This Paper Presents results for a linear Lagrangian entraining parcel model of an overshooting thunderstorm cloud top. The model is based on that of Adler and Mack, but differs in that it achieves analytic, exact solutions (for constant stratospheric lapse rates), by representing mixing with Rayleigh damping instead of nonlinearly. These exact linear solutions are forced additively by the initial parcel rise rate, hydrometeor drag, and initial temperature deviation (if any).
Without drag or an initial temperature deviation, an initially rising parcel decelerates, cools below the ambient temperature, sinks and then becomes warmer than the environment. Depending on mixing strength, the parcel approaches a stratospheric equilibrium height at the ambient temperature or fallS back through the tropopause during the first oscillation.
With drag, the parcel eventually returns to the tropopause. If the parcel remains in the stratosphere sufficiently long, it approaches a constant rate of descent at a constant temperature excess over ambient temperature. Both limiting parameters are directly proportional to drag. The descent rate decreases with ambient stability and increases with mixing strength, and vice versa for the temperature excess (over a suitable range of input parameters).
A parcel initially warmer (colder) than the environment undergoes slightly delayed (advanced) oscillations, with a higher (lower) peak overshoot and minor increases (decreases) in both maximum and minimum temperatures, but without other material alterations.
In some respects, the linear parcel model performs similarly to the Alder–Mack simulation (AM). Strong mixing favors a close-in warm point and a large cold-high offset, neither of which is present with weak mixing. Increased stability lowers the cloud summit and cold point while amplifying the cloud-top thermal couplet. Increased drag lowers the cloud summit, cold point, and warm point without appreciably changing the amplitude.
Due to the contrasting mixing formulations, there are also several significant differences between our model results and those of AM under conditions as analogous as possible. With strong mixing, large cold-high offsets require both an inversion and rapid initial ascent in AM, but can occur in our model without either feature. A downslope warm point requires an inversion in AM, but not in our model. Cold-warm couplets show greater amplitude in our model than in AM, especially if mixing is strong. Parcels overshoot farther in our model than in AM, having undergone weaker mixing during vigorous early ascent. In our model, a parcel can approach a stratospheric equilibrium level only if there is no drag, whereas in AM some parcels with weak drag appear to approach an equilibrium level while superimposing several slowly damping oscillations on gradual overall descent.