## 2.5.2. Sensible and Latent Heat Fluxes for Non-Vegetated Surfaces[¶](#sensible-and-latent-heat-fluxes-for-non-vegetated-surfaces "Permalink to this headline")
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Surfaces are considered non-vegetated for the surface flux calculations if leaf plus stem area index \\(L+S<0.05\\) (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)). By definition, this includes bare soil and glaciers. The solution for lakes is described in Chapter [2.12](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Lake/CLM50_Tech_Note_Lake.html#rst-lake-model). For these surfaces, the surface may be exposed to the atmosphere, snow covered, and/or surface water covered, so that the sensible heat flux \\(H\_{g}\\) (W m\-2) is, with reference to [Figure 2.5.1](#figure-schematic-diagram-of-sensible-heat-fluxes),

(2.5.62)[¶](#equation-5-62 "Permalink to this equation")\\\[H\_{g} =\\left(1-f\_{sno} -f\_{h2osfc} \\right)H\_{soil} +f\_{sno} H\_{snow} +f\_{h2osfc} H\_{h2osfc}\\\]

where \\(\\left(1-f\_{sno} -f\_{h2osfc} \\right)\\), \\(f\_{sno}\\), and \\(f\_{h2osfc}\\) are the exposed, snow covered, and surface water covered fractions of the grid cell. The individual fluxes based on the temperatures of the soil \\(T\_{1}\\), snow \\(T\_{snl+1}\\), and surface water \\(T\_{h2osfc}\\) are

(2.5.63)[¶](#equation-5-63 "Permalink to this equation")\\\[H\_{soil} =-\\rho \_{atm} C\_{p} \\frac{\\left(\\theta \_{atm} -T\_{1} \\right)}{r\_{ah} }\\\]

(2.5.64)[¶](#equation-5-64 "Permalink to this equation")\\\[H\_{sno} =-\\rho \_{atm} C\_{p} \\frac{\\left(\\theta \_{atm} -T\_{snl+1} \\right)}{r\_{ah} }\\\]

(2.5.65)[¶](#equation-5-65 "Permalink to this equation")\\\[H\_{h2osfc} =-\\rho \_{atm} C\_{p} \\frac{\\left(\\theta \_{atm} -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).

The water vapor flux \\(E\_{g}\\) (kg m\-2 s\-1) is, with reference to [Figure 2.5.2](#figure-schematic-diagram-of-latent-heat-fluxes),

(2.5.66)[¶](#equation-5-66 "Permalink to this equation")\\\[E\_{g} =\\left(1-f\_{sno} -f\_{h2osfc} \\right)E\_{soil} +f\_{sno} E\_{snow} +f\_{h2osfc} E\_{h2osfc}\\\]

(2.5.67)[¶](#equation-5-67 "Permalink to this equation")\\\[E\_{soil} =-\\frac{\\rho \_{atm} \\left(q\_{atm} -q\_{soil} \\right)}{r\_{aw} + r\_{soil}}\\\]

(2.5.68)[¶](#equation-5-68 "Permalink to this equation")\\\[E\_{sno} =-\\frac{\\rho \_{atm} \\left(q\_{atm} -q\_{sno} \\right)}{r\_{aw} }\\\]

(2.5.69)[¶](#equation-5-69 "Permalink to this equation")\\\[E\_{h2osfc} =-\\frac{\\rho \_{atm} \\left(q\_{atm} -q\_{h2osfc} \\right)}{r\_{aw} }\\\]

where \\(q\_{atm}\\) is the atmospheric specific humidity (kg kg\-1), \\(q\_{soil}\\), \\(q\_{sno}\\), and \\(q\_{h2osfc}\\) are the specific humidities (kg kg\-1) of the soil, snow, and surface water, respectively, \\(r\_{aw}\\) is the aerodynamic resistance to water vapor transfer (s m\-1), and \\(r \_{soi}\\) is the soil resistance to water vapor transfer (s m\-1). The specific humidities of the snow \\(q\_{sno}\\) and surface water \\(q\_{h2osfc}\\) are assumed to be at the saturation specific humidity of their respective temperatures

(2.5.70)[¶](#equation-5-70 "Permalink to this equation")\\\[q\_{sno} =q\_{sat}^{T\_{snl+1} }\\\]

(2.5.71)[¶](#equation-5-71 "Permalink to this equation")\\\[q\_{h2osfc} =q\_{sat}^{T\_{h2osfc} }\\\]

The specific humidity of the soil surface \\(q\_{soil}\\) is assumed to be proportional to the saturation specific humidity

(2.5.72)[¶](#equation-5-72 "Permalink to this equation")\\\[q\_{soil} =\\alpha \_{soil} q\_{sat}^{T\_{1} }\\\]

where \\(q\_{sat}^{T\_{1} }\\) is the saturated specific humidity at the soil surface temperature \\(T\_{1}\\) (section [2.5.5](#saturation-vapor-pressure)). The factor \\(\\alpha \_{soil}\\) is a function of the surface soil water matric potential \\(\\psi\\) as in [Philip (1957)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#philip1957)

