• Ball, F., 1963: Some general theorems concerning the finite motion of a shallow rotating liquid lying on a paraboloid. J. Fluid Mech.,17, 240–256.

  • Benilov, E. S., 1996: Beta-induced translation of strong isolated eddies. J. Phys. Oceanogr.,26, 2223–2229.

  • Cushman-Roisin, B., 1982: Motion of a free particle on a beta-plane. Geophys. Astrophys. Fluid Dyn.,22, 85–102.

  • Goldstein, H., 1981: Classical Mechanics. Addison-Wesley, 672 pp.

  • Graef, F., 1998: On the westward translation of isolated eddies. J. Phys. Oceanogr.,28, 740–745.

  • Killworth, P. D., 1983: On the motion of isolated lenses on the beta-plane. J. Phys. Oceanogr.,13, 368–376.

  • McDonald, N., 1998: The time-dependent behaviour of a spinning disc on a rotating planet: A model for geophysical vortex motion. Geophys. Astrophys. Fluid Dyn.,87, 253–272.

  • Nof, D., 1981: On the β-induced movement of isolated baroclinic eddies. J. Phys. Oceanogr.,11, 1662–1672.

  • Nycander, J., 1996: Analogy between the drift of planetary vortices and the precession of a spinning body. Plasma Phys. Rep.,22, 771–774.

  • Paldor, N., and P. D. Killworth, 1988: Inertial trajectories on a rotating earth. J. Atmos. Sci.,45, 4013–4019.

  • Ripa, P., 1997: “Inertial” oscillations and the β-plane approximation(s). J. Phys. Oceanogr.,27, 633–647.

  • ——, 2000: Effects of the earth’s curvature on the dynamics of isolated objects. Part II: The uniformly translating vortex. J. Phys. Oceanogr., in press.

  • Stommel, H. M., and D. W. Moore, 1989: An Introduction to the Coriolis Force. Columbia University Press, 297 pp.

  • White, A., 1989: A relationship between energy and angular momentum conservation in dynamical models. J. Atmos. Sci.,46, 1855–1860.

  • View in gallery

    The central grid are isolines of the rotated spherical coordinates (λ′, θ′), constructed so that the pole θ′ = π/2 is located at a given reference point, defined by the (terrestrial) longitude and latitude (λ∗, θ∗), and such the meridians λ′ = 0 and λ = λ∗ coincide (vertical lines in the figure). The polar coordinates of an arbitrary point (r, ϕ) in the new frame are defined through a stereographic projection, so that λ′ = ϕπ/2 and θ′ = π/2 − r/R + O(R−2).

  • View in gallery

    Bottom: The left graph coordinates (x, y) are fixed to the earth, whereas the right graph coordinates (x′, y′) = r(cosϕ, sinϕ) are defined by the transformation of Fig. 1, following the secular drift of an object: λ∗ = λ0 + δΩt and θ∗ = θ0. Both sets of coordinates resemble Cartesian ones near the origin; (x, y) are proportional to longitude and latitude changes (so the grid is made up of parallels and meridians) and the (x′, y′) space is a stereographic projection (the closed curves are circles with center in (x′, y′) = (0, 0). Top: Comparison, in both coordinates, of the inertial oscillation of a particle predicted by the exact equations (solid) and the classic β-plane approximation (dashed). The initial (northward) velocity equals 0.25 RΩ.

  • View in gallery

    The large circle shows the trajectory in the stereographic projection (X′(t), Y′(t)) of the center of a circular object (smaller circles). Notice that when X′ ≠ 0, the orientation of the axes in the (x′, y′) system, shown by the crosses, do not exactly coincide with the eastward and northward directions, shown by the latitude and longitude isolines (dashed). As a result, there are two definitions, σ and ω, of the vertical angular velocity.

  • View in gallery

    Disk on a rotating planet, described by the longitude Λ(t) and latitude Θ(t) of the center of mass, and the intrinsic rotation ϕ(t). Orthogonal components of the instantaneous rotation vector ωI are shown by thick lines.

  • View in gallery

    Inertia momenta In of the disk (with respect to the earth’s center) as a function of its radius a. Notice that I = R2Iz/Ix is very close to ½a2 (dashed line).

  • View in gallery

    Examples of disk evolution. Left and middle panels show the center’s orbit and velocity, whereas right panels show the vertical angular velocity relative to a fixed direction ω (solid) or relative to the local east σ (dashed). The initial center of mass position and velocity are Λ(0) = 0°, Θ(0) = 60°, U(0) = 0, and V(0) = 0.1RΩ. The initial internal rotation is cyclonic ω(0) = Ω (top), null ω(0) = 0 (middle), or anticyclonic ω(0) = −Ω (bottom).

  • View in gallery

    Inertial oscillation of the center of mass of a disk, seen from above and following the secular drift (moving stereographic coordinates). According to f-plane dynamics, the center of mass would make a circular uniform oscillation, with X′ = −f0 × ′ = −f20X′. With curvature effects and up to O(R−1), the orbit is also a circle but with larger speed the closer it is to the pole. The excess acceleration from the f-plane balance + f20X′ (thick arrows) is given by the sum of the excess Coriolis force −f × ′ + f20X′ (pointing radially) and the (poleward) sum of the mean Coriolis force due to the internal motion 〈−f × u′〉 and the geoforce −Φ′; the latter equals the imbalance between a poleward gravitational force (related to the shape and mass distribution of the earth) and the equatorward centrifugal force (see Table 2).

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Effects of the Earth’s Curvature on the Dynamics of Isolated Objects. Part I: The Disk

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  • 1 CICESE, Ensenada, Mexico
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Abstract

A disk over the frictionless surface of the earth shows an interaction between the center of mass and internal motions. At low energies, the former is an “inertial oscillation” superimposed to a uniform zonal drift c and the latter is a rotation with variable vertical angular velocity ω (as measured by a terrestrial observer).

The dynamics is understood best in a stereographic frame following the secular drift. The center of mass has a circular but not uniform motion; its meridional displacement induces the variations of the orbital and internal rotation rates. On the other hand, the temporal mean of the Coriolis forces due to both rotations produces the secular drift.

In spherical terrestrial coordinates geometric distortion complicates the description. For instance, the zonal velocity of the center of mass U is not equal to the average zonal component of the particle velocities 〈u〉, as a result of the earth’s curvature. The drift c and the temporal means U and u are all three different. In addition, ω differs from the local vertical angular velocity σ (as measured by an observer following the disk). The classical“β plane” approximation predicts correctly the value of c but makes order-one errors in everything else (e.g., it makes U = u = c and ω = σ).

The results of this paper set up the basis to study curvature effects on an isolated vortex. This, more difficult, problem is discussed in Part II.

Corresponding author address: Dr. Pedro Ripa, CICESE, P.O. Box 434844, San Diego, CA 92143-4844.

Email: ripa@cicese.mx

Abstract

A disk over the frictionless surface of the earth shows an interaction between the center of mass and internal motions. At low energies, the former is an “inertial oscillation” superimposed to a uniform zonal drift c and the latter is a rotation with variable vertical angular velocity ω (as measured by a terrestrial observer).

The dynamics is understood best in a stereographic frame following the secular drift. The center of mass has a circular but not uniform motion; its meridional displacement induces the variations of the orbital and internal rotation rates. On the other hand, the temporal mean of the Coriolis forces due to both rotations produces the secular drift.

