Components

TrenchR functions are organized into the following categories (Fig 2):

General biophysical functions.

Provide generalized models of heat exchanges between organisms and their environment so that users can build custom biophysical models. The functions allow implementing an energy balance including the following components of heat exchange between organisms and their environment:

• radiative heat exchanges of solar and thermal radiation;
• convective heat exchanges between organisms and their surrounding fluid (air or water) driven by fluid flow;
• conductive heat exchanges between organisms and solid surfaces (generally the ground) due to physical contact; and
• evaporative and metabolic heat exchanges associated with organisms’ evaporative water loss and metabolic heat generation.

Additional functions aggregate these forms of heat exchange into energy balances and to use the energy balances to predict body temperatures.

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Fig 2. An examplar aim of translating from environmental and phenotypic data to organismal conditions in a given environment (grey boxes) can be achieved in TrenchR via several categories of functions (white boxes).

We list example functions for each category used in the example application below.

https://doi.org/10.1371/journal.pclm.0000139.g002

Vignettes and introductory tutorials

We introduce the functions in several vignettes. A good place to start is the Allometry and Conversions vignette, which provides tools for preparing data, such as estimating additional dimensions of organisms from measured dimensions. The Estimating Microclimates vignette provides resources for estimating the environmental conditions experienced by organisms. This includes estimating solar radiation and its components, diurnal variation in temperature and radiation, temperature and wind speed profiles, and soil temperatures and profiles. Finally, the core biophysical modeling functions are described in a tutorial on Using Energy Balances to Estimate Body Temperatures. Components of an energy budget can be estimated using individual functions and then body temperatures can be solved for using either a generic energy balance or taxa specific biophysical models. We additionally offer a List of Symbols used in equations.

For additional background on microclimate and biophysical modeling, we have updated a series of tutorials entitled Physical Processes in Ecosystems. We intend the tutorials to provide a more contained and accessible introduction to microclimate and biophysical modeling than that included in classic textbooks [13, 14]. The tutorials provide less detailed coverage of many of the topics contained in the books. The series of 14 tutorials starts with an overview of the calculus and physics principles underlying the modeling.

Thermodynamics and energy budget modeling are then introduced with detailed examples. Tutorials address the climate space concept, operative temperatures, biophysical models for leaves and sheep, and heat flow in soils. We have revised the tutorials (which originated from an NSF training grant in 1979 and lacked broad distribution) to include R code and utilization of TrenchR functions. We expanded the original series to include tutorials contributed by M. Kearney introducing the Microclim environmental data [27] and integrating the Microclim data with biophysical modeling to examine broad scale climatic limits. The tutorials are provided in html form and available for download as a pdf at the bookdown server (https://bookdown.org/huckley/Physical_Processes_In_Ecosystems/). R markdown files for the tutorials are available in Github (https://github.com/trenchproject/TrenchRmodules).

Environmental data

At the first stage, we prepare the environmental data for analysis. We will estimate hourly air and soil temperatures and radiation using a function describing diurnal temperature variation. We start by estimating the day of year and the timing of sunrise and sunset:

1. # Set up input data as variables

1. lat <- 35.69 # Latitude (degrees)
2. lon <- -105.944 # Longitude (degrees)
3. elev <- 2121 # Elevation (meters)

1. Tmin <- 10 # Minimum air temperature (C)
2. Tmax <- 25 # Maximum air temperature (C)
3. Tmin_s <- 15 # Minimum soil temperature (C)
4. Tmax_s <- 30 # Maximum soil temperature (C)
5. u <- 1 # Wind speed (m/s)

1. # Assumptions

1. tau <- 0.7 # Atmospheric transmissivity
2. rho <- 0.6 # Albedo
3. Tb0 <- 25 # Initial assumption of body temperature (C)

1. doy <- day_of_year("2021-06-01", format = "%Y-%m-%d") # Day of year

1. snoon <- solar_noon(lon = lon, doy = doy) # Estimate solar noon
2. dayl <- daylength(lat = lat, doy = doy) # Estimate day length

1. tr <- snoon—dayl / 2 # Time of sunrise
2. ts <- snoon + dayl / 2 # Time of sunset

Although measured solar radiation is preferable if available, we can estimate hourly solar radiation by discounting incoming solar radiation as it moves through the atmosphere as follows. We use the approach from Cambpell and Norman [14], which uses an empirical relation to partition radiation into direct, diffuse, and reflected components. The partition_solar_radiation() function includes 8 empirical relationships for, and the proportion_diffuse_solar_radiation() includes a more complex numerical approximation for, partitioning radiation components as described in the Estimating microclimates vignette.

