Introduction
Canopy flow occurring within and immediately above vegetation canopies plays
a substantial role in regulating atmosphere–biosphere interaction. The canopy
layer is an interface between land and atmosphere, in which most natural
resources humans need are produced by biochemical reactions. Canopy flow
influences those biochemical processes through the control of gas exchange
between the vegetation and the atmosphere (e.g., influencing reaction rates
by changing gas concentrations), heat exchanges (e.g., influencing reaction
conditions by changing temperature), and momentum exchanges (e.g., changing
turbulent mixing conditions). Better understanding of canopy flow behavior
has many practical implications in accurately determining, for instance,
terrestrial carbon sinks and sources (Sun et al., 2007), the fate of ozone
within and above forested environments (Wolfe et al., 2011), forest fire
spread rate (Cruz et al., 2005), bark beetle management (Edburg et al.,
2010), and others.
The typical patterns of forest canopy turbulent flows are characterized by an
S-shaped wind profile with an exponential Reynolds stress profile rather than
the widely used logarithmic wind profile and constant Reynolds stress
observed over bare ground (Yi, 2008). S-shaped wind profiles have been
observed within forest canopies in numerous studies (Baldocchi and Meyers,
1988; Bergen, 1971; Fons, 1940; Lalic and Mihailovic, 2002; Landsberg and
James, 1971; Lemon et al., 1970; Meyers and Paw U, 1986; Oliver, 1971; Shaw,
1977; Turnipseed et al., 2003; Yi et al., 2005; Queck and Bernhofer, 2010;
Sypka and Starzak, 2013). The S-shaped profile refers to a secondary wind
maximum that is often observed within the trunk space of forests and a
secondary minimum wind speed in the region of greatest foliage density. The
features of S-shaped wind profiles imply that K theory and mixing-length
theory break down within a forest canopy layer (Denmead and Bradley, 1985;
Yi, 2008). Particularly, the assumption of a constant mixing-length within a
canopy is not consistent with the original mixing-length theory. This is
because a mixing-length (lm) must satisfy von Karman's rule (von
Kármán, 1930; Schlichting, 1960; Tennekes and Lumley, 1972), which indicates
that a mixing length is a function of velocity distribution (Schlichting,
1960), as
lm=κdU/dzd2U/dz2,
where κ is von Karman's constant, U is wind speed, and z is height
within the canopy. The mixing length of the S-shaped velocity distribution is
not constant, being minimum at the local extreme values of the wind profile
(dU/dz=0, d2U/dz2≠0) and maximum
at the inflection point of the wind profile (dU/dz≠0,
d2U/dz2=0) (Wang and Yi, 2012). A mixing-length that
varies with height within canopy has been demonstrated by large-eddy
simulations (Coceal et al., 2006; Ross, 2008) and by water tank experiments
(Poggi and Katul, 2007a).
The features of S-shaped wind profiles also dictate the existence of
super-stable layers near levels where wind speed is maximum (or minimum) and
temperature inversion (temperature increasing with height) exists, leading
the Richardson number (Ri) to be extremely large or infinity (Yi et al., 2005). A
super-stable layer acts as a “lid” or “barrier” that separates fluid into
two uncorrelated layers: (1) the lower layer between the ground and the
super-stable layer, and (2) the upper layer above the super-stable layer.
This canopy flow separation was verified by SF6 diffusion observations
(Yi et al., 2005) and carbon isotope experiments (Schaeffer et al., 2008).
The lower layer is sometimes called a “decoupled layer” (Alekseychik et
al., 2013) that is shallow, usually within the trunk space of a forest.
Because the super-stable layer prohibits vertical exchanges, the decoupled
layer channels air in the horizontal direction. The characteristics of the
channeled air are highly dependent on soil conditions, containing a high
concentration of soil respired CO2 and soil evaporated water vapor, and
consisting of colder air cooled by radiative cooling at the ground surface
(Schaeffer et al., 2008). The channeled air is sometimes termed “drainage
flow”, and is a common phenomenon in hilly terrains under stable atmospheric
conditions, such as on calm and clear nights (Yi et al., 2005; Alekseychik et
al., 2013). The drainage flow limits the accuracy of tower-based estimates of
ecosystem–atmosphere exchanges of carbon, water, and energy. Sensors on the
tower above the canopy cannot measure the fluxes conducted by drainage flow
because the layer above the canopy is decoupled from the drainage flow by the
isolating super-stable layer. This advection problem is a well-known issue
that has not yet been solved using eddy-flux measurements (Goulden et al.,
1996; Aubinet et al., 2003; Staebler and Fitzjarrald, 2004; Sun et al., 2007;
Yi et al., 2008; Montagnani et al., 2009; Feigenwinter et al., 2010; Aubinet
and Feigenwinter, 2010; Queck and Bernhofer, 2010; Tóta et al., 2012;
Siebicke et al., 2012).
The concept of a super-stable layer is useful in interpreting data
associated with stratified canopy air (Schaeffer et al., 2008). However,
stratified canopy flows over complex terrain are far too complex to be able
to characterize considering only a super-stable layer. Canopy structure
(quantified by leaf area density profile), terrain slope, and thermal
stratification are three key parameters in understanding the details of
stratified canopy flows over complex terrain. The thermal stratification
plays a leading role in the development of pure sub-canopy drainage flows
(Chen and Yi, 2012): strong thermal stratification favors drainage flow
development on gentle slopes, while weak or near-neutral stratification
favors drainage flow development on steep slopes. We speculate that
interaction between thermal stratification and terrain slopes and vegetation
canopy may result in multiple super-stable layers. The complicated thermal
and flow patterns cause difficulties in understanding the mechanisms and
rates of exchange of mass and energy between the terrestrial biosphere and
the atmosphere (Alekseychik et al., 2013; Burns et al., 2011; Yi et al.,
2005).
