Periodic Thermal Waves

*Updated to add 500mb anomaly reanalysis maps at the bottom *

A few days ago I thought it would be interesting to see what would happen when calculating what day of the week was the warmest in the U.S. It was intended to be a silly exercise to demonstrate that even if one day was the highest it would be statistically trivial. I looked at 'primary' stations only since my poor computer could only handle two years at once before crashing. So I queried 2012 and 2013.  Well, much to my surprise, there were very distinct patterns. Not only were the patterns geographical, but the were temporal and propagated from west to east. Figure 1 shows the warmest days of the week for all primary stations in the U.S. Across areas with minimal longitudinal variation, the daily percentages are staggeringly variable. In Alaska for example (see Figure 2), an amazing 41% of stations were warmest on Thursday and only 4% are warmest on Tuesday or Wednesday. To me, that is a signal indicating a possible climate connection.

Fig 1. All primary stations in the U.S. mapped according to which day of the week during 2012 and 2013 was the warmest (1=Sunday, 2 = Monday, and so on.) Only stations with 95% complete data were used.

Fig 2. All primary and RAWS stations in Alaska mapped according to which day of the week during 2012 and 2013 was the warmest (1=Sunday, 2 = Monday, and so on.). Only stations with 90% complete data were used.

I then set out to check time periods other than seven days. Of course a seven day period corresponds very nicely with a calendar week. I checked time periods between 2 and 30 days. For the 30-day period think of it as grouping Jan 1, Feb 1, March 1, etc. and calling those "Group 1". Then take Jan 2, Feb 2 March 2, etc. and call those "Group 2". 

After a little trial and error, it was pretty obvious that 4-days was a very prominent pattern for the Lower 48 and modestly prominent for Alaska. Figure 3 shows which of the repeating 4 days is the warmest for the Lower 48 and Figure 4 shows the same but for Alaska.

Fig 3. All primary stations in the U.S. mapped according to which day, of a repeating 4-day pattern, during 2012 and 2013 was the warmest. Only stations with 95% complete data were used.

Fig 4. All primary and RAWS stations in Alaska mapped according to which day, of a repeating 4-day pattern, during 2012 and 2013 was the warmest. Only stations with 90% complete data were used.

Looking at the ESRL Reanalysis data (see Figures 5, 6, 7, and 8 below), a very clearly defined 4-day thermal wave is evident. I compared each of the four repeating days with the other days in the 4-day set to track the atmospheric patterns for 2012 and 2013. I am quite amazed that given the number of days used in the analysis (728) that any pattern whatsoever holds up. If the 4-day pattern was 3.5 or 4.5 days it would break down in the reanalysis data after a while. When I ran the same analysis at the 7-day interval (not shown) a noticeable, but less prominent thermal wave was also evident. I can share those images with anyone who may be interested.

So what recurs at 4 and 7 days. One obvious answer is planetary (Rossby) waves. They tend to recur at 7-day intervals according to the literature but their period is pretty variable and is highly dependent on the number of waves. The fact that the 7-day pattern in Figure 1 is more prominent between 40°N and 50°N lends credence to the planetary wave origin for that length time period. But what about the 4-day pattern? It has a much stronger signal. Is it related to planetary waves? What drives the thermal push depicted in Figures 5-8? Also, might the 7-day pattern actually be a 1/2 strength version of the 4-day pattern since 7 is almost, but not exactly, a multiple of 4?

Surely this is not a new discovery. My brief search of the literature didn't lead to any obvious answers beyond the Rossby wave solution. Any ideas would be much appreciated.

Fig 5. Day 1 of the 4-day thermal wave as depicted by the ESRL daily composite reanalysis. A wave axis is superimposed as a red line. This is a compilation of all Day 1s minus the average of Days 2, 3, and 4. 728 days were used in the analysis.

Fig 6. Day 2 of the 4-day thermal wave as depicted by the ESRL daily composite reanalysis. A wave axis is superimposed as a red line. This is a compilation of all Day 2s minus the average of Days 1, 3, and 4. 728 days were used in the analysis.

Fig 7. Day 3 of the 4-day thermal wave as depicted by the ESRL daily composite reanalysis. A wave axis is superimposed as a red line. This is a compilation of all Day 3s minus the average of Days 1, 2, and 4. 728 days were used in the analysis.

Fig 8. Day 4 of the 4-day thermal wave as depicted by the ESRL daily composite reanalysis. A wave axis is superimposed as a red line. This is a compilation of all Day 4s minus the average of Days 1, 2, and 3. 728 days were used in the analysis.

* Update section

After reading a section from a paper from 1976 (Blackmon, M. L. 1976. A climatological spectral study of the 500 mb geopotential height of the Northern Hemisphere. J. Atmos. Sci. 33, 1607 -1623.) that describes the propagation period of Rossby waves as 20° longitude per day, I decided to make reanalysis plots of the 500 mb height anomalies. The 20° value corresponds to 1/4 of the width of the Lower 48 states. Therefore, the 4-day thermal wave may be entirely explained by that.The wave train of anomalies stands out very nicely and clearly propagate from west to east. 

Fig 9. 500mb height anomaly on Day 1 of the 4-day thermal wave as depicted by the ESRL daily composite reanalysis. This is a compilation of all Day 1s minus the average of Days 2, 3, and 4. 728 days were used in the analysis.

Fig 10. 500mb height anomaly on Day 2 of the 4-day thermal wave as depicted by the ESRL daily composite reanalysis. This is a compilation of all Day 2s minus the average of Days 1, 3, and 4. 728 days were used in the analysis.

Fig 11. 500mb height anomaly on Day 3 of the 4-day thermal wave as depicted by the ESRL daily composite reanalysis. This is a compilation of all Day 3s minus the average of Days 1, 2, and 4. 728 days were used in the analysis.

Fig 12. 500mb height anomaly on Day 4 of the 4-day thermal wave as depicted by the ESRL daily composite reanalysis. This is a compilation of all Day 4s minus the average of Days 1, 2, and 3. 728 days were used in the analysis.