### 2.5.3.1. Theory[¶](#theory "Permalink to this headline")

The air within the canopy is assumed to have negligible capacity to store heat so that the sensible heat flux \\(H\\) between the surface at height \\(z\_{0h} +d\\) and the atmosphere at height \\(z\_{atm,\\, h}\\) must be balanced by the sum of the sensible heat from the vegetation \\(H\_{v}\\) and the ground \\(H\_{g}\\)

(2.5.89)[¶](#equation-5-86 "Permalink to this equation")\\\[H=H\_{v} +H\_{g}\\\]

where, with reference to [Figure 2.5.1](#figure-schematic-diagram-of-sensible-heat-fluxes),

(2.5.90)[¶](#equation-5-87 "Permalink to this equation")\\\[H=-\\rho \_{atm} C\_{p} \\frac{\\left(\\theta \_{atm} -T\_{s} \\right)}{r\_{ah} }\\\]

(2.5.91)[¶](#equation-5-88 "Permalink to this equation")\\\[H\_{v} =-\\rho \_{atm} C\_{p} \\left(T\_{s} -T\_{v} \\right)\\frac{\\left(L+S\\right)}{r\_{b} }\\\]

(2.5.92)[¶](#equation-5-89 "Permalink to this equation")\\\[H\_{g} =\\left(1-f\_{sno} -f\_{h2osfc} \\right)H\_{soil} +f\_{sno} H\_{snow} +f\_{h2osfc} H\_{h2osfc} \\ ,\\\]

where

(2.5.93)[¶](#equation-5-90 "Permalink to this equation")\\\[H\_{soil} =-\\rho \_{atm} C\_{p} \\frac{\\left(T\_{s} -T\_{1} \\right)}{r\_{ah} ^{{'} } }\\\]

(2.5.94)[¶](#equation-5-91 "Permalink to this equation")\\\[H\_{sno} =-\\rho \_{atm} C\_{p} \\frac{\\left(T\_{s} -T\_{snl+1} \\right)}{r\_{ah} ^{{'} } }\\\]

(2.5.95)[¶](#equation-5-92 "Permalink to this equation")\\\[H\_{h2osfc} =-\\rho \_{atm} C\_{p} \\frac{\\left(T\_{s} -T\_{h2osfc} \\right)}{r\_{ah} ^{{'} } }\\\]

where \\(\\rho \_{atm}\\) is the density of atmospheric air (kg m\-3), \\(C\_{p}\\) is the specific heat capacity of air (J kg\-1 K\-1) ([Table 2.2.7](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Ecosystem/CLM50_Tech_Note_Ecosystem.html#table-physical-constants)), \\(\\theta \_{atm}\\) is the atmospheric potential temperature (K), and \\(r\_{ah}\\) is the aerodynamic resistance to sensible heat transfer (s m\-1).

Here, \\(T\_{s}\\) is the surface temperature at height \\(z\_{0h} +d\\), also referred to as the canopy air temperature. \\(L\\) and \\(S\\) are the exposed leaf and stem area indices (section [2.2.1.4](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Ecosystem/CLM50_Tech_Note_Ecosystem.html#phenology-and-vegetation-burial-by-snow)), \\(r\_{b}\\) is the leaf boundary layer resistance (s m\-1), and \\(r\_{ah} ^{{'} }\\) is the aerodynamic resistance (s m\-1) to heat transfer between the ground at height \\(z\_{0h} ^{{'} }\\) and the canopy air at height \\(z\_{0h} +d\\).