In this paper, we attempt to use a computational fluid dynamics (CFD)
technique to examine the micro-structure of stratified canopy flows to
provide insight into the role of physical processes that govern drainage
motion and its turbulent characteristics within canopy in complex terrain.
There are many challenges to face when pursuing this goal. First, the
mixing-length theory and K theory that are widely used as closure
approaches to momentum equations (Wilson et al., 1998; Pinard and Wilson,
2001; Ross and Vosper, 2005; Katul et al., 2006) have been shown to have
questionable validity within a forest canopy layer both theoretically (Yi,
2008) and observationally (Denmead and Bradley, 1985). Second, the analytical
model (Finnigan and Belcher, 2004) is limited to neutral condition and hills
of gentle slope. The analytical model is developed based on the linearized
perturbation theory for the flow over a rough hill (Jackson and Hunt, 1975),
which assumes that the mean flow perturbations caused by the hill are small
in comparison to the upwind flow. Poggi and Katul (2007b) and Ross and
Vosper (2005) have shown that the analytical model fails to model the flow
pattern on dense canopies on narrow hills. Third, even though turbulence
closure models and large eddy simulation models have been used to simulate
flow within and above the canopy in numerous published studies, most
numerically reproduced canopy flow is confined to idealized cases: either
neutral (Ross and Vosper, 2005; Dupont et al., 2008; Ross, 2008) or weakly
unstable (Wang, 2010) atmospheric conditions, or flat terrain with a
homogeneous and extensive canopy (Huang et al., 2009; Dupont et al., 2010).
Simulations of stratified canopy flow have received little consideration.
This might be attributed to difficulties in numerical simulations arising
from small scales of motion due to stratification (Basu et al., 2006), and
complex interaction between wind and canopy drag elements (Graham and
Meneveau, 2012). Large eddy simulation has been quite successful in
producing turbulent flow and its related scalar transport in neutral and
unstable cases (Shen and Leclerc, 1997; Wang, 2010; Mao et al., 2008).
However, under stable conditions, due to flow stratification, the
characteristic size of eddies becomes increasingly small with increasing
atmospheric stability, which eventually imposes an additional burden on the
large eddy simulation subgrid-scale (LES-SGS) models (Basu et al., 2010). If resolution is high enough, any
turbulent flow can be simulated accurately by LES. In fact, given
sufficiently fine resolution, LES becomes direct numerical simulation (DNS),
demanding very fine spatial and temporal resolution (Galperin and Orszag,
1993), which is currently beyond the reach of available computational power.
In this paper, we employ the renormalized group (RNG) k–ε
turbulence model to investigate stably stratified canopy flows in complex
terrain. The RNG k–ε turbulence model was developed by Yakhot
and Orszag (1986a) using the renormalized group methods and prescribes the
turbulent-length scale related to transport of turbulent kinetic energy and
dissipation rate (Yakhot and Orszag, 1986b; Smith and Reynolds, 1992).
Compared to a standard k–ε turbulence model, the numerically
derived parameters are not subject to experimental adjustment in the RNG
k–ε turbulence model. The rate of strain term in the dissipate
transport equation is important for treatment of flows in rapid distortion
limit, e.g., separated flows and stagnated flows (Biswas, and Eswaram, 2002)
which commonly occur in vegetated hilly terrain. The initial successes in
applying the RNG k–ε turbulence model to generate airflows in
hilly terrain have been demonstrated by Kim and Patel (2000) and Xu and
Yi (2013).
Method
Numerical implementation
The two dimensional computational domain extends over
1400 m × 130 m in a Cartesian coordinate system, corresponding to
1200 × 157 grid intervals in the x and y directions. A single
hill is 100 m long covered with a 15m tall homogeneous forest canopy, which
extends from 650 m of the domain in horizontal. The mesh spacing in both
horizontal and vertical at the forested hill is 0.5 m and is stretched with
a power law, starting with a grid spacing of 0.5 m throughout the canopy,
with a larger grid spacing stretching outwards from the edge of the forest
and the top of the canopy on the hill crest. The stretch power in both
horizontal and vertical is 1.15. Ground surface roughness height is set to be
0.01 m.
In this study, the topography is specified with a ridge-like sinusoidal hill,
infinite in the unsimulated third dimension. The shape function of the hill
in 2-D is defined as
H(x)=H2cosπx2L+H2,
where H is the hill height, L is the half-length scale (half of the hill
width at mid-slope height), x is longitudinal distance with x = 0 at
the center of the single hill. The variation of the slope (H/L) is
specified by changing H with a constant L = 25 m.
The porous canopy layer (canopy height h = 15 m) is designed
horizontally homogeneous along the slope. The leaf area density profile
a(z) is specified as values from observation of an actual forest (Yi et
al., 2005) with the maximum leaf area density at about 8 m. Leaf area index
(LAI) is 3.3. The ambient temperature is θ0(z)=θ00+γz, where θ00=288 K is the potential
temperature at z=0, γ is ambient lapse rate, set to
-6 ∘C km-1. The cooling rate at ground surface is set to
-15 W m-2. Since we are most interested in calm nighttime
conditions, no wind in the domain is initially specified. The fixed pressure
boundary condition (open boundary) is applied to lateral boundaries and top
boundary, where the pressure is close to 0.0 Pa, relative to the external
pressure.
