J. STEWART California Institute of Technology

T he outstanding features of the general circulation of the atm osphere are the belts of w esterly winds and, on the equatorial side of these, the system of sem i­

permanent sub-tropical high pressure areas. In a previous paper1 the author has dis­

cussed the problem of the formation of such high pressure system s. In particular, it was shown th at these system s probably represent dynam ically stable concentrations of vorticity similar to the Karman “vortex street” which is formed behind any two- dimensional bluff body over a wide range of values of the Reynolds number. It now appears that a further exam ination of the periods of the characteristic oscillations of such system s is of considerable interest. I t is seen th at the period of these oscillations is of the order of m agnitude of years. This indicates th at oscillations of this type may be of importance in the calculation of the long period displacem ents of the Pacific or Azores high pressure system s.

It is believed th at this is the first time that atm ospheric m otions have been dis­

cussed which have a period of the order of magnitude of, but different from, a year.

Since the weather shows large variations from one year to another, it is apparent that such m otions m ust exist; and, since the non-seasonal variation of the only external parameter, the solar energy input, is very small, these long period m otions m ust be explainable in terms of the free oscillations of the earth’s atmosphere.

It seem s th at the horizontal field of motion is of primary importance in determ in­

ing the motion of these large scale system s; so it is assumed th at the atmosphere can be treated as a single layer of fluid of constant density with the vertical velocities being of small importance so th at the pressure can be determined from the hydro­

static equation. It is also assumed th at the apparent acceleration is negligible when compared to the Coriolis acceleration. In addition the effects of friction and of the variation of the Coriolis parameter with latitude are neglected. T his latter factor means th at the fluid m otions considered are those taking place on a rotating disc rather than on a rotating sphere.

T he notation used in the discussion is as follows:

x , y = Cartesian coordinates on a rotating disc, u = velocity in x direction, v — veloc­

ity in y direction, to = angular velocity of the disc, h = depth of the fluid, g== accelera­

tion due to gravity.

If the motion could have been started from rest w ith a uniform depth ho, the principle of conservation of the absolute vorticity states that

dv dll 2w

--- = — (h — ho). (1)

d x d y ho

* Received August 4, 1943.

1 Stewart, H. J., Proc, Nat. Acad, of Sci., 26, 604 (1940).

PER IO D IC PR O PER TIES OF A TM O SPH ERIC PR ESSU RE SY STEM S 263

T his equation can be further simplified by the introduction of dim ensionless variables, X = 2wx/\Zgho, Y = 2u y / y / g h o and ij = (h — ho)/ho. W ith these new variables, Eq. (3) becomes

d2rj d-T]

— + — - V = 0. (4)

d X2 d Y2

In terms of the dim ensionless depth and dim ensionless velocities defined by U = u / \ / g h0 and V = v / y / g h 0, the geostrophic wind equation can be rewritten as

Based on a homogeneous atmosphere having a mean sea-level pressure and density of 1.013 X I 06 d y n e s/c m2 and 1.22 X 1 0~ 3 g m /cm ’ respectively, the same as the stand­

ard atmosphere, the characteristic velocities and distances used above to produce d i­

m ensionless variables are y / g h0 = 2.87 X lO4 cm /se c and Vg h j2 < x = 1.97 X 1 08 cm. A t a distance of 2000 km. from the center of the Pacific or Azores high pressure system s, the characteristic velocity is of the order of 10 m eters/second. Since A fi(l) = 0 .602 , this indicates th at these anticyclones have a strength such th at a is approxim ately 0.06.

If the interaction between the northern and southern hemispheres is neglected, the ring of subtropical anticyclones can be roughly represented by N equal an ti­

cyclones of the type given b y Eq. (5) which are placed on a ring of radius a and spaced 1 Grey, Mathews and MacRobert, Bessel Functions, Macmillan and Co., London, 1931.