1. # Estimate zenith angle (degrees)

1. psi_deg <- sapply(0:23, FUN = zenith_angle, doy = doy, lat = lat, lon = lon)

1. # Convert to radians

1. psi_rad <- degrees_to_radians(psi_deg)

1. # Estimate radiation

1. Srad <- sapply(psi_rad, FUN = solar_radiation, doy = doy, tau = tau, elev = elev, rho = rho)

1. # Separate solar radiation into direct, diffuse, and reflected components

1. Sdir <- Srad[1,] # Direct solar radiation (W/m2)
2. Sdif <- Srad[2,] # Diffuse solar radiation (W/m2)
3. Sref <- Srad[3,] # Reflected solar radiation (W/m2)

We then calculate hourly air and soil surface temperatures based on daily minimum and maximum temperatures. We select the sine-exponential model for air temperature and the sine model for surface temperature [29]:

1. # Air temperature (C)

1. Ta <- sapply(1:24, diurnal_temp_variation_sineexp, T_max = Tmax, T_min = Tmin, t_r = tr, t_s = ts, alpha = 2.59, beta = 1.55, gamma = 2.2)

1. # Soil surface temperature (C)

1. Ts <- sapply(1:24, diurnal_temp_variation_sine, T_max = Tmax_s, T_min = Tmin_s)

At the second stage, we use microclimate models to scale air temperature (T_r) and wind speed (u_r) from weather station height (reference height z_r = 2 m) to lizard height (organism height z = 0.02 m). We assume a surface roughness of z₀ = 0.2 m, which corresponds to bare sand and determines the turbulence of airflow. We implement free air temperature and wind speed profiles driven by density differences but profiles forced by wind speed are also available.

1. # Neutral air temperature profile

1. Ta_liz <- air_temp_profile_neutral(T_r = Ta, zr = 2, z0 = 0.2, z = 0.02, T_s = Ts)

1. # Neutral wind speed profile

1. u_liz <- wind_speed_profile_neutral(u_r = u, zr = 2, z0 = 0.2, z = 0.02)

Energy balance

Finally, we will use our microclimates estimates to solve the following energy balance to estimate T_e: where Q_net is the net energy exchange with the environment (W), Q_abs is the solar radiation absorbed (W), Q_emit is the net thermal radiation emitted (W), Q_conv is energy exchange due to convection (W), Q_cond is energy exchange due to conduction (W), Q_met is the energy generated by metabolism (W), and Q_evap is the energy generated by evaporative water loss (W). We will estimate each term on the right side of the equation in turn. Estimating Q_abs requires the surface area exposed to radiation and the solar absorptivity of the animal surface (a proportion). We use zenith angle psi to estimate the projected (silhouette) area as a portion of the surface area of the organism, which allows estimating absorbed solar radiation. We model a 10 gram Sceloporus lizard with solar absorptivity a = 0.9 [13]. We will initially assume T_b = T_a + 10 to illustrate the calculations before solving for T_b given the environmental conditions.

1. mass <- 10 # Mass (g)
2. svl <- 0.006 # Snout vent length (meters)
3. a <- 0.9 # Solar absorptivity (proportion)

1. #assume 1/3 of surface area is in contact with surface
2. psa_g <- 0.33

1. # Estimate surface area (m^2) and the proportion sihouette area

1. A <- surface_area_from_mass(mass, "lizard")
2. psa <- sapply(psi_deg, proportion_silhouette_area, taxon = "lizard", posture = "prostrate")

1. # Change negative values to zero
2. psa[psa < 0] = 0

We calculate the hourly solar and thermal radiation absorbed (W) as follows:

1. Qabs <- rep(NA, 24)
2. for (hour in 1:24) {

1. Qabs[hour] <- Qradiation_absorbed(a = a, A = A, psa_dir = psa[hour], psa_dif = 0.5, psa_ref = 0.5, S_dir = Sdir[hour], S_dif = Sdif[hour], rho = rho)

1. }

We estimate thermal radiation Q_emit (W) for the lizard outdoors, where psa_dir and psa_ref are the view factors, also refered to as configuration factors, that indicate the proportions of surface area A (m²) exposed to the sky and ground, respectively. We assume the surface emissivity of lizards, epsilon_s = 0.965 [30].