Conservation of mass and momentum
The flow is assumed to be steady and the Boussinesq approximation is applied.
The mass, momentum, and energy balance equations in the canopy sub-layer can
be written as
∂u‾j∂xi=0,u‾j∂u‾i∂xj=-1ρ∂P∗∂xi+ν∂2u‾i∂xixj-∂∂xjui′uj′‾-giβθ‾-θ∞-FDi,u‾j∂θ‾∂xj=Γ∂2θ‾∂xixj-∂∂xjθ′uj′‾+1ρcpQsource,
where u‾i and u‾j are the mean velocity
components along xi and xj direction, respectively;
θ‾ is the mean potential temperature; ui′, uj′,
and θ′ are the fluctuations from their mean value u‾i,
u‾j, and θ‾; ρ is the air density; ν is kinematic viscosity of air; P∗ is the deviation of pressure
from its reference value; β is the thermal expansion coefficient of
air; θ∞ is the reference temperature; gi is the
gravity acceleration in i direction; Γ=ν/Pr is thermal
diffusion coefficient; and turbulent Prandtl number Pr is 0.5 in canopy
layer and 1 above the canopy. Pr = 0.5 is close to the values
used in large-eddy simulations of stably stratified atmospheric boundary
layer turbulence (Basu and Porté-Agel, 2006; Stoll and Porté-Agel,
2008). In most of the region above the canopy (except very near the top of
canopy), turbulence is very weak. In this region, molecular effects are
dominant, especially in conditions without synoptic wind. Qsource
is the energy source. When the atmosphere is stably stratified,
Qsource < 0 indicating radiative cooling of the canopy elements
and ground surface. The constant cooling rate at the surface can drive a
steady-state stable boundary layer on flat and sloped terrain (Brost and
Wyngaard, 1978), so we set Qsource = 0 in the lower-canopy
layer (0–8 m) and then linearly decreased to -8 W m-3 at the top-canopy layer. The thermal conditions are sufficient to drive fully developed
turbulent flows, according to dimensional analysis of the bulk Reynolds
number:
Reb=hiUν=O101m×O10-1m s-1O10-5m2s-1=O(105),
where hi is the depth of boundary layer, U is bulk velocity, and ν
is kinematic viscosity.
The steady-state assumption is satisfied with condition proposed by
Mahrt (1982):
FH^/T^≪1,
where F is the Froude number, H^ is the ratio of the average flow
depth H to the surface elevation drop ΔZs, and T^ is
the ratio of the timescale T to the Lagrangian time L/U. The Froude
number is defined as
F=U2/gΔθθ0H,
where U is downslope velocity scale
(= O(10-1)ms-1), g is gravity acceleration
(= 9.81 m s-2), Δθ is scale value for potential
temperature deficit of the canopy layer (= O(100) K), θ0
is the basic state potential temperature (= O(102) K), and H is the
flow depth scale, chosen to be the depth of significant temperature deficit
which coincides with the layer of enhanced thermal stratification
(= O(101) m). In this simulation setting, F=O(10-2).
H^=H/ΔZs, where ΔZs=Lsinα,
L is downslope-length scale (= O(101) m), sinα
(%) = O(101); thus, H^=O(100). T^=TU/L, where
T=O(104) s is suggested by Mahrt (1982) to represent the order of
magnitude of temporal accelerations associated with the diurnal evolution of
drainage circulations. In our simulation, T^=O(102); thus,
FH^/T^=O(10-4)≪1.
FDi is the drag force exerted by the canopy elements in i
direction:
FDi=12KruiU,
where Kr is the resistance coefficient, which is derived from an
empirical relationship given by Hoerner (1965):
Kr=1232ϕ-12,
where φ is porosity of the canopy layer, which can be obtained from
leaf area density profile a(z) (Gross,
1993):
ϕ(z)=1+4a(z)+12a(z).
FDi is zero above the canopy.
RNG k–ε model
The RNG model was developed by Yakhot and Orszag (1986a, b; Yakhot et al.,
1992) using RNG methods. The RNG k–ε
turbulent model has been successfully applied in reproducing topographic and
canopy related flows (Kim and Patel, 2000; Xu and Yi, 2013; Pattanapol et
al., 2007).
In RNG k–ε model, the Reynolds stress in Eq. (3) and turbulent
heat flux in Eq. (4) are solved by turbulent viscosity, as
-ui′uj′‾=μt∂u‾i∂xj+∂u‾j∂xi-23δijk,-θ′uj′‾=μθ∂θ‾∂xj,
where μt and μθ=μt/Pr are the turbulent
viscosities of momentum and heat, respectively, δij is Kronecker
delta, and k is the turbulent kinetic energy.
RNG k–ε model assumes that turbulence viscosity in Eq. (10) is
related to turbulence kinetic energy k (TKE) and dissipation ε:
μt=ρCμk2ε,
where k and ε are determined from the transport equations for
k and ε; Cμ is a dimensionless constant.
