ON THE RELATION BETWEEN ANNUAL PRESSURE AND TEMPERATURE VARIATIONS

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LIBRARY OP HSW YORK UWIYERSITI

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ON THE RELATION BETWEEN ANNUAL PRESSURE AND TEMPERATURE VARIATIONS

JEROME SPAR ■ NOVEMDfiRj 1 9 *9

A dissertation in the Department of Meteorology submitted to the faculty of the Graduate School of Arts and Science of New York University in partial fulfillment of the requirements for the degree of Doctor of Philosophy.

CONTENTS Section

Page Abstract

1

Introduction

1

2

The perturbation equations

6

3

Solution of the perturbation equation

13

4

The effect of friction

22

5

The integrated temperature pertur­ bation (Q)

23

6

Spherical harmonic analysis of Q

27

7

The theoretical distribution of semi­ annual surface pressure change

31

Discussion of results

3*

8

Acknowledgements References

30\o44

38

ABSTRACT

A quantitative theory is derived showing the relation between temperature and surface pressure oscillations.

By

extension of methods used by Jeffreys and Bartels, a solution is obtained for the linearized hydrodynamic equations in terms of spherical surface harmonic functions.

The solution

gives the surface pressure variation as a function of an integral containing the temperature variation throughout the atmosphere.

The theory is applied to the problem of evalu­

ating the amplitude of the annual surface pressure oscilla­ tion over the whole sphere.

The temperature integral is

computed from mean data for January and July and subjected to spherical harmonic analysis.

The mean semi-annual surface

pressure change is then evaluated from the coefficients of the harmonic series for the temperature integral and compared with observations.

1. 1.

Introduction If averages are taken over a sufficiently long period

of time, the atmospheric pressure at any place is found to exhibit certain marked periodicities.

The best known of

these periods are the diurnal, the semi-diurnal and the annual.

Although these atmospheric oscillations have been

the subjects of numerous investigations ^ , both statis­ tical and theoretical, much work remains to be done on the problem. The theoretical foundation for the study of atmospheric oscillations was established by Laplace

2

who

demonstrated the mathematical analogy between the theory of the oscillations of the atmosphere and the theory of tides in a homogeneous ocean.

Laplace obtained solutions for the

free and tidal oscillations based on the assumptions that the vertical velocity component is zero and that the atmos­ phere is isothermal and experiences only isothermal changes of state.

About one hundred years later Margules [1]

showed that the diurnal and semi- diurnal pressure oscilla­ tions could be explained as a resonance phenomenon between the free Laplacian oscillations of the atmosphere and a forced oscillation connected with the periodic temperature variation.

A survey of the observations and theories of periodic pressure variations may be found in Hann-Surlng, Lehrbuch der Meteoroloeie. Fifth Ed., R. String,Ed. Verlag W. Keller, Leipzig. 1938. 2 Mecanique Celeste,

livre IV.

Ch. 5. 1799

In his work Margules retained Laplace's assumptions. An important contribution to the theory of thermally induced, periodic pressure variations was published in 1926 by Jeffreys [2].

In this paper Jeffreys introduced

the device of integrating the hydrodynamic equations over the whole vertical extent of the atmosphere, thereby elimi­ nating the vertical velocity term in the continuity equation. The solution obtained by Jeffreys for the annual pressure variation was restricted to the cylindrical case and the Coriolis parameter was assumed to be constant.

Jeffreys

calculated the amplitude of the mean annual pressure varia­ tion for one point on the earth's surface (in Central Asia); but his numerical solution was based on an assumed form of the mean annual temperature variation, as a function of height and radial distance, instead of on observations. The theory of atmospheric oscillations was further ad­ vanced the next year with the publication of Bartels 1 paper on atmospheric tides [3]. Bartels employed the method of spheri­ cal harmonics which had been first introduced into the Laplacian theory of tides by Hough [4] and which had later been applied by Chapman [5] to the atmosphere.

Jeffreys'

device of integrating the hydrodynamic equations was used to eliminate the vertical velocity.

The assumption of Laplace

and Margules that changes of state are Isothermal was replaced by the more realistic assumption of an adiabatic law. assumption of an Isothermal atmosphere was eliminated.

The Thus

3.

Bartels' model was the most realistic one that had been in­ vestigated up to that time.

However, Bartels was concerned

primarily with the free oscillations and the oscillations connected with tidal forces.

He discussed the thermally

induced oscillations only briefly. A somewhat different approach to the problem of surface pressure variations was made by Wexler [6] who attempted, on the basis of the Brunt-Douglas theory of the isallobaric wind, to calculate the pressure rise which occurs at the surface of the earth as a result of radiational cooling of the air. Wexler's study was not restricted to periodic variations. The solutions obtained by Wexler are approximate and apply only to the cylindrical case.

Recently F. H. Schmidt [7]

extended Wexler's theory and obtained exact solutions for a plane earth and for a special periodic form of the temperature variation.

Schmidt has shown that his results are in good

agreement with those of Jeffreys. In this paper a dynamical theory of the relation between periodic variations of temperature and surface pressure is derived for a spherical earth by an extension of the methods of Jeffreys and Bartels.

