Fluctuations in the Atmospheric Inertia: 1873–1950

46 Pages·1961·3.862 MB·English

by  El Sayed Mohammed Hassan (auth.)

METEOROLOGICAL MONOGRAPHS
BOARD OF EDITORS
Assistant Editor Editor-in-Chief Assistant to the Editor
THOMAS A. GLEESON WERNER A. BAUM WHEATON M. CoWARD, JR.
Florida State University Florida State University American Meteorological Society
Associate Editors
DAVID ATLAS F. N. FRENKIEL jEROME NAMIAS
A. F. Cambridge Research Center J. Hopkins Applied Physics Lab. U. S. Weather Bureau
w.
GERALD L. BARGER LAWRENCE GATES HANS NEUBERGER
U. S. Weather Bureau University of California at L.A. Pennsylvania State University
LOUIS J. BATIAN JOSEPH J. GEORGE CHESTER W. NEWTON
University of Arizona Eastern Air Lines University of Chicago
FREDERIC A. BERRY MAURICE H. HALSTEAD HANS A. PANOFSKY
Aerometric Research Inc. Navy Electronics Laboratory Pennsylvania State University
RoscoE R. BRAHAM, JR. BERNHARD HAURWITZ NORMAN G. PHILLIPS
University of Chicago University of Colorado Mass. Institute of Technology
RICHARD A. CRAIG SEYMOUR L. HESS RICHARD J. REED
Florida State University Florida State University University of Washington
GEORGE P. CRESSMAN HENRY G. HouGHTON HERBERT RIEHL
U. S. Weather Bureau Mass. Institute of Technology University of Chicago
A. NELSON DINGLE WOODROW C. JACOBS HENRY STOMMEL
University of Michigan Library .of Congress Woods Hole Ocean. Instn.
GORDON E. DUNN HELMUT E. LANDSBERG VERNER E. SUOMI
U.S. Weather Bureau U. S. Weather Bureau University of Wisconsin
ROBERT G. FLEAGLE JAMES E. MILLER HARRY WEXLER
University of Washington New York University U. S. Weather Bureau
•
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METEOROLOGICAL MONOGRAPHS
Volume 4 May 1961 Number 24
FLUCTUATIONS
IN THE
ATMOSPHERIC INERTIA:
1873-1950
by
El Sayed Mohammed Hassan
PUBLISHED BY THE AMERICAN METEOROLOGICAL SOCIETY
4 5 8 E A C 0 N ST., 80S T 0 N 8, MASS.
METEOROLOGICAL MONOGRAPHS
BOARD OF EDITORS
Assistant Editor Editor-in-Chief Assistant to the Editor
THOMAS A. GLEESON WERNER A. BAUM WHEATON M. COWARD, JR.
Florida State University Florida State University American Meteorological Society
Associate Editors
DAVID ATLAS F. N. FRENKIEL JEROME NAMIAS
A. F. Cambridge Research Center J. Hopkins Applied Physics Lab. U. S. Weather Bureau
GERALD L. BARGER W. LAWRENCE GATES HANS NEUBERGER
U. S. Weather Bureau University of California at L.A. Pennsylvania State University
LOUIS J. BATTAN JOSEPH J. GEORGE CHESTER W. NEWTON
University of Arizona Eastern Air Lines University of Chicago
FREDERIC A. BERRY MAURICE H. HALSTEAD HANS A. PANOFSKY
Aerometric Research Inc. Navy Electronics Laboratory Pennsylvania State University
RoscoE R. BRAHAM, JR. BERNHARD HAURWITZ NORMAN G. PHILLIPS
University of Chicago University of Colorado Mass. Institute of Technology
RICHARD A. CRAIG SEYMOUR L. HESS RICHARD J. REED
Florida State University Florida State University University of Washington
GEORGE P. CRESSMAN HENRY G. HOUGHTON HERBERT RIEHL
U. S. Weather Bureau Mass. Institute of Technology University of Chicago
A. NELSON DINGLE WooDRow C. JAcoBs HENRY STOMMEL
University of Michigan Library of Congress Woods Hole Ocean. Instn.
GORDON E. DUNN HELMUT E. LANDSBERG VERNER E. SUOMI
U. S. Weather Bureau U. S. Weather Bureau University of Wisconsin
ROBERT G. FLEAGLE JAMES E. MILLER HARRY WEXLER
University of Washington New York University U. S. Weather Bureau
METEOROLOGICAL MONOGRAPHS, a serial publication of the American Meteorological Society, serves
as a medium for original papers, survey articles, and other material in meteorology and closely related fields;
it is intended for material which is better suited in length or nature for publication in monograph form than
for publication in the journal of Meteorology} in the Bulletin of the American Meteorological Society or in
Weatherwise. A METEOROLOGICAL MONOGRAPH may consist of a single paper or of a group of pa
pers concerned with a single general topic.
