Weather

This page summarizes the air temperature model used by ACE Weather.

Simulation of Air Temperature

The SinExp Model

Air temperature is simulated with the corrected SinExp model from Ref. [1], using a sine function during daytime and an exponential decay at night. The temperature $T$ at hour $h$ for the model is given below:

$$T(h)=\{\begin{array}{cc}{T}_{\min }+(T_{\max }-T_{\min })\cdot \sin \left(\frac{\pi \cdot (h-h_r)}{L+2\alpha }\right)& |h_r\le h<h_s\\ \tau +(T_s-\tau )\cdot \exp (-\gamma \cdot \frac{h-h_s}{24-L})& |h_s\le h<h_r^{\prime }\end{array}$$ $$L=h_s-h_r$$ $$\alpha =h_{\max }-\frac{h_r+h_s}{2}$$ $$\tau =\frac{T_{\min }^{\prime }-T_s\cdot \exp (-\gamma \cdot \frac{h_r^{\prime }-h_s}{24-L})}{1-\exp (-\gamma \cdot \frac{h_r^{\prime }-h_s}{24-L})}$$

where $T_{\min }$ and $T_{\max }$ are the daily minimum and maximum temperatures, $T_s$ is the temperature at sunset and $T_{\min }^{\prime }$ is the next morning’s minimum. $h_r$ is sunrise hour, $h_s$ is sunset hour, $h_{\max }$ is the hour of maximum temperature and $L$ is day length. $h_r^{\prime }$ denotes the hour of the next sunrise. $\gamma$ is the decay coefficient (Weather manager’s m_fACE_SinExp_Gamma). The peak hour $h_{\max }$ is interpolated from from the weather manager’s m_aACE_MonthlyAverageTemperatureAirMaxHours.

Minimum and Maximum Temperatures

The baseline minimum and maximum temperatures are interpolated between the terrain’s monthly averages (Weather manager’s m_aACE_MonthlyAverageDailyTemperatureAirMins and m_aACE_MonthlyAverageDailyTemperatureAirMaxs). Overcast and random variation adjust these values:

$$\mathrm{\Delta }{T}_{\max }=X_{\mathrm{mean}}-X_{\mathrm{range}}-{\lambda }_{\text{ơ}}\cdot \overline{\text{ơ}}$$ $$\mathrm{\Delta }{T}_{\min }=X_{\mathrm{mean}}+X_{\mathrm{range}}+{\lambda }_{\text{ơ}}\cdot \overline{\text{ơ}}$$

where $\mathrm{\Delta }{T}_{\max }$ and $\mathrm{\Delta }{T}_{\min }$ are the shifts that get added, $\overline{\text{ơ}}\in [0,2]$ is the estimated average overcast over the next 12 hours and ${\lambda }_{\text{ơ}}$ is the temperature factor for overcast (Weather manager’s m_fACE_OvercastTemperatureFactor). Overcast generally decreases the diurnal temperature range (i.e difference between daily maximum and minimum). Additional random samples $X_{\mathrm{mean}}$ and $X_{\mathrm{range}}$ shift the mean and diurnal temperature range respectively. Both use first-order autoregressive noise, so successive samples are correlated:

$$X_n=\rho \cdot X_{n-1}+{ϵ}_n{X}_0\sim 𝒩(0,{\sigma }^2)$$ $${ϵ}_n\sim 𝒩(0,{\sigma }^2(1-{\rho }^2))$$

where $\rho$ is the autoregressive coefficient, ${ϵ}_n$ is the innovation, $\sigma$ is the standard deviation of the noise and $𝒩(\mu ,s^2)$ denotes a normal distribution with mean $\mu$ and standard deviation $s$. The sampled temperature extrema are clamped to maintain a minimum diurnal temperature range (Weather manager’s m_fACE_MinDiurnalTemperatureRange).

Lapse Rate (Altitude Dependence)

The effective air temperature $T_{\mathrm{eff}}(h,z)$ is calculated based on the altitude above sea level $z$ in meters:

$$T_{\mathrm{eff}}(h,z)=T(h)-6.5\mathrm{K}\cdot \frac{z}{1000\mathrm{m}}$$

where $T(h)$ is the SinExp-based temperature.

References

[1] T. Brandsma, G. P. Können, Int. J. Climatol. 2006, 26, 75–89.