In spherical terrestrial coordinates geometric distortion complicates the description. For instance, the zonal velocity of the center of mass U is not equal to the average zonal component of the particle velocities 〈u〉, as a result of the earth’s curvature. The drift c and the temporal means U and u are all three different. In addition, ω differs from the local vertical angular velocity σ (as measured by an observer following the disk). The classical“β plane” approximation predicts correctly the value of c but makes order-one errors in everything else (e.g., it makes U = u = c and ω = σ).

The results of this paper set up the basis to study curvature effects on an isolated vortex. This, more difficult, problem is discussed in Part II.

Corresponding author address: Dr. Pedro Ripa, CICESE, P.O. Box 434844, San Diego, CA 92143-4844.

Email: ripa@cicese.mx

1. Introduction

A particle on the surface of the earth is attracted toward the nearest pole by a gravitational force due to the deviation of the geoid from a perfect sphere and to inhomogeneities in the mass distribution within the planet. It is convenient—and quite accurate—to work in a spherical geometry but keeping the main effect of that horizontal force, namely, that it can sustain a constant rotation with angular velocity ±Ω, relative to an inertial frame, in any point on its surface. It follows that this force (per unit mass) is given by −Φ, where is the horizontal gradient operator,
i1520-0485-30-8-2072-eq1
Ω and R are the rotation rate and radius of the earth, and θ is the latitude. On the terrestrial frame (moving with the earth) this gravitational force cancels the horizontal component of the centrifugal force, and one is left with the Coriolis and other forces (e.g., friction, pressure gradient, etc.); the first is characterized by the Coriolis parameter
fθ.
If the kinetic energy E (as measured by a terrestrial observer) of a compact and isolated object is much smaller than the minimum one for equatorial crossing, then it remains in a small latitude band around, say, θ = θ0. Three environmental parameters are needed for the approximate description of such a system (Ripa 1997, hereafter R97):
i1520-0485-30-8-2072-eq3
notice that f0β/τ0 + f20 = 4Ω2, f0/(τ0β) = R2, and f0τ0/β = tan2θ0. [The notation of this paper is actually simpler than that of R97 because it shows explicitly that only three parameters are needed. For instance, the approximation of the Coriolis parameter in Eq. (29) of R97 is equivalent to the much simpler expression, f = f0 + βy/(1 − τ0y), in the notation of the present paper. Variables a and τ0 used in R97 correspond to R and τ0R in this paper, whereas φ2 from R97 should here be set equal to zero.]

Cushman-Roisin (1982), Paldor and Killworth (1988), Stommel and Moore (1989), and R97 showed that at low energies E an isolated particle performing an “inertial oscillation” experiences a secular westward drift. Nof (1981), Killworth (1983), and Benilov (1996) found that an isolated vortex, which in the f-plane approximation (f = f0) is steady, to first order in R−1 also has a slow westward drift. Nycander (1996) related the drifting vortex to a solid disk that rotates and precesses over the (frictionless) surface of the earth: the formula for the drift velocity c is the same in both cases if expressed in terms of the vertical angular momentum. The general motion of such a disk is an interesting problem, because it has more degrees of freedom than the particle and yet it is much simpler than the vortex.

The exact equations for a disk of arbitrary size are obtained and discussed here. For low E there are two distinct contributions to the drift velocity c: that caused by the intrinsic rotation ci (discussed by Nycander) and that due the inertial oscillations c0 (as with the particle), which will be respectively denoted “internal drift” and“orbital drift” for simplicity (both refer to the secular translation of the object, though, the name distinguishes the origin of the drift). Table 1, to be derived below, presents the formulas for the drift c and two other bulk variables: the time mean of the center of mass U and volume-averaged 〈u〉 zonal velocities; these formulas are also valid for homogeneous or stratified isolated vortices, as shown in Part II of this paper (Ripa 2000). All through these papers, the tilde denotes equal modulo o(R−1), that is, results with an O(R−2) error.

Nof (1981), Killworth (1983), Benilov (1996), and Cushman-Roisin (1982) worked in the framework of the classical β-plane approximation of the shallow-water equations, that is, allowing for a linear variation of f with θ but using Cartesian coordinates. This approximation is incorrect because the curvature corrections to a flat geometry are of the same order as the variation of the Coriolis parameter, namely, O(R−1). More precisely, the β-plane approximation uses only two parameters (it is formally equivalent to making τ0 = 0) and thus it gives incorrect results, except in the equatorial β plane (θ0 = 0). Graef (1998) proved that the formula derived by Nof (1981) and Killworth (1983) for the vortex’s internal drift ci is correct. However, this does not mean that the classical β plane gives the right description of all other details of the motion. The inadequacy of this approximation is clearly shown in the misrepresentation of U and u at lowest order (see Table 1). For instance, in the case of a particle performing an inertial oscillation with radius ρ (≪R), the drift is the orbital one co, and the temporal mean of the zonal velocity is u = co(1 + f0τ0/β) ≡ co sec2θ0; the β-plane model predicts correctly the former (due to the fortuitous cancellation of large errors) but subestimates the latter by a factor of cos2θ0 (see Table 1 and R97). Exactly the same result is shown in Part II to be true for an isolated (homogeneous or stratified) vortex governed by the primitive equations: even though the internal drift ci is correctly given by the formula derived by Nof (1981) and Killworth (1983) from the classical β-plane model (Graef 1998), the value predicted for the average zonal velocity 〈u〉, within the vortex, is again a factor of cos2θ0 too small (see Table 1).

Even though the problem of the circular disk can be solved by quadratures, in order to understand the physics it is useful to derive approximate solutions in appropriate coordinates. Two different frames are discussed in section 2: (x, y), which are scaled spherical coordinates, and (x′, y′), which correspond to a stereographic projection following the secular drift of the object. The results of this section can be used with a asymmetric disk or with the vortex studied in Part II;readers interested in the general solution for the circular disk, rather than these geometrical details, may skip directly to the first part of section 3, where the dynamics of that object are first discussed, using the full spherical geometry. Approximations are then obtained expanding in R−1, with either (x, y) or (x′, y′) coordinates. Changes of the local vertical angular velocity are shown to be induced by both meridional displacements and the zonal velocity of the center of mass, parameterized by β and τ0, respectively. The classical β-plane approximation is finally discussed. In particular, the reason why it gives fortuitously the right value of c is clarified here. Summary and conclusions are presented in section 4, whereas some mathematical details are left for appendixes.

Articles on isolated particles or disks might seem out of place in a physical oceanography journal. However, analyzing earth’s curvature effects in simple systems helps to understand them in idealized ocean problems, like the isolated vortex considered in Part II and many other papers, as well as in more complicated studies, like numerical simulations. In particular, it is important to clarify to what extent—and in what sense—the classical β-plane approximation is incorrect since it is used very frequently in the study of ocean physics.

2. Moving stereographic coordinates

Consider an isolated thin “object” over the surface of the earth, not necessarily a symmetric and rigid disk. For simplicity, its density will be considered uniform. Approximations valid when the object remains near a latitude θ0 are developed here. These results are particularly useful for the study of the vortex, done in Part II, but they are introduced here in order give a feel for how they work in a simpler problem. Moreover, even in the case of the circular disk, the approximate solution in the stereographic moving frame is important to understand the physics of the solution.

