The modulated annual cycle: an alternative reference frame for climate anomalies

In climate science, an anomaly is the deviation of a quantity from its annual cycle. There are many ways to define annual cycle. Traditionally, this annual cycle is taken to be an exact repeat of itself year after year. This stationary annual cycle may not reflect well the intrinsic nonlinearity of the climate system, especially under external forcing. In this paper, we re-examine the reference frame for anomalies by re-examining the annual cycle. We propose an alternative reference frame for climate anomalies, the modulated annual cycle (MAC) that allows the annual cycle to change from year to year, for defining anomalies. In order for this alternative reference frame to be useful, we need to be able to define the instantaneous annual cycle: we therefore also introduce a new method to extract the MAC from climatic data. In the presence of a MAC, modulated in both amplitude and frequency, we can then define an alternative version of an anomaly, this time with respect to the instantaneous MAC rather than a permanent and unchanging AC. Based on this alternative definition of anomalies, we re-examine some familiar physical processes: in particular SST re-emergence and ENSO phase locking to the annual cycle. We find that the re-emergence mechanism may be alternatively interpreted as an explanation of the change of the annual cycle instead of an explanation of the interannual to interdecadal persistence of SST anomalies. We also find that the ENSO phase locking can largely be attributed to the residual annual cycle (the difference of the MAC and the corresponding traditional annual cycle) contained in the traditional anomaly, and, therefore, can be alternatively interpreted as a part of the annual cycle phase locked to the annual cycle itself. In addition to the examples of reinterpretation of physics of well known climate phenomena, we also present an example of the implications of using a MAC against which to define anomalies. We show that using MAC as a reference framework for anomaly can bypass the difficulty brought by concepts such as "decadal variability of summer (or winter) climate" for understanding the low-frequency variability of the climate system. The concept of an amplitude and frequency modulated annual cycle, a method to extract it, and its implications for the interpretation of physical processes, all may contribute potentially to a more consistent and fruitful way of examining past and future climate variability and change.