Analytically Modelling Mean Wind and Stress Profiles in Canopies

Abstract

An analytical model for mean wind profiles in sparse canopies (W. Wang, Boundary-Layer Meteorol 142:383–399, 2012) has been further developed, with (1) an explicit solution being derived, and (2) a linear term being added to the $K$-closure scheme to improve the shear-stress parametrization when the contribution of non-local transport is significant. Results from large-eddy simulations and from laboratory experiments are used to evaluate the model and adjust model parameters, showing that the model can well simulate canopy wind and stress profiles not only for sparse-canopy scenarios, but also for dense-canopy scenarios. The analytical solution converges exactly to the standard surface-layer logarithmic wind profile in the case of zero canopy density, and tends to an exponential wind profile for a dense canopy.

Appendices

Appendices

1.1 A.1 LES Experiments

LES experiments were conducted to derive the dependence on canopy density of wind and stress profiles within a canopy. The model described in Bryan and Fritsch (2002) was used to simulate canopy flow over a flat terrain under neutral conditions; this model has been successfully applied to simulating atmospheric boundary-layer turbulence under various conditions (Kang and Davis 2008; Wang and Davis 2008; Wang 2009, 2010; Wang and Rotach 2010). Canopies are assumed to be horizontally homogeneous and uniform in the vertical. To account for the drag imposed by canopy elements, a forcing term, $D_i =-C_\mathrm{d} a_0 Vu_i$, is added to the momentum equation, where $u_{i}$ is the resolved-scale instantaneous velocity, subscript $i$ is taken to be 1, 2, or 3 referring to the velocity or force in the $x$, $y$, or $z$ directions, respectively. $V$ represents the resolved-scale instantaneous wind speed, while $C_\mathrm{d} = 0.25$. A turbulence kinetic energy based subgrid-scale model (Deardorff 1980) is used to parametrize the subgrid-scale turbulence, with an inclusion of the effects of the canopy drag on kinetic energy, such as wake effects (Shaw and Schumann 1992; Dwyer et al. 1997; Shaw and Patton 2003; Patton and Katul 2009). A third-order Runge–Kutta time integration scheme is used, with a third-order vertical advection scheme and fifth-order horizontal advection scheme. The simulation domain is $660 \times 330\;\hbox {m}^{2}$ in the streamwise $(x)$ and spanwise $(y)$ directions, respectively, with a horizontal grid of 1 m. The vertical grid size is 0.5 m below 16 and 3.5 m above 50 m, and varies linearly in between, with the top of the domain at 330 m. Given the limited computational resource, the domain is large enough to capture most of the scales of motions in the canopy boundary layer, according to Finnigan et al. (2009) and Patton and Katul (2009). A logarithmic law is used between the ground and the first model level, with a roughness length of 0.05 m. The lateral boundary condition is cyclic, and a rigid lid is used as the top boundary condition. Turbulence is triggered by randomly adding perturbations to initial velocity fields, while the Coriolis force is not considered for this small-scale simulation. Flow is driven by a pressure gradient of $3.2\times 10^{-5}$ Pa m$^{-1}$ imposed in the $x$ direction above the canopy top (15 m), resulting in $u_{*}$ being approximately 0.1 m $\hbox {s}^{-1}$ after the turbulent flow reaches an equilibrium state. The model is integrated for 15 h for each run, and results from the last 5 h are averaged over time and in the $x$ and $y$ directions to approximate the ensemble averages of atmospheric variables, such as the first and second moments of the turbulent velocity fields. Seven LES runs with $FAI = 0.1, 0.2, 0.4, 0.8, 1.2, 3.0$, and 5.0, ranging from sparse to dense canopies, were made.

1.2 A.2 Turbulence Closure

Ayotte et al. (1999) showed that standard parametrizations (without canopies) for the budget equations of the second moments may be appropriate for canopy flow if the energy dissipation is adjusted to reflect canopy dynamics (Finnigan 2000). Starting from the budget equation for the shearing stress in the time- and volume-averaged flow, Finnigan and Belcher (2004) derived a simple non-local closure parametrization scheme for the stress in the canopy roughness sublayer. For a steady horizontally homogeneous flow, the parametrized budget equation for the shearing stress can be written, after some rearrangement and simplifications, as

$$\begin{aligned} -\overline{{u^{\prime }}{w^{\prime }}} =K(z)\frac{\partial U}{\partial z}- \frac{\partial }{\partial z}\left( {A(z)\frac{\partial \overline{{u^{\prime }}{w^{\prime }}}}{\partial z}}\right) , \end{aligned}$$

