Space and space-time geometries in atmospheric mechanics

Int. ~ Engag Sci. Vo~ 19, No. 12, pp. 1595-1600, 1981

0020-72251811121595-06~)2,00/0

Prinled in Grea~ Britain.

© 1981 Pergamon Pre'is Ltd

SPACE AND SPACE-TIME GEOMETRIES IN ATMOSPHERIC MECHANICS P. DEFRISE Universite libre de Bruxelles, Facult6 des Sciences, CP 218, Campus de la Plaine, Boulevard du Triomphe, 1050 Bruxelles, Belgique (Received 7 November 1980)

Abstract--Brief survey of various geometric structures of space and space-time that can be found in general atmospheric mechanics and in models of atmosphere: world alfine connections of a special type, flows, riemannian metrics and world quadratic forms. 1. DEFINITIONS AND BASIC PROPERTIES ALL THE structures we consider are in a 4-dimensional space-time with absolute time t, where we adopt local coordinates x ~ with x ° = t. By convention, Greek indices take the values 0. I. 2, 3 while Roman indices take the values 1,2, 3 only. The instantaneous spaces t = coast. are equipped with a riemannian metric g~j(x~), euclidian (Section 2) or not (Section 3). We will meet three types of geometric objects: (1) world atfine connections F~,, all s u b m i t t e d to t h e f o l l o w i n g c o n s t r a i n t s : a o a F,,# = F,, = F,~ (symmetry), F°, = 0 (invariance of time component), F ~ is the riemannian connection of g~i

(2)

flows, deformable or rigid (with respect to g~i); (3) world symmetric tensors G~e (or quadratic forms) of rank 3 or 4 s u c h t h a t Go = go. We enounce very briefly some basic properties of these objects (for developments see[l]), A n a.(line c o n n e c t i o n FT,,,, with its covariant derivative V, allows one to define the acceleration of a particle x ~ = x"(t): A ~ = ~uV.~ ~

(~

= dx~/dt)

A ° = O, A i = .fi + Fi yc,,.f,~.

(2) t3)

Since A ° = 0, A" is an instantaneous vector. There exists a decomposition of F~m into a pure strain and a rigid connection fl~,~; the only difference between F~, and f ~ i is •

1

is v.~

l~,, - F~,, = ~ g vog,,,.

(4)

A f l o w R is defined by its world velocity R~(x'r), where R ° -- 1. A connection F~",~is determined by the requirement that F include the dragging along over R" dt: F io,, = - c~,,,R ~ - F ,i, , s R ,~ F ~ = - ?,oR ~ + R ' ( O , , , R ~ + F ,i, , , R ).

These formulas express the fact that V,R ~ = 0. If R is taken as a reference body, the velocity V i of a particle x a = x a ( t ) and its kinetic energy ES Vol. t9, No 12--J

1595

1596

P. DEFRISE

K are V i = fc i - R i

(61

2 K = g~jV~V ~ = g~Y¢~£~.

(7)

The kinetic tensor g~#, of rank 3, is given by gkO : gOk : -- gsk R s ,

goo = giiRiRJ; (R k =

- gmkgmo).

(8)

The covariant acceleration, defined by means of the covariant derivative V of F (5), d OK : ~ V ~ Vi = d t cgxi

aK K = K ( x % ~ i ) --7; ~x

(9)

differs from Ai(3) if R is deformable. From a symmetric world tensor G~a of rank 3 or 4 (Gij = g,i) one can deduce a connection O~: 1

ok~u = ~ g

km

(O~G,.u + 8~Gm~ -@mG..).

(10)

Using a formula similar to (3), one obtains the vector ji

=

.fi

i "~'~ +O.~x x

(11)

which coincides with the vector defined by the Lagrangian derivative: d 8L c~L L L(x",Yc ~) J~ - dt OJc~ ----7; Ox =

(12)

2L = G~.~%f#.

