3. Data & Method
3.7. Model method
To aid in the interpretation of the results of the fourier and PC analysis a simple heat balance model has been developed. The model consists of the Incoming Solar
Radiation (ISR), a clear sky atmosphere, a heat response surface layer which radiates upwelling longwave flux, and a conductive layer that thermally conducts with the surface layer (see fig. 3.5).
Figure 3.5.: Diagram of the model with ISR, clear sky atmosphere and a surface
The model initiates with a set arbitrary temperature (293K) for the surface, the heat capacity is set according to the specific heat capacities for soil, sand and other materials [Arya, 2001]. For example, a desert surface type is interpreted as sand in the model with a specific heat of 835 J kg−1K−1. The ISR time series is taken from GERB shortwave data, a portion of which is reflected by the surface (25%) [DeLiberty, 1999].
The incoming solar flux (F ) is converted into energy (E) using the equation:
The area (A) is defined as 1m2, and the time (dt) is the model time step (15 minutes)
in seconds. The ISR data is taken from the GERB file data and re-sampled to 15 minutes resolution by linear interpolation to synchronise with the model time steps. The input ISR varies according to the solar zenith angle (θ) indicated in Fig. 3.5. This SZA is dependent on the latitude of location, the season and the time of day. This ISR variation with respect to the SZA is already integrated into the GERB ISR file. Therefore, we were only required to take this ISR data and linearly interpolated it to 15 minutes interval for our use. Note that at the illumination end points of the diurnal cycle (i.e. dawn, dusk), we have chosen to simply re-bin the points to eliminate any tailing effects from linear interpolation.
The incoming solar radiation is governed by the SZA in the following expression:
I = S cos θ (3.15)
where I is the incoming solar radiation, S is the solar constant. The ISR (from GERB) then strikes the surface with the assumption of a clear sky scenario which in turn provide energy to the surface. In the clear sky scenario, we have assumed a fixed combine atmospheric absorption and scattering to be a factor of 0.2. We recognise the atmospheric path length changes during the day, and the atmospheric absorption is dependent on it. However, we are only modelling the OLR close to the equator (30◦in latitude), and under the assumption that we do not expect the delay between peak in surface heat response and the peak in ISR to exceed 3 hours (i.e. any change in ISR outside this time limit would not influence the peak time) which is under what we have observed in our surface type PCA. This would give us equivalent of a time angle of 45◦. The SZA will be around 41◦, which gives roughly an extra 32% in atmospheric path length. Thus, the difference of a varying atmospheric path length will only give us a few percent more absorption/scattering in the SW around this time.
The change in temperature (∆T , K) is then calculated using the thermal energy (Q, in J ) equation:
Q = m.c.∆T (3.16)
where m (kg) is the mass of the surface which is given by the density multiplied by unit area (1x1m) and depth (2.5cm); c (J kg−1K−1) represents the specific heat
capacity of the material. The change in temperature (∆T , K) is added onto the original temperature for each time step. For example, the temperature after the first time step would be 293K + ∆T1. The density of the surface layer for the desert
type is 1600kgm−3 with a heat capacity of 800J kg−1K−1[Arya, 2001].
This temperature of the surface and atmosphere is then used to calculate the radiative flux (F ) using the Stefan-Boltzmann law:
Fout = σT4 (3.17)
where Fout signifies the upwelling shortwave flux, T is the temperature in K and
is the emissivity. The flux is converted into energy using equation 3.14 and the net energy is recalculated at the beginning of every model time step. We have used a surface emissivity of 0.85 [Ogawa et al., 2008] and an atmospheric emissivity of 0.68 [Herrero and Polo, 2012].
The conductive layer consists of a slab which bares the same surface properties as the surface and is responsible for conductive heat transfer with the surface layer. The heat transfer follows the Fourier’s law of thermal conduction in single dimension given in equation 3.18.
Q =−kdT
dx (3.18)
The symbols Q, k, dT and dx in eqn. 3.18 refer to heat flux exchange between the layers, conductivity of the layer, temperature difference between the two layers, and the depth of the surface layer respectively. The thermal and radiative exchanges between the layers are demonstrated in fig. 3.6. The heat responsive layer and conductive layer are 3cm and 1m respectively, where a large temperature diurnal cycle (over 20◦C of diurnal variation is observed by the soil layer model in [Arya, 2001]) at the depth of 2.5cm and only small diurnal fluctuations are observed at depths of 1m.
The upwelling flux from the surface is then absorbed by the atmosphere as shown in fig. 3.6 by the factor of emissivity given by the Kirchoff’s Law, where the emis- sivity is equal to the absorptivity. This absorption of the surface upwelling flux warms up the atmosphere and thus radiates in both directions itself. In thermal balance, the atmosphere radiates the same amount as it absorbs, leading to a direct
Figure 3.6.: Diagram of model in separate layers to demonstrate the heat flux in/out of each layer, * denotes the reflection of the atmospheric emitted down- welling radiation by the surface, (1− s)aσTa4
relationship between atmospheric and surface temperature given by the expression (eqn. 3.19)
2aσTa4 = saσTs4+ (1− s)2aσT
4
a (3.19)
This equation assumes a thermal balance in the atmosphere but it does not mean a net energy of 0 into the layer at all time(i.e. constant temperature). It only assumes a thermal balance at every model time step, i.e. it would not take more than a model time step for the atmosphere to heat up and reach the new thermal equilibrium. The combined effect of the atmosphere and the surface gives the OLR output at the TOA a total of:
OLR = aσTa4+ (1− a)sσTs4 + (1− a)(1− s)aσTa4 (3.20)
The feedback longwave flux from the atmosphere is then calculated at the end of every model time step.
The ISR time series contains one month of data, thus the model was allowed to run during the same length of time. All the surface and atmosphere layers in the model are set to a room temperature of 293K, they are allowed to vary until they
reach an equilibrium. The modelled outgoing flux was found to stabilise after a few days from the initial set up.
3.8. Chapter Summary
This chapter has presented information on the GERB instruments and the data products available which are relevant to our analysis. The data analysis methods used in our study have also been explained and discussed in this chapter.
The Fourier transform technique is valuable in identifying and analysing multiple timescale cycles within the time record of the data. On the other hand, the PCA and band pass filter techniques are more orientated in analysing specific timescale cycles. The PCA is especially powerful in investigating the spatial and temporal properties of various climate processes.
A simple heat balance model developed to aid in the interpretation of the results has also been introduced. The model consists of a single layer of atmosphere, a sur- face which radiates according to Stefan-Boltzmann law, a conductive layer beneath the surface and ISR from GERB SW data.