Currently ranked as the fastest supercomputer in the world, the Sunway TaihuLight supercomputer measures in at 125 petaFLOPS (theoretical peak)—five times as fast as the supercomputer in second place. Housed in the National Supercomputing Center in Wuxi, it is comprised of 10.6 million cores and is being used for climate research, earth systems modeling and data analytics. On top of being the fastest supercomputer in the world, the Sunway TaihuLight is currently ranked as the fourth most energy-efficient one as well, requiring substantially fewer megawatts per megaFLOPS. (source: https://futurism.com/the-eight-most-powerful-supercomputers-in-the-world/ )
There are several means by which weather prediction is performed. See
https://en.wikipedia.org/wiki/Numerical_weather_prediction
for the history of the subject.
The usual weather station predictions are based on a variety of measurements, including satellite and radar images, and input from reconnaissance aircraft, besides local temperature, humidity and pressure measurements. It is easy to predict the weather a few hours ahead generally, and today's weather is a good predictor of tomorrow's weather. Thus, you could use a Kalman filtering/state estimation framework based on coupled atmospheric+oceanic dynamics to predict weather over short periods of time. This is the sort of result you see if you track weather predictions--they keep changing, sometimes quite significantly, based on current measurements.
If you wanted to predict accurately, you would need to build an observer for the fluid dynamics of the atmosphere. We really do not have the capacity to simulate the actual Liouville equations to determine the initial condition uncertainty that exists in these prediction methods, even with supercomputers. Hence, our forecasts are limited to 5-6 days, and even these are not very accurate. This is performed by using probability distributions of the ensembles used to initialize atmospheric state (distribution of temperature, pressure, velocity, humidity etc) for the forecast that approximate their probability distribution in the atmosphere. The reason we don't have good samples is that many areas of the earth, like the Pacific are not sampled well.
Thus these practical weather prediction computations using the initial conditions derived through complex calculations would take minuscule time on any cutting edge supercomputer. See some details and links in this article: https://www.top500.org/news/top500-meanderings-supercomputers-for-weather-forecasting-have-come-a-long-way/
The increased power of supercomputers is really used to improve the estimation of initial conditions mentioned above so that predictions are more precise and further ahead of the actual event, as European computers predicted Hurricane Sandy's motion days ahead of the NOAA's computer at the time: https://www.usatoday.com/story/weather/2016/02/22/supercomputer-reston-noaa-cray-ibm/80290546/
For comparison, the computation of a Kalman filter update on a 1990s computer for an entire aircraft was already under 5 milliseconds. While the number of states in the atmospheric simulation is large, the dynamics has a sparse matrix structure that permits parallelization of computation. Now we can solve large dimensional optimization problems (linear or convex programming) on standard PCs in seconds.
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