The thermal environment of the citrus rust mite

Agricultural Meteorology, 20(1979) 411--425 © Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands

411

THE THERMAL ENVIRONMENT OF THE CITRUS RUST MITE*

JON C. ALLEN and CLAYTON W. MCCOY

University of Florida, Institute of Food and Agricultural Sciences, Agricultural Research and Education Center, Lake Alfred, FL 33850 (U.S.A.) (Received March 13, 1978; accepted September 19, 1978)

ABSTRACT Allen, J. C. and McCoy, C. W., 1979. The thermal environment of the citrus rust mite. Agric. Meteorol., 20 : 411--425. The distribution of citrus rust mite on individual fruit and in the whole citrus tree suggests an avoidance of solar exposure. A model of fruit temperatures as affected by solar radiation in different parts of the tree was constructed, and its predictions are compared with field temperatures and rust mite distribution patterns. The fruit temperature model is a function of latitude, time of year, time of day, fruit diameter and reflectance, position in the tree, atmospheric transmissivity and wind velocity. Where model predictions indicate unfavorable conditions, rust mite population levels were low. When the tree was divided into north--south, top-bottom quadrants, the north bottom quadrant had the most favorable temperatures and usually the most rust mites; the south bottom was also favorable and had high mite densities. The south top quadrant was least favorable often having temperatures in the lethal range, and it also had the lowest rust mite populations.

INTRODUCTION P o p u l a t i o n m o d e l s o f e c t o t h e r m i c organisms always d e p e n d heavily o n t e m p e r a t u r e to p r e d i c t d e v e l o p m e n t rates. M u c h has been said a b o u t the w a y in w h i c h t h e d e v e l o p m e n t a l r a t e ' d e p e n d s u p o n t e m p e r a t u r e (Stinner et al., 1 9 7 4 ; Allen, 1 9 7 6 ; Sharpe a n d D e m i c h e l e , 1 9 7 7 ) . In the case o f a t i n y o r g a n i s m such as the citrus rust mite, P h y l l o c o p t r u t a oleivora (Ash.), living in citrus trees, the p r o b l e m o f m i c r o c l i m a t e variation is as serious as the developm e n t rate p r o b l e m . While d e v e l o p m e n t rates m a y be p r e d i c t e d a c c u r a t e l y f r o m t e m p e r a t u r e , t h e t e m p e r a t u r e e x p e r i e n c e d b y t h e mites can be v e r y d i f f e r e n t f r o m t h a t m e a s u r e d in a w e a t h e r shelter. Several a u t h o r s have n o t e d t h a t rust mites on citrus fruit t e n d t o avoid the bright sunlit side in f a v o r o f semishade ( H u b b a r d , 1 8 8 5 ; Y o t h e r s and Mason, 1 9 3 0 ; Albrigo a n d M c C o y , 1 9 7 4 ; Van Brussel, 1 9 7 5 ) . O n t h e o t h e r h a n d , s h a d e d groves and t h e s h a d e d side o f f r u i t do n o t e x h i b i t mite densities as * Florida Agricultural Experiment Stations Journal Series, No. 1050.

412 high as semishade areas (Yothers and Mason, 1930; Muma, 1970; McCoy and Albrigo, 1975). Several hypotheses are suggested by the aggregation of mites around sunlit areas of which the following seem most plausible: (1) the aggregation is due to a temperature gradient; (2) the aggregation is due to a moisture gradient of either dew or water vapor; (3) some combination of these two hypotheses. The picture is further complicated by the fact that the rust mite is attacked by a fungal pathogen, Hirsutella thompsonii (Fisher et al., 1949; McCoy and Knavel, 1969) whose infectivity is dependent on the presence of free water and high h u m i d i t y (McCoy, 1978). It is our purpose here to examine the effects of temperature on the distribution pattern of rust mites in the tree. This is done by developing a model of fruit temperatures as a function of latitude, time of year and position in the tree. The model's predictions are then compared with temperature measurements in the field and with data on rust mite population dynamics from different parts of the citrus tree. A MODEL OF TEMPERATURES ON CITRUS FRUIT We wish to compare surface temperatures at different positions on individual fruit, at different positions in the tree and at different times of year. This means that the model must include the effects of solar elevation and azimuthal angles as well as the heat balance on individual fruit. Our mathematical notation will be as shown in the Notation.