The steady-state transport equations for k and its dissipation ε are written as
u‾i∂k∂xi=∂∂xiμtσk∂k∂xi+Ps+Pb+Pw+Tp-ε,u‾i∂ε∂xi=∂∂xiμtσε∂ε∂xi+Cε1εkPs-ρCε2ε2k-S,
where Ps is shear production, given by
Ps=μt∂u‾i∂xj∂u‾i∂xj+∂u‾j∂xi,
Pb is buoyancy production, given by
Pb=-μθgiβ∂θ‾∂xi.
Pw is wake production caused by canopy elements as (Meyers and
Baldocchi, 1991)
Pw=u‾iFDi=12KrUu‾i2.
Tp is pressure collection term, which is calculated as residual of other
TKE components; S is a volumetric source term which includes the
rate-of-strain, given by
S=Cηη31-ηη0ε21+β0η3k,η=kεPsμt1/2,
where the empirical constants Cμ, σk, σε, Cε1, Cε2, β0, and η0 are
0.0845, 0.7194, 0.7194, 1.42, 1.68, 0.012, and 4.38, respectively (Yakhot and
Orszag, 1986a, b).
Results and discussion
After a quasi-equilibrium condition is approached, all the solved fields in
the studied cases are developed to be near-symmetric horizontally (in the
x direction) with respect to the center of the modeled hill at x=0 due
to the homogeneous boundary conditions and initial settings. We restrict our
discussion to the right half of the hill. Our results show (Fig. 1) that wind
structure is differentiated into down sweep (H/L ≤ 0.6) and
updraft (H/L ≥ 0.8) within canopy. The temperature, wind, and
turbulence characteristics on representative gentle (H/L = 0.6) and
steep (H/L = 1.0) hills are illustrated (see Fig. 1) to explore the
thermal and mechanical processes that govern the airflow structures.
Thermal analysis
In the model, strong stratification develops with distinct thermal
distribution on the slope, subject to heat loss on the slope surface and the
upper-canopy layer. The heterogeneous distribution of heat within the canopy
causes a “fish-head”-shaped temperature distribution on the slope, with the
upper jaw in the upper-canopy layer and the lower jaw attaching to the slope
surface. The jaws consist of cold air while the open mouth shows relatively
warmer air (Fig. 2). In comparison with the upper jaw which is confined to
the middle and lower slope, the lower jaw extends up to the crest of the
hill. As the slope intensity is reduced, the fish-head effect's upper jaw is
diminished. For a very gentle slope (i.e., H/L ≪ 1), the model
produces a horizontal isotherm pattern with cold air at the bottom of the
slope and warm air upslope, as would be expected in real-world conditions. A
significant difference in temperature distribution among varied slopes
results in a different angle of orientation of the fish-head temperature
profile. Isotherms are inclined parallel to the slope surface because they
tend to follow the shape of the slope and the top-canopy layer since the
cooling along the slope surface is uniform. The temperature distribution on a
gentle hill is shown as an angled fish-head shape, while the fish-head is
tilted by the slope on the steep hill, which is shown by the isotherms on the
lower jaws. The different angle of the fish-head profile can explain specific flow
structures in the canopy (see Sect. 3.2). In accordance with the fish-head
temperature distribution, temperature profiles are shown in three layers
(Fig. 3a–d). A strong inversion layer is developed across the lower jaw,
above which temperature slightly decreases with height in a thermal
transition zone and a weak inversion layer is formed across the upper jaw.
The temperature gradient and the depth of the lower inversion layer
increases, since cold air flowing down the slope results in a cool pool on
the lower slope where a single inversion layer extends above the canopy
(Fig. 3e, f). The temperature difference from the hill surface to the top of
the canopy at the hill crest is about 0.8 and 0.4 ∘C for gentle and
steep hills, respectively, while the difference increases to around
3.2 ∘C in the canopy layer at the feet of both hills. The inversion
strength near the surface is larger than in the upper canopy, which is due to
the stronger radiative cooling effect on the surface. The temperature
gradient and inversion on the steep hill are predicted weaker than on the
gentle hill, because at the same horizontal x/L location, the canopy layer
is at a higher elevation on the steep hill. Regardless of the horizontal
location x/L, we find that inversions both near the surface and in the
upper canopy are stronger on the steep hill than on the gentle hill at the
same elevation, which benefits the development of stronger drainage flow on
the steep slope.
Simulated streamlines in the forested hill:
(a) H/L = 0.6; (b) H/L = 1.0. The translucent
green masks indicate the regimes with instability within the canopy. The top
of the
canopy is marked by black-dashed line. The black “WV” marks the region of
wake vortices next to the edge of canopy. The “DS” in (a) and “UD”
in (b) indicate the region of down-sweep wind and updraft wind on
the gentle and steep slopes, respectively.
Contours of potential temperature (K) along the right slope:
(a) H/L = 0.6; (b) H/L = 1.0. The difference
between isotherms is 0.25 K. The numbers on isotherms indicate the
temperature. The x axis is normalized by the half-length scale of the hill
L and y axis is normalized by the height of the canopy h. White-dashed
lines indicate the top of canopy and the isotherms marked with cyan-dashed lines highlight the “fish-head” temperature distribution.
