Equations

\[ \begin{multline} \\ \Large Cattle\space Comfort\space Index\left(^{\circ}\!C\right) = T_{avg} + H_{cor} + W_{cor} + R_{cor}\\[15pt] T = \text{average air temperature $\left(^{\circ}\!C\right)$};\quad H = \text{average relative humidity $\left(\%\right)$}\\[12pt] W = \text{average wind speed $\left(m/s\right)$};\quad R = \text{average solar radiation $\left(W/m^2\right)$}\\[12pt] H_{cor}=\exp\left(0.00182H + 0.000018TH\right)\times \left(0.000054T^2 + 0.00192T - 0.0246\right) \times \left(H - 30\right)\\[12pt] W_{cor}=\dfrac{-6.56}{\exp\left[\frac{1}{\left(2.26W + 0.23\right)^{0.45}} + 2.9 + 0.00000114W^{2.5} - log_{0.3}\frac{1}{\left(2.26W+0.33\right)^2}\right]} - 0.00566W^2 + 3.33\\[30pt] R_{cor}=0.0076R-0.00002RT+0.00005T^2\sqrt{R}+0.1\left(T+0.019R\right)-2 \\ \\ \end{multline} \]

\[ \begin{multline} \\ ET_r=\large\dfrac{0.408\Delta\left(R_n-G\right)+\gamma\dfrac{C_n}{T+273}u_2\left(e_s-e_a\right)}{\Delta+\gamma\left(1+C_du_2\right)}\\[18pt] ET_a=\large K_cET_r\\[8pt] ET_r=\text{reference evapotranspiration on a daily or hourly time step}\left(mm\right)\\[6pt] ET_a=\text{actual evapotranspiration on a daily or hourly time step}\left(mm\right)\\[6pt] R_n=\text{total net radiation at the crop surface}\left(MJ/m^2\right)\\[6pt] G=\text{total soil heat flux}\left(MJ/m^2\right)\\[6pt] u_2=\text{mean wind speed at 2 meters}\left(m/s\right)\\[6pt] T=\text{mean air temperature}\left(^{\circ}\!C\right)\\[6pt] e_s,e_a=\text{mean saturation and actual vapor pressure}\left(kPa\right)\\[6pt] \Delta=\text{slope of the saturation vapor pressure curve}\left(kPa/^{\circ}\!C\right)\\[6pt] \gamma=\text{psychrometric constant}\left(^{\circ}\!C\right)\\[6pt] C_n,C_d=\text{bulk surface resistance and surface aerodynamic roughness parameters}\\[6pt] K_c=\text{crop coefficient} \\ \\ \end{multline} \]

\[ \begin{multline} \\ \Large Growing\space Degree\space Days \left(^{\circ}\!F;\space daily\right) = \dfrac{TMax+Tmin}{2} - T_{low}\\[15pt] TMax,\space TMin=\text{Daily maximum and minimum air temperatures $\left(^{\circ}\!F\right)$}\\[15pt] T_{low} = \text{lower temperature threshold $\left(^{\circ}\!F\right)$}\\[15pt] TMax_{low},\space TMin_{low}=> TMax < T_{low} = T_{low};\space TMin < T_{low} = T_{low}\\[25pt] \Large Modified\space Growing\space Degree\space Days \left(^{\circ}\!F;\space daily\right) = \dfrac{TMax+Tmin}{2} - T_{low}\\[15pt] TMax,\space TMin=\text{Daily maximum and minimum air temperatures $\left(^{\circ}\!F\right)$}\\[12pt] T_{low} = \text{lower temperature threshold $\left(^{\circ}\!F\right)$};\space \space T_{high} = \text{high temperature threshold $\left(^{\circ}\!F\right)$}\\[15pt] TMax_{low},\space TMin_{low}=> TMax < T_{low} = T_{low};\space TMin < T_{low} = T_{low}\\[12pt] TMax_{high},\space TMin_{high}=> TMax > T_{high} = T_{high};\space TMin > T_{high} = T_{high}\\[25pt] \\ \\ \end{multline} \]

