an equilibrium at a value close to its initial value of 290 K, and near the midpoint between the upper and lower boundary values. Figure 2 shows precisely what is meant by coupled and uncoupled in the context of the present study. From now on, the term decoupled refers to the state with very weak turbulence and consequent low temperatures near the surface, while the coupled state has stronger turbulence intensities and higher temperatures. As stated in Sect. 1, this definition is broadly consistent with the more commonly used SBL classification between weakly (coupled) or very stable (decoupled) boundary layers, and also with the weak and strong turbulence regimes found by Sun et al. (2011). The distinction between states is not gradual, but sharp, as seen in Fig. 3, which shows the equilibrium temperature dependence on u h for the 1- and 9-level models. Consistent with the conclusion of Derbyshire (1999), coupling is a feature of all formulations considered, as cold equilibrium is always found at low u h , while a warm solution occurs for high u h . The main differences are related to the mechanical forcing threshold for coupling occurrence, the sharpness of the transition between the two states and the solution behaviour in the decoupled state. In general, the transition occurs over a similar range of u h , from 5 to 7 m s−1 , for all schemes. Both the TKE/SF and long-tail models show a smooth transition between the two

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Fig. 3 Equilibrium temperature at the lowest vertical level after 10 h of simulation as a function of the wind speed at the SBL top for each of the formulations considered, according to legend. Upper panel is the result for a 1-level model, middle panel is for nine levels at the lowest level, and lower panel is for nine levels at the fifth vertical level

states, with intermediate equilibrium temperatures occurring for a wide range of mechanical forcing, while the TKE and short-tail schemes show a more sudden transition from the cold to the warm equilibrium. It is interesting that the long-tail scheme, generally used to represent the average state of an entire area, rather than a local condition, shows the most gradual transition of all schemes, because even if the real-world transition is very abrupt locally, it is likely to be much more gradual over a larger area. For a 1-level grid, the no-turbulence state of the TKE model for weak external forcing shown in Fig. 2 for the 2 m s−1 solution persists for u h as large as 2.5 m s−1 , when cooling slowly starts to occur at the main level. The long-tail scheme also shows a condition of no cooling for very weak mechanical forcing, but limited at u h ≈ 1.5 m s−1 . The consequence is that in both the TKE and long-tail models, the lowest temperatures do not occur at the lowest winds at the SBL top but, rather, at a slightly larger value. Although this may seem a limitation of such schemes, observations show that weak enough winds produce cooling, by restricting the mixing activity to a shallow layer that includes the cold surface, but not the warm SBL top (Acevedo and Fitzjarrald 2003), similarly to what occurs at the low mechanical forcing limit for those two schemes. When nine levels are used (Fig. 3, middle and lower panels), the general patterns follow those for the 1-level case. Near the surface, the lower equilibrium temperatures with nine

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levels are an obvious consequence of the level under consideration being much closer to the surface. In the TKE model, the condition of no turbulence is limited for a narrower range of mechanical forcing (u h < 1.5 m s−1 ). The transition between states is smoother for all schemes when more levels are considered, and at the lowest level there is a larger variability of the higher equilibrium temperature at the connected state among the different formulations (Fig. 3, middle panel). Such variability is reduced in the middle of the SBL (Fig. 3, lower panel). The short-tail SF leads to the highest (291.2 K at 5 m) and the TKE/SF model to the lowest (286.8 K at 5 m) equilibrium temperatures at the high limit of mechanical forcing. Such a 4.4 K discrepancy among the schemes shows that merely resolving the coupling state is not enough to properly forecast the nocturnal near-surface air temperature. The higher equilibrium temperature in the coupled state of the short-tail formulation is a consequence of this formulation having a slower turbulence decay with stability than the others. On the other hand, in the disconnected state the short-tail solution is less turbulent and generally colder than the long-tail case. The model dependence on the number of levels has been analyzed, and shows that with nine levels all variables converge to a profile independent of the number of levels, for all turbulence formulations considered (figure not shown). For this reason, for the remaining of the study, only the 9-level grid will be considered. Probably, the most striking differences among the formulations considered regards the equilibrium vertical potential temperature profiles, especially in the decoupled state (Fig. 4). For very weak mechanical forcing at the SBL top (u h = 2 m s−1 ), only the TKE formulation

Fig. 4 Equilibrium vertical potential temperature profiles for the 9-level simulation as a function of u h , as given by legend. The schemes compared are identified at the top of each panel

