Hydrodynamic Equations of the Sea
3.4 Effects of Spin
Principles of Ocean Physics
There are two primary effects of the earth’s spin, both classified as appar- ent or fictitious forces-the Coriolis force and the centrifugal force. Here we shall use spin to describe the earth’s daily motion about its axis, and ro- tation to describe its annual motion about the sun. In later chapters, howev- er, we shall speak more loosely about rotations, meaning spin all the while.
We will also neglect other very small accelerations caused by nonuniform spin rates brought about by redistributions of mass and hence moments of inertia, both interior to the earth and within the atmosphere. As an aside, it is interesting to note that observed small variations in the length of day are well correlated with seasonal redistributions of the earth’s atmosphere as a result of changes in solar heating and orbital eccentricity throughout the year.
To derive the fictitious forces, let fl be the angular velocity of the earth’s spin relative to the ‘‘fixed” stars, i.e., inertial space. This vector points north along the north-south axis, and the earth’s spin is thus right-handed, having positive helicity. Its scalar value,
fl
= I f lI ,
isfl
= 2741+
1/365.24) rad day-’= 7.292 x rad s - ’
.
(3.15)The correction term in parentheses represents the fact that during a com- plete solar day, in order to bring the sun overhead again, the earth must spin through an additional angle approximately equal to the angular change in- curred during that day’s rotation about the sun. This rotation rate is there- fore measured with respect to inertial space. Motions taking place on the spinning earth, when viewed from a coordinate system fixed on the earth, will clearly be different than when viewed from inertial space. We will there- fore require relationships between derivatives in inertial and rotating coor- dinate systems.
Let A be an arbitrary vector, with Cartesian components A,, A,, and A , in an inertial frame and A:, A;, and A ; in the spinning frame (which is the one from which we shall view our dynamics). Then in the inertial frame we may write
A =
2, +
:Ay+
/&I,, (3.16)while the same vector, when viewed from the frame spinning at angular ve- locity fl has the representation
A = PA; + P A , , ’
+
k ‘ A ; . (3.17)Let ( d A / d t ) i be the total time derivative in the unprimed (inertial) frame, in which Newton's equation holds; then
- T ' -
dA,' dA; dA,'+ k'
-+ J I
-- dt dt dt
(3.18)
dT' dj^' dk'
dt dt dt
+ - A : + - - ; + - A ; ,
where the second expression accounts for the fact that the primed unit vec- tors are rotating in the coordinate system as well. Note that dT'ldt is the velocity of
T'
due to its spin; from the diagram of Fig. 3 . 1 , one concludesa
Fig. 3.1 Diagram for derivation of the rate of change ot a unit vector in a rotating coordinate system.
that di^'/dt = fl x
T ' ,
and similarly for dj^'/dt and dk^'/dt. The first three terms in Eq. 3.18 give just the rate of change of A as observed in the rotat- ing coordinate system. Thus we may write(3.19)
68 Principles of Ocean Physics
or in general, ( d / d t ) ; = ( d / d t ) ,
+ n
x , which is a kinematic relationship giving an inertial derivative in terms of a derivative observed in the rotating frame, plus a temporal change induced by spin. Thus if A = x , the position vector of a fluid element, then(g);
=(z), +
51 x x . (3.20)The left side is the velocity in inertial space while on the right is the velocity as observed on the spinning earth. Thus
u; = u,
+ n
x x . (3.21)If we apply the kinematic relationship to the velocity u; itself, we obtain
(3)
= (%)r+ n
x u ; ,dt j
(3.22)
which, upon substituting Eq. 3.21 into Eq. 3.22, yields
(3)
dt i =(f>,
[u,+ n
x X I+
a2 x [ur+ n
x x1
=
(%), + n
x(g),
+ n x u , + n x ( n x x )=(&),
+ 2 3 x u , + n x0
n x x.
(3.23)The term on the left-hand side of Eq. 3.23 is the acceleration in the inertial frame; on the right side are the accelerations observed in the rotating frame- the apparent acceleration, (du,/dt),
,
the Coriolis acceleration (which acts at right angles to both spin and velocity), and the centripetal acceleration (which acts outward at right angles to the spin axis at a rate quadratic ina).
We may rewrite the latter by inspecting Fig. 3.2, which represents the gravitational and rotational forces acting on an ellipsoidal earth whose radi- us, R e ( @ , varies with colatitude, 8. The negative of the double cross-product in Eq. 3.23 points in a direction perpendicular to the spin axis, as indicated by the unit vector, j, in that direction, and the resultant centripetal accelera- tion has the effect of changing the gravity vector go to a new effective gravi- ty, gel-, as indicated by the force parallelogram in Fig. 3.2:I
C 3
p"
-a x
(i-2x
Fig. 3.2 Force diagram for the effective gravity, g e f f , on an ellipsoidal earth.
&JJ = go Q 2 R e Sill 8 b , (3.24)
where, in terms of unit vectors in the directions of increasing r and 8, 6 = ? s i n 8
+
g c o s 8 . (3.25) At the equator, this value of effective gravity is less than the mean gravita- tional acceleration, go = 9.80 m s - ~ , by approximately 0.35% for an aver- age earth radius, R e , of 6.371 x l o 6 m. The equatorial flattening of 11298.257 implies that the equatorial radius of the earth is about 21 km larger than the polar radius. By the process of isostatic compensation, the plastic spinning earth has apparently adjusted itself to the centrifugal force over geological times by assuming the shape of an approximate ellipsoid of revo-70 Principles of Ocean Physics
lution in which g,,- is very close to the perpendicular at the local surface.
This figure of revolution is termed the reference ellipsoid and is used as the basis for measuring anomalies in both gravity and geoidal heights.
The centripetal acceleration can be derived from a potential function, @(r,O)
= - ( Q 2 r 2 / 2 ) sin2 0 , which can be added t o the spherical potential of Eq.
3.14 to form an effective potential, ,: ,,@.
Gm, n2r2
aef.
= - - - - sin2 8.
r 2 (3.26)
Clearly the centripetal acceleration and the ellipticity of the earth must be included in the specification of the equipotential surface that defines the ma- rine geoid. However, higher order terms must be incorporated into the geo- potential for precision work in marine geodesy (cf. Section 3.6).
During the manipulations involving equations written in inertial and rotat- ing frames, the meaning of the derivative, dldt, as applied to x(t) did not require that a distinction be made between the ordinary and the convective derivative, since x is a function only of time. However, the total derivative of Eq. 3.22, in which the velocity is a function of space and time, requires that distinction to be made. The proper writing of Eq. 3.23 for the accelera- tion is then
(””)
Dt = (%)r+
u , . V U ,+
20 x U,+
il X (f? x x ) , (3.27)i
where all quantities on the right-hand side are specified in the rotating frame.
This form, in which the time derivative contains four terms describing dis- tinct physical effects, will henceforth be used in the fluid dynamics of the ocean.