Snow acts as an excellent insulator. When snow piles up on a roof, it can add additional

insulation to a building. Engineered insulation typically comes with $"R$ values" that

measure the thermal resistance to temperature differences across the insulation. The $R$

value describes the temperature difference across the insulation per unit heat flux, $R=T$

$/Q.$ When a building's $R$ value is increased through adding insulation, the heat flux for a

given temperature difference decreases. $R$ values in the US are given in imperial units

$(f{t}^{2}Fh\frac{r}{B}TU).$ Internationally, $R$ values are in $SI$ units $(m^2\frac{K}{W}),$ and the conversion

between these units is $R-$value $(SI)=R-$value $($US$)/5\mathrm{.}67.$ Typical buildings may have

roofs with $($US$)R$ values of $40.$

a$)$ In SI units, calculate the heat flux through the roof when the outside temperature

in Boulder is $-10C$ and the indoor temperature is set to $18C.(0\mathrm{.}5pts)$

b$)$ Assume that the insulated roof has $40cm$ of snow on top of it$,$ with a density of

$350k\frac{g}{m^3}.$ By what percent does this snow cover reduce the heat flux through

the roof? Use the Sturm et al$.(1997)$ expression for thermal conductivity for

settled snow:

$k_s=0\mathrm{.}138-1\mathrm{.}01{}_s+3\mathrm{.}23{}_s^2$

Reference: Sturm M$,$ Holmgren J$,$ K$$nig M$,$ Morris K$.$ The thermal conductivity of

seasonal snow. Joumal of Glaciology. $1997$;$43(143)$:$26-41.$

doi:$10\mathrm{.}3189/$S$0022143000002781$

c$)$ What can you infer about the insolation of snow for ecosystem processes? For

example, how does snow depth affect pika, a small mountain mammal, habitat?