## 1. Introduction

Despite ongoing intensive study, understanding and parameterizing stably stratified turbulence in geophysical flows and, in particular, in atmospheric and oceanic planetary boundary layers (PBLs) remains at the forefront of geophysical turbulence research. The dynamics of the stable boundary layer (SBL) are still in a state of discovery as the couplings between large-scale forcings, internal waves, and weak turbulence over undulating rough boundaries in the presence of strong stratification continues to resist a generic description; see Fernando and Weil (2010) and Mahrt (2014) for recent reviews. The importance of the SBL is well appreciated in climate and weather modeling as these forecasts show great sensitivity to the form of their SBL parameterization (Holtslag et al. 2013). Intermittent SBL turbulence is also important in numerous application areas—for example, electromagnetic wave propagation (Wyngaard et al. 2001), interpreting observations collected from wind profilers (Muschinskii and Sullivan 2013), and air quality (Weil 2012).

A crucial thread in SBL research for climate and weather applications is quantifying the connecting relationship between the mean wind and temperature fields and the average turbulent fluxes and variances used in large-scale modeling applications; for example, Brost and Wyngaard (1978), Large et al. (1994), Svensson and Holtslag (2009), Sorbjan (2010), and Huang et al. (2013) all propose single-column models for the SBL. However, to further improve these flux–gradient relationships, to better understand mixing in the SBL, and to guide the interpretation of observational data collected in field campaigns requires a more fundamental understanding of the building blocks—that is, the coherent structures, in turbulent SBLs. It is now widely appreciated that coherent structures, loosely defined as spatially organized entities long lived in a Lagrangian frame of reference, are the key flux-carrying structures in most geophysical boundary layers. Observational and numerical studies find that the properties of the organized flow structures vary with stratification and the external driving forces in the PBL. For example, thermal plumes dominate the daytime convective PBL (Deardorff 1970), large-scale rolls are pervasive in mixed shear–convective PBLs (Moeng and Sullivan 1994; Fedorovich et al. 2004), low-speed streaks dominate near-neutral surface layers (Marusic et al. 2010), large eddies in canopy turbulence are determined by an inflection-point instability centered near the canopy top (Finnigan et al. 2009), and surface-wave-generated Langmuir circulations populate the upper ocean (Sullivan and McWilliams 2010), to mention a few.

Because of the challenges in observing and simulating stably stratified weak turbulence, the coherent structures in the SBL are less studied, and likely more variable, than those in the convective and near-neutral PBL. In the very stable boundary layer characterized by large Richardson number (Ri), Mahrt (2014) shows that the morphology of structures in the surface layer is exceedingly diverse with intermittent turbulence mixed with wavelike motions and two-dimensional modes. These surface-layer features differ from the highly intermittent Kelvin–Helmholtz instabilities that appear to dominate the overlying residual turbulence above the SBL top (Balsley et al. 2003). At lower Ri in the weakly stable regime, the turbulence is near continuous and a new collection of turbulent structures emerge in the SBL. For example, in a slightly stratified wind tunnel flow, Chen and Blackwelder (1978) analyze time series from a vertical rake of instruments and find that “the most interesting observation was the existence of a sharp internal temperature front … that extended throughout the entire boundary layer.” Their temperature front appears to be part of the family of cliff–ramp structures observed by Thorpe and Hall (1980) in a lake under moderate wind conditions and by Gao et al. (1989) above a forest under slightly unstable conditions. Chung and Matheou (2012) also find cliff–ramp fronts in direct numerical simulations (DNS) of stably stratified shear flow with no solid boundaries; their flow visualization also reveals that the fronts tilt farther forward (or downstream) and can become intermittent with increasing stratification. The dynamics behind the cliff–ramp fronts in boundary layers is not completely explained but is possibly linked to hairpin packets found in neutral wall-bounded flows as discussed by Adrian (2007). Williams and Smits (2011) speculate that hairpin packets become elongated in the downstream direction in a thermally stratified laboratory flow. Cliff–ramp structures are also ubiquitous features of passive scalars in the turbulent flows described by Warhaft (2000).