The Ri is the ratio of the relative importance
of buoyant suppression to shear production of turbulence, which is used to
indicate dynamic stability and formation of turbulence. Ri is
calculated based on mean profiles of wind and temperature. For different
purposes and data availability, gradient Richardson number
(Rig) and bulk Richardson number (Rib)
are used to predict the stability within canopy. Yi et al. (2005) found that
the Rig,
Rig=g/θ‾∂θ‾/∂z∂U‾/∂z2,
with ∂U‾/∂z=0 and ∂θ‾/∂z≠0 at the inflection points of the S-shaped wind profile
resulted in an infinite Rig, which describes the
super-stable layer. In a forest, wind and temperature are typically only
measured in a few levels, making ∂U‾/∂z and
∂θ‾/∂z impossible to directly
calculate. Therefore, Rib is commonly used to quantify
stability between two levels (z1 and z2) using the measured
temperature and wind speed (Zhang et al., 2010; Burns et al., 2011;
Alekseychik et al., 2013),
Rib=gθ‾θz2-θ(z1)U(z2)-U(z1)2z2-z1.
Potential temperature (K) profiles on the slope for H/L = 0.6
(blue) and H/L = 1.0 (red). The locations of the six sections are
labeled as (a)–(f), and their locations with respect to the hill are presented.
Horizontal distances are normalized by the half-length scale L of the hill.
The cyan blue curves indicate the thermal transition zone with negative
temperature gradient.
In our modeling setting, the gridding space in vertical is Δz=z2-z1, which is 0.5 m in the canopy layer. We define a local
Richardson number to evaluate stability around the forested hill and examine
the local stability in response to the heterogeneous distribution of heat.
The local Richardson number in grid (m,n) is calculated as,
Ril=gθm,nθm,n-θm,n-1zm,n-zm,n-1um,n-um,n-12+wm,n-wm,n-12.
The local Richardson number indicates that, within the canopy, flow is stably
stratified except for an unstable region penetrating from the hill summit
into the middle slope within the thermal transition regime (Fig. 1).
Ril is found to be extremely large (∼ 105) just
above the canopy on the upper to middle slope
(Fig. 4 locations a–d) indicating a thin primary
super-stable layer just above the top of canopy. The primary super-stable
layer is elevated and deepened on the lower slope (Fig. 4 locations e and f),
extended from the height of 1.3–1.4 h to about the height of 2 h. The
deep primary super-stable layer is caused by the strong cooling and
temperature inversion at the base of the hill, regardless of slope intensity.
Within canopy, a secondary super-stable layer with extremely high
Ril is developed below 0.5 h. On the lower slope, the
depth of the secondary super-stable layer extends from the slope surface up
to 0.5 h. The deep secondary super-stable layer is consistent with deep
and strong temperature inversion layer where wind is stagnated. The absence
of a secondary super-stable layer on the summit could be explained by
stronger mixing of warmer air from above-canopy, because stronger drainage
flow promotes the penetration of warm air from aloft when cold air moves down
the slope (Zängl, 2003). Air in the transition region with negative
temperature gradient is unstably stratified. The transition region is
developed by the downwelling of cool air from the upper canopy with
relatively warmer air upwelling from the lower canopy. The results show that
for a sufficiently steep slope, the effects of the hill dominate the
atmospheric profile, while for more gentle slopes the effects of the canopy
dominate the resultant atmospheric profile.
Locations of super-stable layers for H/L = 0.6 and
H/L = 1.0 (left panel). The primary super-stable layers are marked by
dash-dotted lines with yellow solid circles and secondary super-stable layers
are marked by dash-dotted lines with green solid circles. The Ri
numbers at locations indicated by the yellow and green solid circles are
extremely large, which are illustrated on the right panel for the locations
(b) and (e). PSL denotes primary super-stable layer. SSL denotes secondary
super-stable layer. UL denotes unstable layer.
The nocturnal stable canopy layer could be used to explain the occurrence of
within- and above-canopy flow decoupling observed in prior studies. van
Gorsel et al. (2011) reported a very stable nighttime canopy layer
(Rib>1) using the bulk Richardson number, indicating
that the canopy layer is decoupled from air aloft. Decoupling at the top of
the canopy is more likely to occur as the buoyancy is more dominant and air
at the top of the canopy is strongly stable. The canopy top decoupling
weakens vertical exchange of mass and heat between the vegetation and the
atmosphere aloft. The measurement data show large temperature and CO2
gradients (Burns et al., 2011) as decoupling occurs in a strongly stabilized
atmosphere. Decoupling at the top of the canopy produced stronger carbon
dioxide and temperature gradients than within-canopy decoupling (Alekseychik
et al., 2013). The primary super-stable layer in our study is shown as a lid
located at the top and above canopy, which could terminate the vertical
exchange between the canopy and the air above. During nighttime, soil
respiration contributes about 60–70 % (Janssens et al., 2001) of the
total CO2 emission from the terrestrial ecosystem. The soil respired
CO2 could be blocked by the secondary super-stable layer forming a very
shallow pool on the slope surface.
Wind flow structures
Figure 1 shows that air above the canopy sinks and converges towards the hill
and then shifts direction within the canopy. Flow converges to the hill from
all sides, and is then inflected near the top of the canopy. The height of
inflection points increases as the air flows down the slope. The inflection
points are approximately at the bottom of the primary super-stable layer. As
a result of the abrupt convergence in the top of the canopy at the base of
the hill, wake vortices are developed near the forest edge, after the wind
leaves the hillside within the primary super-stable layer. The wake vortices
can extend to about 2.6 L in horizontal and 1.3 h in vertical.
