Bill, thanks for addressing some points in my post, his clarity of thinking is always welcome. I do want to say that PCA does have a multivariatie normality assumption (crack open any introductory multivariate statistics textbook). To the question of violating assumptions. The overwhelming consensus is that it does not matter as long as your intent is data reduction and not inference. In statistics we violate assumptions all the time, it is just a matter of how much and what the effect is. In image analysis, where the intent is pure data reduction, PCA is applied commonly without a second thought to data distributions.
The motivation behind my advice was in specific reference to understanding climate process. I can think of very few instances in climate analysis where, at some point along the line, inference is not necessary. As such, it does not strike me as a data reduction problem. The relationship between temperature or precipitation and timing/duration (e.g., frost free period) can be a driver of many ecological processes. It is important to understand these relationships to fully realize the effects on a given process. These relationships can be non-linear lending themselves to more suitable statistics than PCA. Additionally, it is possible for certain climate variables to have multiple modes, which standardization does not correct, thus resulting in difficult interpretation. PCA is applied commonly to climate data (in both climate science and ecology) and there has recently been considerable "push back" as to how analysis are conducted, particularly in community ecology. I review for several ecological and modeling journals and this trend is becoming clear. I have seen several papers rejected recently because one of the reviewers does not agree with how a PCA was applied and does not believe that the results are supported. You just need to be careful that your question does not unintentionally lead to an ordination type analysis. This is driven by what you intend to do with the resulting reduced-climate PCA results. If your intent is to look at a process (e.g., species richness, productivity, etc...) along an ecological gradient then you start running into issues. This is very well documented in the ecological literature.
However, I do have to admit that I made an overreaching assumption on what the intent of your analysis was, so my advice may be way off base. Just remember that, in statistics, what is technically correct and what you can get away with are often very different and the subject of much debate among statisticians. Because of this, "correct" methodology can be very confusing with much contradictory information.
