In ijlyttle/groundTemp: Make models of ground temperature

library(ggvis)
library(dplyr)
library(lubridate)
library(groundTemp)

Once you have the coefficients for the surface-temperature model, the next step is to determine the reference depth, $z_0$. This reference depth depends on the thermal diffusivity of the ground material, as well as the period, which is defined as a tropical year.

[{z_0} = \sqrt {\frac{{\alpha \,\Delta {t_0}}}{\pi }} ]

For example, a typical value for thermal diffusivity of concrete is 10-6 m2/s, and is a reasonable value to use for an initial investigation (thanks, Patrick Beguery). Following the math, and keeping in mind that is a tropical year, this gives us a reference depth of 3.17 m.

Let's put this in the context of the model for ground-temperature:

[T(t, z) = {T_0} + \Delta {T_0}{e^{ - \frac{z}{{{z_0}}}}}\sin \left( {\frac{{2\pi (t - {t_0})}}{{\Delta {t_0}}} - \frac{z}{{{z_o}}} - \frac{\pi }{2}} \right)]

The physical meaning of this reference depth is that, at this depth, the amplitude of the sinusoid is $1/e$ , or about 40%, of the amplitude of the sinusoid at the surface. Further, at this depth, the phase lags the surface phase by one radian, or about one-sixth of the cycle.

Now we have all the information to build a function that will return an estimate for grounf temperature, given a set of surface-temperature coefficients, and thermal diffusivity.

Let's calculate the surface-temperature coefficients:

surface_temp_coef <- surface_temp_coef(wx_2011$instant, wx_2011$tmp_db)

The next step is to choose a thermal diffusivity. Let's consider concrete:

alpha <- 1.e-6

Now, we will build a function that will take as aruguments the depth, $z$ and the time $t$, and return an estimate for the temperature.

gnd_temp <- 
  get_fn_ground_temperature(
    temp_ref = surface_temp_coef$temp_ref,
    dtemp_ref = surface_temp_coef$dtemp_ref,
    time_ref = surface_temp_coef$time_ref,
    diffusivity = alpha
  )

gnd_temp

We will call the function we created to determine the ground temperature given the depth and the instant of time.

We can show the environment for this function, to show where it gets its values for depth_norm, etc.