# Global Intermittency and Collapsing Turbulence in the Stratified Planetary Boundary Layer

## Abstract

Direct numerical simulation of the turbulent Ekman layer over a smooth wall is used to investigate bulk properties of a planetary boundary layer under stable stratification. Our simplified configuration depends on two non-dimensional parameters: a Richardson number characterizing the stratification and a Reynolds number characterizing the turbulence scale separation. This simplified configuration is sufficient to reproduce global intermittency, a turbulence collapse, and the decoupling of the surface from the outer region of the boundary layer. Global intermittency appears even in the absence of local perturbations at the surface; the only requirement is that large-scale structures several times wider than the boundary-layer height have enough space to develop. Analysis of the mean velocity, turbulence kinetic energy, and external intermittency is used to investigate the large-scale structures and corresponding differences between stably stratified Ekman flow and channel flow. Both configurations show a similar transition to the turbulence collapse, overshoot of turbulence kinetic energy, and spectral properties. Differences in the outer region resulting from the rotation of the system lead, however, to the generation of enstrophy in the non-turbulent patches of the Ekman flow. The coefficient of the stability correction function from Monin–Obukhov similarity theory is estimated as \(\beta \approx 5.7\) in agreement with atmospheric observations, theoretical considerations, and results from stably stratified channel flows. Our results demonstrate the applicability of this set-up to atmospheric problems despite the intermediate Reynolds number achieved in our simulations.

## Keywords

Direct numerical simulation External intermittency Monin–Obukhov similarity theory Stable boundary layer Stratified shear turbulence## 1 Introduction

The characteristics of a planetary boundary layer (PBL) crucially depend on its density stratification. In the absence of humidity and advection, the stratification is caused by heating or cooling at the surface—mostly as a consequence of radiative processes. If the surface cools sufficiently, turbulence is often observed to cease partially or even entirely (Mahrt 1999; Wiel et al. 2012). This cessation of turbulence and the associated decoupling of the PBL from the surface impose challenges for mixing formulations in numerical weather prediction (NWP) and climate modelling. Enhanced mixing formulations need often to be used in the mixing parametrizations for boundary layers of NWP and climate models to prevent a decoupling of the atmosphere from the surface (Wiel et al. 2012). Being heuristically formulated and tuned for the performance of NWP models, these enhanced mixing formulations lack a physical basis and cause warm biases at the surface under very cold conditions (Tjernstrom et al. 2005). A better understanding of the underlying dynamics and physical processes, especially in the very stable limit, could hence contribute to alleviate and ultimately overcome these problems of mixing formulations under stable stratification (Mahrt 1999). Conceptual studies of the very stable boundary layer (McNider et al. 1995; Derbyshire 1999; Wiel and Moene 2002) have led to qualitative models of turbulence collapse and global intermittency. In this work, we address fundamental aspects of wall-bounded stably-stratified turbulence and their implications for the stably stratified PBL (the stable boundary layer—SBL).

Often, SBLs are classified into three regimes (Mahrt et al. 1998; Garg et al. 2000; Sun et al. 2012). First, in the *weakly stable regime*, temperature behaves almost as a passive scalar, and the PBL structure is indistinguishable from the neutral reference: the weakness of the temperature gradient limits the turbulent exchange of heat. Consequently, if stratification is strengthened slightly, the turbulent heat flux increases monotonically as a function of the stratification. Second, in the *intermediately stable regime*, the turbulent heat flux stagnates if stratification is strengthened; an increased temperature gradient is compensated by a decrease of vertical velocity fluctuations. Third, in the *very stable regime*, stratification drastically alters the turbulence structure of the SBL to the degree that the weakness of turbulent motion limits the vertical heat exchange: The turbulent heat flux decreases with strengthening stratification. If strong enough, stratification can—locally or globally—lead to the absence of turbulence.

Although the very stable regime is commonly observed in the atmosphere (Mahrt et al. 1998; Ha et al. 2007), there is still a lack of a general framework for the SBL incorporating that very stable regime (Wiel and Moene 2012; Mahrt 2014). Monin–Obukhov similarity theory (MOST, Obukhov 1971) lacks the ability to properly reproduce turbulent fluxes under weak-wind conditions (Ha et al. 2007), i.e. in the very stable regime. From atmospheric observations, it is unclear if stratification can become strong enough to suppress turbulent mixing entirely (Mauritsen and Svensson 2007), and it proves problematic to locally classify a very stable PBL as turbulent or non-turbulent. If turbulence is treated as an on–off process, a *runaway cooling* at the surface is often seen in PBL models and large-eddy simulations (LES) applied under very stable conditions (Jiménez and Cuxart 2005; Wiel et al. 2012; Huang et al. 2013).

