## Accurate predictions of Earth’s warming require computers that are too expensive for one country or institution.

The root of the problem is one of the most basic assumptions of the computer simulations, the possibility of dividing up the atmosphere and oceans into a “grid” of small pieces. Computers then calculate how the pieces interact with one another in small time increments. While doing so, information that is smaller than the size of the pieces — so-called “sub-grid” information about clouds, ocean eddies and the capacity of soil to retain water — must be approximated.

According to Dr. Palmer, this method of calculation may be overly simplistic and suffers from severe shortcomings. The formula used to calculate changes of the atmosphere and oceans — the Navier-Stokes equation — has what physicists call “scale symmetry,” meaning it works the same on all distances. However, as Dr. Palmer points out, this symmetry is violated when calculations approximate the sub-grid information.

The consequences for climate predictions are serious:We underestimate the durability of extreme weather situations and, at the same time, overestimate how likely our predictions are to be correct.The Navier-Stokes equation, central to predicting Earth’s climate, is famously difficult to solve and has caused mathematicians and physicists headaches for 200 years. To this day, turbulences and eddies have remained challenging to understand. The Clay Mathematics Institute has named the Navier-Stokes equation one of its millennium problems and will award progress toward solving it with a $1 million prize.

.. There are two possible ways to arrive at better climate predictions. The best way would be to use a higher resolution for the models: to divide up the land and oceans into much smaller pieces. But doing this with existing computing facilities would take too long to be of any use.