Applications of solutions to non-linear energy budget equations

Agricultural and Forest Meteorology, 43 (1988) 121-145

121

Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands

A P P L I C A T I O N S OF S O L U T I O N S TO N O N - L I N E A R E N E R G Y BUDGET EQUATIONS

KYAW THA PAW U and WEIGANG GAO

Department of Land, Air and Water Resources, University of California, Davis, CA 95616 (U.S.A.) (Received July 27, 1987; revision accepted January 6, 1988 )

ABSTRACT Paw U, K.T. and Gao, W., 1988. Applications of solutions to non-linear energy budget equations. Agric. For. Meteorol., 43: 121-145. A fourth-order (quartic) and second-order equation is developed to solve the energy budget equation for latent energy flux density. The solutions are compared with conventional iterative techniques and the linearized Penman-Monteith form. The fourth-order and second-order equations are then used to show that the generally accepted expression for equilibrium evapotranspiration is theoretically incorrect, both for infinite aerodynamic resistance and for zero vapor pressure deficit. The new methods give new insights for the analysis of canopy-air temperature differences, interactions between canopy resistance and vapor pressure deficit, the Priestley-Taylor equation, and the Jarvis-McNaughton concept of atmospheric coupling.

INTRODUCTION

The Penman and Penman-Monteith equations for evapotranspiration (ET) and latent energy flux density have been used as predictive and diagnostic tools during the last quarter century. The beauty of these equations arises from their simultaneous solution of the energy budget and flux-resistance equations coupled with linear solutions which are easy to interpret physically. The linearity of the equations has been employed in the estimation of evapotranspiration, leaf temperature and water stress (Tanner, 1963; Verma et al., 1976; Verma and Rosenberg, 1977; Idso et al., 1981; Jackson et al., 1981; Hatfield, 1983; Paw U and Daughtry, 1984; Hatfield et al., 1984, Choudhury et al., 1987), estimation of evapotranspiration from satellite data (Carlson et al., 1981; Taconet et al., 1986) and analysis of the interaction between plants and the atmosphere (Idso, 1983; Jarvis and McNaughton, 1986; Choudhury and Monteith, 1986). Over a decade ago, Brown and Rosenberg (1973) reported an accurate iterative solution to the latent energy and energy budget equations, without the 0168-1923/88/$03.50

© 1988 Elsevier Science Publishers B.V.

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first-order Taylor linearization technique used in the original Penman and Penman-Monteith equations (also see Verma and Rosenberg, 1977). In the present paper we represent the saturation vapor pressure function as a quartk' (fourth-order) polynomial, and then use the analytically exact solution to the quartic equation. We also analyze the energy budget as a function of absorbed (incoming) radiation Ri instead of Rn. These procedures are used to analyze the theoretical accuracy of the Penman-Monteith formulation, the accuracy of other linearized relationships based on surface temperature-air temperature differences as functions of vapor pressure deficits and the implications of the quartic solution to some plant-atmosphere interactions (stomatal openingvapor pressure deficit relationships, canopy-atmosphere coupling, and Pries tley-Taylor evapotranspiration estimates). A second-order Taylor approximation is examined and compared to the Penman-Monteith first-order approximation. THEORY

It is well known that when the saturation vapor pressure function is linear~ ized, one can arrive at a relatively simple formulation of the latent energy flux density, LE, as a function of net radiation, Rn, ground heat flux, G, canopy resistance, re, aerodynamic resistance, ra, air temperature, Ta, and the vapor pressure, ea (Penman, 1948; Monteith, 1965, 1973)

LE_A(Rn-G) +pCp~e/r~ J+7(l+r,./r~ where ~e is the vapor pressure deficit defined as (es(Ta)-ea), e,( T., i is the saturation vapor pressure evaluated at air temperature, A is the derivative o~ the saturation vapor pressure function with respect to temperature, p is the air density, Cp is the atmospheric specific heat per unit mass, and 7 is the psy chrometric constant. Because the linearization used to obtain eq. 1 involves a Taylor approximation for the saturation vapor pressure function centered on the air temperature, T~

e~(T~)~e~(T~ )+A(T~-T~)

~2)

where T~ is the surface temperature, eq. 1 can be expected to have increasin~ error as the surface temperature departs from the air temperature. Such conditions might be expected with very low transpiration rates and high radiation levels, or under high transpiration rates. Another possible error can occur in eq. 1 if it is assumed that the incoming radiation flux Ri is a fixed quantity and the net radiation Rn is therefore a variable, being a function of R~ and the surface temperature, T~. For some theoretical evaluations of the energy budget, using R~ as a fixed entity would be more logical than using Rn. If R~ is assumed known, then eq. 1 becomes

123

L E - A (Ri - eaT4a - G) + pCpge/rp A+y(ra +rc/rp)

(3)

where the new terms are the resistance %, defined as the parallel resistance of the aerodynamic resistance ra and the so-called 'radiative resistance' rr, with rr = pCp ~4eaT a3, e is the emissivity, and a is the Stefan-Boltzmann constant. However, for cases where Rn is actually measured, the analysis should use Rn and not Ri because it is Rn that is now a fixed quantity, and the emitted long-wave radiation is included in the Rn measurement. Brown and Rosenberg ( 1973 ) combined the latent energy flux density equation with the energy budget using an iterative solution instead of the PenmanMonteith linearized formulation, and identified some of the errors that would occur from linearization. Because of the nature of iterative solutions, it was not clear that all of their solutions were the most physically realistic ones. The highly non-linear saturation vapor pressure function e~(T) can be approximated in many ways; the easiest methods are to use the Taylor approximation (eq. 2 represents a first-order Taylor approximation) or to fit a polynomial to e~(T). Formulations involving transcendental functions of temperature and inverse temperature have been used (Goff and Gratch, 1945; Tabata, 1973; Hull, 1974; Riegel, 1974; Wigley, 1974), in addition to sixth-order polynomials using temperature (Lowe, 1977 ). If these are applied to the energy budget, however, no well-known analytical solution exists to the transcendental functions and the sixth-order polynomials. Analytical quartic solutions are well known and can be used if the latent energy flux density can be formulated as a fourth-order polynomial. Paw U (1987) demonstrated that the saturation vapor pressure function could be fit by a fourth-order polynomial with acceptable accuracy (0.1% maximum error from 5 to 45 ° C; with most of the largest errors at the extremes of this temperature range) in the temperature range of common biological phenomena. Because this function is more accurate than eq. 2, energy budget analysis using the polynomial should be more accurate. Using Paw U's (1987) notation, the saturation vapor pressure function may be expressed as

e ~ ( Ts ) = ~ + o~T~ + flT ~ + ~ T ~3+ /~T ~4

(4)

where Ts is the surface temperature (this could be any temperature in a general saturation vapor pressure function formulation) and the Greek symbols are polynomial coefficients (see Table 1 ). This expression can be substituted into the gradient-resistance formulation for latent energy flux density (LE) and the general energy budget equations

