Kurihara 1965
Kurihara 1965
Kurihara 1965
xi
4I a=-1
a=O
METHOD A METHOD B
I n this case, d is to be obtained as a solution of starting from the given initial value wo=uo=l, and as-
suming f = r / 9 (hr.-l). The period of oscillation is then
-sin 9/( 1+cos 3)=a, 18 hr. Figure 2.4 shows single-step predictions of u for
and R= 1. various values of At. To use long time steps in Method
The trace of X is shown in figure 2.3. This scheme is A gives complete damping, i.e., ul=O; while with Method
neutral in the sense that it neither amplifies nor damps a B, u1=--u0="1 and d= --vo=O. Although there exists
wave. The eigenvalue for large la1 approaches -1. no damping effect in Method B, the error in the phase
Thus, the phase of a wave will be shifted by T, i.e., half velccity will make a prediction meaningless if the time
a wavelength, in one time step if la1 is infinitely large. step chosen is larger than about one-sixth of a period of
We shall now give examples of numerical integration thewave. I n figure 2.5 are shownpredictions of u, in
with the useof the above two implicit methods. A system which MethodsA, B, and explicit leapfrog method
of equations admitting only inertia oscillations is (Method 1) were repeatedly used, respectively,with a
bU =j. time step of one hour. The damping effect in Method A
{
- is clearly seen.
The general case of (2.2) will be referred to as MethodC.
Method C. (partlyimplicitmethod) .-For convenience,
(2.2) is repeated:
where u is an eastward mind velocity, 8 is a northward hr+l-hr=--iahr+l-iphr (2.2)
windvelocity,and j is the Coriolis parameter.These
equations are rewritten in the form of (2.1), In this case the real and imaginary partsof the eigenvalue
are
(2.5)
Areat= (1-&>l(1+a2)
where
w=u-kiu. A*nza,=-(a+P)l(1+aZ)
A time integration of (2.5) was done by the formulas: respectively. Accordingly, the magnitude of X is
36 MONTHLY WEATHERREVIEW Vol. 9 3 , No. 1
10"
36 100 500
X METHOD A
0
- 1.0 *..r"-
FIGURE 2.4.-Singlc-stcp prediction of inertia oscillation
with
various time intervals ( A t in hr.). ZL a t t=At is plotted.
METHOD 1
FIGURE 2.6.-Tracc of eigenvalue for Method 1 (leapfrog method).
U Eigenvalues of t h e amplification matrix of (2.8) are shown on
t h e complex plane. p.is a parameter.Theright half of t h e
unit circle corresponds to the computed physical mode and thc
left half tothecomputationalmode. If Ipl<2, one of t h e
t,wo eigenvaluesrepresents theformermodeandtheother
does thelattermode. If lp1>2, twoeigenvaluesareonthe
axis of X,=O and one of them is outside the unit circle, Le.,
t h e scheme is computationally unstable.
0<iY,<;
if IPI<I.I.
T<&<$
1
3 v2(U-c1)
u = x ui, ut=+{
R 2 = l / ( l - a tan 6 ) . i=l y- (U-c,)2v2
t
As R is real, 6 is undefined within some ranges, i.e. (3.2)
1 1 3s
tan" -<6<; and tan" -+s<6<- if a> 0
a a 2 1J
s tan" -1 and--<d<
3s 1 if a<O.
tan" --"T
--<6< where v=2s/L,L is the wavelength, c t ( i = l , 2 , 3) are the
2 a 2 a
threephasevelocitiesand Si areamplitudes of three
waves. The c z should be obtained as roots of the equation
Furthermore, it is seen that when a>O
+$c=o.
(u-c)3--gu(u--c)
For the ordinary values of f, g, H , U , and v, Bhese are
written
~
TABLE 3.1.-Phase
velocities (c,,c2, and CQ) and vei. I n ( 3 . 3 ) ,
U=50 m./sec., yH=8XlO4 m.2/sec.2 and f at the 45' latitude are
asslcmed.
