Mesoscale structure of the atmospheric boundary layer across a natural roughness transition

Significance

We live within the Atmospheric Boundary Layer (ABL), where air flow feels the friction of the planet’s surface, producing turbulence. When ABLs encounter changes in roughness—from sea to land, or rural to city—a near-surface region with distinct turbulence characteristics develops. The structure within influences transport of heat and dust; yet, data in this region are sparse, with existing models unable to capture important behaviors. We use advanced computational tools at a massive scale to identify patterns in the turbulence structure over a roughness transition in a dune field, validating our model against rare field observations. Our results explain feedback between flow and topography that influence this landscape, revealing behaviors that may be common across natural roughness transitions.

Abstract

The structure and intensity of turbulence in the atmospheric boundary layer (ABL) drive fluxes of sediment, contaminants, heat, moisture, and CO₂ at the Earth’s surface. Where ABL flows encounter changes in roughness—such as cities, wind farms, forest canopies, and landforms—a new mesoscopic flow scale is introduced: the internal boundary layer (IBL), which represents a near-bed region of transient flow adjustment that develops over kilometers. Measurement of this new mesoscopic scale lies outside present observational capabilities of ABL flows, and simplified models fail to capture the sensitive dependence of turbulence on roughness geometry. Here, we use large-eddy simulations, run over high-resolution topographic data and validated against field observations, to examine the structure of the ABL across a natural roughness transition: the emergent sand dunes at White Sands National Park. We observe that development of the IBL is triggered by the abrupt transition from smooth playa surface to dunes; however, continuous changes in the size and spacing of dunes over several kilometers influence the downwind patterns of boundary stress and near-bed turbulence. Coherent flow structures grow and merge over the entire $\sim$10 km distance of the dune field and modulate the influence of large-scale atmospheric turbulence on the bed. Simulated boundary stresses in the developing IBL counter existing expectations and explain the observed downwind decrease in dune migration, demonstrating a mesoscale coupling between flow and form that governs landscape dynamics. More broadly, our findings demonstrate the importance of resolving both turbulence and realistic roughness for understanding fluid-boundary interactions in environmental flows.

Whenever turbulent flows impinge on a surface, a boundary layer develops—a thin region near the surface where the shear stress of the boundary is felt by the flow, characterized by a steep velocity gradient (1, 2). The stress exerted by the flow on the boundary depends sensitively on the geometry of boundary roughness itself; thus, there is a feedback between flow and form (3, 4). Even for the canonical case of uniform sand grains glued to a pipe, surface drag is a complex and nonmonotonic function of the roughness Reynolds number $k_s^{+}\equiv u_{\tau }{k}_s/\nu$, where $u_{\tau }$ is the friction velocity, $\nu$ is the kinematic viscosity of the fluid and $k_s$ is the equivalent sandgrain roughness, a hydraulic length-scale defined by Nikuradse (3, 5). Seemingly minor changes to the geometry and spacing of roughness elements can produce drastically different drag effects (3). Additionally, although effective due to their low-cost Reynolds averaged Navier–Stokes (RANS) turbulence closure models have been shown to provide variable results for quantities of interest for simpler roughness configurations, despite resolving the roughness (6). A next level of complexity involves spatial changes in roughness, which trigger the growth of a near-bed region of transient flow adjustment called an internal boundary layer (IBL) (7–10). There are many applications that involve turbulent flows encountering complex and spatially varying roughness, and where it is critical to know the boundary stress—from wind turbines, to aircraft wings, to marine infrastructure (3, 11).

We live within the atmospheric boundary layer (ABL), the roughly 1-km thick (${\delta }_{\mathit{ABL}}\sim {10}^3$ m) surface layer where flow interacts with topography and human-built structures (12–14). Due to the large length scales and highly heterogeneous roughness involved, ABL flows are highly turbulent with friction Reynolds numbers $R{e}_{\tau }\equiv u_{\tau }{\delta }_{\mathit{ABL}}/\nu \sim O({10}^6-{10}^7)$. Step changes in roughness occur in many places and have important consequences. Public health in cities is influenced by the interaction of ABL flows with buildings and the associated fluxes of soot and heat carried by those flows (15, 16). Effluxes of CO₂ and dust from landscapes—critical components of the climate system (17, 18)—are also influenced by roughness transitions; sea to land, agricultural field to forest, or flow encountering mountains and dunes (19, 20).