The specific goals of this article are to identify and characterize coherent structures, and in particular temperature fronts, in large-eddy simulations (LESs) of a rough-wall weakly stable PBL. Previous pioneering work by Mason and Derbyshire (1990), Derbyshire (1999), and Saiki et al. (2000) used LES with some success to investigate the SBL under weak stratification. However, in these simulations the combination of small-scale turbulence and coarse mesh, ^{9} grid points follows recent trends in the use of high-resolution LES (Bou-Zeid 2015) and provides an opportunity to examine in a systematic manner the sensitivity of the solutions to the mesh spacing similar to our work with the convective PBL (Sullivan and Patton 2011). The problem posed is a canonical SBL with a homogeneous lower boundary. We are aware that coupling with nonhomogeneous or time-varying surface conditions (Van de Wiel et al. 2002; Nieuwstadt 2005; Flores and Riley 2011; Ansorge and Mellado 2014; Mironov and Sullivan 2016) can lead to a different family of structures in the SBL and possibly global turbulence collapse at least in low–Reynolds number DNS. Van de Wiel et al. (2012) and Donda et al. (2015) using a combination of analytic models and DNS provide evidence that turbulence collapse is likely a transient state even with high surface cooling. A discussion of the turbulent-to-laminar transitional regime at large Ri is beyond the scope of the present article. The road map of the manuscript is as follows: a brief description of the LES equations and numerical algorithm is given in section 2, an outline of the numerical experiments is provided in section 3, results from the grid resolution tests and experiments with surface cooling variations are given in section 4, the identification of temperature fronts and coherent structures are presented in sections 5 and 6, and a summary of the findings is provided in section 7.

## 2. LES equations

*e*. In (1d) the divergence-free (incompressible) condition determines the elliptic pressure variable

*f*, unit vector

*g*is gravity and

*ρ*, which do not appear explicitly in (1). The overbar notation denotes a spatially filtered quantity.

In our LES, the sidewall

We utilize well-established algorithms to integrate the LES equations in (1). The equations are advanced in time using an explicit fractional step method that enforces incompressibility at every stage of the third-order Runge–Kutta scheme. Dynamic time stepping with a fixed Courant–Fredrichs–Lewy (CFL) number is employed, which we have found naturally adapts to a wide spectrum of dynamical processes. The spatial discretization is second-order finite difference in the vertical direction and pseudospectral in horizontal planes. For the present application, the advective terms in the momentum equations are written in rotational form, while a flux-conserving form is used for the advective terms in the scalar (temperature) equation. The vertical velocity equation is solved for the deviation of *w* from its horizontal mean value at each height. The flow variables are explicitly filtered at each time step, or dealiased, using the 2/3 rule (Moeng and Wyngaard 1988). Further algorithmic details are given by Moeng (1984), Sullivan et al. (1994, 1996), McWilliams et al. (1999), Sullivan and Patton (2011), Moeng and Sullivan (2015), and the references cited therein.

To streamline the notation and text in the following discussion, we now drop the overbar symbol on all spatially filtered resolved variables and simply refer to virtual potential temperature *θ* as “temperature.”

## 3. Design of LES experiments

The SBL flow examined here is the first GEWEX Atmospheric Boundary Layer Study (GABLS1) described by Beare et al. (2006). This high-latitude SBL is a benchmark intercomparison case for canonical stable LES and, in addition, serves as a clean test case for the evaluation of single-column PBL schemes used in climate and weather models (Cuxart et al. 2006). GABLS1 does not include potentially important feedbacks from a land surface as investigated by Holtslag et al. (2007). The GABLS1 PBL is driven by steady geostrophic winds ^{−1} below the boundary layer top

The present suite of LES experiments extends the original GABLS1 problem design in the following ways. First, four different cooling rates are applied at the surface^{1} that increase the overall level of stratification in the SBL compared to GABLS1 and, second, three levels of grid refinement are considered, namely grid spacing *M*^{3} =

The simulations are integrated for more than 9 physical hours, which equates to more than 40 nondimensional times ^{6} core hours or ~1130 wall clock hours on the NCAR peta-scale machine Yellowstone. The code parallelization and performance is described by Sullivan and Patton (2011).

