1. Introduction
Turbulent flows over the forest canopy and forest gaps have been studied in wind-tunnel, field, and numerical experiments, leading to a good understanding of turbulence development mechanisms and a better knowledge of turbulence structures in homogeneous canopies and at forest edges [
1]. A variety of numerical models are now used to simulate wind fields with a reasonable degree of accuracy. Among the different techniques used to simulate wind flows, one of the most fruitful approaches is the Large-Eddy Simulation (LES), which enables explicit computation of the turbulent structures larger than grid size [
2,
3,
4,
5,
6,
7,
8,
9,
10,
11]. This technique has been applied to a wide variety of investigations. For instance, simulations enable researchers to disentangle the mechanisms associated with interactions between canopy and wind flow [
7,
8] and to determine the spatial extent and other important properties of transition zones in heterogeneous canopies [
2,
4,
9]. They enable simulation of seed [
10] and pollen [
11] dispersal. In the context of wildfire, they can be used to develop precursor windflows for physics-based-wildfire-behavior modeling (e.g., References [
2,
3]) or firebrand transport [
12], and even to analyze wind measurement accuracy or representativeness in the context of fire experiments [
13]. Finally, they help understand discrepancies in observations, e.g., for eddy-covariance measurements [
14] or to compute precursor wind flow for wind turbine modeling [
15]. The development of resolved canopy-shear-induced turbulence requires fetch distances that are much greater than the desirable domain size for the studies described above (hundreds to thousands of meters). Promising methods based on stochastic perturbations have been proposed to accelerate this transition [
16,
17], but periodic boundary conditions are still widely used. A periodic domain essentially serves as a Lagrangian window that moves downwind as the shear profile develops and allows the modeler to side-step explicit representation of vast distances (thereby reducing computational resource requirements). The obvious limitation of this approach is that the influences of structures or patterns in the atmosphere that are larger than the domain size are not explicitly represented. Additional steps must be taken to account for the large-scale forcing that counterbalances the momentum sink resulting from vegetation drag in the lower part of the boundary layer. Unfortunately, the existing forcings associated with periodic boundary conditions have some practical limitations under particular configurations. These are detailed below.
Modeling phenomena affected by ambient winds in real events (e.g., reproducing fire experiments [
18], particle or scalar dispersion and transport) requires representing the upwind atmospheric conditions. Nudging and data assimilation techniques have been developed to force atmospheric models with meteorological observations at regional (e.g., References [
19,
20]) or local [
21] scales. At the forest canopy or fire experiment scale, observations are often limited to wind speeds and directions at given heights at a limited number of points. Reproducing the real three-dimensional boundary-layer flows is impossible with such limited description, forcing modelers to make assumptions or simplifications in their boundary and initial conditions leading to somewhat idealized simulations. The Ekman balance [
22], referred to later as EKB, is the theoretical balance between the mesoscale pressure gradient, the Coriolis force, and the vegetation drag. EKB is quite often used in idealized LES of the atmospheric boundary-layer (e.g., Reference [
23]), but far less often for the atmospheric surface layer [
9]. The magnitude and direction of this pressure gradient, generally assumed constant, are set according to the expected geostrophic wind. LES of EKB reported streaks [
23,
24], which are streamwise vortical structures as large as several hundreds of meters, can be observed in the simulated atmospheric boundary-layer [
23,
25,
26]. They build, evolve, and dissipate over periods of several tens of minutes [
26,
27,
28]. With EKB, the wind velocity and direction at a given height close to the canopy depend on vegetation characteristics and the Coriolis parameter, and is only controlled by the geostrophic wind. This is not practical when the reference conditions (wind speed and direction) are only known close to the ground, as in most wildfire or wind energy applications, since the appropriate value for the coinciding geostrophic wind is difficult to determine a priori. Additionally, the rotation with height induced by the Coriolis force complicates the analysis of simulations when vegetation or topography is not homogeneous over the domain. The Coriolis force either supports or destroys the planetary boundary-layer structures, depending on whether their vorticity is parallel or counter parallel to the planetary vorticity [
29].