Energy balance on individual fruit Equations for the dynamic energy exchange for irradiated citrus fruit have been obtained by Poppendiek (1953), but for simplicity we have adopted the equilibrium state equations of Thorpe (1974) and particularly those of Smart and Sinclair (1976). At equilibrium the energy balance can be written as: Energy absorbed = Energy lost by + Energy lost by + Energy lost by from radiation convection transpirational long wave cooling radiation

(1)

The last two terms, transpirational cooling and long wave radiation, wilt be ignored as they were by Smart and Sinclair {1976}. Their arguments show that these quantities are normally quite small compared to convection losses for grapes under field conditions. Citrus fruit (Valencia orange) has a water vapor resistance comparable to (Albrigo, 1977) or perhaps even greater than (Nobel, 1975} the grapes used by Smart and Sinclair. Thus, the transpirational cooling should be no greater on citrus than on grapes. On the input side, only direct beam solar radiation will be considered. For the temperature distribution at equilibrium within the spherical fruit, we can use the fact that the rate of change in the temperature gradient in any

NOTATION a

a l b e d o ( r e f l e c t a n c e ) of spherical f r u i t surface ~ 0.6 for Valencia oranges (Gaffney, 1973)

A

area o n t h e surface o f spherical tree

m

AA

small finite e l e m e n t of tree surface area

m2

day

J u l i a n day n u m b e r (1 = Jan. 1 )

day

D

d i a m e t e r of spherical f r u i t

m

h

convective heat transfer coefficient

Wm-1 °C-I

H

solar h o u r angle m e a s u r e d f r o m solar n o o n

radians

Io

solar b e a m i n t e n s i t y n o r m a l to f r u i t surface

Wm-2

IoA

average solar b e a m e n e r g y f l o w r a t e t h r o u g h a n area, A, o f t h e tree

W

I 0 QUAD

average solar b e a m f l u x t h r o u g h a p a r t i c u l a r tree q u a d r a n t

Win-2

2

u n i t v e c t o r s in t h e n o r t h , east a n d vertical d i r e c t i o n s ks

t h e r m a l c o n d u c t i v i t y of t h e f r u i t (~--0.46 W m -1 ° C - l ) ( B e n n e t t e t al., 1 9 7 0 )

Wm-10C-1

ha

t h e r m a l c o n d u c t i v i t y of air ( 0 . 0 2 5 5 W m -1 ° C - 1 )

Wm-1 °C-1

n o r m a l v e c t o r to t h e tree surface distance from center of fruit

m

radius o f f r u i t

m

solar b e a m v e c t o r

Win-2

solar c o n s t a n t ( 1 3 6 0 Wrn-2)

Wm-2

t

hour of the day

hr

T

e l e v a t i o n o f fruit t e m p e r a t u r e a b o v e a m b i e n t

°C

Tcen

t e m p e r a t u r e e l e v a t i o n a b o v e a m b i e n t at f r u i t c e n t e r

°e

Tmax

temperature elevation above ambient at ~ = 0 and r = R ("hotspot")

°C

Train

t e m p e r a t u r e e l e v a t i o n a b o v e a m b i e n t at/~ = 1 8 0 ° a n d r = R

°C

U

wind velocity

m sec

R

So

-1

solar a z i m u t h angle m e a s u r e d clockwise f r o m N o r t h

radians

angle b e t w e e n a p o i n t in or o n t h e f r u i t a n d t h e solar b e a m

radians

7

angle b e t w e e n t h e solar b e a m a n d a n o r m a l v e c t o r t o t h e tree surface

radians

5

solar d e c l i n a t i o n angle: (--) w i n t e r , (+) s u m m e r

radians

A

b o u n d a r y layer t h i c k n e s s

m

c

solar e l e v a t i o n angle

radians

l a t i t u d e : (+) N o r t h , (--) S o u t h (-~ + 0 . 4 9 r a d ( 2 8 ° N ) at our latitude)

radians

e l e v a t i o n angle in t h e tree c o o r d i n a t e s y s t e m

radians

0

a z i m u t h angle in t h e tree c o o r d i n a t e s y s t e m

radians

P T

r a d i u s o f spherical tree

m

zenith atmospheric transmissivity

4l 4 direction is zero, i.e. : V2T = 0 where

(2) 2 is the Laplacian operator. The solution to eq. 1 can be written as: --/°(-1--a)fe0s~--"

