The integration of the governing equations of hydrodynamics by numerical methods subject to specified initial conditions.

Numerical approximations are fundamental to almost all dynamical weather prediction schemes since the complexity and nonlinearity of the hydrodynamic equations do not allow exact solutions of the continuous equations.

Haltiner, G. J., and R. T. Williams 1980. Numerical Prediction and Dynamic Meteorology. Wiley, New York, 477 pp.

(via AMS Glossary, "Numerical forecasting")

El Niño-Southern Oscillation: El Niño and the Southern Oscillation, also known as ENSO is a periodic fluctuation in sea surface temperature (El Niño) and the air pressure of the overlying atmosphere (Southern Oscillation) across the equatorial Pacific Ocean. El Niño is so termed because it generally reaches full strength toward the end of the year, and early Christian inhabitants of western equatorial South America equated the warm water current and the resulting impacts with their holiday celebrating the birth of Jesus Christ (known as El Niño in Spanish).

The Southern Oscillation describes a bimodal variation in sea level barometric pressure between observation stations at Darwin, Australia and Tahiti. It is quantified in the Southern Oscillation Index (SOI), which is a standardized difference between the two barometric pressures. Normally, lower pressure over Darwin and higher pressure over Tahiti encourages a circulation of air from east to west, drawing warm surface water westward and bringing precipitation to Australia and the western Pacific. When the pressure difference weakens, which is strongly coincidental with El Niño conditions, parts of the western Pacific, such as Australia experience severe drought, while across the ocean, heavy precipitation can bring flooding to the west coast of equatorial South America.

Extended Reconstructed Sea Surface Temperature: The Extended Reconstructed Sea Surface Temperature (ERSST) dataset is a global monthly sea surface temperature dataset derived from the International Comprehensive Ocean–Atmosphere Dataset (ICOADS).

Oceanic Niño Index: 3-month average SST departures

in the Niño-3.4 region

based on the latest ERSST dataset

Sea-surface temperature

A forecast based on a systematic statistical examination of data representing past observed behavior of the system to be forecast, including observations of useful predictors outside the system.

In short-term climate forecasting, CCA (canonical correlation analysis), as described by Barnston (1994), is a good example of a statistical forecast. Depending on method and scope, the limitations of statistical forecasts are related to shortness of record, danger of overfitting, assumptions of linearity (often), absence (often) of physical considerations, etc. Purely statistical forecasts in weather forecasting have become rare; however, a combination of dynamical model output and statistics is very common in weather forecasting. Some statistical methods are guided by physical principles to such an extent that they resemble dynamical models. An example of the latter is empirical wave propagation described by Qin and van den Dool (1996).

Barnston, A. 1994. Linear statistical short-term climate predictive skill in the Northern Hemisphere. J. Climate. 7. 1513–1564.

Qin, J., and H. M. van den Dool 1996. Simple extensions of an NWP model. Mon. Wea. Rev.. 124. 277–287.

(via AMS Glossary)