I
"CZ
-232.9
250332.949.99 1.256 10-3 8.366 10-3 5.853 10-3
-232.9
500333.049.96 6.278 lo" 4.184
2.927
-233.2
1000333.449.83 1.465 2.095 3.131
-234.4
2000335.149.34 7.364
1.053
1.550 10"
4000 47.46 -239.0341.6 7.455 10-5 5.366 10-4 3.755
-257.2
8000366.041.17 3.233 2.020 2.875
-323.4
16000 446.526.84 1.270 1.751 1.054
-524.9
32000 663.811.12 2.184 10" 1.303
1.031
b I I
1.0
I ,
.
I
2.0
I
\" ,
' I
3.0 ' b
-
c,=u+2J-; cos 3
I
(:-+-
3 .)- u
4
1
I
0.5
"""
-"" M ----""".
ETHOD A
at
@=F1+F2 This is the same as (2.3) with a=vci(Al). The amplifica-
(3.5)
tion rate of hi,i.e., R,, and a measureof the fictitiouschange
where of phasevelocity'areshowninfigure 3.1 againstthe
3'1 = - Udh/dx parameter b=vci(At). Fromthis figure andtable 3.1,
F2=the right-hand side of (3.1).
it is easy to see that for a specified wavelength and At,
damping of the wave is highly selective for gravitational
The problemis, now, to do thetimeintegration of waves, forwhich vcf isseveraltimeslarger than fora
(3.5) with various methods and to emmine their character- low-frequency wave. This is themerit of this method.
istics. The names of the methocls in the following shoulcl It should be noted, however, that damping of the meteoro-
correspond to those in section 2. logical wave is also unavoidable, however small At may
fllethod A.-Time integration of (3.5) takes the form be.Consequently, successive use of (3.6) will atlast
cause a noticeable da.mping of the lorn-frequency wave.
h,rkl-p=At,F;+'+At.P;+1 (3.6)
1 As mentioned before, the amplification rate and phase velocityof the computed mode
where 7,r+ 1, and At are two time-levels and the interval are given by RI and -81/v(At) respectively, where RI and 91 are the magnitude and phase
angle of the eigenvalue for an amplification matrix of (3.7). Accordingly, -&/.ei(At)=
betweenthemrespectively. 3';;;' means that F1, should -&/b is theratio of phase velocity of the computed value to the t,rue phase velocity.
January 1965 Yoshio Kurihara 39
As for the error in phase velocity, it is large when b is This isequivalenttoh4ethod C insection 2. From
large. (2.2) and (3.11) it follows that
hl+'-k=-
At (Ff+'+lq)+,
At (F;+'+Zq. (3.5) (3.12)
2 i
p=v -.
The form of (3.4) corresponding to (3.S) is
ci+u
2
(3.10)
TABLE3.2-Theshortestwavelengthto be treated ( L ) andrnaximunt
value of the timeincre?nent(At)whichsatisJiesthecomputational
Hence a corresponding formula for each wave is derived, stabilitycondition for ilfethod 1 . I t is assztntedthat a wave i s
treated analytically, L e . , a spectrummethod is u s e d .
(3.11)
1 250
500
1000
2000
4000
119
950
1804
40 REVIEWWEATHER
MONTHLY Vol. 93, No. 1
wavesshorterthan
critical
a wavelengthshould be largerthan 1/31, andthose waves will bedamped.Con-
truncated from the functional form. sequently,thismarchingschemecannotbe used for a
The most troublesome deficiency in this method is the long-range time integration.
occurrence of the computational mode. If the amplitude However, since the amplification ra.te of the met,eorolog-
of this mode becomes large it is meaningless to continue ical wave is very small, this method may be used in short-
the time integration. range integrations. A test computation of this kind was
attemptedby usingsimple
a linearized model. The
Method D.-This method is written as follows: model adopted is the same as (3.1), u = 5 0 m.sec.-l, j is
hr+'--hr"=2.AtF:+2.AtF~+' (3.15) taken a t 45' latitude,andgH=8X104 m.2sec.-2. The
wavelength of the sinusoidal wave we treated is 4500 km.