(13)

F ° = 1, F k = -gk~G~o

(14)

where

The flow F

is orthogonal to the instantaneous spaces with respect to G~. If G~a is of rank 3, it is the kinetic tensor relative to F. In any case, Ok~ coincides with the rigid part fl~. (4) of F determined by F according to (5). One can say that G~, if it is of rank 4, endows the space-time with a riemannian structure, but the riemannian connection of G~ does not satisfy the second condition (1). Another interpretation, well known in general Lagrangian dynamics[2], consists in considering that ds = L(x% Yc~) dt makes the space-time a Finsler space. 2. A P P L I C A T I O N S

TO A T M O S P H E R I C

MECHANICS

In meteorology the atmosphere is generally referred either to the so-called absolute reference frame (following the Earth in its orbital motion without participating in its rotation) or to the Earth (relative reference frame). For the sake of simplicity, let us adopt as coordinates X 1 = A, X 2 = ~, X 3 = r

where ),, ~ are the geocentric longitude and latitude, r the distance to the Earth's center. The

Space and space-timegeometriesin atmospheric mechanics

159~

instantaneous (euclidean) metric is ds 2 = r 2 COS2 ~(dA )2 + r2(dq~)2 + (dr)2.

(15)

The "terrestrial" connection F, associated with the Earth according to (5), is given by F~2- F,iI = - t a n ~, F,3 - F~, = F~3 = F~2 = l/r, I

_

I

__

~

( 16t

F]l = sin ~ cos ~, F]l = - r cos 2 ~, F~2 = - r the other F~, being zero. This connection is flat and rigid (with respect to &). Referred to the Earth, the velocity of the air is (A, 4, ?) and the equations of motion (neglecting friction) are: 1 r 2 cos z ~ £ + 2r)t' cos ~o(/"cos ~o- r~b sin ~o) = - ~ a~p - a6I' - 2rfl cos ~o(~ cos ~ - r,~ sin ~)

r295 + 2rt:~b + r2A2 sin ~ cos ¢ = 1

a2p - a 2 ~ - 2r2~(D. sin ~ cos W

117)

/

1 ~:- r~bz - r~ 2 cos 2 ~ = - ~ a~p - a3~ + 2rail cos 2

where p is the atmospheric pressure, • the geopotential, p the density of the air and fl the angular speed of rotation of the Earth; the last term in the right member represents the Coriolis force. The first members of (17) are the covariant components of the acceleration, deduced from (3) and (16). It is easy to geometrize the gravity by incorporating it in the structure of space-time: ene replaces F'~ by another world connection F~,, the only difference being

i i = FooFoo ai~, hence F& -- ;#qb.

(18)

Indeed, in this new connection F, the acceleration of the air. in virtue of (3), is A~ = A~ + ai~.

tl9)

Another procedure, using (11) and (12), consists in passing from the kinetic tensor g,~ (15)to the tensor G~, with the only change Goo = goo - 2~, hence Goo = - 2qb.

i20)

In other words the kinetic energy K is replaced by the Lagrangian L = K - ~. Let us note the fact that the flow F associated either with g,,B or with G,,B by (14) is the same. The connection O deduced from G~B by (I0) is the same as F (18); it is not flat. Instead of the terrestrial connection (16), one could introduce the world connection associated with the absolute reference frame. 3. APPLICATIONSTOATMOSPHER1CMODELS In some atmospheric models, thanks to the shallowness approximation ( r - a ~ a ) . the metric (15) is replaced by a 2 cos 2 ~ (dA)2 + a 2 (d~)2 + (dr)2 where a is the (meanj radius of the Earth.

i21)

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P. DEFRISE

The instantaneous space becomes a riemannian space; in its riemannian connection, the only F~,~ different from zero are ]"112 = r~l = -

tan ~, F 2, = sin q~cos

(22)

which are the connection coefficients of the sphere. The curvature tensor (notations as in [3]) has components different from zero, namely Ri~ = -R~i~ = l, R~i] = - Ri2] = cos z ¢.

(23)

Contrary to (15), the metric (21) is decomposable (definition in[3]). One should note that the Earth is rigid with respect to (21). In the space-time, the terrestrial connection F~,,, similar to (16), has now (22) as only coefficients different from zero; it is not flat. With the metric (21), the equations of motion of the air with respect to the Earth, instead of (17), become 1

a 2 COS2 q~(,~"- 2A~btan ~0) = - o Olp - Ol¢ - 2al) cos ¢(f cos ~o- a~b sin ~o) 1

a2(ff + ,~2 sin ¢ cos ~) = - ~ ~: = _ 1

p

Ozp -

02(Y~

-

2a2J(f~ sin q~cos

(24)

03p - 03qb + 2 a , ~ f l c o s 2 ~.

