Roughness sublayer model
Wind profile above canopy
The wind profile, \(u\), above canopy the is given as
\[\begin{equation}
\label{eqn_du_dz_above}
\frac{\kappa (z-d)}{u_* }\frac{\partial u }{\partial z} = \Phi_m\left( z \right)
\end{equation}\]
where \(\Phi_m\) [-] is an effective similarity function that is given by
\[\begin{equation}
\label{eqn_phi_m}
\Phi_m(z) = \phi_m \left( \frac{z-d}{L_{MO}}\right) \hat{\phi}_m \left( \frac{z-d}{l_m /\beta }\right)
\end{equation}\]
\[\begin{equation}
u(z) = \frac{u_*}{\kappa}
\left[
\ln \left( \frac{z - d}{z_{0m}} \right)
- \psi_m\left( \frac{z - d}{L_{MO}} \right)
+ \psi_m\left( \frac{z_{0m}}{L_{MO}} \right)
+ \hat{\psi}_m(z)
\right]
\end{equation}\]
where
\[\begin{equation}
\label{eqn:wind_profile_rsl_above}
\hat{\psi}_m = \int_z^\infty \phi_m \left( \frac{z' - d}{L_{MO}} \right)
\left[ 1 - \hat{\phi}_m \left(\frac{z' -d }{l_m/\beta} \right) \right]
\frac{dz'}{z' - d}
\end{equation}\]
Given equation \eqref{eqn:wind_profile_rsl_above}, the wind at canopy height is given by
\[\begin{equation}
\label{eqn:wind_at_hc}
u_{hc} = \frac{u_*}{\beta}
\end{equation}\]
\[\begin{eqnarray}
u(z) &=& \frac{u_*}{\kappa}
\left[
\ln \left( \frac{z - d}{h_c - d} \right)
- \psi_m\left( \frac{z - d}{L_{MO}} \right)
+ \psi_m\left( \frac{h_c - d}{L_{MO}} \right) \right. \nonumber \\
& &
\label{eqn_u_with_hc}
\left.
+ \hat{\psi_m}(z)
- \hat{\psi_m}(h_c)
+ \frac{\kappa}{\beta}
\right]
\end{eqnarray}\]
Wind profile within canopy
HF-2007 assumed an exponential wind profile within the canopy that is given as
\[\begin{equation}
\label{eqn_du_dz_within}
u(z) = u(h_c)\exp\left( \frac{z - h_c}{l_m/\beta} \right)
\end{equation}\]
and the derivative of the wind profile is
\[\begin{eqnarray}
\frac{\partial u(z)}{\partial z} &=& \frac{u(h_c)}{l_m/\beta} \exp\left( \frac{z - h_c}{l_m/\beta} \right) \nonumber \\
&=& \frac{u(z)}{l_m/\beta}
\end{eqnarray}\]
\[\begin{equation}
h_c - d = \frac{l_m}{2\beta} = \beta^2 L_c
\end{equation}\]
Enforcing the continuity of derivative of wind at canopy height (\(h_c\)) from
equation \eqref{eqn_du_dz_above} and \eqref{eqn_du_dz_within} leads to
\[\begin{eqnarray}
\frac{u_*}{\kappa (h_c - d)} \Phi_m(h_c) &=& \frac{u(h_c)}{l_m/\beta} \nonumber \\
\Phi_m(h_c) &=& \frac{u(h_c)}{u_*} \times \frac{\kappa (h_c - d)}{l_m/\beta} \nonumber \\
\Phi_m(h_c) &=& \frac{1}{\beta} \times \frac{\kappa l_m/(2\beta) }{l_m/\beta} \nonumber \\
\label{eqn_phi_m_at_hc}
\Phi_m(h_c) &=& \frac{\kappa l_m}{2\beta}
\end{eqnarray}\]
Wind similarity function
For the above canopy wind profile, HF-2007 assumed the similarity function for momentum to be
\[\begin{equation}
\hat{\phi}_m \left( \frac{z-d}{l_m /\beta} \right) =
1 - c_1 \exp \left[ -c_2 \left( \frac{z-d}{l_m/\beta} \right)\right]
\end{equation}\]
where