Since the interest is for θθ0, it is useful to change to rescaled spherical coordinates:
xλλ0Rθ0yθθ0R,
which resemble Cartesian ones for θθ0. The squared arc element is exactly given by dr2 = γ2dx2 + dy2, where γ = cosθ secθ0. Since γ = 1 − τ0y + O(y2/R2) and f = f0 + βy + O(y2R−2), non-Cartesian terms are of the same order as the β term as y/R → 0.
For some calculations it is better to use coordinates such that non-Cartesian terms are O(R−2). Let (λ∗, θ∗) be the terrestrial coordinates of some reference point, as yet unspecified, and construct the polar coordinates (r, ϕ) defined in Fig. 1, by means of a stereographic projection. The transformations (r, ϕ) ↔ (λ, θ) of the coordinates of an arbitrary point are given in Eqs. (A.1)–(A.4) of appendix A. Cartesian-like coordinates in the stereographic projection are defined by
xyrϕ,ϕ
and the arc element is given by |dr| = γ̃dr2 + r22 = γ̃dx2 + dy2, with
i1520-0485-30-8-2072-e22b
Consequently (r, ϕ) and (x′, y′) resemble polar and Cartesian coordinates as r/R → 0. The bottom panels in Fig. 2 compare both systems of coordinates for θ∗ = 60°: the grid is uniform in (x, y), whereas the closed curves are the circles r = const with uniform spacing. Notice that for x ≠ 0 or x′ ≠ 0 the directions in both frames are not equivalent.
Two cases of this transformation will be used here: either (λ∗, θ∗) are the coordinates of the center of mass or the transformation is to an uniformly rotating frame moving with the secular drift of the object, corresponding to λ∗ = (δΩ)t and θ∗ = θ0 (as in Fig. 2). In the second case, and in any coordinates, the velocity in the new system is1
uuθλ̂
and a particle is subject to two forces: the Coriolis one −f × u′, where
fδθ
is the corresponding Coriolis parameter, and the imbalance −Φ′ between the equatorward centrifugal force and the poleward gravitational one, discussed in the introduction, where
i1520-0485-30-8-2072-e25
is an “effective potential.” For lack of a better name, −Φ′ will be called the “geoforce.” If the frame is subrotating, |Ω + δΩ| < Ω (suprarotating, |Ω + δΩ| > Ω), then the geoforce is poleward (equatorward). The parameters for the terrestrial (inertial) frame are recovered for δΩ = 0 (δΩ = −Ω), which in particular makes Φ′ ≡ 0 (f′ ≡ 0). In order to write f′ and Φ′ in terms of (x′, y′), we use sinθ = (2γ̃ − 1) sinθ0 + γ̃yR−1 cosθ0, from (A.5).
In general, the position of an arbitrary particle will be indicated by the coordinates of the center of mass (capital letters) and relative ones (lower case) as
i1520-0485-30-8-2072-e26
where c = Ω cosθ0. A quite surprising result is that the stereographic coordinates (x′, y′), of the particle relative to the center of mass, are not the same in both cases, as explained next and derived in appendix B.
The full coordinates transformation (x′, y′), → (x, y) is given by (A.6); if the interest is near r = 0, making an expansion in r/R, it follows that
i1520-0485-30-8-2072-e27
whose inverse is x′ ∼ xx∗ − τ0(xx∗)y and y′ ∼ yy∗ + ½τ0(xx∗)2 where, recall, the tilde means with an O(R−2) error. Even though both transformations look quite symmetric, the variation of the geometric terms is O(R−1) for (x, y) and O(R−2) for (x′, y′), namely, γ = 1 − τ0y + O(y2R−2), whereas γ̃ = 1 − ¼r2R−2 + O(r4 R−4).
Directions in the (x, y) and (x′, y′) planes are clearly not equivalent, as appreciated in the bottom panels of Fig. 2; (2.7) shows that the differences are O(R−1), that is, of the same order of the phenomena of interest here. Thus, in case 2 the local eastward and northward directions are x ∼ (1 + τ0y′)′ + τ0xŷ′ and y ∼ −τ0x′ + ŷ′, respectively: directions in both frames differ when τ0x′ ≠ 0. Using transformation (2.7) for the coordinates of an arbitrary particle in both cases shown in (2.6) results in (B.1) and (B.2), respectively. Subtracting from both representations the center of mass coordinates (B.3), it follows that the relative coordinates are not the same in both cases but are related by
i1520-0485-30-8-2072-eq4
Since τ0X′ = O(R−1) ≪ 1, this transformation is just a rotation, which can be appreciated more clearly in polar coordinates. Writing
i1520-0485-30-8-2072-e28
it is αϕ + τ0X′, whose time derivative gives
ωστ0Uωα̇,σϕ̇
and U′ ∼ ′, as shown in Fig. 3, ω and σ are the object vertical angular velocity relative to a fixed direction and the local east, respectively; the description of the inner rotation of object elements is not the same in both frames; which description is simpler can be assessed by the form of the energy and momenta integrals.

a. Energy and momenta

The components A and Ae of the angular momentum (per unit mass) in the directions of the vertical (at the position of the center of mass) and earth’s axis ê and the relative kinetic energy E (per unit mass and measured by a terrestrial observer) are defined by
i1520-0485-30-8-2072-eq5
where xI(t) is the trajectory of a generic particle (measured by an inertial observer), the angle brackets represent an average over the volume of the object, and I − Ωê × xI is the velocity measured by a terrestrial observer. Notice that the energy can also be written as E = ½〈2I + (Ωê × xI)2〉 − ΩAe, where the term ½(Ωê × xI)2 is the potential Φ mentioned in the introduction and the term −ΩAe arises from the transformation from the inertial to the terrestrial frame (White 1989).

Approximations of A, Ae, and E for the object can be calculated using x1 = R(cosθ cos(λ + Ωt), cosθ sin(λ + Ωt), sinθ), with the transformation (x′, y′) → (λ, θ) derived in appendix A, and expanding in R−1. The average 〈 · · · 〉 can easily be approximated in the stereographic coordinates x′ = (x′, y′), because non-Cartesian terms are O(R−2). For instance, if x′ are the particle coordinates relative to the center of mass, then 〈x′〉 ∼ 0 and 〈u′〉 ∼ 0, with u′ ∼ ′. For case 1 we also needed 〈Ry′〉 ∼ [bu1034]R−1yx2 + y3〉, which follows from (A.7) with (X′, Y′) = (0, 0). The results for case 2 can be obtained in a similar way or using (B.3) and (2.9) in those of case 1.

In order to compare it with the x momentum in the f plane, M := Ae/(R cosθ0) − RΩ cosθ0 will be used instead of Ae. In the first case of (2.6), that is, describing the center of mass in spherical coordinates and the relative motion in the stereographic projection, we find
i1520-0485-30-8-2072-e210
In the second case, that is, describing both the center of mass and the relative motion in the stereographic projection, we find
i1520-0485-30-8-2072-e211a
where
i1520-0485-30-8-2072-e211b
is the vertical angular momenta of the center of mass relative to the point (x∗, y∗) = (ct, 0). Notice that the energy in the frame rotating with velocity Ω + δΩ is
i1520-0485-30-8-2072-e211c
where f0cY′ is the leading term of Φ′ from (2.5).
In order to stress the difference between the vertical angular velocity measured in the terrestrial frame ω and that observed following the object σ, the expressions for A are calculated using (2.8), where (r, ϕ, α), for a particle, and (ρ, κ), for the center of mass, are functions of time. Neither σ = ϕ̇ nor ω = α̇ are assumed to be the same for all particles in the object. The contributions of and ρ̇ are found to vanish and the final result is
i1520-0485-30-8-2072-eq6
in the first case and
i1520-0485-30-8-2072-eq7
in the second [the term τ0c is O(R−2) and is not included at this level of approximation]. Similarly, the kinetic energy of the relative motion is ½〈2 + r2(σ + τ0U)2〉 in case 1 and ½〈2 + r2ω2〉 in case 2. The expressions for A and E are simpler using (u′, υ′) from case 2 or, equivalently, using ω rather than σ. The distinction between ω and σ is crucial to understand the evolution of the disk, studied next.