Namely, the advection term is estimatedexplicitlyand To give the initial values of u, v, and 4, S l = l O O O gpm.,
the other terms implicitly. The corresponding formulafor S2=50 gpm. and S3=50 gpm. were taken in (3.2). Then,
each mode is computations were repeated with A t = l hr. by the scheme
(3.15), mere h stands for u, v, and 4. I n computing
&+'-@"=-2i~(~i-U) *(At)@+"2ivU* ( A t ) h l (3.16) @+I, we nmde a slightchange in the scheme.Namely,
vr was used i n s t e d of ~ + forl evaluating the first term on
This is identical with (2.6) if we put the right hand side of the third equation of (3.1). Then
substituting UT+' inthethirdequationfromthefirst
( Y = ~ v ( c ~ - U( )A.t )
(3.17) equation, in which vr+l was substituted from the second
/3=2vU* ( A t ) equation, a one-dimensional Helmholtz-type equation for
@+l was obtained. I n our test,, a finit,e difference compu-
If we suppose ci= U in (3.16), it takes the form of (3.14). tation with a 300 km. grid was used and a Helmholtz-type
If we neglect the second term on the right hand side of equation was solved bymatrix inversion.Withthe
(3.16), assumingthat Icil> U, then we have a form similsr solution of @ + l , both a r + l and vr+l were easily computed.
t80 (3.7). Hencethismethod looks favorablefromthe In this way calcu1:itions were continued up to five days,
viewpoint of effective damping of gravitational waves. i.e., 120 t,ime steps.In figure 3.2 the values of 4 and
Strictly speaking, however, this method is not computa- bulbx a t x=O are plotted together with the true variations.
t'ionally stable.This will beexplained as follows. The Effective damping of gravitational waves is clearly seen.
conclusion from the previous section was that the condi- Chm~gesin amplitucle of themeteorologicalwaveare
tion of computationalstability of (3.16) is IaI>IpI. I n negligible so far as thisexample is concerned. A rough
the case of a meteorologicalwave, a takes a small and estimate for our test case shows that the amplification
non-zero valueandthisconditioncannotbe satisfied. rates for the meteorologicalwave is 1+0(10-3)and
On the other hand for gra.c.itationn1 waves, la1 is much those for gravitational waves a,re 0.6 or thereabout,.
I I 1 1 I
0 1 2 3 4 5
DAYS
FIGURE
3.2.-Prediction of 0 and au/dx with a system of equations (3.1). Method D was used with A t = l hr. Time variation of 4 and
3u/dx at x = O is shown (true value is shown by continuous line and computed value by asterisks).
January 1965 Yoshio Kurihara 41
t h*-hr"=2.At
hrfl-hr=At. F*
Fr (leapfrog method)
(backward correction)
(4.7)
in a high ra,te of damping of the computational mode,
especially that corresponding to the meteorological wave.
Only Method 1 is neutral, provided the stability condition
is satisfied. Consequently, it seems a good design t o use
h;+'=(1-2b2)h:-J=ibh;" (4.8) Method 1 a t most time, steps but t,o utilize some kind of
42 MONTHLY
REVIEW WEATHER Vol. 93, No. 1
/*
METHOD
1.. . .---
0.1
4 "
....... / . 4"
5 .............
5 ..............
b
0.90 ' ' ' 1 I I I n l a
0.5 1.0 Q 1.5 b
0.0 0.5 1.0 V 7 1.5
METHOD
1
METHOD
1
0.90 - 0.5 - 2-
3 ""
4 "
5 ............
0.85 ' ' ' I I I I I t I I 1 - 4
b
0.0 - 0.5 1.0 ," 1.5
/'
- I
- 0.5
/
, III
- 1.5-
FIGURE
4.1.-Amplification rate of computed physical mode (the upper leftfigure) and thatof computational mode (the upper rightfigure)
are shown against parameter 6. b=vcAt, if a spectrum method is used in treating a wave. When a centered difference grid method is
used, b.=vc'At.(vc and vc' are listed in tables 3.1, A.l, and A.2.) The lowcr left figure shows ratio of phase velocity of computed
physical mode to c (or c', if the grid method is used). Ratio of phase velocity of computational moclc to c (or c') is shown in the lower
right figure. In three figures, vertical scale is changed at b = 0 . 4 . Suppose that 6=0.5 and Method 4 is used. Then, an amplitude
of computed physical mode at a time level T + 1 is 0.99 times t h a t at T . It moves with a speed 0.99Xc, if a spectrum method is used,
or with a speed 0.99Xc', if a centered difference grid method is used. An amplitude of computational mode atT + 1 is 0.25 times that
at T . Its moving spced is 2.15Xc or.2.15Xc'.
