A new nonlinear analytical model for canopy flow over a forested hill

Abstract

A new nonlinear analytical model for canopy flow over gentle hills is presented. This model is established based on the assumption that three major forces (pressure gradient, Reynolds stress gradient, and nonlinear canopy drag) within canopy are in balance for gentle hills under neutral conditions. The momentum governing equation is closed by the velocity-squared law. This new model has many advantages over the model developed by Finnigan and Belcher (Quart J Roy Meteorol Soc 130: 1–29 2004, hereafter referred to as FB04) in predicting canopy wind velocity profiles in forested hills in that: (1) predictions from the new model are more realistic because surface drag effects can be taken into account by boundary conditions, while surface drag effects cannot be accounted for in the algebraic equation used in the lower canopy layer in the FB04 model; (2) the mixing length theory is not necessarily used because it leads to a theoretical inconsistency that a constant mixing length assumption leads to a nonconstant mixing length prediction as in the FB04 model; and (3) the effects of height-dependent leaf area density (a(z)) and drag coefficient (C _d ) on wind velocity can be predicted, while both a(z) and C _d must be treated as constants in FB04 model. The nonlinear algebraic equation for momentum transfer in the lower part of canopy used in FB04 model is height independent, actually serving as a bottom boundary condition for the linear differential momentum equation in the upper canopy layer. The predicting ability of the FB04 model is largely restricted by using the height-independent algebraic equation in the bottom canopy layer. This study has demonstrated the success of using the velocity-squared law as a closure scheme for momentum transfer in forested hills in comparison with the mixing length theory used in FB04 model thus enhancing the predicting ability of canopy flows, keeping the theory consistent and simple, and shining a new light into land-surface parameterization schemes in numerical weather and climate models.

Appendices

Appendix

1.1 Wind over a flat forested surface

Over a flat forested surface, the turbulent stress in the layer above the canopy (Fig. 1a) is governed (Sutton 1953; Wyngaard 1973; Yi 2008) by

$$ \frac{{\partial {\tau_B}}}{{\partial z}} \approx 0, $$

(22)

where ${\tau_B} = - \overline {u\prime w\prime }$ and is the kinematic turbulent stress, u′ and w′ are the fluctuations of velocity components in the horizontal and vertical, respectively. The mixing length model, in which

$$ {\tau_B}(z) = {\left( {\kappa \left( {z + d} \right)\frac{{\partial {U_B}}}{{\partial z}}} \right)^2}, $$

(23)

is valid in this layer, where d is the displacement height associated with the canopy and the origin of the vertical coordinate is taken at the canopy top. Based on (23), the mean velocity profile, U _B , can be derived from (22), i.e,

$$ {U_B}(z) = \frac{{{u_{*}}}}{\kappa }\ln \left( {\frac{{z + d}}{{{z_0}}}} \right), $$

(24)

where, z ₀ is the roughness length of the canopy, u _* is the friction velocity.

Within the canopy layer (Fig 1a, z = −h to 0, where h is the canopy depth), according to the hypothesis in Yi (2008), the governing equation of the kinematic turbulent stress for a dense canopy can be given by

$$ \frac{{\partial {\tau_B}(z)}}{{\partial z}} = {F_d} \approx a(z){\tau_B}(z), $$

(25)

where a(z) is LAD and F _d is the drag forcing exerted by the canopy. In Eq. 25, we have assumed that the drag on flow with a dense canopy is attributed largely to canopy elements except near the ground, i.e.,

$$ {\tau_B}(z) = {C_d}(z)U_B^2(z), $$

(26)

where, C _d (z) is a bulk drag coefficient and is a function of height and canopy morphology. The analytical solution of the kinematic turbulent stress derived from Eq. 25 is

$$ {\tau_B}(z) = {\tau_B}(0){e^{{ - \left( {LAI - L(z)} \right)}}}, $$

(27)

where, ${\tau_B}(0) = - \overline {u\prime w\prime } (0) = u_{*}^2$ is the kinematic turbulent stress at the canopy top, LAI is the leaf area index, and

$$ L(z) = \int_{{ - h}}^z {a\left( {z\prime } \right)} dz\prime, $$

(28)

is the cumulative leaf area per unit ground area below height z. Equation 27 indicates that the turbulent stress can be predicted by LAD profile alone, which is in excellent agreement with observations (Yi 2008).

The mean wind profile within the canopy can be derived from Eqs. 26 and 27 as

$$ {U_B}(z) = {U_h}{\left( {\frac{{{C_d}(0)}}{{{C_d}(z)}}} \right)^{{\frac{1}{2}}}}{e^{{ - \frac{1}{2}\left( {LAI - L(z)} \right)}}}, $$

(29)

where, U _h is the wind speed at the top of canopy.

