Hydrodynamic Equations of the Sea
3.5 The Coriolis Force
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.
action of both terms. Behind their dynamics are three distinguishable but closely interrelated physical phenomena: (1) the rotation of the earth under a moving particle during its time of flight; (2) changes in the centripetal ac- celeration with variation in the particle's distance from the spin axis; and (3) conservation of the particle's absolute angular momentum, i.e., the sum of its planetary-induced momentum and its relative fluid angular momen- tum. We shall discuss each of these in turn.
Consider a freely rotating, frictionless pendulum of the type developed by Foucault, swinging from a suspension at the North Pole; take the plane of the swing t o coincide initially with the Greenwich Meridian, (9 = 0") at t = 0 (see Fig. 3.3). During its rotation at the rate Q (which is 15" h - ' as measured in solar time), the earth will turn under the pendulum, so that af- ter time t , the reference meridian will have advanced through an hour angle, Qt. An observer in inertial space will see the pendulum continuing to swing in the same plane in which it started, but to an observer on the rotating earth near the pole it will appear that the plane of the pendulum's motion has ro- tated t o the right, i.e., clockwise. After 12 hours, the plane of rotation will have rotated through 180" and will once again coincide with the original plane of oscillation, passing through the Greenwich Meridian. Thus the spin of the earth apparently displaces a particle in the clockwise direction (in the Northern Hemisphere) for this purely meridional motion, and in the coun- terclockwise direction in the Southern Hemisphere. The period for 180" of rotation (not oscillation) of the pendulum is termed the haw-pendulum day, and varies with latitude, as will be discussed below, but in this case is one- half a sidereal (or stellar) day (approximately 11 h 58 min of solar time).
A somewhat different gedanke (thought) experiment will demonstrate the Coriolis effect away from the Pole. Consider a circular pan of water, steadi- ly rotating at angular speed Q, after the transients associated with its spin- up have damped out (Figs. 3.4a and 3.4b). Under the combined influences of gravity and centripetal acceleration, the surface of the water will assume the shape of a paraboloid of revolution, which for small displacements from equilibrium will be almost spherical, with the downward normal to the sur- face lying along the resultant of the vectors g and Q:Rj. At the point O1 on the surface of the water at a distance R from the axis (Fig. 3.4b), allow a small floating object initially t o co-rotate with the water, but to move across the surface without friction. Let the direction from O1 to the axis of rota- tion be called north. The forces at O1 are in equilibrium, with the poleward slope of the surface just balancing the outward centripetal acceleration. Start- ing at 0, (Fig. 3.4a), if we now impart a small southward (outward) im- pulsive velocity, 6ur, to the particle in the purely radial direction, it will initially move in that direction. However, upon arriving at 0 2 , one-eighth of a period later, it will have a larger radius, R
+
ri, from the spin axis than(a)
180'
>@
O0t = O
t = 2 h
i
Inertial system
(b)
180'
180'
t = 2 h
t = 1 2 h * Rotating system
Flg. 3.3 Rotations of a Foucault pendulum at the North Pole as viewed (a) from inertial space and (b) from an earthbound coordinate system.
ning pan of water, demonstrating the Coriolis effect. The particle is arranged on a very long pendulum of length L , and co-rotates with the pan at an angular speed of Q,. Un- der the combined influences of gravity and centripetal acceleration, the water’s sur- face is nearly a spheroid of revolution, across which the pendulum swings tangentially.
A small radial velocity, 6u,, is imparted impulsively to the particle at point 0,, where an observer is located. The particle then describes an elliptical path as seen from in- ertial space, but appears to an observer on the spinning pan to rotate clockwise in an inertial circle of radius T i , lying first to the east of O,, then south at 02, west at 03, and north at 0,. As viewed by the observer, the particle makes two complete revo- lutions in the inertial circle about the points O i for one revolution of the pan, thereby giving rise to the factor of 2 in the Coriolis acceleration. (c) Positions of the particle relative to the observer located at the center of the compass rose. As applied to the earth, R, is the local vertical component of the earth’s angular velocity, and R the dis- tance from the spin axis.
74 Principles of Ocean Physics
at equilibrium; however, it will have retained its azimuthal velocity, ub =
RQ,, which is lower than that of its surroundings, u; = ( R
+
r i ) Q z . Thus the particle will fall behind the rotating water, i.e., it will appear to be deflect- ed clockwise to the right. At this point, the particle lies to the south of the observer at 0, and its linear velocity, ub = ( R+
r i ) Q, - 6u, is purely azimuthal; however, its effective angular velocity around the spin axis has been reduced fromQ,
toQ,
- 6u/(R+
T i ) . As a consequence of this lowered spin rate, the centripetal acceleration at O2 will be reduced from~ Z ( R +
r i ) toa, =
(a,
- - y ( R 6U+
r i ).
