Tuesday, May 8, 2012

The "simplest" kind of weather model

When we talk about our weather models these days we're talking about complex systems of equations coded into computer programs that are expensive and time-consuming to run.  There's so much detail in what goes on in the atmosphere that it takes so much memory storage and processing power to compute forecasts for the time period we're interested in and at the detail we want.  Models that I show on this blog like the GFS or the WRF model contain hundreds of thousands of lines of code and can take supercomputers to run them efficiently (particularly in the case of the GFS).  But do we really need all of this power?  If we're willing to settle for a bit less detail in our models and trust our meteorological instincts some more, just how well can we make a forecast?

Do we really need big supercomputers and fancy, complex codes to make decent weather predictions on a global scale?

By doing a lot of simplifying, we can actually get a fairly good idea of the general flow in the atmosphere from some very basic models.  Today I'm going to talk about what is probably the "simplest" model that can realistically model atmospheric flow around the globe--the barotropic model.

The barotropic model was the first kind of numerical weather model ever successfully implemented--it was based on work by Charney in the 1940s.  You can see their original paper describing the work at this link.  So what goes into a barotropic model?

First, what does barotropic even mean?  Meteorologists use that word to describe an environment where pressure is only a function of temperature or density.  This means that if we were to look at, say, the 500mb surface in a barotropic environment, we'd be assuming that the temperature (or density) on that 500mb surface is everywhere exactly the same.  In other words, there are no temperature gradients on constant pressure surfaces.  Because of that, we don't have to worry about the affects of cooling or heating or temperature advection.  Immediately you'll notice that we run into problems--the entire atmospheric circulation is driven by heating by the sun, and yet we're ignoring heating?  How are we going to make things happen in this model?

Well, in some ways, we don't make things happen in this model--there are no features that add energy to the atmosphere in this model, or in other words there are no forcings.  In the barotropic model, since there are no external forcings, all we're really doing is taking the current state of the winds in the atmosphere and letting them blow until they blow themselves out.  The only thing we keep around is the fact that the earth is turning, so admittedly there is a Coriolis "force" present.  But there are no mountains causing lift, no ocean/land differences, and no solar heating.

"Well this is silly," you might think.  "How can we make any kind of decent weather forecast without having any impact from the sun or the land or anything?"  It turns out that, on the large scale, atmospheric motions in the relatively short term (the first few days or so) are pretty well dominated by this barotropic motion--the simple continuation and evolution of the flow without external forcing.  This has been known since the work of Carl Rossby in the 1930s.  I'm going to show examples of how well this actually works here today.

Based on a description of a modern-day implementation of the barotropic model from Issac Held and collaborators at Princeton, I coded up a "simple" global barotropic model in Python.  You can see animations of the output from the model on my webpage:


Though I warn you it's not updated that regularly, as my initialization data comes in several days late.  Still, it's there...and kind of fun to watch.

All this model does is predict the future wind, vorticity and height anomalies of the 500mb pressure surface.  That's it.  Here's how the barotropic model basically works:

  • We start with a global field of vorticity--that is, a measure of how much rotation there is in the atmosphere.  We're talking rotation on large scales--not small scale tornado-style rotation.  Remember in our barotropic assumptions above that we assumed that density or temperature was the exact same everywhere on a constant pressure surface.  If density is everywhere the same, we can't have any place where the winds are pushing air together to increase the density.  This means that we make an assumption of incompressibility which leads to an assumption of non-divergence--the winds everywhere can never be divergent or convergent.  If we make this assumption and we know the degree of rotation in the air (the vorticity), we can back out a wind field.  Here's an example of the global vorticity and non-divergent wind field I use to initialize my model:

Focus on the northern hemisphere in the image above.  Blue areas are areas with positive absolute vorticity--wind around them blows counterclockwise--think troughs or low centers.  Red areas are areas with negative absolute vorticity--wind around them blows clockwise--think ridges or high centers.  By going through and looking at how strong the vorticity gradients are, we can figure out how strong the winds should be, and we know what directions they're going based on the shape and orientation of these vorticity areas.  It's actually pretty simple to figure out if you know what you're doing.
  • Now that we have our wind field and vorticity field at the current time, we can use the wind field to advect the vorticity field.  That is, we know what the winds are right now, so we use the winds right now to push the vorticity values around for a certain amount of time.  In my model, I have a 15 minute time step, so I assume the winds are constant and use them to push along those blobs of vorticity for 15 minutes.
  • After doing this, I have a new vorticity field that has been advected around some.  However, I still have the old wind field.  Here, the model stops, takes that new vorticity field and computes the new wind field that corresponds with that new vorticity field, just like we did in the very first step. Now I have a new wind field that matches the new vorticity field.
  • From here, the model just repeats those same steps over--it takes these new winds and uses them to push the new vorticity around again for another 15 minutes.  Then we stop, recompute the wind field from the next vorticity field, then go on.  We can keep going for as long as we want...
So you see, all the barotropic model does is use the winds to basically push vorticity around, which gives new winds that push the vorticity around some more, and so on.  Once again, to reiterate, this model has:
  • No temperature or density gradients
  • No fronts
  • No mountains or terrain
  • No oceans
  • No heat sources from the below
  • No heat from the sun
  • No divergence or convergence
  • No rising or sinking motion
So how well does it do?

Well, here are some comparisons.  One thing we can also back out from the wind and vorticity fields is a field called the "streamfunction", which, for all we need to talk about here, is like a height anomaly map.  It highlights where there would be troughs and ridges on the 500mb map, were I actually forecasting the 500mb height.

On the left below is the 18 hour forecast of 500mb winds and that streamfunction/height anomaly field (the black solid and dashed contours).  On the right is the actual 500mb map from that time.
As always, you can click on the image to make it pop up bigger.  I've highlighted the major trough axes on both of the images just for reference.  You can see that, at least in terms of the placement of ridges and troughs, it's actually doing very well in the 18 hour forecast.  It has a deep trough over the western Gulf of Alaska, and the tightly-packed height contours on the actual 500mb map do imply that there are strong winds through the base of that trough, just like the barotropic model is predicting.  The model also does a good job forecasting the elongated trough over the western US and the compact upper-level low over northern Quebec.  It's a surprisingly good forecast for a model that has no real external forcing.

Let's go a bit further out into the forecast.  Here's a comparison of the 44 hour forecasts:
In many ways the forecast is still very good.  The Gulf of Alaska trough is in about the right place with strong winds right where they should be.  That little upper-level low over northern Quebec is also still being handled rather well.  However, we can start to see differences creeping in.  Notice that the elongated trough over the western US doesn't look quite right in the model.  The barotropic model has kept the trough much more compact and has it centered over the western plains whereas, in reality, the trough still extended quite far back off to the southwest.  Remember that in real life the Rocky Mountains stretch across the western US, but the barotropic model has no concept of mountains or topography.  So, any impact the high terrain would have on the flow is totally beyond anything the barotropic model would predict...

Finally, the 66 hour forecast comparison:

Now at almost three days out, things are beginning to fall apart.  The barotropic model still has that northern Quebec low in about the right position with a wind maximum in the right place, so at least that's good.  But now that Gulf of Alaska trough isn't in the right spot--the barotropic model has moved the trough to fast to the east, placing the trough axis over the coast whereas in reality the trough stayed cut off and sort of hung back in the central Gulf of Alaska.   That western US trough is almost totally absent from the barotropic model--the barotropic model kept its more compact, lower amplitude trough and moved it slightly east, centering it over the Mississippi valley.  In reality, the trough stayed elongated to the west and even formed a cutoff low off the coast of California--something not at all present in the barotropic model.

So we can see that, at least for the first day or two, the barotropic model with no real forcing actually does a fairly good job at predicting the major upper air pattern--not that bad for a very simplistic model. However, since there is no vertical motion, no divergence, no heating, etc., the barotropic model really cannot capture the formation and development of new troughs very well.  That cut-off low that formed by 66 hours is a prime example--the forcings that led to that cut-off low being formed were just not present in the barotropic model, so it missed it.

So we'll stick with our far more complicated global models that actually include all those things we ignored for now.  They may be a lot more expensive to run, but they're a bit more reliable beyond 48 hours.  As an interesting comparison, looking back at the history of those first barotropic numerical weather models, the first 24-hour global barotropic model forecasts on the ENIAC computer took 24 hours to run in the 1950s--not much of a forecast if you don't get it until right when it's happening!  By comparison, my barotropic model integrates out to 120 hours in about 20 seconds on my desktop.  Amazing how far computers have come...

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