The statistics presented here are generated in flight and also by further postprocessing of archived data volumes. As is customary practice, statistics are computed by averaging in horizontal *x*–*y* planes at each *z* and also over time; this is the LES approximation to an ensemble (mean) average. The averaging time window is from hours 8 to 9 as in the original GABLS1 LES comparisons. Averages are denoted by angle brackets with resolved turbulent fluctuations indicated by primes; for example, for variable *f* its mean is *x*–*y* plane: *u*_{*} and *Q*_{*}, respectively, and the Monin–Obukhov length

Bulk simulation properties. Entries are as follows: “run” is the simulation name with mesh listed under “grid points,” *L* is the Monin–Obukhov stability length,

## 4. Interpretation of low-order moments

*N*and shear frequency

*S*:The boundary layer top

*t*. However, in the present application with very fine meshes we find the local gradient method often finds a false low estimate of

### a. Varying resolution with fixed cooling rate

Profiles of the mean wind speed, mean temperature, gradient Richardson number, and shear and buoyancy frequencies are displayed in Figs. 1 and 2 for varying mesh resolution at a fixed cooling rate *u*_{*} and *Q*_{*} are, respectively, 13% and 32% larger on the coarse mesh compared to the finest mesh (see Table 1) and in the SBL interior Ri is largest on the coarse grid (see Fig. 2). Further reexamination of the GABLS1 intercomparisons also shows the same overall trend for boundary layer depth as the mesh decreases from 12.5 to 2 m for all LES codes (Beare et al. 2006, p. 253). A broadly similar effect is also found in LES of a convective PBL where ^{3} m is adequate for capturing the largest scales of motion in our SBL and that the solution mesh sensitivity is due entirely to small scales. DNS of stratified homogeneous shear flow show a dependence on the computational domain size (Chung and Matheou 2012) and large scales are also reported in DNS of a stably stratified Ekman layer with zero buoyancy gradient aloft by Ansorge and Mellado (2014). We believe that their results are a consequence of the problem posing; these DNS do not contain a stably stratified capping inversion or a LLJ. In our SBL,

Our mesh resolution tests indicate that the small scales *ϵ* is the viscous dissipation; see Doughtery (1961) and Ozmidov (1965). Recall eddies with vertical scale *z* depending on the mesh resolution; the viscous dissipation is calculated from the SGS model

### b. Impacts of cooling rate with fixed resolution

The impacts of the surface cooling rate

Table 1 and the sequence of Figs. 4–8 summarize the impacts of surface cooling on the bulk statistics and the vertical profiles of typical low-order moments in the SBL. First, we notice that with increasing surface cooling (or stratification) *h* based on a stress minimum, which tends to occur near

Inspection of the wind profiles shows that, with increasing stratification, the SBL is shallower, the height of the LLJ descends, the winds turn more sharply with height, and the surface wind stress decreases; see Fig. 4. At the same time, the mean temperature profile develops sharper vertical gradients in the lower boundary layer and weaker gradients aloft—especially so for the simulation with the highest cooling rate. The SBL appears to divide into two regimes with increasing stratification, which is particularly apparent in the *θ* profile for simulation F and in the mean vertical gradients of wind and temperature shown in Fig. 5. Above the LLJ

Below the LLJ, *z* and with stratification; *z* and stratification but is always less than 0.25. For example, at