264 H. J. STEW ART [Vol. I, N o. 3

From Eq. 5 the dimensionless velocities in the radial and tangential directions, u r and m respectively, are given distributed cyclonic vorticity to th e north of the w esterly winds. I t is easily seen that this cyclonic vorticity tends to produce an eastward displacem ent of the subtropical highs. It appears th at these two displacem ents cancel one another so th at the sy s­

The velocities of the vortices m ay be calculated as before by Eq. (9). If the displace­

ments are small, the changes in velocity of the iVth vortex from the equilibrium value indicated by Eq. (10) m ay be written as

Expressions similar to Eq. (1 2) for the velocities of the other vortices could be w ritten from sym m etry. T hese would form a set of sim ultaneous differential equations for the displacem ents. discussed by considering the normal modes of oscillation. From the sym m etry condi­

tions, the displacem ents in each normal mode m ust be of the form A r n = A r Ne in'f and

3 Lamb, H., Hydrodynamics, 6th edition, Cambridge University Press, London, 1932, 220.

266 H. J. STEW ART [Vol. I, N o. 3 also be seen that for the specified values of <p, A is always a purely imaginary quantity, B is always real and not less than zero. From Eq. (17), the condition th at the frequen­

cies be real is th at C be real and non-negative. Complex frequencies, of course, charac­

terize system s in which the am plitudes increase with tim e and are thus unstable. N ow C is alw ays real and is always positive for N < 7. If N = 7, C is alw ays positive if a > 71.

Conclusion. T he present calculations cannot be considered as a quantitative theory of the oscillations of the sem i-permanent high pressure system s; th ey m ust be con­

sidered rather as an existence proof. Since the essential features of the model, vorticity concentrations at distances of roughly 1 0 , 0 0 0 km., are also found in the atmosphere, motions of this type m ust exist in the atmosphere. I t m ight be expected th at the effects of coupling between the system s of the Northern and Southern Hemispheres and of any cyclonic vorticity concentrations on the polar sides of the westerly winds

1943] PER IO DIC PR O PE R TIES OF ATM O SPH ERIC PR ESSU RE SYSTEM S 267

Table 2.

Normal Modes of Oscillation for N = 3 , a -3.0,a=0.06.

2ir/3 4 tt/3

— i A 0.002248 -0.002248

B 0.00414 0.00414

C 0.001182 0.001182

P 0.000268 0.000002 -0.000268 -0.000002

T-days 1,865 2.5X106 1,865 2.5X105

(iSOy

Ar.v -0.53» +0.53» +0.53» -0.53»'

Table 3.

Normal Modesof Oscillationfor 7V=3, a=3.5, <*= 0.06.

2x/3 47r/3

— i A 0.000859 -0.000859

B 0.001467 0.001467

C 0.000375 0.000375

P 0.000096 0.000007 -0.000096 -0.000007

T - days 5,200 7.1X10* 5,200 7.1X10*

oAO.v

A rN -0.51» 0.51» 0.51» -0.51»'

would be to decrease the period of the shortest oscillation and to introduce additional natural frequencies. N o attem p t has as y e t been made to estim ate the magnitude of these effects or of the errors involved in using velocity distributions corresponding to vortices on a rotating disc rather than to vortices on a rotating sphere and in neglect­

ing the seasonal variations in the strength of the semi-permanent high pressure sys­

tems. I t is suggested that the present calculations m ay prove useful as a guide for the statistical analysis of empirically obtained data.

268

— N O T E S —

ON HERZBERGER’S DIRECT METHOD IN GEOMETRICAL OPTICS*

By J. L. SYNGE (Ohio Slate University)

1. Introduction. In recent papers M . H erzberger1'2 has developed a “direct m eth od ” for analytical ray-tracing through an instrum ent of revolution. A t the end of the first paper he refers to H am ilton ’s m ethod, which he says “leads to an elim ina­

tion problem, hitherto unsolved.” Nevertheless the question arises: W hat is the connection between Herzberger’s approach and th a t of H am ilton? T his question is b est answered b y attacking Herzberger’s problem b y the m ethod of H am ilton.

As we shall see, this is quite feasible. Indeed, if we combine Herzberger’s “direct m ethod” with H am ilton ’s character function we obtain a very powerful technique.

Section 2 contains the formulation of the problem of determining the Herzberger