In fact, it is a well-accepted hypothesis that the cessation of turbulence is not an on–off process but rather a complex transition beginning with the local absence of turbulence in an otherwise turbulent boundary layer. Such a local break-down of turbulence is referred to as *global intermittency* (Mahrt 1999). It has been shown from observations that in a sufficiently stable environment, globally intermittent turbulence can be triggered by a variety of external disturbances including orographic obstacles (Acevedo and Fitzjarrald 2003), solitary and internal gravity waves (Sun et al. 2004) and non-turbulent wind oscillations, such as nocturnal low-level jets (Sun et al. 2012). Whether global intermittency can occur in a SBL without these external triggering mechanisms remains, however, unclear.

A quantitative understanding of strongly stable and globally intermittent turbulence from observations has proved difficult. In particular, accurate flux measurements are hard to obtain with standard methods, and the above mentioned processes often interact: under atmospheric conditions, the entire range of scales, orographic complexity, interaction with the surface, and radiative processes are always present. Thus, it is hard to isolate the signal of a single process as is sometimes necessary for a basic physical understanding. We choose here Ekman flow over a smooth wall—a much simplified configuration. This choice enables a systematic and quantitative study of the intermediately and very stable regimes of turbulence in a simplified and well-defined set-up.

While the study of the weakly stratified limit of simplified SBL configurations is well accomplished by LES (Beare et al. 2006; Huang and Bou-Zeid 2013), the study of very stable cases remains a challenge (Saiki et al. 2000; Jiménez and Cuxart 2005). In particular, the treatment of quasi-laminar patches under very stable conditions is problematic within the conceptual framework of LES. Hence, we consider direct numerical simulation (DNS). Following the pioneering work by Coleman et al. (1990), neutrally stratified Ekman flow has been subject to a number of studies (Coleman 1999; Shingai and Kawamura 2004; Miyashita et al. 2006; Spalart et al. 2008, 2009; Marlatt et al. 2010). Since Coleman et al. (1992) and Shingai and Kawamura (2002), who studied Ekman flow under weak to moderate stratification in rather small domains, however, no studies have come to the authors’ attention.

Instead, it is common practice to use stratified channel flow as a surrogate for the stratified Ekman boundary layer, which is possible due to the analogy between the surface layer of channel flow and that of Ekman flow. In channel flow, oscillations on a period that is large when compared with the large-eddy turnover time, were observed; but no intermittency was found at moderate Reynolds numbers (Nieuwstadt 2005). More recently, global intermittency is simulated in channel flow, and it was proven that too-small domains lead to the formation of artificial flow regimes manifest for instance in the low-frequency oscillation of global statistics (García-Villalba and del Álamo 2011; Flores and Riley 2011). Whereas García-Villalba and del Álamo 2011 use a fixed-temperature boundary condition, Flores and Riley (2011) impose a constant buoyancy flux at the surface. Flores and Riley (2011) observe large-scale intermittency linked to the collapse of turbulence. In contrast to channel flows, Ekman flow has no symmetry in the spanwise direction, which is known to cause large-scale structures in the neutrally stratified limit (Shingai and Kawamura 2004). Whether these structures affect the collapse of turbulence remains unclear. Moreover, Jiménez et al. (2009) found in a non-rotating configuration that *the outer flow of boundary layers and channel flows are intrinsically different*. Hence, we expect Ekman flow to differ from channel flow as well—we address here the question of how much.

- (1)
Does our simplified set-up reproduce all three regimes of turbulence in terms of a bulk Richardson number as the only control parameter? If so: which are the differentiating bulk properties among these regimes?

- (2)
What is the character of the very stable regime? Can we study the turbulence collapse and global intermittency with our simplified set-up – even if the forcing usually made responsible as a trigger for intermittent events is absent?

- (3)
Does Ekman flow differ from other configurations used to study the PBL such as channel-flow surrogates? How do large scales in the flow interact with stratification?