LE

pCp ( e s ( T ~ ) - e , ) 7 re +ra

(5)

124 TABLE1 Table of coefficients ibr saturation vapor pressure curve ~c~= ]~= qJ= g=

6.174 4.222 1.675 1.408 5.818

(10=') (101) (10 °) (10 -~) (10 -4 )

Pa Pa°C Pa ~C Pa °C Pa°C"

H pCp(Ts-Ta)

~i

ra

LE=Rn-H-(;

"t?

where e~(T~) is the saturation vapor pressure at the surface temperature, and the other terms have been previously defined. Equation 6 is rearranged to solve for T~; then a form of eq. 7 is substituted for H in the rearranged eq. 6. This expression for 7"~ is then substituted into eq. 4 and finally eq. 4 is combined with eq. 5. This eliminates T~, allowing the equation to be solved for LE. The following quartic equation results

kLE4 +a'LE:~ +b'LE2 +c' LE +d ' = 0

(b

where

k= l(pC, a' = [411(Rn-G)r~/ (pCp ) +4~T~ + q/]/ (pC,/ra ):~

(Sa lSb

b' = [6~(Rn-G)2r~/ (pCp )2+ 6 T ~ p + 1 2 p ( R n - G ) Tara/ (pCp ) + 3~,Ta +3q/(Rn-G)ra/(pC, ) +fl]/(pCp/r a )2

(go

c' = [4p(Rn-G)3r2/ (pCp ):~+ 4#T~ + 12t~(Rn-G) T ~r~ / (pC,,) + 12,u( R n - G)2T~ r~ / (pCp)~ + 3 ~ ( R n - G)2r~ / (pCp)2 + 3q/T

+6@(Rn-G)r~T~/ (pCp ) + 2fl(Rn-G)r~/ (pC, ) + 2flT~ + o~+'/ir, + r~,)/r~ ]ra/ (pC~,)

8d

d' = # ( R n - G P r ~ / (pCp P + gT~ + 41~(Rn-G)3ra~ T~/ (pC, )~ + 4/~(Rn -G)r.Ta~/ (pC~, ) +6#(Rn-G)'2r~T~/ (pC, )'2+ ~(Rn a)ar~ / (pC.)a + q/T ~ + 3gc(Rn- G )2r2 T~ / (pCp)'~+ 3q/(Rn -

-G)r~ T2 / (pCp ) + fl(Rn-G)2r~/ (pCp )2 + flT~ + 2fl(Rn-G)r~, 1,,, (pC,, +o~ ( R n - G ) r ~ / ipCp i + ozT~ + ~-e~

(Be !

125

The general solution to quartic equations is given in the appendix. This solution will have four roots, which can be checked to see if they are real (usually some of the roots will be complex) and physically credible. For all of the variations of typical daytime values from dusk until dawn (dry, temperate and tropic humidities; low and high resistances, etc.), the same root was always real and physically credible. Thus, in practice, only this root need be calculated. An iterative solution such as that proposed by Brown and Rosenberg ( 1973 ) would converge around only one root, and it would not be clear if that root was the most physically realistic one. The quartic method has the advantage that all four roots are found so that the most realistic one may be chosen. If the net radiation term is decomposed into the incoming absorbed radiation Ri and the outgoing long-wave radiation, the above quartic equations can be modified. Unfortunately, an elegant direct substitution procedure such as that used for eq. 8 cannot be derived if Ri is used, because of the long-wave emission term eaTS. Instead, a two-stage equation may be used, where T~ is solved analytically with a quartic solution (Paw U, 1987) and then LE is obtained from T~ (eqs. 5 and 4 combined). A simpler method to estimate the latent energy flux density is to use a second-order Taylor approximation in place of the first-order one, and then carry out the same substitution process as used in the P e n m a n - M o n t e i t h equation. First, the saturation vapor pressure is approximated using a second-order Taylor series 1

d2es

e~ (T~)=e~ (T~) + A(T~- T~) + ~--~(T~-Ta) 2

(9)

where A is the first derivative ofes (T~) with respect to T~ and (d2es)/dT 2= d ~ / d T evaluated at Ta. T h e n the surface temperature T~ is eliminated in the same manner as in the P e n m a n - M o n t e i t h equation, to yield a quadratic equation for LE

aLEe +bLE +c=O

(10a)

with J d2e~

2

ra

( lOb )

a= ~ - ~ pCpy(r~ + r~ ) b= -1

r~A

d2es (Rn-G)r2.

y(ra+rc)

dT2pCpy(ra+rc)

ld2es r~(Rn-G) 2 raA pCp c=7(r~ +rc )ge4 ?(r~ +rc) (Rn-G)A 2dT2 pCpT(ra +rc)

(10c) (10d)

where ge is the vapor pressure deficit. This equation may be trivially solved

126

using the quadratic formula. The solution to eq. 10a should involve less error than the P e n m a n - M o n t e i t h equation, because the saturation vapor pressure function is approximated by a quadratic curve instead of a straight line If Ri is considered the fixed quantity instead of Rn, a quadratic equation similar to eq. 10 can be derived. A second-order Taylor approximation cannot be used for the long-wave emission term ~aT ~ if a direct substitution method is to be used; instead a first-order Taylor approximation for the long-wave emissions can be combined with eq. 9 to yield an equation with the same ibrm as eq. 10a. The quadratic coefficients a, b and c are now 2 d2es a=rp-~/[2ypCp rp

lOe,

2 d2es

,,(r~+r,,) r,~-~(R,-e~T~)/[TpC,(ra+r~)J

b=-i

c --

(r~ +re )]

rpzJ ?'(r a

+r,: )

(R~-eaT; ~,)+a(R~-6ffT4a)2-~ pCp6e

)'(ra +r~. )

10f, t log

In addition to being used for latent energy flux density equations, the same quartic and quadratic methods may be used for solving for surface temperature (Paw U, 1987 ). The quadratic solution for surface temperature has advantages over previous iterative methods used by Gates and Papian (1971), Tracy et al. ( 1984 ) and Bristow (1987) in two ways: ( 1 ) the tack of iterations can yield a faster computational method; ( 2 ) the solution may be differentiated and limits of solution may be readily determined. MATERIALS AND METHODS

