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Effects of Temperature Anomalies on the Palmer
Drought Severity Index in the Central United States

A Theoretical Analysis

The PDSI is defined as

  GIF -- Equation 1: PDSI equation (1)

Here, subscripts i and i - 1 indicate current and previous months at some arbitrary time, respectively, and PDSI0 = 0. The Zi in Equation (1), called the monthly Z-index, is defined as

  GIF -- Equation 2: Monthly Z-index equation (2)

where K is a coefficient and

  GIF -- Equation 3: Equation for d (3)

In the above, P is actual monthly precipitation. The terms in the parentheses on the right-hand-side (r.h.s.) of Equation (3) combine to yield monthly 'climatologically appropriate rainfall'. In particular, PE is potential evapotranspiration, PR potential water recharge to soil, and PRO potential runoff. Palmer (1965) used a two-layer soil model consisting of a surface layer, 'plow layer', and an underlying layer, 'root zone', and defined PL as the sum of soil water of the two layers available for evapotranspiration. He called this term 'potential loss of soil water to evapotranspiration'. The four coefficients in Equation (3) are defined as

  GIF -- Equation 4: Coefficients of equation 3 (4)

where the overbar denotes a long-term average of the parameter values for the ith month. The numerators in the above four expressions are long-term monthly averages of evapotranspiration, soil water recharge, runoff and available soil water amount, respectively.

Using the recursive relation of PDSIi and PDSIi-1 we can rewrite Equation (1) as

  GIF -- Equation 5: Rewrite of equation 1 for soil dryness/wetness anomalies (5)

This new formula of PDSI indicates how soil dryness/wetness anomalies of previous months affect the PDSI value of the ith month. Because c = 0.897 < 1 and m is a positive integer, the effect of previous months on the index gradually decreases as (i - m) increases. Soil wetness anomaly in the ith month has a full impact on the index value for that month. The simple form of Equation (5) for Z and the PDSI also indicates that the task of examining separate effects of precipitation and temperature anomalies on the PDSI can be carried out by examining the relationship between Z and anomalies of precipitation and temperature.

Rearranging the terms in Equation (3), we have

  GIF -- Equation 6: Rearrangement of equation 3 (6)

Here, we have combined the last three terms in the parentheses on the r.h.s of Equation (3) into term S. Therefore, (P - S) yields actual available soil water for evaporation and transpiration from rainfall events in a specified time interval, e.g. one month. The term αPE is total monthly loss of soil water by evapotranspiration. When this term is larger than the first term, the monthly rainfall is inadequate to maintain the 'climatologically required' soil water and the soil suffers a net loss of water by an amount of αPE - (P - S). In this situation, the available soil water decreases to meet evapotranspiration demand and a drought condition develops. Because S is determined by topography and physical and hydraulic properties of soils at a site and is independent of surface air temperature, it can be represented further as a fraction of total precipitation of the month, that is, S = κP. In this expression, 0 ≤ κ ≤ 1, and 0 ≤ κ came from the fact that κ defines the fraction of monthly precipitation used in runoff, recharge to soil and contribution to the storage of the available soil water, and these contributions are zero when monthly rainfall is zero. Mathematically, this is justified by the fact that the undefined expression S/P approaches zero when both P and S are zero.

Only PE in Equation (6) is dependent on air temperature. The coefficient of this term, α, satisfies 0 < α < 1, and is nondimensional. Applying Thornthwaite's (1948) calculation method for PE, which has been used in calculation of the PDSI (Palmer, 1965; Karl, 1986), we have PE = ηTa, where T is monthly mean surface air temperature in degrees Celsius, η = 1.6(10/I)a(I)Ta(I) with I being the heat index (Thornthwaite, 1948) and a a function of I given in Thornthwaite (1948). Note that η has a dimension [L][K]-1, where L is length and K is Kelvin, and ηT has a length unit similar to the precipitation unit. Combining these terms, we can rewrite Equation (6) as

  GIF -- Equation 7: Rewrite of equation 6 (7)

Differentiating Equation (7) with T and P and writing the result in a finite difference form yields

  GIF -- Equation 8: Equation after differentiating equation 7 with T and P (8)

This equation shows effects of temperature and precipitation anomalies on changes of d and, hence, the Z-index. In Equation (8), change of d is linearly dependent on precipitation anomaly with the coefficient (1 - κ). The dependence of d on temperature anomalies is a nonlinear function of temperature itself. According to Thornthwaite (1948), η = 0.18 in the central United States. In a temperature range of 0-40C, T[a(I)-1] ranges from 0 to 5. In a more realistic temperature range of 20-40C for summer, T[a(I)-1] varies between 3.5 and 5. With these values, the coefficient of the temperature perturbation term in Equation (8) varies between 0.5 and 0.8. Finally, because (1 - κ) varies in a similar value range, the variation of d and, hence, monthly Z is about equally dependent on air temperature anomaly and rainfall anomaly. In other words, monthly contributions of temperature and precipitation anomalies to total PDSI are nearly equal when they have similar magnitudes of anomalies.

Equation (8) separates the effect of temperature and precipitation on the PDSI. It shows that positive anomalies of monthly rainfall will yield a positive d perturbation. If there coexists a negative anomaly in monthly temperature, a temperature effect will enlarge this positive anomaly of d. Together these will yield a positive value of Z and contribute to a positive PDSI. On the other hand, a large positive anomaly of temperature could override the effect of a smaller positive rainfall anomaly on variation of d to result in a net negative value of Z and cause a negative PDSI. Similarly, for a negative monthly rainfall anomaly, the Z-index of the month can still be positive if temperature is significantly cooler than the 'appropriate' temperature corresponding to 'appropriate rainfall' of the month.

The above analysis shows the effect of monthly temperature and precipitation anomalies on monthly Z-index values. The linear relationship of Equation (5) indicates that this effect will be linearly added over a dry period to determine the PDSI for the period.

The relationship between the PDSI and temperature and precipitation anomalies is reasonable for a drought monitoring index, because droughts develop from combined effect of both precipitation and temperature anomalies on soil water availability to plants. However, because the PDSI variations can be significantly affected by temperature anomalies and are not necessarily consistent with precipitation fluctuations, using the PDSI to interpret precipitation variations in past climates could lead to ambiguous conclusions, especially when variations of either precipitation or temperature are unknown.

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