Ar(r, f l ) -

2(ks+hR)

R'I

V

(2n + 1)Pn(O)Pn(cOsfl)

I' r ~"

-o , , . - - a ) 7-0 2(n+2i(n--1}(nks+hR)~R-! (3)

after Thorpe (1974) and Smart and Sinclair (1976). Pn(O)and Pn(cos fl) are Legendre polynomials of degree n, and n takes only even values in the summation, r is the distance from the center of the fruit, and fl is the polar angle between a point and the solar beam. By assuming a linear temperature gradient along the solar beam, Smart and Sinclair (1976) were able to use the energy balance equation (1) to derive simplified expressions for the increase above air temperature at the center, hotspot and backside of a spherical fruit. These expressions are: Tcen = I 0 (1 -- a)/4h

(4)

Tmax=Io(1

a)(ks + 4hR)/{(Rh + ks)4h}

(5)

Tmin = I0(1

a)ks/{(Rh + ks)4h}

(6)

The heat transfer coefficient, h, is calculated from:

h = ka/A + 2ka/D

(7)

(Nobel, 1975; Smart and Sinclair, 1976) and the boundary layer thickness, A, (in meters) adapted from Nobel's eq. 4 is obtained from: D A : 2-8~.87~vD}~:~

(8)

where ka is the thermal conductivity of air (0.0255 Wm -1 °C-1), D is the diameter of the sphere (m) and v is the wind velocity in m sec -~ . We have found t h a t computer calculations of Tmax using eq. 5 and the exact solution (eq. 3) for 6 cm diameter oranges involved disagreements of less than 1%. Smart and Sinclair (1976) also f o u n d that eqs. 5 and 6 gave values very close t o eq. 3. For this reason, we have used eqs. 5 and 6 to describe the range of temperatures on the surface o f citrus fruit as a function of I0, the solar beam intensity.

Average energy flux in a tree quadrant The intensity o f the solar beam, I0, in eqs. 4, 5 and 6 is dependent on several variables. Some of these, such as latitude, time of year, time of day and position in the tree can be easily predicted or measured with considerable accuracy. Others such as cloudiness and wind velocity may have predictable

415 S.

/~/

S?U~H/AAf ~ P

SOUT~ BOTTOM

N

NORTH

EAST

Fig.1. Spherical s h a p e used t o r e p r e s e n t a citrus tree. P o r t i o n m o r e t h a n 35 ° b e l o w c e n t e r is n o t used. H e a v y lines i n d i c a t e q u a d r a n t d e s i g n a t i o n s , ~ a n d ~ are a z i m u t h a n d e l e v a t i o n angles at the c e n t e r of t h e area e l e m e n t AA.

effects but be entirely whimsical in time and space. As a starting point for an energy flux model, some sort of citrus tree geometry must be assumed. We have used two different shapes to represent citrus trees: (1) spherical and (2) icosahedron (virus particle-like polygon having t w e n t y equilateral triangular faces). Both of these shapes gave virtually the same results with the sphere having some computational advantages. For this reason we utilize the spherical model t h r o u g h o u t as represented in Fig. 1. The complete sphere is n o t used, only that portion down to a plane 35 ° below the center since this seemed to approximate the observed profile of most citrus trees. The " s o u t h t o p " quadrant is bounded on the north by the east-west plane and on the b o t t o m by a plane 10 ° above center; the " s o u t h b o t t o m " quadrant is bounded on the north by the east--west plane and on the top and b o t t o m by the +10 ° and --35 ° planes, respectively. The north top and b o t t o m quadrants are analogously defined. The total direct beam solar energy flow rate through any area, A, of the spherical tree is given by the integral of the dot product between the solar and normal vectors taken over the area, i.e.: I0 A

= f f S " N dA

(9)

A

This integral is difficult to evaluate analytically because of complicated limits involving both the time varying solar terminator and the fixed quadrant

416

boundaries. We have, therefore, chosen to use the approximation:

IOA -~ ~ S "

Ni~Ai

A

or:

Io A ~

ISI IN I cos TiAAi

(10)

