Finally...how do we calculate snow ratios?

A while ago I said that I would look into how we calculate snow ratios, as these have big impacts on determining the amount of snowfall to expect.  Today I'm finally getting around to looking at that particular subject....

First, what is a snow ratio?  Technically, it's the ratio of inches of snow on the ground to the liquid water content that the snow contains.  For example, if you had ten inches of snow on the ground, then all of it melted and you were left with one inch of water, that would be a 10:1 snow ratio.  Once the snow falls, it's easy to measure snow ratio (once you correct for the surface area from which you gathered the snow).  However, it's difficult to develop a purely physical basis for calculating snow ratios.  Most of our current knowledge on the subject comes from empirical test results.  The basic pattern of thinking behind snow ratios is this:

1. How deep the snow on the ground becomes depends on the shape and size of the ice crystals in it.  Big, fluffy snowflake crystals (called "dendrites") tend to stack up deeper than small, compressed ice needles or plates.  It's simply a factor of their geometry.
2. It has long been established that different types of snow crystals form at different temperatures.  Therefore it should be possible to relate snow crystal type to the temperature at which it forms and consequently relate the snow ratio to the temperature at which the snow crystals are forming.

With this in mind, lots of empirical studies were done and the result was a chart that looks like this:

Fig 1-  Empirical graph relating snow ratios to temperature of snow crystal source region.  Graph from a powerpoint presentation by Daniel Cobb.

This handy little graph forms the basis for almost all of our snowfall forecasting techniques.  A few things to note from this graph:

1. If we take the average of all those snow ratios, it works out to be around 10:1.  This is why we typically assume a 10:1 snow ratio when we want to make quick calculations--it splits the difference and shouldn't be that far off...usually.
2. Note that the peak of the graph is between -12 degrees Celsius and -18 degrees Celsius.  Not coincidentally, this is the region of the most rapid dendrite snow crystal growth (also known as the "dendritic growth zone").  Remember that big, fluffy dendrites tend to make deeper snow for a given water content.  Therefore it's no coincidence that the temperatures that coincide with the highest snow ratios are also the temperatures where dendrite growth is the strongest.

Just for a refresher, here's a diagram I randomly found on the web (actually, according to the website I took it from, adapted from a 1954 book by Ukichiro Nakaya called Snow Crystals, Natural and Artificial) that relates the crystal type to the temperature of formation.

Fig 2 -- Diagram relating the snow crystal type to the temperature of formation.  Adapted from Snow Crystals, Natural and Artificial by Ukichiro Nakaya (1954).  Found on this website.

And once again we see above that dendrites are favored in the temperature region of maximum snow ratios.  Why is snow crystal growth so rapid in this temperature region?  As seen in the line on the graph above, this temperature region is also where the saturation vapor pressure with respect to ice is the most different (lower) than the saturation vapor pressure with respect to liquid water.  This means that ice crystals will form preferentially to liquid water droplets, greatly speeding ice crystal growth.

So we see that the snow ratio graph above (in figure 1) makes a lot of sense.  The next question is--what temperature are we going to assume for the snow ratio?  There are two main methods for looking at this (if we're only looking at temperature with no other information):

1. One basic way is to just use the temperature at the surface as the temperature to look up the snow ratio on the chart above.  Since surface temperature is well-represented in both models and observations, it's commonly available and can perhaps provide a more refined measurement than simply guessing at a 10:1 ratio.  However, it's clear that most of our snow crystals are not forming right before the surface--they're forming much higher.  I suppose if there's a deep isothermal layer from the surface to, say, 900mb, perhaps this might be better.  But in general, this won't be very accurate.
2. Another temperature that can be used is the maximum temperature in the profile.  The thought is that any ice crystals forming above the maximum temperature will have to fall through the zone of maximum temperature and this might cause some melting or changing of crystal habit.  However, below that level, the ice crystals are "frozen" in what ever state they left the warmest layer, and therefore whatever reaches the surface should reflect the properties of that warmest layer.  This provides a distinct improvement over just using the surface temperature, as this temperature is probably closer to the temperature at which snow would form.  Of course, sometimes the surface temperature IS the max temperature in the profile, so then both temperatures would be the same.

Here's an example of a model snow sounding annotated with both of these temperatures.

Fig 3 -- GFS 24-hour forecast sounding for KRFD at 12Z, Dec. 24, 2010. 