R
+
ri (3.28)The difference between the centripetal acceleration at radius R
+
rj for thewater and the particle is
Aa, = 2QZ6u
+
6 u 2 / ( R+
r i ).
(3.29) It will be recognized that the first term is the Coriolis acceleration, which is, after all, the source of the deflections to the right. The remaining term is of second order and is negligible for velocities in the ocean.By the time the particle has advanced with the earth to 0, , the clockwise deflection has resulted in a northward radial velocity, so that the particle now appears to be west of the observer at 0 3 . The motion, however, car- ries it inward past its equilibrium radius, R , and this will again result in deflec- tion to the right, since the particle’s azimuthal component of total velocity for rj
<
R will be greater than the nearby water and it will overtake its local environment. At position O,, the particle is to the north and its increased angular velocity will result in an increased centripetal acceleration and a south- ward deflection. At position O5 (one half-period from the start), the veloc- ity will have become purely radial again with the conditions the same as at 0, and with the particle once again located to the east of the observer.Thus, in a half-period of rotation of the pan of water, the particle, as seen by an observer in the rotating coordinate system, will appear to have traversed a full circle whose radius, ri , is called the inertial circle. The associated peri- od is also a half-pendulum day, and is termed an inertial period, Ti, Its val- ue ranges from approximately 12 hours at the pole, through 24 hours at about 30°, to infinity at the equator. The relationship to the pendulum comes from the fact that a pendulum whose geometry is the same as that shown in Fig.
3.4 (which is not the same one as the Foucault pendulum of Fig. 3.3) will oscillate with the inertial period. To show this, recall that the period of a pendulum, T p , of length L is
Tp = 27r(L/g)
‘ .
(3.30) From Fig. 3 . 4 , the balance of forces and the geometry give, for small angles from the vertical,L / R = g / Q : R , or
and thus the period of the rotation is equal to the period of the pendulum:
T = 27r/Q, = 27r(L/g)‘ = Tp
= 2 T ; . (3.31)
Therefore, an imaginary pendulum of length L that undergoes small oscilla- tions of radius r, about 0 as an equilibrium point would have a period given by twice the inertial period. For the earth, the length of the pendulum is very large, depending on the local value of the inertial period; at the pole, this length is approximately 1.84 x l o 6 km.
The radius of oscillation, r , , can be calculated by equating the Coriolis force t o the centrifugal force that maintains the motion in the small circle.
Since the small velocity, 6u = 6u,, is measured with respect to the rotating system, this results in
or
r; = ( 6 u / 2 Q z ) (3.32)
as the radius of the inertial circle. Thus the radius is proportional to the ve- locity imparted to the particle.
Returning to the upper diagram of Fig. 3.4, the path of the particle viewed from inertial space is an ellipse, while that of the observer is a circle of radi- us R , about which the ellipse is centered. It is clear that the observer will see the particle rotating clockwise about him in the inertial circle, first t o the south, then west, north, and east, t o return to the south within one-half a rotation period, 27r/Q,: this is suggested by the small compass rose dia-
76 Principles of Ocean Physics
gram of Fig. 3 . 4 ~ . This doubling of the apparent rotation rate is the origin of the factor of 2 in the formula for the Coriolis frequency (Eq. 3.23).
If small to moderate friction exists, the radius of oscillation of the parti- cle in the inertial circle will gradually be reduced as it loses its initial veloci- ty, 6u, but the period will remain nearly unchanged. In the rotating coordinate system, the particle’s path will be viewed as an inward spiral that converges on the observer’s position.
An alternative view of the dynamics is that expressed by conservation of angular momentum. The particle’s total angular momentum, L, can be con- sidered as the sum of its momentum about the spin axis and the angular momentum of its small oscillation. The movement in the inertial circle can be thought of as the particle’s attempt to conserve its angular momentum as it oscillates between the extremes of its moment arm, R f r i .
On a spherical earth, the situation is more complicated and more subtle.
Now fl and g are no longer antiparallel, and the spin vector at any colati- tude, 8, has both horizontal and vertical components (Fig. 3.5). As a conse- quence, any large-scale motions that involve appreciable north-south excursions will see a varying component of Coriolis frequency and force.
This will turn out to have far-reaching implications for mesoscale and plane- tary-scale motions, a fact that has been mentioned earlier and whose conse- quences will be developed in Chapter 6.
For the present, let us consider the Coriolis force as viewed in a local right- handed Cartesian coordinate system called the tangent plane system (Fig.
