By: Jian-Feng Gu

Moist convection plays a fundamental role in large-scale circulations and climate, ranging from cumulus clouds smaller than 100m to organized weather systems of several thousands of kilometers. Limited by their grid spacing, numerical models are not able to fully resolve moist convection across its broad range of scales, therefore, we represent moist convection in numerical models using a simplified process, called parameterization.

**Figure 1**:. A schematic diagram of bulk mass flux approximation. The left panel shows many individual different clouds. The right panel shows how the clouds are represented using the top-hat assumption.

In current climate models, convection is mostly parameterized using bulk plume models, which estimate the vertical transport of heat, moisture and momentum of a large group of clouds. To understand the general idea, imagine a grid box of around (100×100) km^{2} – that’s about half the size of Wales. Imagine a large number of different clouds randomly scattered across the domain (see the left panel of figure 1). Representing each cloud individually is very computationally expensive and hence simplification is necessary. The simplest idea to describe the overall behavior of these clouds is to consider them as a single entity, that is, a bulk cloud (see the right panel of figure 1).

A further simplification is to assume that each property of the bulk cloud is the averaged value of all the individual clouds, and that these values are distributed evenly within the bulk cloud. This is called the top-hat assumption (Randall et al. 1992). This means that the overall vertical transport by these clouds can be described as the transport of mean properties by the bulk cloud, which is called the bulk mass flux approximation. It approximates the sub-grid vertical flux of a quantity as being the product of the convective mass flux with the departure from the grid-box average of the transported quantity. However, this approximation can underestimate the vertical fluxes by 30-50% (Yano et al. 2004), depending on the variables considered and the resolution of the model. Therefore, a parameterization of the neglected contributions to the vertical flux is necessary. How might this be achieved without sacrificing computational efficiency?

The vertical flux underestimation arises from the physical assumptions we make in the bulk plume model. In Gu et al. (2020), we show both the mean properties of the clouds, and the departures from these mean properties contribute toward the total vertical flux. By representing many clouds as a single bulk cloud, we are removing differences between individual clouds. By assuming a top-hat distribution, we are neglecting the inhomogeneity within each cloud. These two neglected variabilities are called inter-object and intra-object variability, respectively. As a result, the bulk mass flux approximation underestimates the total vertical transport by moist convection. However, it can be improved by relaxing these assumptions. For example, a spectral model that deals with clouds of different sizes is able to minimize the inter-object variability because it takes into account the differences of mean properties between different types of clouds. But it still underestimates the vertical heat fluxes because of neglecting the intra-object variability.

**Figure 2**: A schematic of the core-cloak representation of convection. Both updrafts and downdrafts are represented as the combination of a strong core surrounded by a weak cloak.

To improve the representation of both the vertical heat and water fluxes, we proposed the “core-cloak” conceptual model (Figure 2, Gu et al. 2020). In this model, we decompose the flow into different types of drafts depending on the strength of vertical motions. More specifically, we collect the strong updrafts together as the updraft “core” and the weak updrafts together as the updraft “cloak”. This core-cloak structure can also be applied to downdrafts. This flow decomposition partly includes the inter-object and intra-object variability and therefore better represents the vertical heat and water transport.

To evaluate our conceptual model, we performed simulations of shallow convection (dx=25 m, 50 m, 100 m) and deep convection (dx=100 m, 200 m, 400 m), and cloud resolving simulations of organized deep convection (dx=1 km). Our results show that the “core-cloak” conceptual model can significantly improve the representation of vertical heat and moisture fluxes, compared to the bulk mass flux approximation. The improvement can be seen in both shallow and deep convection, and even in organized convection.

We also found that the clouds which have a “core-cloak” structure contribute most of the vertical fluxes. Therefore, the “core-cloak” conceptual model provides simply a possible decomposition of the flow that gives a reasonable and efficient description of turbulent fluxes using a mass flux approximation. Parameterization of this core-cloak model would need careful treatment of exchanges between the different types of drafts. We intend to pursue the practical implications of this conceptual model within the future development of a convection parameterization.