Assuming that mean wind velocity and shear stress are continuous at the canopy top (z = 0), i.e., wind speed and its derivative with respect to z from (29) at the canopy top are equal to those from (24), and the shear stress from (27) at z = 0 is equal to that from (23), we have

$$ u_{*}^2 = {C_d}(0)U_h^2, $$

(30)

$$ {U_h} = \frac{{{u_{ * }}}}{\kappa }\ln \left( {\frac{d}{{{z_0}}}} \right), $$

(31)

$$ d = \frac{{2\sqrt {{{C_d}(0)}} }}{{\kappa \left[ { - \frac{1}{{{C_d}(0)}}\frac{{\partial {C_d}(0)}}{{\partial z}} + a(0)} \right]}}, $$

(32)

$$ {z_0} = d\exp \left( { - \frac{\kappa }{{\sqrt {{{C_d}(0)}} }}} \right) $$

(33)

If both a(z) and C _d (z) are constant, Eqs. 32 and 33 are reduced to those in Eq. 6 of FB04.

Wind above the canopy over a gentle hill

As in FB04, the shape of a sinusoidal hill (Fig. 1b) is described in the rectangular coordinate system (X, Z) as

$$ {Z_s} = \frac{1}{2}H\cos \left( {kX} \right) - h $$

(34)

where, Z _s is the surface height, H is the hill height, k is equal to π/(2L _h ), L _h is the hill half-length.

To obtain an analytical solution, two assumptions about the prescribed hill are made. First, the hill slope is sufficiently low, and perturbations to the background wind (U _B) above the canopy can be solved with linearized equations. Second, the hill is long enough. This means that L _h should be greater than 2L _c (Poggi et al. 2008), where L _c is a canopy adjustment length scale which is equal to 1/(C _{d 0} a ₀), C _{d 0} and a ₀ are the characteristic values for the canopy drag coefficient and LAD, respectively. In this case, the advection terms in the momentum equation may be negligible for a dense canopy over gentle terrain. This assumption has been supported by numerical experiments (Ross and Vosper 2005).

The same displaced coordinate system as in FB04 is used. The displaced (x, z) and the rectangular (X, Z) coordinate systems are related by,

$$ x = X + \frac{H}{2}\sin \left( {kX} \right){e^{{ - kZ}}}, $$

(35)

$$ z = Z - \frac{H}{2}\cos \left( {kX} \right){e^{{ - kZ}}}. $$

(36)

with this displaced (streamline) coordinate system, extra terms appear in the momentum equations (compared with those in a rectangular coordinate system), which are O(H ² /L _h ²) or smaller and, hence, may be negligible for the low slope hill (see FB04 for details). As a result, the streamwise (x direction) momentum equation can be written as,

$$ u\frac{{\partial u}}{{\partial x}} + w\frac{{\partial u}}{{\partial z}} = - \frac{{\partial p}}{{\partial x}} + \frac{{\partial \tau }}{{\partial z}} $$

(37)

where, u and w are the wind components in the x and z directions; receptively, p is the kinematic pressure, τ is the kinematic turbulent shear stress above the canopy, which is parameterized using the mixing length theory.

Under neutral conditions, the pressure perturbation in the inner region induced by the gentle sinusoidal hill is represented by

$$ \Delta p(x) = - \frac{1}{2}U_0^2Hk\exp \left( {ikx} \right), $$

(38)

(Jackson and Hunt 1975; Finnigan and Belcher 2004) and the horizontal PG forcing, driving the flow throughout the depth of the inner region (and canopy), is

$$ {\text{PG}} = {\rm Re} \left( {\frac{{\partial \Delta p}}{{\partial x}}} \right) = \frac{1}{2}U_0^2H{k^2}\sin \left( {kx} \right), $$

(39)

where, U ₀ is the characteristic wind velocity in the outer region and is estimated as the background wind at the middle layer height h _m . According to Hunt et al. (1988a, b), h _m is given by,

$$ \frac{{{h_m}}}{{{L_h}}}{\left( {\ln \left( {{h_m}/{z_0}} \right)} \right)^{{1/2}}} = 1, $$

(40)

provided that L _h is less than the boundary layer depth. The height of the inner region, h _i , is defined by,

$$ \frac{{{h_i}}}{{{L_h}}}\ln \left( {{h_i}/{z_0}} \right) = 2{\kappa^2}. $$

(41)

