A Closer Look at Critical Thickness

Last time I talked about the impending "arctic outbreak" that was being forecast to occur at the end of this week (I also have a brief update on that at the end of the post).  I mentioned that one product we could look at was a plot of 1000-500 mb thickness to serve as a rough guide to a dividing line between snowfall and rainfall in our precipitation.  On a side note, I also jumped ahead last week (kind of) and made a statement that for much of the upper midwest this arctic outbreak could bring the first snows for much of the upper midwest.  I neglected to look in the near term and, as many people in Minnesota, Iowa and parts of Wisconsin now have seen, the first decent snowfall occurred this weekend:

Fig 1 -- 2-day snow accumulations as of 1500Z, Nov 14, 2010.  From NCDC.

However, more of the midwest, including points east of that snow swath, could see their first snow later this week.  I based that on using an analysis product that contoured the 1000-500 mb thickness and talked about how there was this "magical" rule of thumb where the dividing line between snow and rain was typically at the 5400 m 1000-500 mb thickness contour.  However, that's not the only thickness rule of thumb that exists.  Below is a plot of critical thicknesses for several different layers (more than just 1000-500 mb) from the College of DuPage's model output.

Fig 2 -- Critical thickness analysis values from an operational WRF-NMM model at 12Z, Nov 14, 2010. From the College of DuPage website.

And below is their guide to what each of the contours represent (the shading is (I think, because it's not labelled) 850 mb relative humidity, a rough proxy for where precipitation may be falling):

RED = 1000-700mb 2840m Thickness Contour	CYAN = 850-700mb 1540m Thickness Contour	YELLOW = 1000-850mb 1300m Thickness Contour
MAGENTA = 700-500mb 2560m Thickness Contour	GREEN = 850-500mb 4100m Thickness Contour	WHITE = 1000-500mb 5400m Thickness Contour
	BLUE = 850mb 0 degree Isotherm

Why are these called "critical" thicknesses?  The "critical" part comes from their association with the rain-snow dividing line.  For each of these layers, empirical experience has shown that these particular thickness values usually correspond to the approximate rain-snow dividing line.  However, critical thickness values are different in different locations.  There are also many, many examples where critical thickness levels did not correspond to the actual rain-snow line.  But, in general, these provide a good first guess for estimating precipitation type.  By looking at several different layers and their critical thicknesses (like in figure 2 above), we can gain a reasonable degree of confidence about what kind of precipitation will fall.  For example, if you happen to be north of every single critical thickness line (assuming it's colder to he north, which it almost always is), you're pretty sure to receive snow.  If you're in the middle of the spread of lines, that's much less certain.

But what do these critical thickness values really mean?  Sure they came from "years of observation", but what can they tell us about the difference between rain and snow?  I mentioned in my last post that the thickness of a layer is related to the mean temperature in that layer, with a colder mean temperature corresponding to  a "thinner" thickness.  There's actually a (relatively simple) equation to describe this, known to most meteorology students as the hypsometric equation:

--From the Wikipedia page for the hypsometric equation

Where h is the thickness, R is the gas constant for dry air (287 J/kg/K), g is the acceleration due to gravity (9.81 m/s^2), and T is the mean (virtual) temperature (in Kelvin) of the layer.  P1 is the pressure at the bottom of the layer and P2 is the pressure at the top of the layer.  We know the pressures at the top and bottom of our layer and the critical thickness of the layer, so we can solve this for the mean temperature we would expect to find in each layer when the layer's thickness is at the critical thickness.  The results are shown below:

Mean Temperature Calculated from Layer Thickness
Pressure Levels (mb)	Thickness (meters)	Mean Temperature (degrees Celsius)
1000-700	2840 m	-0.8 ⁰C
700-500	2560 m	-13 ⁰C
850-700	1540 m	-2 ⁰C
850-500	4100 m	-9 ⁰C
1000-850	1300 m	0.4 ⁰C
1000-500	5400 m	-7 ⁰C

Table 1 --Calculated mean temperatures based on layer thicknesses via the hypsometric equation.

What can we see from these numbers?  A couple things stand out:

1. The 1000-850 mb critical thickness of 1300 m corresponds to a mean temperature of 0.4 degrees Celsius.  That's an average temperature above freezing.  This tells us that snow can fall even when the temperature is above freezing at the surface.  We see this quite often, actually, and usually the snow that falls is very wet snow.  Frozen snowflakes falling from above need time as they fall to melt, so if the layer above freezing is relatively shallow, the snowflakes simply don't have time to fully melt before they hit the ground.
2. The mean temperature in the 700-500 layer should be cooler than -13 degrees Celsius.  This represents a temperature on the upper bound of the so-called "dendritic growth zone".  It turns out that the most vigorous production of snowflakes tends to occur where there are temperatures between -12 to -18 degrees Celsius (the exact numbers will vary depending on what study you look at or who you ask).  Therefore, it would make sense that we need to have a layer that is at least that cold to be confident in seeing snowflakes (if snow is forming).
3. The mean temperatures never get above freezing except in the lowest layer.  This implies that if we ever see the temperature on our profile get above freezing (except for in a very near-surface layer, but even then...) we must begin to seriously question whether or not snow will fall.  A small layer above freezing may not be enough to fully melt the snow crystals.  However, a relatively deep layer above freezing will start pulling the mean values in layers spanning that particular layer closer to the freezing point.  This will in turn warm the mean temperatures in those layers beyond these "critical" values.
4. If we actually plot these mean temperature values at the average pressure levels they represent and calculate some rough lapse rates (not shown here), we see that below ~800 mb the lapse rate represented by this profile is absolutely stable and above ~800 mb the lapse rate becomes conditionally unstable.  I believe this implies that critical thickness values may be more representative in atmospheres where the lower part of the troposphere (i.e. below ~800 mb) is statically stable, since the further our lapse rates stray from the idealized lapse rates in this profile, the less representative of the atmosphere this critical thickness idealization will be.  (I was initially dubious when I had this idea until I saw this paper by Paul Heppner (1992).  It's an excellent review of how accurate critical thickness values are based on a statistical analysis.  He also confirms a tendency for the values to be more applicable in a stable environment).

So, remember--critical thickness plots are fun tools that can help provide an initial guess at checking precipitation type.  It's always best to check multiple critical thickness values for different layers to get a clearer picture of what's going on.  And, we can see from the simple calculations above what these critical thickness values can tell us about a typical snow vs a typical rain environment.  There's a lot more analysis that could be done, but this is just a flavor of what critical thickness implies.

***UPDATE: The upcoming "arctic outbreak"***

Remember last time I mentioned how we would want to see a buildup of really cold arctic air on our side of the globe if we were to have an "arctic outbreak" here later this week?  Here is the hemispheric plot of 500 mb heights (or, in proxy form, temperature) from 48 hours ago:

Fig 3 -- Northern Hemispheric plot of 500 mb heights (shaded) and mean sea-level pressure (contoured) from 12Z, Nov. 12, 2010. From the HOOT website.

We can see that the coldest air (represented by the lowest 500 mb heights) was just about centered over the North Pole two days ago.  Now look at this morning's plot:

Fig 4 -- Northern Hemispheric plot of 500 mb heights (shaded) and mean sea-level pressure (contoured) from 12Z, Nov. 14, 2010. From the HOOT website.

The center of the cold air has shifted off the pole!  Not only that, but it has shifted toward the North American side.  Could this be the beginning of our arctic outbreak air?  Possibly.  Remember this air has a long way to go before it gets down here, and interactions with the land can warm the air considerably.  (Though this weekend's snow cover over the upper midwest won't do anything to help warm the air mass, if the snow sticks around...)