## 2 Formulation

### 2.1 Scaling of the Neutrally Stratified System

In contrast to channel flows, \(u_\star \) in the Ekman layer cannot be known a priori but only a posteriori, and \(u_\star \) depends weakly on \(Re\) (Spalart 1989). We follow common practice, and study the flow in terms of an inner and an outer layer. In the inner layer, i.e. the surface layer, we choose the wall unit \(\nu /u_\star \) and the friction velocity \(u_\star \) for normalization; normalized quantities are denoted by a superscript \(+\). In the *outer layer* quantities are normalized by \(u_\star \) and \(\delta \), and correspondingly normalized quantities are denoted by a superscript \(-\). In the outer layer we choose, because of its physical meaning, the inertial period \(2\pi f^{-1}\) as the reference time scale.

### 2.2 Imposing Stratification: Initial and Boundary Conditions

### 2.3 The Numerical Method

The governing equations are integrated using a high-order finite-difference algorithm on a structured, collocated grid. The time advancement is carried out with a low-storage fourth-order Runge–Kutta scheme (Williamson 1980). Derivatives are computed using a compact scheme described in Lele (1992) such that the accuracy in the interior of the domain is sixth order with overall fourth-order accuracy. We solve the pressure-Poisson equation using a Fourier decomposition along the horizontal directions, which results in a set of second-order differential equations along the vertical coordinate (Mellado and Ansorge 2012). Boundary conditions are no-slip at the wall and free-slip with a Rayleigh damping layer of 10 points and an e-folding time of one inertial period at the top. The grid is stretched moving upwards, and the top boundary is placed at \(z\approx 3\delta _{\mathrm{neutral}}\). Adequacy of both horizontal and vertical resolution has been assured for the neutrally stratified simulations using approximately twice the vertical resolution and twice the horizontal resolution for small test simulations at each \(Re\) (not discussed here).

### 2.4 Set-up of Numerical Simulations

A series of simulations is carried out as shown in Table 1 to identify the parameter range of \(Ri_B\) in which global intermittency occurs. From Coleman et al. (1992), a configuration of parameters within the weakly stable regime is known, and we use this as a starting point in the weakly stable regime for our case \(\mathtt W015S \). With the series of cases W031–S620 the stratification is increased until turbulence collapses.

Set-up of numerical simulations and their parameters

Case | | | \({ Ri}_B\) | \(L_{xy}/D\) | \(L_{xy}/\delta \) | \(L_{z}/\delta \) | \(\Delta x^{+}\) | \(\Delta y^{+}\) | \(\Delta z^{+}{\vert }_{z=0}\) | |
---|---|---|---|---|---|---|---|---|---|---|

N500S | A | 500 | \(\infty \) | 0 | 135 | 8.8 | 2.8 | 4.1 | 4.1 | 1.05 |

N500 | B | 500 | \(\infty \) | 0 | 270 | 17.5 | 2.8 | 4.1 | 4.1 | 1.05 |

N750 | C | 750 | \(\infty \) | 0 | 270 | 18.2 | 3.1 | 5.9 | 11.7 | 1.7 |

N1000 | E | 1,000 | \(\infty \) | 0 | 270 | 20.5 | 3.5 | 4.6 | 9.1 | 1.32 |

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## 3 The Neutrally Stratified Ekman Layer

This section introduces the neutrally stratified cases. These simulations are used as initial condition and reference for the stratified Ekman layer discussed in Sects. 4 and 5. We show that, despite the moderate Reynolds number \(Re\), we can distinguish between the inner and outer layer, the latter being characterized by external intermittency. In the overlap region where the velocity profile is approximately logarithmic, we find both a large-scale structure originating from the outer layer and small-scale hairpin vortices stemming from the buffer layer. We also ascertain the degree of \(Re\)-independence to better differentiate the effect of stratification by considering three Reynolds numbers \(Re=\{500,\,750,\,1000\}\).

### 3.1 Conventional Statistics

Global statistics as a function of the Reynolds numbers for the neutrally stratified configuration

Case | N500L | N750L | N1000L |
---|---|---|---|

\(u_\star /G\) | 0.0618 | 0.0561 | 0.0531 |

\(\alpha \) | 25.5 | 21.0 | 19.2 |

\(\delta _{95}/\delta \) | 0.668 | 0.650 | 0.631 |

\(Re_t\) | 203 | 407 | 655 |

\(Re_\tau =\delta ^+\) | 478 | 898 | 1399 |

\(fG^{-3}\int _{0}^{\infty } e\mathrm{d}z\) | \(2.18\times 10^{-4}\) | \(1.64\times 10^{-4}\) | \(1.44\times 10^{-4}\) |