\(c_1\) and \(c_2 (= 0.5)\) are parameters. The parameter \(c_1\) is found by evaluating \(\hat{\phi}_m\) at \(z=h_c\)
that gives
\[\begin{eqnarray}
\hat{\phi}_m \left( \frac{h_c - d}{l_m /\beta} \right) &=& 1 - c_1 \exp \left[ -c_2 \left( \frac{h_c-d}{l_m/\beta} \right)\right] \nonumber \\
&=& 1 - c_1 \exp \left[ -c_2 \left( \frac{l_m/(2\beta)}{l_m/\beta} \right)\right] \nonumber \\
&=& 1 - c_1 \exp \left( -0.5c_2\right) \nonumber \\
c_1 &=& \left[ 1 - \hat{\phi}_m \left( \frac{h_c - d}{l_m /\beta} \right)\right] \exp (0.5c_2)
\end{eqnarray}\]
In order to compute \(c_1\) from the above equation, an expression of \(\hat{\phi}_m\) at \(z = h_c\) is needed,
which is obtained using equation \eqref{eqn_phi_m} at \(z = h_c\) and equation \eqref{eqn_phi_m_at_hc} as
\[\begin{equation}
\hat{\phi}_m \left( \frac{h_c-d}{l_m/\beta}\right) = \frac{\kappa}{2\beta} \phi_m^{-1} \left( \frac{h_c-d}{L_{MO}}\right)
\end{equation}\]
Wind beta term
The critical unknown in the roughness sublayer parameterization is \(\beta\).
HF-2007 derived an expression for \(\beta\) as
\[\begin{equation}
\beta \phi_m \left( \frac{h_c - d}{L_{MO}} \right) = \beta_N
\end{equation}\]
or
\[\begin{equation}
\label{eqn_beta}
\beta \phi_m \left( \frac{\beta^3 L_c}{L_{MO}} \right) = \beta_N
\end{equation}\]
The solution of equation \eqref{eqn_beta} depends on if \(\phi\) is evaluated for unstable or stable condition
and leads to following equations
\[\begin{eqnarray}
(\beta^2)^2 + 16 \frac{L_c}{L_{MO}}\beta^4\beta^2 - \beta_N^4 &=& 0 L_{MO} < 0 \\
5 \frac{L_c}{L_{MO}} \beta^3 + \beta - \beta_N &=& 0 L_{MO} \geq 0
\end{eqnarray}\]
The correct solution is the larger root for the sable condition, while the
unstable case has only one real root.
Temperature profile
Similar to the equations for wind profiles, the equations describing profiles
of heat (or scalar) above and within the canopy are given below.
\[\begin{eqnarray}
\label{eqn_theta_du_dz_1}
\frac{\kappa (z-d)}{\theta_*}\frac{\partial \theta}{\partial z} &=& \Phi_c\left( z \right) \\
\Phi_c(z) &=& \phi_c \left( \frac{z-d}{L_{MO}}\right) \hat{\phi}_c \left( \frac{z-d}{l_m /\beta }\right) \\
\theta(z) - \theta_s &=& \frac{\theta_*}{\kappa}
\left[
\ln \left( \frac{z - d}{z_{0c}} \right)
- \psi_m\left( \frac{z - d}{L_{MO}} \right)
+ \psi_m\left( \frac{z_{0c}}{L_{MO}} \right)
+ \hat{\psi}_c(z)
\right]
\end{eqnarray}\]
where
\[\begin{equation}
\hat{\psi}_c = \int_z^\infty \phi_c \left( \frac{z' - d}{L_{MO}} \right)
\left[ 1 - \hat{\phi}_c \left(\frac{z' -d }{l_m/\beta} \right) \right]
\frac{dz'}{z' - d}
\end{equation}\]
An exponential profile of air temperature is assumed within the canopy that is given by
\begin{equation}
\theta(z) - \theta_s = (\theta(h_c) - \theta_s) \exp\left[ \frac{f(z - h_c)}{l_m/\beta} \right]
\end{equation}