A where ~ A i is a (small) finite area element and 7i is the angle between the solar and normal vectors at the " c e n t e r " of this element. Computer evaluation of eq. 10 is relatively easy since cos 7 becomes negative for 7 > 90°, and: these negative parts can be excluded from the summation thus avoiding the shaded part of the tree. In order to evaluate eq. 10 we must first know 7, the angle between the solar vector, S, and the normal vector, N, at any point on the spherical tree surface (Fig.l). If the finite area element A A i is b o u n d e d by the spherical coordinate angles Oi, 0_i+1 and ¢i, ¢i+1 and has average coordinates_ _ -Oi = (0i + 0i+1)/2 and ¢i = (¢i + ~i+1)/2, then the normal vector at 0i, ¢i can be written as: Ni

~

-"

~ l = i cos ~i sin 0i +J cos ¢i cos 0i + k sin ¢i

(11)

where, for the moment, J/VI will be taken as 1. In the same coordinate system (Fig.l), the solar vector can be written as: S

Isl

icosesina+

cosecosa+

sine

(12)

where, 181 will also be taken as 1 and a and e are the solar azimuth (measured clockwise from north) and elevation angles. If we momentarily shorten the notation of 11 and 12 to:

N=A1i+A

+A a

(13)

and:

then since INI = IS] = 1, cos 71can be written as: cos T = AIB1 + A2B2 + A3B3

(15)

The solar elevation angle, e, can be calculated from the well-known formula: sin e = sin k sin 5 + cos k cos 5 cos H

(16)

(Gates, 1962, pp. 58--59; Stapleton et al., 1973; Schultze, 1976). The solar hour angle, H, measured as time from solar noon in radians is given by: H = (12 -- t)Tr/12

(17)

417 where t is hour on a 24-h cycle. Corrections for non-central time zone location and elliptical earth orbit are available in Stapleton et al. (1973). The declination angle, 5, may be found in Solar Ephemeris Tables or, more simply, calculated from: 5 = --0.40773 cos [(11.36 + day) 2n/365]

(18)

where day is Julian day number (A. Green, personal communication, 1978). The solar azimuth angle, ~, can be calculated from: sin ~ = cos 5 sin H/cos e

(19)

(Gates, 1962; Stapleton et al., 1973). In this expression, we have removed a negative sign from in front of Stapleton's formula (A33, p. 40) to put the azimuth angle in our coordinate system (measured clockwise from north, e.g., 0 in Fig.l). In addition, Gates' formula (eq. 2.13, p. 59) in terms of zenith angles should be divided by sine rather than cosine of zenith angle. Eqs. 11--19 allow us to calculate cos 3'i in eq. 10. The spherical area element, AAi, in eq. 10 may be calculated from: -Oi+ 1

AAi=P2 jl

cPi+l

re) i

cos~)d@dO

i

or

AAi = p2 (Oi+' _ Oi) (sin

@i+1--sin ~i)

(20)

Using eqs. 11, 12 and 20, eq. 10 becomes:

IOA ~-- ISI

[NIp2 ~ {(cos ~i sin0i cos e sin a + c o s T / c o s 0i cos e cos a + A

sin ~i sin e) (Oi+, -- Oi) (sin ~i+, -- sin ¢i)}

(21)

where the summation is to be taken over a "quadrant" of the tree and negative values (shaded portions) are excluded. Since assumingthat p2 = 1 is equivalent to assuming a spherical tree of 1-m radius, IOA must be divided by the area of the quadrant to obtain average incident energy flux per quadrant. The two top quadrants have an area of: ATO P = p2 ( ~ - - 2 )

{sin(90°)-- sin(10°)}

p2 2.596 m2

(22)

and the b o t t o m quadrants have an area of:

ABOT = p2 ( ~ _ _ ~ ) {sin(10o)_ sin(_35o)} = p2 2.347 m2

(23)

from eq. 20. The sum in eq. 21 must be divided by eq. 22 or 23 for top and

418

b o t t o m quadrants respectively. In addition the magnitude of the solar vector is given approximately by: I SI =

So rcsce

(24)

(Allen, 1974; Schulze, 1976). So is the solar constant, 1360 W m -2, (Monteith, 1973), v is the zenith path atmospheric transmissivity (about 0.8 in central Florida) and e is the solar elevation angle given by eq. 16. From eqs, 22, 23 and 24, eq. 21 can be written as: IOQuAD ~ p2 (SoTCSCe/AQUAD)

~

{(cos ~i sin 0i cos e sin a +

QUAD COS ~i COS 0i COS e COS O~+ sin ~i sin e)

(0i+1- -

Oi)

