An Explication of the Coriolis Effect

Norman A. Phillips Merrimack, New Hampshire

ABSTRACT

Inertia circle is a familiar concept demonstrating the behavior of a horizontally moving particle on the rotat- ing earth in response to the Coriolis . Three dynamical principles valid in nonrotating are used to explain the principal properties of this motion without the need for solving differential equations. The motion is an oscillation about a central latitude selected by the particle’s angular , in which the restoring force is set by the relative magni- tudes of the poleward attraction of gravity and the equatorward attraction of the as the particle oscil- lates about the central latitude.

1. Goal and principles rotating earth, has its horizontal displacements subject only to the . This makes it a simple ex- The object of this note is to provide an interpreta- ample from which to understand the Coriolis effect on tion of how motion on the rotating earth is affected by motion. that rotation. The physical principles to be used are ex- For horizontal motion, that force is equal to the tremely well known, but as far as I know, have not been product of the , C, and the Coriolis parameter, f: applied to this problem in precisely this style, a style designed to avoid the solution of differential equations. Coriolis force per unit = fC, f = 2Ωsinφ. (1) In spite of its long history, the subject is very much alive, especially in a pedagogical context (Stommel Here Ω is the angular of the earth and φ is and Moore 1989; Duran 1993). Persson (1998) pro- latitude. The force is directed 90° to the right of the vides many references and a historical perspective. motion in the Northern Hemisphere and 90° to the left Two examples are discussed here, circle mo- in the Southern Hemisphere. It therefore does not tion and geostrophic flow. It is suggested that the re- change the speed. In the absence of other , it sults might provide a simple physical understanding gives rise to clockwise in the North- to supplement the elementary manipulations in the ern Hemisphere and counterclockwise motion in the conventional derivation of the Coriolis force, which, Southern Hemisphere. (For brevity, further discussion according to Persson (1998) and Stommel and Moore will focus on the Northern Hemisphere.) (1989), remain a mystery to many. The circular motion requires an inward accelera- Inertia circle motion is the idealized behavior of a tion equal to C 2 divided by the radius. Equating this particle that is confined to horizontal motion on the to fC from (1), we find that the radius of the inertia earth and, when viewed in a framework fixed to the circle is

C/f. (2)

Corresponding author address: Norman A. Phillips, 18 Edward This solution is idealized by the convenience of Lane, Merrimack, NH 03054. E-mail: [email protected] considering only small changes in latitude. A similar In final form 14 July 1999. assumption will be taken for granted in the remainder ©2000 American Meteorological Society of this note. Ripa (1997) has carefully considered the

Bulletin of the American Meteorological Society 299 effect of variations in latitude and approximate ways Here u is the eastward velocity relative to the rotating to allow for this element. earth.

The above is the explanation of an inertia circle Durran (1993) has shown how FG gives rise to in- from the viewpoint of an observer in the rotating sys- ertia circle motion. His demonstration is based on solv- tem who is familiar with the Coriolis force. But physi- ing the for a parcel in nonrotating cal principles are often most familiar when viewed coordinates. Like Durran, I present my arguments in from a coordinate system fixed in space. We will there- the context of nonrotating coordinates. However, the fore use a nonrotating coordinate system for a view major aspects of inertia circle motion can be obtained of the physical processes underlying this simplest ex- without solving differential equations. This will ex- ample of the Coriolis force. pose the dynamical principles that are involved. Three In a nonrotating coordinate system, however, it is familiar principles will be sufficient. necessary to recognize two related effects that do not appear explicitly when the equations of motion are 1) Conservation of . From (6), we expressed relative to the rotating earth. As a result of see that this determines the change of U as a par- its rotation, the earth is an oblate spheroid, such that ticle changes its distance from the rotation axis. gravitational attraction at the earth’s surface has a hori- 2) The principle: zontal component directed toward the pole. This com- × ponent, FG , equals the horizontal component of the change in = force . (7) earth’s centripetal : This will determine the speed of the particle. Ω 2 φ FG = Rsin , (3) 3) The meridional equation of motion, (4). This de- termines changes in the poleward velocity of the where R is the distance to the axis of the earth. (The R parcel. Only the sign of dV/dt is necessary in the at the surface of the earth is closely equal to the ra- following discussion. A slightly more detailed ex- dius of the earth the cosine of the latitude.) This component of gravitational attraction is what keeps everything from sliding to the equator, as shown in Fig. 1. Both effects are contained in the meridional component of the horizontal equations of motion when expressed in nonrotating spherical coordinates:

