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I'm new to climate science - starting to get familiar with the field by reading some published articles. Terminology is always hard when starting into a new domain.

In this paper, Dai - Increasing drought under global warming in observations and models, what does the author mean by the term "forcing," in relation to the PDSI?

Here is an example of usage:

The PDSI is calculated from a water-balance model forced with observed precipitation and temperature...

Another example:

Figure 1a,b shows that the broad patterns of the linear trends from 1950 to 2010 in observed annual precipitation and calculated sc_PDSI_pm using observation-based forcing2

Here is the citation:

Dai, Aiguo. "Increasing drought under global warming in observations and models." Nature Climate Change 3, no. 1 (2013): 52-58.

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What does 'forcing' mean?

The PDSI is calculated from model output data. Models do not predict all relevant variables but some data need to be provided externally from the user. These externally provided data are denoted as "forcing" because they "force" the model to move in one direction (colloquially speaking). You might also call it "drive" a model -- but "to force" is the established expression in science.

In this case

In this particular case, the authors use observational data as input data for the model calculations. Therefore, the authors write

The PDSI is calculated from a water-balance model forced with observed precipitation and temperature...

Other situations

One might force a model also with data from other models. A river run-off model might get input data from an atmospheric/meteorological model, which predicts precipitation and solar radiation at surface level. We could write

The river run-off model ABC is forced by modeled precipitation and radiation of the XYZ meteorological model.

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