The Coriolis force is neglected in most simulations of the surface-boundary-layer (e.g., References [
2,
4,
7,
30,
31,
32,
33]), as well as in many simulations of the atmospheric boundary-layer (e.g., References [
34,
35,
36]). In these simulations, a spatially-constant-pressure-gradient force most often compensates the momentum sink induced by surface roughness elements. Several variants of such a pressure gradient are used among studies cited above. The pressure-gradient can either be constant, or evolve with time to keep the mass flow constant. The forcing can be applied to the whole domain or in the upper part only. This type of forcing, later referred to as CPGF, avoids the complexity induced by the Coriolis-induced spiraling, and is generally presented as a relevant representation of atmospheric surface and boundary-layer flows [
34]. However, even if the wind velocity at a given height can be roughly estimated with the law of the wall, it cannot be specified accurately, as for the EKB. Moreover, Reference [
31] reported an unusually-large forcing, due to limited domain height and overestimations of the mean streamwise velocity. Regarding applied studies, the streaky structures simulated with CPGF tend to be persistent and “locked”, when domain extents are less than some tens of kilometers [
36,
37], which is not appropriate in applied canopy flow studies where velocities at specific locations are important, such as wildfire-precursor computations. CPGF is often used in incompressible codes using spectral solvers with a free-slip upper boundary condition. With compressible codes, a frequent alternative to EKB for canopy-flow simulations is the use of a geostrophic wind corresponding to a base state at the top of the domain that drives the flow, in combination with a Rayleigh damping layer (e.g., References [
5,
6]). This type of forcing is typically used for wildfire simulations, in which domain height is generally large with respect to domain length (to include the plume) and the grid resolution is fine near the ground, although it also leads to streaks locking [
2].
This manuscript presents a forcing for capturing the effects of large-scale-pressure-gradient forces that has been developed for wildfire-precursor simulations, to avoid practical limitations of EKB and CPGF. Building on a modified version of CPGF for applied canopy-flow simulations, our approach aims at (1) using a pressure profile closer to the one of the EKB, (2) allowing the forcing of the velocity at a given height, (3) mitigating streak locking in precursor simulations. Here, simulations with various pressure forcing methods are compared to idealized atmospheric boundary-layer simulations, for configurations typically used for precursor computations in wildfire simulations. In addition, the sensitivity to the grid size is investigated in the manuscript, so that the use of the methods to other applications is discussed (canopy flows, idealized boundary-layer research, comparison with observed events or wind turbine simulations).
3. Numerical Experiments
The following numerical experiments aim at testing and comparing wind-flow simulations obtained with different pressure gradient forces in two typical reference configurations. First, a domain with relatively coarse resolutions is used, as an idealized representation of the lower atmospheric boundary-layer (“reference boundary-layer simulation”). A second simulation is carried out at a finer resolution (), in a high domain (615 m), but with a relatively short horizontal extent (512 m), and serves as a typical “wildfire-precursor simulation”. Its setting is a compromise between the resolution (that is fine to represent vegetation shear and flow details near the fire front), the domain height (that is high enough to model the plume development), and computational costs, resulting in domain length generally shorter than typical boundary-layer simulations. In addition, for a more comprehensive understanding of domain effect, a sensitivity to 7 other computational grids is carried out, which are ran with various pressure forcing, resulting in a total of 21 wind simulations.
3.1. Vegetation Scenarios
Simulations are performed over four different canopies (
Table 2). Canopy Can4 is extrapolated from the wind-tunnel study described in Reference [
40], as done in References [
6,
9]. Additional details about these canopies and data collected in the field can be found in the reference paper cited for each scenario (
Table 2).
3.2. Numerical Details
Simulations are done on several domain sizes and various grid spacing (
Table 3). Our reference grid for the
boundary-layer simulation is a 1568 m × 1568 m × 800 m domain, with a horizontal resolution
dx = 8 m and a moderate vertical stretching. The vertical resolution of the lowest cell is 4 m, and the cell height is less than 16 m below 540 m, so that the aspect ratio between horizontal and vertical is kept between 0.5 and 2 in the part of the domain of interest. Our second reference grid is a 512 m × 512 m × 615 m domain with a horizontal resolution of
dx = 2 m and a strong vertical stretching. The resolution of the lower vertical cell is 1.5 m, and the cell height is close to 30 m at 400 m. This set-up, referred to as “
wildfire-precursor simulation”, is typically used in wildland-fire simulations with HIGRAD-FIRETEC and enables a vertical resolution of about 1/10th the canopy height in the surface layer (e.g., Reference [
6]). Its vertical domain extent is similar to a boundary-layer simulation, which is much higher than typical canopy-flow simulations (e.g., Reference [
32]), so that the fire plume can develop in the domain. However, its horizontal extent is much shorter than a typical boundary-layer simulation, to limit computational costs.
