### 2.12.5.4. Eddy Diffusivity and Thermal Conductivities[¶](#eddy-diffusivity-and-thermal-conductivities "Permalink to this headline")

The total eddy diffusivity \\(K\_{W}\\) (m2 s\-1) for liquid water in the lake body 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))

(2.12.34)[¶](#equation-12-29 "Permalink to this equation")\\\[K\_{W} = m\_{d} \\left(\\kappa \_{e} +K\_{ed} +\\kappa \_{m} \\right)\\\]

where \\(\\kappa \_{e}\\) is due to wind-driven eddies ([Hostetler and Bartlein (1990)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#hostetlerbartlein1990)), \\(K\_{ed}\\) is a modest enhanced diffusivity intended to represent unresolved mixing processes ([Fang and Stefan 1996](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#fangstefan1996)), \\(\\kappa \_{m} =\\frac{\\lambda \_{liq} }{c\_{liq} \\rho \_{liq} }\\) is the molecular diffusivity of water (given by the ratio of its thermal conductivity (W m\-1 K\-1) to the product of its heat capacity (J kg\-1 K\-1) and density (kg m\-3), values given in [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 \\(m\_{d}\\) (unitless) is a factor which increases the overall diffusivity for large lakes, intended to represent 3-dimensional mixing processes such as caused by horizontal temperature gradients. As currently implemented,

(2.12.35)[¶](#equation-12-30 "Permalink to this equation")\\\[\\begin{split}m\_{d} =\\left\\{\\begin{array}{l} {1,\\qquad d<25{\\rm m}} \\\\ {10,\\qquad d\\ge 25{\\rm m}} \\end{array}\\right\\}\\end{split}\\\]

where _d_ is the lake depth.

The wind-driven eddy diffusion coefficient \\(\\kappa \_{e,\\, i}\\) (m2 s\-1) for layers \\(1\\le i\\le N\_{levlak}\\) is

(2.12.36)[¶](#equation-12-31 "Permalink to this equation")\\\[\\begin{split}\\kappa \_{e,\\, i} =\\left\\{\\begin{array}{l} {\\frac{kw^{\*} z\_{i} }{P\_{0} \\left(1+37Ri^{2} \\right)} \\exp \\left(-k^{\*} z\_{i} \\right)\\qquad T\_{g} >T\_{f} } \\\\ {0\\qquad T\_{g} \\le T\_{f} } \\end{array}\\right\\}\\end{split}\\\]

where \\(P\_{0} =1\\) is the neutral value of the turbulent Prandtl number, \\(z\_{i}\\) is the node depth (m), the surface friction velocity (m s\-1) is \\(w^{\*} =0.0012u\_{2}\\), and \\(k^{\*}\\) varies with latitude \\(\\phi\\) as \\(k^{\*} =6.6u\_{2}^{-1.84} \\sqrt{\\left|\\sin \\phi \\right|}\\). For the bottom layer, \\(\\kappa \_{e,\\, N\_{levlak} } =\\kappa \_{e,N\_{levlak} -1\\, }\\). As in [Hostetler and Bartlein (1990)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#hostetlerbartlein1990), the 2-m wind speed \\(u\_{2}\\) (m s\-1) (eq. ) is used to evaluate \\(w^{\*}\\) and \\(k^{\*}\\) rather than the 10-m wind used by [Henderson-Sellers (1985)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#henderson-sellers1985).

The Richardson number is

(2.12.37)[¶](#equation-12-32 "Permalink to this equation")\\\[R\_{i} =\\frac{-1+\\sqrt{1+\\frac{40N^{2} k^{2} z\_{i}^{2} }{w^{\*^{2} } \\exp \\left(-2k^{\*} z\_{i} \\right)} } }{20}\\\]

where

(2.12.38)[¶](#equation-12-33 "Permalink to this equation")\\\[N^{2} =\\frac{g}{\\rho \_{i} } \\frac{\\partial \\rho }{\\partial z}\\\]

and \\(g\\) is the acceleration due to gravity (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)), \\(\\rho \_{i}\\) is the density of water (kg m\-3), and \\(\\frac{\\partial \\rho }{\\partial z}\\) is approximated as \\(\\frac{\\rho \_{i+1} -\\rho \_{i} }{z\_{i+1} -z\_{i} }\\). Note that because here, _z_ is increasing downwards (unlike in [Hostetler and Bartlein (1990)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#hostetlerbartlein1990)), eq. contains no negative sign; this is a correction from CLM4. The density of water is ([Hostetler and Bartlein (1990)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#hostetlerbartlein1990))

(2.12.39)[¶](#equation-12-34 "Permalink to this equation")\\\[\\rho \_{i} =1000\\left(1-1.9549\\times 10^{-5} \\left|T\_{i} -277\\right|^{1.68} \\right).\\\]

The enhanced diffusivity \\(K\_{ed}\\) is given by ([Fang and Stefan 1996](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#fangstefan1996))

(2.12.40)[¶](#equation-12-35 "Permalink to this equation")\\\[K\_{ed} =1.04\\times 10^{-8} \\left(N^{2} \\right)^{-0.43} ,N^{2} \\ge 7.5\\times 10^{-5} {\\rm s}^{2}\\\]

where \\(N^{2}\\) is calculated as in eq. except for the minimum value imposed in.

The thermal conductivity for the liquid water portion of lake body layer _i_, \\(\\tau \_{liq,i}\\) (W m\-1 K\-1) is given by

(2.12.41)[¶](#equation-12-36 "Permalink to this equation")\\\[\\tau \_{liq,i} =K\_{W} c\_{liq} \\rho \_{liq} .\\\]

The thermal conductivity of the ice portion of lake body layer _i_, \\(\\tau \_{ice,eff}\\) (W m\-1 K\-1), is constant among layers, and is given by

(2.12.42)[¶](#equation-12-37 "Permalink to this equation")\\\[\\tau \_{ice,eff} =\\tau \_{ice} \\frac{\\rho \_{ice} }{\\rho \_{liq} }\\\]

where \\(\\tau \_{ice}\\) ([Table 2.2.7](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Ecosystem/CLM50_Tech_Note_Ecosystem.html#table-physical-constants)) is the nominal thermal conductivity of ice: \\(\\tau \_{ice,eff}\\) is adjusted for the fact that the nominal model layer thicknesses remain constant even while the physical ice thickness exceeds the water thickness.

The overall thermal conductivity \\(\\tau \_{i}\\) for layer _i_ with ice mass-fraction \\(I\_{i}\\) is the harmonic mean of the liquid and water fractions, assuming that they will be physically vertically stacked, and is given by

(2.12.43)[¶](#equation-12-38 "Permalink to this equation")\\\[\\tau \_{i} =\\frac{\\tau \_{ice,eff} \\tau \_{liq,i} }{\\tau \_{liq,i} I\_{i} +\\tau \_{ice} \\left(1-I\_{i} \\right)} .\\\]

The thermal conductivity of snow, soil, and bedrock layers above and below the lake, respectively, are computed identically to those for vegetated land units (Chapter [2.6](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Soil_Snow_Temperatures/CLM50_Tech_Note_Soil_Snow_Temperatures.html#rst-soil-and-snow-temperatures)), except for the adjustment of thermal conductivity for frost heave or excess ice ([Subin et al., 2012a, Supporting Information](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#subinetal2012a)).