\[ \begin{multline} \\ \large HI^{\dagger\ddagger*}\left(^{\circ}\!F\right)=-42.379 + 2.04901523T + 10.14333127H - 0.22475541TH - 0.00683783T^2 - 0.05481717H^2 + 0.00122874HT^2 + 0.00085282TH^2 - 0.00000199T^2H^2\\[10pt] T = \text{air temperature at 1.5 - 2 meters}\left(^{\circ}\!F\right)\\[6pt] H = \text{relative humidity}\left(\%\right)\\[15pt] ^{\dagger} \text{If} \space H\le 13\%\space\&\space80^{\circ}\!F \le T \le 112^{\circ}\!F,\space\space HI=HI - \frac{13-H}{4}\sqrt{\frac{17-\lvert T-95\rvert}{17}}\\[15pt] ^{\ddagger} \text{If} \space H\ge 85\%\space\&\space80^{\circ}\!F \le T \le 87^{\circ}\!F,\space\space HI=HI + \left(\frac{H-85}{10}\right)\left(\frac{87-T}{5}\right)\\[15pt] ^*\text{The Heat Index is not calculated for temperatures $\lt 80^{\circ}\!F$} \\ \\ \end{multline} \]

\[ \begin{multline} \\ Sea\space Level\space Pressure \left(mb\right)=\left(P-0.3\right)\times \left(1 + \left(\dfrac{1013.25^{0.190284} \times 0.0065}{288} \times \dfrac{z}{\left(P - 0.3\right)^{0.190284}}\right)\right)^\dfrac{1}{0.190284}\\[15pt] P = \text{station pressure}\space (mb);\space z = \text{station elevation}\space (m) \\ \\ \end{multline} \]

\[ \begin{multline} \\ \Large Stress\space Degree\space Days\left(^{\circ}\!F;\space daily\right)= TMax - T_{high}\\[15pt] TMax\space = \text{Daily maximum air temperature $\left(^{\circ}\!F\right)$}\\[12pt] TMax_{high} => TMax > T_{high} = T_{high};\space T_{high} \space = \space 86^{\circ}\!F \space \left(Corn\right) \\ \\ \end{multline} \]

\[ \begin{multline} \\ \large Ft^{\dagger}\left(minutes\right)=\left(\left(-24.5 \times \left(\left(0.667 \times \left(V \times 8/5\right)\right)+4.8\right)\right) + 2111\right) \times \left(-4.8-\left(\left(T-32\right)\times 5.9\right)\right)^{-1.668}\\[10pt] V = \text{wind speed at 10 meters}\left(mph\right)\\[6pt] T = \text{air temperature at 1.5 - 2 meters}\left(^{\circ}\!F\right)\\[15pt] \large Ft^{\dagger}\left(minutes\right)=\left(\left(-24.5\times\left(\left(0.667\times V\right)+4.8\right)\right)+2111\right)\times \left(-4.8-T\right)^{-1.668}\\[10pt] V = \text{wind speed at 10 meters}\left(km/hr\right)\\[6pt] T = \text{air temperature at 1.5 - 2 meters}\left(^{\circ}\!C\right)\\[10pt] \small^{\dagger}\!\textit{Valid for 10-meter wind speeds $\ge$ 16mph (25 km/hr) and Ft $\le$ 30 minutes} \\ \\ \end{multline} \]