## 5. Temperature fronts

### a. Flow visualization

Extensive visualization of the LES flow fields *x*–*y*, *x*–*z*, *y*–*z*) and 3D volumes for all mesh resolutions. From this large database we concentrate on the temperature field *θ* from simulations using a mesh of 1024^{3} grid points. One of the most ubiquitous features common to all simulations is displayed in Figs. 9 and 10. The top panel of Fig. 9 shows a grayscale image of the temperature difference *θ* minus *θ*_{o} in an *x*–*z* plane late in the simulation *t* ≈ 9 h from simulation C; the top panel of Fig. 10 shows the same field but displayed as a family of contour lines. The bottom panel of Fig. 10 displays results for higher stratification from simulation F. Inspection of these figures reveals an impressive array of locally compacted temperature lines; that is, sharp temperature fronts tilted in the streamwise (or downstream) direction. Each front marks a sharp boundary between warm upstream and cool downstream air with more uniformly well-mixed air between fronts. In the lower boundary layer, the fronts tilt upward often with an apparent origin near the surface. Near the low-level jet, the fronts are weaker with values of tilt angle that transition from positive, to zero, to negative (downward tilt) as *z* moves across the LLJ. Animations show that the images in Fig. 9 are not isolated special instances and that the fronts meander and evolve in time but maintain clear coherence as they propagate with the winds. The temperature fronts are mainly tilted into the mean wind direction with finite spatial extent in the crosswind direction—that is, the fronts are not 2D sheets as shown in the bottom panel of Fig. 9. Increasing stratification reduces their vertical tilt—the fronts are tipped more toward the downstream direction—and also narrows the spacing between individual fronts; see the bottom panel of Fig. 10. We emphasize that in the current problem posing the surface boundary condition for temperature varies in time but is spatially homogeneous, constant across the *x*–*y* domain, and thus the observed fronts are internally generated by the dynamical interaction between well-resolved turbulence and a stably stratified temperature field.

*x*–

*z*location in the domain. For example, consider the two fronts marked by dashed white lines in the top panel of Fig. 9. The front that starts and ends at

*x*axis. It is illuminating to estimate the tilt of a constant-

*θ*surface in the presence of turbulence. The gradient vector

*θ*surface has instantaneous tilt angles

*θ*fluctuations aligned with the mean wind is comparable to or larger than

*θ*. Evidently the fronts at the top of the boundary layer in Fig. 10 are nearly flat, level surfaces because of weak horizontal temperature fluctuations while those in the lower boundary layer are tilted upward when the temperature fluctuations are vigorous with strong local horizontal and vertical gradients.

A horizontal *x*–*y* slice of *x*–*y* spatial coherence of a front with increasing distance from the surface. The frontal boundary, although irregular in *x*–*y*, remains sharp and clearly identifiable. It appears to rotate slightly in a clockwise direction and is found at a larger downstream *x* distance with increasing *z*. This apparent propagation with increasing *z* reflects the downstream tilt observed in Fig. 9, while the front’s rotation appears to track the rotation of the mean wind vector with height. The positive correlation between the wind and temperature fluctuations upstream and downstream of a frontal boundary in *x*–*y* planes leads to positive horizontal temperature fluxes

Figure 13 displays a 2D image of the fluctuating vertical temperature gradient *θ* field shown in the top panel of Fig. 9. The most intense positive values of this statistic nicely overlap with the front locations found previously, and the compression of the vertical gradients into thin streaky filaments provides a sense of the front sharpness. Notice that the largest fluctuating vertical gradients are often more than 10 times larger than the local mean gradient. Overall the image shows that at a fixed *x*–*y* location the temperature progression from the surface (cool) to the SBL top (warm) occurs in a series of jumps or in staircase fashion, and between jumps the temperature is relatively well mixed despite the overall bulk stable stratification; see Fig. 4. The probability density functions (PDFs) shown in Fig. 14 quantify the skewed character of the vertical and horizontal temperature gradients. Near the surface

We are aware that warm–cool temperature fronts are also found in idealized low-Re direct numerical simulations of homogeneous shear flows (no walls); for example, see Gerz et al. (1994), Warhaft (2000), and Chung and Matheou (2012). The warm–cool fronts found here are cousins to those in homogeneous shear flows but differ because of the presence of a rough wall, the vertically varying stratification *z* especially in the region of the LLJ.