3. The disk

Consider a uniform density thin disk (actually, a thin spherical cap) of radius a. (In principle, it is a < πR, where for a = πR/2 the “disk” covers a hemisphere and for aπR it is a whole sphere.) The exact evolution equations can be derived from the Lagrangian
i1520-0485-30-8-2072-e31
where I is the velocity of a particle of the disk as measured by an inertial observer (but expressed in whichever coordinates are desired), θ is the latitude of that particle, and the angle bracket denotes an average over the entire area of the disk. Three coordinates are enough to describe the motion of the disk: the longitude Λ(t) and latitude Θ(t) of its center, measured by an observer fixed to the earth, and the angle ϕ(t) it has rotated around its symmetry axis (the local vertical), with respect to the local east.
The Lagrangian could be calculated by brute force, as follows. Spherical coordinates in the inertial frame are λI(t) = Λ(t) + Ωt + F1(λ′, θ′, Θ(t)) and θ(t) = F2(λ′, θ′, Θ(t)), where (λ′, θ′) are spherical coordinates with their “North Pole” θ′ = π/2 at the center of the disk, and the functions F1 and F2 are defined in Eq. (A.2). The coordinates of an arbitrary particle in the disk are λ′ = ϕ0 + ϕ(t) and θ′, where ϕ0 is a label. The average on the disk then takes the form
i1520-0485-30-8-2072-eq8
This operation could be used to evaluate the Lagrangian directly, as L = ½R2θ̇2 + (λ̇2I − Ω2) cos2θ〉, but this is a quite cumbersome calculation. Instead, there is a much simpler and physically meaningful way to obtain L, as described next.
Since the disk is over the surface of the earth, its motion is just a rotation around the center of the spherical planet, with an angular velocity ωI(t) made up of three contributions (see Fig. 4): 1) an angular velocity component Λ̇ + Ω around the earth’s axis; 2) a component −Θ̇ around an axis in the equatorial plane, pointing in the direction of the longitude Λ + π/2; and 3) the intrinsic rotation ϕ̇ around the symmetry axis of the disk. In a frame moving with the center of the disk, the third component is vertical (see Fig. 4), the second one is horizontal in the zonal direction, whereas the first one can be projected into the horizontal meridional and vertical directions. The total angular velocity is then given by
i1520-0485-30-8-2072-eq9
where (see Fig. 4)
i1520-0485-30-8-2072-e32
are the zonal and meridional components of the horizontal velocity of the center of the disk, and the vertical angular velocity, all of them measured by an observer fixed to the earth. Notice that ω is different from both the rotation rate ϕ̇ with respect to the local eastward direction (i.e., that measured by an observer following the motion of the disk) and the total vertical angular velocity
ωzωσUR−1
(i.e., the rotation rate measured by an inertial observer), where σ = ϕ̇. This exact relationship between ω and σ corresponds to the approximate formula (2.9).
The corresponding inertia momenta per unit mass relative to the center of the planet around any horizontal and the vertical axes are calculated as
i1520-0485-30-8-2072-eq10
[As shown in Fig. 5, it is IzIz except for huge disks, e.g., Iz = R2 + O(a2) and Iz = ½a2 + O(a4R−2) as a/R → 0.] Recall that the Ij are referred to the center of the planet, not the disk’s center of mass. The kinetic energy term in (3.1), T = ½ Σ3j=1Ijω2j, equal to
i1520-0485-30-8-2072-eq11
is a second-order polynomial in Ω, say, T = T0 + ΩT1 + Ω2T2. The potential energy part of L is evaluated by the following reasoning: If the disk is put motionless on the surface of the earth then it remains in that state, for any Θ. This means that it must be ∂L/∂Θ = 0 for Λ̇ = Θ̇ = ϕ̇ = 0 and any Θ, which implies2 V = ½IxΩ2 cos2Θ + ½IzΩ2 sin2Θ ≡ Ω2T2. Consequently, the Lagrangian (3.1) is simply equal to the expression derived for T, without the terms proportional to Ω2, namely, LT0 + ΩT1. Finally, recall that any multiple of L gives the same equations of motion; therefore the actual values of Ix and Iz are not as important as their ratio, which Fig. 5 shows to be very close to the inertia momentum of a flat disk with the same radius, namely,
i1520-0485-30-8-2072-eq12
as a/R → 0. The case of the particle, studied in R97, is recovered setting I = 0.
The normalized Lagrangian L(Θ, Λ̇, Θ̇, ϕ̇) = (T0 + ΩT1)R2/Ix gives second-order equations in Λ(t), Θ(t), and ϕ(t). A first-order set can be obtained using (U, V, ω), defined in (3.2); the extended Lagrangian takes the form
LΛ̇Θ̇,ϕ̇,U, V, ωΛ̇AeΘ̇RVϕ̇AE
where the symbols (A, Ae, E) in this expression must be understood as the right-hand sides of
i1520-0485-30-8-2072-e35
These are also the three integrals of motion of the problem, whose conservation laws are related to three symmetries of the system, manifested by the absence of ϕ, Λ, and t in L. Namely, invariance under a fixed rotation of the disk around its vertical axis gives dA/dt = 0, under a change in the reference longitude gives dAe/dt = 0, and under a change of the origin of time gives dE/dt = 0. Note that the three terms on the right-hand side of (3.3) are included in the vertical angular velocity A. Variations of σ are induced by changes in Θ and by the zonal velocity U, whereas variations of ω are due to only the first effect. Conservation of A for the disk, f + 2ω = const, is very similar to conservation of potential vorticity of a fluid element (in nondivergent setting). The first part of Ae is the same as that of a particle, whereas the second one represents the projection of the vertical angular momentum into the direction of ê. Recall that E is the kinetic energy measured by a terrestrial observer, which is why in the zonal kinetic energy appears U and not U + RΩ cosΘ, and in the internal kinetic energy appears ω and not ω + Ω sinΘ.
Variation of L with respect to U, V, and ω gives the definitions (3.2) of these variables. The other three equations of motion are dAe/dt = 0, the one obtained from the variation of L with respect to Θ, and dA/dt = 0, which take the form
i1520-0485-30-8-2072-e36
where
i1520-0485-30-8-2072-eq13
The variable χ parameterizes the interaction between the center of mass (U, V) and internal (ω) motions, giving a first hint that it is not possible to split the dynamics of an isolated vortex into both motions, as done by Ball (1963) in the f plane.

The equations of motion can be set in the Hamiltonian form, dz/dt = J · ∂H/∂z, in three different interpretations: 1) z groups the generalized coordinates [Λ, Θ, ϕ, . . . and their conjugate momenta . . . , Ae, RV,A]; H equals the total energy E but with U and ω expressed as functions of (Ae, Θ, A); and J is the canonical 6 × 6 Poisson tensor. 2) z = [Θ, U, V, ω]; H = E; and J(z) is a singular matrix of corank two. The null space of J is spanned by the gradients of Ae and A, which are the Casimir integrals of motion (J · ∂Ae/∂z = 0 = J · ∂A/∂z). 3) For given values of Ae and A, determined by the initial conditions, z = [RΘ, V]; H = E; and J is the canonical 2 × 2 Poisson tensor.

