### 2.7.2.2. Surface Water Storage[¶](#surface-water-storage "Permalink to this headline")

A surface water store has been added to the model to represent wetlands and small, sub-grid scale water bodies. As a result, the wetland land unit has been removed as of CLM4.5. The state variables for surface water are the mass of water \\(W\_{sfc}\\) (kg m\-2) and temperature \\(T\_{h2osfc}\\) (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)). Surface water storage and outflow are functions of fine spatial scale elevation variations called microtopography. The microtopography is assumed to be distributed normally around the grid cell mean elevation. Given the standard deviation of the microtopographic distribution, \\(\\sigma \_{micro}\\) (m), the fractional area of the grid cell that is inundated can be calculated. Surface water storage, \\(Wsfc\\), is related to the height (relative to the grid cell mean elevation) of the surface water, \\(d\\), by

(2.7.28)[¶](#equation-7-66 "Permalink to this equation")\\\[W\_{sfc} =\\frac{d}{2} \\left(1+erf\\left(\\frac{d}{\\sigma \_{micro} \\sqrt{2} } \\right)\\right)+\\frac{\\sigma \_{micro} }{\\sqrt{2\\pi } } e^{\\frac{-d^{2} }{2\\sigma \_{micro} ^{2} } }\\\]

where \\(erf\\) is the error function. For a given value of \\(W\_{sfc}\\), [(2.7.28)](#equation-7-66) can be solved for \\(d\\) using the Newton-Raphson method. Once \\(d\\) is known, one can determine the fraction of the area that is inundated as

(2.7.29)[¶](#equation-7-67 "Permalink to this equation")\\\[f\_{h2osfc} =\\frac{1}{2} \\left(1+erf\\left(\\frac{d}{\\sigma \_{micro} \\sqrt{2} } \\right)\\right)\\\]

No global datasets exist for microtopography, so the default parameterization is a simple function of slope

(2.7.30)[¶](#equation-7-68 "Permalink to this equation")\\\[\\sigma \_{micro} =\\left(\\beta +\\beta \_{0} \\right)^{\\eta }\\\]

where \\(\\beta\\) is the topographic slope, \\(\\beta\_{0} =\\left(\\sigma\_{\\max } \\right)^{\\frac{1}{\\eta } }\\) determines the maximum value of \\(\\sigma\_{micro}\\), and \\(\\eta\\) is an adjustable parameter. Default values in the model are \\(\\sigma\_{\\max } =0.4\\) and \\(\\eta =-3\\).

If the spatial scale of the microtopography is small relative to that of the grid cell, one can assume that the inundated areas are distributed randomly within the grid cell. With this assumption, a result from percolation theory can be used to quantify the fraction of the inundated portion of the grid cell that is interconnected

(2.7.31)[¶](#equation-7-69 "Permalink to this equation")\\\[\\begin{split}\\begin{array}{lr} f\_{connected} =\\left(f\_{h2osfc} -f\_{c} \\right)^{\\mu } & \\qquad f\_{h2osfc} >f\_{c} \\\\ f\_{connected} =0 &\\qquad f\_{h2osfc} \\le f\_{c} \\end{array}\\end{split}\\\]

where \\(f\_{c}\\) is a threshold below which no single connected inundated area spans the grid cell and \\(\\mu\\) is a scaling exponent. Default values of \\(f\_{c}\\) and \\(\\mu\\) are 0.4 and 0.14, respectively. When the inundated fraction of the grid cell surpasses \\(f\_{c}\\), the surface water store acts as a linear reservoir

(2.7.32)[¶](#equation-7-70 "Permalink to this equation")\\\[q\_{out,h2osfc}=k\_{h2osfc} \\ f\_{connected} \\ (Wsfc-Wc)\\frac{1}{\\Delta t}\\\]

where \\(q\_{out,h2osfc}\\) is the surface water runoff, \\(k\_{h2osfc}\\) is a constant, \\(Wc\\) is the amount of surface water present when \\(f\_{h2osfc} =f\_{c}\\), and \\(\\Delta t\\) is the model time step. The linear storage coefficent \\(k\_{h2osfc} = \\sin \\left(\\beta \\right)\\) is a function of grid cell mean topographic slope where \\(\\beta\\) is the slope in radians.