(sin ¢i+, -- sin ¢i)} (25)

w h e r e IOQuAD is the average solar energy flux in W m -2 incident upon a

particular tree quadrant, AOUAD is the area of the quadrant from eq. 22 or 23 and the summation is to be taken over the sunlit part of the quadrant in question. In all our computer calculations, (Oi+l -- Oi) and (¢i+1 -- ¢i) = 5° ~0.087266 rad. To simulate average fruit surface temperatures in a tree quadrant,/0QUA D is substituted for I0, the solar beam intensity, in eqs. 4, 5 or 6. COMPARISON OF THE MODEL WITH FIELD DATA

Three exposed Valencia oranges in each tree quadrant were fitted with thermistor temperature probes wired to a twelve-point recorder. The probes were held tightly against the outside surface with rubber bands and a drop of heat sink c o m p o u n d to insure good thermal contact. The three fruit in each quadrant were positioned " a r o u n d " the quadrant dividing it into three equal parts. This was done to obtain a more representative average " h o t s p o t " temperature for the quadrant than would have resulted from random spatial selection. Air temperatures were measured with a hygrothermograph in a standard weather shelter in the same grove. Two days of particular interest are shown graphically in Fig.2. The average fruit temperatures by quadrant for September 28, 1977 are shown in the four right hand panels of Fig.2. The main purpose here is to demonstrate the effect of clouds on fruit temperatures. This particular day had clear sky followed by cloud cover followed by clear sky again, and the depressing effect of clouds on fruit temperatures is quite obvious. Cloud cover will reduce direct solar intensity by about 80%, (Brooks, 1964; Schulze, 1976); i.e., eq. 22 would be multiplied by 0.20. This fact and the data in Fig.2 suggest that cloud cover is a major factor in the thermal environment of the rust mite on exposed citrus fruit. The effects of solar radiation on fruit temperatures on a clear day, September 30, 1977, are shown in the four left-hand panels of Fig.2. The south top quadrant has average hotspot temperatures 10°C above air temperature, while

419 CLEAR (9-30-77)

CLOUDY (9-28-77)

--Model z.lll& SOUTH

[

•

Fruit

*

Air SOUTH TOP

,,% 30 20 SOUTH

SOUTH

A

BOTTOM

30

o

20

i

,

,

I

NORTH TOP

^I

uF

I

I

I

I

|

TOP NORTH

It

w

L~J I--

20

j

nL

NORTH

vt

BOTTOM

20

I

I

I

i

•~

I

I

NORTH BOTTOM

~ 6

12

6 12 18 24 6 18 24 6 SOLAR TIME

Fig.2. Temperature cycles on fruit (u) compared with air temperature (+) for cloudy and clear days. The solid line represents simulated temperature using eqs. 5 and 25. the n o r t h b o t t o m s h o w s o n l y a 1 - - 2 ° C increase. S i m u l a t i o n of t h e s e t e m p e r a t u r e s requil'es w i n d v e l o c i t y to calculate the h e a t t r a n s f e r c o e f f i c i e n t , h, f r o m eqs. 7 a n d 8. Wind v e l o c i t y o n t h a t d a y was a p p r o x i m a t e l y 3 m sec -1 at 18.3 m as m e a s u r e d b y t h e N O A A N a t i o n a l W e a t h e r Service at L a k e l a n d , Florida. A v e l o c i t y o f 3 m sec -1 at 18.3 m will be r e d u c e d b y a p p r o x i m a t e l y 2 / 3 at t h e c a n o p y t o p (Albrigo, 1976}. In a d d i t i o n , a n o t h e r 50% r e d u c t i o n w o u l d s e e m t o be in o r d e r in t h e b o t t o m h a l f o f t h e tree ( V a n E i m e r n et al., 1 9 6 4 ) . T h u s w i n d v e l o c i t y was a s s u m e d to be 1 m sec -1 in t h e t o p half o f the t r e e a n d 0.5 m sec -1 in t h e b o t t o m half. Eq. 25 was u s e d t o c a l c u l a t e IOQUAD, t h e average solar e n e r g y f l u x f o r e a c h t r e e q u a d r a n t , at 15 m i n intervals. This e n e r g y was t h e n u s e d as I 0 in eq. 5 to o b t a i n t h e e l e v a t i o n o f average h o t s p o t t e m p e r a t u r e o v e r air t e m p e r a t u r e . T h e

420

temperature increase from eq. 5 was then added to observed air temperature to produce the solid lines in Fig.2. In spite of a rough wind velocity estimate and a sample size of only three fruit per quadrant, the model agrees remarkably well with the observed data, RUST MITE OUTBREAKS IN TREE QUADRANTS

Two studies were done in which rust mite densities on exposed fruit (Valencia oranges) were followed during unsprayed outbreaks in different tree quadrants. In this section we will compare the behavior of these populations with simulated fruit temperatures. The actual fruit temperature during these outbreaks cannot be reproduced because of the absence of solar radiation data, but fruit temperatures can be simulated for different times of year and different wind velocities.