+ 2 φ dV/dt (U /R)sin = FG. (4)

Here V is the northward component of the horizontal velocity, U is the absolute eastward component of velocity, and t is . For a particle at rest on the earth, U equals the product ΩR. For clarity, the expression + (U2/R)sinφ will be referred to as the centripetal ac- Ω celeration. It equals FG when U = R. The other horizontal equation of motion can be written

dU/dt − (UV/R)sinφ = 0. (5) FIG. 1. Orientation of forces at a point P on the surface of the earth. Here, tt is a tangent plane. PC points to the center of the Since there is no force in the zonal direction, and V earth. Pg is local gravity and defines the local vertical, perpen- sinφ is equal to −dR/dt, (5) can be manipulated to show dicular to the tangent plane. Pg is gravity, the sum of PN, the that the angular momentum of a particle is not Newtonian gravitation vector, and Ng or Pr, the centrifugal force. changed: PG is the horizontal component of the gravitation vector PN, given by projection of PN on the tangent plane. PG is equal and opposite to the projection of Pr on the tangent plane. φ is the angular momentum = RU = R(u + ΩR) = constant. (6) geodetic latitude, the angle between the equatorial plane and the local vertical.

300 Vol. 81, No. 2, February 2000 amination of (4) will be needed in the analysis of Here ∆s = −∆R/sinφ is meridional arc length along geostrophic flow, however. the earth’s surface. − 3) For small (R0 R)/R the bracket term is approxi- Principles 1 and 2 are specialized integral state- mately equal to 2. Using the value of f defined in ments of the equations of motion that are valid for in- (1), we can write for later reference, ertial motion. Stommel and Moore have formulated two constraints governing inertial motion on a rotat- du = f∆s. (8b) ing equipotential surface (1989, 245–246.) Principle 1 is one of their constraints. The other is similar to 4) This increases the total eastward velocity U = AM/ principle 2, but in the more elegant form of conserva- R, and hence the U2 term in principle 3. This in turn tion of the kinetic energy of the absolute motion and makes dV/dt negative. the associated with FG. This yields a 5) But the particle will continue moving poleward simple deduction of the extrema of R. The form 2 is until the negative dV/dt has reduced V to zero. How used here instead because it focuses more explicitly far will it move, using arguments available to an on the role of FG and because it leads to evaluation of observer in nonrotating space? Principle 2 provides the relative . Principle 3 is necessary to ex- the answer. The initial kinetic energy is amine the interplay of FG with the centripetal accel- 2 + Ω 2 eration that determines the sign of dV/dt. KE0 = (1/2)[v0 ( R0) ]. (9)

The kinetic energy when the parcel reaches its φ 2. A northward-moving particle polemost location at latitude 1 and R = R1, with zero meridional motion, is We begin by observing a parcel as it crosses lati- φ 2 + Ω 2 tude 0 at radius R0 in a northward direction with a KE1 = (1/2)U1 = (1/2)(u1 R1) . (10) purely meridional velocity,

Here u1 is given by (8a) with R = R1. The differ- > V0 0. ence must be due to the done by the force (per unit mass) times the displacement. In differential At this initial point, identified by subscript zero, the form this is relative motion is dWork = (Ω2Rsinφ) × ∆s = −(1/2) Ω 2dR2. (11)

u0 = 0, v0 = V0 > 0, 6) Evaluation of the terms in (9) and (10), and inte- and its constant absolute angular momentum is grating (11) from R0 to R1, shows that this balance reduces to the simple result