(2.5.73)[¶](#equation-5-73 "Permalink to this equation")\\\[\\alpha \_{soil} =\\exp \\left(\\frac{\\psi \_{1} g}{1\\times 10^{3} R\_{wv} T\_{1} } \\right)\\\]

where \\(R\_{wv}\\) is the gas constant for water vapor (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)), \\(g\\) is the gravitational acceleration (m s\-2) ([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 \_{1}\\) is the soil water matric potential of the top soil layer (mm). The soil water matric potential \\(\\psi \_{1}\\) is

(2.5.74)[¶](#equation-5-74 "Permalink to this equation")\\\[\\psi \_{1} =\\psi \_{sat,\\, 1} s\_{1}^{-B\_{1} } \\ge -1\\times 10^{8}\\\]

where \\(\\psi \_{sat,\\, 1}\\) is the saturated matric potential (mm) (section [2.7.3.1](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Hydrology/CLM50_Tech_Note_Hydrology.html#hydraulic-properties)), \\(B\_{1}\\) is the [Clapp and Hornberger (1978)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#clapphornberger1978) parameter (section [2.7.3.1](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Hydrology/CLM50_Tech_Note_Hydrology.html#hydraulic-properties)), and \\(s\_{1}\\) is the wetness of the top soil layer with respect to saturation. The surface wetness \\(s\_{1}\\) is a function of the liquid water and ice content

(2.5.75)[¶](#equation-5-75 "Permalink to this equation")\\\[s\_{1} =\\frac{1}{\\Delta z\_{1} \\theta \_{sat,\\, 1} } \\left\[\\frac{w\_{liq,\\, 1} }{\\rho \_{liq} } +\\frac{w\_{ice,\\, 1} }{\\rho \_{ice} } \\right\]\\qquad 0.01\\le s\_{1} \\le 1.0\\\]

where \\(\\Delta z\_{1}\\) is the thickness of the top soil layer (m), \\(\\rho \_{liq}\\) and \\(\\rho \_{ice}\\) are the density of liquid water and ice (kg m\-3) ([Table 2.2.7](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Ecosystem/CLM50_Tech_Note_Ecosystem.html#table-physical-constants)), \\(w\_{liq,\\, 1}\\) and \\(w\_{ice,\\, 1}\\) are the mass of liquid water and ice of the top soil layer (kg m\-2) (Chapter [2.7](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Hydrology/CLM50_Tech_Note_Hydrology.html#rst-hydrology)), and \\(\\theta \_{sat,\\, 1}\\) is the saturated volumetric water content (i.e., porosity) of the top soil layer (mm3 mm\-3) (section [2.7.3.1](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Hydrology/CLM50_Tech_Note_Hydrology.html#hydraulic-properties)). If \\(q\_{sat}^{T\_{1} } >q\_{atm}\\) and \\(q\_{atm} >q\_{soil}\\), then \\(q\_{soil} =q\_{atm}\\) and \\(\\frac{dq\_{soil} }{dT} =0\\). This prevents large increases (decreases) in \\(q\_{soil}\\) for small increases (decreases) in soil moisture in very dry soils.

The resistance to water vapor transfer occurring within the soil matrix \\(r\_{soil}\\) (s m\-1) is

(2.5.76)[¶](#equation-5-76 "Permalink to this equation")\\\[r\_{soil} = \\frac{DSL}{D\_{v} \\tau}\\\]

where \\(DSL\\) is the thickness of the dry surface layer (m), \\(D\_{v}\\) is the molecular diffusivity of water vapor in air (m2 s\-2) and \\(\\tau\\) (_unitless_) describes the tortuosity of the vapor flow paths through the soil matrix ([Swenson and Lawrence 2014](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#swensonlawrence2014)).

The thickness of the dry surface layer is given by

(2.5.77)[¶](#equation-5-77 "Permalink to this equation")\\\[\\begin{split}DSL = \\begin{array}{lr} D\_{max} \\ \\frac{\\left( \\theta\_{init} - \\theta\_{1}\\right)} {\\left(\\theta\_{init} - \\theta\_{air}\\right)} & \\qquad \\theta\_{1} < \\theta\_{init} \\\\ 0 & \\qquad \\theta\_{1} \\ge \\theta\_{init} \\end{array}\\end{split}\\\]

where \\(D\_{max}\\) is a parameter specifying the length scale of the maximum DSL thickness (default value = 15 mm), \\(\\theta\_{init}\\) (mm3 mm\-3) is the moisture value at which the DSL initiates, \\(\\theta\_{1}\\) (mm3 mm\-3) is the moisture value of the top model soil layer, and \\(\\theta\_{air}\\) (mm3 mm\-3) is the ‘air dry’ soil moisture value ([Dingman 2002](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#dingman2002)):