Simulation of fruit temperature Typical temperature regimes on fruit for our latitude (28°N) are shown for the summer and winter solstices in Fig.3 and for the vernal and autumnal equinoxes in Fig.4. Air temperature was assumed to be sinusoidal (5.5 ° amplitude) around a 35-yr average for the day with a peak at 15h00. Tmax or Train of eq. 5 or 6 was added t o this ambient temperature cycle to produce the upper and lower limits of temperature on the fruit as indicated by the pairs of lines in Figs.3 and 4. All fruit temperatures must lie between these lines for the given set of conditions. Wind velocities of I and 4 m sec -1 (top quadrants) and 0.5 and 2 m sec -1 ( b o t t o m quadrants) are illustrated.

Rust mite population dynamics Rust mite population densities by tree quadrant in two different groves are shown in Fig.5. Waverly grove has a wide tree spacing (6.1 x 9.1 m) with small trees having considerable open space around t h e m and little shading of one tree by another. Bay Lake grove is a hedgerow spacing (4.6 x 7.6 m) of large overlapping trees with rows running east--west which would be expected to accentuate any north--south differences. Both groves were 4.05 ha blocks, and eight fruit were sampled f r o m each quadrant of four randomly selected trees on each sample date. Both groves had a summer mite o u t b r e a k with the b o t t o m of the tree showing the highest mite densities (Fig.5). For unknown reasons, Bay Lake grove had a m u c h higher popuiation on the north b o t t o m at one c o u n t than occurred in any other sample in either ~ o v e . This may simply b e d u e to our random sample including a few unusually high density trees. It cannot be easily explained on the basis of temperature differences. North and south b o t t o m show nearly identical temperature regimes on June 21 (Fig.3), and it seems unlikely than an east--west hedgerow would drastically alter this

421

JUNE2[

DECEMBER21

SOUTH TOP --I msec -L

35

....

f

SOUTH TOP ~1 msec -I

35

-- I

......

4

m

s ec - I

2G

t5 i

L

h

i

L

L

L

SOUTH BOTTOM --05

35

m sec t

.....

~

25

i

I"

SOUTH BOTTOM

35I

--0.5

C - I

e

i

......

2

m see"~ m $ec -t

25

15 i

A

i

i

i

i

"~-

i

i

NORTHTOP --I .....

~ 35

mse¢ "l -I

i

NORTH TOP ~ 1 msec -I ..... 4 m sec -I

35

25 co

15

15 i

NORTHBOTTOM --0.5 .....

35

m sec "~ -i

i

~

i

h

h

NORTH BOTTOM 35

~ 0 . 5 lit SeC"1 ...... 2 m sec-I

2G

15

15 i

4

i

i

B ,2 ,~ 2'0 2'4

,~ ,'~ 2'0 2'4

,~

SOLAR

TIME

Fig.3. S i m u l a t e d t e m p e r a t u r e s o n fruit f o r t h e s u m m e r a n d w i n t e r solstices in c e n t r a l F l o r i d a . The t w o solid lines i n d i c a t e t h e m a x i m u m a n d m i n i m u m t e m p e r a t u r e s for 1 m sec- 1 w i n d ( t o p q u a d r a n t s ) a n d 0.5 m sec- 1 w i n d ( b o t t o m q u a d r a n t s ) . T h e t w o d o t t e d lines i n d i c a t e t h e r a n g e f o r 4 m sec -1 a n d 2 m sec -1 w i n d t o p a n d b o t t o m , respectively.

situation. (In fact, the temperatures reported in Fig.2 are from the Bay Lake grove and fit the model quite well.) Hobza and Jeppson (1974) reported an o p t i m u m temperature for citrus rust mite at 24.5°C. Reed et al. (1964) f o u n d that temperatures of 32.2 ° _ 37.8°C were lethal to citrus rust mite in 2--3 h. Therefore it is not surprising t h a t the n o r t h and south top quadrants with an average m a x i m u m temperature range between 33 ° and 38°C (1 m sec -1 wind) show very low rust mite population densities. Even 4 m sec-1 winds do not reduce average temperatures out of the lethal range in the top of the tree on June 21 (Fig.3).