The new equations (8 and 10a-d) for latent energy flux density were tested by using selected data of Pruitt et al. (1968), Stenmark and Drury (]970) and Morgan et al. ( 1971 ). These data were gathered over a well-watered rescue crop for a total of five days in 1966 (June 2, June 3, July 13, July 14 ) and 1967 (May 4). The Davis 6.1-m diameter lysimeters were used to measure the surf'ace stress (and thus the friction velocity) and the latent energy term LE. Net radiation was determined using Fritschen style net radiometers and by measuring radiation components. Ground heat flux density was estimated with Thornthwaite heat flux plates and soil thermocouples. Wind speed was measured at five heights above the ground with Thornthwaite sensitive-cup and Casella (three heights) anemometers; at these same heights the air temperature and humidity were determined with aspirated psychrometers. Leaf tern perature was directly determined with thermocouples on the leaves; the overall

127

canopy temperature, Ts, was radiometrically determined with a Barnes PRT5 infra-red thermometer. Data quality analysis and the experimental set up are discussed in Pruitt et al. (1968) and Morgan et al. (1971). A reliable sensible heat flux density, H, was determined from these data as a residual of the other energy budget terms. Although eddy flux measurements of H were also taken, these data sets were not as complete or reliable, so they were not used. Along with the friction velocity u* and mean temperature, the Monin-Obukhov length, L, could be estimated. With u* and L, an aerodynamic resistance for each data set could be estimated, using the similarity relationships for the transfer of sensible heat taken from Brutsaert ( 1982 ). This will be referred to as 'method I'. We also estimated ra, based on the energy-budget-derived sensible heat, H, measured Ts and measured Ta; we refer to this as 'method II'. Method II is circular in its definition of ra, because H is derived from the energy budget. A perfect mathematical solution will therefore yield absolute accuracy with method II, for any given data set. The canopy resistance rc was then estimated by using the latent energy flux density based on the lysimeter data, the vapor pressure measured at 2.0 m above the surface, the radiometric surface temperature, and the aerodynamic resistance, with the standard flux-resistance form,

re-

flCp [e~ ( T s ) - e a ] LE

ra

(11)

Errors of the linear Penman-Monteith equation were determined using these data, and compared to the quadratic and quartic errors. The same technique was used with ranges of input variables chosen to represent a wide range of possible daytime microclimates (net radiation ranging from 250 to 750 W m - 2; air temperature ranging from 10 to 40 ° C; vapor pressure ranging from 0 Pa to saturation conditions; canopy resistance ranging from 0 to 25 s m-1; aerodynamic resistance ranging from 5 to 50 s m - 1). The quartic and quadratic solutions were also applied to surface temperature-air temperature differences, to examine the widely-used method of relating this difference to vapor pressure deficit. The physiological implications of relating the canopy resistance to the vapor pressure deficit, as reported by Choudhury and Monteith (1986), were examined in the light of the more accurate higher-order polynomial depiction of latent energy flux density. Finally, the different solution methods were applied to the concepts of equilibrium evapotranspiration (Slatyer and McIlroy, 1961; Priestley and Taylor, 1972) and atmospheric coupling (Jarvis and McNaughton, 1986). RESULTS AND DISCUSSION

Accuracy of LE estimation The quartic solution yielded more accurate estimates of LE than the quadratic and Penman-Monteith (linear) equations, although the quadratic ap-

128

proximation was always within several percent of the quartic solution, based on the field data from Davis. Table 2 summarizes the root-mean-square ( r.m.s. error and mean errors associated with the different LE solutions based on an ra calculated with method I; the quartic method produced an overall r.m.s, error of 26.6 W m -2, compared to the quadratic method, which was nearly as accurate with a r.m.s, error of 27.3 W m -2. The linear P e n m a n - M o n t e i t h method had the greatest r.m.s, error of 35.3 W m-2. A similar analysis of the mean absolute errors again showed the quartic and quadratic equations to yield less error {20.9 and 21.5 W m -2, respectively) than the linearized P e n m a n - M o n teith formulation error (28.6 W m -2). The relative mean absolute errors e x pressed as percentages of the latent energy flux density were 6.5, 6.7 and 8.8%, for the quartic, the second-order Taylor approximation, and the P e n m a n Monteith formulation, respectively. An example of a diurnal profile is shown in Fig. 1. Table 2 demonstrates that when the field data were used in a consistent manner (r, estimated with method II), such that the quartic (or an iterative ! solution would exactly give the measured latent energy flux density, the quadratic equation showed much smaller errors ( < 1 W m - 2 } than the P e n m a n Monteith formulation (errors of 8-9 W m - ~ ) . Because the range of m e t e o r o logical variables occurring naturally in the field was limited to a semi-arid Davislike climate, a better general idea of the error was obtained by using ranges oi possible values. W h e n artificial ranges of the variables were used for the above comparison, the P e n m a n - M o n t e i t h equation yielded errors near 20% during some conditions when the surface temperature would be greatly different from the air temperature. The quadratic (second-order Taylor) approximation, on the other hand, generally exhibited errors of < 5% (see Fig. 2). Because the measurement error for the lysimeter, when compared to eddyTAB LE 2 Total errors between predicted and measured latent energy flux density

Method I Penman-Monteith Quadratic Quartic Method II Penman-Monteith Quadratic

Mean absolute errors

M e a n relative errors

R.m.s. errors

(Win :)

(%)

(Wm :~)

28.6 2 ! .5 20.9

8.8 6.7 6.5

35.3 27.,2 26.6

(i,6 ~).55

2.3 0.18

8.5 0.74

Sample ~umber~

Kt5 I ~)5 105

t04 ~t~4

129

400

300

'

>~ 200

100

\

,

. j

•f /

L

- - 4 - - 1 700 900

I

1100

'

I 1500

~

I 1500

'

1700

Time(hours) Fig. 1. Sample latent energy flux densities estimated by linear equation (~ - - ), second-order Taylor approximation (- - - ) and quartic equation ( - - ) compared to actual lysimeter measurements (©), as a function of time-of-day. Data taken on M a y 4, 1967 for a well-watered rescue crop.