Fluxes and exchange rates of CO₂, dust, water vapor, and heat are quantified by a global network of Eddy Covariance towers (20, 21). These towers offer measurements in a single spatial location, and are often placed in areas of heterogeneous roughness, which can cause measurement uncertainty (21). Examples of their use include monitoring the removal of CO₂ from the atmosphere by forests (20, 21), or the efficacy of farming practices aimed at reducing emissions (22). The formation and evolution of aeolian landscapes are governed by the feedback between wind and topography through sediment transport (23). Our previous work has shown that sand dunes can trigger IBL formation and in turn the developing IBL controls the spatial patterns of dune migration (24), with cascading impacts on vegetation growth and the water balance in arid environments (25–27). In their recent review paper, Bou-Zeid and colleagues (12) categorized ABL flows based on the complexity of roughness, and noted that the category of irregular, heterogeneous roughness “remains understudied, and a formal approach to understand the complex flow patterns over such surfaces and their regionally averaged characteristics is critically lagging.” Resolving the structure and dynamics of the IBL, which develops from spatial changes in roughness, is hard to do with observations alone. Collecting time and height-resolved flow data is expensive and technically challenging, making spatial coverage sparse (14, 28). Many field studies rely on at-a-point time series data, using Taylor’s hypothesis to shift observations to the spatial domain (29); this approach is questionable for the highly nonuniform flow conditions of IBLs, as many downstream stations are required to elucidate spatial variation due to its growth (10, 13, 30, 31). More, natural ABL flows are nonstationary and often influenced by buoyancy effects (32, 33), making it difficult to isolate the influence of roughness. Existing analytical formulas for describing the development of an IBL do a reasonable job of predicting the time-averaged scale of IBL thickness (7, 34–39). Schemes for computing the velocity profile within the IBL—and the resultant boundary stress—however are limited to regular, homogeneous surfaces (12). Due to these challenges, large-eddy simulation (LES) has emerged as the tool of choice for examining spatially resolved mesoscopic dynamics of the ABL (40). LES parameterizes smaller-scale (inertial) turbulence while resolving larger eddies, reducing computational cost, which allows for the exploration of much higher Reynolds numbers than direct numerical simulation (41–43). Using LES for ABL simulations over natural topography, however, can be prohibitively expensive computationally (40); this means that compromises must sometimes be made. For example, the immersed boundary method (IBM) is often chosen to approximate the solid surface (40, 44). In addition, coarse computational meshes may limit flow resolution in the critical near-bed region, and field data are often insufficient to validate model choices.

This background leads us to ask: what parts of the IBL development over a natural roughness transition are site-specific, and what of the development is general; similarly, what about the trends in the boundary stress? Here, we use LES to examine the flow across a natural roughness transition at White Sands National Park: sand dunes that emerge abruptly from a smooth playa. While previous LES studies have examined the variation in surface winds over individual dunes and dune clusters (45, 46), they did not examine an entire dune field. More, we undertake a uniquely large-scale study which directly resolves a heterogeneous topography at high resolution, and we have rare field data to validate against. We choose White Sands because it has been proposed that IBL development, triggered by the dunes, drives a downwind change in boundary stress that controls the migration of the dunes themselves—with knock-on consequences for the hydrology and ecology of the region (24–27). White Sands is also a significant source of dust in the region (47, 48). Our previously collected lidar velocimetry has shown how diurnal forcing drives a daily rhythm of near-surface flow due to buoyancy effects (33), and our analysis of high-resolution topographic data has documented a spatial trend in dune migration that is consistent with IBL development (24, 26). We perform a numerical experiment in which a steady and neutrally buoyant flow is introduced over White Sands topography (Fig. 1A). This allows us to isolate the effect of spatially varying roughness on IBL development over $\sim$6 km, and to elucidate the nature of the hypothesized coupling between topography and boundary stress at the mesoscopic scale. Our simulation uses a surface-conforming mesh to directly resolve topography, and a highly resolved and large numerical domain to capture a wide range of turbulence scales. Steady flow simulations reproduce the observed time-averaged velocity profile of wind over the smooth playa surface. Simulated IBL thickening downwind of the roughness transition is consistent with classic scaling models; however, changes to the flow are not as smooth or abrupt as these classic models and correlations, which are used to predict the development of the IBL and boundary stress response, would suggest. The flow responds continuously to changes in the spacing and geometry of dunes throughout the field, and the resulting boundary stress, ${\tau }_b$, is inconsistent with simplified composite schemes. Nevertheless, the vertical structure of turbulence within the developing IBL exhibits a robust self-similar structure. Simulated patterns of boundary stresses explain the downwind slow-down of dunes previously reported (24, 26), and reveal how mesoscale turbulent structures grow across the entire 6-km dune field. Our study shows the profound influence of complex roughness geometry on boundary stress, and how this can be isolated from other complicating factors in ABL flows using LES. These findings have relevance for other roughness transitions in natural ABL flows.

Fig. 1.

Topography and numerical setup. (A) Topographic lidar scan of White Sands National Park Dune Field, New Mexico, annotated with LiDAR wind velocity measurement locations (magenta triangles) and simulation domain (magenta dot-dashed box). (B and C) Topographic scans representative of the roughness levels at the velocity measurement locations upstream (B) and downstream (C) of the roughness transition. (D) Setup of the computational domain, with a synthetic turbulent inflow at the inlet, a sponge outflow at the outlet, symmetry conditions on the side and top walls, and an algebraic wall model is applied to the bottom wall, which is synthesized from the topographic scan of the dune field. Dimensions of the domain are given (orange). (E) The LES mean streamwise velocity profile, normalized by the freestream velocity from the LES (blue line) at the upstream field observation location is compared to the time-averaged experimental data with 5% measurement uncertainty (black, red-filled circles). The FATE data are normalized with the freestream velocity of the experiment, and the wall elevation is normalized by the height of the domain, $h$. The Inset shows a similar comparison of the downstream field data and the LES mean velocity profile on the stoss side of a dune representative of the downstream FATE data location.

Results

Field Data.

