  We present the design and scalable implementation of an exascale climate
emulator for addressing the escalating computational and storage requirements
of high-resolution Earth System Model simulations. We utilize the spherical
harmonic transform to stochastically model spatio-temporal variations in
climate data. This provides tunable spatio-temporal resolution and
significantly improves the fidelity and granularity of climate emulation,
achieving an ultra-high spatial resolution of 0.034 (approximately 3.5 km) in
space. Our emulator, trained on 318 billion hourly temperature data points from
a 35-year and 31 billion daily data points from an 83-year global simulation
ensemble, generates statistically consistent climate emulations. We extend
linear solver software to mixed-precision arithmetic GPUs, applying different
precisions within a single solver to adapt to different correlation strengths.
The PaRSEC runtime system supports efficient parallel matrix operations by
optimizing the dynamic balance between computation, communication, and memory
requirements. Our BLAS3-rich code is optimized for systems equipped with four
different families and generations of GPUs, scaling well to achieve 0.976
EFlop/s on 9,025 nodes (36,100 AMD MI250X multichip module (MCM) GPUs) of
Frontier (nearly full system), 0.739 EFlop/s on 1,936 nodes (7,744 Grace-Hopper
Superchips (GH200)) of Alps, 0.243 EFlop/s on 1,024 nodes (4,096 A100 GPUs) of
Leonardo, and 0.375 EFlop/s on 3,072 nodes (18,432 V100 GPUs) of Summit.