According to the flow location within the canopy, we identify the drainage
flow as two streams: the majority air mass within the upper-canopy inversion
layer is called the upper-canopy drainage flow (UDF) layer, and the majority
air mass within the inversion layer in the lower-canopy is called the
lower-canopy drainage flow (LDF) layer. The UDF is developed as the air above
the canopy sinks from lateral sides towards slopes of the hill. However,
instead of further descending into the canopy, the sinking motion is diverted
to follow the shape of the top-canopy layer as it reaches the top of the
canopy (Figs. 1 and 6). The UDF accelerates down the slope between the top of
the unstable layer and the bottom of the primary super-stable layer, reaching
its maximum wind speed of 0.3 meters per second (m s-1) at location
(Figs. 5d and 6a) on the gentle slope and 0.35 m s-1 at location
(Figs. 5e and 6b) on the steep slope, and then decelerates down to the feet
of the hills. The air sinking over the crest can directly reach the surface
of the crest and flow along the slope to form the LDF. The maximum wind speed
of the LDF is at location (Fig. 5d) for a gentle slope (0.18 m s-1)
and at location (Fig. 5c) for a steep slope (0.29 m s-1). The maximum
wind speed in LDF occurs on the slope surface, below the secondary
super-stable layer. Deceleration of the flow towards the base of the hill
should occur for a number of reasons. The pool of cool, dense air at the base
of the hill resists incoming flow. Also, the drag force acting against the
wind is dependent on the speed of the air flow squared.
Profiles of streamwise velocity (u; m s-1; top panel) and
vertical velocity (w; m s-1; bottom panel) for H/L = 0.6 (blue)
and H/L = 1.0 (red). The locations of the six sections are labeled as
(a)–(f), and their locations with respect to the hill are marked in Fig. 3 with
the same letters. Note that wind velocity on the slope surface is not zero
because the centers of bottom grid cells in the numerical calculation are not
exactly at the surface.
UDF and LDF show different patterns within canopy for different slopes, which
essentially regulates the direction of wind shifting within canopy (Figs. 1
and 6). On the gentle slope (H/L = 0.6), UDF is much thicker compared
with LDF (Fig. 6a). Air in UDF accelerates within the regime of the upper
inversion layer reaching its maximum at the top of thermal transition region
and then decelerates to a minimum (u = 0 and w = 0, Fig. 5) at
the top of the slope surface inversion layer. Then, UDF sweeps horizontally
to join the shallow LDF on the slope surface, which is shown as negative
streamwise velocity and near-zero vertical velocity in Fig. 5 (down-sweep).
When the slope is steep (H/L = 1.0), UDF is much shallower than LDF on
the upper slope. Air in LDF accelerates on the upper slope (Fig. 5a–c),
followed by deceleration and stagnation. The stagnated flow jumps
perpendicularly from the deep canopy layer to join the shallow UDF in the
upper-canopy layer (the updraft, with u>0 and w>0, is visible in
Figs. 1 and 5). The shifting winds on both gentle and steep slopes are
parallel to the isotherms in the warm and open fish-mouth
region of
the profile. Rotational vortices are formed below the shifting winds.
The generation and direction of the shifting-wind structure are primarily
driven by the slope and stratification. Under calm and stably stratified
conditions, the dominant driving force of sinking drainage flow on the slope
is the hydrostatic buoyancy force which is given as Fhs=gΔθ/θ0sinα, where α is
the slope angle, Δθ is the potential temperature difference
between the ambient air and the colder slope flow, and θ0 is the
ambient potential temperature. The drainage flow on both the gentle and steep
slopes is initiated by the dominant Fhs as the air is calm and
stably stratified (Froude number ≪ 1; Belcher et al., 2008). The
magnitude of Fhs increases with slope angle α so that
Fhs is much larger on a steep slope than a gentle slope, leading to
a stronger sinking motion above the crest. The sinking air penetrates to the
lower part of the canopy at the hilltop. Thus, the LDF layer is deeper than
the layer of UDF for a steep slope. However, the sinking motion above the
crest on the gentle slope is diverted to follow the shape of the slope in the
upper canopy due to smaller Fhs, which is not strong enough to
completely penetrate the canopy. As a result, UDF is deeper than the LDF on
gentle slopes, in contrast to that on steep slopes. The heterogeneous cooling
in the canopy layer causes two baroclinic zones consistent with the UDF and
LDF: the upper-canopy layer and slope surface layer. The strong baroclinicity
on the steep slope surface causes the deep LDF wind to rotate
counterclockwise (i.e., turning upwards on the lower slope, perpendicular to
the hill slope). However, the rotated wind is forced to shift down when
hitting the top-canopy UDF. The wind at the baroclinic zone with a deep UDF
on a gentle slope rotates clockwise, but shifts downslope when hitting the
layer of the LDF.
Wind velocity (U, m s-1) on the slopes for
(a) H/L = 0.6 and (b) H/L = 1.0. The white
solid lines are streamlines as shown in Fig. 1. The black-white-dashed lines
denote the top of the canopy.