Several other grid sizes and resolutions are also used for the sensitivity analysis (
Table 3). Thanks to an adequate choice of stretching parameters, vertical grid spacings are identical below 106 m for all runs with vertical extensions smaller than 615 m, with a first cell on the order of 1.5 m tall, so that these runs only differ by their extent, not their grid spacing. Runs with a domain size of 800 m, however, are less resolved close to the ground (first cell on the order of 4 m, which is about 1/3rd of canopy height) to limit computational cost.
Various conditions are used in the upper part of the domain. Except for CPGF, the boundary-layer simulations with 800 m domain heights use a capping inversion between 550 and 700 m. The temperature increase rate is 0.08 Km
−1 in this layer and 0.003 Km
−1 in the geostrophic layer above 700 [
41]. The wind velocity is forced with the geostrophic wind at the top of the domain. The simulations done with 615-m domain heights use a Rayleigh damping layer, starting at 450 m above the ground, to avoid the reflection of gravity waves and to damp the fire plume, as in most HIGRAD-FIRETEC simulations without topography.
For all runs, initial wind velocities and the geostrophic wind are set using empirical profiles described in
Appendix A, using a target
3 m s
−1 at 40 m, resulting in a geostrophic wind of 6.7 m s
−1. These empirical profiles depend on
LAI and fuel height
h. They use a log profile above 2
h and an exponential profile below
h. Depending on the grid spacing, modeled scenarios are run with timesteps
dt of 0.002 or 0.004 s, with the method of averages applied over 10 small timesteps. A drag coefficient
of 0.26 is chosen for Can1 and Can2 [
9]; whereas, 0.15 is used for Can3 and Can4 [
6].
3.3. Simulation Set
The model is run for a total of 21 simulations (
Table 4), with the EKB using a capping inversion (EKBS) or a damping layer (EKBD), with the constant pressure-gradient forcing (CPGF) and with the four versions of PGF described above. The update period of PGF is
= 200 s. Several domains, resolutions, and canopy types are used.
EKBS_1568x800dx8 and EKBS_3136x800dx8 are canonical boundary-layer simulations. EKBD_1568x615dx4 has similar domain extents, but has a higher resolution in the surface layer and uses a steeper vertical stretching to limit computational costs. CPGF_256x106_dx2 is a canonical canopy-flow simulation, but the extents of the domain are very limited. EKBD_512x615_dx2 is an intermediate configuration between the boundary-layer and the canopy-flow simulations, typical of a precursor-wildfire simulation.
3.4. Data Analysis
Instantaneous vertical wind profiles are calculated by computing the horizontally-average mean wind velocity at a given height, for a given time. Following the evolution of these profiles provides valuable information regarding the convergence and the stability of each simulation. The profiles of several wind statistics (mean velocity, momentum flux, turbulent kinetic energy, the standard deviation of velocity components) are calculated using averages over space (horizontally), and time.
We also compute turbulent spectra. They are derived from the Fast-Fourier Transform of the
u-velocities calculated for different
x positions along the
y-direction and averaged over one hour [
41]. The number of Fourier modes and the wavelengths vary with the physical distance (domain width) and the grid resolution, the smallest wave number being
.
5. Discussion
The vertical profiles of turbulent statistics and turbulent spectra, shown in the present manuscript, enable comparison of major aspects of EKB, CPGF, and PGF and the influences of domain size and resolution when used to simulate wind flows within and above forest canopy for applied studies. The comparison of the different coherent structures is not as detailed as what could be achieved using multivariate methods of statistical analysis, such as the proper orthogonal decomposition (POD). However, in this context, they enable us to (1) clearly demonstrate some limits of EKB and CPGF in this context, (2) examine to which degree the proposed method solves these identified limits, (3) illustrate the consequences of domain size and resolution on main features. These points are developed in the following subsections.