### b. Instantaneous vertical profiles of θ

We attempt to use our LES results to help interpret outdoor observations by simply asking the following: What is the signature of a coherent warm–cool temperature front in a fixed-tower-based measurement? To expand on this idea, we position a “virtual tower” in the LES domain at a fixed horizontal location and monitor the instantaneous temperature profile as function of time. Typical results are shown in Fig. 15 for a virtual tower placed at ^{−1}, the system of fronts in Fig. 9 propagates more than 100 m over this time period. In this example, three temperature fronts are found in profile 1, near *θ* at these locations stiffens, relaxes, and stiffens, but generally a vertically well-mixed region is maintained between the fronts. Note that the horizontal advection speed increases with height and thus the fronts appear at different times in our tower measurement. Because of the front’s forward tilt, the location of the sharp vertical gradient in *θ* descends with increasing time when observed at a fixed location. We emphasize that this apparent descent is not a consequence of downward vertical advection as we are sampling the same frontal structure as time advances but at different locations along its tilted boundary because of streamwise advection. Mahrt (2014) discusses the very stable boundary layer where downward advection of turbulence from the LLJ occurs above the surface layer. Inspection of numerous profiles shows that the process sketched in Fig. 15 is generic with the vertical temperature profile often taking on a staircase shape with increasing *z*.

The LES findings shown in Fig. 15 are tantalizing targets for observations. Thus, we next search for SBL temperature fronts in the observational database collected during the 1999 Cooperative Atmosphere–Surface Exchange Study (CASES-99) field campaign [for an overview, see Poulos et al. (2002)]. Typical temperature profiles collected from the 55-m-tall tower are displayed in Fig. 16 under conditions broadly similar to the LES. At 0730:00 UTC 21 October 1999, the observations reported (*u*_{*}, *Q*_{*}) ~ (0.16 m s^{−1}, −0.025 m s^{−1} K), a LLJ with winds 12–13 m s^{−1} at a height *z*_{j} ~ 100–150 m (Balsley et al. 2003), and winds from 5 to 6 m s^{−1} at *z* = 45 m. Although qualitative, inspection of the profiles clearly shows the main features predicted by the LES; that is, multiple vertically well-mixed regions bounded by sharp temperature gradients aloft. Also, the maximum temperature gradient shows a steady descent with increasing time—a key feature of a streamwise advecting tilted temperature front shown in Fig. 15. Note the temperature jump across the front can approach 2 K because of stronger surface cooling during the observation period. However, because of the limited tower height, we are only able to observe fronts that propagate into and across the surface layer; Fig. 15 shows multiple fronts distributed vertically for *z* > 60 m. Finescale measurements collected from a slowly ascending kite during CASES-99 (Sorbjan and Balsely 2008; see their Fig. 1a) also support the LES predictions of intermittent temperature jumps in the middle and upper regions of the SBL.

## 6. Linear stochastic estimation

The flow visualization described in section 5 paints an image of many randomly distributed but locally organized warm–cool temperature fronts populating the weakly stable boundary layer. These sharp fronts are tilted in the downstream direction, exhibit spatial spanwise and vertical coherence, and propagate in time as organized entities. We seek to identify in an average sense the flow structures and dynamics that couple with the formation of these many warm–cool temperature fronts.

To establish a connection between flow structures and temperature fronts in our simulations, we use event-based conditional sampling—specifically, linear stochastic estimation (LSE); for an excellent essay on the topic of stochastic estimation, see Adrian (1996). LSE has several advantages over other eduction techniques. Conditional averages are obtained using unconditional variances, covariances, and two-point spatial correlations, and thus LSE conditional estimates are robust since all available data are used in forming the averages. Also, with LSE it is straightforward to use complicated detection events at multiple locations in the estimation procedure. As LSE conditional estimates depend linearly on the event data once the underlying statistical functions are computed and stored, parametric variations in the event data are easily formed. Finally, numerous tests using experimental and simulation data for a variety of turbulent flows show that LSE is an excellent approximation of a conditional average; for example, see Adrian et al. (1987), Guezennec (1989), Adrian et al. (1989), Adrian (1996), Christensen and Adrian (2001), and Richter and Sullivan (2014).