\[ \begin{multline} \\ \Large VP_{sat\left(w\right)}^{\dagger,\ddagger,*}=f\times 22064000\times \exp\left(\frac{647.096}{T} \times \left(-7.85951783\vartheta+1.84408259\vartheta^{1.5} - 11.7866497\vartheta^3+22.6807411\vartheta^{3.5}-15.9618719\vartheta^4+1.80122502\vartheta^{7.5}\right)\right)\\[15pt] VP_{sat\left(w\right)}=\text{saturated vapor pressure over water $\left(Pa\right)$};\quad f = \text{enhancement factor}^\bigstar\\[12pt] T=\text{air temperature $\left(K\right);\quad t=$ air temperature $\left(^{\circ}\!C\right);\quad \vartheta = 1 - \dfrac{T}{647.096};\quad P_{atm}=$ station pressure $\left(Pa\right)$}\\[12pt] f\left(t\gt0^{\circ}\!C\right) = \exp\left(3.53624\text{e-}04 + 2.9328363\text{e-}05t + 2.6168979\text{e-}07t^2 + 8.5813609\text{e-}09t^3\times \left(1 - VP_{sat\left(w\right)}/P_{atm}\right) + \\ \exp\left(-10.7588 + 6.3268134\text{e-}02t - 2.5368934\text{e-}04t^2 + 6.3405286\text{e-}07t^3\right)\times \left(P_{atm}/VP_{sat\left(w\right)} - 1\right)\right)\\[25pt] f\left(t\le0^{\circ}\!C\right) = \exp\left(3.62183\text{e-}04 + 2.6061244\text{e-}05t + 3.866777\text{e-}07t^2 + 3.8268958\text{e-}09t^3\times \left(1 - VP_{sat\left(w\right)}/P_{atm}\right) + \\ \exp\left(-10.7604 + 6.3987441\text{e-}02t - 2.6351566\text{e-}04t^2 + 1.6725084\text{e-}06t^3\right)\times \left(P_{atm}/VP_{sat\left(w\right)} - 1\right)\right)\\[12pt] ^{\dagger} \text{The vapor pressure calculation over water is used even when temperatures reach as cold as $-40^{\circ}\!C$}. \\[12pt] ^{\ddagger} VP_{sat\left(w\right)}\space \text{is first calculated without $f$, then $f$ is calculated and applied.} \\[12pt] ^{*} \text{Typical values for f range from 0.3% to 0.5%.}\\[25pt] \Large VP_{sat\left(i\right)}^{\dagger,\ddagger}=f\times \exp\left(\ln\left(611.57\right) - 13.928169\left(1-\left(\frac{273.16}{T}\right)^{1.5}\right) + 34.7078238\left(1-\left(\frac{273.16}{T}\right)^{1.25}\right)\right)\\[15pt] VP_{sat\left(i\right)}=\text{saturated vapor pressure over ice $\left(Pa\right)$};\quad f = \text{enhancement factor}^\bigstar\\[12pt] T=\text{air temperature $\left(K\right);\quad t=$ air temperature $\left(^{\circ}\!C\right);\quad P_{atm}=$ station pressure $\left(Pa\right)$}\\[12pt] f\left(t\le0^{\circ}\!C\right) = \exp\left(3.61345\text{e-}04 + 2.9471685\text{e-}05t + 5.2191167\text{e-}07t^2 + 5.019421\text{e-}09t^3\times \left(1 - VP_{sat\left(w\right)}/P_{atm}\right) + \\ \exp\left(-10.7401 + 7.3698447\text{e-}02t - 2.6890021\text{e-}04t^2 + 1.5395086\text{e-}06t^3\right)\times \left(P_{atm}/VP_{sat\left(w\right)} - 1\right)\right)\\[12pt] ^{\ddagger} \text{The NE Mesonet uses the vapor pressure over ice to determine frost point only.}\\[25pt] \large VP_{act} = \text{Actual Vapor Pressure $\left(Pa\right)$};\quad VP_{act} = \dfrac{RH}{100}\times VP_{sat\left(w\right)}, VP_{sat\left(i\right)}\\[15pt] \large VPD = \text{Vapor Pressure Deficit $\left(Pa\right)$};\quad VPD = VP_{sat\left(w\right)},\space VP_{sat\left(i\right)} - VP_{act}\\[20pt] \Large \text{The $\bf{Dew\space Point\left(T_d\right)}$ is calculated by iterating $T$ in $VP_{sat\left(w\right)}$ until $VP_{sat\left(w\right)} = VP_{act}$}\\[15pt] \Large \text{The $\bf{Frost\space Point\left(F_p\right)}$ is calculated by iterating $T$ in $VP_{sat\left(i\right)}$ until $VP_{sat\left(i\right)} = VP_{act}$}\\[15pt] ^\bigstar \text{$f$ improves the estimation of $VP_{sat\left(w\right)}$ and $VP_{sat\left(i\right)}$, but has no effect on $T_d$ and $T_f$} \\ \\ \end{multline} \]