![Image 1: ../../_images/image12.png](https://escomp.github.io/ctsm-docs/versions/master/html/_images/image12.png)

Figure 2.5.1 Figure Schematic diagram of sensible heat fluxes for (a) non-vegetated surfaces and (b) vegetated surfaces.[¶](#id8 "Permalink to this image")

![Image 2: ../../_images/image2.png](https://escomp.github.io/ctsm-docs/versions/master/html/_images/image2.png)

Figure 2.5.2 Figure Schematic diagram of water vapor fluxes for (a) non-vegetated surfaces and (b) vegetated surfaces.[¶](#id9 "Permalink to this image")

Equations [(2.5.89)](#equation-5-86) - [(2.5.92)](#equation-5-89) can be solved for the canopy air temperature \\(T\_{s}\\)

(2.5.96)[¶](#equation-5-93 "Permalink to this equation")\\\[T\_{s} =\\frac{c\_{a}^{h} \\theta \_{atm} +c\_{g}^{h} T\_{g} +c\_{v}^{h} T\_{v} }{c\_{a}^{h} +c\_{g}^{h} +c\_{v}^{h} }\\\]

where

(2.5.97)[¶](#equation-5-94 "Permalink to this equation")\\\[c\_{a}^{h} =\\frac{1}{r\_{ah} }\\\]

(2.5.98)[¶](#equation-5-95 "Permalink to this equation")\\\[c\_{g}^{h} =\\frac{1}{r\_{ah} ^{{'} } }\\\]

(2.5.99)[¶](#equation-5-96 "Permalink to this equation")\\\[c\_{v}^{h} =\\frac{\\left(L+S\\right)}{r\_{b} }\\\]

are the sensible heat conductances from the canopy air to the atmosphere, the ground to canopy air, and leaf surface to canopy air, respectively (m s\-1).

When the expression for \\(T\_{s}\\) is substituted into equation [(2.5.91)](#equation-5-88), the sensible heat flux from vegetation \\(H\_{v}\\) is a function of \\(\\theta \_{atm}\\), \\(T\_{g}\\), and \\(T\_{v}\\)

(2.5.100)[¶](#equation-5-97 "Permalink to this equation")\\\[H\_{v} = -\\rho \_{atm} C\_{p} \\left\[c\_{a}^{h} \\theta \_{atm} +c\_{g}^{h} T\_{g} -\\left(c\_{a}^{h} +c\_{g}^{h} \\right)T\_{v} \\right\]\\frac{c\_{v}^{h} }{c\_{a}^{h} +c\_{v}^{h} +c\_{g}^{h} } .\\\]

Similarly, the expression for \\(T\_{s}\\) can be substituted into equations [(2.5.92)](#equation-5-89), [(2.5.93)](#equation-5-90), [(2.5.94)](#equation-5-91), and [(2.5.95)](#equation-5-92) to obtain the sensible heat flux from ground \\(H\_{g}\\)

(2.5.101)[¶](#equation-5-98 "Permalink to this equation")\\\[H\_{g} = -\\rho \_{atm} C\_{p} \\left\[c\_{a}^{h} \\theta \_{atm} +c\_{v}^{h} T\_{v} -\\left(c\_{a}^{h} +c\_{v}^{h} \\right)T\_{g} \\right\]\\frac{c\_{g}^{h} }{c\_{a}^{h} +c\_{v}^{h} +c\_{g}^{h} } .\\\]

The air within the canopy is assumed to have negligible capacity to store water vapor so that the water vapor flux \\(E\\) between the surface at height \\(z\_{0w} +d\\) and the atmosphere at height \\(z\_{atm,\\, w}\\) must be balanced by the sum of the water vapor flux from the vegetation \\(E\_{v}\\) and the ground \\(E\_{g}\\)

(2.5.102)[¶](#equation-5-99 "Permalink to this equation")\\\[E = E\_{v} +E\_{g}\\\]

where, with reference to [Figure 2.5.2](#figure-schematic-diagram-of-latent-heat-fluxes),

(2.5.103)[¶](#equation-5-100 "Permalink to this equation")\\\[E = -\\rho \_{atm} \\frac{\\left(q\_{atm} -q\_{s} \\right)}{r\_{aw} }\\\]