The field data used for this study have been extensively described elsewhere, and so are only briefly outlined here. Open-source topographic data (49) for White Sands were gridded at 1 m spatial ($x-y$) resolution, and have a vertical ($z$) resolution of $\sim$0.1 m (24). Topography begins ($x=0$) on a smooth playa surface known as the Alkali Flat, and rises relatively rapidly ($x=1.8$ km) into a region we consider to be the start of the roughness transition where large transverse dunes abruptly emerge ($x\approx 1.9$ km). Within a kilometer of this transition, transverse dunes break up into isolated and heterogeneous barchan dunes, whose migration rate and amplitude decline gradually over several kilometers until the dunes are immobilized by vegetation (25–27). Flow velocity data come from the field aeolian transport events (FATE) campaign, collected from a fixed position on the smooth playa upwind of the dune field, with additional data collected on the stoss side of a dune downstream (33) (Fig. 1A–C). A Campbell Scientific ZephIR 300 wind lidar velocimeter collected vertical velocity profiles every 17 s over approximately 70 d during the spring windy season of 2017 at White Sands, with a vertical resolution of 10 log-spaced bins from $z=10$ m to $z=300$ m above the surface and an additional point at $z=36$ m. Due to stratification effects, night-time winds produce a nocturnal jet aloft that skims over a surface layer of cool stagnant air (33); as a result, boundary roughness effects and sand transport are suppressed at night. We average over all daytime measurements, a 12-h window from 06:00 to 18:00 local time, to produce a time-averaged daytime velocity profile, that is used to validate the inflow conditions for our LES study.

Numerical Setup and Validation.

We perform wall-modeled LES (WMLES) using the CharLES code from Cascade Technologies (Cadence Design Systems), which has been used for many complex flows (50, 51), including studies of ABL flows with urban buildings (52–54), but has yet to be deployed in an ABL study with such complex, spatially varying heterogeneous roughness. CharLES is an unstructured grid, body-fitted finite-volume LES flow solver (Materials and Methods). We numerically analyze a neutrally buoyant and steady ABL flow over an 8.6- by 0.5-km domain of the White Sands topographic data (Fig. 1D), oriented in the direction of dominant winds and dune migration ($\sim$15 degrees N of E). The length of the domain is chosen to capture the mesoscopic scale of IBL development, and the width is chosen to be much larger than an individual dune. Although the surface of the dunes at White Sands have an inherent roughness from the sandgrains, they are represented as smooth in our simulations. Prior field studies at White Sands have estimated the ABL thickness, ${\delta }_{\mathit{ABL}}$, to fluctuate daily between $O({10}^2-{10}^3$ m) (24, 33). The height of the domain is chosen to be $h=$1,000 m. The height of the ABL is chosen to be $300$ m, which lies in the middle of the previously observed range, and is the highest available point with data collected in the FATE campaign; no field data are available above this elevation for validation. Simulations presented have a resolution of $0.75$ m closest to the boundary, and $28$ m in the outer region of the flow (Materials and Methods). A sensitivity analysis confirmed that our results are insensitive to domain size and resolution choices (SI Appendix, Figs. S3–S5). To minimize numerical effects, we deploy a numerical sponge outflow condition (55, 56) and at the top and sidewalls we use a symmetry boundary condition. A synthetic turbulent inflow generation based on digital filter techniques (57) is implemented at the inlet, and the numerical domain of the smooth Alkali Flat is extended to ensure that the inflow achieves a uniform condition before encountering the roughness transition (SI Appendix). On the Alkali Flat and dune field we employ an algebraic wall model, which solves the simplified boundary layer equations and only accounts for wall-normal diffusion (58). Simulations are run for over 1,000 large-eddy turnover times, $T\equiv {\delta }_{\mathit{ABL}}/U_{\infty }$ (SI Appendix), to achieve steady conditions and allow convergence of various time-averaged flow quantities presented below.

Streamwise, spanwise, and wall-normal coordinates within the domain correspond to $x$, $y$, and $z$, respectively, and have instantaneous velocity components $U$, $V$, and $W$. Using the Reynolds decomposition, $U=⟨U⟩+u$, instantaneous velocities may be decomposed into time-averaged (bracketed) and fluctuating (lower-case) quantities (2). Averaging over the spanwise direction is represented with barred notation, i.e., $\overline{·}$. Quantities with normalization based on friction velocity and kinematic viscosity are signified by ${·}^{+}$. We denote streamwise location in relation to the roughness transition with $\widehat{x}=x-x_0$, $x_0$ being the approximate location of the start of the dune field. The FATE data allow us to validate simulation results following the procedure outlined in Hu et al. (58); such validation is rare for ABL flows, given the limited field data (59). We first check that the inflow conditions have reached a fully developed state, matching a canonical zero-pressure-gradient turbulent boundary layer, by comparing the skin friction coefficient to the momentum thickness Reynolds number, $R{e}_{\theta }$, leading up to the transition (SI Appendix, Fig. S1). We then test the validity of our model choices by comparing the time-averaged horizontal velocity profiles, $⟨U⟩$, of our simulation and the observations of FATE on the Alkali Flat, normalized by their respective freestream values, $U_{\infty }$. The two agree over the entire measured elevation range to within 5% (Fig. 1E). This is remarkable, considering that the field data average over nonstationary forcing and buoyancy effects that are not modeled in the simulation. Additionally, we compare the FATE data to the time-averaged horizontal velocity profile at a downstream location on the stoss side of a dune, normalized by a local value of $U_{\infty }$. We find good agreement to 5% over the elevation range (Inset in Fig. 1E). The downstream data were collected a year after the data in the Alkali Flat, so magnitudes of velocity are expected to be different, while profile shapes in each location should remain the same, justifying the use of local freestream velocity values. Overall, this agreement indicates that treating the ABL flow at White Sands as steady and neutrally buoyant is appropriate for describing time-averaged behavior.

