### 2.6.2.1. Soil and Snow Layers[¶](#soil-and-snow-layers "Permalink to this headline")

Upon update, the snow/soil temperatures are evaluated to determine if phase change will take place as

(2.6.50)[¶](#equation-6-53a "Permalink to this equation")\\\[\\begin{array}{lr} T\_{i}^{n+1} >T\_{f} {\\rm \\; and\\; }w\_{ice,\\, i} >0 & \\qquad i=snl+1,\\ldots ,N\_{levgrnd} \\qquad {\\rm melting} \\end{array}\\\]

(2.6.51)[¶](#equation-6-53b "Permalink to this equation")\\\[\\begin{split}\\begin{array}{lr} \\begin{array}{lr} T\_{i}^{n+1} <T\_{f} {\\rm \\; and\\; }w\_{liq,\\, i} >0 & \\qquad i=snl+1,\\ldots ,0 \\\\ T\_{i}^{n+1} <T\_{f} {\\rm \\; and\\; }w\_{liq,\\, i} >w\_{liq,\\, \\max ,\\, i} & \\quad i=1,\\ldots ,N\_{levgrnd} \\end{array} & \\quad {\\rm freezing} \\end{array}\\end{split}\\\]

where \\(T\_{i}^{n+1}\\) is the soil layer temperature after solution of the tridiagonal equation set, \\(w\_{ice,\\, i}\\) and \\(w\_{liq,\\, i}\\) are the mass of ice and liquid water (kg m\-2) in each snow/soil layer, respectively, and \\(T\_{f}\\) is the freezing temperature of water (K) ([Table 2.2.7](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Ecosystem/CLM50_Tech_Note_Ecosystem.html#table-physical-constants)). For the freezing process in soil layers, the concept of supercooled soil water from [Niu and Yang (2006)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#niuyang2006) is adopted. The supercooled soil water is the liquid water that coexists with ice over a wide range of temperatures below freezing and is implemented through a freezing point depression equation

(2.6.52)[¶](#equation-6-54 "Permalink to this equation")\\\[w\_{liq,\\, \\max ,\\, i} =\\Delta z\_{i} \\theta \_{sat,\\, i} \\left\[\\frac{10^{3} L\_{f} \\left(T\_{f} -T\_{i} \\right)}{gT\_{i} \\psi \_{sat,\\, i} } \\right\]^{{-1\\mathord{\\left/ {\\vphantom {-1 B\_{i} }} \\right.} B\_{i} } } \\qquad T\_{i} <T\_{f}\\\]

where \\(w\_{liq,\\, \\max,\\, i}\\) is the maximum liquid water in layer \\(i\\) (kg m\-2) when the soil temperature \\(T\_{i}\\) is below the freezing temperature \\(T\_{f}\\), \\(L\_{f}\\) is the latent heat of fusion (J kg\-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 \_{sat,\\, i}\\) and \\(B\_{i}\\) are the soil texture-dependent saturated matric potential (mm) and [Clapp and Hornberger (1978)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#clapphornberger1978) exponent (section [2.7.3](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Hydrology/CLM50_Tech_Note_Hydrology.html#soil-water)).

For the special case when snow is present (snow mass \\(W\_{sno} >0\\)) but there are no explicit snow layers (\\(snl=0\\)) (i.e., there is not enough snow present to meet the minimum snow depth requirement of 0.01 m), snow melt will take place for soil layer \\(i=1\\) if the soil layer temperature is greater than the freezing temperature (\\(T\_{1}^{n+1} >T\_{f}\\) ).

The rate of phase change is assessed from the energy excess (or deficit) needed to change \\(T\_{i}\\) to freezing temperature, \\(T\_{f}\\). The excess or deficit of energy \\(H\_{i}\\) (W m\-2) is determined as follows

(2.6.53)[¶](#equation-6-55 "Permalink to this equation")\\\[\\begin{split}H\_{i} =\\left\\{\\begin{array}{lr} \\frac{\\partial h}{\\partial T} \\left(T\_{f} -T\_{i}^{n} \\right)-\\frac{c\_{i} \\Delta z\_{i} }{\\Delta t} \\left(T\_{f} -T\_{i}^{n} \\right) & \\quad \\quad i=snl+1 \\\\ \\left(1-f\_{sno} -f\_{h2osfc} \\right)\\frac{\\partial h}{\\partial T} \\left(T\_{f} -T\_{i}^{n} \\right)-\\frac{c\_{i} \\Delta z\_{i} }{\\Delta t} \\left(T\_{f} -T\_{i}^{n} \\right)\\quad {\\kern 1pt} {\\kern 1pt} {\\kern 1pt} {\\kern 1pt} & i=1 \\\\ -\\frac{c\_{i} \\Delta z\_{i} }{\\Delta t} \\left(T\_{f} -T\_{i}^{n} \\right) & \\quad \\quad i\\ne \\left\\{1,snl+1\\right\\} \\end{array}\\right\\}.\\end{split}\\\]