\(fu_{\star }^{-3}\int _{0}^{\infty } e\mathrm{d}z\) | \(0.0570\) | \(0.0521\) | \(0.0511\) |

\(G^{-3}\int _{0}^{\infty } \varepsilon \mathrm{d}z\) | \(1.31\times 10^{-3}\) | \(1.34\times 10^{-3}\) | \(1.32\times 10^{-3}\) |

\(u_{\star }^{-3}\int _{0}^{\infty } \varepsilon \mathrm{d}z\) | \(5.53\) | \(7.58\) | \(8.85\) |

Velocity hodographs are shown in Fig. 2a. The dependency on \(Re\) is also relatively small, in particular the change from \(Re=750\) to \(Re=1{,}000\) is much smaller than the change from \(Re=500\) to \(Re=750\).

Attempts have been undertaken to obtain the logarithmic law for Ekman flow and associated constants describing the mean wind-speed profile within the surface layer at Reynolds numbers as high as \(Re=2{,}828\) (Spalart et al. 2008, 2009). We find an approximately constant slope of the velocity \(U^+\) around \(z^+=30\) for the three Reynolds numbers considered here (Fig. 2b); this agrees with Coleman (1999) and Miyashita et al. (2006). In accordance with Tennekes (1973), the height of departure from the logarithmic law for the three cases coincides with the height at which the velocity begins to turn significantly (see markers at \(z^-=0.12\) in Fig. 2b). This logarithmic variation supports the analogy with channel flows, which have been studied in great detail.

Contributions to the TKE budget in the outer layer, normalized such that at any level the sum of their squares equals one, are shown in Fig. 3b. The change from the production-dominated to the transport- dominated regime, both balanced by viscous dissipation, occurs at \(z^-\approx 0.5\). This is about 20 % lower than in a non-rotating boundary layer (Pope 2000, Fig. 7.34), and might hint at the fact that the outer scale \(\delta \) has to be adapted by an order-unity constant when quantitatively comparing Ekman flow to other non-rotating flows. Otherwise, the normalized profiles qualitatively agree with those in a non-rotating boundary layer.

### 3.2 External Intermittency and Conditional Statistics

For Ekman flow, it is readily seen from the enstrophy equation that vortex tilting of planetary vorticity \(f\hat{e}_z\) constitutes a source \(f \partial u/\partial z\) of streamwise vorticity \(\omega _x\) (and hence a source \(2f\omega _x \partial u/\partial z\) of streamwise enstrophy \(\omega _{x}^2\)). This source is absent in a non-rotating reference frame. Hence, small gradients in the mean velocity profile \(\partial \langle u \rangle / \partial z\) will, even in a purely non-turbulent flow, generate mean vorticity that grows until balanced by dissipation. In comparison to the non-rotating flows mentioned above, this enstrophy generation leads to a larger variation in \(\gamma (z)\) if the threshold is doubled and halved with respect to \(\omega _0\) (Fig. 4a). The \(Re\)-independency of \(\langle u(z)\rangle \) in the outer layer suggests that this mechanism is independent of the Reynolds number since the term \(\langle \omega _x\rangle f \partial \langle u\rangle /\partial z\) scales inviscidly. This \(Re\)-independency is corroborated by the small sensitivity of the profile to \(Re\) seen in Fig. 4. We conclude that this vortex tilting, irrespective of \(Re\), is a fundamental mechanism in Ekman flow that renders the outer layer different from non-rotating external flows.

### 3.3 Flow Visualizations

The change of organization in the turbulent flow when moving upwards is illustrated by horizontal cross-sections of wind magnitude (Fig. 5c–e): In the buffer layer (Fig. 5c), the flow is dominated by surface streaks aligned with the mean wind at that level, which is anti-parallel to the force exerted on the fluid by surface shear stress \(\tau _\mathrm{wall}\). In the fully turbulent part of the outer layer (Fig. 5d), the turbulence is modulated at large scale that is rotated by about \(20^\circ \)–\(30^\circ \) clockwise with respect to the geostrophic wind. At this height, the small-scale structure appears as noise. At higher levels (Fig. 5e), the boundary layer is externally intermittent, because turbulence at those levels is mainly provided by strong ejections from lower levels happening only sporadically. Such generated turbulent structures in the outer layer of the flow are long-lived because of their relatively large extent and the weakness of turbulent dissipation at these large scales.