where parameter \(f\) relates the length scale of heat (scalar) to that of momentum and is given by
\[\begin{equation}
f = \frac{1}{2} (1 + 4r_c\text{Pr})^{1/2} - \frac{1}{2}
\end{equation}\]
and
\[\begin{equation}
\text{Pr} = 0.5 + 0.3\tanh(2L_c/L_{MO})
\end{equation}\]
The derivatives of the profile within the canopy is
\[\begin{equation}
\label{eqn_theta_du_dz_2}
\frac{\partial \theta (z)}{\partial z} = \frac{(\theta(h_c) - \theta_s) f}{l_m/\beta}\exp\left[ \frac{f(z - h_c)}{l_m/\beta} \right]
\end{equation}\]
Enforcing continuity of derivative at \(z = h_c\) using equation \eqref{eqn_theta_du_dz_1} and \eqref{eqn_theta_du_dz_2}
\[\begin{equation}
\left. \frac{\partial \theta}{\partial z}\right|_{z=h_c} = \frac{\theta_*}{\kappa (h_c - d)}\Phi_c(h_c) = \frac{f[\theta(h_c) -\theta_s]}{l_m/\beta}
\end{equation}\]
so that
\[\begin{equation}
\frac{\theta(h_c) - \theta_s}{\theta_*} = \frac{\text{Pr}}{f\beta}
\end{equation}\]
An equation similar to equation \eqref{eqn_u_with_hc} can be written for \(\theta\)
\[\begin{eqnarray}
\theta(z) - \theta_s &=& \frac{\theta_*}{\kappa}
\left[
\ln \left( \frac{z - d}{h_c - d} \right)
- \psi_c\left( \frac{z - d}{L_{MO}} \right)
+ \psi_c\left( \frac{h_c - d}{L_{MO}} \right)
\right. \nonumber \\
\label{eqn_q_with_hc}
& & \left.
+ \hat{\psi_c}(z)
- \hat{\psi_c}(h_c)
+ \frac{\kappa\text{Pr}}{f\beta}
\right]
\end{eqnarray}\]
For the above canopy wind profile, HF-2007 assumed the similarity function for momentum to be
\[\begin{equation}
\hat{\phi}_c \left( \frac{z-d}{l_m /\beta} \right) =
1 - c_1 \exp \left[ -c_2 \left( \frac{z-d}{l_m/\beta} \right)\right]
\end{equation}\]
\[\begin{eqnarray}
c_1 &=&
\left[ 1 - \hat{\phi}_c \left( \frac{h_c - d}{l_m /\beta} \right)\right] \exp (0.5c_2) \\
\hat{\phi}_c \left( \frac{z-d}{l_m/\beta}\right) &=&
\frac{\kappa \text{Pr}}{2\beta} \phi_c^{-1} \left( \frac{z-d}{L_{MO}}\right)
\end{eqnarray}\]
Aerodynamic conductance
The aerodynamic conductances for scalar, \(g_{ac}\), is given as
\[\begin{equation}
\frac{1}{g_{ac}} = \int_{z_1}^{z_2} \frac{dz}{\rho_m K_c}
\end{equation}\]
where \(K_c\) is the eddy diffusivity for scalar based on K-theory.
The aerodynamic conductance above canopy is
\begin{equation}
g_{am,i+1/2} = \rho_m \kappa^2 u_*
\left[
\ln \left( \frac{z_{i+1} - d}{z_i - d} \right) +
\psi_{i+1} - \psi_i
\right]^{-1}
\end{equation}
where
\[\begin{equation}
\psi_i =
- \psi_c\left( \frac{z_i - d}{L_{MO}} \right)
+ \psi_c\left( \frac{h_c - d}{L_{MO}} \right)
+ \hat{\psi_c}(z_i)
- \hat{\psi_c}(h_c)
\end{equation}\]
The aerodynamic conductance within canopy is
\[\begin{equation}
g_{ac,i+1/2} = \frac{\rho \beta u_*}{\text{Pr}}
\left[
\exp
\left(
\frac{-(z_{i+1}-h_c)}{l_m/\beta}
\right)
-
\left(
\frac{-(z_i-h_c)}{l_m/\beta}
\right)
\right]^{-1}
\end{equation}\]