The solution of the sets (3.2) and (3.6) give the full evolution of the disk, from an arbitrary initial condition. In practice, the solution is reduced to a quadrature by use of the three integrals of motion: From the momenta conservation it is found ω = A/I − ΩsinΘ and U = R−1Ae secΘ − R−1A tanΘ − RΩ cosΘ; then from energy conservation it follows
Θ̇2R−2ER−2AeR−2A2IR−2AI2
Therefore t(Θ) can be obtained by a simple integration;U(t) and ω(t) are then calculated from Θ(t) using conservation of A and Ae; and finally Λ(t) and ϕ(t) are integrated using definitions (3.2a,c). Four nondimensional parameters determine the solution: one related to the size of the disk IR−2 (=Iz/Ix; see Fig. 5) and the integrals of motion E/R2Ω2, A/IΩ, & Ae/R2Ω for a particular orbit. Unlike the case of the particle (Cushman-Roisin 1982; Paldor and Killworth 1988; R97), equator-crossing orbits, which correspond to 2E > (AeR−1 − ΩR)2 + A2/I, are not necessarily symmetric under change of sign of Θ. The generic solution Θ(t) is periodic, just as in the case of the particle; U(t), V(t) and ω(t) are also periodic (since they are calculated from Θ and Θ̇), whereas Λ(t) and ϕ(t), being the indefinite integral of a periodic function, are the sum of a periodic part and a secular drift. This problem is analogous to that of the “heavy symmetrical top with one point fixed,” with potential ½R2Ω2 cos2Θ instead of gR sinΘ;the method developed in Goldstein (1981, section 5.7) can be used to obtain the solution seen by an inertial observer.
Consider first the case studied by Nycander (1996), with no oscillations whatsoever, say, Θ(t) = θ0 (and thus V = 0). Making = 0 in (3.6b) and using (3.2c) it follows that the rotations around the vertical axis and along a latitude circle must be related by
i1520-0485-30-8-2072-eq14
and ω = ϕ̇ + Λ̇ sinθ0. This equation can be written as (Ω + Λ̇)N = −∂V/∂Θ, where N = Iyωy sinθ0Izωz cosθ0 is the component of the angular momentum in the equatorial plane (ωy and ωz are the meridional and vertical components of ωI). The balance then expresses that the torque −∂V/∂Θ sustains the uniform rotation of component N with angular velocity Ω + Λ̇ as shown in Fig. 4 (the other equatorial component, −IzΘ̇, vanishes, and Ae is constant). There are two values for Λ̇ (or U) for each ϕ̇; the slow solution (Λ̇ ≪ Ω) is given by
i1520-0485-30-8-2072-eq15
where the second approximation corresponds to Iz ≪ 2Ix (or a2 ≪ 4R2). For this pure precession solution, it follows from (B.4)u〉 ∼ U + τ0xυ′〉 ∼ U sec2θ0. Since the center of the disk is at a fixed latitude, the drift velocity c = R cosθ0Λ̇ coincides with U. These are contributions proportional to ci in Table 1. The coincidence U = c is no longer true when meridional oscillations are allowed, as discussed next and derived in R97 for the case of a particle (see the contributions proportional to co in Table 1).

Consider then more general solutions. Figure 6 show a few examples of evolution of a small disk (a/R = 0.1, which implies R−2I = 0.0050), for three different values of the initial intrinsic rotation ω(0). The time span equals two periods T of the motion of Θ(t), which in the cases shown is very close to the inertial period 2π/|f0| corresponding to the latitude θ0. The center of mass motion is clearly influenced by the value of ω (∼ω(0) = σ(0)):On one hand, the zonal drift (left graphs in Fig. 6) increases with this parameter; it can even be eastward (top panel). Moreover, the extreme values of U are shifted from those of V (middle graphs) by an amount that grows with ω (this is an evidence that U does not vanish, even though it must be V = 0). On the other hand, the internal motion, depicted by the time evolution of σ(t) and ω(t) in the right graphs in Fig. 6, is influenced by the motion of the center of mass: Both decrease (increase) when the center moves north V > 0 (south V < 0), but the oscillations of σ (dashed) are larger than those of ω (solid); indeed (σσ) ≈ 7(ωω).

These results are generalized next with solutions for which the oscillations of Θ(t) are very small, which correspond to the vicinity of a minimum of E for fixed A and Ae. Table 1 shows the lowest-order term of c, U, and u as a function of the radius of the inertial oscillation ρ ∼ |V(0)/f0| and ω. The form of solutions such that Θ(t) − θ0 is small can be entirely derived from the integrals of motion (see appendix C). However, the physics of the phenomenon is revealed more clearly by making approximations, with O(R−2) errors, of the Lagrangians and equations of motion. This is done next in both (x, y) and (x′, y′) coordinates because the results for the disk are useful to interpret those of the vortex, which is clearly a more complicated system. It is more difficult to solve the vortex equations in the (x, y) frame. In the (x′, y′) frame, on the other hand, the β term is the main effect of the earth’s curvature, whereas τ0 terms appears in the transformation back to (x, y) coordinates.

a. Approximate solution in spherical coordinates

Denoting by (X, Y) the scaled spherical coordinates of the center of the disk, the Lagrangian (3.4) takes the form
LY,Ẋ,Ẏ,ϕ̇U,V,ωMẎVϕ̇AE
So far the treatment of the problem is exact; approximations are developed expanding in Y the trigonometric functions in the definitions of A and M. For instance,
i1520-0485-30-8-2072-e38b
Variation of L with respect to (U, V, ω) gives
i1520-0485-30-8-2072-eq16
whereas variation with respect to (X, Y, ϕ) gives the same equations of motion (3.6), where now
i1520-0485-30-8-2072-e39
Absence of ϕ, X, and t in L implies conservation of A, M, and E, which coincide with those of (2.10). [The constant terms −¼ has been subtracted from the definitions of M, whereas the last term, namely, ⅓τ0βY3, is actually incorrect but of O(R−2); it is included here in order to simplify the equations of motion since, without it, it is φ = f0 + βY/(1 − τ0Y) + τ0 (as used in R97 for the particle).]
Solutions are found making an expansion in R−1. The lowest-order solution, an f-plane inertial oscillation with constant intrinsic rotation,
i1520-0485-30-8-2072-e310
is enough to calculate the results of Table 1. First, from = 0 it follows
i1520-0485-30-8-2072-eq17
and thus
i1520-0485-30-8-2072-eq18
where I ∼ ½a2. Second, since U(1 − τ0Y) = cτ0YẊ, the secular drift cU + τ0Y00 is given by
i1520-0485-30-8-2072-eq19
Finally, the time mean of the particles zonal velocity average is calculated using (B.14), that is,
i1520-0485-30-8-2072-eq20
taking the temporal average it follows that
i1520-0485-30-8-2072-eq21
These are the results are presented in Table 1: all three are different, c is a factor cos2θ0 smaller than u (both are proportional to the relative vertical angular momentum), whereas U coincides with u in the term due to the inertial oscillation and with c in the term due to the intrinsic rotation.
Using conservation of A and the relationship between σ and ω, namely, σωτ0U, to next order rotation functions are given by
i1520-0485-30-8-2072-eq22
Notice that variations of the local rotation are larger than those of the total vertical rotation, namely, σ1/ω1 = 1 + 2 tan2θ0; in the case of Fig. 6, it is 1 + 2 tan2θ0 = 7.

