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I am seeking information on how weather models use randomness (if at all). From what I can gather daily weather forecasting uses partial differential equations with initial and boundary conditions plugged in from data. Do they use randomness to vary the initial conditions (to account for uncertainty)? Or is there some point in the modeling process where randomness is inserted (e.g. drawing from a normal distribution)? Or are weather models completely deterministic?

I read a thread on EarthScienceSE for probability of precipitation (PoP), and it still isn't clear to me if there is any randomness at all in the model and how its output is averaged over a given area. How are the "confidence in any precipitation" and "coverage area" generated (which are multiplied to get the overall PoP)?

I know that "chance of rain" is a very specific example, but I am interested more generally where randomness is used in common weather models. E.g. "this model uses the normal distribution to estimate this quantity" etc. Specific information is especially appreciated, e.g. name of model and where it uses random numbers. Any help is appreciated!

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    $\begingroup$ Due to the lack of time just a comment: Randomness is used. You should probably check "Ensemble forecasting". As for the PoP. Depending on the literature you get different explanations on "certainty of precipitation". This is mainly due to counter intuitive behavior of models. The more resolved a model is, the more likely it is to fail with predictions. Additionally probability density functions are used for parameterizations (how much rain is in one grid cell?). If for example the average humidity is 80% it still is likely that there is rain somewhere in the grid cell. $\endgroup$ – J. Fregin Jun 15 at 21:51
  • $\begingroup$ @J.Fregin thank you for the info. This is a good start for me! I just did some reading on ensemble forecasting and it is definitely on the right track. I think scientists use the term "ensemble" where I just call it "a bunch of simulations of the same model" (my background is math/stats). $\endgroup$ – jdods Jun 15 at 22:45
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Probability is used in weather forecasting. I will only highlight some examples due to my lack of knowledge in some areas.

Before initializing a forecast model the data needs to be assimilated. That means we need to somehow put measurements from ground based stations, satellites etc. into the model and fit this data to the grid. We then have an "atmospheric state estimate". This usually involves the minimization of cost functions. Typically the uncertainty of this "atmospheric state estimate" is used in the process of ensemble forecasting. An introduction to data assimilation can be found here at ECMWF. Every operational weather forecasting model has to do this data assimilation process. Since you specifically asked for names, here are a few: Global Forecasting System (GFS, probably the model you weather app uses), ECMWF model (European Center for Medium Range Weather Forecasts), Icosaheadral Nonhydrostatic Model (ICON, operational model of german weather service),High Resolution Local Area Modelling for numerical weather prediction (HIRLAM, operational model of some Scandinavian countries).

In an ensemble forecast the equations are solved as you stated in the question. However, there are certain aspects of the model that are varied to account for uncertainties. These aspects can include the initial conditions (including time, sometimes time-lag is used), variations of parameterizations (and there are a lot of them, basically for every process that can not be resolved by the model grid) and sometimes even multi-model ensembles. The model is then initiated many times to obtain multiple and different forecasts. Again, there is a nice introduction at ECMWF. Some ideas on how to visualize ensemble forecasts can be found here.

All of the above focussed on probability in the course of initializing a model. However, even if no ensemble is used and if we'd know the state of the atmosphere without error - we'd still have to rely on probability due to sub grid scale processes. A prominent example is cloud cover. Suppose we have a mean humidity $q$ in a grid cell (which is what the model gives us). Clouds will only form if $q > q_s$, where $q_s$ is the saturation humidity. Depending on other parameters like temperature a probability density function $Q(T,...)$ is assumed. A possible parameterization for cloud cover could be that the fraction of $Q(T,...)$ larger than $q_s$ defines cloud cover. I will not give examples for typical distributions used, since this is an active area of research. Once more I'd like to refer to ECMWF if you wish to see some details and examples of pdf's used.

Finally, I'd like to add that a main source of uncertainty in models (there is not necessarily probability used) is the parameterization of of convection (shallow and deep). Precipitation is highly related to humidity and convection and therefore large errors are often the result of parameterization.

I hope this helps somewhat.

Edit: Regarding the precipitation problem. What you see at google or the typical weather app is based on model simulations but most certainly not a model output. If google tells you "30% rain probability at your place" then this will most likely refer to a 30% chance that precipitation in an (by google unspecified) area exceeds a certain threshold. The certainty is is generated by looking at the ensemble. The threshold is exceeded in 30% of the simulations. This is also the reason why a high resolution model is less accurate. It is easier to predict that it's raining 0.1 l somewhere in New York tomorrow than to predict that it rains 0.1 l in the central park tomorrow.

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  • $\begingroup$ Good answer, but I'm confused by "based on model simulations but most certainly not on a model output", clearly before the model run (simulation) has finished running and produced an output, the result cannot be used by any users (such as Google)? Or do you mean that probability is itself not a diagnostic variable of an individual model run? Would Google need to download all the individual ensemble members to derive the 30%, or would ECMWF (or others) provide a summary of the ensemble members from which Google can extract the probability directly? $\endgroup$ – gerrit Jun 16 at 7:38
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    $\begingroup$ @gerrit thanks for showing flaws in my answer. What i mean is that PoP is something that is obtained later from the (multiple) model output data. Actually I don't know if websites like weather.com process the data themselves or e.g NCEP or DWD offers a product and therefore, doing it for them. So yes probability is not a diagnostic variable of an individual model run. $\endgroup$ – J. Fregin Jun 16 at 10:09
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    $\begingroup$ @gerrit these days the probability is available from ECMWF directly - meteologix.com/in/model-charts/euro/central-europe/… $\endgroup$ – gansub Jun 16 at 12:18
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    $\begingroup$ @J.Fregin One can have ensembles of high resolution mesoscale models - aoml.noaa.gov/hwrf-hurricane-ensemble/.But the forcing model will still be a GCM. $\endgroup$ – gansub Jun 17 at 2:56

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