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results in a simple profile, with an almost linear increase with height. All other formulations, which according to Fig. 3, show a cold equilibrium at this situation, present an unusual profile with two nearly adiabatic layers, one next to the surface and the other next to the top, with a sharp transition near the middle of the SBL. This is a consequence of the fact that, in such formulations, the small levels of turbulence that result under these conditions drive the temperature at each level to the closest boundary value. The strong stability that results at the middle of the domain destroys turbulence locally, causing a local friction velocity minimum (Fig. 5) and further decoupling the two sublayers from each other. As u h increases in the TKE/SF, short- and long-tail schemes, a deeper layer next to the surface becomes turbulent, and this entire layer cools to a value close to the lower boundary temperature, but still leaving a sharp thermal gradient above it. In the TKE scheme, on the other hand, when the near-surface temperature decreases, a smoother thermal gradient is maintained aloft. Once connection occurs (u h ≥ 8–10 m s−1 , depending on the scheme) all profiles tend to a similar shape. However, important differences exist that may, as discussed above, lead to as much as a 4.4 K discrepancy in the near-surface equilibrium temperature. The friction velocity profiles vary substantially among the different schemes (Fig. 5). In particular, the TKE formulation leads to very large turbulence intensities in the coupled state. This is avoided in the other schemes, as a consequence of the reduction imposed by the SFs used. However, in these cases, the quantitatively more realistic friction velocities occur along with qualitatively unrealistic profiles, in the sense that, in all cases except the TKE formulation, there are occurrences of a u ∗ increase with height, which would correspond to momentum flux convergence and consequent acceleration of the local flow due to friction.

Fig. 5 Same as in Fig. 4, but for friction velocity

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Fig. 6 Same as in Fig. 4, but for wind speed

The schemes with the warmer equilibrium are the TKE and short-tail, which show a more curved, nearly logarithmic, potential temperature profile in the connected state. In fact, these two schemes, and only these two, also lead to a nearly logarithmic wind profile at the connected state (Fig. 6, notice the logarithmic vertical axis). These two formulations that lead to higher temperatures in the coupled state have in common the lack (for the TKE model) or a diminished (for the short-tail SF) imposed turbulence reduction due to stability, so that the wind speed is allowed to naturally converge to its logarithmic vertical profile, with the temperature profile following accordingly, ultimately driving the resulting temperature difference near the surface. The vertical friction-velocity profiles also differ appreciably among the formulations (Fig. 5). Besides the difference already discussed that occurs in the decoupled state, important variations occur at the connected state, specially concerning the u ∗ magnitude. The TKE and TKE/SF models show larger values of the friction-velocity, but in the case of the TKE/SF model the effective exchange coefficients are not proportionally as large, because of the imposed stability-based reduction. For this reason, the TKE model has the largest effective exchange, followed by the short-tail SF formulation and, not surprisingly, in these two cases a nearly logarithmic wind profile results in the coupled state. 3.2 Constantly Cooling Surface To address how the schemes resolve the coupling state of the SBL in a more realistic, temporally evolving situation, a case that starts with an adiabatic 300-K potential temperature profile, for which the surface cools at an intense rate of 2 K h−1 is considered.

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Fig. 7 Same as in Fig. 3, but for the simulation with a prescribed 2 K h−1 cooling rate at the surface, and a 9-level grid

The simulations were integrated for 10 h, so that in the end the upper and lower boundary temperatures are the same as in the case discussed in the previous sub-section. Decoupling still happens for all schemes, and the final near-surface air temperature dependence on the mechanical forcing at the SBL top resembles that shown in Fig. 3 for the constant lower boundary condition (Fig. 7). The most important difference is in the decoupled state, the fact that for all formulations the lowest temperatures occur at u h ≈ 3–4 m s−1 , implying that for low mechanical forcing the surface cools as u h increases. With a constant surface temperature, such a result only occurs for the TKE model. As before, the TKE/SF and long-tail schemes have a smoother transition between states, and the final temperature variability in the connected state is the same as that with a constant temperature at the surface. With very weak mechanical forcing at the SBL top (u h = 1 m s−1 , Fig. 8), the TKE model once more produces a discrepant solution with respect to the others. In the TKE case, the surface cooling is more strongly felt at lower levels, but the entire SBL is affected, despite the very weak winds aloft. For the other three schemes, on the other hand, the surface cooling is only felt in a shallow 15-m layer, which cools drastically while the temperature remains constant at the initial value above it. When u h = 5 m s−1 (Fig. 9), the turbulent mixing is sufficient so that the low surface temperature affects a deeper layer in the TKE/SF, short- and long-tail formulations. An entire layer develops, almost as deep as the SBL, with temperatures close to those at the surface, and a sharp thermal gradient is created just below the SBL top, similarly to that shown in Fig. 6 for the decoupled state. Such mixing near the ground is responsible for the cooling with respect to the u h = 1 m s−1 case. In the TKE model, on the other hand, similar lower temperatures occur near the surface with u h = 5 m s−1 than with u h = 1 m s−1 , but a smoother temperature gradient aloft accompanies it. In the connected state, all schemes lead to a curved potential temperature vertical profile along the entire simulation and the final profiles after 10 h are very similar to those obtained and discussed in the previous sub-section, with a constant lower boundary condition (figures not shown).

