Mathematical Problems of Climate Science? Are there any interesting research-level mathematics problems that have come out of climate science specifically or that are abstractions of some of the issues and models pertaining to climate science?
When I say "specifically" I mean to try to exclude things like "climate science deals with fluid dynamics and therefore any problem from fluid dynamics counts"
 A: Absolutely!  Climate models are incredibly complex coupled systems of nonlinear PDE, with boundary forcing due to solar output, anthropogenic factors (like C02 emissions), volcanic activity, ocean involvement, and so on.  The models involve multiple time and spatial scales and uncertainties in model parameters.  Broadly speaking the mathematical challenges in climate science are related to the general set of mathematical problems in "complex systems", where there are many unsolved problems. 
A lot of the major issues that arise in climate science are related to numerical simulation and uncertainty quantification.  Formulating good numerical schemes for solving coupled PDE with multiple scales is already very challenging.  And, since not all model parameters are known exactly, they must be assigned probabilistic descriptions, which leads to further computational challenges since then Monte Carlo methods must be used to sample from such distributions.  There are lots of "unsolved problems" here - though in this kind of applied mathematics, it is much less typical to see something like a "conjecture" that will have a definitive proof or answer.  Usually the questions are more broad, like "understand the role of noise on the efficiency of Monte Carlo sampling" or "derive reduced order models that make simulation more efficient".
I would give this article by Majda a read.  There is also a good, short overview of weather and climate modeling in Chapter 2 of the book "Uncertainty Quantification" by Smith.