If the melting criteria is met [(2.6.50)](#equation-6-53a) and \\(H\_{m} =\\frac{H\_{i} \\Delta t}{L\_{f} } >0\\), then the ice mass is readjusted as

(2.6.54)[¶](#equation-6-56 "Permalink to this equation")\\\[w\_{ice,\\, i}^{n+1} =w\_{ice,\\, i}^{n} -H\_{m} \\ge 0\\qquad i=snl+1,\\ldots ,N\_{levgrnd} .\\\]

If the freezing criteria is met [(2.6.51)](#equation-6-53b) and \\(H\_{m} <0\\), then the ice mass is readjusted for \\(i=snl+1,\\ldots,0\\) as

(2.6.55)[¶](#equation-6-57 "Permalink to this equation")\\\[w\_{ice,\\, i}^{n+1} =\\min \\left(w\_{liq,\\, i}^{n} +w\_{ice,\\, i}^{n} ,w\_{ice,\\, i}^{n} -H\_{m} \\right)\\\]

and for \\(i=1,\\ldots,N\_{levgrnd}\\) as

(2.6.56)[¶](#equation-6-58 "Permalink to this equation")\\\[\\begin{split}w\_{ice,\\, i}^{n+1} = \\left\\{\\begin{array}{lr} \\min \\left(w\_{liq,\\, i}^{n} +w\_{ice,\\, i}^{n} -w\_{liq,\\, \\max ,\\, i}^{n} ,\\, w\_{ice,\\, i}^{n} -H\_{m} \\right) & \\qquad w\_{liq,\\, i}^{n} +w\_{ice,\\, i}^{n} \\ge w\_{liq,\\, \\max ,\\, i}^{n} {\\rm \\; } \\\\ {\\rm 0} & \\qquad w\_{liq,\\, i}^{n} +w\_{ice,\\, i}^{n} <w\_{liq,\\, \\max ,\\, i}^{n} {\\rm \\; \\; }\\, \\end{array}\\right\\}.\\end{split}\\\]

Liquid water mass is readjusted as

(2.6.57)[¶](#equation-6-59 "Permalink to this equation")\\\[w\_{liq,\\, i}^{n+1} =w\_{liq,\\, i}^{n} +w\_{ice,\\, i}^{n} -w\_{ice,\\, i}^{n+1} \\ge 0.\\\]

Because part of the energy \\(H\_{i}\\) may not be consumed in melting or released in freezing, the energy is recalculated as

(2.6.58)[¶](#equation-6-60 "Permalink to this equation")\\\[H\_{i\*} =H\_{i} -\\frac{L\_{f} \\left(w\_{ice,\\, i}^{n} -w\_{ice,\\, i}^{n+1} \\right)}{\\Delta t}\\\]

and this energy is used to cool or warm the snow/soil layer (if \\(\\left|H\_{i\*} \\right|>0\\)) as

(2.6.59)[¶](#equation-6-61 "Permalink to this equation")\\\[\\begin{split}T\_{i}^{n+1} = \\left\\{\\begin{array}{lr} T\_{f} +{\\frac{\\Delta t}{c\_{i} \\Delta z\_{i} } H\_{i\*} \\mathord{\\left/ {\\vphantom {\\frac{\\Delta t}{c\_{i} \\Delta z\_{i} } H\_{i\*} \\left(1-\\frac{\\Delta t}{c\_{i} \\Delta z\_{i} } \\frac{\\partial h}{\\partial T} \\right)}} \\right.} \\left(1-\\frac{\\Delta t}{c\_{i} \\Delta z\_{i} } \\frac{\\partial h}{\\partial T} \\right)} & \\quad \\quad \\quad \\quad \\, i=snl+1 \\\\ T\_{f} +{\\frac{\\Delta t}{c\_{i} \\Delta z\_{i} } H\_{i\*} \\mathord{\\left/ {\\vphantom {\\frac{\\Delta t}{c\_{i} \\Delta z\_{i} } H\_{i\*} \\left(1-\\left(1-f\_{sno} -f\_{h2osfc} \\right)\\frac{\\Delta t}{c\_{i} \\Delta z\_{i} } \\frac{\\partial h}{\\partial T} \\right)}} \\right.} \\left(1-\\left(1-f\_{sno} -f\_{h2osfc} \\right)\\frac{\\Delta t}{c\_{i} \\Delta z\_{i} } \\frac{\\partial h}{\\partial T} \\right)} & \\qquad i=1 \\\\ T\_{f} +\\frac{\\Delta t}{c\_{i} \\Delta z\_{i} } H\_{i\*} & \\quad \\quad \\quad \\quad \\, i\\ne \\left\\{1,snl+1\\right\\} \\end{array}\\right\\}.\\end{split}\\\]

