JANUARY 2015 P H I L L I P S 409 Inertial Motion Viewed from a Potential Well NORMAN PHILLIPS Merrimack, New Hampshire (Manuscript received 18 June 2014, in final form 5 September 2014) ABSTRACT The motion of a mass undergoing free horizontal motion on a planet’s surface is analyzed in an inertial frame by using a potential well presentation. The following results from the conventional analysis in rotating coordinates are also found in inertial coordinates. Under typical circumstances the mass oscillates about the latitude where its angular momentum per unit mass matches that of the planet’s surface. For masses with nonzero angular momentum, conservation of angular momentum bounds the mass away from the pole. Motion with zero angular momentum is confined to a fixed meridian plane on which the mass may oscillate across the pole. Motion across the equator is possible only for masses with sufficient energy. Masses with angular momentum per unit mass greater than that of the planet at the equator oscillate about the equator with a limiting period determined by the excess angular momentum. 1. Presentation in the rotating frame called the ‘‘inertial circle.’’ The corresponding period is 2pr/C,or2p/(2v sinu). Free horizontal motion is a common topic when dis- Inertial motion is seen most clearly in the ocean, cussing dynamics of a point mass confined to the surface perhaps because the atmosphere does not experience of a planet. In a rotating reference frame, four forces act the external forcing such as surface drag, and the much in the absence of friction: Newtonian gravitation, cen- larger background velocities seem to mask any inertial trifugal force, surface reaction, and Coriolis force. The motion. Figure 1 shows a spectral analysis from 598N sum of the first two is ‘‘gravity,’’ which is perpendicular [taken from Weller (1982)]. to the surface. In the absence of friction, the surface This is the traditional analysis in the rotating reference reaction is also perpendicular to the surface. For free frame. However, when viewed from a nonrotating frame, horizontal motion, this leaves only the Coriolis force. the forces are quite different; they reveal the roles of cen- The horizontal component of this force on a mass m trifugal energy and conservation of angular momentum. moving horizontally at speed C is mC2v sinu, where v is The organization into a logical whole is greatly enhanced the angular speed of the planet’s rotation and u is the by the use of the potential well concept. Such an approach latitude of the mass (the angle between the surface leads naturally into a classification of several modes of normal and the normal to the equatorial plane). The free horizontal motion. force acts at right angles to the motion and thus pro- The appendix defines the potential well properties of the duces a centripetal acceleration C2/r, where r is the ra- trajectories pictured by Paldor and Killworth (1988).(Their dius of curvature of the path (clockwise in the Northern calculations are based in rotating coordinates.) The present Hemisphere and counterclockwise in the Southern article does not delve into the physics of the subject, but an Hemisphere, when viewed from above). excellent description of the physics behind the Coriolis If this is equated to the force per unit mass, we get force is found in the book by Stommel and Moore (1989). r 5 C/(2v sinu). (1) When C is sufficiently small, and thus sinu is sensibly 2. A potential well formulation constant, the path approximates a reentrant circle, Marvin (1915) seems to have been one of the first Corresponding author address: Norman Phillips, 18 Edward Lane, Americans after Ferrel to recognize that inertia motion Merrimack, NH 03054. is partly due to a component of Newtonian attraction E-mail: [email protected] that is parallel to the surface of the earth. He carried this DOI: 10.1175/JAS-D-14-0185.1 Ó 2015 American Meteorological Society Unauthenticated | Downloaded 10/01/21 10:00 AM UTC 410 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 72 FIG. 2. Gravity components acting on a mass at point P. The solid line t–t represents the horizontal tangent plane, normal to Pg (gravity). PN is Newtonian gravitation, with a component PG in the plane t–t. Ng is the centrifugal force per unit mass. PZ is the local vertical at an angle u (geocentric latitude) from the plane of the equator. PG has a magnitude equal to the negative component