Evaluation methodologies in visualization have mostly focused on how well the tools and techniques cater to the analytical needs of the user. While this is important in determining the effectiveness of the tools and advancing the state-of-the-art in visualization research, a key area that has mostly been overlooked is how well established visualization theories and principles are instantiated in practice. This is especially relevant when domain experts, and not visualization researchers, design visualizations for analysis of their data or for broader dissemination of scientific knowledge. There is very little research on exploring the synergistic capabilities of cross-domain collaboration between domain experts and visualization researchers. To fill this gap, in this paper we describe the results of an exploratory study of climate data visualizations conducted in tight collaboration with a pool of climate scientists. The study analyzes a large set of static climate data visualizations for identifying their shortcomings in terms of visualization design. The outcome of the study is a classification scheme that categorizes the design problems in the form of a descriptive taxonomy. The taxonomy is a first attempt for systematically categorizing the types, causes, and consequences of design problems in visualizations created by domain experts. We demonstrate the use of the taxonomy for a number of purposes, such as, improving the existing climate data visualizations, reflecting on the impact of the problems for enabling domain experts in designing better visualizations, and also learning about the gaps and opportunities for future visualization research. We demonstrate the applicability of our taxonomy through a number of examples and discuss the lessons learnt and implications of our findings.