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I have a climate model that has 8km resolution and I am not sure what it means, does it mean that every cell is showing an equivalent of 8 km in real life?

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Depending on the model the 8km resolution would be the output from the model assigning that particular modelled value to a cell ~8km by ~8km.

As an example, the ECMWF atmospheric models are interpolated to approximately 0.1° or 9km grids.

https://www.ecmwf.int/en/forecasts/documentation-and-support

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