### 2.12.4.1. Surface Properties[¶](#surface-properties "Permalink to this headline")

The fraction of shortwave radiation absorbed at the surface, \\(\\beta\\), depends on the lake state. If resolved snow layers are present, then \\(\\beta\\) is set equal to the absorption fraction predicted by the snow-optics submodel (Chapter [2.3](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Surface_Albedos/CLM50_Tech_Note_Surface_Albedos.html#rst-surface-albedos)) for the top snow layer. Otherwise, \\(\\beta\\) is set equal to the near infrared fraction of the shortwave radiation reaching the surface simulated by the atmospheric model or atmospheric data model used for offline simulations (Chapter [2.32](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Land-Only_Mode/CLM50_Tech_Note_Land-Only_Mode.html#rst-land-only-mode)). The remainder of the shortwave radiation fraction (1 \\({-}\\) \\(\\beta\\)) is absorbed in the lake body or soil as described in section [2.12.5.5](#radiation-penetration).

The surface roughnesses are functions of the lake state and atmospheric forcing.

For unfrozen lakes (\\(T\_{g} > T\_{f}\\)), \\(z\_{0m}\\) is given by ([Subin et al. (2012a)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#subinetal2012a))

(2.12.3)[¶](#equation-12-3 "Permalink to this equation")\\\[z\_{0m} =\\max \\left(\\frac{\\alpha \\nu }{u\_{\*} } ,C\\frac{u\_{\*} ^{2} }{g} \\right)\\\]

where \\(\\alpha\\) = 0.1, \\(\\nu\\) is the kinematic viscosity of air given below, _C_ is the effective Charnock coefficient given below, \\(u\_{\*}\\) is the friction velocity (m/s), and _g_ is the acceleration of gravity ([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 kinematic viscosity is given by

(2.12.4)[¶](#equation-12-4 "Permalink to this equation")\\\[\\nu =\\nu \_{0} \\left(\\frac{T\_{g} }{T\_{0} } \\right)^{1.5} \\frac{P\_{0} }{P\_{ref} }\\\]

where \\(\\nu \_{0} =1.51\\times 10^{-5} {\\textstyle\\frac{{\\rm m}^{{\\rm 2}} }{{\\rm s}}}\\) , \\(T\_{0} ={\\rm 293.15\\; K}\\), \\(P\_{0} =1.013\\times 10^{5} {\\rm \\; Pa}\\) , and \\(P\_{ref}\\) is the pressure at the atmospheric reference height. The Charnock coefficient _C_ is a function of the lake fetch _F_ (m), given in the surface data or set to 25 times the lake depth _d_ by default:

(2.12.5)[¶](#equation-12-5 "Permalink to this equation")\\\[\\begin{split}\\begin{array}{l} {C=C\_{\\min } +(C\_{\\max } -C\_{\\min } )\\exp \\left\\{-\\min \\left(A,B\\right)\\right\\}} \\\\ {A={\\left(\\frac{Fg}{u\_{\*} ^{2} } \\right)^{{1\\mathord{\\left/ {\\vphantom {1 3}} \\right.} 3} } \\mathord{\\left/ {\\vphantom {\\left(\\frac{Fg}{u\_{\*} ^{2} } \\right)^{{1\\mathord{\\left/ {\\vphantom {1 3}} \\right.} 3} } f\_{c} }} \\right.} f\_{c} } } \\\\ {B=\\varepsilon \\frac{\\sqrt{dg} }{u} } \\end{array}\\end{split}\\\]

where _A_ and _B_ define the fetch- and depth-limitation, respectively; \\(C\_{\\min } =0.01\\) , \\(C\_{\\max } =0.01\\), \\(\\varepsilon =1\\) , \\(f\_{c} =100\\) , and _u_ (m s\-1) is the atmospheric forcing wind.

The scalar roughness lengths (\\(z\_{0q}\\) for latent heat and \\(z\_{0h}\\) for sensible heat) are given by ([Subin et al. 2012a](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#subinetal2012a))

(2.12.6)[¶](#equation-12-5a "Permalink to this equation")\\\[\\begin{split}\\begin{array}{l} {R\_{0} =(\\frac{z\_{0m} u\_{\*} }{\\nu })^{0.5} ,} \\\\ {z\_{0h} =z\_{0m} \\exp \\left\\{-\\frac{k} {Pr} (4 R\_{0} ^{0.5} -3.2) \\right\\},} \\\\ {z\_{0q} =z\_{0m} \\exp \\left\\{-\\frac{k} {Sc} (4 R\_{0} ^{0.5} - 4.2) \\right\\}}\\end{array}\\end{split}\\\]

where \\(R\_{0}\\) is the near-surface atmospheric roughness Reynolds number, \\(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)), \\(Pr = 0.713\\) is the molecular Prandt number for air at neutral stability, \\(Sc = 0.66\\) is the Schmidt number for water in air at neutral stability. \\(z\_{0q}\\) and \\(z\_{0h}\\) are restricted to be no smaller than \\(1 \\times 10^{-10}\\).

For frozen lakes ( \\(T\_{g} \\le T\_{f}\\) ) without resolved snow layers ( \\(snl = 0\\) ), \\(z\_{0m} =z\_{0m\_{ice}} =2.3\\times 10^{-3} {\\rm m}\\) ([Meier et al. (2022)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#meieretal2022)).

For frozen lakes with resolved snow layers ( \\(snl > 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}\\)

(2.12.7)[¶](#equation-12-5b "Permalink to this equation")\\\[z\_{0m} =\\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}\\)

(2.12.8)[¶](#equation-12-5c "Permalink to this equation")\\\[z\_{0m} =\\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

(2.12.9)[¶](#equation-12-5d "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).

For frozen lakes without and with resolved snow layers, an initial guess for the scalar roughness lengths is derived by assuming \\(\\theta\_{\*} = 0 {\\rm }\\) ([Meier et al. (2022)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#meieretal2022))

(2.12.10)[¶](#equation-12-5e "Permalink to this equation")\\\[z\_{0h}=z\_{0q}=\\frac{70 \\nu}{u\_{\*}}\\\]

where \\(\\nu=1.5 \\times 10^{-5}\\) is the kinematic viscosity of air (m2 s\-1), and \\(u\_{\*}\\) is the friction velocity in the atmospheric surface layer (m s\-1). Thereafter, the scalar roughness lengths are updated within the stability iteration described in section [2.12.4.2](#surface-flux-solution-lake) as

(2.12.11)[¶](#equation-12-6 "Permalink to this equation")\\\[z\_{0h}=z\_{0q}=\\frac{70 \\nu}{u\_{\*}} \\exp (-\\beta {u\_{\*}} ^{0.5} |{\\theta\_{\*}}| ^{0.25} )\\\]

where \\(\\beta\\) = 7.2, and \\(\\theta\_{\*}\\) is the potential temperature scale (section [2.12.4.2](#surface-flux-solution-lake)).

