## 1. Introduction

[2] Many properties of cumulus convection, including its predictability, depend on the large-scale (synoptic and mesoscale) environment in which it is embedded. To some degree the convective activity can be regarded as under the control of the larger scales, and indeed this is the basis of most cumulus parameterisation schemes in numerical models, but this control is only partial, and the degree of control is likely to vary with time and place [see, e.g., *Arakawa*, 2004].

[3] The initiation and lifecycle of a convective cloud is directly a result of processes local to the cloud itself: the conditional instability of the column, the absence of a capping inversion or other inhibiting factors, and the boundary layer variability that can trigger an updraft. If these factors are present a cloud will rapidly form, developing within about a half hour.

[4] Conversely there are two ways in which convection can be prevented. First, in the absence of processes such as large-scale ascent that cools the troposphere and creates conditional instability, the Convective Available Potential Energy (CAPE) may be rapidly exhausted. Second, if the triggering processes in the boundary layer are not strong enough to overcome the energy barrier of Convective Inhibition (CIN), convection will not occur even in the presence of large CAPE. Either of these processes may control convection, leading to two scenarios for the influence of large scales on convection. If convection is limited by the availability of CAPE, the amount of convection (mass flux, precipitation etc.) will be controlled by the rate at which the larger scale flow creates new CAPE. Since individual clouds still respond to local influences, only the average amount of convection is constrained, leading to a statistical equilibrium (also called quasi-equilibrium or in this paper simply equilibrium). If, on the other hand, the amount of convection is limited by the interaction of triggering processes with CIN, large amounts of CAPE may build up, and there is no reason to expect a close relationship between the amount of convection and the large-scale flow.

[5] The two regimes, *equilibrium* and *triggered convection*, represent dramatically different modes of interaction between the convection and larger scales, and it would be desirable to be able to distinguish which is dominant in a given meteorological situation. One possibility would be to look at CAPE, which would be expected to be small in equilibrium situations [*Emanuel et al.*, 1994]. However the amount of CAPE is variable, even in equilibrium, and it is impossible to identify a threshold value that distinguishes the two regimes. A more fundamental approach, introduced by *Arakawa and Schubert* [1974] would be to compare the rate at which conditional instability is created by the large-scale flow with the rate at which it is destroyed by convection. The synoptic scale flow evolves on a timescale of a day or so, while in the absence of inhibiting factors, convection could be expected to remove CAPE with a turnover time of about an hour. If this is the case, the rate of convection will closely follow the large-scale forcing. On the other hand, a longer convective timescale would indicate that the convection is not constrained by the forcing and equilibrium is not present.

[6] In this work, we will estimate the timescale for removal of conditional instability by convection, *τ*_{c}, defined schematically as

where *CAPE* is given by

with *T* the environmental temperature, *T*_{a} the temperature of a pseudo-adiabatically lifted boundary layer parcel, and *T*_{0} a constant reference temperature.

[7] Following *Done et al.* [2006], we note that the *CAPE* can be removed by supplying enough heat to eliminate the difference between *T* and *T*_{a} through the column. The vertically integrated latent heat release can be determined from the precipitation rate *P* (kg s^{−1} m^{−2})

so that

[8] The convective timescale can then be estimated as

The factor of 1/2 is introduced because this calculation ignores convective modification of the boundary layer, and will thus over-estimate the convective timescale significantly. The value of 1/2 corresponds to the assumption that tropospheric heating and boundary cooling (and drying) contribute equally to the reduction of CAPE [e.g., *Betts*, 1986].

[9] The equilibrium limit occurs when *τ*_{c} is short in comparison with the time over which the large-scale flow evolves and creates CAPE, ranging from 24 hrs (diurnal cycle) to several days (synoptic cyclone). In the radiative-convective equilibrium simulations of *Cohen and Craig* [2004], convection responded to changes in large-scale forcing on a time-scale of about an hour, but the value depended on the large scale forcing. Triggered convection occurs in the limit of large values of the convective timescale. In the case considered by *Done et al.* [2006], values of tens or hundreds of hours were found, with a tendency to decrease systematically through the duration of the event as precipitation increased and CAPE decreased once convection was initiated. Indeed, a transition to equilibrium is possible, with values of *τ*_{c} not corresponding to either extreme.

[10] Previous studies have sometimes found it useful to define a threshold to distinguish the equilibrium regime (controlled by the large scales), from the remaining non-equilibrium events (triggered convection and transitional events). *Keil and Craig* [2011] examined forecast uncertainty of convective precipitation of a convection-permitting ensemble prediction system and identified a weather regime dependence. *Molini et al.* [2011] found that large or small values of *τ*_{c}, corresponding to equilibrium or non-equilibrium convection, correlate to changes in the morphology of observed precipitation events. An arbitrary threshold value of 6 hours was used in these studies, but given the uncertainties in estimating both *τ*_{c} and the timescale of the large-scale processes, only order of magnitude differences in *τ*_{c} should be regarded as significant. On the other hand G. C. Craig et al. (Constraints on the impact of radar rainfall data assimilation on forecasts of cumulus convection, submitted to *Quarterly Journal of the Royal Meteorological Society*, 2011) showed that the length of time that a high resolution model retains information from assimilation of radar reflectivity is proportional to *τ*_{c} over values ranging from 0.5 to 100 hours.

[11] Each of the studies cited above relied on model or reanalysis data for the calculation of *τ*_{c}. The first aim of this paper is to present calculations of *τ*_{c} based purely on observational data, and thus provide a first estimate for the relative frequency of equilibrium and non-equilibrium convection in nature. This will be done using a multi-year data set of summertime radiosonde ascents over Germany, combined with a high-resolution precipitation data set. The second goal of the paper is to demonstrate the utility for verification of numerical models of considering equilibrium and non-equilibrium events separately, by showing that for a particular numerical model the error characteristics are quite different in the two regimes.