(2.5.78)[¶](#equation-5-78 "Permalink to this equation")\\\[\\theta\_{air} = \\Phi \\left( \\frac{\\Psi\_{sat}}{\\Psi\_{air}} \\right)^{\\frac{1}{B\_{1}}} \\ .\\\]

where \\(\\Phi\\) is the porosity (mm3 mm\-3), \\(\\Psi\_{sat}\\) is the saturated soil matric potential (mm), \\(\\Psi\_{air} = 10^{7}\\) mm is the air dry matric potential, and \\(B\_{1}\\) is a function of soil texture (section [2.7.3.1](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Hydrology/CLM50_Tech_Note_Hydrology.html#hydraulic-properties)).

The soil tortuosity is

(2.5.79)[¶](#equation-5-79 "Permalink to this equation")\\\[\\tau = \\Phi^{2}\_{air}\\left(\\frac{\\Phi\_{air}}{\\Phi}\\right)^{\\frac{3}{B\_{1}}}\\\]

where \\(\\Phi\_{air}\\) (mm3 mm\-3) is the air filled pore space

(2.5.80)[¶](#equation-5-80 "Permalink to this equation")\\\[\\Phi\_{air} = \\Phi - \\theta\_{air} \\ .\\\]

\\(D\_{v}\\) depends on temperature

(2.5.81)[¶](#equation-5-81 "Permalink to this equation")\\\[D\_{v} = 2.12 \\times 10^{-5} \\left(\\frac{T\_{1}}{T\_{f}}\\right)^{1.75} \\ .\\\]

where \\(T\_{1}\\) (K) is the temperature of the top soil layer and \\(T\_{f}\\) (K) is the freezing temperature of water ([Table 2.2.7](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Ecosystem/CLM50_Tech_Note_Ecosystem.html#table-physical-constants)).

The roughness lengths used to calculate \\(r\_{am}\\), \\(r\_{ah}\\), and \\(r\_{aw}\\) are \\(z\_{0m} =z\_{0m,\\, g}\\), \\(z\_{0h} =z\_{0h,\\, g}\\), and \\(z\_{0w} =z\_{0w,\\, g}\\). The displacement height \\(d=0\\). The momentum roughness length is \\(z\_{0m,\\, g} =0.0023\\) for glaciers without snow (\\(f\_{sno} =0) {\\rm }\\), and \\(z\_{0m,\\, g} =0.00085\\) for bare soil surfaces without snow (\\(f\_{sno} =0) {\\rm }\\) ([Meier et al. (2022)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#meieretal2022)).

For bare soil and glaciers with snow ( \\(f\_{sno} > 0\\) ), the momentum roughness length is evaluated based on accumulated snow melt \\(M\_{a} {\\rm }\\) ([Meier et al. (2022)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#meieretal2022)). For \\(M\_{a} >=1\\times 10^{-5}\\)

(2.5.82)[¶](#equation-5-81a "Permalink to this equation")\\\[z\_{0m,\\, g} =\\exp (b\_{1} \\tan ^{-1} \\left\[\\frac{log\_{10} (M\_{a}) + 0.23)} {0.08}\\right\] + b\_{4})\\times 10^{-3}\\\]

where \\(M\_{a}\\) is accumulated snow melt (meters water equivalent), \\(b\_{1} =1.4\\) and \\(b\_{4} =-0.31\\). For \\(M\_{a} <1\\times 10^{-5}\\)

(2.5.83)[¶](#equation-5-81b "Permalink to this equation")\\\[z\_{0m,\\, g} =\\exp (-b\_{1} 0.5 \\pi + b\_{4})\\times 10^{-3}\\\]

Accumulated snow melt \\(M\_{a}\\) at the current time step \\(t\\) is defined as

(2.5.84)[¶](#equation-5-81c "Permalink to this equation")\\\[M ^{t}\_{a} = M ^{t-1}\_{a} - (q ^{t}\_{sno} \\Delta t + q ^{t}\_{snowmelt} \\Delta t)\\times 10^{-3}\\\]

where \\(M ^{t}\_{a}\\) and \\(M ^{t-1}\_{a}\\) are the accumulated snowmelt at the current time step and previous time step, respectively (m), \\(q ^{t}\_{sno} \\Delta t\\) is the freshly fallen snow (mm), and \\(q ^{t}\_{snowmelt} \\Delta t\\) is the melted snow (mm).

