Parameterizations

The Canopy-App model includes various parameterizations for physical and chemical processes within and above forest canopies. This section provides detailed descriptions of the scientific formulations and parameters used.

Turbulence Parameterization

K-Theory Approach

The model uses K-theory to parameterize turbulent transport:

\[ \overline{w'\phi'} = -K_{\phi} \frac{\partial \phi}{\partial z} \]

where: - \(\overline{w'\phi'}\) is the turbulent flux of scalar \(\phi\) - \(K_{\phi}\) is the eddy diffusivity for scalar \(\phi\) - \(z\) is height above ground

Mixing Length Model

The eddy diffusivity is calculated using a mixing length approach:

\[ K_m = l_m^2 \sqrt{\left(\frac{\partial U}{\partial z}\right)^2 + \left(\frac{\partial V}{\partial z}\right)^2} \]

\[ K_h = \frac{K_m}{\Pr_t} \]

where: - \(K_m\) is momentum diffusivity - \(K_h\) is heat/scalar diffusivity - \(l_m\) is mixing length - \(\Pr_t\) is turbulent Prandtl number (typically 0.7-1.3)

Mixing Length Formulation

Within the canopy, the mixing length is parameterized as:

\[ l_m(z) = \begin{cases} \beta h \left(\frac{z}{h}\right)^n & \text{for } z < h \\ \kappa z & \text{for } z \geq h \end{cases} \]

where: - \(h\) is canopy height - \(\beta\) is scaling parameter (typically 0.1-0.3) - \(n\) is shape parameter (typically 1-3) - \(\kappa\) is von Karman constant (0.41)

Radiation Parameterization

Leaf Area Distribution

The cumulative leaf area index from canopy top:

\[ LAI(z) = LAI_{total} \exp\left(-\alpha \left(\frac{h-z}{h}\right)^{\beta}\right) \]

Parameters: - \(\alpha\): extinction coefficient (typically 0.5-2.0) - \(\beta\): shape parameter (typically 0.5-2.0)

Biogenic Emissions

Isoprene Emissions

Following Guenther et al. (2012):

\[ E_{iso} = \epsilon_{iso} \cdot \gamma_{CE} \cdot \gamma_{age} \cdot \gamma_{SM} \cdot D \cdot LAI \]

where: - \(\epsilon_{iso}\): emission factor - \(\gamma_{CE}\): canopy environment activity factor - \(\gamma_{age}\): leaf age activity factor - \(\gamma_{SM}\): soil moisture activity factor - \(D\): light distribution factor

Temperature and Light Dependence

\[ \gamma_{CE} = C_T \cdot C_L \]

\[ C_T = \frac{\exp\left(\frac{C_{T1}(T-T_s)}{RT_sT}\right)}{1 + \exp\left(\frac{C_{T2}(T-T_{M})}{RT_sT}\right)} \]

\[ C_L = \frac{\alpha C_L1 PPFD}{\sqrt{1 + \alpha^2 PPFD^2}} \]

Parameters: - \(C_{T1} = 95,000\) J/mol - \(C_{T2} = 230,000\) J/mol - \(T_s = 303\) K (standard temperature) - \(T_M = 314\) K (maximum temperature) - \(\alpha = 0.0027\) (empirical coefficient) - \(C_{L1} = 1.066\) (empirical coefficient)

Dry Deposition

Resistance Model

Total deposition velocity:

\[ v_d = \frac{1}{r_a + r_b + r_c} \]

where: - \(r_a\): aerodynamic resistance - \(r_b\): quasi-laminar boundary layer resistance - \(r_c\): canopy resistance

Canopy Resistance

\[ \frac{1}{r_c} = \frac{1}{r_s + r_m} + \frac{1}{r_{lu}} + \frac{1}{r_{dc}} + \frac{1}{r_{cl}} \]

where: - \(r_s\): stomatal resistance - \(r_m\): mesophyll resistance - \(r_{lu}\): resistance of upper canopy - \(r_{dc}\): resistance of lower canopy/ground - \(r_{cl}\): resistance of exposed surfaces

Parameter Tables

Vegetation-Specific Parameters

Parameter	Conifer	Deciduous	Grass	Units
\(V_{c,max}\) (25°C)	60	80	40	μmol/m²/s
\(J_{max}\) (25°C)	120	160	80	μmol/m²/s
\(g_1\)	3.0	4.0	5.0	kPa^0.5
\(\epsilon_{iso}\)	0.1	10.0	0.0	μg/g/h
\(c_d\)	0.15	0.20	0.10	-

Temperature Response Parameters

Parameter	Value	Units	Description
\(Q_{10,V}\)	2.0	-	\(V_{c,max}\) temperature sensitivity
\(Q_{10,J}\)	1.9	-	\(J_{max}\) temperature sensitivity
\(Q_{10,R}\)	2.0	-	Respiration temperature sensitivity
\(H_a\)	72000	J/mol	Activation energy
\(H_d\)	200000	J/mol	Deactivation energy
\(\Delta S\)	650	J/mol/K	Entropy term

Sensitivity Analysis

Key sensitive parameters identified through sensitivity analysis:

Leaf area index (±20% → ±15% flux change)
Drag coefficient (±50% → ±25% wind change)
Maximum carboxylation rate (±20% → ±18% photosynthesis change)

References

Key scientific references for parameterizations:

Farquhar et al. (1980): Photosynthesis model
Ball et al. (1987): Stomatal conductance
Guenther et al. (2012): Biogenic emissions
Raupach & Thom (1981): Canopy turbulence
Dai et al. (2004): Two-stream radiation

For implementation details, see the API Reference.