o

a

°

~

o

_

o

-

~,,

o o , , . , , . ~ o o _

o

©

-

8

~

~

oo

~o~ ;,

2

o

-2°L 30 -12

-

o

o

-8

•

I

I

"4

0

4

(r s-Te)(°

i

I

8

i

!~--

~

6

C}

Fig. 2. Percentage errors of latent energy flux density equations as a function of canopy-air temperature difference. Quadratic estimates are denoted by triangles, and linear by circles. Estimates based on wide ranges of possible weather conditions (r a = 10-1000 s m - 1; rc = 0-1000 s m - 1; Ta = 20 40°C; relative humidity = 0 - 1 0 0 % ) . (a) For ( R i - G ) = 1000 W m -2. (b) For R n = 5 0 0 W m -2.

correlation and Bowen-ratio energy budget methods is 10% or less, the Penman-Monteith equation is reasonable for the real data tested, as expected. However, the artificial ranges of variables reveal the Penman-Monteith equation could yield errors significantly greater than expected measurement errors. In addition, the quartic and quadratic methods represent relative accuracy im-

130

provements of 25-30% over the P e n m a n - M o n t e i t h equation, for the cases when real data were used. Statistical tests, such as the Student's t-test, are not used to examine the relationship between the P e n m a n - M o n t e i t h , quadratic second-order Taylor. and quartic equations. This is because the equations are analytically differenl. and the errors of the P e n m a n - M o n t e i t h equation compared to the other e q u a tions are therefore true errors, as opposed to inferrential errors deduced directly from observational data sets, which usually involve statistical analysis. It might be argued that the numerous assumptions required by the flux resistance equations and the energy budget imply that the P e n m a n - M o n t e i t h equation is sufficiently accurate, especially if' T ~ - T , is <5~C. However, i~ many areas, including semi-arid and arid regions, irrigated crops exhibit canopy-air temperature differentials of 5°C or greater (see Idso et al.. 1981). in addition, the above analysis (modified suitably) can be applied to leaves, and many cases exist of leaf-air temperature differentials of 5 ~C or greater (Ehlers, 1915; Clum, 1926; Linacre, 1964; Smith, 1978; Sumayao and Kanemasu. 1979; Geller and Smith, 1982; Smith et al., 1983). One should also consider that in order to use the P e n m a n - M o n t e i t h equation, many variables such a~ r,: and ra must be estimated; once one has worked so hard to use this generai type of equation, why not use a slightly more complicated equation (the q u a dratic, second-order Taylor approximation) to obtain an answer up u~ 15~i more accurate? The quartic solution was careihlly tested during this run with a more conventional N e w t o n - R a p h s o n iteration technique similar to the method for warded by Brown and Rosenberg (1973), with the saturation vapor pressure function determined from the standard Goff-Gratch formulation; howeveL this iteration method yielded results that were found to match the realistit: root from the quartic solution within 0.1%. These two solutions are practically identical, so in the rest of' the paper no further comparisons are p r e s e n t e d T, - T~ relation~ Extensive work has been published in the last five years concerning the r c lationship between the canopy-air temperature difference and the vapor pressure deficit. The most interesting hypotheses concerning the leaf:air temperature difference are the somewhat empirical findings of a linear relationship (Idso et al., 1981; Idso, 1983 ) with the vapor pressure deficit and other theoretical analyses showing possible non-linearities in this relationship (Jackson et al., 1981; Paw U and Daughtry, 1984; Hipps et al., 1985). The relationship between leaf-air or canopy-air temperature differences and vapor pressure deficit was examined by either changing the water vapor pressure under constant air temperature conditions or changing the air temperature under constant vapor pressure conditions. These two variables were expected

131

to provide the most dramatic changes in T s - Ta; however, other energy budget variables clearly could influence the relationship between T s - T a and 6e, as described by Paw U and Daughtry (1984) and Hipps et al. (1985). We used the quartic, second-order Taylor and linearized (first-order Taylor ) methods (Paw U, 1987) to examine the relationship between the canopy/ leaf-air temperature difference and the vapor pressure deficit 6e. The analysis was based on both Rn and R i a s known input variables; little difference was apparent between the two methods, when approximately equivalent values were used. If the air temperature is constant, and the vapor pressure deficit variation is caused by atmospheric vapor pressure changes, the canopy/leaf-air temperature difference is linear with ~e only for the linearized solution (see Fig. 3a). The quartic solution and the second-order Taylor approximation yield a concave-down curve whether Rn or Ri is held constant. Under certain conditions, the concavity could be mild enough that the exact quartic solution curve would not be visibly different from a linear one for practical (experimental) applications. If the vapor pressure is constant and the air temperature is allowed to vary,

(o) 10

-20

(P 0I o

(D)

0 -10

I

GO F- -20

t

-

@

-100

L

1

2

,3

4

................ 5

6

Vapor pressure deficit(kpG) Fig. 3. (a) Canopy-air temperature difference as a function of vapor pressure deficit, when air temperature is constant and vapor pressure is allowed to vary, with ( R i - G ) = 1000 W m -2, approximately equivalent to (Rn-G) =500 W m -2. Ta--30°C, ra=30 s m-1, and rc=O. The linear model is denoted by (- - -), the second order Taylor approximation by (- - - ), and the quartic by ( - - ) . (b) Canopy-air temperature difference as a function of vapor pressure deficit, when vapor pressure is constant and air temperature is allowed to vary. ( Ri - G) = 1000 W m - 2, rc = 0.0 s m - 1, ea = 670 Pa, ra = 10 s m - 1. The linear model is denoted by (- - -), the second-order Taylor approximation by (- - - ), and the quartic by ( - - ) . (c) Canopy-air temperature difference as a function of vapor pressure deficit, when vapor pressure is c o n s t a n t and air temperature is allowed to vary, with canopy resistance d e p e n d e n t on canopy vapor pressure deficit. Conditions for a forest canopy ( R i - G) = 1000 W m-2, rco = 10 s m - 1 , Dr, = 7 kPa, ea = 670 Pa, r~ = 1 s m-1. The linear model is denoted by (- - -) and the quartic by ( - - ) .

