New wetbulb temperature algorithm for HadISDH

 In short:

I have found that the previous method of calculating wetbulb temperature (T_w) resulted in spurious very high (> 35 °C) values when dewpoint temperature (T_d) was very low but temperature (T) very high. Such conditions can occur in hot, arid regions. High T_w is an indicator of severe heat stress and so we want to make sure these errors are not included in any analyses. A new method has been found that, although not perfect - no algorithm will be for T_w because it is an empirical thermodynamic quantity - removes the erroneous high T_w values and generally performs better. This new method (Stull, 2011) has the following caveats:

• The valid range of RH is 5 - 99 %rh
• The valid range of T is -20 to 50 °C
• The algorithm is optimised at a surface pressure of 1013 hPa and T = 20 °C 
• Its reported error ranges from -1 to 0.65 °C
• T_w will be overestimated when T > 20 °C, especially at low RH
• T_w will be underestimated when T < 20 °C
• The algorithm is optimised at a surface pressure of 1013 hPa and T = 20 °C 
• Differences of 200 hPa make negligible difference when RH > 50 %rh with errors up to 1 ^oC when RH < 10 %rh. 
• T_w will be overestimated in very low P (and low RH, high T)
• T_w will be underestimated in very high P (and low RH, high T)

• The algorithm is not valid for very cold dry conditions
• Note that at 100 %rh the bias remains very small, reducing to 0 when T = 20 °C
• T_w is too low (-0.2 °C) when T = -20 °C
• T_w is too high (0.1 °C) when T = 30 °C. 

This new method is now used to produce HadISDH T_w for HadISDH.land, HadISDH.marine, HadISDH.blend and HadISDH.extremes. This is for versions 4.5.0 (land), 1.4.0 (marine) and 1.4.0 (blend) onwards.

The problem:

Previously, for HadISDH I calculated T_w from T and T_d using an equation from Jensen et al., (1990):

T_w = (((a*T) + (b*T_d)) / (a + b))

a = 0.000066*P

b = ((409.8*e) / ((T_w + 237.3)^2))

Here, P is surface pressure in hPa and e is vapour pressure in hPa. This paper is no longer accessible online unfortunately. The full reference is:

Jensen, M.E., R.D. Burman, and R.G. Allen (ed). 1990. “Evapotranspiration and Irrigation Water Requirements.” ASCE Man. and Rep. on Engineering Pract. No. 70, New York, 332 p.

At the time of developing HadISDH I thought that this was sufficient, and T_w was not the primary focus of the dataset. However, now we are more interested in T_w from a human heat health perspective - particularly values close to 35 °C. On deeper investigation, I have realised that this equation results in very large errors, especially in hot dry conditions that would undermine any analysis of high T_w occurrences. Figure 1 shows T_w calculated using the Jensen et al., (1990) method for a range of T and T_d between -30 - 40 °C. The extremely large errors when T_d < 0 °C are clear - reaching T_w > 30 °C!!!

Figure 1. Wetbulb temperature calculated for a range of temperature and dew point temperatures using the Jensen et al., (1990) equation. Note spurious extremely high wetbulb temperatures at bottom right of plot.

  

The solution:

The equation from Stull (2011) fits an optimal solution to obtain T_w from RH and T at standard surface pressure 1013 hPa:

T_w = (T * arctan(0.151977 * ((RH + 8.313659)^0.5))) + arctan(T + RH) - arctan(RH - 1.676331) + (0.00391838 * ((RH)^(3./2.)) * arctan(0.023101 * RH)) - 4.686035

Although there are caveats with this approach (see above list), the errors are relatively small and certainly present an improvement over the Jensen et al., (1990) equation. Figure 2 shows the range of T_w resulting from the Stull equation for a range of T and T_d between -30 - 40 °C. Clearly, at very low T_d and high T the Stull T_w remains stable. Very high values of T_w (>30 °C) only occur when both the T and T_d are close to or above 30 °C.

Figure 2. Wetbulb temperature calculated for a range of temperature and dew point temperatures using the Stull (2011) equation. 

The details:

Figure 3 shows the difference in T_w between the Stull and Jensen equations (Stull minus Jensen) for a range of T and T_d between -30 - 40 °C. For the majority of the range the Stull equation gives lower values of T_w, especially in low T_d and high T conditions. For warm and humid or cold and dry conditions the two equations are similar, within +/- 1 °C. 

Figure 3. Wetbulb temperature differences calculated for a range of temperature and dew point temperatures using the Stull (2011) and Jensen et al., 1990 equations. Brown regions show where the Stull equation gives lower wetbulb temperatures and blue regions show where the Stull equation gives higher wetbulb temperatures.

As stated in the Stull (1990) paper, this equation is not valid for cold and dry conditions. Therefore, users of HadISDH should be cautious if their interests lie in T_w in cold conditions. The main interest at present is for high extremes and a new dataset, HadISDH.extremes, is currently under development. This dataset provides monthly gridded indices based on daily maximum T and T_w for assessing changes in frequency and magnitude of humid and dry heat events.