The theory is tested by calculating

the amplitude of the mean annual surface pressure variation from the amplitude of the observed mean annual temperature variation over the whole earth and comparing it with observa­ tions. It appears physically reasonable that the annual

periodic component of the pressure variation is controlled by the annual periodic temperature variation.

The latter

is determined primarily 3 by the seasonal variation of in­ solation and the distribution of continents and oceans, factors which are external to the atmosphere and therefore independent of the pressure variation. ment about causality is offered here.

However, no state­ The pressure and

temperature variations are considered to occur simultaneously. It is impossible to construct a realistic model of the thermally induced pressure oscillation without talcing into account the irregular temperature distribution resulting from the asymmetrical arrangement of land and water masses

4

The analytical representation of the observed temperature distribution which was necessary for the solution of the problem was accomplished by spherical harmonic analysis. Although it has found wide application in other branches of geophysics, particularly geomagnetism, spherical harmonic representation has rarely been utilized in meteorology

5.

3 The temperature field is also affected by the motion of the atmosphere and is therefore, to some extent, depen­ dent on the pressure field. 4

5

The importance of the continents and oceans for the annual pressure oscillation has been demonstrated in a statistical study by Wahl [8] who showed that the mean annual mass exchange between continents and oceans must be much larger than the interhemisphere exchange. Spherical harmonic analysis and its application to the problem of atmospheric osclllattns is the subject of a comp^rehensive but unpublished report by B. Haurwitz and R. Craig on research conducted at the Massachusetts Institute of Technology.

One of the major advantages of spherical harmonics is the feasibility of subjecting the observed distribution of meteorological variables to mathematical analysis rather than having resort to oversimplified and unrealistic models. In the derivation of the theory the hydrodynamic equa­ tions for an inviscid atmosphere are linearized by the per­ turbation method.

A single second-order, inhomogeneous

partial differential equation of the type encountered in the theory of linear forced vibrations is then derived for the pressure perturbation.

The temperature perturbation

appears in the equation as a kind of force function.

The

particular solution obtained for the differential equation represents the forced pressure oscillation which has the same period as the temperature oscillation.

In the case that

was studied this period is one year. The use of perturbation method always raises the per­ tinent question concerning the validity of the method in the case of oscillations of finite amplitude.

The empirical

verification of the solution may be regarded as a test of the perturbation method.

However, it is not a completely

conclusive test since failure of the theory to provide a correct solution may be due to several other important as­ sumptions.

For example, friction and topographic irregu­

larities are neglected and an equilibrium state of no motion is assumed.

Also, certain assumptions and approximations

are employed to describe the temperature disturbance.

A

more direct, although still not entirely adequate, check on the applicability of the perturbation method to this problem, based on certain properties of the spherical harmonic coeffi­ cients, is discussed in section 3» 2.

The perturbation equations. If vertical accelerations are neglected, the equations

of motion for an inviscid atmosphere may be written

_

-

I

Tip* ax

M

o>-°=

% + -^il

The continuity equation in spherical coordinates takes the following form? (4) —

+ - 4 --- — (fVreosf) +— —

c

In the equations above u is the zonal wind component (positive toward east);

v is the meridional wind component

(positive toward north); w is the vertical wind component (positive toward zenith);

is the angular

7.

velocity of rotation of the earth; p and p are density and pressure; g is the acceleration of gravity.

The distance

from the center of the earth, r, is assumed throughout to he equal to the mean radius of the earth. The asterisks denote instantaneous values of the quan­ tity and the subscript zero is used to denote the equilibrium value.

Terms without subscript or asterisk are deviations

from the equilibrium value and will be treated as perturbation quantities.

Thus

■u* = u*+ u . V V = V 0+

w* = w0 + w P* = P0+ P

P* w - P + P H

The simplest possible equilibrium state, namely that of rest, was selected for the model to be studied.

The

writer had hoped to be able to Investigate more complicated models, for example, one with some kind of zonal motion in the equilibrium state.

However, it soon became obvious that

altering the equilibrium conditions in this way greatly in­ creased the mathematical difficulty of the problem. project was therefore abandoned.

This

At the present time it

is not possible to estimate the extent to which a change in the equilibrium conditions affects the final solution. The equilibrium state is defined

by the following set

of identities: 3-fc = 94» " ^ = 3t ~ a* "

" 3-ts U o - vo-ur0» °

Upon substituting (5) and (6) In equations (1), (2) and (4) the following linear perturbation equations are obtained: (la)

—

—

2 to i/s/* /

zfr

--- -—

'

(2a)

r/f

coi/

3^ P A

y- Zcou_$/i, d = - -3 J t

r

fo US'

(4a)

£t 3

'

r^

Cosyf

rS

9f

¥? 3

rtosd

= O

Equations (la), (2a) and (4a) were derived under the usual assumption that quadratic perturbation terms may be neglected Also the term 2anrcos

- t

V I

I

PT

s (J .+ T ?

3*

or (9b)

0

9-t

' *0 r

*

t ’ST v -- d"i J

T 2- 3±.

If quadratic and higher order perturbation quantities are neglected, (9b) reduces to (10)

4-

■3ft

\

'a?

^

3t

f * J i----3' “-