132

changing the vapor pressure deficit, the curve is concave-up for the linearized case ( see Fig. 3b ). The exact quartic solution and the quadratic approximation both yield relationships with much lesser curvature, which could be indistinguishable from a straight line in some cases (see Fig. 3b). These results imply that the observed linear relationship between the leaf/ canopy-air temperature difference and the vapor pressure deficit is not the result of some special relationship between the canopy resistance or stomatal resistance and the net radiation R n or the absorbed radiation Ri, but rather simply due to the climatic conditions under which the experiments have been carried out. Proper analysis of the energy budget equations shows that onl? limited restrictions on some variables are needed to arrive at relatively linear curves. Relationships between stomataI resistance and deficit Recent work has combined linear P e n m a n - M o n t e i t h solutions with obser vations of stomatal response to leaf' vapor pressure deficit to infer canopy level response (Choudhury and Monteith, 1986). Our results were applied to the hypothesis that the surthce (in this case, canopy} resistance is a function ,~l the canopy vapor pressure deficit in this manner r,=r~,,(1-I)' /D'm)

!l-'~

where re,, is a reference value of canopy resistance based on crop type. I ) i ~ the canopy vapor pressure deficit (the vapor pressure at canopy temperature minus the vapor pressure in the air) and D ' m is a response parameter based on crop type. This relationship was substituted into the quadratic solution; a quartic resulted that was solvable by either the quartie solution method or by more traditional iterative methods, for D". The linearized P e n m a n - M o n t e i t h equation yielded a quadratic solution when a similar substitution ]'or r,. was made, which placed stringent restrictions on the possible real values of D ' m (Choudhury and Monteith, 1986). The analysis was made for both constant Rn and R~; no significant difference was found t'or consideration of LE as .~ function of vapor pressure deficit. The limiting value of D' m was labeled l)* m. This limit was not always confirmed with the quartic equation solution results ing from using the quadratic L E equation combined with eq. 12 above; in at least several eases D'rn values were found that were 40-70 Pa less than the D*m restriction derived from P e n m a n - M o n t e i t h theory. Of related interest was the relationship between canopy-air temperature difference and the air vapor pressure deficit ~e. Errors from linearization are apparent in the canopy temperatures up to large 3e, and the curves tend to have the same concave-up shape (Fig. 3c). It should be noted that the values given by Choudhury and Monteith (1986) for a crop canopy result in an unusual curve of the canopy-air temperature difference as a function of vapor

133

2

Q)

0

o

(s

2 Coniferous

I

fores:

4

2 ¸

\ -2

i

--

Vgpor pressure

2

5

4

deficit(kpa)

Fig. 4. C a n o p y - a i r temperature difference as a function of vapor pressure deficit, when air temperature is c o n s t a n t a n d vapor pressure is allowed to vary, with canopy resistance dependent on canopy vapor pressure deficit. T~ = 25 ° C. (a) For a n n u a l crops, with ra = 20 s m - 1, rco = 100 s m - i, Din=7 kPa. (b) For coniferous forest, with ra=5 s m -1, rco= 1000 s m -1, Din=3.3 kPa. Linear model is denoted by ( - - - ) a n d quartic by ( - - ) .

pressure deficit (Fig. 4a), the crop temperatures generally being greater than the air temperatures. Also, the canopy-air temperature differentials for the forest is almost independent of vapor pressure deficit (Fig. 4b). It is possible that the parameter values chosen by Choudhury and Monteith (1986) are not appropriate in this case. When the rco value for crops is decreased by an order of magnitude, the canopy-air temperature differentials are less than zero, as generally found by other researchers (Idso et al., 1981 ). It is interesting to note that the curve for the quadratic analysis is linear using the Choudhury-Monteith relationship between canopy resistance and canopy vapor pressure deficit. Canopy vapor pressure deficit control of canopy resistance becomes one possible explanation for some of the linearity observed experimentally for canopy-air temperature differentials.

Equilibrium E T and atmospheric coupling The linearized P e n m a n - M o n t e i t h equation has been used by many authors for theoretical analyses of evapotranspiration and plant ecosystem-atmosphere linkages. The quadratic and quartic equations were used to analytically and numerically determine the limits of estimated ET for such extremes as a saturated atmosphere (one form of equilibrium evapotranspiration, see Slatyer and McIlroy, 1961 ), a 'decoupled' plant-atmosphere system (with ra approaching infinity, another form of equilibrium evapotranspiration), and a

134 perfectly coupled plant-atmosphere system (ra approaching zero). With the linearized Penman-Monteith equation, based on Rn, LE for the first two forms of equilibrium evapotranspiration are identical, when the canopy resistance r approaches zero J LE=~-z-(Rn- G ~

i:~

The LE for ra approaching zero is

LE=PCP (~e )'r,:

14

where the LE in eq. 14 is sometimes called the 'imposed' LE. It should also be noted that the linear approximation insures that the actual LE value must lie between the two limits given above; i.e., the curve is monotonic. This allows the elegant analysis of McNaughton and Jarvis ( 1983 ), Jarvis ( 1985 ) and J a r vis and McNaughton (1986) for defining a coupling parameter ~, placing all surfaces within these two limits. In our analysis we modify eq. 3 of McNaughton and Jarvis ( 1983 ) to their .Q equation form eq. 10, but maintain the usual definition of de, instead of using their Dm. The quadratic (second-order Taylor) eq. 10a-d, with a given Rn may be readily analyzed for limits of high and low aerodynamic resistance r~, zero vapor pressure deficit, and for any possible minimum/maximum points with partial differentiation. The quadratic limit of LE for aerodynamic resistance approaching zero is the same as for the limit arrived at with the P e n m a n Monteith equation (eq. 14). However, the limit for LE with aerodynamic resistance approaching infinity, analyzed with the quadratic equation, is the available energy to the plant community, ( R n - G ) , which is quite different from the equilibrium LE derived from the Penman-Monteith equation (eq. 13 ). This result may be interpreted as the saturation vapor pressure term dominating at high r, due to the non-linearity of the saturation vapor pressure curve. As the surface temperature rises with increasing aerodynamic resistance, sensible heat is virtually shut off by the increased resistance while the saturation vapor pressure rises so quickly that water vapor flux is forced through the increased resistance such that LE matches the available energy. The Penman-Monteith model cannot predict this occurrence due to its linearized treatment of the saturation vapor pressure curve. A similar analysis of the quartic solution (based on Rn) for LE is difficult, but the limits can be estimated by plotting values of LE versus r~ for the quartic, quadratic, and linear solutions (see Fig. 5). Clearly, the limit at higher resistances is the available energy for all non-linear solutions. The quadratic solution is always close to the quartic root. The error between the linear and quadratic limits is the ratio, (J + 7)/A, which can be relatively large. The implication that arises initially is that the coupling factors of Jarvis and Mc-

135

600

(°)

400

~-.:

-

2O0

400

~ :'t--~.

8

~ ' - ~

200

........ ... (t,)

400

200

---"''-

~

\ -.

-1

0

1

2

5

\

-..