(2.5.104)[¶](#equation-5-101 "Permalink to this equation")\\\[E\_{v} = -\\rho \_{atm} \\frac{\\left(q\_{s} -q\_{sat}^{T\_{v} } \\right)}{r\_{total} }\\\]

(2.5.105)[¶](#equation-5-102 "Permalink to this equation")\\\[E\_{g} = \\left(1-f\_{sno} -f\_{h2osfc} \\right)E\_{soil} +f\_{sno} E\_{snow} +f\_{h2osfc} E\_{h2osfc} \\ ,\\\]

where

(2.5.106)[¶](#equation-5-103 "Permalink to this equation")\\\[E\_{soil} = -\\rho \_{atm} \\frac{\\left(q\_{s} -q\_{soil} \\right)}{r\_{aw} ^{{'} } +r\_{soil} }\\\]

(2.5.107)[¶](#equation-5-104 "Permalink to this equation")\\\[E\_{sno} = -\\rho \_{atm} \\frac{\\left(q\_{s} -q\_{sno} \\right)}{r\_{aw} ^{{'} } +r\_{soil} }\\\]

(2.5.108)[¶](#equation-5-105 "Permalink to this equation")\\\[E\_{h2osfc} = -\\rho \_{atm} \\frac{\\left(q\_{s} -q\_{h2osfc} \\right)}{r\_{aw} ^{{'} } +r\_{soil} }\\\]

where \\(q\_{atm}\\) is the atmospheric specific humidity (kg kg\-1), \\(r\_{aw}\\) is the aerodynamic resistance to water vapor transfer (s m\-1), \\(q\_{sat}^{T\_{v} }\\) (kg kg\-1) is the saturation water vapor specific humidity at the vegetation temperature (section [2.5.5](#saturation-vapor-pressure)), \\(q\_{g}\\), \\(q\_{sno}\\), and \\(q\_{h2osfc}\\) are the specific humidities of the soil, snow, and surface water (section [2.5.2](#sensible-and-latent-heat-fluxes-for-non-vegetated-surfaces)), \\(r\_{aw} ^{{'} }\\) is the aerodynamic resistance (s m\-1) to water vapor transfer between the ground at height \\(z\_{0w} ^{{'} }\\) and the canopy air at height \\(z\_{0w} +d\\), and \\(r\_{soil}\\) ([(2.5.76)](#equation-5-76)) is a resistance to diffusion through the soil (s m\-1). \\(r\_{total}\\) is the total resistance to water vapor transfer from the canopy to the canopy air and includes contributions from leaf boundary layer and sunlit and shaded stomatal resistances \\(r\_{b}\\), \\(r\_{s}^{sun}\\), and \\(r\_{s}^{sha}\\) ([Figure 2.5.2](#figure-schematic-diagram-of-latent-heat-fluxes)). The water vapor flux from vegetation is the sum of water vapor flux from wetted leaf and stem area \\(E\_{v}^{w}\\) (evaporation of water intercepted by the canopy) and transpiration from dry leaf surfaces \\(E\_{v}^{t}\\)

(2.5.109)[¶](#equation-5-106 "Permalink to this equation")\\\[E\_{v} =E\_{v}^{w} +E\_{v}^{t} .\\\]

Equations [(2.5.102)](#equation-5-99) - [(2.5.105)](#equation-5-102) can be solved for the canopy specific humidity \\(q\_{s}\\)

(2.5.110)[¶](#equation-5-107 "Permalink to this equation")\\\[q\_{s} =\\frac{c\_{a}^{w} q\_{atm} +c\_{g}^{w} q\_{g} +c\_{v}^{w} q\_{sat}^{T\_{v} } }{c\_{a}^{w} +c\_{v}^{w} +c\_{g}^{w} }\\\]

where

(2.5.111)[¶](#equation-5-108 "Permalink to this equation")\\\[c\_{a}^{w} =\\frac{1}{r\_{aw} }\\\]