1. epsilon_s <- 0.965 # Surface emissivity of lizards

1. Qemit <- rep(NA, 24)
2. for (hour in 1:24) {

1. Qemit[hour] <- Qemitted_thermal_radiation(epsilon = epsilon_s, A = A, psa_dir = 0.5, psa_ref = 0.5, T_b = Ta_liz[hour] + 273.15, T_g = Ts[hour] + 273.15, T_a = Ta_liz[hour] + 273.15, enclosed = FALSE)

1. }

We next estimate convection Q_conv (W) and conduction Q_cond (W). We will estimate the lizard’s heat transfer coefficient, H_L (Wm^-2K^-1) using an empirical relationship for lizards (heat_transfer_coefficient()). We average thermal conductivity and kinematic viscosity across the day for simplicity and since there is not substantial diurnal variation. We also illustrate a function estimating H_L using a spherical approximation (heat_transfer_coefficient_approximation()) and a simplified approximation (heat_transfer_coefficient_simple()) for cases when taxon specific relationships for estimating heat transfer coefficients are not available. We estimate the characteristic dimension, which determines exposure to convective heat exchange as the cube root of volume, assuming the animal density approximates that of water [31].

These coefficients assume convection is forced by the wind. TrenchR includes approaches for free convection and a function (free_or_forced_convection()) that evaluates whether free or forced convection is appropriate. The function uses dimensionless numbers, which have been developed to describe heat transfer coefficients associated with convection over different geometries and can be estimated using TrenchR (e.g., Grashof, Nusselt, and Reynolds numbers).

1. # Use DRYAIR from NicheMapR to estimate the thermal conductivity of air and kinematic viscosity.

1. ap <- airpressure_from_elev(elev) * 1000 # Barometric pressure (pascal)

1. DRYAIRout <- DRYAIR(db = Ta, bp = ap, alt = elev)

1. K <- mean(DRYAIRout$thcond) # Thermal conductivity (Wm^-2K^-1)
2. nu <- mean(DRYAIRout$viskin) # Kinematic viscosity (m2 s-1)

1. # Estimate the characteristic dimension as cube root of volume, assuming density of water as 1000kg/m^3

1. D <- ((mass / 1000) / 1000) ^ (1 / 3)

1. # Estimate the heat transfer coefficient using an empirical relationship for lizards

1. H_L <- heat_transfer_coefficient(u = u_liz, D = D, K = K, nu = nu, taxon = "lizard_surface")

1. # Estimate the heat transfer coefficient using a spherical approximation

1. H_L2 <- heat_transfer_coefficient_approximation(u = u_liz, D = D, K = K, nu = nu, taxon = "lizard")

1. # Estimate the heat transfer coefficient using a simplified version of the approximation

1. H_L3 <- heat_transfer_coefficient_simple(u = u_liz, D = svl, type = "Gates")

We estimate convective heat exchange between the animal and surrounding air using the following relationship: where an enhancement factor, ef, multiplier can be incorporated to account for increases in heat exchange resulting from air turbulence in field conditions. We implement the function in R assuming that 2/3 of the lizard’s surface area is exchanging heat through convection.

1. Qconv <- rep(NA, 24)
2. for (hour in 1:24) {

1. Qconv[hour] <- Qconvection(T_a = Ta_liz[hour] + 273.15, T_b = Ta_liz[hour] + 10 + 273.15, H = H_L, A = A, proportion = 0.67, ef = 1.3)

1. }

We estimate conductive heat flow (W) from the lizard to the surface assuming conductance through the animal tissue is the rate limiting step as follows:

where K_skin in the thermal conductivity of lizard skin (Wm^-2K^-1). We implement the estimate assuming that conductive heat exchange occurs down to a soil depth of 2.5cm. We use this value rather than skin thickness, which results in rapid conduction and does not readily reach steady state conditions.

