Understanding the various sources of heating in the atmosphere, like convection and radiation, is critically important in Tropical meteorology. Unfortunately we cannot just go out and directly measure the temperature tendency from convection. However, we can measure the temperature and estimate how it changes as a result of air flowing over an area, and then calculate the “leftover” residual temperature tendency to get an idea of how diabatic processes (i.e. convection and radiation) are heating the atmosphere. In practice, there are two widely used approaches to produce these estimates, and the goal of this article is to provide a brief comparison of them.

Yanai et al. (1973) were one of the first to explore this idea with data collected from a field campaign. Their approach was to use the dry static energy (DSE),

which is conserved for an air parcel as long as no diabatic heating occurs. In other words, as long as there is no condensation, evaporation, or radiative cooling, then the DSE value will stays the same, even if the parcel moves vertically and experiences adiabatic heating or cooling.

Using the DSE budget, the net diabatic heating can be diagnosed as follows,

I usually cite Yanai et al. (1973) when doing this calculation, however I recently realized that this idea had been used in earlier studies. One example is Reed and Recker (1971), in which they calculate the heat budget from a triangular sounding array in the West Pacific.

Similar to Yanai’s work, Reed and Recker also parse out the details of convective vs. non-convective regions in terms of their mass, heat, and moisture variations. It’s pretty interesting to compare this paper with the work of others that followed.

Anyway, a slightly different approach is used by other studies such as Nigam (1994), Nigam et al. (2000), and Hagos et al. (2010), use potential temperature (θ), defined as

Potential temperature has the similar conservation properties as DSE. The budget can then be written as,

The ratio in the front simply represents the conversion from θ to absolute temperature, although the units stay the same.

So to compare these two methods I grabbed some 6-hourly ERA-interim data for the month of August 2016, and calculated the heating estimates with both method and averaged them over the entire Tropics (30°S-30°N). Here’s the result:

Both methods give a similar average profile with small differences. As Darrell Huff said in his book “*How to Lie with Statistics*“:

“A difference is a difference only if it makes a difference.”

So is this difference anything to worry about?

The heating near the surface is mostly associated with boundary layer turbulent fluxes that communicate the surface latent and sensible heat fluxes vertically. The heating at the top of the troposphere is partly due to shortwave heating of ozone. The mid-tropospheric heating maximum is mostly due to heating from convection. There is also a small longwave radiative cooling throughout the troposphere, which is overwhelmed by the other processes. In spite of the non-zero heating the Tropics does not continue to get hotter because the general circulation exports a good deal of heat toward the poles.

The difference in the right panel above shows that the method using θ gives slightly more warming on average, which seems really small, but I was surprised that difference was concentrated near the surface and top of the troposphere.

If we look at the differences in just the horizontal and vertical advective components of each equation, the difference of horizontal advection explains most of the difference shown above.

If we do a zonal average we can see these differences are largest away from the equator and away from the regions with the most convective activity, particularly around 25°-30°S.

Although the average difference is small (<0.1 K/day), the difference can be quite large for any given local 6-hourly estimate. This is illustrated by the profiles of the minimum and maximum difference below, as well as the standard deviation of the difference.

So the differences between the two methods can be quite large at times, and it seems this could make a difference in our interpretation of the data. But *why* are they different? Or more importantly, which one is more accurate? Ultimately, I’m not sure how to tell.

Part of the answer lies in the linearity of DSE and non-linearity of θ and how this influences the horizontal gradients of DSE and θ. This can be seen by using some basic derivative rules to expand the horizontal advection terms, although it gets messy.

I’m still pondering why these differences exist. I’m inclined to speculate that there may be times when hydrostatic balance is invalid, which could point to the θ method being superior, but I need to think about this more.