(2.5.112)[¶](#equation-5-109 "Permalink to this equation")\\\[c\_{v}^{w} =\\frac{\\left(L+S\\right)}{r\_{b} } r''\\\]

(2.5.113)[¶](#equation-5-110 "Permalink to this equation")\\\[c\_{g}^{w} =\\frac{1}{r\_{aw} ^{{'} } +r\_{soil} }\\\]

are the water vapor conductances from the canopy air to the atmosphere, the leaf to canopy air, and ground to canopy air, respectively. The term \\(r''\\) is determined from contributions by wet leaves and transpiration and limited by available water and potential evaporation as

(2.5.114)[¶](#equation-5-111 "Permalink to this equation")\\\[\\begin{split}r'' = \\left\\{ \\begin{array}{lr} \\min \\left(f\_{wet} +r\_{dry} ^{{'} {'} } ,\\, \\frac{E\_{v}^{w,\\, pot} r\_{dry} ^{{'} {'} } +\\frac{W\_{can} }{\\Delta t} }{E\_{v}^{w,\\, pot} } \\right) & \\qquad E\_{v}^{w,\\, pot} >0,\\, \\beta \_{t} >0 \\\\ \\min \\left(f\_{wet} ,\\, \\frac{E\_{v}^{w,\\, pot} r\_{dry} ^{{'} {'} } +\\frac{W\_{can} }{\\Delta t} }{E\_{v}^{w,\\, pot} } \\right) & \\qquad E\_{v}^{w,\\, pot} >0,\\, \\beta \_{t} \\le 0 \\\\ 1 & \\qquad E\_{v}^{w,\\, pot} \\le 0 \\end{array}\\right\\}\\end{split}\\\]

where \\(f\_{wet}\\) is the fraction of leaves and stems that are wet (section [2.7.1](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Hydrology/CLM50_Tech_Note_Hydrology.html#canopy-water)), \\(W\_{can}\\) is canopy water (kg m\-2) (section [2.7.1](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Hydrology/CLM50_Tech_Note_Hydrology.html#canopy-water)), \\(\\Delta t\\) is the time step (s), and \\(\\beta \_{t}\\) is a soil moisture function limiting transpiration (Chapter [2.9](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Photosynthesis/CLM50_Tech_Note_Photosynthesis.html#rst-stomatal-resistance-and-photosynthesis)). The potential evaporation from wet foliage per unit wetted area is

(2.5.115)[¶](#equation-5-112 "Permalink to this equation")\\\[E\_{v}^{w,\\, pot} =-\\frac{\\rho \_{atm} \\left(q\_{s} -q\_{sat}^{T\_{v} } \\right)}{r\_{b} } .\\\]

The term \\(r\_{dry} ^{{'} {'} }\\) is

(2.5.116)[¶](#equation-5-113 "Permalink to this equation")\\\[r\_{dry} ^{{'} {'} } =\\frac{f\_{dry} r\_{b} }{L} \\left(\\frac{L^{sun} }{r\_{b} +r\_{s}^{sun} } +\\frac{L^{sha} }{r\_{b} +r\_{s}^{sha} } \\right)\\\]

where \\(f\_{dry}\\) is the fraction of leaves that are dry (section [2.7.1](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Hydrology/CLM50_Tech_Note_Hydrology.html#canopy-water)), \\(L^{sun}\\) and \\(L^{sha}\\) are the sunlit and shaded leaf area indices (section [2.4.1](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Radiative_Fluxes/CLM50_Tech_Note_Radiative_Fluxes.html#solar-fluxes)), and \\(r\_{s}^{sun}\\) and \\(r\_{s}^{sha}\\) are the sunlit and shaded stomatal resistances (s m\-1) (Chapter [2.9](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Photosynthesis/CLM50_Tech_Note_Photosynthesis.html#rst-stomatal-resistance-and-photosynthesis)).