I I None
Damping
DRmping of gravity
wave
and weak nmplif:iinp of
meteoroloaical wave
1 1 lcayfrog
(centered)
I hr+l-h,-l=ZAtFr I 3 I Conditionallg
@<I)
stable No change No change Modcrate accrleration
Little
damping
Little crror V e r g e f f e c t i v ~damping
(in particular of mc-
tcorological wave)
I
ModeratelyselectireModerate
damping
-I
acceleradion Damping
The equation,for which theiterativemethods mere and the computntion by Method 5 became unstable. All
npplied, is the same ns ( 2 . 5 ) . j=.lr/9 (hr.-') was assumed. of these coincide quite well withwhat we observed in
Hence,the period of oscillation is IS hr. As a starting figure 4.1.
value, wo=uo=l was givenforh4ethods 2 and 3. For 5 . SUMMARY
Methods 1, 4, and 5, it is also necessary to give the value
of w a t a time-level nearest to the initial, i.e., w' or w-l, Themainproperties of themethods considered in
to start the calculation. If we estimate w1 from wo by a sections 2 to 4 are shown in table 5.1.
modified Euler nlethod which we used to start thecalcula- The properties of Method A (two time-levels, backward
tion by Method 1 in figure 2.5, we cannotdetectthe inlplicit method) , &letbod B (two time-levels, trapezoidal
existence of computationalmode. I n ordertoiorce a implicit method),Method C (two time-levels, partly
large initid amplitude of thecomputationalmodethe implicit method), andMethod 1 (three time-levels,
inte,grationswithMethods 1, 4, and 5 were begunwith leapfrog method) have been discussed so far, more or less.
wl=wo. Figure 4.2 shows the predictions of u in the case They are confirmed in section 2, where the characteristics
o l At=1 hr. For this case we have b=2r(At)/(period)= of these methods in case of wave equation in simple form
0.35. On the other hand, the ordinate values against this are described. In section 3, we considerthese methods
value of b in figure 4.1 suggest damping of the physical especially from the viewpoint of their applicability to the
mode of oscillation by Methods 2, 3, and 5, the consider- int'egration of the primitive equations.
able damping of the computational mode by IMethod 4 Methods A and B are conlputationally
absolutely
(52 percent a t each step) and by Method 5 (63 percent), stable. I nt h e use of thesemethods,theanlount of
the conservation of both modes byMethod 1, andthe computationrequired to solve thenon-trivialequations
fictitiousincrease of phasevelocity by Method 2. It is for the quantities at a, new time-level and the decrerme
seen that the features of the curves in figure 4.2 are the of accurttcy of the predicted low frequency wave should
sit~ne withthese suggestions. The predictions made with be weighed against the advantage of a long time interval
A t = 2 hr., for which the corresponding value of 6 is 0.70, in a marching process. The :bmplitude of any wave will
showed the fast damping of the conlputational mode by not be changed with Method B. Method A results in a
Met,hod 4 and slow clanlping by h4ethod 5. I n case of damping which increases with the increasing value of the
At=2.7 hr., for which b=0.94, h/Iethod 2 yielded a very parameter b. ( b = v c A t if a spectrummethod is used in
slow damping of the physical mode and a large fictitious treating a wave. When we use a centered difference grid
decrease in periodof oscillation; Method 3 rapidly damped method, b=vc'At where e' is a modified phase velocity.)