5.1. Limitations of CPGF and EKB Approaches for Flow Simulations within and above Canopies
Our simulations show that the S-shape profile, already observed with CPGF by Reference [
42], occurs in configurations when near-ground drag elements are resolved. Hence, simulated flows with CPGF differs from the expected features of the upper part of the atmospheric boundary layer. Moreover, the magnitude and locking of the streaky structure are clearly stronger with CPGF than with EKB simulations, which can be taken as a reference to idealized atmospheric flow. This finding is in agreement with the results of Reference [
29], who identify a stabilizing role of Coriolis force on streamwise coherent structures. As explained in
Section 2, CPGF only induces a convergence of the spatially-averaged profile, whereas EKB is a stable equilibrium that has a damping effect on persistent lateral heterogeneities. This different nature of the two forcing types explains why streaks are locked when using CPGF, because they can persist and grow over time with a spatially-constant forcing, while they are damped by the EKB.
The EKB enables a rapid convergence of the flow, satisfactory turbulent statistics close to the canopy when the resolution is fine enough, and evolving streaky structures. However, the magnitude of the wind velocity in the canopy neighborhood is only controlled by the geostrophic wind value, which is not practical when working on canopy flows or trying to establish a turbulent profile that matches observations, such as those measured from towers. Even when realistic initial profiles are used, the convergence in the lower part of the canopy can be far from the expected value (2.24 m s
−1 instead of 3 m s
−1), which is a drawback for several applications, particularly for wildfire-precursor simulations. This limitation can eventually be solved by trial and error methods at the price of the user and computational times. However, this process is considerably complicated by the angle of rotation of the wind direction, which also varies with geostrophic wind magnitude, elevation reference, and vegetation type. In the simulations of the present paper, this rotation varies between 25 and 40°. This rotation complicates modeling applications even more, as the direction can vary abruptly near the canopy top. Indeed, wind velocities are so small below the canopy, that the Coriolis and drag forces become negligible compared to the mesoscale pressure gradient [
43]. As a consequence, the rotation between the top and the bottom of the canopy can be important. Although neglected in most canopy-flow or wildfire simulations, such a rotation has been reported in deep and dense canopies [
43], and observed in simulations [
44], in which the wind direction tends to align with the mesoscale pressure gradient below the canopy, i.e., orthogonal to the geostrophic wind [
9]. At coarser scales, simulations over-idealized topography become more complex in the presence of this rotation. Even if the EKB probably leads to more realistic flows within and above canopies than CPGF, this physical framework should not be seen as a fully realistic scenario, since it relies on some strong assumptions, such as the mesoscale pressure gradient taken constantly over the domain. In practice, it is not necessarily constant neither over the vertical extent of the domain, nor along the distance traveled by the flow in a periodic simulation of several hours. Considering how sensitive the rotation is to vegetation and wind velocity, it is likely that its direction strongly depends on local variations of the pressure gradient, which could lead to either overestimation or underestimation of the rotation near the canopy. As an example, the authors of Reference [
9] noticed that the rotation that they observed in their simulations was not present in the field dataset. Another point is the fact that it is unclear to the authors if these steep changes in direction in the lower part of the domain are consistent with the use of periodic boundary conditions when vegetation is heterogeneous. Indeed, the wind direction at a given height can be significantly different inside a forest and in a clearing, for example, leading to a confusing situation. We think that further investigations are required to evaluate the consequences of this constant mesoscale-pressure-gradient assumption in EKB scenarios. For these reasons, EKB forcing has never been used in wildfire or wind turbine precursor computations to the best of our knowledge.
5.2. Applicability of PGF-Methods to Idealized Boundary-Layer, Canopy-Flow, and Wildfire-Precursor Simulations
The PGF methods developed aim at providing a pressure-gradient forcing that avoids the complexity and uncertainty induced by the spiraling of the EKB, but which (1) is based on a more realistic pressure profile than the CPGF, (2) does not lead to locked streaky structures as CPGF, and (3) enables the prescription of a target velocity vector.