When the expression for \\(q\_{s}\\) is substituted into equation [(2.5.104)](#equation-5-101), the water vapor flux from vegetation \\(E\_{v}\\) is a function of \\(q\_{atm}\\), \\(q\_{g}\\), and \\(q\_{sat}^{T\_{v} }\\)

(2.5.117)[¶](#equation-5-114 "Permalink to this equation")\\\[E\_{v} =-\\rho \_{atm} \\left\[c\_{a}^{w} q\_{atm} +c\_{g}^{w} q\_{g} -\\left(c\_{a}^{w} +c\_{g}^{w} \\right)q\_{sat}^{T\_{v} } \\right\]\\frac{c\_{v}^{w} }{c\_{a}^{w} +c\_{v}^{w} +c\_{g}^{w} } .\\\]

Similarly, the expression for \\(q\_{s}\\) can be substituted into [(2.5.87)](#equation-5-84) to obtain the water vapor flux from the ground beneath the canopy \\(E\_{g}\\)

(2.5.118)[¶](#equation-5-115 "Permalink to this equation")\\\[E\_{g} =-\\rho \_{atm} \\left\[c\_{a}^{w} q\_{atm} +c\_{v}^{w} q\_{sat}^{T\_{v} } -\\left(c\_{a}^{w} +c\_{v}^{w} \\right)q\_{g} \\right\]\\frac{c\_{g}^{w} }{c\_{a}^{w} +c\_{v}^{w} +c\_{g}^{w} } .\\\]

The aerodynamic resistances to heat (moisture) transfer between the ground at height \\(z\_{0h} ^{{'} }\\) (\\(z\_{0w} ^{{'} }\\) ) and the canopy air at height \\(z\_{0h} +d\\) (\\(z\_{0w} +d\\)) are

(2.5.119)[¶](#equation-5-116 "Permalink to this equation")\\\[r\_{ah} ^{{'} } =r\_{aw} ^{{'} } =\\frac{1}{C\_{s} U\_{av} }\\\]

where

(2.5.120)[¶](#equation-5-117 "Permalink to this equation")\\\[U\_{av} =V\_{a} \\sqrt{\\frac{1}{r\_{am} V\_{a} } } =u\_{\*}\\\]

is the magnitude of the wind velocity incident on the leaves (equivalent here to friction velocity) (m s\-1) and \\(C\_{s}\\) is the turbulent transfer coefficient between the underlying soil and the canopy air. \\(C\_{s}\\) is obtained by interpolation between values for dense canopy and bare soil ([Zeng et al. 2005](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#zengetal2005))

(2.5.121)[¶](#equation-5-118 "Permalink to this equation")\\\[C\_{s} =C\_{s,\\, bare} W+C\_{s,\\, dense} (1-W)\\\]

where the weight \\(W\\) is

(2.5.122)[¶](#equation-5-119 "Permalink to this equation")\\\[W=e^{-\\left(L+S\\right)} .\\\]

The dense canopy turbulent transfer coefficient ([Dickinson et al. 1993](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#dickinsonetal1993)) is

(2.5.123)[¶](#equation-5-120 "Permalink to this equation")\\\[C\_{s,\\, dense} =0.004 \\ .\\\]

The bare soil turbulent transfer coefficient is

(2.5.124)[¶](#equation-5-121 "Permalink to this equation")\\\[C\_{s,\\, bare} =\\frac{k}{a} \\left(\\frac{z\_{0m,\\, g} U\_{av} }{\\upsilon } \\right)^{-0.45}\\\]

where the kinematic viscosity of air \\(\\upsilon =1.5\\times 10^{-5}\\) m2 s\-1 and \\(a=0.13\\).

