Please find my comments attached.

Drawing inspiration from recent advancements in understanding atmospheric stability and stratification in heatwaves, and leveraging the Lagrangian framework developed by Roethlisberger and Papritz (2023), this paper delivers the first meticulous Lagrangian analysis of the 3-D atmosphere during three extreme heatwaves. This study is technically robust and discusses properly the existing body of literature. It provides a fresh perspective on heatwaves and holds the potential to expand its findings to analyze more events in future research. I believe this manuscript is deserving of publication following some minor revisions.

Line 174: The negative values are likely due to strong entrainment, which dilutes the specific humidity of air parcels as they ascend through the dry boundary layer, as noted by Zhang and Boos. 

Line 215: I’m not entirely convinced that the conclusion of the PNW heatwave was primarily due to convective damping rather than changes in advection. Utilizing an Eulerian viewpoint might offer a clearer understanding of the decline in boundary layer temperature. If the Eulerian advection term continues to rise when the surface temperature is already declining, then it is reasonable to infer that advection is not the primary cause. Additionally, the T’ values utilized here represent accumulations, not tendencies. Thus, discussions regarding changes in total T’ should center around the changes in advective T’. I am not requesting additional analysis given the Lagrangian focus of this paper, but anticipate that the authors will address these reservations and refine their discussion accordingly.

Line 271: This paragraph is somewhat unclear to me. It appears that by 29 June, the peak adiabatic T’ has already reached 850 hPa, well below the PBL top. My interpretation is that the positive adiabatic T’, rather than the peak, only descended to the PBL top around 29-30 June. Could the authors clarify whether they intend to convey that the peak of the surface temperature aligns with the time when adiabatically heated air stopped to mix into the boundary layer?

Line 325: Might the RU heatwave exhibit more similarities to the PNW heatwave if evaluated over a longer time frame? There is no need for the authors to change the figures, but a brief comment on this would be beneficial.