From a functional perspective, the task of predicting future weather and climate may be reduced to the following iterative procedure. First, given the state of the atmosphere, ocean, and other Earth-system components at any time (the input), use the governing equations to compute the state at a slightly later time (the output). Then, repeat the loop as many times as required, always using the previous output as the next input.
The above prediction framework presents three main challenges, each of which potentially degrades the reliability of the forecast. First, Earth observations, which always contain measurement errors, are required to serve as the initial state. Second, the vast array of active physical processes and interactions is incompletely known and imperfectly represented in the spatially truncated governing equations. Third, the discrete stepping from one time level to the next is merely an approximation to the exact time-continuous evolution. I have recently been working on a possible avenue for progress with the third of these three challenges, which has received scant attention compared to the extensive research efforts devoted to the first two.
Many different time-stepping methods have been proposed over the years, but the leapfrog scheme has emerged as the method of choice in weather and climate models, because it is easy to implement, computationally inexpensive, and has low run-time storage requirements. Just like the game played in school playgrounds, the model ‘leapfrogs’ over a middle time to get from the past into the future. However, a major problem with the leapfrog scheme is that it admits spurious computational modes, in which the even and odd time steps decouple unphysically and split apart. The conventional solution to prevent the decoupling is to apply a Robert-Asselin filter after each time step, as has been done for decades in many weather and climate models. Unfortunately, this filter introduces artificial damping and reduces the accuracy.
Figure: The essential time-stepping lines of a typical leapfrog computer code, imagined to appear inside a time-stepping loop. The unshaded code performs a leapfrog step and applies the Robert-Asselin filter. The upgrade to RAW filtering is achieved via the trivial insertion of the shaded code.
I have recently proposed a modification to the Robert-Asselin filter, which eliminates the damping and improves the accuracy. The modification has become known as the Robert-Asselin-Williams (RAW) filter. The attractive features of the RAW filter are that it is very easy to implement in an existing computer code (see figure), and that it does not increase the computational expense. Therefore, the filter offers a convenient and simple method for improving the performance of predictive computer models of the atmosphere and ocean, and it is currently being tested in many different models around the world.
To give an example, I have recently been collaborating with Javier Amezcua and Eugenia Kalnay from the University of Maryland, to test the RAW filter in the Simplified Parameterizations, Primitive Equation Dynamics (SPEEDY) atmospheric general circulation model. We examined whether the new filter improves the skill of short- and medium-term forecasts. January 1982 data from the NCEP-NCAR reanalysis are used to evaluate the forecast skill. Improvements due to the RAW filter are found in all the model variables (except the relative humidity, which is hardly changed). The improvements increase with lead time and are especially evident in medium-range forecasts (96-144 hours). For example, in tropical surface pressure predictions, 5-day forecasts made using the RAW filter have approximately the same skill as 4-day forecasts made using the Robert-Asselin filter. The results of this work are encouraging for the implementation of the RAW filter in other models, and have been reported in the media.