The leaf boundary layer resistance \\(r\_{b}\\) is

(2.5.125)[¶](#equation-5-122 "Permalink to this equation")\\\[r\_{b} =\\frac{1}{C\_{v} } \\left({U\_{av} \\mathord{\\left/ {\\vphantom {U\_{av} d\_{leaf} }} \\right.} d\_{leaf} } \\right)^{{-1\\mathord{\\left/ {\\vphantom {-1 2}} \\right.} 2} }\\\]

where \\(C\_{v} =0.01\\) ms\-1/2 is the turbulent transfer coefficient between the canopy surface and canopy air, and \\(d\_{leaf}\\) is the characteristic dimension of the leaves in the direction of wind flow ([Table 2.5.1](#table-plant-functional-type-aerodynamic-parameters)).

The partial derivatives of the fluxes from the soil beneath the canopy with respect to ground temperature, which are needed for the soil temperature calculations (section [2.6.1](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Soil_Snow_Temperatures/CLM50_Tech_Note_Soil_Snow_Temperatures.html#numerical-solution-temperature)) and to update the soil surface fluxes (section [2.5.4](#update-of-ground-sensible-and-latent-heat-fluxes)), are

(2.5.126)[¶](#equation-5-123 "Permalink to this equation")\\\[\\frac{\\partial H\_{g} }{\\partial T\_{g} } = \\frac{\\rho \_{atm} C\_{p} }{r'\_{ah} } \\frac{c\_{a}^{h} +c\_{v}^{h} }{c\_{a}^{h} +c\_{v}^{h} +c\_{g}^{h} }\\\]

(2.5.127)[¶](#equation-5-124 "Permalink to this equation")\\\[\\frac{\\partial E\_{g} }{\\partial T\_{g} } = \\frac{\\rho \_{atm} }{r'\_{aw} +r\_{soil} } \\frac{c\_{a}^{w} +c\_{v}^{w} }{c\_{a}^{w} +c\_{v}^{w} +c\_{g}^{w} } \\frac{dq\_{g} }{dT\_{g} } .\\\]

The partial derivatives \\(\\frac{\\partial r'\_{ah} }{\\partial T\_{g} }\\) and \\(\\frac{\\partial r'\_{aw} }{\\partial T\_{g} }\\), which cannot be determined analytically, are ignored for \\(\\frac{\\partial H\_{g} }{\\partial T\_{g} }\\) and \\(\\frac{\\partial E\_{g} }{\\partial T\_{g} }\\).

The roughness lengths used to calculate \\(r\_{am}\\), \\(r\_{ah}\\), and \\(r\_{aw}\\) from [(2.5.55)](#equation-5-55), [(2.5.56)](#equation-5-56), and [(2.5.57)](#equation-5-57) are \\(z\_{0m} =z\_{0m,\\, v}\\), \\(z\_{0h} =z\_{0h,\\, v}\\), and \\(z\_{0w} =z\_{0w,\\, v}\\).

The vegetation roughness lengths and displacement height \\(d\\) are from [Meier et al. (2022)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#meieretal2022)

(2.5.128)[¶](#equation-5-125 "Permalink to this equation")\\\[z\_{0m,\\, v} = z\_{0h,\\, v} =z\_{0w,\\, v} = z\_{top} (1 - \\frac{d} {z\_{top} } ) \\exp (\\psi\_{h} - \\frac{k U\_{h}} {u\_{\*} } )\\\]

where \\(z\_{top}\\) is canopy top height (m) ([Table 2.2.2](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Ecosystem/CLM50_Tech_Note_Ecosystem.html#table-plant-functional-type-canopy-top-and-bottom-heights)), \\(k\\) is the von Karman constant ([Table 2.2.7](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Ecosystem/CLM50_Tech_Note_Ecosystem.html#table-physical-constants)), and \\(\\psi\_{h}\\) is the roughness sublayer influence function

(2.5.129)[¶](#equation-5-125a "Permalink to this equation")\\\[\\psi\_{h} = \\ln(c\_{w}) - 1 + c\_{w}^{-1}\\\]

where \\(c\_{w}\\) is a pft-dependent constant ([Table 2.5.1](#table-plant-functional-type-aerodynamic-parameters)).