Horizontal planes in the quasi-logarithmic layer of the flow (at \(z^+\approx 40\), Fig. 5f–g) show—consistent with the intermittency function \(\gamma (z^+=40)\approx 1\)—that the field is homogeneously turbulent, and that the dominating small-scale structures are hairpin vortices typical of the logarithmic layer (Adrian 2007). Their intensity is modulated at a large scale that is rotated about \(20^\circ \)–\(30^\circ \) clockwise with respect to the geostrophic wind vector, similarly to the large-scale organization observed before in Fig. 5d. This large-scale organization is typical of wall-bounded flows (Marusic et al. 2010; Adrian 2007), and there remains considerable controversy about the role of these large-scale structures in the inner layer (Jiménez 2012). In our case, they have a clear organization that can be attributed to some large-scale instability inherent to the flow (Barnard 2001). We consider the existence of such large-scale structures a fundamental property of turbulent Ekman flow and expect that they are crucial when the flow is exposed to stable stratification, as discussed in Sect. 5.

## 4 Turbulence Regimes and Stability

### 4.1 Classification

- (1)
Weakly stable: integrated TKE changes slightly (10–20 %, cases W015S, W031S) with respect to the neutral configuration

- (2)
Intermediately stable: integrated TKE significantly (50 %) decreases and subsequently recovers (case I150, S310S)

- (3)
Very stable: integrated TKE is diminished nearly entirely, and subsequently recovers (case S620).

After this initial decrease or even breakdown of turbulence, such as that simulated in case S620, the turbulence intensity recovers because the buoyancy difference across the boundary layer is fixed (Sect. 2.2). Hence, the stratification close to the surface decreases to compensate for a deepening of the stratified layer. Concomitantly, the fluctuation kinetic energy reaches and exceeds that of neutral stratification; such a recovery happens in all our cases (Fig. 6a). The magnitude of this recovery is even larger if the energy is normalized with the instantaneous friction velocity \(u_\star (t)\) as \(u_\star (t)\) decreases when the flow is exposed to stratification (Fig. 6a). The reasons for this recovery are complex, and for an explanation we refer the reader to Sect. 5. None of the cases reaches an equilibrium on time scales of one or two inertial periods. In the Appendix we study the associated time scales, and we conclude that, even under weak stratification, no equilibrium of the entire turbulent boundary layer is reached over the course of a night.

A possible origin of this increase in fluctuation kinetic energy with respect to the non-stratified configuration is non-vortical or, when compared to turbulent eddies, weakly vortical large-scale modes—possibly gravity waves—in the flow. This is confirmed by the fact that, unlike the fluctuation kinetic energy, the integrated streamwise vorticity r.m.s. remains below the neutral reference (Fig. 6b). We investigate this and the organization of the flow further in Sect. 5 by means of the intermittency function, conditional statistics, and spectra. The remainder of this section is devoted to a more detailed description of the turbulence regimes that we have identified.

### 4.2 The Weakly Stable Regime

### 4.3 The Intermediately Stable Regime

In the intermediately stable regime at \(Ri_B=0.15\), TKE is reduced by \(\approx 50~\%\) throughout the initial transient, and the integrated buoyancy flux at \(t^-=1\) is the maximum of all simulations carried out within this study (Fig. 6c). The increase in the integrated buoyancy flux is one order of magnitude smaller than the reduction in TKE (Fig. 6a) and shear production (Fig. 6c) with respect to the neutral reference. This illustrates that the main impact of buoyancy on the flow is not the direct destruction of TKE but a decrease in the shear-induced production, in particular of the stress \(\langle u'w'\rangle \) (Jacobitz et al. 1997, p. 243). Profiles of shear production (not shown) indeed confirm this explanation, in agreement with Jacobitz et al. ’s study of stably stratified shear flow.

In contrast to the simulations attributed to the weakly stable regime, the hodograph in the intermediately stable regime (\(Ri_B=0.15\), case I150) departs significantly from the neutral reference case. It lies in between the hodographs from the neutrally stratified case and a laminar one (Fig. 7c).

After an initial decay, the fluctuation kinetic energy recovers slowly on a time scale of a few inertial periods (blue curve in Fig. 7a). If expressed in terms of \(f^{-1}\), the time scale of this slow oscillation matches the time scale for recovery observed by Nieuwstadt (2005) in a stably stratified channel flow. Turbulence intensity recovers across the entire boundary layer, and concomitantly the depth of the stratified layer increases (sequence of blue lines in Fig. 7b). This increase in depth of the stratified layer is compensated by weakening stratification in the surface layer (\(z^-\lesssim 0.1\)). Eventually, during this recovery, the fluctuation kinetic energy increases beyond the neutral reference both above \(z^-\approx 0.5\) and in the production region (Fig. 6a) as also observed by Nieuwstadt (2005).