@i _] 4

5

Log(to) Fig. 5. Latent energy flux density LE as a function of the logarithm of the aerodynamic resistance r~. With constant (Rn-G)=500 W m -z, (-..) denotes linear model solutions, (--) denotes quartic and quadratic solutions (these are within a few percent of each other so they are not plotted separately); with constant Ri, (- - -) denotes linear solution and (- - -) denotes the quartic and quadratic solutions. (a) For Ta = 30 ° C, relative humidity = 50%, rc = 50 s m- 1, (Ri - G) = 1000 W m -2. (b) Same as (a), but relative humidity 95%. (c) (Ri-G) =860 W m -2, T.= 10°C, relative humidity--- 100%, and rc = 0 s m-1 (equilibrium potential ET). N a u g h t o n (1986) m u s t be q u a n t i t a t i v e l y m o d i f i e d to a c c o u n t for this error. N o t o n l y is t h e n o n - l i n e a r L E limit for high ra d i f f e r e n t t h a n t h a t p r e d i c t e d f r o m linear analysis, b u t t h e m i n i m u m L E is n o t usually t h e equilibrium evap o t r a n s p i r a t i o n level, a n d t h e curve is n o t necessarily m o n o t o n i c (Fig. 5a). A similar a n a l y s e s leading to r e m a r k a b l y d i f f e r e n t results m a y be o b t a i n e d for Ri t a k e n as a k n o w n variable in place of Rn. I n s t e a d o f a m i n i m u m , t h e curves of L E versus ra e i t h e r h a v e a m a x i m u m value or decrease with increasing ra (Fig. 5). T h e limiting value o f L E for ra a p p r o a c h i n g i n f i n i t y is n o w zero (this limit can be derived a n a l y t i c a l l y using eqs. 3 a n d 10a,e-g, a n d n u m e r i c a l l y for the iterative a n d quartic m e t h o d s ), because b o t h H a n d L E are s h u t off b y i n c r e a s i n g 'decoupling' of t h e a t m o s p h e r e , with t h e o u t g o i n g long-wave radiation e v e n t u a l l y b a l a n c i n g Ri ( r a d i a t i v e e q u i l i b r i u m ) . T h i s case differs f r o m t h a t a s s u m i n g a c o n s t a n t Rn; for c o n s t a n t R n as t h e surface t e m p e r a t u r e increases, the Ri m u s t c o m p e n s a t e a n d increase such t h a t radiative equilibrium ( R n = 0 ) c a n n o t be achieved. At r~ a p p r o a c h i n g zero, eq. 14 m a y be derived a n a l y t i c a l l y f r o m eqs. 3 a n d 10a,e-g or n u m e r i c a l l y with t h e quartic a n d iterative m e t h o d s ; this lower limit is t h e same w h e t h e r one considers Rn or R~ as a given variable. A relative m a x i m u m c a n exist b e c a u s e o f t h e u p p e r limit of L E = 0; at low r,, L E will increase w i t h a n i n c r e a s i n g ra. T h e rise in t h e surface t e m p e r a t u r e with i n c r e a s i n g r~ creates a v e r y large rise in t h e s a t u r a t e d v a p o r p r e s s u r e at t h e

1'36

surface, which overcompensates the increased resistance. However, as the increased r~ continues to cause surfhce temperature rises, the long-wave emissions begin to become a dominant part of the energy balance, sufficiently so, that the saturation vapor pressure rise no longer compensates for the increased Our results may be analyzed in the light of Parkhurst and Loucks t -t972 and Smith and Geller ( 1980 I. in which leaf size was examined from an ecological perspective. A minimum in transpiration was found using an iterative s~.lution method, with respect to changes in leaf size and thus r~ {Smith and Geller, 1980). This analysis raises the issue that leaves might either evolve larger r~ (and thus sizes) or smaller sizes for low LE under certain condition~ when a relative maximum of' LE exists (see Fig. 5b). The above results imply that although the linear equilibrium evapotranspir ation equation is simple and easily interpreted, it is theoretically incorrect. The coupling factor ~2 (Jarvis and McNaughton, 1986) is not an ideal conceptual framework for analysis of plant biometeorology, because the LE from a plato community does not have to lie between the linear equilibrium evapotranspiration LE (eq. 13) and the imposed linear (eq. 14) limit; rather the LE wilt lie between the minimum LE possible as calculated by non-linear models and the upper limit if Rn is constam. At if2 close to 1, the linear ~2 factor model can be greatly in error compared to the more accurate solution methods (see Fig. 6). For one case the linear

{,

Fig. 6. (a) L a t e n t e n e r g y flux d e n s i t y as a f u n c t i o n of t h e c o u p l i n g p a r a m e t e r 62, for (R n - G ) = 501) W m - 2, T~ = 30 ° C, relative h u m i d i t y = 50%, a n d rc = 90 s m - 1 L i n e a r r e s u l t s i n d i c t e d by ~- - ), a n d q u a d r a t i c by ( - - ) . ( ...... ) i n d i c a t e s s a m e c o n d i t i o n s as above, linear results, b u t (Ri - G ) = 500 W m -e, a n d ( - - - - ) i n d i c a t e s q u a d r a t i c r e s u l t s w i t h ( R i - G ) = 5 0 0 W m -2. T h e q u a r t i c i t e r a t i o n c u r v e s are w i t h i n a few p e r c e n t of t h e q u a d r a t i c c u r v e s a n d are n o t s h o w n . ( b ) R e l a t i v e errors h)r above, n o r m a l i z e d to L E , w i t h s a m e k e y code for lines. Relative error d e f i n e d as ( L E . . . . . - L E q . . . . . ) / LE~u~rtlc. A d d i t i o n a l line, ( O O ) , is for a large l i n e a r e r r o r case w i t h ( R n - G ) = 2 5 0 W m "' T,, = 30 ~ C, 5e = 0.0 Pa, a n d r~ = 5.0 ~ m ~,