Some arguments lead to a further simplification: the term containing fl cos2 ~o in the first and third equation (24) is deleted. The final equations can be directly obtained by modifying the Lagrangian, hence the associated ® (10). With respect to the absolute reference frame, the kinetic energy is Ka = ~1 [r 2 cos 2 ~o(/+ 0) 2 + r2~b2 +/21

(25)

lr 2 L,, = K. - qb,, = ~tr cos 2 ~¢(a'2 + 2 0 / ) + r242 + t;21 - qb

(26)

and the Lagrangian

where ~o is the gravitational potential I 2 ,.-x2

¢o=¢+~rs~

2

(27)

cos ~,.

Applying (11) or (12) to (26), Lagrange's equations give the equations of motion (17). Now one introduces the shallowness approximation as in (21) and takes lr 2 cos 2 ¢(/2 + 2I~X) + aZq~2 + t 2] - qb La = ~ta

(28)

instead of (26). With this new Lagangian, the equations of motion become, as desired, 1

a2cos 2 ~o(,('- 2~'ff tan ~) = - o ~jp - c ~ + 2a2flff sin ~ cos a 2 ( ~ + ,(2

1

sin ~ cos ~) = - ~ 32p ~:=

32~

1 __

p

33p

-

c93dp.

--

2a2fl/sin ~ cos

(29)

Space and space-time geometries in atmospheric mechanics

1599

If, in (28), the term 1/2f "2 is neglected, the Lagrange's equations leave unchanged the first t,~o equations (29), but the third one is the hydrostatic equation 0 = - 1 asp + a3qb . p

(~())

These are the so-called primitive equations[4]; one adds the spherical approximation ¢b = qblr), or ill(D= c92~ = 0 ,

O3~=g.

17'1)

When the hydrostatic approximation (30) is introduced, p is very often used as vertical coordinate, instead of r:

p = p(t, A, ¢, r).

I~2)

In some models the following metric is adopted a 2 c o s 2 ¢ (dA)2 + a 2 (d¢)2 + p 2g 2 (dp)2: g = c o n s t .

(33)

where the density p is taken as a function of p alone: p = p(p). The space is then riemannian, decomposable and not flat; in its riemannian connection, the F~,,, are zero except F{2 = F~l = - tan ¢, F~ = sin ¢ cos ¢, F~3 = - p 'p'; (p' = dp/dp).

34)

The world velocity of the Earth is

l,O,O, R 3 = ~ p ( t , A , ¢ , r ) . For the terrestrial world connection, the F,~,, different from zero besides (34) are r~o = ro,~ .

.a,R3, . r~o.

F32 =

a,R3, r~o - Fo3 3 -- - ~)~R 3 + p tp'R3, . !36)

F~, = - aoR 3 + R3(a~R 3 - p Ip'R3). The relative kinetic energy K is given by 2K = a 2 cos 2 ¢1'2 + a2q~2 +/9 2g 2(]~ _ R3)2.

t37)

The first two contravariant components of the relative acceleration, in virtue of (3) and (36), reduce to £ - 21.~b tan ¢, 4; + a'2 sin ¢ cos ~o.

!38)

They are also the first two components of the horizontal component of the acceleration, in the (~, ¢, p) as well as in the (A, ~#, r) coordinate system. With p as vertical coordinate, the first two equations of motion, corresponding to (29), are a2cos 2 ¢(,(' - 2A'4, tan ¢) = - (?rqb+ 2a21)~b sin ¢ cos ¢ a:(4; + 1." sin ¢ cos ¢) = - i~2qb- 2a2~,( sin ¢ cos ¢. One should note that the Earth is not rigid with respect to (33).

!39)

1600

P. DEFRISE

REFERENCES [1] P. DEFRISE, Bull. Soc. Math. Belgique 13, 6 (1961) and 23, 377 (1971). [2] J. L. SYNGE, Classical dynamics. Handbuch der Physik (Edited by S. Fliigge), III/I, Springer-Verlag, Berlin (1960). [3] J. A. SCHOUTEN, Ricci Calculus. Springer-Verlag, Berlin (1954). [4] K. H. HINKELMANN, Primitive equations (Lectures on numerical short-range weather prediction. World Meteor. Org. Sem.), Leningrad 1969.