Editorial Type:
Article
Article Type:
Research Article

Can a Simple Dynamical System Describe the Interplay between Drag and Buoyancy in Terrain-Induced Canopy Flows?

Frederik De Roo Atmospheric Environmental Research, Institute for Meteorology and Climate Research, Karlsruhe Institute of Technology, Garmisch-Partenkirchen, Germany

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Tirtha Banerjee Atmospheric Environmental Research, Institute for Meteorology and Climate Research, Karlsruhe Institute of Technology, Garmisch-Partenkirchen, Germany

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Print Publication:
01 Mar 2018
DOI:
https://doi.org/10.1175/JAS-D-17-0161.1
Page(s):
775–786
Full access

Abstract

Under nonneutral stratification and in the presence of topography, the dynamics of turbulent flow within a canopy is not yet completely understood. This has, among other consequences, serious implications for the measurement of surface–atmosphere exchange by means of eddy covariance: for example, the measurement of carbon dioxide fluxes is strongly influenced if drainage flows occur during night, when the flow within the canopy decouples from the flow aloft. An improved physical understanding of the behavior of scalars under canopy turbulence in complex terrain is urgently needed. In the present work, the authors investigate the dynamics of turbulent flow within sloped canopies, focusing on the slope wind and potential temperature. The authors concentrate on the presence of oscillatory behavior in the flow variables in terms of switching of flow regimes by conducting linear stability analysis. The authors revisit and correct the simplified theory that exists in the literature, which is based on the interplay between the drag force and the buoyancy. The authors find that the simplified description of this dynamical system cannot exhibit the observed richness of the dynamics. To augment the simplified dynamical system’s analysis, the authors make use of large-eddy simulation of a three-dimensional hill covered by a homogeneous forest and analyze the phase synchronization behavior of the buoyancy and drag forces in the momentum budget to explore the turbulent dynamics in more detail.

Current affiliation: Applied Terrestrial, Energy, and Atmospheric Modeling, Earth and Environmental Sciences Division (EES-16), Los Alamos National Laboratory, Los Alamos, New Mexico.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Frederik De Roo, frederik.deroo@kit.edu

Abstract

Under nonneutral stratification and in the presence of topography, the dynamics of turbulent flow within a canopy is not yet completely understood. This has, among other consequences, serious implications for the measurement of surface–atmosphere exchange by means of eddy covariance: for example, the measurement of carbon dioxide fluxes is strongly influenced if drainage flows occur during night, when the flow within the canopy decouples from the flow aloft. An improved physical understanding of the behavior of scalars under canopy turbulence in complex terrain is urgently needed. In the present work, the authors investigate the dynamics of turbulent flow within sloped canopies, focusing on the slope wind and potential temperature. The authors concentrate on the presence of oscillatory behavior in the flow variables in terms of switching of flow regimes by conducting linear stability analysis. The authors revisit and correct the simplified theory that exists in the literature, which is based on the interplay between the drag force and the buoyancy. The authors find that the simplified description of this dynamical system cannot exhibit the observed richness of the dynamics. To augment the simplified dynamical system’s analysis, the authors make use of large-eddy simulation of a three-dimensional hill covered by a homogeneous forest and analyze the phase synchronization behavior of the buoyancy and drag forces in the momentum budget to explore the turbulent dynamics in more detail.

Current affiliation: Applied Terrestrial, Energy, and Atmospheric Modeling, Earth and Environmental Sciences Division (EES-16), Los Alamos National Laboratory, Los Alamos, New Mexico.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Frederik De Roo, frederik.deroo@kit.edu
Keywords: Complex terrain; Forest canopy; Differential equations; Large eddy simulations

1. Introduction

The turbulence dynamics of complex canopy flows is not only interesting from a fluid-mechanical or dynamical systems perspective, but it also has implications for flux measurements in complex terrain. Aubinet et al. (2010) stated that the lack of knowledge of the spatiotemporal variability of scalar gradients in the canopy makes the computation of nighttime carbon dioxide emissions from forested sites difficult, leading to an underestimation. Vickers et al. (2012) showed the existence of a nocturnal subcanopy flow regime decoupled from the flow aloft, in which case the subcanopy flow is driven by a competition between friction and buoyancy. Drainage flows can strongly influence turbulent measurements of the surface fluxes. In general, over complex forested sites, advective effects have to be understood to link above-canopy fluxes to the sources and sinks within the canopy (Katul et al. 2006; Ross and Harman 2015). Furthermore, drainage flows on vegetated slopes have implications for air quality (Pardyjak et al. 2009), transport of heat (Monti et al. 2002), frost formation (Laughlin and Kalma 1987), and cold-air pooling (Whiteman et al. 2001).

In the absence of topography, the turbulence structure within and above homogeneous canopies is reasonably well understood: the presence of a drag force inside the canopy leads to an inflection point in the wind profile and thus to roll instabilities, which develop into hairpin vortices that induce a pressure wave in the canopy (Finnigan et al. 2009). However, whether the hairpins are the primary mechanism or only a secondary effect that drives the commonly observed sweep–ejection cycle is a question still under research (Bailey and Stoll 2016). Furthermore, in complex terrain, the physical picture is much less clear. The two “workhorse” study cases for canopies in complex terrain are the flow across a sharp forest edge and the flow over an idealized forested hill under neutral conditions. For the sharp forest edge, analytic solutions have been presented (Belcher et al. 2003; Kroeniger et al. 2018). For the gentle topography, Finnigan and Belcher (2004) developed a first-order closure analytical wind-field model based on linearized perturbation theory for flow over a rough hill (Jackson and Hunt 1975). For stably stratified flow instead, Belcher et al. (2008) offer a qualitative picture of the canopy turbulence: decoupling of motions can happen when the turbulence collapses within the canopy and drainage flows become possible.