137

model can be made to have the lower and upper L E limits equal, with judicious choice of the atmospheric and surface variables; a coupled and decoupled system would give the same L E according to the Jarvis and McNaughton ( 1986 ) model. However, the more accurate L E solutions show a 100 W m -2 difference between the lower and upper limits. In another case (Fig. 6), the linearized theory has the LE limit for low ra (small ~2) as higher than the limit for high ra (i.e., the perfectly coupled canopy has a higher L E than a decoupled canopy); however, the non-linear solutions show the coupled canopy has the lower L E with the completely decoupled canopies having a much higher L E (by 100 W m - 2). Also, the linear ~ model has a range of approximately five times smaller than the higher-order models. One may argue that the value of ra for minimum LE is relatively large (of the order of 102 s m - ~corresponding to ~2> 0.80), such that realistically plant communities reach their minimum (or in some cases maximum) L E at this practical upper limit. If this is so, the concept of a completely 'decoupled' canopy should not be used at all, and should be replaced with a 'practically decoupled' canopy. At this practical limit, LE can be quite different from 'equilibrium' LE. Also, the same analysis used for communities has been applied to individual leaves, with minor modifications for a two-sided leaf surface and possible differential resistances (McNaughton and Jarvis, 1983; Jarvis and McNaughton, 1986). Under such conditions the higher ra and ~ values noted above are possible. For example, leaves with dimensions greater than 10 cm will often exhibit ~2 > 0.8 (Jarvis and McNaughton, 1986). Under low wind conditions, leaves may have aerodynamic resistances of 350 s m-1 (Slatyer and Bierhuizen, 1964). Furthermore, conditions can exist when L E errors are significant at ~2as low as 0.5 (Fig. 6). These cases indicate that caution is necessary when applying the linearized ~2 concept of decoupling, especially when ~ > 0.8. Similar results arise when Ri is considered to be known or constant instead of Rn, with increasing error of the linear analysis at high r a and high ~2. At high ~2, the error between the linear solution and the non-linear solutions can be significant (Fig. 6) for certain conditions, and is always large as ~2 approaches 1.0. The difference between the non-linear solution for Ri constant and the linear omega analysis (with Rn constant) is very large at high ~ (Fig. 6). This indicates problems with the linear ~2 analysis for application to agricultural crops, grasses, or any surfaces with relatively high ~. The results for Ri confirm the great accuracy of linear analysis at low ~, such as for forests, because of the Ri and Rn solutions, non-linear and linear alike, all having the same low ra limit (eq. 14). In most of the analyses discussed above (for example, the canopy-air temperature differentials), the use of Ri instead of Rn does not greatly change the results. However, as shown in Figs. 5 and 6, the curve shape of L E as a function of r~ or ~2 is significantly different at high values of the independent variable,

138

indicating for this form of analysis that careful choice must be made of whether Ri or Rn should be considered 'constant'. It could be argued that even R~ does not remain constant if r~ is changed because an increase in emitted long-wave radiation will result in a somewhat compensating increase of downward longwave radiation due to the greenhouse/atmosphere effect, but such details are beyond the scope of our analysis. Another interesting result occurs at the limit for zero vapor pressure deficit and rc = 0; the linear P e n m a n - M o n t e i t h equation yields the same equation as for the infinite r~ limit (equilibrium evapotranspiration). The limit for zer(~ vapor pressure deficit de, using the quadratic solution, is

L E = ( R n _ G ) 4 . pC~,

A

pCp'/(rc+r~)

)'(r,.+ra)

p(2~.i~

L r.~(~-~) ),(r, + r.~ )

dT 2

which is different from the non-linear solutions' limits of infinite r,~by the tour terms to the right of ( R n - G ) . An interesting dependence is predicted by the non-linear eq. 15 for zero de and re; the aerodynamic resistance r, is important in determining the value of LE, unless the value of ra is such that LE is at the relative minimum (see Fig. 5c). Linear theory predicts equilibrium LE inde pendent of G. Equilibrium LE for a zero vapor pressure deficit and re.--0 is similar to eq. 13 if one uses a first-order Taylor approximation (eq. 3) for given Ri instead of Rn, except now the radiative resistance can enter if significant

LE-

A(Ri -~aT4a ) 7(1 + r~/rr ) +,:]

(16~

If radiative resistance r~ is much higher than r,, then eq. 16 is identical to eq. 13 except for the radiation term being R~- eaT 4 in place of Rn. However, if the second-order Taylor (quadratic) approximation is used (eq. 10a,e-g), the s o lution is not independent of r~ as it is with eq. 13 (or eq. 16 if r~ is much greater than r~). The quadratic solution for LE steadily decreases as r~ increases ( for example, at T~ = 30 ° C, r~ = 10 s m - ~, the LE is ~ 562 W m - 2; at r~ = 200 s m LE drops to 507 W m -2). The iterative and quartic solutions for LE confirm the variation of LE with r,, and usually differ from the quadratic solution by under 10%. Equation 15 lends itself to an analysis of the sensitivity of canopy E T to canopy resistance (presumably influenced by stomatal opening). Jarvis and M c N a u g h t o n (1986) concluded that canopy E T is quite insensitive to canopy resistance when t2 is large. Table 3 summarizes the results from a partial dif-

139 TABLE 3 Comparison of LE sensitivity to canopy resistance as estimated by linear and quadratic methods ( R n - G ) =500 W m-2; Ta=30°C, ra= 10 s m -1 Relative humidity = 0% rc (sm-~)

OLE rc Ore LE - ( 1 - t 2 )

OLE rc arc LE =quadratic

0.01 0.10 0.20 0.40 0.75 1.00 2.00 5.00 10.00 100.00

-2.2 -2.2 -1.1 -4.3 -1.6 -2.1 -4.2 -9.8 -1.8 -6.8

-2.7 -2.6 -1.1 -3.4 -7.7 -9.1 -1.3 -2.1 -2.8 -6.9

(10 -~) (10 -4) (10 -:~) (10 3) (10 -2 ) (10 -2 ) (10 -2 ) (10 -2) (10 -1) (10-')

(10 -4) (10 -3) (10 -2) (10 -~) (10 -~) (10 -2 ) (10-') (10 -~) (10-') (10 -1)

ferential analysis of the P e n m a n - M o n t e i t h equation of the Jarvis and McNaughton ( 1 - ~ ) form and of the more accurate quadratic equation. Under absolutely dry conditions and low canopy resistances, the 'real' sensitivity is over 10 times greater than that estimated using linearized theory. At moderate relative humidity (50%), the sensitivities are more equal with the error being only 26% at low canopy resistances. These results imply stomatal function and canopy resistance has more impact on E T when t2 is close to 1 than indicated by Jarvis and McNaughton (1986). Fortunately, even though the sensitivity is over 10 times higher at low values of re, the magnitude remains small (10-2 to 10-4). The concept of equilibrium evapotranspiration is also used in the PriestleyTaylor empirical formulation, which has a factor ~ times the equilibrium transpiration. Well-watered crops and forests can thus be expected to have ~ > 1, with many authors reporting values ranging from 1.0 to 3.6, but most frequently averaging near 1.26 (Priestley and Taylor, 1972; Wilson and Rouse, 1972; Davies and Allen, 1973; McCaughey, 1985; Choudhury and Idso, 1985; Singh and Taillefer, 1986). Soils and lakes also exhibited a similar range (Stewart and Rouse, 1977; Marsh et al., 1981; Walker, 1984; Steenhuis et al., 1985; Roulet and Woo, 1986; Matthias et al., 1986). These reported ~ values may be analyzed with two methods. In the first, the minimum value of latent energy flux density, as a function of ra, may be estimated using the second-order Taylor approximation solution for LE. Our results show that the difference between the minimum L E and the linear equilibrium L E is a function of air temperature; the lower the air tem-