(20)

where we have used the same symbols as in Finnigan and Belcher (2004), the dimension of coefficient $A(z)$ is the squared length, $A(z)\propto l^{2}$, where l is a height-dependent length scale. The first term on the right-hand side is the product of an eddy diffusivity and the local vertical gradient of mean wind speed and is termed a local term. The second term is associated with the parametrized turbulence/pressure transport and pressure-strain processes, which may be called a non-local term. If the non-local term is negligible, Eq. 20 can be approximated as Eq. 4. To use Eq. 20 in the analytical model, we have to simplify the second term. Here a scaling method is used, and by expanding the non-local term, we have,

$$\begin{aligned} \frac{\partial }{\partial z}\left( {A(z)\frac{\partial \overline{{u^{\prime }}{w^{\prime }}}}{\partial z}}\right) =\frac{\partial A(z)}{\partial z}\frac{\partial \overline{{u^{\prime }}{w^{\prime }}}}{\partial z}\;+A(z)\frac{\partial ^{2}\overline{{u^{\prime }}{w^{\prime }}}}{\partial z^{2}}. \end{aligned}$$

(21)

Finnigan and Belcher (2004) postulated that the length scale is a constant for a dense canopy, but for sparse and open canopies, the ground stress is not negligible, and therefore we assume that the length scale to first order is proportional to height above the ground, i.e., $A(z)\propto z^{2}$. As a result, with $A(z)\propto z^{2}$, the stress ($\overline{{u^{\prime }}{w^{\prime }}})$ scaled by $u_*^2$ and height by h, the first term on the right-hand side of Eq. 21 can be scaled with $u_*^2 z/h$, and the second term can be scaled with $u_*^2 \left( {z/h}\right) ^{2}$, which is smaller than $u_*^2 z/h$ for $z<$ h within the canopy. Therefore, with the second derivative term being neglected, the non-local term can be scaled with $u_*^2 z/h$. To a first-order approximation, we can estimate the non-local term to be proportional to $u_*^2 z/h$, i.e. ${f}_{n}u_*^2 z/h$, where ${f}_{n}$ is a height-independent coefficient. Despite the simplification, the parametrized stress (Eq. 9) can reasonably reproduce the stress profiles within canopies, with errors being within 10–20 % compared with laboratory measurements (Fig. 4) and LES results (Fig. 1).

1.3 A.3 Wind Profile as FAI Approaches Zero

Here we show that Eq. 11 reduces to the standard logarithmic wind profile in the case of canopy density being zero. As the order ($\nu$) is fixed and $x\rightarrow 0$, the limiting forms of the modified Bessel functions are,

$$\begin{aligned} I_{\nu } (x)&\rightarrow \frac{1}{2}\frac{x^{\nu }}{\Gamma (\nu +1)},\end{aligned}$$

(22)

$$\begin{aligned} K_0 (x)&\rightarrow -\ln (x),\end{aligned}$$

(23)

$$\begin{aligned} K_1 (x)&\rightarrow \frac{\Gamma (1)}{x}, \end{aligned}$$

(24)

where $\Gamma (n)=(n-1)!$ is the gamma function. As FAI decreases to zero, $f_n$ approaches zero and $s_h$ goes to 1. In this case, $g(h)$ tends to zero. With the above properties, and neglecting the pressure gradient forcing term, the integration constants (Eq. 16) become,

$$\begin{aligned} C_1&\rightarrow -\frac{4u_*}{\kappa }\ln (g(z_0)),\end{aligned}$$

(25)

$$\begin{aligned} C_2&\rightarrow -\frac{2u_*}{\kappa },\end{aligned}$$

(26)

$$\begin{aligned} I_0 (g(z))&\rightarrow \frac{1}{2}, \end{aligned}$$

(27)

and

$$\begin{aligned} K_0 (g(z))\rightarrow -\ln (g(z)). \end{aligned}$$

(28)

Therefore, substituting Eqs. 25–28 into Eq. 11, leads to,

$$\begin{aligned} U(z)&= C_1 I_0 (g(z))+C_2 K_0 (g(z))\nonumber \\&\rightarrow -\frac{2u_*}{\kappa }\ln (g(z_0))+\frac{2u_*}{\kappa }\ln (g(z))=\frac{u_*}{k}\ln \left( {\frac{z}{z_0}}\right) . \end{aligned}$$

(29)