For complex heterogeneous canopies combined with a real topography, the assumption of constant vertical profiles of scalar and momentum fluxes becomes invalid. The elliptic nature of the Poisson equation that connects the pressure perturbations to the velocity makes the solutions of the incompressible Navier–Stokes equations sensitive to nonlocal effects. For example, even a local perturbation due to the presence of a small slope will influence the pressure in the whole domain. Hence, an analytic expression for the flow field becomes intractable, and one has to resort to modeling to advance the understanding. The standard modeling approaches are large-eddy simulation (LES) or first-order (mixing length) closure schemes. The latter are a type of Reynolds-averaged Navier–Stokes (RANS) modeling and have the advantage of being computationally cheaper. Grant et al. (2016) made use of mixing-length modeling to study the flow above a Scottish island, with satisfactory results for near-neutral stability. Indeed, under neutral conditions, LES confirms the results of the simpler first-order closure modeling (Ross and Vosper 2005). However, an LES approach has the advantage of making fewer assumptions about the underlying turbulence structure and is more readily applicable to stratified flows, even though LES under stable conditions is a challenging topic (Beare et al. 2006). For neutral conditions, Ross (2008) found that the turbulence of forested ridges is similar to that observed in homogeneous canopies, though the turbulent structure is of course modified across the hill. It is still dominated by sweep–ejection events as in the homogeneous case. Nevertheless, even when analytic models become intractable, a dynamical systems approach can still yield qualitative insights (Alligood et al. 1996; Strogatz 1994). For canopy flows on sloped topography, Yi (2009) proposed a simple model that takes buoyancy and drainage forces into account. He predicted the presence of an oscillation below a critical temperature. The presence of recirculating flow for sloped canopies was also investigated by a two-tower system at the Black Rock Forest of New York (Kutter et al. 2017) and (neutral) flume experiments in the laboratory (Poggi and Katul 2007). The latter study found that the recirculation region is ambiguous for sparse canopies but well delineated within the canopy on the lee side of the hill for dense canopies. Their conclusions were supported by high-resolution LES carried out by Patton and Katul (2009).

In this work, we investigate the turbulence within a homogeneous canopy in the lee of a hill. We revisit the dynamical system proposed by Yi (2009) and independently verify the structure of the phase space spanned by slope wind and temperature. This theoretical section primarily consists of a classical stability analysis of the dynamical system put forward by Yi (2009). Afterward, we turn to large-eddy simulation to explore the dynamics in more detail. For the LES, we investigate the flow on the leeward side of a gentle Gaussian hill.

2. Theory

a. Governing equations

To investigate a dynamical system that can describe canopy thermal-slope flows, we consider the same equations as Yi (2009). These equations describe the horizontally homogeneous flow on a uniform slope with a canopy in a terrain-following coordinate system. Yi (2009) based his formulation on the Fleagle–McNider model, the difference being the quadratic drag force, instead of the original linear relation in the Fleagle–McNider model (Fleagle 1950; McNider 1982), yielding
e1
e2
Here, is the velocity parallel to the slope within the canopy; is the potential temperature, where the overbars indicate time averaging; and is the reference temperature at the surface. The other essential parameters are the surface sensible heat flux , the angle of the slope α, and the temperature lapse rate γ, which is always positive because of the environmental stratification assumption of the model (1) and (2); see, for example, Chen and Yi (2012). Because of the homogeneity of the formulation, after a suitable transformation of the velocity and the time variable, two other parameters can be effectively removed from the problem, that is, the gravitational acceleration g and the canopy length scale , with a the plant-area density (PAD) and the drag coefficient, the former of which in general depends on the vertical coordinate z. We have introduced instead of Yi’s (2009) to avoid confusion with the canopy length scale .

The hypothesis that the turbulent stress gradient can be neglected in these equations has been confirmed by Banerjee et al. (2013) also for the case of homogeneous slope canopies, so we will indeed work with the hypothesis that buoyancy and drag forces are the primary components in the force balance. A remaining issue is at which vertical level the equation is written, and this has to be answered in the context of the expression for . Yi (2009) refers to experiments that show that the velocity and temperature are quite uniform within the drainage layer (i.e., where the flow is katabatic). If that is indeed the case, we do not have to interpret (1) and (2) at a specific height —for example, commonly at the first moment of the canopy density . Instead, we can integrate the equations over the canopy depth and obtain from . Finally, because the vertical turbulent diffusion of potential temperature is not considered either, (1) and (2) do not hold anymore when the surface heat flux becomes very large compared to the slope of the hill, suitably weighed. With the meaning of the equations elucidated, we are now ready to investigate their dynamical behavior in a qualitative manner.