The scalar roughness lengths (\\(z\_{0q,\\, g}\\) for latent heat and \\(z\_{0h,\\ g}\\) for sensible heat) are calculated as ([Meier et al. (2022)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#meieretal2022))

(2.5.85)[¶](#equation-5-82 "Permalink to this equation")\\\[z\_{0h,\\, g}=z\_{0q,\\, g}=\\frac{70 \\nu}{u\_{\*}} \\exp (-\\beta {u\_{\*}} ^{0.5} |{\\theta\_{\*}}| ^{0.25} )\\\]

where \\(\\beta\\) = 7.2, and \\(\\theta\_{\*}\\) is the potential temperature scale.

The numerical solution for the fluxes of momentum, sensible heat, and water vapor flux from non-vegetated surfaces proceeds as follows:

1.  An initial guess for the wind speed \\(V\_{a}\\) is obtained from [(2.5.24)](#equation-5-24) assuming an initial convective velocity \\(U\_{c} =0\\) m s\-1 for stable conditions (\\(\\theta \_{v,\\, atm} -\\theta \_{v,\\, s} \\ge 0\\) as evaluated from [(2.5.50)](#equation-5-50) ) and \\(U\_{c} =0.5\\) for unstable conditions (\\(\\theta \_{v,\\, atm} -\\theta \_{v,\\, s} <0\\)).
    
2.  An initial guess for the Monin-Obukhov length \\(L\\) is obtained from the bulk Richardson number using [(2.5.46)](#equation-5-46) and [(2.5.48)](#equation-5-48).
    
3.  The following system of equations is iterated three times:
    
4.  Friction velocity \\(u\_{\*}\\) ([(2.5.32)](#equation-5-32), [(2.5.33)](#equation-5-33), [(2.5.34)](#equation-5-34), [(2.5.35)](#equation-5-35))
    
5.  Potential temperature scale \\(\\theta \_{\*}\\) ([(2.5.37)](#equation-5-37) , [(2.5.38)](#equation-5-38), [(2.5.39)](#equation-5-39), [(2.5.40)](#equation-5-40))
    
6.  Humidity scale \\(q\_{\*}\\) ([(2.5.41)](#equation-5-41), [(2.5.42)](#equation-5-42), [(2.5.43)](#equation-5-43), [(2.5.44)](#equation-5-44))
    
7.  Roughness lengths for sensible \\(z\_{0h,\\, g}\\) and latent heat \\(z\_{0w,\\, g}\\) ([(2.5.82)](#equation-5-81a) , [(2.5.83)](#equation-5-81b) , [(2.5.85)](#equation-5-82))
    
8.  Virtual potential temperature scale \\(\\theta \_{v\*}\\) ( [(2.5.17)](#equation-5-17))
    
9.  Wind speed including the convective velocity, \\(V\_{a}\\) ( [(2.5.24)](#equation-5-24))
    
10.  Monin-Obukhov length \\(L\\) ([(2.5.49)](#equation-5-49))
    
11.  Aerodynamic resistances \\(r\_{am}\\) , \\(r\_{ah}\\) , and \\(r\_{aw}\\) ([(2.5.55)](#equation-5-55), [(2.5.56)](#equation-5-56), [(2.5.57)](#equation-5-57))
    
12.  Momentum fluxes \\(\\tau \_{x}\\) , \\(\\tau \_{y}\\) ([(2.5.5)](#equation-5-5), [(2.5.6)](#equation-5-6))
    
13.  Sensible heat flux \\(H\_{g}\\) ([(2.5.62)](#equation-5-62))
    
14.  Water vapor flux \\(E\_{g}\\) ([(2.5.66)](#equation-5-66))
    
15.  2-m height air temperature \\(T\_{2m}\\) and specific humidity \\(q\_{2m}\\) ([(2.5.58)](#equation-5-58) , [(2.5.59)](#equation-5-59))
    

The partial derivatives of the soil surface fluxes 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

(2.5.86)[¶](#equation-5-83 "Permalink to this equation")\\\[\\frac{\\partial H\_{g} }{\\partial T\_{g} } =\\frac{\\rho \_{atm} C\_{p} }{r\_{ah} }\\\]

(2.5.87)[¶](#equation-5-84 "Permalink to this equation")\\\[\\frac{\\partial E\_{g} }{\\partial T\_{g} } =\\frac{\\beta \_{soi} \\rho \_{atm} }{r\_{aw} } \\frac{dq\_{g} }{dT\_{g} }\\\]

where

(2.5.88)[¶](#equation-5-85 "Permalink to this equation")\\\[\\frac{dq\_{g} }{dT\_{g} } =\\left(1-f\_{sno} -f\_{h2osfc} \\right)\\alpha \_{soil} \\frac{dq\_{sat}^{T\_{soil} } }{dT\_{soil} } +f\_{sno} \\frac{dq\_{sat}^{T\_{sno} } }{dT\_{sno} } +f\_{h2osfc} \\frac{dq\_{sat}^{T\_{h2osfc} } }{dT\_{h2osfc} } .\\\]

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} }\\).