the oscillation; Method 4 damped t,he computational mode; The property of selective dumping of wave is useful for
44 REVIEWWEATHER
MONTHLY Vol. 93, No. 1
z(z+A)-z(z-A) .sin V A We shall call these phase velocities modified phase veloci-
2A
= 2 ___
A 4 x 1, ties. I t is seen, from the comparison of the above equa-
tionwiththe corresponding one in section 3, that cf
is the same with c in the case where U and gH are modified
where A is the space-increment. The similar one fora vf
five-point method is (Y
to - U and - gH, respectively. As v’lv is nearly equal
t o one for large n, the fictitious change of phase velocities
~.z(x+A)-~.z(x-A)-z(x+~A)+z(x-~A) due to space finite differencing is s~nnllfor relatively long
12.4 waves. On thecontrary, v f / v is smaller thanabout 0.9
=i (sin v A ) . (4-cos v A ) for n 1 8 (in the case of thethree-pointmethod)or for
3A (x) n 1 5 (five-point method),andanerrorinthephase
velocity of waves corresponding to these n becomes large.
The above two finite difference formulas take a common An importantformula which is derivedfrom (3.1-A)
form, namely b z j b x = i v f z instead of analytical vdue i v z . and (3.2-A) and is equivalent to (3.4) is
Now, with the use of the above expression for a hori-
zontalgradientand an assumption of an equal wave ah,
length for u, v, and 4, (3.1) is modified as follows: -+iv’Uh,=-~(vc;-v’U)h,. (3.4-A)
bt
January I965 Yoshio Kurihara 45
TABLE
4.1.-Ratio(c,'/ci) of themodifiedphasevelocity(c,')tothe TABLE A.B.-Ratio (c'i/ci) of themodijiedphasevelocity (c'i) tothe
analytacalphasevelocity(ci)and vci', in case of athree-point analyticalphasevelocity(c;)and vc'i, in thecase of a five-point
Jinite difference scheme with a grid size of 250, 500, and 1000 k m . jinite difference scheme with a grid size of 250, 500, and 1000 k m .
n i s the number of grid points within a wavelength, Le., nX(5rid Refer to table A.1 for further explanation.
size)=wavelength.Assumedvalues of U, gH,f arethe sameas
those shown in table 3.1.
I1
250-km. grid
2 0.0 2. 5 3. 5 .......... 1.0% 10" I. 0% 10-4
3 61.8 62. 1 62. 1 2.585 1o;r 1.734 10-3 1.213 10-3
250-km. grid 4 84. 4 84. 9 84.9 2. 654 1:779 1. 245
2 ....... - 1.028 1 0 - 4 1.028 10-4 5 93. 0 93. 2 93. 2 2.326 1.563 1.093
3
4
1. 713 1 0 - 4
1.9%
1 . 1 ~ 910-3 8.110
1.337 9.352 ; 96. 4
98.0
96. 5
98. 1
96. 5
98. 1
2.005
1.741
1.351
1. 178
9.448 10-4
8.239
5
6
1.885
1.713
1.2i2
1.159
8.898
8.110
a 98.8 98.8 98.8 1.531 1.040 7.278
9 99. 2 99. 3 99. 3 1.362 9.304 1 0 - 4 6.509
7 1. 543 1. 048 7.331 10 99. 5 99. 5 99. 5 1.225 8.411 5.884
8 1.391 9.492 lo" 6. 640 20 100.0 100.0 100.0 5.796 10-5 4.351
3.046
9 1.260 8. 643 6.046 40 100.0 loo. 0 1.698loo. 0
2.406 2.352
10 93.3 93. 7 93. 7 1.148 7.918 5. 540
20 98. 1 98. 5 98. 5 5.688 1 0 - 5 4. 287 3.001
40 99.4 99. 7 99. 7 2. 338 2.398 1.698 5O(tkm, grid
2 o.. n- -
4.. 9 7. 0I ....._ 1.028
._
~. 10-4 1.028 10-4
3 61. 3 62. 4 62.4 I. 274
10-4 8.732 6.109
500-km. grid 4 84. 4 85. 1 85.