In the above sounding, note that we are saturated or nearly saturated up to around 500 mb--lots of moisture there.  We're also saturated between -12 to -18 degrees Celsius (where the temperature profile is highlighted in yellow), so there are probably dendrites forming.  However, the max-temp-in-profile theory assumes that the crystals will have properties as if they formed around -5.4 degrees Celsius.  From our chart above, this is about a 9:1 snow ratio.  The surface temperature isn't that much colder at -6.2 degrees Celsius.  This results in a snow ratio of about the same--9:1 or 10:1.

However, this doesn't necessarily reflect everything that's going on in the atmosphere.  It has been shown in many studies that areas of intense snow crystal growth and formation can be collocated with the areas of maximum vertical velocity (when the air is saturated).  If there is a lot of vertical motion within a saturated environment, lots of moisture is moving through that particular region and as a result lots of snow crystals can grow there.  This has given rise to the well-known "cross-hairs" technique for looking for areas of particularly heavy snowfall.  (Still looking for a good graphic that shows that clearly...).  If we are saturated, areas of greater upward vertical velocity tend to produce more snow crystals than areas with weaker upward vertical velocity.

This can be applied to give two more methods for getting snow ratios:

One way to choose the temperature at which most of our snow crystals are forming is to find the maximum vertical velocity within the saturated parts of the sounding.  Then, we find the temperature at that level.  Since snow crystal formation is enhanced in areas of high vertical velocity, we'll assume that we'll see more crystals from this particular level than any other level.  Therefore the temperature at this level should define the geometry of a good portion of our snow crystals.

Fig 4 -- 25 hour NAM forecast for KRFD at 13Z, Dec. 24, 2010.  Vertical velocity is shown in white with "zero" vertical velocity shown as the white vertical dotted line.

In the example profile above (from about the same time as the GFS model image in figure 3--the NAM has better vertical velocities) we see that the peak vertical velocity occurs at around 828mb.  We are near saturation (particularly with respect to ice) and the temperature is -8 degrees Celsius at this level.  Using our chart above, this would translate to a snow ratio of about 10:1.

Of course, this is just choosing one level and assuming that the majority of our snow crystals will come from this level of maximum upward vertical velocity.  But we know that snow crystals are forming above and below that layer.  We also know that since we're below freezing in our profile, there's a very good chance that all the crystals are managing to fall through somehow.  So can we improve upon our forecast of snow ratio even more by taking into account snow crystals forming at other levels?

Daniel Cobb, when he worked at the Caribou WFO of the National Weather Service, developed an algorithm to do just that based on model data. This algorithm, now commonly called the COBB algorithm or the Caribou method, calculates a weighted average of snow ratios at all levels of model output where snow crystals could form (i.e., it's cold enough and the air is saturated, etc...).  The weighting depends on the vertical velocity--you sum all of the vertical velocities at each level of the model where there could be snow crystal growth and then divide each level's vertical velocity by that total to get the "weight" to apply to the snow ratio at that level.  In this way, the "crosshairs" technique described above can be expanded to all levels where snow crystals can be growing--including those layers where it's saturated in the dendritic growth zone.  As such, this technique (a form of which is used in Bufkit's "zone omega" snow ratio option) usually gives higher estimates of the snow ratio than the other techniques.  But often, it can be more accurate...

So there we have a summary of several techniques that can be used to obtain snow ratios.  All are based upon that critical graph in figure 1--that temperature-snow-ratio relation is the key to our snowfall forecasting.  To find a forecast snow depth?  Simply multiply the quantitative liquid precipitation forecast by the snow ratio.  For instance, if the model is telling you that .25 in. of liquid precipitation is expected and you have a snow ratio calculated at 10:1, to find the snow depth take .25 x 10 = 2.5 inches of snow.  It's that simple.

There are lots of excellent resources online for learning more about snow ratios.  For example, Daniel Cobb has a powerpoint presentation where he very nicely outlines the basic COBB algorithm--it can be found at:
http://cstar.cestm.albany.edu/nrow/NROW6/Cobb.ppt

The Warning Decision Training Branch also has a good teaching presentation as part of their AWOC winter weather course on snow ratios.  It goes into far more detail than I do here.This presentation is available at:
http://www.wdtb.noaa.gov/courses/winterawoc/IC6/lesson5/part1/player.html

Of course, in closing, many of our advanced models (like WRF) can forecast snow amount explicitly if they have a microphysics scheme that computes ice crystal concentration.  So in the future we may have models explicitly try to determine snowfall amounts and use those instead of snow ratios.  But I still think snow ratios are rather fun...

Fig 5 -- Calculated 24-hour snowfall accumulations from the UW-WRF model at 12Z, Dec. 24, 2010.