ISBN 978-1-940033-50-1 (eBook)
DOI 10.1007/978-1-940033-50-1
FLUCTUATIONS IN THE ATMOSPHERIC INERTIA: 1873-1950
El Sayed Mohamed Hassan
Cairo University'
TABLE OF CO~TE!\'TS will vary because of the variable storage of atmospheric
water vapor. Other effects are comparatively small.
1. INTRODUCTION ................. .
2. THE VARIABLE LOAD ............. . According to Bannon and Steele (1957), the seasonal
3. OCEAN YIELDING ................ . 1 variation is given by
4. THE ATMOSPHERIC INERTIA ...... . 1
5. THE DISTRIBUTION OF STATIONS .. 2 V(t) = - 0.17 cos 0 - 0.08 sin 0 (1)
6. DATA ............................ . 4
7. TABULATED VALUES ................ . 5 where 0 is the sun's longitude, varying from 0 deg on
8. HARMONIC EXPRESSIONS ........... . 27 1 January to 360 deg on 31 December.
APPENDIX .......................... . 27
A. List of all stations used in the computation. 27
B. Sea level correction. . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3. Ocean yielding
C. The mean of the pressure record ...... . 39
D. Rejected data ................................ . 40 In the frequency range under consideration here, it
would appear that the ocean responds to atmospheric
pressure as an inverted barometer (M unk and Mac
1. Introduction
Donald, 1960; §9.3). Consequently, departures of the
In a study of the meteorological contribution to total load on the sea floor are everywhere the same at
ward irregularities in the rotation of the earth, Munk a fixed time, but the departures vary in time because
and Hassan (1961) have evaluated the seasonal terms of the variation in the fraction of the atmosphere
in the moments and products of atmospheric inertia that lies above oceans. Let q (t) designate the value
0
and the power spectrum of the non-seasonal products of q(O, A; t) for oceanic areas. Then
of inertia. For that purpose, mean monthly values r
f.
were computed from all available station-level pres +
SV(t) q(O, A; t)ds q(O, A; t)ds
sures since 1873. Calculations were extensive and per
~ Land Oceans
formed on the IBM 709 computer at the University
(2)
of California at Los Angeles. The resulting time series
constitute a description of some integral properties of
Here Anm, Bnm designate the amplitudes of the
the global atmosphere-in particular, of the coeffi
spherical harmonics of an expansion of a function
cients of the second spherical harmonic of the variable F = q(O, X; t) over land and F = 0 over oceans.
surface pressure. In publishing the computed values,
Similarly, anm, bnm refer to the expansion of a function
we hope that this description of the atmosphere might
equal to 1 over oceans and zero over land. The latter
be useful to other investigators.
coefficients have been evaluated by Munk and Mac
Donald (1960, §A.1).
2. The variable load
Let q(O, A; t) designate the departure, from the 4. The atmospheric inertia
mean, of atmospheric load per unit surface area, as a
It is convenient to represent the inertial tensor in
function of colatitude 0, east longitude A, and time t.
terms of the dimensionless quantities 1/;1, 1/;2, 1/;a. Here
Let S = 411"a2 designate the surface area of the earth,
f/;1, 1/;2 are components of the atmospheric products of
and let ds = a2 sin OdOdA designate a differential sur
inertia divided by ( C - A), where A, A, C are the
face element. Then
three principal moments of inertia of the planet earth.
