I've had my eye on properly understanding climate, climate change and the wide array of phenomena related. But, as a physics grad student, I'd fancy more exact, math-based bibliography, that treats all these interesting topics in a more formal manner, rather than with vague, high-school level explanations.
Recommendations are very much appreciated.
I definitely do not want to claim to give a one-stop canonical answer. So let that be a forewarning before reading my answer. I am just giving a few well-received options.
A fairly easy introduction to the subject can be found in Dennis Hartmann’s book, which covers both the principles of climate science across earth’s many systems and the foundational principles appropriate to asking coherent questions on the subject. It is not terribly advanced, but might be suitable for an introduction. Unfortunately, it does a poor job of actually deriving the material presented, and thus may be better used to expose you to the things you would find derivations for elsewhere.
The book by Raymond Pierrehumbert seems to fill a good role for students with strong physics and chemistry backgrounds and little climate science exposure. It includes fairly detailed explanations in terms of thermodynamics and radiation, scattering, and a tiny bit of fluid dynamics. It is not as advanced as a graduate physics text, of course, but it is far more rigorous than a typical text may be.
A much math-heavier treatment of topics in climate modeling can be found in Zhang and Moore’s book, which includes several chapters of mathematical topics like Fourier analysis and signal processing filter design. Since you are looking for rigor, this book may be a great place to start since it explains in a fairly deep way how many of the computational models of modern climate science are built. It is, however, not as sweeping as the previous book, and therefore may not address every aspect of climate science you are interested in.
It might also be worth your time to read the IPCC’s reports on the physical basis of climate change, as these are written by many highly-active researchers and policymakers in the field. I do not claim these will address your particular needs, but it s definitely a more active source than a textbook.
Here are options for a Physics Grad student like you, you might like these:
Intermediate Level:
"Dire Predictions: Understanding Climate Change" by Michael E. Mann and Lee R. Kump This book offers a more in-depth exploration of climate science, including discussions on climate modeling, historical climate data, and predictions for the future.
"The Sixth Extinction: An Unnatural History" by Elizabeth Kolbert While not exclusively focused on climate science, this Pulitzer Prize-winning book delves into the interconnectedness of species extinction and climate change, providing a deeper understanding of their effects.
Advanced Level:
"The Physics of Climate Change" by Lawrence M. Krauss This book delves into the physics behind climate change, explaining the scientific principles that underpin the climate system. It's recommended for readers with a strong background in physics and science.
"Climate Change Science: A Modern Synthesis" edited by G. Thomas Farmer and John Cook This comprehensive book provides an overview of climate science from various perspectives, making it suitable for those looking for an in-depth exploration of the subject.
Remember that climate science is a multidisciplinary field, and books on the topic often touch on aspects of physics, chemistry, biology, and more. Choose a book that matches your current knowledge level and interests, and don't hesitate to further explore specific areas that intrigue you.