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Fig. 8 Vertical potential temperature profile evolution for simulations with u h = 1 m s−1 and a 2 K h−1 surface cooling rate. Each panel corresponds to a different turbulence scheme, identified at panel top. The lines are the profiles for different times, according to legend

4 Discussion The results indicate that the TKE model has a distinctive behaviour in the four formulations considered. Moreover, it can be argued that this scheme resolves some aspects of the coupling phenomenon in the SBL better than the others, namely: – In the TKE model, the lowest temperatures always occur at an intermediate value of mechanical forcing, in agreement with observations that show that winds below a given threshold tend to cool the surface (Acevedo and Fitzjarrald 2003; Acevedo et al. 2006), so that the lowest temperature do not occur with the weakest winds. – It leads to more realistic thermal profiles in the decoupled state, without sharp temperature gradients in the middle or at the top of the SBL. – In the connected state, the wind profile converges to a logarithmic form at lower values of mechanical forcing than in the other formulations (except the short-tail SF). – It is the only formulation for which the friction velocity never increases with height. On the other hand, the TKE model produces larger turbulence intensities than the other schemes, indicated by the friction velocity profiles in Figs. 4 and 7, and this is precisely why it has the advantages described above. This, in turn, leads to large sensible heat fluxes that, in the present analysis, with prescribed ground temperature, have little effect on the

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Fig. 9 Same as in Fig. 8, but for u h = 5 m s−1

overall performance of the scalar profiles. This, however, may change if the surface responds dynamically to the energy budget. The mere idea of using a TKE model with no imposed stability dependence on the exchange coefficients may seem inappropriate. However, this was considered here under the expectation that using a TKE prognostic equation could naturally lead to realistic turbulence decay with stability. Indeed, a turbulence reduction with stability is, on average, achieved in the simulations with a cooling surface (Fig. 10), for all simulations. Figure 10 also indicates that for all simulations, the average response of turbulence intensity to stability is affected by the coupling state of the SBL. All formulations show a slope transition around the equilibrium solution for u h = 5 m s−1 , which is also when the transition to coupling generally happens.

5 Conclusion Cuxart et al. (2006) and Svensson et al. (2011) have provided detailed comparisons of a large number of turbulent formulations actually implemented in numerical weather forecast schemes used around the world. Their comparisons also aimed to reproduce observed cases, thus having a realistic character. In general, they found that schemes that solve prognostic equations for TKE tend to perform better under nocturnal conditions.

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Fig. 10 Average friction velocity dependence on the Richardson numbers for simulations with u h from 1 to 15 m s−1 and a 2 K h−1 surface cooling rate for the different turbulent schemes, according to legend. The numbers 1, 5 and 12 identified the final values after 10 h of integration for cases with u h = 1, 5 and 12 m s−1 , respectively. The number colours follow those shown in legend

The present study, on the other hand, has an idealized character and aimed at characterizing the response of the turbulent schemes to a specific, although rather critical, feature of the SBL, namely the strength of the vertical coupling. The advantage of such an approach is that the specific response of the phenomenon analyzed to the formulations under comparison can be isolated. The general character of the analysis also dictated the choice of schemes compared. Each belong to a main general class, and all of them consist of the simplest formulation that characterizes that given type of description. The analysis showed that a simple TKE scheme with no imposed turbulence dependence on the stability has qualitative advantages on both decoupled and coupled states of the SBL, which were identified in Sect. 4. On the other hand, the scheme leads to larger turbulence intensities than typically observed, a deficiency that becomes critical when non-idealized situations are simulated. Therefore, the main conclusion of the present study is that, ideally, one must seek a formulation that retains the features of the TKE model described here, where the only dependence of turbulence intensity on stability is that naturally imposed by the buoyancy term of the TKE equation, while overall reducing the turbulence intensity. That can be achieved in a simple manner by merely reducing the constant c in Eq. 3 or, more elaborately, by introducing a prognostic equation for the dissipation rate (Wyngaard 1975; Duynkerke 1988). The study also showed that there is a general direct relationship between model temperature and its turbulent intensity. This explains why short-tail schemes tend to be colder than long-tail ones in the decoupled state, and warmer when the state is coupled. However, such a direct relationship is broken when the transition between states is considered. The shorttail scheme, for which turbulence is almost totally suppressed in the decoupled state, has a sharper transition than the long-tail ones, but interestingly the TKE scheme has the sharpest transition of all formulations. Some studies suggest that the transition may be very sharp in the real world (Acevedo et al., 2003), but a detailed analysis of observations is necessary to characterize the sharpness of the transition between states in a more definitive way.

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Costa et al. (2011) showed that the use of a TKE formulation favours the representation of intermittent solutions in a simplified SBL scheme. In the present study, the idealized characteristics of the SBL prevented any such type of complexity to arise. However, it is unlikely that such could be achieved with a SF formulation in which the imposed connection between atmospheric stability and turbulence intensity arbitrarily reduces the number of degrees of freedom of the system, even in non-idealized conditions. The identification of the minimal necessary changes for the idealized scheme presented here to allow intermittent solutions to arise, in the manner shown by Costa et al. (2011), is an important task for future work. Acknowledgment

This work was partially supported by CNPq, Brazilian Research Agency.

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