b. Stability analysis

First of all, we rescale the dynamical variables, the time variables, and the parameters by the following transformation:
e3a
e3b
By demanding stationarity of the solution, that is, setting (1) and (2) equal to zero, we find the location of the fixed point:
e4
There is only one fixed point for each triple of the parameter set . This contradicts the result of Yi (2009), where the existence of two solutions was claimed: more precisely, one stable fixed point and one unstable fixed point. However, the negative root of (6) in Yi (2009) has to be discarded because it is not a solution in the downslope case. After setting the quadratic equation in (1) equal to zero,
e5
(2) has to be satisfied as well. Because it is linear in , it can only support one solution for :
e6
The solution for follows by applying (5). In other words, (6) picks the sign of the solution for . After expression of the fixed point (4) in the original coordinates, we obtain the same form as the one of Yi’s (2009) solutions that is correct (as the other solution does not actually exist).
The transformation formulas (3) allow us to express the location of the fixed point in terms of the Froude number of the flow. The Froude number expresses the ratio between the inertia and an external force (usually gravity). Though the convective term does not appear as such, it can be related to the drag term, and the buoyancy term takes the role of external force. The relevant length scale obtained from the drag term is . In the context of the Froude number, the other length scales are not relevant: the height of the trees does not appear because of the bulk averaging to obtain the dynamical equations, and for the infinite slope, the height or extent of the hill does not appear either. With as the length scale for the “inertia,” the absolute value of the rescaled velocity represents the Froude number expressing the ratio between drag and buoyancy, and by inserting the solution for the fixed point,
e7
At the fixed point, the drag and buoyancy are not necessarily balanced, but their proportionality is encoded in Frs. The drag force will be stronger for a denser canopy and a stronger surface flux, leading to supercritical flow. The buoyancy force is stronger when the slope is steeper and the stratification more pronounced, leading to subcritical flow. For the case , the fixed point corresponds to critical flow.
Because there is only one fixed point, we can make a change of parameters and put the governing equations in a form where their dynamical structure stands out:
e8
e9
Substituting the parameter set for the parameter set is without loss of generality, as the transformation of the parameters can be inverted:
e10
The critical temperature is given by
e11
Again, this corresponds to the critical temperature of Yi (2009) when moving to the original variables. We have the freedom to choose the direction of the x axis, and therefore, we shall restrict sinα to positive values. In that case, using (4), and must always be of opposite sign. Furthermore, from (11), it follows that .
We now study the stability of the fixed point. The Jacobian at any point in the space is given by
e12
The eigenvalues at the fixed point (Lyapunov frequencies) are obtained from the characteristic equation
e13
Hence,
e14
where we made use of . For the stability analysis, there are three cases to consider, depending on whether the square root in (14) is imaginary, real and smaller than one, or real and larger than one. This leads to the following cases:

  • : In this case, the fixed point is stable, but there is an oscillatory decay toward the fixed point due to the imaginary part of the eigenvalues.

  • : The fixed point is stable.

  • : The fixed point is unstable but still Lyapunov stable because one Lyapunov frequency is real and negative (saddle point).

In Fig. 1, the phase portrait near the fixed point is shown. The diagram of the steady states in Fig. 2 is relatively simple compared to Yi (2009), where branching is suggested within the us–θs plane. However, the only change in the phase diagram arises from lowering the critical temperature below zero and turning the stable node into a saddle point.
Fig. 1.
Fig. 1.

Phase portrait with eigendirections. The slow direction is the manifold that changes direction when crosses zero.

Citation: Journal of the Atmospheric Sciences 75, 3; 10.1175/JAS-D-17-0161.1

Fig. 2.
Fig. 2.

Bifurcation diagram of the dynamical system (1) and (2). There is no standard bifurcation, but the stability of the fixed point changes when crosses zero.

Citation: Journal of the Atmospheric Sciences 75, 3; 10.1175/JAS-D-17-0161.1

Since there is only one fixed point and there are no orbits, we have to conclude that this dynamical model does not predict oscillations. Nevertheless, three remarks are in place. The first is that the behavior of the dynamical system (1) and (2) critically depends on the sign of the lapse rate through (11). Furthermore, the Lyapunov frequencies in (14) diverge for , leading to a quick approach to (or runaway from) . When is on average close to zero, a local change of its sign will thus complicate the dynamical behavior, which suggests that a two-dimensional spatial framework is needed (and perhaps even three-dimensional). However, to incorporate horizontal advection and the related possibility of spatiotemporal chaos exceeds the scope of this work. In addition, the simplifications that led to the formulation of (1) and (2) neglect the influence of the PAD profile. The latter’s shape directly impacts the stability: for canopies that absorb most radiation at the canopy top, the turbulence within the canopy can be stably stratified even during daytime conditions (Thomas 2011). The last point is that dynamical systems are sensitive to the number of dynamical variables (Strogatz 1994).

c. Two-dimensional hill flow

Because of the sensitivity to the number of dynamical variables, we have to verify to what extent a wind component across the slope and a geostrophic wind (which can be along or across the slope) can alter the previous stability analysis, even though we would intuitively presume that the Rossby number in the canopy is too low for the crosswind flow to change the results. When considering the two-dimensional surface of topography with a homogeneous slope, we will choose the coordinate system such that u is the wind component aligned with the hill slope and υ is the component perpendicular to the hill slope and parallel to the hill surface. The coupling between u and υ appears through the geostrophic wind and the canopy drag. The equation for the potential temperature is independent of the crosswind; hence, the value of remains unchanged, and we can use the same notation for the fixed point as in the previous paragraph. However, in the present case, it is more convenient to keep the parameter α explicit instead of moving to . In scaled variables [(3)], the dynamical equations become
e15a
e15b
e15c
Although the scaled Coriolis parameter is very small, the presence of the drag is crucial, as υ would never reach a steady state because, in general, . Because of the smallness of , we can expect the asymptotic value of to be of order or smaller unless strongly deviates from .
To investigate the dynamics, we will make an expansion in and neglect terms higher than first order. This yields for the component of the fixed point
e16
The shift in due to the inclusion of in the dynamical system is second order in and can be neglected. Hence, we retain the original expression (4) for and . The characteristic equation for the Lyapunov frequencies up to first order in is
e17
Comparing with (13) and (10), we recognize that we have the same roots as in (14) plus an additional root . The latter is due to the presence of the extra dynamical variable υ. Because the extra root is always negative (ultimately because of the drag term for υ), it cannot alter the previous findings: a stable node will remain stable, and a Lyapunov stable node will remain Lyapunov stable. Hence, the presence of υ, or in other words, considering a two-dimensional slope, does not qualitatively alter the dynamical structure, and we can restrict the analysis to two dynamical variables. This qualitative picture of the dynamics will be supplemented by a modeling approach that is not restricted to the simplifying assumptions that led to the dynamical system equations in (15).