The ratio of wind speed at canopy height to friction velocity, \\(\\frac{U\_{h}} {u\_{\*}}\\) is derived from an implicit function of the roughness density \\(\\lambda\\)

(2.5.130)[¶](#equation-5-125b "Permalink to this equation")\\\[\\frac{U\_{h}} {u\_{\*} } =(C\_{S} + \\lambda C\_{R})^{0.5} \\exp(\\frac{\\min \\left(\\lambda, \\lambda\_{\\max}\\right) c U\_{h}} {2 u\_{\*}})\\\]

where \\(C\_{S}\\) represents the drag coefficient of the ground in the absence of vegetation, \\(C\_{R}\\) is the drag coefficient of an isolated roughness element (plant), and \\(c\\) is an empirical constant. These three are pft-dependent parameters ([Table 2.5.1](#table-plant-functional-type-aerodynamic-parameters)). \\(\\lambda\_{max}\\) is the maximum \\(\\lambda\\) above which \\(\\frac{U\_{h}} {u\_{\*}}\\) becomes constant. \\(\\lambda\_{max}\\) is set to the value of \\(\\lambda\\) for which [(2.5.130)](#equation-5-125b), in the absence of \\(\\lambda\_{max}\\), would have its minimum. \\(\\lambda\_{max}\\) is also a pft-dependent parameter ([Table 2.5.1](#table-plant-functional-type-aerodynamic-parameters)). [(2.5.130)](#equation-5-125b) can be written as

(2.5.131)[¶](#equation-5-125c "Permalink to this equation")\\\[X \\exp(-X) =(C\_{S} + \\lambda C\_{R})^{0.5} c \\frac{\\lambda} {2 }\\\]

where

(2.5.132)[¶](#equation-5-125d "Permalink to this equation")\\\[X =\\frac{c \\lambda U\_{h}} {2 u\_{\*} }.\\\]

\\(X\\) and therefore \\(\\frac{U\_{h}} {u\_{\*}}\\) can be solved for iteratively where the initial value of \\(X\\) is

(2.5.133)[¶](#equation-5-125e "Permalink to this equation")\\\[X\_{i=0} =(C\_{S} + \\lambda C\_{R})^{0.5} c \\frac{\\lambda} {2 }\\\]

and the next value of \\(X\\) at \\(i+1\\) is

(2.5.134)[¶](#equation-5-125f "Permalink to this equation")\\\[X\_{i+1} =(C\_{S} + \\lambda C\_{R})^{0.5} c \\frac{\\lambda} {2 } \\exp(X\_{i}).\\\]

\\(X\\) is updated until \\(\\frac{U\_{h}} {u\_{\*}}\\) converges to within \\(1 \\times 10^{-4}\\) between iterations.

\\(\\lambda\\) is set to half the total single-sided area of all canopy elements, here defined as the vegetation area index (VAI) defined as the sum of leaf (\\(L\\)) and stem area index (\\(S\\)), subject to a maximum of \\(\\lambda\_{max}\\) and a minimum limit applied for numerical stability

(2.5.135)[¶](#equation-5-126 "Permalink to this equation")\\\[\\lambda = \\frac{\\min(\\max(1 \\times 10^{-5}, VAI), \\lambda\_{max})} {2 }\\\]

The displacement height \\(d\\) is

(2.5.136)[¶](#equation-5-127 "Permalink to this equation")\\\[d = z\_{top}\\left\[1- \\frac{1-\\exp(-(c\_{d1} 2 \\lambda)^{0.5}} {(c\_{d1} 2 \\lambda)^{0.5} }\\right\]\\\]

where \\(c\_{d1} =7.5\\).

