Regime-based evaluation of cloudiness in CMIP5 models

Abstract

The concept of cloud regimes (CRs) is used to develop a framework for evaluating the cloudiness of 12 fifth Coupled Model Intercomparison Project (CMIP5) models. Reference CRs come from existing global International Satellite Cloud Climatology Project (ISCCP) weather states. The evaluation is made possible by the implementation in several CMIP5 models of the ISCCP simulator generating in each grid cell daily joint histograms of cloud optical thickness and cloud top pressure. Model performance is assessed with several metrics such as CR global cloud fraction (CF), CR relative frequency of occurrence (RFO), their product [long-term average total cloud amount (TCA)], cross-correlations of CR RFO maps, and a metric of resemblance between model and ISCCP CRs. In terms of CR global RFO, arguably the most fundamental metric, the models perform unsatisfactorily overall, except for CRs representing thick storm clouds. Because model CR CF is internally constrained by our method, RFO discrepancies yield also substantial TCA errors. Our results support previous findings that CMIP5 models underestimate cloudiness. The multi-model mean performs well in matching observed RFO maps for many CRs, but is still not the best for this or other metrics. When overall performance across all CRs is assessed, some models, despite shortcomings, apparently outperform Moderate Resolution Imaging Spectroradiometer cloud observations evaluated against ISCCP like another model output. Lastly, contrasting cloud simulation performance against each model’s equilibrium climate sensitivity in order to gain insight on whether good cloud simulation pairs with particular values of this parameter, yields no clear conclusions.

Appendix: Centroid distance relationship

For model histogram data $\left( { x_{i,1} , x_{i,2} , \ldots , x_{i,m} } \right)$ assigned to the kth regime whose reference centroid is $\left( {\overline{y}_{1,k} , \overline{y}_{2,k} , \ldots ,\overline{y}_{m,k} } \right)$, where i and m indicate the member of a population N _k , and the vector dimension (m = 42 in our study), respectively, the model’s own centroid $\left( {\overline{x}_{1,k} , \overline{x}_{2,k} , \ldots , \overline{x}_{m,k} } \right)$ is defined as follows.

$$\overline{x}_{j,k} = \frac{1}{{N_{k} }}\mathop \sum \limits_{i = 1}^{{N_{k} }} x_{i,j} \quad {\text{where}}\quad j = [1,m]$$

(2)

In addition, the squared (Euclidean) distance of $\left( { x_{i,1} , x_{i,2} , \ldots , x_{i,m} } \right)$ from its own centroid $\left( {\overline{x}_{1,k} , \overline{x}_{2,k} , \ldots , \overline{x}_{m,k} } \right)$, the squared distance from the reference (ISCCP) centroid $\left( {\overline{y}_{1,k} , \overline{y}_{2,k} , \ldots ,\overline{y}_{m,k} } \right)$, and the squared distance between the model centroid and the reference centroid are, respectively, as follows:

$${\text{DOC}}_{i,k}^{2} = \mathop \sum \limits_{j = 1}^{m} \left( {x_{i,j} - \overline{x}_{j,k} } \right)^{2}$$

(3)

$${\text{DRC}}_{i,k}^{2} = \mathop \sum \limits_{j = 1}^{m} \left( {x_{i,j} - \overline{y}_{j,k} } \right)^{2}$$

(4)

$$D_{k}^{2} = \mathop \sum \limits_{j = 1}^{m} \left( {\overline{x}_{j,k} - \overline{y}_{j,k} } \right)^{2}$$

(5)

Equation (4) can be rewritten as follows:

$$\begin{aligned} {\text{DRC}}_{i,k}^{2} & = \mathop \sum \limits_{j = 1}^{m} \left( {x_{i,j} - \overline{x}_{j,k} + \overline{x}_{j,k} - \overline{y}_{j,k} } \right)^{2} \\ & { = }\,{\text{DOC}}_{i,k}^{2} + D_{k}^{2} + 2\mathop \sum \limits_{j = 1}^{m} \left( {x_{i,j} - \overline{x}_{j,k} } \right) \cdot \left( {\overline{x}_{j,k} - \overline{y}_{j,k} } \right) \\ \end{aligned}$$

(6)

The mean squared distance from the reference centroid over population N _k is as follows:

$$\overline{{{\text{DRC}}_{k}^{2} }} = \frac{1}{{N_{k} }}\mathop \sum \limits_{i = 1}^{{N_{k} }} {\text{DRC}}_{i,k}^{2}$$

(7)

Equation (7) can be rewritten using Eq. (6) as follows:

$$\begin{aligned} \overline{{{\text{DRC}}_{k}^{2} }} & = \frac{1}{{N_{k} }}\mathop \sum \limits_{i = 1}^{{N_{k} }} \left[ {DOC_{i,k}^{2} + D_{k}^{2} + 2\mathop \sum \limits_{j = 1}^{m} \left( {x_{i,j} - \overline{x}_{j,k} } \right) \cdot \left( {\overline{x}_{j,k} - \overline{y}_{j,k} } \right)} \right] \\ & = \frac{1}{{N_{k} }}\mathop \sum \limits_{i = 1}^{{N_{k} }} DOC_{i,k}^{2} + D_{k}^{2} + 2\mathop \sum \limits_{j = 1}^{m} \left( {\overline{x}_{j,k} - \overline{y}_{j,k} } \right) \\ & \quad \times \frac{1}{{N_{k} }}\mathop \sum \limits_{i = 1}^{{N_{k} }} \left( {x_{i,j} - \overline{x}_{j,k} } \right) \\ \end{aligned}$$

(8)

The third term of right side of Eq. (8) is zero because of Eq. (2). In addition, if we define the first right term of Eq. (8) as the mean squared distance of model histograms from their own centroid for the kth regime, $\overline{{{\text{DOC}}_{k}^{2} }}$, Eq. (8) can be simplified to yield Eq. (1):

$$\overline{{{\text{DRC}}_{k}^{2} }} = \overline{{{\text{DOC}}_{k}^{2} }} + D_{k}^{2}$$

(1)