3. Methods

a. Model description

We employ the parallelized large-eddy simulation model (PALM; Maronga et al. 2015). PALM is capable of computing turbulent flow above topography, which it resolves explicitly. It can also handle canopy turbulence by its internal canopy model. PALM uses the 1.5-order closure developed by Deardorff (1980) as modified by Moeng and Wyngaard (1988), which assumes a gradient-diffusion parameterization. The turbulent diffusivities for momentum and for heat are parameterized as a function of the subgrid-scale turbulent kinetic energy e, and the LES equations include a prognostic equation for e. The canopy in PALM is represented by a fairly standard distributed drag parameterization (Shaw and Schumann 1992; Watanabe 2004), that is, by adding an additional term in the momentum budget equations as . The square brackets indicate implicit volume averaging over an LES grid box; that is, [u] is the resolved turbulent velocity. The effect of the canopy on the subgrid-scale turbulence, the “spectral shortcut” (e.g., Shaw and Patton 2003), is accounted for by adding an additional sink term in the prognostic equation for e as , as suggested by Shaw and Schumann (1992). The topography is explicitly resolved by the grid but without immersed boundaries. The LES relies on a model for the subgrid-scale turbulent kinetic energy and on a model for the vegetation canopy. On the one hand, we resolve the canopy in detail with 18 vertical grid points (with an aspect ratio of 2 compared to the horizontal directions), yet it is still coarse enough to allow for the necessary implicit volume averaging for obtaining the squared velocity canopy drag based on the locally resolved turbulent velocity [u]. On the other hand, we do not consider extreme cases such as waving canopies, for which analytic results are sensitive to the choice of model formulation for the drag force (Luminari et al. 2016).

b. Model setup

We have simulated a domain of approximately 2.4 × 2.4 × 1.2 km3 with a horizontal resolution of 3.92 m and a vertical resolution of 1.96 m; in total, there are (640)3 grid points. The time step is 0.1 s. The boundary conditions are periodic (i.e., the simulated flow sees an infinite series of hills), but for the mild background wind of 2.0 m s−1, the separation is far enough so that the periodic copies do not directly influence themselves. Each hill has a Gaussian shape, described by
e18
where x is the streamwise coordinate, y the spanwise coordinate, and z the vertical coordinate in opposite direction of the gravity vector. The hill parameters are σx = 200 m, σy = 200 m, and hm = 400 m; therefore, the hill is steeper in the y direction. The slope of the hill naturally depends on position, but it is maximally about 10° and 20° for the x and y directions, respectively. Because of the vertical grid resolution of about 2 m, the topography of the hill becomes zero at a distance of about 500 m from the hilltop in the x direction. The topography of the hill is illustrated in Fig. 3.
Fig. 3.
Fig. 3.

Topography of the hill, with indication of the extent of the measurement domain in the x–y plane (red) and directions of the azimuthal cross sections (blue).

Citation: Journal of the Atmospheric Sciences 75, 3; 10.1175/JAS-D-17-0161.1

Our model includes the Coriolis force for ϕ = 55° in the Northern Hemisphere. The geostrophic background wind is constant with height and directed along x (from the west). Therefore, in the free atmosphere, the wind is westerly, but close to the surface, the wind direction also has a positive component along y (from the south). For our results, we focus on a measurement box of 400 × 400 × 100 m3 to the east of the hill, positioned symmetrically around the (free atmosphere) streamwise axis (x) through the center of the hill. For the measurement box, see also Fig. 3. The turbulent data were obtained at every time step (0.1 s). The data were extracted for 2 h after 3 h of spinup time. Although the mean wind aloft is along the x direction, the wind is veering with height, and the near-surface winds do have an additional positive y component because of the Coriolis force inherent in the prescription of the constant geostrophic wind (the background pressure gradient is independent of the height) and the combined effects of hill, canopy, and surface boundary conditions. In Fig. 4, we have plotted the mean wind direction at the level z = 40 m. Near the surface, the main wind comes from the south-southwest and not from the west as above the boundary layer. Therefore, the measurement box does contain windward slopes of the hill (y < 0) as well as the leeward slopes (y > 0). The exact orientation of the mean wind does of course depend on the height as well as on the horizontal position.

Fig. 4.
Fig. 4.

Horizontal wind direction and its alignment with respect to the slope vector along the steepest gradient. (left) The horizontal wind direction (°): 0° is wind from the west; 90° is wind from the south. (right) The inner product between the horizontal wind direction and horizontal component of the slope vector, where 1 means they are parallel and −1 that they are antiparallel. As can be seen, the region of y < 0 is mainly windward; the region y > 0 is mainly leeward.

