# Monte Carlo Simulations in Aviation Contrail Study: A Review
## 4 Statistical Approaches and Monte Carlo Simulations for Global Evaluation of Contrail Effect
As mentioned in the previous section, computational models that treat dynamics of the whole atmosphere have a coarser spatial resolution with respect to the characteristic dimensions of clouds, either natural or artificially induced, which are then not resolved in these large-scale calculations. General circulation models (GCM) cope with this sub-resolution issue by introducing the concept of fractional cloud coverage, which originates from subgrid scale variability in humidity and/or in temperature (on which the water saturation mixing ratio, \(r_{s}\), depends). In advanced GCM, as in ECHAM5 [103] (ECWMF model, Hamburg version), which simulates the atmosphere and its chemical and physical processes, statistical approaches offer a solution for calculating fractional cloud coverage [104]. Focusing on the fluctuations in total water content, only, this class of methods writes the percentage \(C\) of cloud coverage in each model cell as  
\[C=\int\limits_{r_{s}}^{\infty}G(r_{t})dr_{t}, \tag{11}\]  
where \(G(r_{t})\) represents the probability density function of the total water mixing ratio in the cell considered, including vapor, liquid, and ice phases. [PERSON] [104] used a cloud-resolving model for determining the best a priori distribution, the Beta function with boundaries \(a\), \(b\) and parameters \(p,q\) as  
\[G(t)=\frac{1}{B(p,q)}\;\frac{(\mathrm{t-a})^{\mathrm{P}-1}(b-t)^{q-1}}{(b-a)^{ p+q-1}}, \tag{12}\]  
where \(B(p,q)=\frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}\) is a normalizing combination of gamma functions. An example of a fully-coupled calculation of AIC effects with a GCM model was provided in [105; 106], where a study of temporal dynamics and radiative forcing of contrails' cirrus has been performed with a modified version of the GCM ECHAM5, the ECHAM5-CCMod. This code, however, was based on a different class of methods for tackling the fractional cloud coverage, dubbed relative humidity schemes [107], and conceived of the use of Eulerian moments methods in describing the subgrid cloud dynamics. ECHAM5-CCMod was used in Monte Carlo calculations for assessing the effects of atmospheric uncertainties on mitigation options for air traffic scenarios [100], even though in association with the climate-response model AirClim [108], employed for reducing the complexity of the model calculation and making the Monte Carlo simulation feasible. In 2020, a calculation employing the most advanced version of ECHAM5-CCMod [109] evidenced a consistent reduction of the estimated radiative forcing due to AIC. Whether the radiative effect of contrails is really less intense than that estimated up to now, or whether such detailed simulations need to properly include other elements, as the coupling with oceans, still needs to be understood in order to be quantitatively reliable on an absolute scale. Different approaches, based on simpler evaluation of contrails' effects, though uncoupled with the atmospheric system, have been developed. An example is the CERM (contail evolution and radiation model) [110], an evolution of the CoCiP (contail cirrus prediction model) [111]. Beyond evaluation of the accuracy of these simplified calculations with respect to detailed GCM models, the questions to be addressed concern the magnitude of the coupling of cloudiness, in general, and contrail cirrus, in particular, with the other constituents of the planetary climate system. This general question goes well beyond the scope of this review; from one point of view, however, it could be argued that the evolution of the Lagrangian Monte Carlo-based calculations will help to shed some light on physical details and, hopefully, on coupling phenomena in planetary-scale models of the atmosphere.