\[ \begin{multline} \\ \text{Wet Bulb Temperature $\left(T_w;^{\circ}\!C\right)$ requires an iterative calculation. We follow the methodology of Al-Ismaili and Al-Azri (2016),} \\[6pt] \text{with modifications to equations 5 - 8 for consistency with our vapor pressure calculations and improved accuracy.} \\[12pt] \text{Step 1: calculate mixing ratio $\left(\frac{kg\space H_2O}{kg\space dry\space air}\right)$ using equation 1.}\\[12pt] W = 0.62198 \times \dfrac{VP_{sat\left(w\right)}}{P_{atm}-VP_{sat\left(w\right)}};\quad (1)\\[18pt] \text{Step 2: iterate $t_{wb}$ in equation $\left(2\right)$ until equal to $W$ from equation $\left(1\right)$}\\[12pt] W = \dfrac{\left(L_{v,0}-\left(C_{p,w}-C_{p,wv}\right)t_{wb}\right)W_{s,wb}-C_{p,da}\left(t_{db}-t_{wb}\right)}{L_{v,0}+C_{p,wv}t_{db}-C_{p,w}t_{wb}};\quad (2)\\[18pt] t_{wb} = \text{guess value for $T_w \left(^{\circ}\!C\right);\quad t_{mean} = \dfrac{t_{wb}+t_{db}}{2};\quad t_{db} =$ air temperature $\left(^{\circ}\!C\right)$}\\[6pt] L_{v,0} = 2500.85 \left(\frac{kJ}{kg}\right);\quad W_{s,wb} = \text{saturation mixing ratio ${@} \space t_{wb}$}\\[15 pt] C_{p,w} = 4.219878 - 3.398504\text{e-}03t_{wb} + 1.156207\text{e-}04t_{wb}^2 - 2.134986\text{e-}06t_{wb}^3 + 2.243590\text{e-}08t_{wb}^4 - 9.865533\text{e-}11t_{wb}^5\\[12pt] C_{p,wv} = 1.858595 + 1.740971\text{e-}04t_{wb} + 9.959111\text{e-}07t_{wb}^2 + 4.740741\text{e-}09t_{wb}^3 + 8.857349\text{e-}23t_{wb}^4\\[12pt] c_{p,da} = 1.003755 + 3.474427\text{e-}05t_{mean} + 2.603984\text{e-}07t_{mean}^2 + 1.086815\text{e-}10t_{mean}^3 + 5.333333\text{e-}12t_{mean}^4\\[18pt] C_{p,w} = \text{specific heat of liquid water $\left(\frac{kJ}{kg\space^{\circ}\!C}\right)$}\quad\\[12pt] C_{p,wv} = \text{specific heat of water vapor $\left(\frac{kJ}{kg\space^{\circ}\!C}\right)$}\quad \\[12pt] C_{p,da} = \text{specific heat of dry air $\left(\frac{kJ}{kg\space^{\circ}\!C}\right)$}\quad \\[12pt] \\ \end{multline} \]

\[ \begin{multline} \\ \large Wind\space Chill\left(^{\circ}\!F\right)=35.74+0.6215T+35.75V^{0.16}+0.4275TV^{0.16}\\[10pt] \text{T = air temperature at 1.5 - 2 meters}\left(^{\circ}\!F\right)\\[6pt] \text{V = wind speed at 10 meters}\left(mph\right)\\[15pt] \large Wind\space Chill\left(^{\circ}\!C\right)=13.12+0.6215T-11.37^{0.16}+0.3965TV^{0.16}\\[10pt] \text{T = air temperature at 1.5 - 2 meters}\left(^{\circ}\!C\right)\\[6pt] \text{V = wind speed at 10 meters}\left(km/h\right)\\[6pt] \\ \\ \end{multline} \]