Citation: Journal of the Atmospheric Sciences 75, 3; 10.1175/JAS-D-17-0161.1

In addition to the geostrophic background wind, the simulation is driven by buoyancy. At the top of the canopy, there is a prescribed canopy top heat flux of Qtoc = 180 W m−2 representing the radiation incident on the canopy, acting as a source term in the potential temperature equation. Within the canopy, the radiation decays exponentially, with the decay rate determined by the downward cumulative plant-area density. This yields an additional in-canopy heat source given by the vertical gradient of the radiation profile within the canopy:
e19
The surface heat flux is obtained as the value of the before-mentioned in-canopy heat source at the surface ( because of the topography), and this acts as the lower Neumann boundary condition for potential temperature. The canopy is uniform within the whole domain and has a vertical extent of 18 grid points, reaching up to 35 m above the surface. Its plant-area distribution is given by a beta distribution, where the exponents are the same as in Banerjee et al. (2017),
e20
with a plant-area index (PAI) of 5 and hc = 35 m. The radiation scheme of the canopy in PALM is a simplification of radiative cooling within the canopy: it is a first approximation to avoid a detailed energy balance in the canopy, which would require the thermal inertia of the leaves.
To adequately resolve the shear layer at the top of the canopy Ls (Raupach et al. 1996),
e21
From our simulations, we can compute that , and near the top of the hill. Other relevant dimensionless numbers for the problem are and ; see, for example, Finnigan and Belcher (2004) and Ross and Vosper (2005) with l the turbulent mixing length, approximated by . The hill is long , so the problem is relatively linear, and the canopy is dense .

c. Synchronization analysis

For the analysis of the coupling between two dynamical variables, we make use of a synchronization analysis, following the method of Kralemann et al. (2008), whom we refer to for more details. Their method relies on the subtle distinction between the true phase of the signal ϕ and the signal’s protophase ψ. The latter is the phase of the signal that is originally supplied to the analysis, and it can be obtained by, for example, a Hilbert transform on the data. Following that procedure, the signal is decomposed as , with the Hilbert transform of , and the protophase is calculated as
e22
In the first step of the actual synchronization analysis, the protophases of the original signals are transformed into the (reconstructed) true phases by employing the chain rule
e23
which leads to the relation
e24
with the natural frequency of the signal. As advocated by Kralemann et al. (2008), this transformation makes the comparison of the signals more robust, because the protophases depend heavily on the scalar observables available and on the analysis technique used to derive the protophase, and only their average frequency equals that of the true phase: . The transformation from protophases to phases is not a filtering or an interpolation but an invertible transformation that preserves all the relevant information. It simultaneously cleanses all observable-dependent features and leads to a reliable calculation of the synchronization index (Kralemann et al. 2008). The second step of the analysis then consists of computing the synchronicity between the true phases.
We carry out both steps of the synchronization analysis with the Data Analysis with Models Of Coupled Oscillators (DAMOCO) software (Kralemann et al. 2008, 2007; Rosenblum and Pikovsky 2001), which is available as a customized Matlab package. The software performs the initial transformation and then computes a synchronization index (Syn) between weakly to moderately coupled self-sustained oscillators whose dynamics obey
e25
e26
Here, are the true phases of the oscillators, ω the natural frequency, and the coupling functions. The synchronization index is a measure for the phase coupling of the dynamic variables, where a constant phase difference (phase locking) would yield Syn = 1 and uncoupled oscillators have Syn = 0 (Kralemann et al. 2008). Therefore, only the phase relation between the two signals matters; the amplitudes are not relevant for the analysis, in contrast to, for example, correlation analysis. We compute the time average of the overlap between the phases, yielding a synchronization index:
e27
Technically speaking, the synchronicity at higher order can be computed as well, leading to a synchronization index matrix Syn[n, m], but in our case, the lowest-order term (n = m =1) always exhibited the highest synchronization.

For our purpose, the synchronization index can express if the two signals are indeed coupled or if they oscillate independently. Our synchronization index is spatially variable, , because our input signals depend on the position in space. We compute the synchronization index between the drag term and buoyancy term in the momentum budget, that is, up to constant prefactors, between the variables and T. Here, is the wind parallel to the local slope, previously denoted by u to stay close to Yi’s (2009) notation in the theoretical section. It is derived as , with β the unit vector along the (local) steepest gradient of the slope. We choose a different variable for the present discussion also because the slope of the hill in our large-eddy simulations is inhomogeneous. Indeed, precisely because the inclination of the Gaussian hill is neither homogeneous in the streamwise or spanwise dimension, we have to decide upon the definition of the velocity parallel to the slope. There are two obvious candidates: we can either consider a fixed plane for the slope vector (the plane that contains the streamwise direction) and find the velocity component along this slope, or alternatively, we can compute the slope vector from the locally steepest gradient. The latter slope vector also depends on y because it does not discriminate between x and y, and only for y = 0, it will be the same as the former option. For an incoming wind field with constant orientation (i.e., without the presence of the Coriolis force), using a fixed plane for the slope vector oriented along the mean wind appears a valid choice. Yet drainage always acts along the steepest slope; therefore, making use of a variable slope vector along the steepest gradient is meaningful when drainage is investigated. It indeed turns out that this choice for the variable slope wind results in slightly higher synchronization indices between the drag and buoyancy forces; therefore, we will present those results for the slope wind along the locally varying steepest gradient.

4. Numerical results and discussion

Focusing on the slope velocity calculated along the steepest gradient at each position of the hill, in Fig. 5 we present cross sections for the synchronization index in the r–z planes. The azimuth angle of 0° corresponds to the positive x axis and π/2 to the positive y axis; see Fig. 3. We obtained qualitatively very similar results when the drag and buoyancy terms from the turbulent kinetic energy (TKE) budget (e.g., Yue et al. 2008) are compared. The similarity between the two approaches can be explained by the synchronization procedure. The latter involves the transformation of protophases into phases, which likely removes most of the functional difference between the terms in the TKE budget with respect to those in the momentum budget (e.g., T vs 1/T).