### 4.4 The Very Stable Regime

Under very strong stability (case S620), the turbulence dies out nearly completely since the production region is eliminated. The hodograph (Fig. 7a) is close to that of the corresponding laminar Ekman flow. In fact, the eddy diffusivity estimated from the laminar fit to the velocity profiles (not shown) is \(1.01\nu \). This re-laminarization in the inner layer is seen in Fig. 8: the turbulence with relatively high enstrophy magnitudes in panel (a) is replaced in panel (c) by large-scale roll-like structures aligned parallel to the wall-shear stress, i.e. is rotated by \(45^\circ \) counter-clockwise with respect to the geostrophic wind vector. This initial re-laminarization is followed by a recovery of turbulence as seen in the time series of TKE and enstrophy (Fig. 6a, b). The recovery of turbulence is similar to that observed in the intermediately stable case. This recovery, however, takes longer, and while the enstrophy levels off around \(60~\%\) of the neutral value, the normalized TKE grows beyond one.

At the beginning of the recovery of TKE, the maximum of TKE associated with the peak shear production in the buffer layer is eroded (Fig. 7a), that is, the production region of turbulent stress is eliminated. Around \(z^-=0.25\), the turbulence intensity is reduced even more than at the peak of production; this illustrates the absence of vertical turbulent exchange across the buffer layer and a decoupling of the flow inside this surface layer from upper layers of the flow. Above the decoupled surface layer (\(z^-\gtrsim 0.5\)), turbulence is affected less strongly by stratification and decays slowly from its fully-turbulent initial state between \(t^-\approx 1\) and \(t^-\approx 2\). Such slowly-decaying residual turbulence is common to nighttime boundary layers cooled from below (Stull 1988), which illustrates the appropriateness and the relevance of these simulations for the study of such cases.

The decoupling of the outer layer from the surface layer is an important consequence of very strong stratification, and there has been debate on whether the decoupling produced by boundary-layer schemes in NWP models (Derbyshire 1999; Acevedo et al. 2012) and LES (Saiki et al. 2000; Jiménez and Cuxart 2005) is an artifact of the turbulence subgrid model. From our data, we conclude that, at bulk Richardson numbers of order one (\(Ri_B=0.62\) for this particular case), a decoupling is possible—at least for an intermediate Reynolds number. This is in accordance with Wiel and Moene (2012). Our estimate, in contrast to estimates from NWP or LES, is not subject to uncertainties in subgrid schemes, but, similar to stratified shear flow (Jacobitz et al. 1997), the particular value of a critical Richardson number for decoupling might depend on \(Re\).

## 5 Flow Organization

By comparison between the neutral case N500 and the intermediately stable case I150 at \(Ri_B=0.15\), we show in this section that the recovery of turbulence in the intermediate cases is accompanied by a large-scale organization of the flow. This organization efficiently couples the outer and inner layers of the flow and hence is the main mechanism governing the spatio-temporal structure of global intermittency in the intermediately stable regime. We present a measure to quantify this global intermittency. For that purpose, we extend the classical concept of external intermittency, that is, the alternation of turbulent and non-turbulent patches of fluid in the outer layer: If there are laminar patches of fluid extending from the outer layer down to the surface layer, we identify the flow as globally intermittent.

### 5.1 Spatial Variability

We have seen in Fig. 5 how the spatial organization of the outer layer leaves a footprint in the quasi-logarithmic layer even in the neutral case. A conspicuous manifestation of this organization is evident in Fig. 9 for the intermediately stable case I150: strongly convoluted parts of an enstrophy iso-surface penetrate deep into the outer layer of the flow where the wind velocity reaches values comparable to the geostrophic velocity. These bulges coexist with much smoother patches of low velocities marking regions very close to the surface (Fig. 9c), where the enstrophy is almost entirely caused by the background shear. Signatures of streaks are also present in these smoother patches; streaks, however, seem to be pushed downwards, which prevents ejections into the boundary layer and locally limits buoyancy and momentum exchange.

