I have known Krish for 50 years since I joined UCLA in 1961 as a visitor from Japan. Since then, I have always been admiring his originality, breadth and productivity in research, which nicely mingles modeling and observational studies. Throughout his entire career, he has enthusiastically devoted his talent and energy to improve predictions of the tropical atmosphere. His recent idea on the multi-model superensemble forecasts is particularly intriguing. In contrast to the usual ensemble forecasts, this approach takes advantage of the diversity of existing models through the optimal combination of separate models.

The cumulus parameterization problem has been one of his major research efforts since his presentation of the statistical refinement of the Kuo scheme in the early 1980s. More recently, Krishnamurti et al. (2003) demonstrated that the superensemble methodology could provide improved precipitation forecasts by combining existing cumulus parameterization schemes, each of which has some virtue depending on the geographical location. After reviewing the history of cumulus parameterization, Arakawa (2004) pointed out that most of the surviving schemes including the Kuo scheme could be viewed as adjustment schemes. Individual schemes, however, choose time scales for the adjustment more or less arbitrarily, while the optimum time scale should depend on the cloud regime and, therefore, on the geographical location at least statistically. It is possible that this geographical dependence of the optimum time scale is one of the major factors responsible for the geographically-dependent performance of the existing schemes.

The main purpose of this talk is to present a new framework of cumulus parameterization in which the time scale for the relaxed adjustment is physically determined for each realization. It should be noted that the need for parameterizations in numerical models arises from the artificial separation of grid-scale and subgrid-scale processes by truncation, and parameterizations must deal with ONLY the latter processes; otherwise, there is a danger of double-counting grid-scale processes that are supposed to be explicitly simulated by the host model. We can show that the vertical eddy transports of thermodynamical properties, which are responsible for the adjustment, are strongly modulated by the fractional area ƒã covered by subgrid convective clouds. This is because as ƒãƒn increases the thermodynamical properties averaged over the grid-cell become closer to those of clouds reducing the intensity of the eddies. Existing cumulus parameterization schemes ignores this ƒã-dependence assuming ƒã <<1 a priori either explicitly or implicitly. The new framework eliminates this assumption and formulates the ƒã-dependence in a way well supported by statistical analysis of cloud-resolving simulations. The framework also includes a method of determining ƒã from the outputs of the conventional parameterization schemes.

When the resolution is sufficiently high to resolve individual clouds, the probability density distribution of ċ becomes bimodal consisting of ċ =1 and ċ =0. Since the eddy transport formulated in the new framework vanishes for both of these values, as it should, there is no parameterized eddy transport for such high resolutions. In this way, the model with the new framework converges to a cloud-resolving model as the resolution is refined. Arakawa et al. (2011) and Arakawa and Jung (2011) called such a framework "unified parameterization" because it effectively unifies the parameterizations in large-scale models and those in cloud-resolving models. Due to this unification, horizontal resolution can be freely chosen without changing the formulation of model physics. This is especially advantageous when the resolution is highly heterogeneous as in the case of local or adaptive mesh refinement.

The unified parameterization is expected to significantly reduce the uncertainties in cumulus parameterization especially in mesoscale models. Uncertainties will still remain in choosing cloud models that determine cloud thermodynamical properties and choosing the quasi-equilibrium profiles to which the adjustment is made. The approach of the superensemble forecasts will still be effective to further increase the skill of the forecasts.