Characterization of the IBL.

With the validation in hand, we now use the simulations as a numerical experiment to examine how the IBL would develop due to the roughness transition under a steady and neutrally buoyant flow. The qualitative flow behavior we observe is consistent with expectations and previous work (13). At the location where dunes emerge ($\widehat{x}=0$), there is an increase in turbulence and a shift of the high-velocity region farther from the bed (Fig. 2A). We observe a gradual thickening of the perturbed flow region indicating a developing IBL. To characterize IBL growth we implement the method of Li et al. (30), which uses streamwise variations in the streamwise turbulence intensity $⟨uu⟩$, to define the thickness of the IBL, ${\delta }_{\mathit{IBL}}$:

$$\begin{array}{c}\hfill \mathrm{\Delta }[\frac{⟨uu⟩}{U_{\infty }^2}]/\mathrm{\Delta }[{log}_{10}\left(\frac{\widehat{x}}{{\delta }_{\mathit{ABL}}}\right)]\to 0.\end{array}$$ [1]

Fig. 2.

Development of the IBL. (A) Instantaneous streamwise velocity in the spanwise center plane, flow going from left to right. (B) Measured values of ${\delta }_{\mathit{IBL}}$ (black, multicolored diamonds) with the power-law fit (blue line) given by $0.29{(x/z_{02})}^{0.71}$, and a center-line profile of the dune field (black line). The dashed line indicates the height of the atmospheric surface layer (ASL) relative to the ABL height.

This expression defines that, for successive downwind locations, the wall-normal height in which the normalized value of $⟨uu⟩$ divided by the normalized distance between stations tends toward zero is equal to ${\delta }_{\mathit{IBL}}$ at the upstream streamwise station (SI Appendix). For our simulations, we choose a threshold value of ${10}^{-4}$ to represent convergence toward zero in Eq. 1 (SI Appendix, Fig. S8). We verified that our results are insensitive to the choice of method for defining the IBL (SI Appendix, Fig. S10 and Tables S2 and S3). Ten velocity probing stations, logarithmically spaced in the downwind flow direction from $\widehat{x}=50$ m to $\widehat{x}=$5,750 m (SI Appendix, Fig. S2), were placed to capture the growth of the IBL (Fig. 2B). It is common in atmospheric flows to represent the surface by the roughness parameter, $z_0$, and to characterize a roughness transition as an abrupt change in $z_0$, $z_{01}\to z_{02}$, modeling IBL growth downwind of this transition as a smooth and monotonic function (7, 34). Gunn et al. (24) identified the characteristic roughness parameters for the Alkali Flat and the dune field to be $z_{01}={10}^{-4}$ m and $z_{02}={10}^{-1}$ m, respectively. Following previous work (10, 13) we fit simulation data with a power law relation, ${\delta }_{\mathit{IBL}}/z_{02}=a_0{(x/z_{02})}^{b_0}$, and determine $a_0=0.29$ and $b_0=0.71$. This observed thickening of the IBL, induced by the smooth $\to$ rough transition associated with dunes, is consistent with classic scaling models [SI Appendix, Fig. S9, (7, 8, 34–39)], and the values previously inferred for White Sands IBL growth based on observations of dune dynamics (26). However, the downwind growth of ${\delta }_{\mathit{IBL}}$ in our simulations is not as smooth as simplified treatments would suggest (Fig. 2B); there are fluctuations superimposed on the general trend. In fact, the second rise in ${\delta }_{\mathit{IBL}}$, that begins around $\widehat{x}=$3,000 m, coincides with a subtle but persistent topographic rise underlying the dunes that was previously identified (24, 60). Observed changes in flow across the roughness transition (Fig. 2A) are not as abrupt as most models assume. The flow appears to respond continuously to changes in the spacing and geometry of dunes throughout the dune field.

Near-Wall Implications of the Roughness Transition.