In the terminology of turbulent structure identification, we are interested in the average state of the SBL flow fields subject to a particular set of events; that is, we wish to compute the conditional average

*z*as our detection event

*d*, and orientation in an

*x*–

*y*plane; it has zero mean. To capture a horizontal front we specify two temperature events with equal but opposite amplitudes

*E*= +0.1 and −0.1 K on the warm and cool sides, respectively, of the fronts in Fig. 11. Nondimensionalizing this event amplitude in terms of surface-layer variables,

*x*–

*y*plane at the same

*z*. The first linear term in the stochastic estimate of

*θ*and the flow variables

*z*locations using fast Fourier transforms at 1024

^{3}

*x*–

*y*points with further averaging over 64 3D volumes collected over the last hour of the simulation. The correlations are archived for later use in (7).

### a. Conditional fields

Conditional fields *d*, needed by our LSE algorithm, is guided by the flow visualization and the 2D energy spectra for

Figure 18 shows typical 3D vortical structures extracted from the turbulent flow fields using our LSE algorithm with

Inspection of the 3D image displayed in Fig. 18 from many viewing angles shows a familiar but also unexpected pattern of vortices at this vertical location in the SBL. The downstream vortical structure is identified as a head-up hairpin vortex similar to the structures found in neutral flat wall boundary layers. The strong vertical vorticity in its vertical legs generates a Reynolds stress event

The upstream vortical system is slightly weaker in vortical strength compared to its downstream counterpart and even with lengthy averaging its shape is less defined. Based on LSE, we cannot unambiguously claim that it is a head-down or head-up hairpin. Head-down hairpins tend to appear away from boundaries as found by Finnigan et al. (2009) in neutral shear flow over a canopy and also by Gerz et al. (1994) in homogeneous shear flow. Our structure more closely resembles a ring vortex with weak connecting arches at the top and bottom of the structure; Figs. 19 and 20 further support this interpretation. Adrian (1996, see p. 183) found in low–Reynolds number DNS isolated hairpins tend to evolve into ring structures away from the wall. Notice the vertical legs of the upstream structure are rotating in opposite directions compared to its downstream counterpart. Then the induced flow is reversed with strong downward-vertical and positive downstream-horizontal velocities through the center of the vortical structure generating a Q4 Reynolds stress event

We emphasize these vortical structures are outputs from our conditional sampling; that is, they are the average flow fields extracted from fully developed turbulent flow fields for a generic temperature front in the SBL. In this aspect our LSE differs from previous applications where Reynolds stress Q2 and Q4 events are simultaneously used as events to identify strong shear layers (e.g., Guezennec 1989). Also recall that the front scale *d* used in the conditioning event lies in the energy containing range of the turbulence and thus the coherent structures are characteristic of the main energy and flux-carrying turbulent eddies.

The upstream and downstream vortices combine to create complex 3D velocity patterns that result in a positive pressure maximum (stagnation point) near the sharp jump in temperature. At the reference height *x*–*y* view, one also observes a temperature front rotated in the *x*_{h}–*z* plane; see Fig. 20. This plane is first centered at *d* and *E*,

### b. Coherent structure and temperature front variability

Based on the results in section 6a, we conclude that temperature fronts are formed by dynamical coupling with vortical structures located upstream and downstream of a frontal boundary. To further investigate this connection, and also determine the sensitivity to the parameters used in the conditional sampling, we sweep across a broad 2D parameter space spanned by *E* for a given value of *d*. Thus, an isosurface of *d*. Somewhat surprisingly, however, our flow visualization shows that the overall vortical patterns displayed in Fig. 18 remain remarkably robust to varying scale *d*. In other words, even at small and large separations, the upstream and downstream structures are roughly ring and hairpin vortices, respectively. The vortices are tilted at angle of 45° with respect to the horizontal at

*d*–

*E*combination. To show this, consider the analog to (6) for the tilt angle of a temperature front based on conditional fieldsThe prediction from (9) for varying scale

*d*is shown in Fig. 21 with event amplitude as a parameter. The tilt angle is shown at the center of the front

*E*in (7), the ratio of its horizontal to vertical gradients [i.e.,

*d*, the tilt angle

*E*as the large vertical gradients in

*d*limit the horizontal gradient of

## 7. Summary

A canonical stably stratified atmospheric boundary layer (SBL) is simulated using high–Reynolds number large-eddy simulation (LES). The problem design is modeled after the first GEWEX Atmospheric Boundary Layer Study (GABLS1) described by Beare et al. (2006). The present set of LES experiments extend the original GABLS1 problem setup by using four different surface cooling rates *L* is the Monin–Obukhov length.