422

MARCH21

SEPTEMBER 2J

SOUTHTOP - - I m SeC-I

35

SOUTH TO,

3~i

/

--,m,ec"

\

25

IS i

i

i

i

SOUTHBOTTOM ~ O . 5 m sec -~

F 35}

°~*'} ~ 2 , 5

2m

sec -I

[

SOUTH BOTTOM

35~

~

t

~

--0.5

25I/

mSiC

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Fig.4. Same as Fig.3 for the vernal and autumnal equinoxes. Waverly grove exhibited a winter outbreak during the warm winter of 1 9 7 4 / 1 9 7 5 which is of some interest (Fig.5). The quadrant differences in winter tend to diminish somewhat; however, the north bottom is highest, and the south top is lowest in rust mites as one might e x p e c t if overheating is involved. The average curve in Fig,3 f o r December 21 is not into the lethal zone for the south top. The average curve can be a bit misleading, however, since only an occasional day o f lethal temperatures would be required to reduce population levels. Even though the temperatures for November 1974 to January 19,75 are only slightlyabove average, there were 41 days in these months with the maximum air temperature above 26°C and 6 days in November having a maximum of at least 30°C. This puts the fruit temperature

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Fig.5. Rust mite population dynamics in different quadrants in two different Valencia orange groves in central Florida. range for Fig.3 well into t h e lethal z o n e several times and this w o u l d be e n o u g h t o r e d u c e rust mites in the s o u t h t o p o f the tree. SUMMARY AND CONCLUSIONS A m o d e l o f citrus f r u i t t e m p e r a t u r e s was c o n s t r u c t e d based o n average solar e n e r g y flux in tree q u a d r a n t s . This m o d e l is a f u n c t i o n o f air t e m p e r a t u r e ,

424 latitude, time of year, time of day, fruit diameter and reflectance, position in the tree, atmospheric transmissivity and wind velocity. Diffuse radiation, radiation heat loss, latent heat loss and wind direction are ignored. The model was f o u n d to agree remarkably well with t em perat ure data from the field and is useful in simulating a variety of conditions affecting fruit temperatures. When the model's predicted temperatures were com pared with rust mite distribution patterns in the tree~ the rust mite populations were low where excessively high t em pe r at ur e s were predicted. It seems that the general population distribution in the tree can be predicted on the basis of t em perat ure alone. It is true however, t h a t high h u m i d i t y favors the devel opm ent of the rust mite's immature stages (Pratt, 1957; H o bza and Jeppson, 1974). F u r t h e r study is needed to explain the aggregation of rust mites around sunlit areas on individual fruit, perhaps by using the model developed here to calculate water loss rates over the fruit surface. These water flux rates will be strongly affected by temperature, and t h e y will, in turn, affect the b o u n d a r y layer vapor pressure in which the rust mites live. ACKNOWLEDGEMENTS The mathematical advice of Dr. James H. St am per was greatly appreciated. This study was s u p p o r t e d by CSRS Grants PL 89--106 and 701--15--58 and also by NSF Grant G B - - 34718 to the University of California and the University of Florida. REFERENCES Albrigo, L. G., 1976. Influence of prevailing winds and hedging on citrus fruit wind scar. Proc. Fla. State Horti. Soc., 89: 55--59. Albrigo, L. G., 1977. Rootstocks affect "Valencia" orange fruit quality and water balance. Proc. Int. Soe. Citriculture, 1: 62--65. Albrigo, L. G. and McCoy, C. W., 1974. Characteristic injury by citrus rust mite to orange leaves and fruit. Proc. Fla. State Horti. Soc., 87 : 48--55. Allen, J. C., 1976. A modified sine wave method for calculating degree-days. Environ. Entomol., 5: 388--396. Allen, L. H., 1974. Model of light penetration into a wide row crop. Agron. J., 66: 41--47. Bennett, A. H., Chace, W. G., Jr. and Cubbedge, R. H., 1970. Thermal properties and heat transfer characteristics of Marsh grapefruit. U.S. Dep. Agric. Tech. Bull. 1413, 29 pp. Brooks, F. A., 1964. Agricultural needs for special and intensive observations of solar radiation. Bot. Rev., 30: 263--291. Fisher, F. E., Griffiths, J. T. and Thompson, W. L., 1949. An epizootic of Phyllocoptruta oleivora (A~hm.) on citrus in Florida. Phytopathology, 39: 510--512. Gaffney, J. J., 1973. Reflectance properties of citrus fruits. Trans. ASAE, 54: 310--314. Gates, D. M., 1962. Energy exchange in the biosphere. Biol. Monogr., Harper and Row, N.Y., 150 pp. Hobza, R. F. and Jeppson, L. R., 1974. A temperature and humidity study of the citrus rust mite employing a constant humidity air-flow technique. Environ. Entomol.,