J
1/;1 is the component relative to an axis from the center
q(O, X; t)ds = SV(t) of mass through the meridian of Greenwich; 1/;2 is the
component relative to an axis 90 deg east of Green
Is the departure in total atmospheric mass, and this wich. if;a is the atmospheric moment of inertia divided
by C. For further details, we refer to Munk and Mac
1 This work was done while the author was visiting the Scripps
Institution of Oceanography. Donald (1960). We then have
1
2 METEOROLOGICAL MONOGRAPHS VoL. 4, No. 24
::c (p~)z ::c P0I P0o ::c pIl cos Ai p0o :z:: PII sinAi p~ I: P02 P0o
::c P0o P0I ::c (p~)2 ::c p~ coo A; p~ I: PII sin >-.-i p~ I: P20 P0I
::c P0o PII cos\ z: po1 piI cos A; z: (piI coo\) 2 :z:: PII sin Ai p: cos Ai E P20 p: cos A;
z: P0o p1I sin ~i z: p10 p: sinAi ::c p11 coo ~i p1I s1. n A· I: (p: ~in A; )2 I: P20 PII sin ).i
::c P0o P0z z: p10 P0z z: p~ coo A; p~ I: PII sinAi p~ E (p~)2
4= ::c p00 Pz1 cos ~i z: p10 Pz1 coo~; z: p11 cos ~i p2I cos ~i z: p11 a1. n x.i p21 cos Ai E P20 PzI cos "-i
::c Poo PzI s1. n ~i z: P0I pi sin A; z: PII cos A; p2I sm. A; :z:: p1I sin Ai p~ Bin Ai l:: P20 p~ sin >..i
::c p00 p22 cos z ~i z: P0I p~ cos 2 A; z: p11 cos A; p2z cos 2 A; z: PI1 sin ).i pi cos 2 X.i E P0z p~ cos 2 ).i
:t P0o p2z s1. n 2 ~i z: P0I p~ sin Z ~i z: p1I cos >..i Pz2 s1. n 2 Ai z: p1I s1. n Ai Pz2 s1. n Z ).i l:: P20 Pz2 s1. n 2 Ai
FIG. 1. Determinant A arising from the
:& IP~lz :& p10 P0o :& PII cos ~i P0o :z: p1I sin ).i P0o z: P0z P0o
:& P0o P0I I: IP~lz I: p: cos A; P0I I: pIl 81. 0 ).i P0I :& P20 P0I
:& P0o PII cos ~i I: P0I p: cos ~i :& (pIi COB Ai) Z :& PII sin ).i p~ cos ~i :& P20 p: cos Ai
:& Poo PII sin ~i I: P0I p: sin ~i I: p1I cos ~i PII ain ).i :& (p: sin~.1 )2 I: P02 PII sin)..1
:& P0o P02 :& P0I P02 :& pIi cos ~i P02 :& PII sin ).i P02 :z:: (p~l2
(.t.AlzI = :& P0o P21 cos ~i :& P0I pi cos ~i :& p11 cos ~i Pz1 coo A; :& PI1 sin ).i PzI cos ).i E P0z p~ cos A;
p:
:& p~ sin ).i :& p10 Pi sinAi :& pIi cos ).i p~ sin ).i :& PI1 sin ).i p~ sin Ai l:: P20 P2I sin A1.
:& p00 pi coo Z ~.,'1 :z:: P0I p~ cos 2 Ai :& P1I CO& ).i p~ cos 2 A; :& PII sin >..i p~ cos 2 ).i l:: Pz0 p~ cos 2 ';
:& P0o p~ sin 2 ).i :& P0I p~ sin 2 A; :& pIl cos >..i p~ sin 2 ).i l:: PII sin X.1. p~ sin 2 A.i :z:: P0z p~ sin 2. \i
FrG. 2. Determinant (AA)21 arising from the
f
- _!!_____ { 5. The distribution of stations
q(fJ, A; t) sin fJ cos fJ cos Ads
"'' (t) C- A Land The stations are not uniformly distributed (fig. 3).
+ iceans Ads } To evaluate surface integrals, the usual procedure is
qo (t) sin fJ cos fJ cos to contour the required quantities as well as possible,
to read off representative values for each 5-deg square
for example, multiply by sin fJ, and sum. The proce
dure is subjective and not suitable for machine calcu
lations. Assume a network of observation stations not
We now substitute for q0(t) from (2), and this allows uniformly distributed on the earth's surface. Denote
us to evaluate 1/1 (t) from measurements over land only:
1 the coordinates of station i by fJ;, A;, the total number
of stations being I. Let F;(t) be the atmospheric
(3) pressure departure for the continental station i
(F = 0 over oceans). It is required to approximate
F;(t) by a function
Similarly,
N
f;(t) = L L I[Anm(t)cos (mA;) +Bnm(t)sin (mA;)]
(4)
n=O m=O
and
such that
(5) (6)
DECEMBER 1960 METEOROLOGICAL MONOGRAPHS 3
l: p~ coo ki p~ :!: p~ sin >.i p~ l: P22 cos 2 >.i P0o :!: p22 S.l n 2 ki P0o
I: p~ cos ~i p~ :!: p~ sin ).i P0I :!: pi coo 2 ki P0I l: Pz2. n.n 2 >.i P0I
l: p~ cos ki ~ cos ki l: p~ sin >.i p~ coo ki :!: p22 cos 2 >.i PII cos >.i :!: p22 u.n 2 >.i p1I cos >.1
l: p~ cos >.i p~ sin >.i :!: p~ sink; PII sin ).i l: p22 cos 2 >.i pI1 sin >.i :!: P22 n.n 2 >.i PII Bin)..1
l: p~ cos '; p~ l: p~ sin >.i P20 l: p22 cos 2 >.i Pz0. l: P22 n.n 2 ki P20
.E (p~ cos >.i)2 :!: p~ sin >.i p~ cos ki I: p22 cos 2 >.i p2I cos >.; :!: Pz2. S.l n 2 ki p2I coo k;
l: p~ coo >.i p~ sin k; I: (p~ sin >.i )2 .E Pz2. cos 2 >.i p2I S.l n '; :!: P2z. o.m z. >.i p2I sm. ki
l: p~ cos ki p~ cos 2 ki :!: p2I s1. n >.i p~ cos 2 .i I: 1p 22 cos 2 >.i) 2 :!: p22 n.n z. >.i p22 cos 2 >.i
l: p~ cos >.i p~ sin 2 >.1 :!: p~ sin >.i p~ sin 2 >.i :!: p~ cos 2 '; Pz2 st. n 2 X.i :!: ( p22 sm. 2 >.1) 2
right-hand side of the system of eq (7).