14)

perature, the higher the percentage difference. At 5:C, the minimum LE is over 30% greater than equilibrium LE; at 20 ~C, 20% greater, and at..~0 ~C. tO?i greater. Table 4 summarizes estimated ratios of the minimum LE and the linea~ equilibrium evapotranspiration, based on estimated temperatures. The atmo sphere was assumed to be completely dry. There is general agreement betweer. these ratios and cc The implication is that many surfaces, under welt-watered conditions, have evapotranspiration rates similar to the minimum possible un der completely dry atmospheric conditions. In this application, analysis using: R~ as a known variable is not considered because the Priestley-Taylor concep~ is generally applied with a measured or estimated Rn. The second torm of analysis utilizes all of the data required tot the energy budget analysis (i.e.. net radiation, aerodynamic and surface resistances, hu TABLE 4 ( : o m p a r i s o n of r e p o r t e d (~ with ratio LE,,,,,,,LEo,~ Source

•~

LEIn~,,/ LE~,

Comme~-

Wilson a n d Rouse ( 1972 }

1.25

1.17

Well-watered maize {corn I

Davies and Alien ( 1973 )

1 . 2 2 i.29

l. 11 - 1.26

Well-walered sweel

maize (COPII i Choudhury and Idso

1 . 0 3.6"

1, l 4 - 1.18

Well-watered wheat

McCaughey ( 1985

1.5tl

] .21

Mixed f(,res~, wet canop~

Singh and Taillefer ( 1986 !

] .41

[.26

Black Spr~lce fl ~res~

Marsh et al. ( 1981 )

1,26

1.26-1.34

Bare s,~iL

Walker ( 1984 )

1.1 --2.44

1.15- 1.24

Soil beneal h crop

Steenhuis et al. ( 1985 )

1.68 1.24

1.3-1.4 1.18-1.26

1.10

1.10-1.13

W i n t e r NY soils S p r i n g NY soils S u m m e r NY soils

Matthias et al. (1986)

1.1 1.9

1.14-1.20

Bare soil

Roulet a n d Woo ( 1986 )

1.3

1 . 3 1.4

Permafrost

Stewart and Rouse ( 1977 )

1.26

1.2-1.3

High latitude lakes

( 1985 )

aEarliest morning observations omitted because of possible net radiation measurement errors.

141

midity, etc. ). However, the literature generally does not describe these data in sufficient detail to carry out this analysis properly. The above results lessen the need to derive expressions for a based on linearized energy budgets (Barton, 1979; De Bruin, 1983; Choudhury and Idso, 1985; Mawdsley and Ali, 1985 ), because of the inherent errors of linear analysis. The ability to include canopy resistance response to radiation and vapor pressure within non-linear solutions decreases the validity of the criticism of Morton (1975, 1984, 1986a,b) and reduces the need for alternative formulations (Brutsaert and Stricker, 1979) for evapotranspiration equations. CONCLUSIONS

We have developed and verified a quartic solution to the energy budget and confirmed that a second-order Taylor approximation is more accurate than the traditional linear P e n m a n - M o n t e i t h equation. The second-order Taylor (quadratic) solution has been shown to be simple, with the advantage of being rather easily differentiable and simply analyzed when compared to the quartic. It thus represents 'the best of both worlds'. The greatest importance of the more accurate analysis presented here is not in the reduction of error when linearized solutions are used for general agricultural or forest meteorology, but in the changing of the concepts used for evapotranspiration analysis and in the serious modification of hypotheses and models based on the linearized equations. When applied to the relationship between foliage-air temperature difference and vapor pressure deficit, the more accurate energy budget solutions show that both concave and convex-up curvature is possible, along with curves that are almost linear; previously developed explanations for the linearity of the curve are not needed because they were based on the linear approximation to the energy budget equation. Some modification was also found for applications of the energy budget to latent energy flux density as controlled by the stomatal response to canopy vapor pressure deficit. The ubiquitous equilibrium evapotranspiration equation, based on linear theory, has been shown to be in error when compared to the more accurate quadratic result. The concept of plant canopy-atmospheric coupling, when quantified by t h e / 2 factor, must be carefully examined in light of the more accurate results presented here, especially under decoupled conditions (typical grasses and crops). It is hoped that greater caution will be displayed in the future when applying linearized theory to conditions under which the linear approximation becomes poor, but it should also be realized that linear theory has provided a comprehensible and practical framework with which to analyze energy budgets. ACKNOWLEDGMENTS

This research was partially funded by the Research Support Program of the IBM Palo Alto Scientific Center, the Water Resources Center of the University

142

of California, and a Faculty Development Award from the University of Cali fornia, Davis. We gratefully acknowledge the word processing help of Ms. K. Bigbee, Ms. C. Felsch and Ms. S. Cummings and the constructive comments on an early draft from Drs. D. Baldocchi, L. Hipps and T.P. Meyers. Corn ments from two anonymous reviewers also helped greatly. Shuguang Wang checked some of our derivations. APPENDIX (Selby, 1967

Given the quartic for L E k L E 4 + a' LE:~ + b' LE~ + c L E + d ' =l)

~A ~ ~,

one may derive a solution tbr y, the resolvent cubic y:~-by2 + ( a c - 4 d ) y - a " d

+ 4bd-c~=O

I A2

where a=a'/k, b=b'/k, c = c ' / k and d=d'/k. This yields two complex ro()t~ and one real root; only the real root to y is used. The quartic roots are then ~A:~ ;,

LE= -a/4-R/2+E/2

( A3a i

L E = - a / 4 + R / 2 +_I ) / 2

where E = [a 2 / 2 - y - b -

(4ab-Sc-w

~)/4R] ~/2

t A3b~

D= [a2/2- y-b+

( 4 a b - 8 c - a : ~ ) / 4 R ] ~/~

tA;k: i

and R = ( a 2 / 4 - b + y ) ~/,e

~A3d :,

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