3: 813--822. Hubbard, H. G., 1885. Rust of the orange. U.S. Dep. Agri. Rep., pp. 3--10. MCCoy, C. W. and Kanavel, R. F., 1969. Isolation of Hirsutella thornpsonii from the citrus

425 rust mite, Phyllocoptruta oleivora, and its cultivation in various synthetic media. J. Invertebr. Pathol., 14: 386--390. McCoy, C. W. and Albrigo, L. G., 1975. Feeding injury to the orange caused by the citrus rust mite, Phyllocoptruta oleivora (Prostigmata: Eriophyoidea). Ann. Entomol. Soc. Am., 68: 289--297. McCoy, C. W., 1978. Entomopathogens in arthropod pest control programs for citrus. In: G. E. Allen, C. M. Ignoffo and R. P. Jaques (Editors), Microbial Control of Insect Pests: Future Strategies in Pest Management Systems. Univ. of Florida, pp. 211--224. Muma, M. H., 1970. Preliminary studies on environmental manipulation to control injurious insects and mites in Florida citrus groves. Proc. Tall Timbers Conf. Ecol. Anim. Control by Habitat Management, 2 : 23--40. Monteith, J. L., 1973. Principles of Environmental Physics. Arnold, London, 241 pp. Nobel, P. S., 1975. Effective thickness and resistance of the air boundary layer adjacent to spherical plant parts. J. Exp. Bot., 26: 120--130. Poppendiek, H. F., 1953. Transient and steady-state heat transfer in irradiated citrus fruit. Trans. Am. Soc. Mech. Eng., 75: 421--425. Pratt, R. M., 1957. Relation between moisture conditions and rust mite infestations. Proc. Fla. State Horti. Soc., 70: 48--51. Reed, D. K., Burditt, A. K. and Crittenden, C. R., 1964. Laboratory methods for rearing rust mites Phyllocoptruta oleivora and Aculus pelekassi on citrus. J. Econ. Entomol., 57: 130--133. Schulze, R. E., 1976. A physically based method of estimating solar radiation from suncards. Agric. Meteorol., 16: 85--101. Sharpe, P. J. H. and Demichele, D. W., 1977. Reaction kinetics of poikilotherm development. J. Theor. Biol., 64: 649--670. Smart, R. E. and Sinclair, T. R., 1976. Solar heating of grape berries and other spherical fruits. Agric. Meteorol., 17 : 241--259. Stapleton, H. N., Buxton, D. R., Watson, F. L., Nolting, D. J. and Baker, D. N., 1973. Cotton: A computer simulation of cotton growth. Ariz. Agric. Exp. Stn., Tech. Bull. 206. Stinner, R. E., Gutierrez, A. P. and Butler, G. D., 1974. An algorithm for temperaturedependent growth rate simulation. Can. Entomol., 106: 519--524. Thorpe, M. R., 1974. Radiant heating of apples. J. Appl. Ecol., 11: 755--760. Van Brussel, E. W., 1975. Interrelations between citrus rust mite, Hirsutella thompsonii and greasy spot on citrus in Surinam. De Surinaamse Landbouw, 23(3): 119--121. Van Eimern, J., Karschon, R., Razumova, L. A. and Robertson, G. W., 1964. Windbreaks and shelter belts. W.M.O. No. 147.TP.70, Tech. Note, 59: 1--188. Yothers, W. W. and Mason, A. C., 1930. The citrus rust mite and its control. U.S. Dep. Agric. Tech. Bull. 176, 56 pp.