l: Pzl sin >l. .P0o :!: p22 cos 2 >.i p00 z; P22 sin 2 ).l. Po0
l: Fi p~ z; P21 sm. ';PI0 E p22 •cos 2 >.i p01 z; P22 sin 2 ).l. P0I
.E F1 p1I cos >-; .E Pz1 st. n x.i p 1I cos x.i l: p22 cos 2 >.; p1i cos >-; :!: Pz2 st. n 2 ).i PII cos ),i
l: F.l p11 sin ).l. :!: PzI S.l n li p1I s1. n x.i l: p22 cos 2 >.; p-11 sin '; :!: Pz2 s1. n 2 Xi PI1 sin ).i
,!; F.l P20 Z: p~ sin >.i p~ Z: p22 cos 2 li.; p20 :!: Pz2 st. n 2 x.i p20
i: F.l P2I cos,\.l z; Pz1 st. n \i p2I cos ki z; p22 cos 2 >.1 PzI cos ,1 .E P22 sin 2 ).i p2I cos xi
z; F.l P2I sin x.1. Z: p(!p ~ sin},1)2 .<: p~ cos 2 >-; p~ sin >.1 :!: Pz2 s1. n 2 \i PzI sin ).i
i: F.l Pz2 cos 2 x.l I; sin hi p~ cos 2 X.i z: (p-; cos 2 >l. .)2 :!: P22 sin 2 )..i p22 cos ).i
:!: F.l Pz2 sin 2 >.l. 2: p~ sin Ai p~ sin 2 \ i: Pz2 cos 2 A.i p22 st. n 2 \ :!: (p~ sin 2 >.1)2
system of eq (7) when solving for A21•
is a minimum. The calculation is carried out for one
month at a time. To simplify the notation, we write
F;, j;, A,m, Bnm for F;(t), j;(t), A,m(t), Bnm(t). Fur
thermore, Pnm (cos fJ;) is designated by (pnm),. By (7)
to{~::}.
minimizing (6) with respect we obtain
and the coefficients Ak;, Bk; are independent. Hence,
the value of any one coefficient (A 2\ say) does not
depend on the number of terms to which the expansion
is carried.
or For the case of nonuniformly distributed stations,
the entire system of eq (7) has to be solved. The value
of any one coefficient (A 21, for example) now depends
on the number of terms to which the expansion is
carried.
:E LN L" {[Anmcos (mX;)+Bnmsin (mX;)](pnm)} We have chosen to evaluate all terms up to and in
i n=O m=O cluding degree two. This involves the nine coefficients:
X [ { c.o sjX~ ; f (Pk0i ] . Ao0; At0, At1; A2°, A2\ A22.
Slll)Ai Bt1; B2\ B22.
For the case of uniformly distributed stations, the The coefficients A0°, A2°, A2\ B21 are required in the
summation of terms containing Pnmpki will vanish expressions for t/1, and it seems reasonable to choose a
except when n = k and m = j. In that case, system (7) that includes all terms up to degree two.
4 METEOROLOGICAL MONOGRAPHS VoL. 4, No. 24
ANTARCTIC
9()'Yf 0
FIG. 3. Positions of stations used in the computation of excitation functions.
We find that (a) Atlantic Ocean: Spitzbergen, Iceland, Green
land, Resolution Island, Baffin Island, Perrin
(8) Islands, The British Isles, Ireland, Cuba, Haiti
and Trinidad.
where A is a ninth order determinant given by fig. 1. (b) Indian Ocean: Ceylon and Madagascar.
