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2.5.4. Update of Ground Sensible and Latent Heat Fluxes¶
The sensible and water vapor heat fluxes derived above for bare soil and soil beneath canopy are based on the ground surface temperature from the previous time step \(T_{g}^{n}\) and are used as the surface forcing for the solution of the soil temperature equations (section 2.6.1). This solution yields a new ground surface temperature \(T_{g}^{n+1}\). The ground sensible and water vapor fluxes are then updated for \(T_{g}^{n+1}\) as
(2.5.149)¶\[H'_{g} =H_{g} +\left(T_{g}^{n+1} -T_{g}^{n} \right)\frac{\partial H_{g} }{\partial T_{g} }\]
(2.5.150)¶\[E'_{g} =E_{g} +\left(T_{g}^{n+1} -T_{g}^{n} \right)\frac{\partial E_{g} }{\partial T_{g} }\]
where \(H_{g}\), \(E_{g}\), \(\frac{\partial H_{g} }{\partial T_{g} }\), and \(\frac{\partial E_{g} }{\partial T_{g} }\) are the sensible heat and water vapor fluxes and their partial derivatives derived from equations (2.5.62), (2.5.66), (2.5.86), and (2.5.87) for non-vegetated surfaces and equations (2.5.92), (2.5.105), (2.5.126), and (2.5.127) for vegetated surfaces using \(T_{g}^{n}\). One further adjustment is made to \(H'_{g}\) and \(E'_{g}\). If the soil moisture in the top snow/soil layer is not sufficient to support the updated ground evaporation, i.e., if \(E'_{g} > 0\) and \(f_{evap} < 1\) where
(2.5.151)¶\[f_{evap} =\frac{{\left(w_{ice,\; snl+1} +w_{liq,\, snl+1} \right)\mathord{\left/ {\vphantom {\left(w_{ice,\; snl+1} +w_{liq,\, snl+1} \right) \Delta t}} \right.} \Delta t} }{\sum _{j=1}^{npft}\left(E'_{g} \right)_{j} \left(wt\right)_{j} } \le 1,\]
an adjustment is made to reduce the ground evaporation accordingly as
(2.5.152)¶\[E''_{g} =f_{evap} E'_{g} .\]
The term \(\sum _{j=1}^{npft}\left(E'_{g} \right)_{j} \left(wt\right)_{j}\) is the sum of \(E'_{g}\) over all evaporating PFTs where \(\left(E'_{g} \right)_{j}\) is the ground evaporation from the \(j^{th}\) PFT on the column, \(\left(wt\right)_{j}\) is the relative area of the \(j^{th}\) PFT with respect to the column, and \(npft\) is the number of PFTs on the column. \(w_{ice,\, snl+1}\) and \(w_{liq,\, snl+1}\) are the ice and liquid water contents (kg m-2) of the top snow/soil layer (Chapter 2.7). Any resulting energy deficit is assigned to sensible heat as
(2.5.153)¶\[H''_{g} =H_{g} +\lambda \left(E'_{g} -E''_{g} \right).\]
The ground water vapor flux \(E''_{g}\) is partitioned into evaporation of liquid water from snow/soil \(q_{seva}\) (kgm-2 s-1), sublimation from snow/soil ice \(q_{subl}\) (kg m-2 s-1), liquid dew on snow/soil \(q_{sdew}\) (kg m-2 s-1), or frost on snow/soil \(q_{frost}\) (kg m-2 s-1) as
(2.5.154)¶\[q_{seva} =\max \left(E''_{sno} \frac{w_{liq,\, snl+1} }{w_{ice,\; snl+1} +w_{liq,\, snl+1} } ,0\right)\qquad E''_{sno} \ge 0,\, w_{ice,\; snl+1} +w_{liq,\, snl+1} >0\]
(2.5.155)¶\[q_{subl} =E''_{sno} -q_{seva} \qquad E''_{sno} \ge 0\]
(2.5.156)¶\[q_{sdew} =\left|E''_{sno} \right|\qquad E''_{sno} <0{\rm \; and\; }T_{g} \ge T_{f}\]
(2.5.157)¶\[q_{frost} =\left|E''_{sno} \right|\qquad E''_{sno} <0{\rm \; and\; }T_{g} <T_{f} .\]
The loss or gain in snow mass due to \(q_{seva}\), \(q_{subl}\), \(q_{sdew}\), and \(q_{frost}\) on a snow surface are accounted for during the snow hydrology calculations (Chapter 2.8). The loss of soil and surface water due to \(q_{seva}\) is accounted for in the calculation of infiltration (section 2.7.2.3), while losses or gains due to \(q_{subl}\), \(q_{sdew}\), and \(q_{frost}\) on a soil surface are accounted for following the sub-surface drainage calculations (section 2.7.5).
The ground heat flux \(G\) is calculated as
(2.5.158)¶\[G=\overrightarrow{S}_{g} -\overrightarrow{L}_{g} -H_{g} -\lambda E_{g}\]
where \(\overrightarrow{S}_{g}\) is the solar radiation absorbed by the ground (section 2.4.1), \(\overrightarrow{L}_{g}\) is the net longwave radiation absorbed by the ground (section 2.4.2)
(2.5.159)¶\[\vec{L}_{g} =L_{g} \uparrow -\delta _{veg} \varepsilon _{g} L_{v} \, \downarrow -\left(1-\delta _{veg} \right)\varepsilon _{g} L_{atm} \, \downarrow +4\varepsilon _{g} \sigma \left(T_{g}^{n} \right)^{3} \left(T_{g}^{n+1} -T_{g}^{n} \right),\]
where
(2.5.160)¶\[L_{g} \uparrow =\varepsilon _{g} \sigma \left[\left(1-f_{sno} -f_{h2osfc} \right)\left(T_{1}^{n} \right)^{4} +f_{sno} \left(T_{sno}^{n} \right)^{4} +f_{h2osfc} \left(T_{h2osfc}^{n} \right)^{4} \right]\]
and \(H_{g}\) and \(\lambda E_{g}\) are the sensible and latent heat fluxes after the adjustments described above.
When converting ground water vapor flux to an energy flux, the term \(\lambda\) is arbitrarily assumed to be
(2.5.161)¶\[\begin{split}\lambda =\left\{\begin{array}{l} {\lambda _{sub} \qquad {\rm if\; }w_{liq,\, snl+1} =0{\rm \; and\; }w_{ice,\, snl+1} >0} \\ {\lambda _{vap} \qquad {\rm otherwise}} \end{array}\right\}\end{split}\]
where \(\lambda _{sub}\) and \(\lambda _{vap}\) are the latent heat of sublimation and vaporization, respectively (J (kg-1) (Table 2.2.7). When converting vegetation water vapor flux to an energy flux, \(\lambda _{vap}\) is used.
The system balances energy as
(2.5.162)¶\[\overrightarrow{S}_{g} +\overrightarrow{S}_{v} +L_{atm} \, \downarrow -L\, \uparrow -H_{v} -H_{g} -\lambda _{vap} E_{v} -\lambda E_{g} -G=0.\]