Scaling the vertical-velocity variance in the convective boundary layer forced by rapidly decaying surface heat flux after equilibrium breakdown

In the bulk of the convective boundary layer driven by surface heating, the vertical-velocity variance is known to scale with the convective velocity scale. This scaling relies on the quasi-equilibrium assumption that the surface heat flux (H) varies slowly compared to the adjustment time scale of the large scale convective eddies (or the eddy-turnover time scale). When the surface heat flux decays moderately or rapidly compared to the eddy-turnover time scale, a departure from the quasi-equilibrium regime is expected. A recent study proposed a parameter for describing such departure during the late afternoon transition: r = H^-1dH/dt^-1/t_*, where t_* is the eddy-turnover time scale. The quasi-equilibrium assumption applies when r>>1, and breaks down when r~1. Building on these results, we further investigate the scaling for the vertical-velocity variance in the regime r<<1 where the surface heat flux decays rapidly compared to the eddy-turnover time scale. In a first step, we use large-eddy simulations to reveal that the regime r<<1 is characterized by a new velocity scale w_*r = (g/θ dH/dt z_i²)^1/4depending on the actual dH/dt rather than H . We argue that the characteristics of the surface heat flux during the initial-state quasi-equilibrium regime are encoded in the actual boundary-layer depth during the following out-of-equilibrium regime. In a second step, we demonstrate that the new velocity scale can be deduced from scaling arguments applied to the budget-equation of the vertical turbulent heat flux. The new velocity scale aims to specifically describe the vertical-velocity variance during the latest decay-stage of the positive surface heat flux, that is a few minutes only before it turns negative. Further research is needed to understand which part of our results can be generalized to other non-stationary configurations, and challenge the results of this idealized study with observations.