Fig. 5.
Fig. 5.

Azimuthal cross sections of the synchronization index for nine angles based on the simulated data between 3 and 5 h of simulated time. The green lines indicate the extent of the canopy. The hill is steeper for the angles of 90° than in the streamwise direction.

Citation: Journal of the Atmospheric Sciences 75, 3; 10.1175/JAS-D-17-0161.1

To obtain the synchronization index for Fig. 5, we have resampled the data to 1 Hz by block averaging, and we computed the synchronization index for the last 2 h combined (hours 4 and 5). Very similar results are obtained when considering hour 4 or 5 separately and when working with the raw data without resampling. It is remarkable that the structures are not limited to the canopy but also extend in the roughness layer above the canopy; hence, there is an extensive depth persistence of the coupling on the upwind side. Of course, we have to keep in mind that the synchronization index was originally derived for coupled self-sustained oscillators. Because of the nature of the turbulence, our turbulent signals do not represent pure oscillators on their own. Hence, part of the coupling can be due to the irregularity of the signal. In addition, the presence of a spatially variable pressure gradient along the hill leads to a mean advective term and causes the mean temperature and mean horizontal velocity to be in phase with the pressure gradient and out of phase with the vertical velocity. Thus, some of the synchrony between drag and buoyancy may stem from the common forcing of these two forces by the mean pressure gradient. The action of turbulence on these mean advective gradients is usually to smooth them (Poggi et al. 2008; Poggi and Katul 2007). To get a better view of the force balance, in Fig. 6 we plot the ratio of the pressure gradient along the slope of the hill with respect to the drag force, both terms averaged over the canopy depth, that is,
e28
and highlight the region where the two terms balance each other (the concentric artifacts are due to the projection of the pressure gradient along the slope vector, i.e., , which exhibits jumps due to the discrete representation of the topography). It can be seen that pressure gradient and drag are close to balance only for isolated points, which does not overlap with the regions of high synchronization index between drag and buoyancy.
Fig. 6.
Fig. 6.

Estimation of the major forces in the momentum budget along the slope of the hill: ratio between the pressure gradient and the drag term.

Citation: Journal of the Atmospheric Sciences 75, 3; 10.1175/JAS-D-17-0161.1

Furthermore, the asymmetry between Syn(x, y) for y < 0 and y > 0 (negative and positive angles) is striking, and also along the minor axis of the hill x, there is hardly a synchronization between drag and buoyancy. We can ascribe the difference due to the direction of the horizontal wind compared to the orientation of the slope. For y < 0 (the effectively windward direction), the mean wind is almost orthogonal to the slope direction; for y > 0 (the effectively leeward direction), the mean wind is almost parallel to the slope direction (see Fig. 4).

We have shown that the dynamical system (1) and (2) does not exhibit oscillations even in the case of a two-dimensional homogeneous slope with an additional dynamical variable. In contrast, the two-dimensional inhomogeneous slope in the large-eddy simulations does exhibit a strong synchronization for the windward side y < 0, corresponding to us < 0 [see (4)]. For the leeward side, it appears that neither the inhomogeneity of the slope nor including the turbulent Reynolds stresses (in the LES) leads to coupling between the drag and buoyancy terms. A significant difference between the theoretical and the modeled cases is the inhomogeneity of the slope. For inhomogeneous slopes, horizontal gradients cannot be neglected. Therefore, we expect that the addition of advection terms in u and θ to the dynamical system (1) and (2), even when describing one-dimensional slope flow, could be sufficient to yield the oscillatory behavior for us < 0. Because the presence of horizontal derivatives already turns the dynamical system with two dynamical variables in an infinite-dimensional dynamical system, the dimensionality of the slope is not likely to be crucial.

5. Conclusions

Turbulent flow inside sloped canopies is of direct use for the flux measurement community, because eddy covariance in complex terrain can be significantly influenced by drainage flows. We have revisited the model by Yi (2009) that predicts oscillations in the dynamical space spanned by the slope wind and the potential temperature from the interplay between the drag and buoyancy forces in the dynamical system. However, our analysis shows that the former model does not exhibit oscillatory behavior. Nevertheless, by employing a synchronization analysis method on large-eddy simulation results of canopy flow, we do find a dynamical coupling between the drag and buoyancy terms on the windward side of the hill. Therefore, the conclusions from the simple dynamical model are valid on the leeward side of the hill. Future efforts are needed to clarify the nature of the interplay between drag and buoyancy, both from the modeling and theoretical perspective, to come to a better understanding of drainage flows on vegetated slopes and its implications for air quality, frost formation and cold-air pooling, and transport of heat, including a better interpretation of flux measurements in sites located in complex topography.

Acknowledgments

Part of this work was previously presented at the EGU (http://meetingorganizer.copernicus.org/EGU2017/EGU2017-2109.pdf). FDR conducted his work within the Helmholtz Young Investigators Group “Capturing all relevant scales of biosphere–atmosphere exchange—The enigmatic energy balance closure problem,” which is funded by the Helmholtz Association through the President’s Initiative and Networking Fund and by KIT. We thank the PALM group at Leibniz University Hannover for their open-source PALM code and their support. We thank the reviewers for their insightful and constructive comments.

REFERENCES

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    • Crossref
    • Export Citation

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    • Export Citation

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    • Export Citation

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    • Search Google Scholar
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    • Search Google Scholar
    • Export Citation

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    • Search Google Scholar
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    • Search Google Scholar
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Journal of the Atmospheric Sciences

  • Alligood, K. T., T. Sauer, and J. Yorke, 1996: Chaos: An Introduction to Dynamical Systems. Springer, 603 pp.