The summation is all terms of the determinant over i. (c) Pacific Ocean: The Japanese Islands, The East
The terms of (AA)nm can be found from A by replacing Indies, Indonesia, New Zealand, and Tasmania.
the column L: (pnm cos mXi)2 by the left-hand side of
eq (7). As an example, (AA)l is given by fig. 2. (2) Station-level pressure was used whenever
reported.
(3) When only sea-level pressure was reported, the
6. Data
correction to bring it up to station level was calcu
All the data have been taken from the Smithsonian
lated. The known station height and the mean monthly
Miscellaneous Collections, Vols. 79, 90, and lOS and
temperature were used, and the correction was calcu
from the World Weather Records published by the
lated with the help of the Smithsonian Meteorological
Weather Bureau. The latter publication is a continua
Tables (Fifth Revised Edition). The record was used
tion of the Smithsonian volumes. In the majority of
only when the maximum seasonal variation of the
the cases, the pressure in all these volumes has been
correction term was less than 10 per cent of the maxi
published for the station level. All the data were
mum variation of the sea-level pressure. This cri
accorded equal weight, even though there are large
terion limits the use of sea-level pressure to stations
differences in precision. In spite of the obviously
rather near sea level. When the station level changed
enormous effort to produce a homogeneous record for
during the length of the record, the lowest value was
any one station, there are, in some cases, variations
taken for this calculation.
in the level of the instrument. In some cases, correc
(4) Calculations were started for the year 1873, the
tions are entered explicitly. In other cases, it is not
first year when the number of stations exceeded SO
clear whether corrections have been made. A list of
(fig. 4). A graph showing the growth of number of
the large number of minor decisions in the reduction
stations is shown in fig. 4, and a map showing the
of the observations are included in the appendix.
positions of all stations used in the computation is
The following rules are followed in processing the
shown in fig. 3.
data:
(S) The mean pressure was calculated on an IBM
(1) Island stations were not used when the area of 6SO for each station using all published monthly
the island was less than ten degrees square, except means whenever possible. In some instances, the
for islands close to continents. The islands used are: homogeneity of the whole record could not be ascer-
DECEMBER 1960 METEOROLOGICAL MONOGRAPHS 5
500r----,,---,l---yl---yi---,-I---,-I-"-TI---..-., the computation according to the following scheme:
.. · 0 90E 180E 270E 0
.· .
Northern hemisphere Col. 3 4 6 5
Southern hemisphere 7 8 10 9
-
400 f-
....... Thus,
. .
I- ... ······· - column 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8
... ..
.. !' •••••• -
300 t- ..
.. means that in the northern hemisphere there is 1 sta
.. - tion between 0 and 90E, there are 2 stations between
t- ..
.. . 90E and 180E, 3 stations between 0 and 90W, and 4
..
stations between 90W and 180W; in the southern
200 - -
hemisphere there are 5 stations between 0 and 90E, 6
.. · - stations between 90E and 180E, 7 stations between 0
I- and 90W, and 8 stations between 90W and 180W.
._ .. .·· The remaining columns give
100 -
I- -
These have been evaluated according to eq (3) and
I I _l _j_ I I I (5) for the following four models:
1880 1900 1920 1940
FIG. 4. Number of stations used in the computation of I. Col. 11-13 Anm, Bnm, from (7), V=O
excitation functions. II. 14-16 Anm, Bnm, from (7), v~o
III. 17-19 Anm, Bnm, from (8), V=O
tained, and the mean was calculated for different IV. 20-22 Anm, Bnm, from (8), v~ o.
portions separately. In most cases, the mean was
based on more than ten years of record. Only in 60 out
Thus, for Models I and II, no allowance is made for
of 567 stations was the record shorter than a decade.
the uneven distribution of stations. In Models III
(6) The reported data were used without correc
and IV, this has been taken into account. For I and
tions. Data were rejected only in cases of obvious
III, the variation in water vapor is neglected. The
errors. No attempts were made to smooth the records
time series plotted in fig. 5 (pp. 26-27) correspond to
or to correct the data beyond the corrections that
Model IV.
have already been published.
The year 1999 at the end of the table gives if/ calcu
lated from the means of all station records. For com
7. Tabulated values parison, we include lit calculated by taking the means
The first two columns refer to year and month. of allif/(t). This mean has been taken over two periods,
Columns 3 to 10 give the number of stations used in 1873 to 1950 and 1900 to 1950.