In this thesis we present a two-dimensional thermodynamic model for cloud edge dynamics. We use the incompressible Euler equations to describe the atmospheric fluid dynamics and link these to the theory of moist thermodynamics through a constitutive law. This leads to a free boundary model for the interface separating clear and cloudy air. The model is further specialized to linearized disturbance equations about conditions that are critically saturated with zero liquid cloud water. Due to the presence of discontinuous derivatives induced by the clear/cloudy interface we adapt the immersed interface method (IIM) for computing the pressure in the system to second order accuracy. In addition, we investigate the unforeseen second order convergence of a "naive" method without the IIM. We conduct an analysis of local and global errors for the naive method leveraging off our analysis of the IIM and present numerical verification of the results when necessary.
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