10  LaTeX equations

In the notebooks we use Markdown to write text, but equations are rendered using LaTeX syntax. LaTeX is a typesetting that allows technical writers and scientists to focus on the content without worrying abouT the format. LaTeX is free under the terms of the LaTeX Project Public License (LPPL).

!>A nice feature of LaTeX is that there is a specific syntax for equations. For instance, you can define inline equations so that this: $y=mx+b$, get’s converted into this: y=mx+b

It is also possible to write equations in separate lines using $$y = e^{-x}$$:

y = e^{-x}

Examples

Below is a set of equations obtained from the FAO 56 manual to calculate reference evapotranspiration. Use this equations as templates to learn how to implement your own equations.

Reference Evapotranspiration Equation

$$ETo = \frac{0.408\Delta(Rn-G)+\gamma\frac{900}{T+273}u2(es-ea)}{\Delta+\gamma(1+0.34u2)}$$

ETo = \frac{0.408\Delta(Rn-G)+\gamma\frac{900}{T+273}u2(es-ea)}{\Delta+\gamma(1+0.34u2)}

ETo = reference evapotranspiration (mm/day)

Rn = net radiation at the crop surface (MJ/m2/day)

G = soil heat flux density (MJ/m2/day)

T = mean daily air temperature at 2 m height

u2 = wind speed at 2 m height (m/s)

es = saturation vapor pressure (kPa)

ea = actual vapor pressure (kPa)

es-ea = saturation vapor pressure deficit (kPa)

\Delta = slope vapor pressure curve (kPa/°C)

\gamma = psychrometric constant (kPa/°C)

Psychrometric constant

$$\gamma = \frac{Cp P}{\epsilon \lambda}$$

\gamma = \frac{Cp \ P}{\epsilon \lambda}

\gamma = psychrometric constant (kPa/°C)

\lambda = latent heat of vaporization, 2.45 (MJ/kg)

Cp = specific heat at constant pressure (MJ/kg/°C)

\epsilon = ratio of molecular weight of water vapour/dry air = 0.622

P = atmospheric pressure (kPa)

Wind speed at 2 meters above the soil surface

$$u2 = uz\frac{4.87}{\ln(67.8z-5.42)}$$

u2 = uz\frac{4.87}{\ln(67.8z-5.42)}

u2 = wind speed at 2 m above ground surface (m/s)

uz = measured wind speed at z m above ground surface (m/s)

zm = height of measurement above ground surface (m)

Mean saturation vapor pressure

$$es = \frac{eTmax+eTmin}{2}$$

es = \frac{eTmax+eTmin}{2}

es = mean saturation vapor pressure (kPa)

eTmax = saturation vapor pressure at temp Tmax (kPa)

eTmin = saturation vapor pressure at temp Tmin (kPa)

Slope of vapor pressure

$$\Delta = \frac{4098\bigg[0.6108\exp\bigg(\frac{17.27 Tmean}{Tmean+237.3}\bigg)\bigg]}{(Tmean+237.3)^2}$$

\Delta = \frac{4098\bigg[0.6108\exp\bigg(\frac{17.27 Tmean}{Tmean+237.3}\bigg)\bigg]}{(Tmean+237.3)^2}

\Delta = slope of saturation vapor pressure curve at air temp T (kPa/°C)

Tmean = average daily air temperture

Actual vapor pressure

$$ea = \frac{eTmin\frac{RHmax}{100}+eTmax\frac{RHmin}{100}}{2}$$

ea = \frac{eTmin\frac{RHmax}{100}+eTmax\frac{RHmin}{100}}{2}

ea = actual vapor pressure (kPa)

eTmax = saturation vapor pressure at temp Tmax (kPa)

eTmin = saturation vapor pressure at temp Tmin (kPa)

RHmax = maximum relative humidity (%)

RHmin = minimum relative humidity (%)

Extraterrestrial solar radiation

$$Ra=\frac{24(60)}{\pi}\hspace{2mm}G\hspace{2mm}dr[\omega\sin(\phi)\sin(\delta)+\cos(\phi)\cos(\delta)\sin(\omega)]$$

Ra = \frac{24(60)}{\pi} \hspace{2mm}G \hspace{2mm} dr[\omega\sin(\phi)\sin(\delta)+\cos(\phi)\cos(\delta)\sin(\omega)]

Ra = extraterrestrial radiation (MJ/m2/day)

G = solar constant (MJ/m2/min)

dr = 1 + 0.033 \cos(2\pi J/365)

J = number of the day of the year

\phi = \pi/180 decimal degrees (latitude in radians)

\delta = 0.409\sin((2\pi J/365)-1.39)\hspace{5mm} Solar decimation (rad)

\omega = \pi/2-(\arccos(-\tan(\phi)\tan(\delta)) \hspace{5mm} sunset hour angle (radians)