The organization of the globally intermittent flow is qualitatively summarized in Fig. 9d: the surface streaks (red, dashed) are parallel to the wall shear stress; their separation distance is much smaller than the boundary-layer depth (Fig. 8). Even adjacent to the wall, the flow is not fully turbulent everywhere but the turbulence is patchy. There are laminar patches extending through the entire vertical column of the flow. The fully turbulent and quasi-laminar patches in the inner and outer layer are organized along lines oriented as indicated by the solid blue line in Fig. 9d. The angle with the mean flow is \({} \approx {23}^\circ \) clockwise, which is similar to the neutral case (Sect. 3). That is, under stable stratification, the orientation of the large-scale structures in the outer layer sets the spatial organization of the flow even in the inner layer by determining the patterns of global intermittency. The wavelength of these large-scale structures, which are already present under neutral conditions, is of the order of several boundary-layer depths (2–5\(\delta \), a fraction \(1/10\,\hbox {to}\,1/4\) of the Rossby radius); these structures need to be resolved for a realistic representation of this complex flow.

### 5.2 External and Global Intermittency

A quantitative measure of external intermittency in the flow is provided by the intermittency factor \(\gamma \) introduced in Sect. 3.2. Compared to the neutral case (Fig. 12, grey line), the intermediate case (blue line) has a much lower intermittency factor in the outer layer, which indicates the absence of turbulence in this region. This absence occurs between \(z^-\approx 0.5\) and \(z^-\approx 1.0\), that is, precisely at those levels where the TKE increases beyond the neutral reference (Fig. 7a) while the integrated streamwise enstrophy r.m.s. (Fig. 6b) does not. This apparent contradiction is resolved if the flow field is inspected visually: in the outer layer of the neutral case, the turbulent field is characterized by a variety of small vortices with rather small vertical velocities (Figs. 5, 8d). In the intermediate case, after one inertial period, these structures are replaced by the large-scale structures discussed in Sect. 5.1 above. We show streamwise-vertical intersects approximately perpendicular to the large-scale structure in the flow (red and white lines in Fig. 9) in Fig. 11. This large-scale structure carries TKE by means of the comparatively large values of vertical velocity: there are large regions with vertical velocities as large as \({} \approx {\pm }\,\,0.1G\) (Fig. 11d). Regions of large vertical velocities are connected by corresponding positive and negative streamwise (Fig. 11b) and spanwise (not shown) anomalies in between them, which indicates a roll-like structure approximately aligned with the large-scale organization discussed above. These large-scale structures are responsible for the large increase of TKE (about 25 %, cf. Fig. 6a) beyond the neutral reference and thus resolve the above noted apparent contradiction.

The TKE conditioned to turbulent and non-turbulent patches of the flow at \(Ri_B=0.15\) (case I150) is plotted in Fig. 12b. Beyond \(z^-\approx 0.7\) the contribution of the non-turbulent region in the stable case increases with respect to the neutral reference, and this increase partly explains the overshoot in the integrated TKE. We also note that, according to our partitioning, the turbulent contribution to the fluctuation kinetic energy \(e\) does not only increase above \(z^-\approx 0.7\) but also in the lower parts of the outer layer, around \(z^-\approx 0.2\). This is likely due to the problems of the intermittency function with the recognition of global intermittency close to the surface as described in the previous paragraph: contributions of the cross term in Eq. 7 are aliased onto the turbulent partition. Further work on conditioning methods to study global intermittency is necessary.

## 6 Discussion

### 6.1 Relation to Monin–Obukhov Similarity Theory

### 6.2 Global Intermittency

In the light of our results, the occurrence of global intermittency might be explained as follows: once turbulence cannot be fully sustained, the turbulence decays and the relative size of the non-turbulent region is determined by a bulk property of the system, like a bulk Richardson number in the case studied here. We thus proved that global intermittency can arise from a global constraint on the flow, for instance the exceedance of the maximum sustainable heat flux (Wiel et al. 2012; Wiel and Moene 2012). Local perturbations, such as surface heterogeneities—or large-scale dynamics as in our case—, simply determine the spatio-temporal distribution of global intermittency (Sun et al. 2012, 2004; Acevedo and Fitzjarrald 2003), but they are not needed as a trigger.

While previous work mainly considers local or external mechanisms as a trigger for global intermittency in the SBL, we show that this process is intrinsic to a stratified atmosphere. This has important implications for turbulence models applied under very stable stratification: global intermittency effects could be incorporated into turbulence closures for Reynolds-averaged NWP models once its dependency on \(Ri\) and \(Re\) is quantitatively understood—without depending strongly on the local details of the flow such as surface heterogeneities.