Sand transport in dune fields is driven by the near-surface winds. In particular, sediment transport equations relate sediment flux, $q_s$, to the local boundary stress, ${\tau }_b$, in excess of the entrainment threshold, ${\tau }_c$; commonly used equations have the form $q_s=K\sqrt{{\tau }_c}({\tau }_b-{\tau }_c)$, where $K$ is a parameter related to sediment properties (24, 61–63). The presence of dunes is known to cause spatial variations in ${\tau }_b$ due to speed up and slow down of near-surface winds, which in fact drives the stoss-side erosion and lee-side deposition, respectively, that migrates dunes (23, 64–67). We examine spatial ($x$) variations in ${\tau }_b$ along the centerline of our model domain over the length of the modeled dune field (Fig. 3A). The first-order observation is that topography and boundary stress covary as expected; dune crests are regions of high stress due to speedup, where ${\tau }_b$ is as much as four times as large as shielded troughs. To examine any systematic downwind change in ${\tau }_b$ that results from IBL development, we must average the variations in stress over individual roughness elements (dunes). Here, we perform a spanwise averaging over the model domain, ${\overline{\tau }}_b$, in order to suppress the contribution of individual dunes and enhance the signal of the mesoscale IBL pattern (Fig. 3A). Models predict that there should be a spike in ${\tau }_b$ at the location of a smooth $\to$ rough transition, followed by a gradual stress relaxation as the IBL develops downwind of the transition [SI Appendix, Fig. S11, (68)]. The observed pattern in our simulations, however, is more complex and subdued than the idealized models. Starting from the roughness transition ($\widehat{x}=0$ m), we use a moving average of $⟨{\overline{\tau }}_b⟩$ (Fig. 3A) to better display its gradual increase downwind over the first $\approx$1.5 km of the dune field. This may be because the dunes are superimposed on an underlying topographic ramp; i.e., the spanwise-averaged elevation rises significantly over the first 1 km of the dune field (Fig. 3A). In addition, dunes grow in size over the first 1 km of the dune field. Together, these factors likely drive an increase in boundary stress over this region. After $\widehat{x}=1.5$ km, $⟨{\overline{\tau }}_b⟩$ slowly, if irregularly, relaxes over several kilometers, resembling the spanwise-averaged elevation—even though local stress peaks on individual dunes continue to be large. This gradual decline in $⟨{\overline{\tau }}_b⟩$ must be the result of the developing IBL. This simulated pattern is consistent with the measured decline in time-averaged 10-m wind speed reported by Gunn et al. (24) from three meteorological towers along a transect at White Sands (their figure 2). We cannot directly compare simulation results to sediment flux determined from dune migration. This is because our simulations use daytime-averaged wind conditions—which produce boundary stress values that are less than the entrainment threshold—whereas sand transport at White Sands only occurs (on average) for several hours per day during the peak windy season (26, 33). Nevertheless, the simulated reduction in ${\tau }_b$ due to IBL development is compatible with the observed decline in sand flux of about a half over the first $\sim$6 km of the dune field (24, 26).

Fig. 3.

Changes to the near-boundary characteristics of the flow due to the development of the IBL. (A) Evolution of the centerline time-averaged boundary stress normalized by the boundary stress upstream of the roughness transition (orange line) and the spanwise- and time-averaged boundary stress (black, blue-filled triangles). A moving average (black line) is applied to the spanwise- time-averaged boundary stress to better represent its trend. An outline of the centerline profile of the dune field (gray shaded area) is given with spanwise mean dune heights (black, gray-filled circles) and RMS values (gray bar-lines). (B) Instantaneous streamwise velocity fluctuations at the first off-wall cell ($z/{\delta }_{\mathit{ABL}}\approx 0.001$), projected onto the surface of the dune field. (C) A wall-parallel plane at $z/{\delta }_{\mathit{ABL}}=0.1$ showing instantaneous streamwise velocity fluctuations. Flow is moving from left to right in (B and C).

Even though simulated IBL growth roughly follows classic scaling behavior, the observed boundary stress pattern does not. In particular, a common parameterization used for estimating ${\tau }_b$ in idealized IBL models overpredicts the stress response to roughness changes for White Sands [SI Appendix, Fig. S11, (68)]. This suggests that heterogeneity of roughness, and the sensitivity of turbulence to that roughness, produces a first-order departure from theory developed for idealized conditions. We look to the structure of the flow across the developing IBL—in particular, the near-bed velocity fluctuations—to better understand how the simulated downwind changes in boundary stress occur. We observe a systematic growth in the scale of coherent turbulent structures, that coincides with the growing IBL (Fig. 3B). Initially, the length of these structures scale as the ASL height, which correlates well with previous observations of low-speed streak length in the log-layer (69, 70). Importantly, the initial length of these low-speed streaks indicates the flow was a fully developed turbulent boundary layer prior to entering the dune field; this implies the growth of structures downstream from the roughness transition is a result of the growing IBL, and not further development of the turbulence from the inflow. Qualitatively similar behavior is observed at other wall-normal elevations within the IBL (Fig. 3C), where an absence of turbulence exists outside of the IBL suggesting that IBL growth may set the scale of growing turbulent structures within.

Self-Similarity of Turbulence within the IBL.