1. Qcond <- rep(NA, 24)
2. for(hr in 1:24) {

1. Qcond[hr] <- Qconduction_animal(T_g = Ts[hr] + 273.15, T_b = Ta_liz[hr] + 10 + 273.15, d = 0.025, K = 0.5, A = A, proportion = psa_g)

1. }

We assume, as is generally done for lizards, that heat exchange associated with metabolism and evaporation is negligible. However, functions for estimating both forms of heat exchange available in TrenchR.

1. Qmet <- 0
2. Qevap <- 0

The full heat budget can be calculated as follows [13]:

1. Qnet <- Qnet_Gates(Qabs = Qabs, Qemit = Qemit, Qconv = Qconv, Qcond = Qcond, Qmet = Qmet, Qevap = Qevap)

We now use a function based on the Gates energy balance above to estimate body temperature given the environmental conditions:

1. Te <- rep(NA, 24)
2. for (hour in 1:24) {

1. Te[hour] <- Tb_Gates(A = A, D = svl, psa_dir = psa[hour], psa_ref = 1—psa[hour], psa_air = 0.67, psa_g = 0.25, T_g = Ts[hour], T_a = Ta_liz[hour], Qabs = Qabs[hour], epsilon = epsilon_s, H_L = H_L, ef = 1.3, K = K)

1. }

We also implement a similar but simplified energy balance [14]. The energy balance omits conduction with the ground:

1. Te2 <- rep(NA, 24)
2. for (hr in 1:24) {

1. # S is solar radiation flux (W m^-2), so we divide by surface area, A

1. Te2[hr] <- Tb_CampbellNorman(T_a = Ta_liz[hr], T_g = Ts[hr], S = Qabs[hr] / A, a_l = 0.96, epsilon = epsilon_s, c_p = 29.3, D = D, u = u_liz)

1. }

We additionally estimate T_b using a specialized function for lizards [32], where F_d, F_r, F_a, and F_g are the view factors between the surface of the lizard and diffuse solar radiation, reflected solar radiation, atmospheric thermal radiation, and ground thermal radiation, respectively:

1. Te3 <- rep(NA, 24)
2. for (hour in 1:24) {

1. Te3[hour] <- Tb_lizard(T_a = Ta_liz[hour], T_g = Ts[hour], u = u_liz, svl = svl * 1000, m = mass, psi = psi_deg[hour], rho_s = rho, elev = elev, doy = doy, sun = TRUE, surface = TRUE, a_s = a, a_l = 0.965, epsilon_s = epsilon_s, F_d = 0.8, F_r = 0.5, F_a = 0.5, F_g = 0.5)

1. }

The microclimate models indicate that air temperatures at lizard height are similar to surface temperatures (Fig 3). The biophysical models indicate that solar radiation can elevate lizard body temperatures far above air temperatures and that the lizard will face thermal stress if it is unable to seek shade (Fig 3). The three biophysical models predict different body temperatures during peak period of solar radiation because they model interactions with the ground differently and users are encouraged to review the details of each biophysical model and perform empirical validations before selection. Differences in estimated body temperatures are accentuated by the high level of solar radiation. Tb_Gates is a general and comprehensive model that is appropriate for many applications. Taxa-specific biophysical models often best account for details of organism environment interactions and have generally been well tested.

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Fig 3. Body temperatures (Te) are predicted to drastically exceed air temperature when lizards are exposed to high levels of solar radiation.

Air temperatures (Ta, °C) at lizard height (0.02 m) are predicted to exceed air temperatures at 2 m and to be similar to surface temperatures (Ts). We estimate body temperatures using two general energy budgets [solid: Tb_Gates(); dotted: Tb_CampbellNorman()] and a lizard specific biophysical model [dashed: Tb_lizard()] that differ in how they model heat exchanges.

https://doi.org/10.1371/journal.pclm.0000139.g003