For the special case when snow is present (\\(W\_{sno} >0\\)), there are no explicit snow layers (\\(snl=0\\)), and \\(\\frac{H\_{1} \\Delta t}{L\_{f} } >0\\) (melting), the snow mass \\(W\_{sno}\\) (kg m\-2) is reduced according to

(2.6.60)[¶](#equation-6-62 "Permalink to this equation")\\\[W\_{sno}^{n+1} =W\_{sno}^{n} -\\frac{H\_{1} \\Delta t}{L\_{f} } \\ge 0.\\\]

The snow depth is reduced proportionally

(2.6.61)[¶](#equation-6-63 "Permalink to this equation")\\\[z\_{sno}^{n+1} =\\frac{W\_{sno}^{n+1} }{W\_{sno}^{n} } z\_{sno}^{n} .\\\]

Again, because part of the energy may not be consumed in melting, the energy for the surface soil layer \\(i=1\\) is recalculated as

(2.6.62)[¶](#equation-6-64 "Permalink to this equation")\\\[H\_{1\*} =H\_{1} -\\frac{L\_{f} \\left(W\_{sno}^{n} -W\_{sno}^{n+1} \\right)}{\\Delta t} .\\\]

If there is excess energy (\\(H\_{1\*} >0\\)), this energy becomes available to the top soil layer as

(2.6.63)[¶](#equation-6-65 "Permalink to this equation")\\\[H\_{1} =H\_{1\*} .\\\]

The ice mass, liquid water content, and temperature of the top soil layer are then determined from [(2.6.54)](#equation-6-56), [(2.6.57)](#equation-6-59), and [(2.6.59)](#equation-6-61) using the recalculated energy from [(2.6.63)](#equation-6-65). Snow melt \\(M\_{1S}\\) (kg m\-2 s\-1) and phase change energy \\(E\_{p,\\, 1S}\\) (W m\-2) for this special case are

(2.6.64)[¶](#equation-6-66 "Permalink to this equation")\\\[M\_{1S} =\\frac{W\_{sno}^{n} -W\_{sno}^{n+1} }{\\Delta t} \\ge 0\\\]

(2.6.65)[¶](#equation-6-67 "Permalink to this equation")\\\[E\_{p,\\, 1S} =L\_{f} M\_{1S} .\\\]

The total energy of phase change \\(E\_{p}\\) (W m\-2) for the snow/soil column is

(2.6.66)[¶](#equation-6-68 "Permalink to this equation")\\\[E\_{p} =E\_{p,\\, 1S} +\\sum \_{i=snl+1}^{N\_{levgrnd} }E\_{p,i}\\\]

where

(2.6.67)[¶](#equation-6-69 "Permalink to this equation")\\\[E\_{p,\\, i} =L\_{f} \\frac{\\left(w\_{ice,\\, i}^{n} -w\_{ice,\\, i}^{n+1} \\right)}{\\Delta t} .\\\]

The total snow melt \\(M\\) (kg m\-2 s\-1) is

(2.6.68)[¶](#equation-6-70 "Permalink to this equation")\\\[M=M\_{1S} +\\sum \_{i=snl+1}^{i=0}M\_{i}\\\]

where

(2.6.69)[¶](#equation-6-71 "Permalink to this equation")\\\[M\_{i} =\\frac{\\left(w\_{ice,\\, i}^{n} -w\_{ice,\\, i}^{n+1} \\right)}{\\Delta t} \\ge 0.\\\]

The solution for snow/soil temperatures conserves energy as

(2.6.70)[¶](#equation-6-72 "Permalink to this equation")\\\[G-E\_{p} -\\sum \_{i=snl+1}^{i=N\_{levgrnd} }\\frac{c\_{i} \\Delta z\_{i} }{\\Delta t} \\left(T\_{i}^{n+1} -T\_{i}^{n} \\right)=0\\\]

where \\(G\\) is the ground heat flux (section [2.5.4](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Fluxes/CLM50_Tech_Note_Fluxes.html#update-of-ground-sensible-and-latent-heat-fluxes)).

