  Motivated by recent advances in operational weather forecasting, we study the
efficacy of low-precision arithmetic for climate simulations. We develop a
framework to measure rounding error in a climate model which provides a
stress-test for a low-precision version of the model, and we apply our method
to a variety of models including the Lorenz system; a shallow water
approximation for flow over a ridge; and a coarse resolution global atmospheric
model with simplified parameterisations (SPEEDY). Although double precision (52
significant bits) is standard across operational climate models, in our
experiments we find that single precision (23 sbits) is more than enough and
that as low as half precision (10 sbits) is often sufficient. For example,
SPEEDY can be run with 12 sbits across the entire code with negligible rounding
error and this can be lowered to 10 sbits if very minor errors are accepted,
amounting to less than 0.1 mm/6hr for the average grid-point precipitation, for
example. Our test is based on the Wasserstein metric and this provides
stringent non-parametric bounds on rounding error accounting for annual means
as well as extreme weather events. In addition, by testing models using both
round-to-nearest (RN) and stochastic rounding (SR) we find that SR can mitigate
rounding error across a range of applications. Thus our results also provide
evidence that SR could be relevant to next-generation climate models. While
many studies have shown that low-precision arithmetic can be suitable on
short-term weather forecasting timescales, our results give the first evidence
that a similar low precision level can be suitable for climate.