Turbulent fluxes of momentum and heat
Figure 7 shows profiles of shear stress u′w′‾. Shear stress is
most significant in the region near the top of the canopy where wind impinges
on the canopy resulting in strong wind shear. Another region of large shear
stress is in the lower canopy. This is related to the wind shifts which lead
to strong wind shear. Shear stress is small on the upper slope but increases
down the slope. The maximum shear stress at the top of the canopy is located
at the wake region (Fig. 7e, f), where the wake vortices are formed. Shear
stress is positive above the canopy indicating a downward transfer of
momentum that is different from the usually observed downward transport of
momentum in the upper canopy. It could be explained by the strong stability
above the top of canopy, because strong stability substantially reduces the
downward transport of momentum (Mahrt et al., 2000). The momentum transfer is
reversed to upward (u′w′‾<0) when approaching the top of the
canopy where airflow is diverted into canopy layer because of the UDF and
shear production of turbulence. Strong upward momentum transfer near the top
of canopy on the lower slope is associated with the wake generation behind
the hill. In the upper canopy at mid-slope and downslope, shear stress decays
rapidly as z decreases, because of the momentum absorption by the dense
crown. The upward momentum (u′w′‾ < 0) in the lower-canopy
indicates momentum sources in the LDF on steep slope. The LDF was recognized
as a jet-like flow in the lower canopy, which has important effects on momentum
transfer within the canopy (Mao et al., 2007). Upward momentum transport in the
canopy is very common, occurring in stable atmospheric conditions (Zhang et
al., 2010). The opposite sign in momentum transfer near the slope surface on
a steep and gentle slope can be explained by the strength of LDF on the slope.
Profiles of shear stress, u′w′‾
(10-3 m2 s-2) on the slope for H/L = 0.6(blue) and
H/L = 1.0 (red). The locations of the six sections are labeled as (a)–(f),
and their locations with respect to the hill are marked in Fig. 3 with the
same letters.
The dominant positive turbulent heat flux, -w′θ′‾ indicates
downward heat transfer above and within the canopy (Fig. 8). Heat transfer on
the upper slope (Fig. 8a, b) is weak because the temperature difference
between the canopy and the atmosphere above is small. The downward heat
transfer is much stronger on the lower slope, where the air is cooled as a
“cool pool” with the greatest temperature gradient. Turbulent heat flux
increases towards the top of the canopy indicating increasing downward heat
transfer (-w′θ′‾>0) but the downward heat transfer
decreases in the upper-canopy layer. The peak of turbulent heat flux near the
top of the canopy is due to the strong radiative cooling in the upper canopy.
Below that the near-zero and slightly upward turbulent heat flux (Fig. 8) is
due to the near-neutral and negative temperature gradient in the thermal
transition zone. As a result of the strong cooling in the ground surface,
there are significant downward heat flux transfers in the lower canopy.
Profiles of turbulent heat flux, -w′θ′‾
(10-2 K m s-1) on the slope for H/L = 0.6 (blue) and
H/L = 1.0 (red). The locations of the six sections are labeled as (a)–(f),
and their locations with respect to the hill are marked in Fig. 3 with the
same letters.
Turbulent kinetic energy budget
In steady state, the TKE budget Eq. (13) can be written as
0=Ta+Tt+Tp+Ps+Pb+Pw-ε,
where Ta is the advection of TKE by the mean wind, Tt
represents the turbulent transport of TKE, Tp represents the
transport of TKE by pressure perturbation, Ps is the shear
production of TKE, Pb is buoyancy production of TKE, Pw
is wake production of TKE, and ε is viscous dissipation of TKE.
We calculate all the terms in the TKE budget equation individually except
Tp which is treated as the residual of other terms.
TKE is examined to show the intensity of turbulence along the slope (Fig. 9).
TKE is usually low within the canopy implying a low turbulence flow under
strongly stable atmospheric conditions. TKE is available near the top of
canopy on the mid-slope and downslope. The region with strongly shifting winds
is on the lower slope where the wind shear is strong. The largest TKE is
found in the region of wake vortices across the canopy edge. The TKE value is
larger on the gentle slope than on the steep slope.
Contours of turbulent kinetic energy (m2 s-2):
(a) H/L = 0.6; (b) H/L = 1.0. The black-dashed
lines indicate the top of canopy.
Contributions from transport and production terms of TKE are complicated.
Pb is a principal sink of TKE under stable conditions (Figs. 10 and
11). Pb exhibits negative values near the top of the canopy and
slope surface, where flow is stably stratified, which suppresses the
turbulence around the top of the canopy and within the deep canopy. In the
thermal transition zone, the contribution of Pb is minimal
(Pb ≈ 0 or slightly positive). Buoyancy production is
neglected in some studies because Pb is (1) unimportant compared
with other terms in TKE budget (Lesnik, 1974) and (2) difficult to measure
(Meyers and Baldocchi, 1991), restricting the modeling and measurement
studies to near-neutral conditions. Shen and Leclerc (1997) showed that near
the top of the canopy, the buoyancy production increases as instability
increases, although it is smaller than 10 % of shear production in
unstable conditions. Leclerc et al. (1990) illustrated a strong positive
correlation between buoyancy production and stability (Pb<0) or
instability (Pb>0) both within and above the canopy, which is
confirmed in our modeling results.
Profiles of TKE components (10-3 m2 s-3) for
H/L = 0.6. Ta is the advection of TKE by the mean wind,
Tt represents the turbulent transport of TKE, Tp represents
the transport of TKE by pressure perturbation, Ps is the shear
production of TKE, Pb is buoyancy production of TKE, Pw
is wake production of TKE and ε is viscous dissipation of TKE.
The locations of the six sections are labeled as (a)–(f), and their locations
with respect to the hill are marked in Fig. 3 with the same letters.