The IBL acts as a mechanically distinct feature, which delineates the portion of the flow that retains a memory of the upstream wall condition, and the part that begins to gradually adapt to the new condition. Previous work on a rough $\to$ smooth transition revealed that the IBL may “shield” the outer region of the flow and that as the flow progresses downstream the spectral energy contained outside the IBL is lost (31). Through analysis of the Reynolds shear stress, $⟨uw⟩$, a strong indicator of turbulence production, it becomes clearer just how much the IBL shields the outer region from the momentum flux induced by the roughness (Fig. 4A). Gul and Ganapathisubramani (13) showed that the IBL height, ${\delta }_{\mathit{IBL}}$, corresponds to the location in the flow where the Reynolds shear stress diminishes to (near) zero for a smooth $\to$ rough transition (their figure 1F). We examine the evolution of $⟨uw⟩$, as a function of wall-normal position, on the Alkali Flat and at multiple stations downstream of the roughness transition. As the flow progresses past $\widehat{x}=0$, we see that the region of elevated Reynolds shear stress thickens (Inset in Fig. 4B). For the smooth Alkali Flat, the region associated with elevated Reynold shear stress should be the ASL, ${\delta }_{\mathit{ASL}}$, which is typically considered to be roughly 1/10 the thickness of the ABL (71–73). Using our assumed ${\delta }_{\mathit{ABL}}=300$ m, we estimate ${\delta }_{\mathit{ASL}}=30$ m. This value is comparable to the estimated ${\delta }_{\mathit{ASL}}=60$ m determined from observations in the well-studied desert of Western Utah—an environment similar to White Sands. Normalizing $z$ by the relevant length scale $\widehat{\delta }$, ${\delta }_{\mathit{ASL}}$ on the Alkali Flat and the first dune station, and ${\delta }_{\mathit{IBL}}$ for each downwind station, we find a decent collapse of the wall-normal Reynolds shear stress profiles, which have been normalized by the upstream friction velocity (Fig. 4B), and that $⟨uw⟩/u_{\tau ,0}^2$ approaches zero at roughly $z/\delta =1$. Two downstream stations depart from the general collapse; we attribute this to the significant fluctuations in IBL height around the overall downstream trend. Nevertheless, the general pattern we observe is a self-similar Reynolds shear stress profile within the IBL, and that the height of the IBL corresponds to the location where turbulence production becomes negligible.

Fig. 4.

Self-similarity of turbulence within the IBL. (A) Schematic of a smooth $\to$ rough transition. Flow is from left to right, and the IBL (dashed line) forms at the interface, $x_0$, delineating the transfer of momentum by the new surface and the outer flow region. The blue pathlines represent the turbulent flow within the ABL, and the green pathlines represent flow within the IBL, which do not necessarily originate within. The added turbulence further from the wall within the IBL is reflected by the thickening of $⟨uw⟩$, and the negative $R_{\mathit{AM}}$ peak can be seen by the interactions between flow at the edge of the IBL and the ABL interface. (B) The Reynolds shear stress, normalized by average upstream friction velocity, through the dune field, with wall-normal location $z$ normalized by the relevant length scale $\widehat{\delta }$, where $\widehat{\delta }\equiv {\delta }_{\mathit{ASL}}=30$ m for ${\widehat{x}}_{-1}$ (Alkali Flat station) and ${\widehat{x}}_1$ (first dune station), and $\widehat{\delta }\equiv {\delta }_{\mathit{IBL}}$ for all other downstream stations, determined using methods described in the text. Colors correspond to locations indicated in Fig. 2B. The Inset shows the same data, not normalized. (C) Profiles of amplitude modulation (AM) coefficients where $z$ is normalized in the same manner as B; the Inset shows same data, not normalized. Colors and locations are the same as (B).

Prior work has demonstrated that the large-scale (low-frequency) motions that exist in the outer region of the boundary layer can influence the near-wall, small (high-frequency) scales (29, 74). This influence may be quantified using an AM correlation coefficient, $R_{\mathit{AM}}$, following (74):

$$\begin{array}{c}\hfill R_{\mathit{AM}}(z)=\frac{⟨u_L^{+}(z,t)E_L(u_s^{+}(z,t))⟩}{\sqrt{⟨{u_L^{+}(z,t)}^2⟩}\sqrt{⟨{E_L(u_s^{+}(z,t))}^2⟩}}.\end{array}$$ [2]

Here, $u_L^{+}$ is the large-scale component of the velocity fluctuations, $u_s^{+}$ is equivalently the small-scale component, and $E_L(u_s^{+})$ represents the filtered envelope of the small-scale velocity fluctuations. The process of AM is outlined in Mathis et al. (74) and is included in more detail in Materials and Methods and SI Appendix, but a brief overview is presented here. A velocity signal, taken to be $u^{+}$, is decomposed into a large-scale and small-scale component using a spectral cutoff filter (Materials and Methods and SI Appendix, Fig. S12). Next, a Hilbert transformation is conducted on the small scales to create an envelope of the signal, that is then subjected to an additional filtering step. The equivalent of a Pearson coefficient (2) is created using the large-scale signal and the filtered envelope of the small-scale signal to find $R_{\mathit{AM}}$ (Materials and Methods).

We calculate a single-point $R_{\mathit{AM}}$ as a function of wall-normal distance at multiple streamwise locations, both preceding and following the roughness transition (Fig. 4C). We first examine the vertical $R_{\mathit{AM}}$ profile over the smooth Alkali Flat. The most notable feature is the large negative correlation, which occurs at a wall-normal elevation of $z\approx 25$ m. Mathis (74) suggests that this is the result of intermittency arising in the outer region of the boundary layer, due to shear with the fluid above the ASL. Indeed, the $R_{\mathit{AM}}$ profile in our simulations is in good qualitative agreement with their experimental observations (74). In their study, which included wind tunnel data and observations from an atmospheric flow, they found that the negative peak occurs between $z/\delta =0.7$ and $z/\delta =1.0$, where $\delta \equiv {\delta }_{\mathit{ASL}}$ was used for the atmospheric case. Using ${\delta }_{\mathit{ASL}}=30$ m for our data suggests a relative height of $z/{\delta }_{\mathit{ASL}}\approx 0.83$, within the range of the results of ref. 74. The same qualitative structure of the $R_{\mathit{AM}}$ is seen across the dune field. The magnitude of the prominent negative correlation is more or less preserved; however, its location, $z$, systematically shifts (Inset in Fig. 4C). At the start of the roughness transition (near $\widehat{x}=0$), the negative peak appears closest to the bed; moving downwind (increasing $\widehat{x}$), the peak consistently shifts away from the bed toward higher elevations (Inset in Fig. 4C). This suggests that the migration in this peak is set by the growing height of the IBL itself. We normalize the wall-normal height following the same procedure used for the Reynolds shear stress profiles and find that the $R_{\mathit{AM}}$ profiles collapse onto a reasonably similar master curve. These results suggest that modulation of large-scale atmospheric turbulence within the IBL occurs in a self-similar manner, that scales with IBL height.