Owing to the absence of heterogeneities in our numerical set-up, the spatio-temporal pattern of global intermittency close to the surface is caused by a large-scale structure in the outer layer of the flow. Indeed, Chung and Matheou (2012) and Brethouwer et al. (2012) observe a very similar phenomenon in homogeneously stratified sheared turbulence respectively rotating Couette flow, in both of which no local or coherent external perturbations are present. This occurrence of global intermittency and its linkage to large-scale forcing from the outer layer is consistent with Mauritsen and Svensson (2007) who, based on observational data, suggest internal gravity waves as a cause of non-zero turbulent fluxes under very stable stratification.

The agreement with observations in terms of \(Ri_B\) and the occurrence of global intermittency suggests that this mechanism is relevant at the atmospheric scale. If our simplified set-up is considered in the phase space spanned by its non-dimensional parameters \(Ri_B\) and \(Re\), there are two ways in which a transition from turbulent to laminar can happen: either through stronger stratification, that is changing only the parameter \(Ri_B\), or via a decrease in the mean shear, affecting both \(Ri_B\) and \(Re\). Once in the fully turbulent regime, it is unlikely that the fundamental character of this transition changes. Such a change would in fact require turbulence in the SBL to be caused by a different instability than in our simulations. The quantitative dependency of this mechanism on \(Re\) has, however, to be elucidated in future work. We expect certain properties of this transitions such as a critical Richardson number to depend on \(Re\) similarly to \(u_\star \) and \(\alpha \) (Sect. 3) and as observed in stably stratified shear and channel flow (Jacobitz et al. 1997; Flores and Riley 2011). Nonetheless, we have shown that this simplified set-up is suited to study dynamics of the stable and very stable stratified PBL. This analogy over a cascade of complexity—ranging from the canonical flow problem of stable channel flow via rotating Cuoette and stable Ekman flow to an atmospheric boundary layer—encourages further investigation of the fundamental aspects of stably stratified turbulence in rotating reference frames.

## 7 Conclusions

We have defined a framework to investigate the SBL using direct numerical simulation of turbulent Ekman flow over a smooth surface with a fixed temperature. This set-up is described in terms of a Reynolds and a Richardson number solely.

In the neutral limit, the flow depends only weakly on the Reynolds number as we demonstrate by a comprehensive flow description including hodographs, the integral value of the turbulence dissipation rate as well as vertical profiles of velocity, TKE, and external intermittency. We thereby show that the analogy between the surface layer in Ekman flow and channel-flow also applies to the budget of TKE, and not only to the logarithmic law of the mean velocity. Through a conditioning of the TKE to the turbulent and non-turbulent patches, we demonstrate that the alternation between those contributes significantly to the velocity variance in the outer layer of the flow.

- (1)
*Stably stratified Ekman flow is suited to study aspects of the SBL.*The regimes of stratified turbulence are reproduced varying a single parameter: the bulk Richardson number. Characteristics of these turbulence regimes, such as hodographs and TKE profiles, compare well with atmospheric observations and theoretical considerations. We estimate the Monin–Obukhov stability correction for stable stratification as \(\varPhi _M-1 \approx 5.7z/L_O\) in agreement with data from channel-flow and atmospheric observations. - (2)
*The analogy of the surface layer with channel flow holds beyond qualitative aspects.*The recovery and overshoot of integrated TKE as well as the laminar patches observed in our simulations are similar to their equivalents in channel flow. - (3)
*A large portion of fluctuation kinetic energy is carried by velocity fluctuations associated with regions of potential flow, i.e. non-turbulent patches.*These fluctuations associated with potential-flow regions occur at a large scale, and are possibly coherent motions associated with a time scale of several large-eddy turnover times. If such coherency occurs in the atmosphere, it has implications both for the flux measurements in the field and modelling of such flows with LES: the averaging period of flux measurements, respectively the grid size of an LES, would have to be chosen to allow for either resolution or complete filtering of these structures. - (4)
*Global intermittency can occur without external perturbations of the flow*: in our cases global intermittency is simulated despite the absence of finite-size triggers from synoptic conditions, low-level jets, or surface heterogeneities. Global intermittency is hence intrinsic to a stable atmospheric boundary layer beyond a certain stability, and we suggests it cannot be treated as an on–off process in time, but should rather be seen as happening in time*and*space.

## Notes

### Acknowledgments

Support from the Max Planck Society through its Max Planck Research Groups program is gratefully acknowledged. Computing resources were provided by the Jülich Supercomputing Centre.

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