# Monte Carlo Simulations in Aviation Contrail Study: A Review
## 3 Monte Carlo Simulations for Contrails Formation and Evolution  
Contrails' formation and transformation into cirrus is believed to be linked to the mutual thermodynamic state, the water-vapor content of the atmosphere, and the engine exhaust, as discussed above. This section is devoted to analyzing the role of computational, stochastic approaches in simulating the formation of the microscopical ice aggregates that characterize a trail's evolution into a persistent cirrus. From a chemical point of view, a contrail originating in the exhaust-atmosphere mixing is a dispersed phase system, containing aerosols and solid particles in a continuum phase. In practice, simulating the origin and evolution of a contrail is strongly linked to the capabilities of computationally reproducing the lifecycle of an (ice) cloud. Historically, clouds were first simulated using Eulerian, density-based approaches [71; 72; 73; 74; 75; 76] with equations that described the time evolution of bulk properties, including mass (one-moment scheme), number concentration (two-moments scheme), and, recently, radar-reflectivity (three-moments scheme). These models are still at the foundation of planetary scale and climate simulations [77; 78; 79; 80; 81; 82; 83; 84; 85; 86; 87; 88; 89; 90; 91; 92; 93; 94], given their relatively low computational complexity, which is suitable for insertion into codes with a high spatial resolution with respect to their length scale, which usually covers the entire planet. The starting point for a theoretical study of contrail formation is the stochastic collection equation (SCE) (see, for examples, the introduction given in [79] and references therein), which describes the time evolution of a system of particles (representing droplets, ice crystals, or both) colliding and subsequently forming a larger structure (_coalescence_):  
\[\frac{\partial n(v,t)}{\partial t}=\frac{1}{2}\ \int\limits_{0}^{v}n\big{(}v-v^{ \prime},t\big{)}n\big{(}v^{\prime},t\big{)}K\big{(}v-v^{\prime},v\big{)}dv^{ \prime}-n(v,t)\int\limits_{0}^{\infty}n\big{(}v^{\prime},t\big{)}K\big{(}v,v ^{\prime}\big{)}dv^{\prime}=I_{1}-I_{2}, \tag{5}\]  
Figure 1: (**a**) Icing Wind Tunnel aerial view; (**b**) IWT main test section. Taken from [70]. where \(n(v)\) is the number density function of the particle with volume \(v\) (equivalent formulation in terms of mass is possible), and \(K(v,v^{\prime})\) is the coalescence kernel of particles with volumes \(v\) and \(v^{\prime}\). In Equation (5), \(I_{1}\) and \(I_{2}\) refer to the integral source and sink term for the coalescence process. Moment methods, in fact, provide a solution to the SCE by modeling each integral term as a whole. Beyond the simplistic early studies on clouds, Equation (5) has been exploited since the 1970s in a class of methods called bin approaches [81, 82], which are based on the discretization of the particle spectra. The discretized form of Equation (5) reads as:  
\[\frac{\partial N_{i}(t)}{\partial t}=\sum_{j,k}^{j>k}(1-\frac{1}{2}\;\delta_{j,k})\;K_{j,k}N_{j}(t)N_{k}(t)-N_{i}(t)\;\sum_{k}K_{i,k}N_{k}(t), \tag{6}\]  
where \(N_{i}(t)=\int_{v_{j}}^{v_{i}+1}n(v,t)dv\) is the number of particles in the \(i\)-th bin, and \(K_{j,k}\) is the discretized form of the coalescence kernel. In fact, this wide class of computational methods is still Eulerian in nature because the concept of a particle is not introduced except for populating the different bins in which the spectrum has been divided. Nonetheless, bin models allow calculation to a good approximation of the development in time of the particle concentrations, which usually exploits a finite difference approach for describing the time variations of Equation (6). Bin models were extremely appealing at the beginning of the electronic computer era since they were capable of producing a reasonable time-evolution calculation even with a limited number of bins. Their accuracy, however, is strongly linked to the chosen number of bins, whose increase is linked quadratically to the complexity of the problem to be solved.