Assuming that the wind perturbation induced by terrain is small compared to the background wind (i.e., wind over the corresponding flat surface), Eq. 37 can be linearized. The resulting approximate solution for the streamwise velocity in the inner region above the canopy is,

$$ u\left( {x,z} \right) = {U_B}(z) + \Delta u\left( {x,z} \right), $$

(42)

where,

$$ \Delta u(x,z) = {\rm Re} \left\{ { - \frac{{\Delta p(x)}}{{{U_B}({h_i})}}\left[ {1 + \delta \left( {1 - \ln (\frac{{z + d}}{{{h_i}}}) - c{K_0}(2\sqrt {{ik{L_h}\frac{{z + d}}{{{h_i}}}}} )} \right)} \right]} \right\}, $$

(43)

δ = 1/ln(h _i /z ₀), and K ₀ is the modified Bessel function of order zero.The turbulent stress is,

$$ \tau \left( {x,z} \right) = {\tau_B} \left( {z} \right) + \Delta \tau \left( {x, z} \right), $$

(44)

where

$$ \Delta \tau \left( {x,z} \right) = 2\kappa {u_{ * }}\left( {z + d} \right)\frac{{\partial \Delta u\left( {x,z} \right)}}{{\partial z}}. $$

(45)

The integration constant c is determined by coupling (42) and (43) to the solutions for flow within the canopy at z = 0 (canopy top). Assuming that turbulent stress and velocity are continuous at z = 0, respectively, we have,

$$ {C_d}(0){\left[ {{U_B}(0) + \Delta u\left( {x,0} \right)} \right]^2} = {\tau_B}(0) + \Delta \tau \left( {x,0} \right). $$

(46)

It is noticed that the exact value of constant c (that should be independent of position x and z) may not be achieved because the wind speed perturbation above the canopy is linear in PG (a function of x) while it is nonlinear within the canopy. This is due to different simplifications of the governing equation above and within the canopy. An approximate solution is provided here. Since the velocity perturbation is small compared with U _B, the left side of the above equation can be approximated as ${C_d}(0)\left[ {U_B^{{\,2}}(0) + 2{U_B}(0)\Delta u\left( {x,0} \right)} \right]$. Substituting (23), (24), (43), and (45) into (46), we have,

$$ c = \frac{{ - {C_d}(0){U_B}(0){U_B}\left( {{h_i}} \right)\left[ {1 + 1{\text{n}}\left( {{h_i}/{z_0}} \right) - 1{\text{n}}\left( {d/{h_i}} \right)} \right] - u_{ * }^21{\text{n}}\left( {{h_i}/{z_0}} \right)}}{{u_{ * }^2d\left( {{{{\partial {K_0}(g)}} \left/ {{\partial z}} \right.}} \right)1{\text{n}}\left( {{{{{h_i}}} \left/ {{{z_0}}} \right.}} \right) - {C_d}(0){U_B}(0){U_B}\left( {{h_i}} \right){K_0}(g)}}, $$

(47)

where $g = 2\sqrt {{{\text{ik}}{L_h}\frac{{z + d}}{{{h_i}}}}}$. With the above approximate value of c, the resulting vertical profiles of wind and turbulent stress are approximately continuous but may not be smooth at the canopy top in some locations (i.e., their first derivatives with respect to z are not continuous at z = 0).

C _d near the ground

For the no-canopy case under a neutrally stratified atmosphere, the drag coefficient is given by

$$ {C_d}(z) = {\left[ {\frac{\kappa }{{1{\text{n}}\left( {{{z} \left/ {{{z_{{g0}}}}} \right.}} \right)}}} \right]^2}, $$

(48)

where, z _g0 is the roughness length of the ground. Equation 48 indicates that C _d is infinite on the ground and decreases dramatically with height near the ground. Variations in C _d are smaller at higher levels. For example, variation in C _d is smaller than 0.03 for z between 10z _g0 and 10³ z _g0. To account for significant variations in C _d near the ground, where the drag effect exerted by the ground is superior to that by canopy, we assume that C _d follows (48) below a level z _L . Thus, we can rewrite (48) as

$$ {C_d}(z) = {C_d}\left( {{z_L}} \right){\left( {\frac{{1{\text{n}}\left( {{{{{z_L}}} \left/ {{{z_{{g0}}}}} \right.}} \right)}}{{1{\text{n}}\left( {{{z} \left/ {{{z_{{g0}}}}} \right.}} \right)}}} \right)^2}, $$

(49)

where, C _d (z _L )is the drag coefficient at z _L and z _g0 is taken to be 0.1 m in the study.