Table 2.5.1 Plant functional type aerodynamic parameters[¶](#id10 "Permalink to this table")       
| Plant functional type
 | \\(d\_{leaf}\\) (m)

 | \\(c\_{w}\\)

 | \\(C\_{S}\\)

 | \\(C\_{R}\\)

 | \\(c\\)

 | \\(\\lambda\_{max}\\)

 |
| --- | --- | --- | --- | --- | --- | --- |
| NET Temperate

 | 0.04

 | 9

 | 0.003

 | 0.05

 | 0.09

 | 4.55

 |
| NET Boreal

 | 0.04

 | 9

 | 0.003

 | 0.05

 | 0.09

 | 4.55

 |
| NDT Boreal

 | 0.04

 | 9

 | 0.003

 | 0.05

 | 0.09

 | 4.55

 |
| BET Tropical

 | 0.04

 | 3

 | 0.01

 | 0.14

 | 0.01

 | 7.87

 |
| BET temperate

 | 0.04

 | 3

 | 0.01

 | 0.14

 | 0.01

 | 7.87

 |
| BDT tropical

 | 0.04

 | 1

 | 0.013

 | 0.13

 | 0.06

 | 8.88

 |
| BDT temperate

 | 0.04

 | 1

 | 0.013

 | 0.13

 | 0.06

 | 8.88

 |
| BDT boreal

 | 0.04

 | 1

 | 0.013

 | 0.13

 | 0.06

 | 8.88

 |
| BES temperate

 | 0.04

 | 20

 | 0.001

 | 0.05

 | 0.12

 | 3.07

 |
| BDS temperate

 | 0.04

 | 20

 | 0.001

 | 0.05

 | 0.12

 | 3.07

 |
| BDS boreal

 | 0.04

 | 20

 | 0.001

 | 0.05

 | 0.12

 | 3.07

 |
| C3 arctic grass

 | 0.04

 | 19

 | 0.001

 | 0.05

 | 0.08

 | 4.61

 |
| C3 grass

 | 0.04

 | 19

 | 0.001

 | 0.05

 | 0.08

 | 4.61

 |
| C4 grass

 | 0.04

 | 19

 | 0.001

 | 0.05

 | 0.08

 | 4.61

 |
| Crop R

 | 0.04

 | 3.5

 | 0.001

 | 0.05

 | 0.04

 | 5.3

 |
| Crop I

 | 0.04

 | 3.5

 | 0.001

 | 0.05

 | 0.04

 | 5.3

 |
| Corn R

 | 0.04

 | 3.5

 | 0.001

 | 0.05

 | 0.04

 | 5.3

 |
| Corn I

 | 0.04

 | 3.5

 | 0.001

 | 0.05

 | 0.04

 | 5.3

 |
| Temp Cereal R

 | 0.04

 | 3.5

 | 0.001

 | 0.05

 | 0.04

 | 5.3

 |
| Temp Cereal I

 | 0.04

 | 3.5

 | 0.001

 | 0.05

 | 0.04

 | 5.3

 |
| Winter Cereal R

 | 0.04

 | 3.5

 | 0.001

 | 0.05

 | 0.04

 | 5.3

 |
| Winter Cereal I

 | 0.04

 | 3.5

 | 0.001

 | 0.05

 | 0.04

 | 5.3

 |
| Soybean R

 | 0.04

 | 3.5

 | 0.001

 | 0.05

 | 0.04

 | 5.3

 |
| Soybean I

 | 0.04

 | 3.5

 | 0.001

 | 0.05

 | 0.04

 | 5.3

 |
| Miscanthus R

 | 0.04

 | 3.5

 | 0.001

 | 0.05

 | 0.04

 | 5.3

 |
| Miscanthus I

 | 0.04

 | 3.5

 | 0.001

 | 0.05

 | 0.04

 | 5.3

 |
| Switchgrass R

 | 0.04

 | 3.5

 | 0.001

 | 0.05

 | 0.04

 | 5.3

 |
| Switchgrass I

 | 0.04

 | 3.5

 | 0.001

 | 0.05

 | 0.04

 | 5.3

 |