The next-order center of mass solution (X1, Y1) is quite complicated in spherical coordinates (see R97). In stereographic coordinates, though, the expression is much more simple and the physics more transparent; the complication is then introduced in the transformation (x′, y′) → (x, y), given in Eq. (3.13) below.

b. Approximate solution in moving stereographic coordinates

Making the transformation (B.3) of the center of mass coordinates (X, Y) → (X′, Y′) in the Lagrangian (3.8), using α̇ = ϕ̇ + τ0′ from (2.9), and neglecting terms of o(R−1), we obtain
LYα̇UVωMVα̇AE
where
i1520-0485-30-8-2072-eq23
are the integrals of motion. (The term proportional to τ0 is an exact time derivative and was subtracted from the definition of L because it does not contribute to the dynamics.) Comparing with (2.11) it follows that A′ = A + O(R−2) and E′ = EcM + O(R−2), whereas M′ = Mτ0A + O(R−1) because it is lacking the term τ0ACM. However, from the equations of motion it follows that dACM/dt = O(R−1) and therefore dM/dt = O(R−2). It is not surprising that M′ + τ0A′ is a poorer approximation of the angular momentum around the earth’s axis than M from (3.8b), because this integral of motion is related to invariance under changes in the origin of X, which is a complicated function of (X′, Y′);see (A.6) or its approximation (B.3). In fact, if c were not chosen correctly, the solution would experience a secular drift in X′ instead of in X, that is, along a great circle (Y′ = 0 ⇒ Y ∼ ½τ0X2) instead of a latitude circle (Y = 0); even though the errors are O(R−2), they would not be uniformly so with time.
The equations of motion derived from this Lagrangian are (U′, V′, ω) = (′, ′, α̇) and
i1520-0485-30-8-2072-e312
The terms proportional to βY′ come from the dependence of f′ from (2.4) on θ, whereas the term 2δΩ sinθ0 is O(R−2) and therefore is not included at this level of approximation. The term −f0c is the leading one in the geoforce −∂Φ′/∂Y′ from (2.5) and can be interpreted as a change of the local vertical (White 1989).
Expanding in powers of R−1, to lowest order it follows the inertial oscillation (3.10), but in the variables (X0, Y0, ω0). The eigenvalue c is then chosen so that there is no secular drift in (X′, Y′), namely, = 0, which yields c = ci + co with ci ≡ ½f−10βIω and co = f−10β = −½βρ2. Finally (B.5) gives Uc + τ0 = ci + co sec2θ0 and uU + τ0υ̃′〉U + ½τ0Iω = (ci + co) sec2θ0. The bulk results c, U, and u (presented in Table 1) are calculated using the lowest-order solution but, in order to understand the physics of the curvature effects, the next-order term must be analyzed. An interesting solution with O(R−2) error is
i1520-0485-30-8-2072-eq24
in the notation of (2.8). This solution is interesting because the orbit is circular, to the given order. The speed is |′| ∼ ρ|f0 + ½βY′|, and thus the particle travels slowly in the half of the orbit closer to the equator. Table 2 and Fig. 7 show how the acceleration is driven by the forces − f × ′ and −(f0cY′). There are two O(R−1) variations of the orbital Coriolis force due to the change of speed and to the change of latitude, respectively. The time mean of the first one vanishes, whereas the second one produces a net equatorward force = −½f0βρ2ŷ′, simply because it is stronger when it is closer to the pole (when it is pointing, an average, toward the equator). A similar analysis applied to the internal Coriolis force gives = ½βIωŷ′. This time-mean Coriolis force is balanced by −(f0cY′) (the time mean of the acceleration is, of course, zero), giving c = ½βIω − ½βρ2. Up to O(R−1), the dynamics of a particle is equal to that of the center of the disk (see Fig. 7), with just the orbital drift co (<0) and the corresponding part of the geoforce − f0coŷ. In the case of the disk, c = co + ci might be positive, for sufficiently fast cyclonic inner rotation.
The motion translated to terrestrial coordinates, obtained using (B.3) and (2.9),
i1520-0485-30-8-2072-e313
is clearly more complicated to describe. Other solutions, like the one presented in R97 in (x, y) coordinates, are obtained adding an arbitrary shift in X′ and/or an f-plane inertial oscillation with amplitude proportional to ρ2/R.

c. The classical β plane

This classical approximation corresponds to setting τ0 = 0 in the Lagrangian (3.8) and therefore has the integrals of motion
i1520-0485-30-8-2072-eq25
which are representations of the true (A, M, E) with an O(R−1) error. The corresponding β-plane equations in the spherical coordinates (X, Y) perform much worse than the correct approximation (3.12) in the moving stereographic coordinates (X′, Y′), and furthermore d(A, M, E)/dt = O(R−1) for the former, whereas d(A, M, E)/dt = O(R−2) for the latter.

Making the transformation (X", Y") = (Xct, Y) in the classical β-plane equations yields exactly the system (3.12), but for (X", Y", ϕ) instead of for (X′, Y′, α) and with f0c replaced by f0c + βY"c; the term βY"c is O(R−2) and therefore can be ignored. This classical approximation has then the right equations in the wrong coordinates and, as a consequence, lacks the terms proportional to τ0 in the solution (3.13). This introduces large errors in both the center of mass and internal motions: On one hand, the top panels in Fig. 2 show the difference with the correct solution for one inertial oscillation; a more detailed analysis of the error is presented in R97 (for the limit of a particle). On the other, the classical β approximation makes no distinction between σ and ω, whereas Fig. 6 shows that the variations of σ are a factor 1 + 2 tan2θ0 (=7 in that example) larger than those of ω. However, the value of c was obtained demanding that there is no drift in (X′, Y′) coordinates: condition ′ = 0 in (3.12) is equivalent to the condition = 0 in the incorrect (X", Y") equations and that is why the classical β-plane equations fortuitously give the right value of c, even though its solution has O(R−1) errors.

4. Conclusions

Ball (1963) studied the dynamics of a finite volume of liquid with free boundaries, over a rotating planet but in the framework of the f-plane approximation (constant Coriolis parameter and Cartesian geometry), showing that the center of mass performs a circular inertial oscillation, independently from the relative motion, which in turn satisfies the full, nonlinear, shallow-water equations. Conservation of the energy E, measured in the terrestrial frame, and the vertical component of the angular momentum A were found to play an important role for this inner motion. Allowing for effects of the planet’s curvature (parametrized by the inverse of the earth’s radius R−1) changes substantially Ball’s scenario:1) the inertial oscillations are not circular but experience a secular drift, 2) they are coupled with the internal motions, and 3) A is no longer an integral of motion. (Conservation of A is linked to the property that a rotation around the vertical axis transforms a solution of the problem into another solution; this is clearly true for any object on the f plane but only for symmetric ones on the sphere or the β plane.)

The motion of the volume of fluid in a spherical planet, addressed in Part II (Ripa 2000), constitutes a difficult problem. As a necessary prologue, a much simpler case was studied here: a rigid circular disk over the frictionless surface of a rotating planet. The inner motion is but a solid-body rotation ω, and the vertical angular momentum A = I(ω + ½f) is conserved (where I and f are the inertia momentum and Coriolis parameter) because of the symmetry of the object. The exact solution is periodic (the time mean in a period is indicated by an overbar) and can be easily found for any initial condition. However, it is more illustrative to analyze low E cases, which remain in a small zonal band, near a reference latitude θ0.