The major findings from the study are as follows:

- For the posed problem, over the range of stratification considered continuous weakly stratified turbulence is maintained in the SBL both above and below the low-level jet (LLJ) with no global turbulence collapse on the finest LES mesh.
- The SBL splits into two regions depending on the height of the LLJ and the surface cooling. Above the LLJ, the turbulence is very weak and the gradient Richardson number is nearly constant at
. Below the LLJ, small scales are found to be dynamically important as the shear and buoyancy frequencies squared change with mesh resolution. Both and Ri decrease with increasing mesh resolution. The largest changes are found when the mesh spacing decreases from 2 to 0.78 m. - With increasing stratification the SBL is shallower, the height of the LLJ descends, the winds turn more sharply with height, and the surface wind stress decreases. Also, the mean temperature profile develops sharper vertical gradients in the lower boundary layer and weaker gradients aloft especially so for the simulation with
. - Vertical profiles of the Ozmidov scale
show a rapid decrease for the strongest cooling rate and over a large fraction of the SBL at high cooling. As a consequence, LES with meshes are likely too coarse to capture the turbulent flow dynamics near the LLJ and near the surface as the cooling increases. - Flow visualization identifies numerous warm–cool temperature fronts in the SBL. The fronts occupy a large vertical fraction of the SBL and tilt forward with increasing stratification. They propagate as coherent entities in space and time. The horizontal and vertical gradients of fluctuating
*θ*are often 10 times larger than the local mean gradient. - The LES results can be used to interpret measurements collected from a fixed observational tower. In a height–time (
*z*–*t*) frame of reference, instantaneous vertical profiles of*θ*appear intermittent. Temperature increases in a staircase pattern from the surface up to the location of the LLJ, and between the fronts the temperature is nearly well mixed. Analysis of observations shows these patterns are also found in the temperature profiles in CASES-99 (Poulos et al. 2002). Althoughis highly intermittent, under heavy space and time averaging, takes on a very uniform vertical structure. Similarly, the average buoyancy frequency *N*^{2}is also smooth and the companion Ri profile varies continuously over the bulk of the SBL. Because of the intermittent character of*θ*, a local Ri(*z*) is inadequate to describe the flow dynamics at short time scales. - Conditional sampling based on linear stochastic estimation (Adrian 1996) is used to identify coherent structures in the SBL. The conditioning event is a two-point model of a horizontal temperature front with varying amplitude and scale. For a weakly stratified SBL at
*z*/*z*_{i}= 0.2, we find that the coherent structures are ring and head-up hairpin vortices upstream and downstream, respectively, of the frontal boundary similar to those in neutrally stratified boundary layers. The scale of these structures lies in the energy and flux containing range of the turbulence. Although the vortical structures are oriented at an angle of 45° from the horizontal, they generate temperature fronts of varying amplitude, spatial distribution, and tilt.

PPS and EGP were supported by the National Science Foundation through the National Center for Atmospheric Research (NCAR). PPS also recognizes support from the Office of Naval Research through the Physical Oceanography Program. HJJJ and DVM acknowledge past support from the NCAR Geophysical Turbulence Program. This research benefited greatly from computer resources provided by the NCAR Strategic Capability program managed by the NCAR Computational Information Systems Laboratory (http://n2t.net/ark:/85065/d7wd3xhc). We thank the three anonymous reviewers for their constructive comments.

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Huang and Bou-Zeid (2013) also vary the surface cooling rate.