8.958
1 1.309 6.267
2 0. 0 4. 9 7. 0 1. 0% 10-4
.__.__.. 1.028 10-4 5 92. 8 93.3 93.3 I. 143 7. 882 5.515
3 39.3 42. 1 42. 1 8.295 10-3 5.890 4.121 6 90. 3 96. 6 96. 6 9.787 10-5 6.833 4.781
4 62. 5 64.3 64.3 9.680 6.766 4.734 7 97. 9 98. 1 98. 1 8.439 5.981 4 185
5 74. 5 76. 3 76. 3 9.175 6.445 4.510 8 98. 7 Y8. 9 98.9 7.358 5.306 3.713
6 81. 6 83. 3 83.3 9 99. 2 99. .? 99. R.
~~ 6.483 4.767 3.337
7 SF. 0 87. 6 87. 7 10 99.4 99.5 99. 5 5.764 4.333 3.033
8 89. 0 90.6 90. 6 20 100.0 100.0 100.0 2.351 2.405 1.698
9 91. 0 92. 6 92. 6 100.040 100.0 100.0 6.674 10-0 1.559 1.155
10 92. 5 94.1 94. 1
20 97. 5 98. 8 98.8
40 99. 1 99.8 99.9 6.615 10- 1.556
1.153 1000-km grid
2 0.0 9. 8 14.0 1.028
_.....".. 10-4 1.028 10-4
1000-km. grid
3 59. 3 w.4 63.5 6.023 1 0 - 5 4.488 3.142
4 83. 2 85. 7 85. 7 6.205 4.598 3.219
2 0
0.14.0 9.8 """" 1. 028 10" 1.028 10" 5 92. 0 93. 7 93. 7 5.335 4.076 2.855
3 36. 2 44. 1 44.3 3.681 10-5 3.122
2.191 6 95. 7 96.9 96. 9 4.472 3.571 2.503
4 59. 2 65.9 OF. 0 4.416 3.538 2.480 7 97. 5 98.3 98. 3 3. 758 3.165 2. 220
5 71. 6 77. 8 77. 9 4.149 3.385 2.374 8 98.4 99.0 99. 0 3.182 2.846 2.000
6 78.8 84. 7 84. 8 3 . w 3.122 2.191 9 98. 9 99. 4 99. 4 2.716 2.596 1.828
7 K?.4 .. n
RQ. ~ R9.
.~ ~1 3.213 2.864 2.012 10 99. 3 99. 6 99. 6 2.335 2.397 1.692
8 SG. 5 91. 8 92.0 2.795 2.639 1.85i 20 99.9 100.0 100.0 6.669 10-6 1.559 1.155
9 88. 7 93. 7 93.9 2.434 2.448 1.72i
10 90.3 95. 1 95. 3 2.124 2.258 1.619
20
99.4 99.2 96. 5 6.439 1o-S 1.547 1.148
ACKNOWLEDGMENTS
The ratio of modified phase velocity to an analyt,ical
The author wishes t o express his thanks to Drs. J. Smagorinsky,
one for some specified cases is shown in tables A . l and S. Manabe,and 3i. Bryanwhohave given himencouragement
A.2. vc; is a useful parameter for examining properties throughoutthisstudy.Particularly, J. Smagorinskyreadthe
of a timeintegrntion scheme of (3.1-A). Thesevalues manuscript very carefully. He wishes to thank also Mrs. J. Snydcr,
are listed in the same tables also. Mrs. R.Brittain,and Mr. E. Rayfield fortheassistance he has
received in preparing this paper.
APPENDIX 2.-SOLUTIONS OF REFERENCES
X2+(A+Bi)X+(C+Di)=0 1. S. A. Bortnikov, "Experience WithShortRangeWeather
Prediction on the Basis of the Solution of a Complete System
I n sections 2 and 4 we had to solve the abovetype of Thermohydrodynamic Equations," [Opyt kratkosrochnogo
equation frequently. A, B, C, and D are real values and prognoza pogody na osnove reshenil% pole1 sistemy uravnenil
termogidrodinamiki) Meteorologic2 i Gidrologic2, Moscow,
i=d- 1. The two solutions are given by
_.