Discussion

For most studies examining IBL development in response to changes in roughness, the idealized analytical solutions derived from classic scaling arguments are still the go-to model. While such closed-form solutions are convenient, they are inadequate for determining the near-bed turbulence and boundary stresses in ABL flows that are critically important for heat and water flux (evaporation), CO₂ (eddy covariance), dust emission, and sediment transport. Our study shows how relaxing the assumption of a step change in roughness, and explicitly modeling natural heterogeneous topography, is essential for capturing the mesoscale flow behavior in the ABL that is of central importance for the evolution of landscapes and the activities of humans living within them. By resolving ABL turbulence using wall-modeled LES, while carefully treating inlet/boundary effects and using a surface-conforming mesh for the topography, we were able to produce simulated flows that were validated against field lidar velocimetry data. Our results are consistent with what has been measured and inferred about IBL dynamics at White Sands from previous studies, while providing qualitative and quantitative insight into the mescoscopic feedback between flow and form that cannot be seen from field data alone.

While resolving large-scale fluid motions and heterogeneous boundary roughness is clearly important, our results also suggest that there may yet be some generic behaviors in the spatially growing IBL. In particular, the self-similar profiles of Reynolds shear stress and AM within the developing IBL indicate that, when present, the IBL is the relevant mesoscopic length scale governing turbulence in the near-surface flow. These findings suggest that if the IBL thickness is known, then aspects of the turbulence structure within it can be predicted. Qualitatively, the growing size of large-scale coherent flow structures downwind of the roughness transition coincides with the growing thickness of the IBL. It is sensible that the mechanically distinct IBL somehow sets the scale for the largest eddies contained within it. This last point warrants further study.

It is important to make clear the limitations of our present study—in their application to White Sands, and for the potential extrapolation to other settings. There are two important factors in wind dynamics that were neglected here. The first is buoyancy; the FATE campaign (33) showed that nonequilibrium buoyancy effects drive the sand-transporting winds at White Sands, and that typical frameworks like Monin–Obukhov similarity theory cannot account for the strength of convection effects on surface winds. The agreement of our simulation results with the time-averaged daytime winds from FATE indicates that i) time averaging removes the buoyancy effect, and/or ii) winds in the near-surface layer are sufficiently mixed by turbulence that buoyancy effects can be neglected—at least in the lower 10s of meters (71–73). We were able to isolate the influence of roughness on driving relative changes in the boundary stress; this suggests that the roughness effect is, to first order, decoupled from the buoyancy effect. However, the magnitudes of our simulated stresses are lower than the sand-transporting winds at White Sands, which can only be resolved with transient and nonlinear buoyancy effects. The second neglected factor is nonstationarity of the flow. Winds in the ABL, including White Sands, are highly variable in magnitude and direction. Because desert sand dunes evolve over decades (75), a steady flow approximation is reasonable for examining feedback between dune roughness and the near-surface winds. This approximation may not be acceptable in other situations, however, where event-scale weather phenomena are of interest.

Our study demonstrates how a previously proposed mesoscale coupling between flow and form (26) arises due to IBL development. The coevolution of dunes and the IBL likely governs landscape evolution in other desert and coastal environments. The emergence of larger-scale structures indicates that modeling dunes in isolation, as is typically done, will not produce the correct stress profile. We suggest that a useful next step will be to examine how the presence of the IBL influences the wind stress profile over individual dunes. The mesoscopic interactions of ABL flows with heterogeneous roughness are also of central importance for cities, forests, and the ocean–land interface where wind may be carrying aerosols, dust, or wildfire smoke (17, 18, 76–78). The transport, deposition, or bypass/ejection of these particulates from landscapes depends on the interaction between flows within the developing IBL and the flow outside of it. This is especially critical when considering the location of Eddy Covariance towers, as their placement near or within a developing IBL will introduce uncertainties into the data collection; this may inadvertently influence farming practices or policy decisions (21, 22). This consideration is not limited to Eddy Covariance towers; the placement of many types of measurement tools, such as the LiDAR used in the FATE campaign to collect velocity data, or traps used to measure sediment fluxes, will be impacted by their proximity to roughness transitions. More informed positioning of these measurement tools can be made with better understanding of the self-similarity that exists after a roughness transition. Recent improvements for modeling particles in LES (79) could be introduced to simulations like ours, to track how the IBL modulates the transport of aerosols across and out of landscapes. Finally, the lidar velocimetry data that allowed validation of our model is rare. We suggest that carefully deployed field campaigns, coupled with well-resolved LES simulations, can allow researchers and practitioners to create 3D flow fields for many complex environmental flows.