of Ng in the horizontal plane. The force of reaction needed to FIG. 1. Spectrum of clockwise motion near the inertia frequency P PZ (solid line) and counterclockwise motion (dashed line) in the At- support a mass at would be directed upward along . lantic Ocean near 598N. Copied from Fig. 7 in Weller (1982). dV/dt 1 UU sinu/r 5 v2r sinu and (3a) idea far enough that in a later paper (Marvin 1920)he introduced the phrase ‘‘the law of the geoidal slope’’ as dU/dt 2 UV sinu/r 5 0, (3b) part of the Coriolis force. More recently, Durran (1993) has revived this consideration. where v2r sinu is the horizontal component of Newtonian Let u denote latitude (increasing northward from the attraction. It is directed poleward and arises from the equator) and l denote longitude (increasing eastward) in a rotational flattening of the planet’s figure, as pictured in nonrotating frame. For a mass confined to the surface of the Fig. 2. This term vanishes at the pole and equator and is planet the northward and eastward velocity components are independent of the sign of v. A mass at rest with respect to the rotating planetary V 5 Rdu/dt and (2a) surface will have V 5 0 and U 5 rv. Under these con- U 5 rdl/dt, (2b) ditions (3) shows that the mass will remain at rest, since then the right-hand side of (3a) is balanced by the U2 where r denotes the perpendicular distance of the mass term on the left-hand side. For convenience, the symbol from the axis of the planet and R denotes the radius of m for mass will be omitted, inasmuch as no trans- the planet. (For planets of small oblateness, such as formations involving mass are involved (i.e., as if m 5 1). Earth, r can be replaced by cosu times R.) Changes in Equations (3a) and (3b) imply energy invariance r are related to changes in latitude by dE/dt 5 0, where Rdu/dt 5 (dr/dt)/sinu. (2c) E 5 (1/2)(U2 1 V2 1 v2r 2), (4) The inertial (nonrotating) equations of motion on the where (1/2)v2r2 is the ‘‘centrifugal energy,’’ or the po- planet’s surface are tential energy associated with the poleward Newtonian Unauthenticated | Downloaded 10/01/21 10:00 AM UTC JANUARY 2015 P H I L L I P S 411 invert the relation to express r2 in terms of V. Our principal interest will be in the turning points of the path taken by the mass—that is, the points where and when V 5 0andE 5 W. For these points, the roots of the quadratic equation are 2 2 2 2 2 1/2 v r2 5 E 2 (E 2 v M ) and (8a) 2 2 2 2 2 1/2 v r1 5 E 1 (E 2 v M ) , (8b) where the minus sign and plus sign subscripts respectively denote the turning points closest to the pole and equator. Since E is nonnegative, the right-hand sides of (8) are nonnegative, as required by the left-hand side. However, the left-hand side of (8b) must not be greater than v2R2 since R is the maximum possible value of r.Wewillfind that if E is sufficiently large, the latter restriction makes it 0 FIG. 3. The nondimensional effective potential W for four values necessary to exclude the root (8b)—or in other words, the of the nondimensional angular momentum: M0 5 0, 0.5, 1.0, and 1.5 0 02 turning point at low latitudes is ignored when E is suffi- [see (8)]. The curves satisfy the relation W 5 x 1 M /x, where x 5 (r/R)2 is the abscissa. (The abscissa is shown for both hemispheres.) ciently large. attraction. In addition, (3b) implies angular momentum 3. Paths limited to one hemisphere invariance 0 5 dM/dt 5 0, (5a) The curve for M 0.5 in Fig. 3 shows that a fixed value of E can have two intersections with that curve within one where hemisphere, so the mass’s path can be confined entirely to one hemisphere. (Since the potential is symmetric with 5 M rU (5b) respect to the equator, the same statement also applies to each hemisphere.) This requires both (8a) and (8b) to be is the angular momentum about the planet’s axis. admissible, and therefore in (8b) we must have E , v2R2. U After eliminating between (4) and (5b), we can write Further, in any case the radical requires jMj , E/jvj,so 2 V2 5 2(E 2 W) and (6a) we must have jMj , jvjR as an additional condition for this case. Still another condition is that E must be greater W 5 (1/2)(M2/r2 1 v2r 2), (6b) than the minimum potential in the well, since according to (6a) there is otherwise no solution. where W is the effective potential—in this case, the The effective potential is a function of v and the mass current sum of azimuthal kinetic energy and gravita- property M, plus the radius r.