AM = U0R0. 2 2 u1 = v0 . (12) The detailed arguments proceed as follows. This equality is consistent with the uniform speed 1) Since u is zero, the particle’s absolute zonal veloc- of inertia circle motion found to exist in the rela- ity U is equal to RΩ and its centripetal accelera- tive system.1

tion is equal to FG. According to principle 3, dV/dt Furthermore, reference to (8b) and (2) shows is zero. that the northward displacement is equal to the ra- 2) But as the inertia of the particle carries it to higher dius of an inertia circle: latitudes, conservation of angular momentum ac- ∆ cording to principle 1 will assign it an eastward ve- s = u1/f = v0 /f. (13) locity component:

+ Ω − u = [(R0 R)/R] (R0 R) 1Conservation of the kinetic energy of relative motion is charac- + Ω φ∆ = [(R0 R)/R] sin s. (8a) teristic of inertial motion in general.

Bulletin of the American Meteorological Society 301 3. Return to the initial latitude able inference that the motion has a constant speed of relative motion in a circle of radius C/f on the earth. Returning now to our nonrotating framework, we The principles 1, 2, and 3 have each contributed to this φ consider the parcel at its new latitude 1, with zonal behavior, behavior that in the rotating system of co- 2 u1 and zero meridional velocity, u1 ordinates is due solely to the Coriolis force.

= v0, and v1 = 0.

1) Its angular momentum is 4. Geostrophic motion

AM = U1R1. (14) The most pervasive influence of the Coriolis force for the atmosphere and ocean is not encountered in 2) But the total zonal motion is still too large for inertia circle motion, but in the relatively steady flow 2 φ (U /R)sin to be balanced by FG at that latitude, and that is characterized by a balance, in the rotating co- the parcel now begins to move equatorward in obe- ordinate system, between the horizontal pressure gra- dience to principle 3. dient force and the Coriolis force. This is referred to 3) Its eastward relative zonal velocity will now de- as geostrophic motion, and it too can be explicated crease according to principle 1. Eventually it will with the principles developed above for inertia flow. reach a lower latitude where u vanishes. Equation (8b) describes the zonal velocity u that φ 4) This latitude, 2, is determined again by principle is generated by conservation of angular momentum for 1: a particle as it moves northward. Division of (8b) on each side by a time interval dt produces the equation Ω 2 Ω 2 AM = U2R2 = R2 = U0R0 = R0 . du/dt = f ∆s/dt = fv. (15) φ We are back at the initial latitude 0. 5) A calculation of the work and kinetic energy This shows that the particle could continue to move φ φ changes in going from 1 to 2 is similar to that for directly northward, that is, with u always zero, if an φ φ the progression from 0 to 1, except that the work additional force (per unit mass), FW, of magnitude fv is now negative, and the equatorward motion were applied toward the west:

means that v2 is negative. The result is − FW = fv. (16) − − u2 = 0, v2 = u1 = v0. This is the geostrophic relation for steady northward This is as it should be, since if we had been watch- flow as expressed in the rotating coordinate system. ing the particle in the Northern Hemisphere from Angular momentum is no longer conserved, however. the rotating frame starting with relative velocity It remains to examine how steady zonal motion that

u1 to the east, it would begin to describe a clock- is faster or slower than the earth can be maintained. wise inertia circle, turning equatorward. When it From the inertia circle discussion, we saw that when φ is one-quarter of the way around, it will have zero the particle had reached latitude 1 with relative zonal