Materials and Methods

Details of WMLES.

In this study, we use CharLES, from Cascade Technologies (Cadence Design Systems), which is an unstructured grid, body-fitted finite-volume LES flow solver. CharLES solves the compressible Navier–Stokes equations in a low-Mach isentropic formulation, using a second-order central discretization in space, and a second-order implicit time-advancement scheme (53). The solver has been deployed for many high Reynolds number turbulent flow cases, including LES over the Japanese Exploration Agency Standard Model (50) and ABL flows over buildings (52, 53), as well as wall-bounded flows with roughness (80). The entire code is written in C$++$ and deploys Message Passing Interface for parallelization. Part of this work used Anvil at Purdue University through allocation MCH230027 from the Advanced Cyberinfrastructure Coordination Ecosystem: Services & Support (ACCESS) program, which is supported by NSF grants #2138259, #2138286, #2138307, #2137603, and #2138296 (81).

CharLES uses an isotropic Voronoi meshing scheme which allows for highly accurate body-fitted meshes suitable for complex, irregular geometries. Mesh design requires an outer, far-field grid spacing, ${\mathrm{\Delta }}_{\mathit{FF}}$, in which subsequent refinement levels are built off. This is based on ${\mathrm{\Delta }}_{\mathit{FF}}/2^n$, where $n$ is the desired number of refinement levels (80). More details on the meshing technique may be found in Lozano-Durán et al. (51).

The grid used for the study contained approximately $85\times {10}^6$ control volumes, with ${\mathrm{\Delta }}_{\mathit{FF}}=28$ m and a finest cell-spacing of ${\mathrm{\Delta }}_{\mathit{min}}=0.75$ m. In viscous units, the near-wall spacing is ${\mathrm{\Delta }}_{\mathit{min}}^{+}\approx$3,200, due to the high $R{e}_{\tau }$, and is the same for the streamwise, spanwise, and wall-normal grid spacing due to isotropy of the cells. We adequately resolve the height of the ABL with nearly 67 control volumes, and the IBL using between approximately 20 control volumes near the roughness transition, to 46 control volumes at the end of the dune field. A refinement study of the mesh was conducted to ensure convergence of quantities of interest (SI Appendix, Figs. S6 and S7).

The domain is sized to be $8.6\times 0.5\times 1$ km³, using a domain study to ensure the spanwise width of the domain had no effect on the flow, due to the symmetry boundary conditions placed on the side walls (SI Appendix, Figs. S3–S5). The top wall of the domain also deploys a symmetry boundary condition, and we observe no influence on the flow. The inflow and outflow regions were artificially extended to allow for development of the incoming turbulent boundary layer and the placement of a numerical sponge outlet condition. A sponge region is used at the outflow to prevent pressure waves from reflecting back into the domain and causing numerical instabilities (55, 56). The sponge region is not considered in the analysis. To reduce computational cost of the simulation, an algebraic wall model boundary condition (41) is imposed on the topography, where the matching height for the wall model is placed at the center of the first cell. More information on the inflow method to generate turbulence is provided in (SI Appendix).

AM.

For the calculation of $R_{\mathit{AM}}$, we probe for $U$ at the same logarithmically spaced streamwise stations within the dune field, at multiple wall-normal locations. A fluctuating signal, $u^{+}$, is then found at each probe point, and filtered. To conduct the filtering technique, a spectral cutoff filter is deployed. This filter uses a cutoff wavelength ${\lambda }_{x,c}={\delta }_{\mathit{ABL}}$, where ${\lambda }_x\equiv U_c/\omega$ is recovered using Taylor’s hypothesis, selecting the mean velocity at each wall-normal position as the convective velocity (82). The signal is transformed into the frequency domain using the Fourier transform, and the cutoff filter is applied at this stage to get the large-scale features of the flow, ${\widehat{u}}_L^{+}$. The signal is transformed back to the physical space, and the small-scale features are found by subtracting the filtered signal from the raw signal, $u_s^{+}=u^{+}-u_L^{+}$. Next, a Hilbert transformation is conducted on the small-scale signal. This transformed signal is again filtered using the technique described above, and we are left with the filtered envelope of the small-scale signal. The process is repeated for every wall-normal location at every streamwise station.

To find $R_{\mathit{AM}}$ and complete the process described above, long time-series data are required. Generally, for $R_{\mathit{AM}}$ to be converged, it is recommended the flow experience 5,000 $\le T{U}_{\infty }/\delta \le$14,000 (74, 82), where $T{U}_{\infty }/\delta$ is a nondimensional large-eddy turnover time. For our calculation, we define $\delta \equiv {\delta }_{\mathit{ABL}}$. Due to the stringent computational cost of this analysis, it is unfeasible to conduct the minimum suggested turnover times, so only $\sim$1,000$T{U}_{\infty }/\delta$ were completed for this analysis. We regard this number as a reasonable time for convergence of $R_{\mathit{AM}}$, especially in the context of ABL flows, and we show a lower threshold of $T{U}_{\infty }/\delta$ may be allowable (SI Appendix, Fig. S13). Given this, the results presented are a means to represent what might be expected of AM at mesoscopic scales.

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

Simulation data have been deposited in Dune_AM (Available at time of publication). Previously published data were used for this work (33). Link to data at: https://github.com/algunn/dune-impinge.