    • Crossref
    • Export Citation

  • Aubinet, M., and Coauthors, 2010: Direct advection measurements do not help to solve the night-time CO2 closure problem: Evidence from three different forests. Agric. For. Meteor., 150, 655–664, https://doi.org/10.1016/j.agrformet.2010.01.016.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Bailey, B. N., and R. Stoll, 2016: The creation and evolution of coherent structures in plant canopy flows and their role in turbulent transport. J. Fluid Mech., 789, 425–460, https://doi.org/10.1017/jfm.2015.749.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Banerjee, T., G. Katul, S. Fontan, D. Poggi, and M. Kumar, 2013: Mean flow near edges and within cavities situated inside dense canopies. Bound.-Layer Meteor., 149, 19–41, https://doi.org/10.1007/s10546-013-9826-x.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Banerjee, T., F. De Roo, and M. Mauder, 2017: Explaining the convector effect in canopy turbulence by means of large-eddy simulation. Hydrol. Earth Syst. Sci., 21, 2987–3000, https://doi.org/10.5194/hess-21-2987-2017.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Beare, R. J., and Coauthors, 2006: An intercomparison of large-eddy simulations of the stable boundary layer. Bound.-Layer Meteor., 118, 247–272, https://doi.org/10.1007/s10546-004-2820-6.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Belcher, S. E., N. Jerram, and J. C. R. Hunt, 2003: Adjustment of a turbulent boundary layer to a canopy of roughness elements. J. Fluid Mech., 488, 369–398, https://doi.org/10.1017/S0022112003005019.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Belcher, S. E., J. J. Finnigan, and I. N. Harman, 2008: Flows through forest canopies in complex terrain. Ecol. Appl., 18, 1436–1453, https://doi.org/10.1890/06-1894.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Chen, H., and C. Yi, 2012: Optimal control of katabatic flows within canopies. Quart. J. Roy. Meteor. Soc., 138, 1676–1680, https://doi.org/10.1002/qj.1904.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Deardorff, J. W., 1980: Stratocumulus-capped mixed layers derived from a three-dimensional model. Bound.-Layer Meteor., 18, 495–527, https://doi.org/10.1007/BF00119502.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Finnigan, J. J., and S. E. Belcher, 2004: Flow over a hill covered with a plant canopy. Quart. J. Roy. Meteor. Soc., 130, 1–29, https://doi.org/10.1256/qj.02.177.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Finnigan, J. J., R. H. Shaw, and E. G. Patton, 2009: Turbulence structure above a vegetation canopy. J. Fluid Mech., 637, 387–424, https://doi.org/10.1017/S0022112009990589.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Fleagle, R. G., 1950: A theory of air drainage. J. Meteor., 7, 227–232, https://doi.org/10.1175/1520-0469(1950)007<0227:ATOAD>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Grant, E. R., A. N. Ross, and B. A. Gardiner, 2016: Modelling canopy flows over complex terrain. Bound.-Layer Meteor., 161, 417–437, https://doi.org/10.1007/s10546-016-0176-3.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Jackson, P., and J. Hunt, 1975: Turbulent wind flow over a low hill. Quart. J. Roy. Meteor. Soc., 101, 929–955, https://doi.org/10.1002/qj.49710143015.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Katul, G. G., J. J. Finnigan, D. Poggi, R. Leuning, and S. E. Belcher, 2006: The influence of hilly terrain on canopy–atmosphere carbon dioxide exchange. Bound.-Layer Meteor., 118, 189–216, https://doi.org/10.1007/s10546-005-6436-2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Kralemann, B., L. Cimponeriu, M. Rosenblum, A. Pikovsky, and R. Mrowka, 2007: Uncovering interaction of coupled oscillators from data. Phys. Rev., 76E, 055201, https://doi.org/10.1103/PhysRevE.76.055201.

    • Search Google Scholar
    • Export Citation

  • Kralemann, B., L. Cimponeriu, M. Rosenblum, and A. Pikovsky, 2008: Phase dynamics of coupled oscillators reconstructed from data. Phys. Rev., 77E, 066205, https://doi.org/10.1103/PhysRevE.77.066205.