The best frame to describe the problem is a stereographic projection (x′, y′), whose origin moves along the reference latitude with the drift velocity c. The Coriolis parameter is f = f0 + βy′ + O(R−2) and, unlike in spherical coordinates, the geometry may be taken as flat with an O(R−2) error. A very simple solution has the center of the disk X′ = (X′, Y′) = ρ(cosκ, sinκ) moving in a circle, ρ = const + O(R−2), with κ̇ = −f0 − ½βY′ + O(R−2) and ω = ω − ½βY′ + O(R−2). In order for this solution to be uniformly valid in time,
i1520-0485-30-8-2072-e41
must be chosen, which expresses an equilibrium between the Coriolis force (averaged in time and over the volume of the disk) and the imbalance −(f0cy′) of the equatorward centrifugal and poleward gravitational forces (present because the frame is rotating, relative to an inertial one, with velocity Ω + cR−1 secθ0 instead of Ω); see Fig. 7.
The rotation κ̇ of the center of mass around (x′, y′) = (0, 0) as well as the rotation ω of the disk around its center are equally affected by the meridional displacements, so that both f + 2κ̇ and f + 2ω remain constant. Consequently the inner rotation with respect to the orientation of 0X is uniform: ωκ̇ = ω + f0 + O(R−2). The inner rotation with respect to the local north (i.e., that measured by an observer moving with the disk) is
i1520-0485-30-8-2072-eq26
The parameter τ0 (=R−1 tanθ0) is ignored by the classical “β-plane” approximation, which therefore incorrectly predicts σω.
The solution transformed to spherical coordinates is complicated by the presence of geometric terms, proportional to τ0. For instance, the time-mean zonal velocity of the center of mass is not equal to c but, rather, given by
i1520-0485-30-8-2072-e42
(NB: f0τ0 = β tan2θ0). Also, the time and space average of the particles zonal velocity is
i1520-0485-30-8-2072-e43
The classical β-plane approximation fortuitously predicts the correct value of c but, because it makes τ0 = 0, incorrectly gives U = u = c.

It will be shown in Part II that the bulk formulas (4.1), (4.2), and (4.3) are also valid for the vortex solutions, when the time mean is well defined. The physics is, of course, more complicated because the water volume can distort and there are pressure forces related to its instantaneous shape.

This work was done unaware of that of McDonald (1998) on the same system, where the disk equations are derived from the Lagrangian, just like in this paper. McDonald (1998) analyzes the solutions corresponding to, in the present notation, the initial condition ϕ̇(0) = −½f0, U(0) = 0—or, more generally, U(0) = O(R−1)—and V(0) = 0, which yields a small inertial oscillation with radius ρ = c/f0 = O(R−1); that is, they are closer to the uniformly translating solution than to the general case considered here, for which it is assumed that ρ = O(R0). In the small oscillation case, ρ = O(R−1), the parameters of (3.6) can be approximated as φ = f0 and χ = ½β, instead of (3.9), and thus the system reduces to f0V = 0 & + f0U = ½βIω (≡f0c), which has a simple analytical solution.

Acknowledgments

This work has been supported by CICESE core funding and by CONACyT (Mexico) under Grant 26670-T. Critical reading of the manuscript by Federico Graef and Javier Beron is sincerely appreciated. Prof. Christopher Hughes called my attention to the paper by McDonald (1998).

REFERENCES

  • Ball, F., 1963: Some general theorems concerning the finite motion of a shallow rotating liquid lying on a paraboloid. J. Fluid Mech.,17, 240–256.

  • Benilov, E. S., 1996: Beta-induced translation of strong isolated eddies. J. Phys. Oceanogr.,26, 2223–2229.

  • Cushman-Roisin, B., 1982: Motion of a free particle on a beta-plane. Geophys. Astrophys. Fluid Dyn.,22, 85–102.

  • Goldstein, H., 1981: Classical Mechanics. Addison-Wesley, 672 pp.

  • Graef, F., 1998: On the westward translation of isolated eddies. J. Phys. Oceanogr.,28, 740–745.

  • Killworth, P. D., 1983: On the motion of isolated lenses on the beta-plane. J. Phys. Oceanogr.,13, 368–376.

  • McDonald, N., 1998: The time-dependent behaviour of a spinning disc on a rotating planet: A model for geophysical vortex motion. Geophys. Astrophys. Fluid Dyn.,87, 253–272.

  • Nof, D., 1981: On the β-induced movement of isolated baroclinic eddies. J. Phys. Oceanogr.,11, 1662–1672.

  • Nycander, J., 1996: Analogy between the drift of planetary vortices and the precession of a spinning body. Plasma Phys. Rep.,22, 771–774.

  • Paldor, N., and P. D. Killworth, 1988: Inertial trajectories on a rotating earth. J. Atmos. Sci.,45, 4013–4019.

  • Ripa, P., 1997: “Inertial” oscillations and the β-plane approximation(s). J. Phys. Oceanogr.,27, 633–647.

  • ——, 2000: Effects of the earth’s curvature on the dynamics of isolated objects. Part II: The uniformly translating vortex. J. Phys. Oceanogr., in press.

  • Stommel, H. M., and D. W. Moore, 1989: An Introduction to the Coriolis Force. Columbia University Press, 297 pp.

  • White, A., 1989: A relationship between energy and angular momentum conservation in dynamical models. J. Atmos. Sci.,46, 1855–1860.

APPENDIX A

Stereographic Coordinates

The terrestrial coordinates (λ, θ) of an arbitrary point are calculated from (λ′, θ′) defined in Fig. 1 as follows. The array R(cosθ′ cosλ′, cosθ′ sinλ′, sinθ′)T contains the Cartesian coordinates of the point, with origin at the center of the earth and in the rotated system; the components R(cosθ cosλ, cosθ sinλ, sinθ)T, in the original system, are obtained by making a rotation of θ∗ − π/2 around the second axis and then a rotation of −λ∗ around the third axis. This gives
i1520-0485-30-8-2072-eq27
Consequently, the transformations (λ′, θ′) → (λ, θ) and (λ, θ) → (λ′, θ′) have the form
i1520-0485-30-8-2072-ea1
where
i1520-0485-30-8-2072-ea2
The arc element is given by |dr| = R2 + cos2θ2; the polar coordinates of Fig. 1 are obtained making
ϕλπ
and seeking a transformation θ′ → r such that |dr| = γ̃(r)dr2 + r22. The solution is given by the stereographic projection
i1520-0485-30-8-2072-ea4
and γ̃ = (1 + (r/2R)2)−1 ≡ cos2(π/4 − θ′/2). Notice that
θγ̃rR,θγ̃
Therefore, transformation (A.1)–(A.4) is equivalent to
i1520-0485-30-8-2072-ea6a
or, in the notation of (x, y) and (x′, y′) coordinates,
i1520-0485-30-8-2072-ea6b
Let (X′ + x′) and X′ be the stereographic coordinates of a generic particle and the center of mass of a thin object, respectively. Let μR be the distance from the center of mass to that of the planet. The values of X′ and μR are obtained making an average of the three-dimensional position vector R(cosθ′ cosλ′, cosθ′ sinλ′, sinθ′)γ. From (A.5) it follows 〈γ̃(X′ + x′)〉 = μΓ̃X, 〈2γ̃ − 1) = μ(2Γ̃ − 1)γ, where Γ̃ is the value of γ̃ at X′; eliminating μ it is found 〈γ̃(X′ + x′)〉/〈2γ̃ − 1〉 = Γ̃X′/(2Γ̃ − 1) and, therefore, 〈γ̃x′〉 = X′〈γ̃Γ̃〉/(2Γ̃ − 1) (and not 〈x′〉 = 0, as a naive use of the x′ coordinates might suggest). Finally using 〈 · · · 〉 = ∫∫ ( · · · )γ̃2 d2x′/∫∫ γ̃2 d2x′ and expanding in R−1 it follows that
i1520-0485-30-8-2072-ea7