zonal relative motion and will have progressed velocity u1, its zonal velocity U1 was too large for can- equatorward a distance equal to the radius of an cellation in the meridional equation of motion (4) be- 2 φ inertia circle. tween FG and the centripetal term (U /R)sin ; whereupon the particle had a negative value for dV/dt. The further progress of the particle below latitude φ 0 will, to the extent we ignore major changes in lati- tude, be a mirror image of what has happened in the φ φ φ 2Readers who like to see differential equations in spite of the progression from 0 through 1 to 2. This reasoning has shown the particle to have the avowed goal of this note can take comfort in the following. The same speed at four locations in its return to the initial meridional equation of motion (4), with V = ds/dt, s being surface distance in the northward meridional direction measured from lati- latitude. During this motion it has traveled northward tude φ 0, can be rewritten with the aid of (8a) and the decomposi- a distance equal to the radius of an inertia circle and it tion preceding (18), in the standard form of the simple oscillator has traveled southward a like amount. It is a reason- equation, d2s/dt2 + f 2s = 0.

302 Vol. 81, No. 2, February 2000 This negative dV/dt can be cancelled by applying a 5. Conclusions suitable northward-directed force. The size needed to balance the extra is determined by ex- The analysis in earlier sections provides the follow- panding the centripetal acceleration term in (4): ing picture of inertial circle motion from the standpoint of a nonrotating coordinate system. It is a stable os- (U2/R)sinφ = [(ΩR + u)2/R]sinφ = Ω 2Rsinφ cillation of a particle about a central latitude on the surface of the ellipsoidal rotating earth, in which there + 2Ωu sinφ + u2sinφ/R. (17) is a contest between the equatorward centrifugal force and the poleward attraction of gravity. The central lati- The Ω 2 term is needed to cancel the gravitational force tude is defined by the angular momentum possessed

FG. It is the second term that primarily determines the by the particle. Poleward of the central latitude the an- excess centripetal acceleration; the necessary north- gular momentum of the particle prescribes an eastward ward force is then zonal velocity greater than that of the earth, which al- lows the centrifugal force to overbalance the poleward + Ω φ FN = 2 usin = fu. (18) gravitational attraction. Conversely, when the particle is on the equatorial side of the central latitude, the Equations (16) and (18) show that the rotation- zonal velocity is slower than that of the earth and the induced of a moving particle on the ro- gravitational attraction exceeds the centrifugal force. tating earth can be canceled by applying a force per This force imbalance accelerates the particle poleward. unit mass of magnitude f × speed. The force should The presence of a horizontal pressure force can be applied toward the left of the motion in the North- cancel the acceleration found in inertial motion and ern Hemisphere and toward the right in the Southern allow motion to occur in a straight horizontal path on Hemisphere. When the force is supplied by a pressure the earth. Angular momentum is no longer conserved gradient force per unit mass, this is referred to as geo- in this situation. strophic flow. It might appear peculiar that different approaches Acknowledgments. This paper materialized as a result of e- have been taken to explain geostrophic balance in the mail discussions with Anders Persson and George Platzman on zonal and meridional directions; conservation of an- problems of rotary motion. Persson’s persistence in seeking in- terpretations that are both accurate and easily understood was gular momentum led to (8b) for u, whereas the geo- especially stimulating. Professor Platzman’s perseverance intro- strophic equation for v came from the expansion of the duced not only clarity but meaning into this presentation. centripetal acceleration in (4). However, the conser- vation of angular momentum principle is derived from the zonal equation of motion (5), which treats the bal- References ance of zonal forces, which include FW. Similarly, the derivation of the geostrophic relation between u and Durran, D., 1993: Is the Coriolis force really responsible for the inertial oscillation? Bull. Amer. Meteor. Soc., 74, 2179–2184. FN is obtained from expanding the centripetal accel- eration in (4), which describes the balance of meridi- Persson, A., 1998: How do we understand the Coriolis force? Bull. Amer. Meteor. Soc., 79, 1373–1385. onal forces, which include F . Viewed at this deeper N Ripa, P., 1997: “Inertial” oscillations and the β-plane approxima- level, there is no peculiarity. tions. J. Phys. Oceanogr., 27, 633–647. Stommel, H., and D. Moore, 1989: An Introduction to the Corio- lis Force. Columbia University Press, 297pp.

Bulletin of the American Meteorological Society 303