    • Search Google Scholar
    • Export Citation

  • Kroeniger, K., T. Banerjee, F. De Roo, and M. Mauder, 2018: Flow adjustment inside homogeneous canopies after a leading edge—An analytical approach backed by LES. Agric. For. Meteor., https://doi.org/10.1016/j.agrformet.2017.09.019, in press.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Kutter, E., C. Yi, G. Hendrey, H. Liu, T. Eaton, and W. Ni-Meister, 2017: Recirculation over complex terrain. J. Geophys. Res. Atmos., 122, 6637–6651, https://doi.org/10.1002/2016JD026409.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Laughlin, G. P., and J. D. Kalma, 1987: Frost hazard assessment from local weather and terrain data. Agric. For. Meteor., 40, 1–16, https://doi.org/10.1016/0168-1923(87)90050-5.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Luminari, N., C. Airiau, and A. Bottaro, 2016: Drag-model sensitivity of Kelvin-Helmholtz waves in canopy flows. Phys. Fluids, 28, 12103, https://doi.org/10.1063/1.4971789.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Maronga, B., and Coauthors, 2015: The Parallelized Large-Eddy Simulation Model (PALM) version 4.0 for atmospheric and oceanic flows: Model formulation, recent developments, and future perspectives. Geosci. Model Dev., 8, 2515–2551, https://doi.org/10.5194/gmd-8-2515-2015.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • McNider, R. T., 1982: A note on velocity fluctuations in drainage flows. J. Atmos. Sci., 39, 1658–1660, https://doi.org/10.1175/1520-0469(1982)039<1658:ANOVFI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Moeng, C.-H., and J. C. Wyngaard, 1988: Spectral analysis of large-eddy simulations of the convective boundary layer. J. Atmos. Sci., 45, 3573–3587, https://doi.org/10.1175/1520-0469(1988)045<3573:SAOLES>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Monti, P., H. J. S. Fernando, M. Princevac, T. A. Chan, W. C. Kowalewski, and E. R. Pardyjak, 2002: Observations of flow and turbulence in the nocturnal boundary layer over a slope. J. Atmos. Sci., 59, 2513–2534, https://doi.org/10.1175/1520-0469(2002)059<2513:OOFATI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Pardyjak, E. R., H. J. S. Fernando, J. C. R. Hunt, A. A. Grachev, and J. Anderson, 2009: A case study of the development of nocturnal slope flows in a wide open valley and associated air quality implications. Meteor. Z., 18, 85–100, https://doi.org/10.1127/0941-2948/2009/362.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Patton, E., and G. G. Katul, 2009: Turbulent pressure and velocity perturbations induced by gentle hills covered with sparse and dense canopies. Bound.-Layer Meteor., 133, 189–217, https://doi.org/10.1007/s10546-009-9427-x.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Poggi, D., and G. G. Katul, 2007: An experimental investigation of the mean momentum budget inside dense canopies on narrow gentle hilly terrain. Agric. For. Meteor., 144, 1–13, https://doi.org/10.1016/j.agrformet.2007.01.009.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Poggi, D., G. G. Katul, J. J. Finnigan, and S. E. Belcher, 2008: Analytical models for the mean flow inside dense canopies on gentle hilly terrain. Quart. J. Roy. Meteor. Soc., 134, 1095–1112, https://doi.org/10.1002/qj.276.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Raupach, M. R., J. J. Finnigan, and Y. Brunet, 1996: Coherent eddies and turbulence in vegetation canopies: The mixing length analogy. Bound.-Layer Meteor., 78, 351–382, https://doi.org/10.1007/BF00120941.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Rosenblum, M., and A. Pikovsky, 2001: Detecting direction of coupling in interacting oscillators. Phys. Rev., 64E, 045202, https://doi.org/10.1103/PhysRevE.64.045202.

    • Search Google Scholar
    • Export Citation

  • Ross, A. N., 2008: Large-eddy simulations of flow over forested ridges. Bound.-Layer Meteor., 128, 59–76, https://doi.org/10.1007/s10546-008-9278-x.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Ross, A. N., and S. B. Vosper, 2005: Neutral turbulent flow over forested hills. Quart. J. Roy. Meteor. Soc., 131, 1841–1862, https://doi.org/10.1256/qj.04.129.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Ross, A. N., and I. N. Harman, 2015: The impact of source distribution on scalar transport over forested hills. Bound.-Layer Meteor., 156, 211–230, https://doi.org/10.1007/s10546-015-0029-5.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Shaw, R. H., and U. Schumann, 1992: Large-eddy simulation of turbulent flow above and within a forest. Bound.-Layer Meteor., 61, 47–64, https://doi.org/10.1007/BF02033994.

    • Crossref
    • Search Google Scholar
    • Export Citation

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  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    Phase portrait with eigendirections. The slow direction is the manifold that changes direction when crosses zero.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    Bifurcation diagram of the dynamical system (1) and (2). There is no standard bifurcation, but the stability of the fixed point changes when crosses zero.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    Topography of the hill, with indication of the extent of the measurement domain in the x–y plane (red) and directions of the azimuthal cross sections (blue).

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    Horizontal wind direction and its alignment with respect to the slope vector along the steepest gradient. (left) The horizontal wind direction (°): 0° is wind from the west; 90° is wind from the south. (right) The inner product between the horizontal wind direction and horizontal component of the slope vector, where 1 means they are parallel and −1 that they are antiparallel. As can be seen, the region of y < 0 is mainly windward; the region y > 0 is mainly leeward.

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    Azimuthal cross sections of the synchronization index for nine angles based on the simulated data between 3 and 5 h of simulated time. The green lines indicate the extent of the canopy. The hill is steeper for the angles of 90° than in the streamwise direction.

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    Estimation of the major forces in the momentum budget along the slope of the hill: ratio between the pressure gradient and the drag term.

Fig. 1.

Phase portrait with eigendirections. The slow direction is the manifold that changes direction when crosses zero.
Fig. 2.

Bifurcation diagram of the dynamical system (1) and (2). There is no standard bifurcation, but the stability of the fixed point changes when crosses zero.
Fig. 3.

Topography of the hill, with indication of the extent of the measurement domain in the x–y plane (red) and directions of the azimuthal cross sections (blue).
Fig. 4.

Horizontal wind direction and its alignment with respect to the slope vector along the steepest gradient. (left) The horizontal wind direction (°): 0° is wind from the west; 90° is wind from the south. (right) The inner product between the horizontal wind direction and horizontal component of the slope vector, where 1 means they are parallel and −1 that they are antiparallel. As can be seen, the region of y < 0 is mainly windward; the region y > 0 is mainly leeward.
Fig. 5.

Azimuthal cross sections of the synchronization index for nine angles based on the simulated data between 3 and 5 h of simulated time. The green lines indicate the extent of the canopy. The hill is steeper for the angles of 90° than in the streamwise direction.
Fig. 6.

Estimation of the major forces in the momentum budget along the slope of the hill: ratio between the pressure gradient and the drag term.
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