APPENDIX B

Center of Mass and Relative Motion

In case 1 of (2.6), that is, describing the center of mass coordinates in the spherical frame (X, Y) and the relative ones in the stereographic projection (x′, y′), using (2.7) with (x∗, y∗) = (X, Y) it follows that
i1520-0485-30-8-2072-eb1
On the other hand, in case 2 of (2.6), that is, describing both the center of mass and relative coordinates in the stereographic frame (X′ + x′, Y′ + y′), using (2.7) with (x∗, y∗) = (ct, 0) and (x′, y′) replaced by (X′ + x′, Y′ + y′), it follows that
i1520-0485-30-8-2072-eb2
Note that it is used c = O(R−1), as shown in Table 1, that is, the drift vanishes in f-plane dynamics. Finally, the coordinates of the center of mass in both frames are related by
i1520-0485-30-8-2072-eb3
The velocity transformations are obtained from the time derivative of either (B.1) or (B.2), recalling that u ∼ (1 − τ0y)ẋ, υ = ẏ, u′ ∼ ′, and υ′ ∼ ′,
i1520-0485-30-8-2072-eb4
are obtained in the first case (B.1), and
i1520-0485-30-8-2072-eb5
in the second one (B.2).

APPENDIX C

Disk’s Oscillations from Integrals of Motion

If the energy is a bit larger than that required for this uniform precession, then the latitude of the disk will experience small oscillations, say,
θ0ρR−1fatOρ2
and therefore
VρfafatOρ2
where the frequency fa has yet to be determined. Substituting in A = const and requiring the coefficient of sinfat to vanish yields
i1520-0485-30-8-2072-eq30
Similarly, from M = const it is found that
i1520-0485-30-8-2072-eq31
Finally, E = const gives
i1520-0485-30-8-2072-eq32
Now, if one assumes IωR2Ω and IR2 then3 it follows
fbf0faf0
namely, the motion of the center of the disk is a particle’s inertial oscillation with radius ρ, given by the equations (3.10).

Fig. 1.
Fig. 1.

The central grid are isolines of the rotated spherical coordinates (λ′, θ′), constructed so that the pole θ′ = π/2 is located at a given reference point, defined by the (terrestrial) longitude and latitude (λ∗, θ∗), and such the meridians λ′ = 0 and λ = λ∗ coincide (vertical lines in the figure). The polar coordinates of an arbitrary point (r, ϕ) in the new frame are defined through a stereographic projection, so that λ′ = ϕπ/2 and θ′ = π/2 − r/R + O(R−2).

Citation: Journal of Physical Oceanography 30, 8; 10.1175/1520-0485(2000)030<2072:EOTESC>2.0.CO;2

Fig. 2.
Fig. 2.

Bottom: The left graph coordinates (x, y) are fixed to the earth, whereas the right graph coordinates (x′, y′) = r(cosϕ, sinϕ) are defined by the transformation of Fig. 1, following the secular drift of an object: λ∗ = λ0 + δΩt and θ∗ = θ0. Both sets of coordinates resemble Cartesian ones near the origin; (x, y) are proportional to longitude and latitude changes (so the grid is made up of parallels and meridians) and the (x′, y′) space is a stereographic projection (the closed curves are circles with center in (x′, y′) = (0, 0). Top: Comparison, in both coordinates, of the inertial oscillation of a particle predicted by the exact equations (solid) and the classic β-plane approximation (dashed). The initial (northward) velocity equals 0.25 RΩ.

Citation: Journal of Physical Oceanography 30, 8; 10.1175/1520-0485(2000)030<2072:EOTESC>2.0.CO;2

Fig. 3.
Fig. 3.

The large circle shows the trajectory in the stereographic projection (X′(t), Y′(t)) of the center of a circular object (smaller circles). Notice that when X′ ≠ 0, the orientation of the axes in the (x′, y′) system, shown by the crosses, do not exactly coincide with the eastward and northward directions, shown by the latitude and longitude isolines (dashed). As a result, there are two definitions, σ and ω, of the vertical angular velocity.

Citation: Journal of Physical Oceanography 30, 8; 10.1175/1520-0485(2000)030<2072:EOTESC>2.0.CO;2

Fig. 4.
Fig. 4.

Disk on a rotating planet, described by the longitude Λ(t) and latitude Θ(t) of the center of mass, and the intrinsic rotation ϕ(t). Orthogonal components of the instantaneous rotation vector ωI are shown by thick lines.

Citation: Journal of Physical Oceanography 30, 8; 10.1175/1520-0485(2000)030<2072:EOTESC>2.0.CO;2

Fig. 5.
Fig. 5.

Inertia momenta In of the disk (with respect to the earth’s center) as a function of its radius a. Notice that I = R2Iz/Ix is very close to ½a2 (dashed line).

Citation: Journal of Physical Oceanography 30, 8; 10.1175/1520-0485(2000)030<2072:EOTESC>2.0.CO;2

Fig. 6.
Fig. 6.

Examples of disk evolution. Left and middle panels show the center’s orbit and velocity, whereas right panels show the vertical angular velocity relative to a fixed direction ω (solid) or relative to the local east σ (dashed). The initial center of mass position and velocity are Λ(0) = 0°, Θ(0) = 60°, U(0) = 0, and V(0) = 0.1RΩ. The initial internal rotation is cyclonic ω(0) = Ω (top), null ω(0) = 0 (middle), or anticyclonic ω(0) = −Ω (bottom).

Citation: Journal of Physical Oceanography 30, 8; 10.1175/1520-0485(2000)030<2072:EOTESC>2.0.CO;2

Fig. 7.
Fig. 7.

Inertial oscillation of the center of mass of a disk, seen from above and following the secular drift (moving stereographic coordinates). According to f-plane dynamics, the center of mass would make a circular uniform oscillation, with X′ = −f0 × ′ = −f20X′. With curvature effects and up to O(R−1), the orbit is also a circle but with larger speed the closer it is to the pole. The excess acceleration from the f-plane balance + f20X′ (thick arrows) is given by the sum of the excess Coriolis force −f × ′ + f20X′ (pointing radially) and the (poleward) sum of the mean Coriolis force due to the internal motion 〈−f × u′〉 and the geoforce −Φ′; the latter equals the imbalance between a poleward gravitational force (related to the shape and mass distribution of the earth) and the equatorward centrifugal force (see Table 2).

Citation: Journal of Physical Oceanography 30, 8; 10.1175/1520-0485(2000)030<2072:EOTESC>2.0.CO;2

Table 1.

Lowest-order contribution to the drift c, mean particle u, and center of mass U zonal velocities, where ω is the intrinsic rotation rate, r is the distance to the center of mass, and ρ is the radius of the inertial oscillation. The bar indicates a temporal mean and the angle brackets denote an average over the volume of the disk or the vortex. An inertial oscillation has the anticyclonic rotation ω = −f0 and thus the formulas for both components of the drift, ci and co, are similar.

Table 1.
Table 2.

Acceleration and minus the forces responsible for the inertial oscillation of a disk, as seen in a stereographic projection following the secular drift; and ŷ′ are unit vectors in the radial and meridional directions.

Table 2.

1

Bold symbols denote vectors. In particular, â is the unit vector in the direction of a.

2

As a curiosity, for a/R = π/2 it is Ix = Iz and therefore V = const: the real effects of earth’s rotation are not felt by such a hemispherical disk.

3

This includes the possibility of very large ω even if E is small, e.g., I = O(ρa) and ω = O(ρb) with a > 0 but b > −a/2.

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