Wake production (Pw) is a principal source of TKE in the upper half
of the canopy where the canopy is dense (i.e., for large values of a and
Kr) on both steep and gentle slopes. Although the magnitude of
Pw is very small on a steep slope, the relative contribution of
Pw is very large in comparison with other TKE components. Even in
the lower-canopy layer on the upper slope, Pw is a dominant source
of TKE. This unusual phenomenon is induced by the deeper and stronger
drainage flow on the slope surface.
The positive shear production Ps indicates the net transfer of
kinetic energy from the mean flow to the turbulent component of the flow
(Figs. 10 and 11). Ps is smaller than Pw except near the
top of the canopy, which is consistent with the observations in soybeans
(Meyers and Paw U, 1986), deciduous forests (Shi et al., 1987; Meyers and
Baldocchi, 1991), and an artificial canopy (Raupach, 1987). Ps
peaks at the top of the canopy, due to strong wind shear. Shear production is
not as important as buoyancy and wake production in the canopy because of
strong stability. Observational data also showed that shear production
decreases with increasing stability in the lower two-thirds of the canopy
(Leclerc et al., 1990).
The same as in Fig. 10, but for H/L = 1.0.
Transport terms are the dominant source to maintain turbulent kinetic energy
near the top of the canopy where strong buoyancy suppression occurs (Figs. 10
and 11). TKE is weakly transported by turbulence upward near the canopy top
(Tt<0) and downward (Tt>0) in the canopy, because
turbulence is limited by strong stability above the canopy. TKE transport by
advection and turbulence is unimportant at all levels and all slopes in
comparison to pressure transport. The field measurement of pressure transport
Tp is difficult and the behavior of Tp in the TKE budget
is uncertain (Raupach et al., 1996; Finnigan, 2000). Maitani and Seo (1985),
Shaw et al. (1990) and Shaw and Zhang (1992) have confirmed that Tp
is not small enough to be neglected according to the surface pressure
measurements. Pressure diffusion is recognized as an important sink of TKE in
the upper canopy and source of TKE below (Dwyer et al., 1997), under unstable
conditions. Our results show that the contribution of pressure transport to
the overall TKE budget is significant when it is identified as a residual of
other TKE components. Tp, which is of the same order as the
production terms, supplies TKE in areas where the buoyancy suppression is
very strong and extracts TKE where wake production is dominant. On gentle
slopes, Tp is important for the compensation of TKE loss by
buoyancy near the top of the canopy and in the lower part of the canopy, and
it compensates TKE gain by wake motion in the upper half of the
canopy (Fig. 10). On steep
slopes, Tp on the lower half of the slope plays the same role as on
gentle slopes to compensate the TKE loss by buoyancy and gain by wake
(Fig. 11d–f), but the relative significance of wake production becomes more
prominent. On the upper slope (Fig. 11a–c), pressure transport is important
in the whole canopy to work against wake production. Our results suggest that
the pressure perturbation is stronger compared with other terms on steep
slopes. In addition, thermal effects on the upper steep slope are diminished
and the canopy effect is magnified since the air is warm and the temperature
gradient is small on the elevated topography.
Concluding remarks
Stably stratified canopy flows in complex terrain are investigated by a RNG
turbulent model, with emphasis on strong boundary effects, including
persistent thermal forcing from ground and canopy elements, damping force
from canopy drag elements, and buoyancy effects from temperature
stratification and topographic character.
The fundamental characteristics of nighttime canopy flow over complex terrain
are addressed by this numerical simulation as follows:
Multiple layering of thermal stratification. The stability around the
canopy is characterized by stratification with super-stable layers above the
top of the canopy and in the lower canopy, and an unstable layer within the
canopy (Figs. 2, 3, 4).
Bifurcation of thermal-driven drainage flows. The drainage flow above
the canopy is mainly driven by thermal stratification, being separated into
two streams in the canopy: the upper-canopy drainage flow (UDF) layer and the
lower-canopy drainage flow (LDF) layer (Figs. 1, 5, 6).
Buoyancy suppression of turbulence. The downward transport of momentum
and heat flux in the canopy is reduced due to strong stability and reversed
to be upward in the deep canopy (Figs. 7, 8). Buoyancy production suppresses
turbulence significantly near the top of the canopy and in the deep canopy
(Figs. 10, 11).
The thermal stratification and nocturnal drainage flows are interactive. The
drainage flows, initiated by thermal stratification, result in the formation
of super-stable layers. In addition, the drainage flows intensify the
temperature inversion down the slope, thus intensifying the stability of
super-stable layers. The properties of momentum and heat transfer may be
related to the “shear-driven” and “buoyancy-driven” coherent structures
that can lead to decoupling between the lower and upper canopy (Dupont and
Patton, 2012). Although an unstable layer is more likely to occur during the
foliated period (Dupont and Patton, 2012) and may only have influence on the
small-scale motions within the canopy (Jacob et al., 1992), the super-stable
layers associated with flow decoupling have direct influence on a larger
scale soil, within- and above-canopy exchange processes (Alekseychik, et al.,
2013).
The canopy flow behavior presented in Fig. 1 is expected to be measurable
directly by multiple eddy-flux towers that are equipped with multi-level
micrometeorological instruments (Feigenwinter et al., 2010; Baldocchi, 2008).
Some turbulent exchange processes remain uncertain and require further study,
including (i) how the varied vegetation structure, strength of background
wind, and ambient stability influence the within-canopy stratification and
turbulence, and (ii) how the complicated flows regulate scalar transfer
within the canopy and scalar exchange between the vegetation and atmosphere
aloft.