PGF1 used a more realistic vertical pressure gradient forcing than CPGF, which is derived from the EKB and enables to avoid the S-shape profile. PGF2 and PGF4 enable the control of velocity magnitude and direction. PGF1 and PGF2 have the same drawback as CPGF in terms of production and locking of strong streaks (compared to the EKB), since the local forcing does not possess the damping features required to suppress the intensification of locked streaks. Therefore, their application should be limited to very long domains, in which locking is limited. For example, Reference [
45] showed that very large streamwise-elongated coherent structures, with a length of the order of tens of the boundary-layer thickness, contribute significantly to an atmospheric boundary-layer simulated with CPGF. PGF3 and PGF4 include a streak-damping or streak-blending mechanism based on a pressure gradient that is not uniform in the crosswise direction, to limit the magnitude of streaky structures and their trend to get locked. A valuable alternative to PGF3 and PGF4 could be to combine PGF1 or PGF2 with shifted boundary conditions, which can mitigate streak locking [
36]. This will be tested in the future. We would like to point, however, that the stabilization mechanism of PGF3 and PGF4 mimics the combination of the Coriolis force and the mesoscale pressure gradient, that acts to speed up or speed down the flow that would be locally slower or faster than the equilibrium value at a given height, contrary to the shifted boundary condition that simply acts as a numerical solution to artificially increase domain length. In the present study, we found that the streaky structure over one hour is similar to the one of the EKB. Also, the turbulence spectra are very similar for PGF and EKB (except for the lowest wavenumbers), suggesting that the contribution of the Coriolis force in the turbulence cascade is probably limited. As a consequence, the wind flows developed with PGF3 and PGF4 methods appear as a non-spiraling version of the canonical EKB simulation. In scenarios in which a rotation with height is desired, but expected to be controlled (for example, prescribing a given angle at a reference height), PGF methods can be adapted to force such a rotation (following Ekman spiral or any other equations providing the desired spiraling).
In our PGF approaches (as for CPGF), a value must be given to the parameter
, the period at which an update of pressure gradient forcing is calculated. In the simulations presented here, we choose
= 200 s as stated in
Section 2. This value was taken because periodic simulations with no pressure gradient led to a decay rate of integrated momentum, which is typically about 5% every 200 s in the lower part of the domain (not shown); therefore, 200 s can be seen as a relevant characteristic time for the action of the large-scale pressure gradient on the mean flow. When elaborating the method, we also tested values 100 and 400 s and found the results to be relatively insensitive within this range of parameter values. However, some high-frequency oscillations of the wind flow with time appear in the lower part of the domain when
is too small (
10 s), because
F is modified much faster than the time required for a change in mean flow. In contrast, when choosing
too large (
2000 s), some slow oscillations of the integrated momentum occur. The value of 200 s was chosen for all simulations presented here, showing that PGF is relatively insensitive to domain size or canopy type.
5.3. Comparison of PGF Wildfire-Precursor Simulations to Observations and Application to Heterogeneous Scenarios
Several authors have already observed that turbulence statistics obtained with LES without pressure gradient, or forced with CPGF or EKB [
4,
5,
6,
9,
33,
46] are close to empirical data. We obtain good agreement with PGF1–4 for experimental datasets corresponding to canopies Can1 to Can4, as illustrated with PGF4 as an example in
Appendix C. This shows that PGF methods do not degrade the predictions near the canopy, which is not surprising as corresponding features are shear-dominated.
Canopies Can2 and Can4 corresponds to heterogeneous landscapes (clearing to forest transition). Using PGF4 as done in
Appendix C remains relevant because the vegetation distribution is constant in the crosswise direction. More generally, forcing the integrated momentum for the streamwise direction only makes sense when the integral of leaf-area density (LAD) in the streamwise direction is nearly uniform along the crosswise direction. Potential issues that may arise from heterogeneous vegetation along the crosswise direction are not specific to PGF and concern any simulations using periodic boundary conditions, whatever the type of forcing. For example, periodic boundary conditions are not appropriate to simulate wind flows blowing parallel to a forest edge, because the mass flow cannot be infinitely faster along some streamwise path (clearing) than others (forest) without breaking the hypothesis of a constant mesoscale pressure gradient in the domain. The challenge associated with domains containing large-scale heterogeneity, can be solved by using the nested domains [
47,
48]. With this technique, an ambient wind flow over a much larger domain with homogenized vegetation and topography (or with heterogeneities that are small compared to domain extents) can be simulated using PGF4. The results of this simulation can then be used as inlet and outlet precursors on a more refined domain with substantial heterogeneous vegetation or topography, as done by Reference [
49]. Another consideration for the application of PGF to heterogeneous landscapes (such as canopy Can2 and Can4) is the question of where the target velocity is defined. More specifically, field data are often collected in an open area. To account for that, the calculation of the integrated momentum presented in
Table 1 can be limited to an appropriate subdomain, for instance, the clearing area in which the collected data is representative, as done by Reference [
49].