J. K. Augell, “Some Velocity andMomentumFlux national Association for Quaternary Research,” Bulletin
Statistics Derived from Transosonde Flights,”Quurterby of the American Meteorological Society, vol. 45, No. 11,
Journal of the Boyal Meteorological Society, 1701. 90, NOV.1964, pp. 688-690.
NO. 3S6, Oct. 1964, pp. 472-477. Ad. J. Rubin, “Antarctic Weather and Climate,” pp. 461-
J. K. Angel1 and J. Korshover, “Quasi-Biennial Variations 476 in Research in Geophysics, col. 2, “Solid Earth and
in Temperature, Total Ozone, and Tropopause Height,” Interface Phenomena,” H. Odishaw, Ed., The h/I. I. T.
Journal of the AtmosphericSciences, 1701. 21, No. 5, Press, Cambridge, Mass., 1964.
Sept. 1964, pp. 479-492. h4. J. Schroeder, D. W. Krueger, et al., Synoptic Weather
M . L. Blanc (with I,. P. Smibh), “International Agricul- Types AssGciated with CriticalFireWeather, Pacific
tural Meteorology,” Agricultural Meteorology, vol. 1) Southwest Forestand
RangeExperiment Station,
No. 1, Mar. 1964, pp. 3-13. Berkeley, Calif., 1964.
H . R. Glahn and J. 0. Ellis, “Note on the Determination R . H .Simpson (with J. S. Malkus), ‘LHurricaneModifi.ca-
of Probability Estimates,” Journal of Applied Meteor- tion,” ScientiJfc American, vol. 211, No. 6, Dec. 1964,
ology, V O ~3,. NO. 5, Oct. 1964, pp. 647--650. pp. 27-37.
A. V . Hardy, “Low Temperature Probabilities in North S. F. Singer, “High Energy Radiat,ion Near the Earth,”
Carolina,” Bulletin 423, Agricultural Experiment, Strt- Astronautica Acia, vol. X, Fasc. 1, 1964, pp. 66--80.
tion, University of North Carolina, Raleigh, Sept.1964. W . Spulrler (with R . F. Pengra and D. L. Moe), “Bibli-
A. J. Kish,‘’South Carolina, GrowingDegree Days,” ography of South Dakota Climate with Annotations,”
Agricultural Weather Research, &’pries No. 3, Sout,h Circular 162, SouthDakota AgriculturalExperiment
Carolina Agricultural ExperimentStation, Clemson Station, Brookings, S.Dalr., 1964, 24 pp.
University, Oct. 1964, 25 pp. Staff, Stratospheric Meteorology Research Project, “Im-
R . J. List, K. Telegadas, and G. J. Ferber, “Meteorologicd plementation of the WMO-IQSU STRATW’ARM
E d u a t i o n of the Sources of Iodine 131 Contamination Programme,” TVMO Bullelin, rol. 13, No. 4, Oct. 1964,
inPasteurizedMilk,” Science, 1701. 146, No. 3640, pp. 200-205.
Oct. 2, 1964, pp. 59-64. K. Telegadns and R. J. List, “Global History of the 195s
31.D. Magnuson, “Accidents and Deaths from Weather Nuclear Debris and Its Meteorological I m p l i ~ a t i o n ~ , ’ ~
extreme^,'^ pp. 519- 532 of Me(lica1 Climatology, Sidney Journal of Geophysical Reseatch, 1101. 69, No. 22, Nov.
Licht, Ed., E. LichtPublisher, New Haven, 1964. 15, 1964, pp. 4741-4753.
J. h4. Mitchell, Jr., “An Analysis of the Fluctuations in S. Teweles,“Application of Meteorological Rocket Net-
the Tropical St,ratospheric Wind,” [Letter to Editor], work Data,” ~ ~ e t e o r o ~ o g i Obseruations
ca~ A.bove SO
Quarterly Journal of the Royal Meteorological Society, Kilometers, NtttionalAeronautics andSpace Adminis-
vol. 90, No. 386, Oct. 1964, p. 481. tration, 1964, pp. 15-35.
J. hiI. Mitchell, Jr., “Focus on INQ,UA-The Inter- (Continued on page 66)