3 . 9 , which touches the earth’s surface at polar angles 0 and q5,with
x
t o the east, y to the north, and z upward; .such a coordinate system will be used extensively in this book. An expansion of the Coriolis term in this coordinate system is= T(2w0 sin
e
- 2vO cos 0 )+
j ( 2 u 0 cos 8 ) - k(2u0 sine )
, (3.33)where
(t k)
are unit vectors in the tangent plane system. Of these four terms, only two are important in geophysical fluids. First, the vertical com- ponent of velocity, w , is generally very small compared with the horizontalt
N y (North)Fig. 3.5 Tangent plane coordinate system touches the earth at the point at which the radius vector at (i3,4) intersects the surface; Cartesian coordinates constitute a right-handed system. The local Coriolis frequency varies linearly as f = f,
+
By inthe north-south direction.
components, u and u , and the term involving w is therefore negligible. Next, the vertical or i component of Eq. 3.33 (which is called the Eotvos correc- tion in marine gravity) is many orders of magnitude smaller than its compet- ing vertical forces, i.e., gravity and fluid buoyancy, and can also be neglected.
This leaves as the effective components of Coriolis acceleration the quantities - ( 2 ~ x u), = 2uo COS
e
(3.34)-(23 x u)v = -2uo cos 8 , (3.35) and
where we have written the Coriolis term as it will appear on the right-hand side of the momentum equation ahead, Eq. 3.83, so that it now acts as a
78 Principles of Ocean Ph ysics Y
Z
T X
2R v cos 0
I
Fig. 3.6 Force diagram showing the components of Coriolis and centrifugal forces at the earth’s surface. These behave as real forces to an observer on the rotating earth.
forcing term. In Fig. 3.6, this redefinition in terms of horizontal force com- ponents makes clear the origin of the inertial motion: In the Northern Hemi- sphere, an initially northward-directed particle ( u
>
0) is deflected eastward by the x component of Coriolis force, and thereby acquires an eastward ve- locity component (u>
0). This component then interacts with southward- directed meridional component of Coriolis force to continue the clockwise deflection. In the Southern Hemisphere, cos 8<
0 and the direction of the force is opposite: The inertial circle rotates counterclockwise.We next expand the centripetal term of Eq. 3.27 evaluated for r = R , , and also consider it as a forcing function on the right-hand side of the momen- tum equation:
(3.36) -!I x (!I x X) = -Q’Re sin 8(j^cos 8 -
k
sin 8) ,= Q’R, sin 8 6
which has both vertical and horizontal components in the tangent-plane sys- tem. Here the unit vector,
6,
is directed outward at right angles from the spin axis and is, from Eq. 3.36,where
8
and f are unit vectors in the 8 and r directions, respectively.I t is now clear how t o interpret the motion shown in Fig. 3.4 on a rotating planet. The local vertical component of the planet’s angular velocity vector, Qz = Q cos 8, interacts with the horizontal components of particle veloci- ty, u, to form the fictitious horizontal Coriolis force at right angles to both.
The horizontal component of planetary angular velocity causes small, es- sentially negligible vertical forces. Thus the angular velocity,
a,,
in Fig. 3.4is to be re-written as Cl cos 8, and the lever arm, R , becomes R , sin 8. The local vertical component of centripetal acceleration, Q2Re sin’ 8, changes the value of g by only a small percentage, while the local horizontal compo- nent, 3’R, sin 8 cos 8, provides the same restoring force as for the rotating disk. We can then write for the local inertial period on an approximately spherical earth,
T. = - 27r
’
2 3 cos 8 ’ and for the radius of the inertial circle,(3.38)
(3.39)
If the north-south particle motions are extensive enough to range through significant changes in latitude, A, the effective value of the earth’s angular velocity also changes. In the tangent plane approximation, the angular ve- locity is expanded in a series about some value fo and only linear terms are retained. Denote the important Coriolis parameter (also called the planetary vorticity) by f:
f
=
2 3 cos 8 = 2 3 sin A , (3.40)80 Principles of Ocean Physics
where A = ( ~ 1 2 ) - 0 is the latitude. For small changes in 0 (or A) about a point where f = f o , one writes
f = f o
+
-af
y + . , aY= f a
+
PY * (3.41)Here the tangent plane touches the earth where the Coriolis parameter has the value f a . Denote by the derivative at that point, which then becomes (3.42)
At mid-latitudes,. say A, = 45", numerical values o f f and
P
are:f = 20 sin A,, = 1.031 x l o p 4 s-I , (3.43) and
= 1.619 x l o - ' ' (m s ) - '
.
(3.44)The interpretation of these values is as follows. The Coriolis parameter is a measure of the planetary vorticity, or the earth's rotation rate, and is ap- proximately rad s