5.4. Domain Size and Resolution of Canopy-Flow LES
LES have been applied to a wide range of domain sizes and resolutions, depending on the objectives of the simulations. Our sensitivity study demonstrates that the choice of the set-up is critical regarding the features of interest. The authors of Reference [
45] used LES to simulate Very-Large-Scale-Motions in 25-km domains and showed that 27% of the energy is available in scales larger than 10 km. However, the resolution of such simulations (50 m) is obviously not well-suited to reproduce canopy-flow statistics and wildfire precursors for detailed physics-based models, such as HIGRAD-FIRETEC. In particular, a grid finer than ~1/3 of canopy height is required to reproduce wind profiles inside canopies, contrary to ~1/10th canopy height resolution, for which turbulent structures of size in the order of canopy height can be resolved.
These fine-scale turbulent structures are of great importance in the context of canopy functioning, seed or pollen transport, windthrow, or wildland fires. For these specific applications and for a limited time of simulation (e.g., one hour), it is probably more important to accurately simulate these small structures than the largest ones, which evolve slowly over a short time line. On the other hand, our sensitivity analysis shows that the simulations done with 106-m domain heights fail to simulate structures larger than canopy height, in particular, the turbulent kinetic energy and the momentum flux above the canopy. These simulated configurations proved very helpful to understand the details of shear dominated canopy flow, but larger domains are probably required for applications potentially affected by slightly larger structures, such as wildfires.
When the focus shifts from the surface and canopy winds to those above the upper part of the plume of a large or intense wildfire, much larger horizontal domains must be used, and neglecting Coriolis force or assuming a constant mesoscale pressure gradient is likely, not satisfactory. Instead of running the model under idealized scenarios with periodic boundary conditions, it is better to prescribe ambient variables (potential temperature, pressure, wind velocities) modeled with mesoscale models [
19].
6. Conclusions
This paper’s main objective was to present approaches developed to capture the effects of pressure-gradient forcing for canopy-flow LES simulations, such as those performed with HIGRAD/FIRETEC or a broader class of LES boundary-layer simulations. In such simulations, the focus resides in the near-surface flow features, which is especially the case of wildfires. These approaches (PGFs) overcome some drawbacks of existing forcing (Ekman balance EKB and constant pressure gradient forcing CPGF), including the lack of control on wind speed and direction close to the ground. Our analysis demonstrates that the use of CPGF with high domain systematically results in unrealistic locked streaky structures, due to the lack of local forcing to damp them and the short domain horizontal extents. Also, turbulent statistics and spectra obtained with CPGF applied in a vertically-short domain—which is typical of CPGF usage—are fairly different from those obtained with the more realistic Ekman balance and a boundary-layer simulation set up. In particular, such simulations in small domains fail to adequately simulate larger structures. On the other hand, the simulations over large domains with low resolution fail to explicitly calculate turbulent structures on the order of canopy height or to reproduce the wind profile below the canopy.
The typical grid used for wildfire-precursor simulation forced with PGF with HIGRAD/FIRETEC is a compromise between computational cost and realism of simulated wind field. Regarding the different PGF methods, PGF2 and PGF4 enable mechanisms to control the velocity magnitude at a given height, whereas PGF3 and PGF4 are designed to damp the development of streaky structures and produce lateral flow heterogeneities, similar to those of EKB (beyond the rotation due to Coriolis)—at least in the lower part of the domain. Comparison with experimental data collected near the canopy showed that PGF methods perform similarly in terms of turbulence statistics to existing approaches and studies. PGF4 can be used directly when vegetation and topography are statistically homogeneous in the crosswind direction. When it is not the case, it can be used on a homogenized landscape to generate precursors for the proper heterogeneous domain.