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,

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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):

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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 c_i (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 c_i 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 c_o, and the temporal mean of the zonal velocity is u = c_o(1 + f₀τ₀/β) ≡ c_o 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 c_i 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.

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(2.2a)

and the arc element is given by |dr| = γ̃dr² + r²dϕ² = γ̃dx′² + dy′², with

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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

uuRδθλ̂(2.3)

and a particle is subject to two forces: the Coriolis one −f′ẑ × u′, where

fδθ(2.4)

is the corresponding Coriolis parameter, and the imbalance −∇Φ′ between the equatorward centrifugal force and the poleward gravitational one, discussed in the introduction, where

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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 + γ̃y′R⁻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

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where c = RδΩ 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

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whose inverse is x′ ∼ x − x∗ − τ₀(x − x∗)y and y′ ∼ y − y∗ + ½τ₀(x − x∗)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′)x̂′ + τ₀x′ŷ′ and ∇y ∼ −τ₀x′x̂′ + 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

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Since τ₀X′ = O(R⁻¹) ≪ 1, this transformation is just a rotation, which can be appreciated more clearly in polar coordinates. Writing

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it is α ∼ ϕ + τ₀X′, whose time derivative gives

ωστ₀Uωα̇,σϕ̇(2.9)

and U′ ∼ Ẋ′, 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 A_e of the angular momentum (per unit mass) in the directions of the vertical ẑ (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

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where x_I(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 ẋ_I − Ωê × x_I is the velocity measured by a terrestrial observer. Notice that the energy can also be written as E = ½〈ẋ²_I + (Ωê × x_I)²〉 − ΩA_e, where the term ½(Ωê × x_I)² is the potential Φ mentioned in the introduction and the term −ΩA_e arises from the transformation from the inertial to the terrestrial frame (White 1989).

Approximations of A, A_e, 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′ ∼ ẋ′. For case 1 we also needed 〈Ry′〉 ∼ [bu1034]R⁻¹〈y′x′² + y′³〉, 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 := A_e/(R cosθ₀) − RΩ cosθ₀ will be used instead of A_e. 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

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In the second case, that is, describing both the center of mass and the relative motion in the stereographic projection, we find

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where

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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

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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 ṙ and ρ̇ are found to vanish and the final result is

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in the first case and

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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〈ṙ² + r²(σ + τ₀U)²〉 in case 1 and ½〈ṙ² + 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 a → πR it is a whole sphere.) The exact evolution equations can be derived from the Lagrangian

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where ẋ_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 λ_I(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

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This operation could be used to evaluate the Lagrangian directly, as L = ½R²〈θ̇² + (λ̇²_I − 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 ω_I(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

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where (see Fig. 4)

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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

ω_zωσUR⁻¹(3.3)

(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

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[As shown in Fig. 5, it is I_z ≪ I_z except for huge disks, e.g., I_z = R² + O(a²) and I_z = ½a² + O(a⁴R⁻²) as a/R → 0.] Recall that the I_j are referred to the center of the planet, not the disk’s center of mass. The kinetic energy term in (3.1), T = ½ Σ³_j=1 I_jω²_j, equal to

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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 = ½I_xΩ² cos²Θ + ½I_zΩ² sin²Θ ≡ Ω²T₂. Consequently, the Lagrangian (3.1) is simply equal to the expression derived for T, without the terms proportional to Ω², namely, L ≡ T₀ + ΩT₁. Finally, recall that any multiple of L gives the same equations of motion; therefore the actual values of I_x and I_z 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,

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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/I_x 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, ωΛ̇A_eΘ̇RVϕ̇AE(3.4)

where the symbols (A, A_e, E) in this expression must be understood as the right-hand sides of

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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 dA_e/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 A_e 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 dA_e/dt = 0, the one obtained from the variation of L with respect to Θ, and dA/dt = 0, which take the form

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where

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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 . . . , A_e, RV, A]; H equals the total energy E but with U and ω expressed as functions of (A_e, 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 A_e and A, which are the Casimir integrals of motion (J · ∂A_e/∂z = 0 = J · ∂A/∂z). 3) For given values of A_e and A, determined by the initial conditions, z = [RΘ, V]; H = E; and J is the canonical 2 × 2 Poisson tensor.

Consider first the case studied by Nycander (1996), with no oscillations whatsoever, say, Θ(t) = θ₀ (and thus V = 0). Making V̇ = 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

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and ω = ϕ̇ + Λ̇ sinθ₀. This equation can be written as (Ω + Λ̇)N = −∂V/∂Θ, where N = I_yω_y sinθ₀ − I_zω_z cosθ₀ is the component of the angular momentum in the equatorial plane (ω_y and ω_z are the meridional and vertical components of ω_I). 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, −I_zΘ̇, vanishes, and A_e is constant). There are two values for Λ̇ (or U) for each ϕ̇; the slow solution (Λ̇ ≪ Ω) is given by

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where the second approximation corresponds to I_z ≪ 2I_x (or a² ≪ 4R²). For this pure precession solution, it follows from (B.4) 〈u〉 ∼ U + τ₀〈x′υ′〉 ∼ 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 c_i 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 c_o 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 (σ − σ) ≈ 7(ω − ω).

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 A_e. 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,ωẊMẎVϕ̇AE(3.8a)

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,

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Variation of L with respect to (U, V, ω) gives

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whereas variation with respect to (X, Y, ϕ) gives the same equations of motion (3.6), where now

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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 −¼Iβ 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) + τ₀Ẋ (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,

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is enough to calculate the results of Table 1. First, from V̇ = 0 it follows

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and thus

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where I ∼ ½a². Second, since U ∼ (1 − τ₀Y)Ẋ = c − τ₀YẊ, the secular drift c ∼ U + τ₀Y₀Ẋ₀ is given by

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Finally, the time mean of the particles zonal velocity average is calculated using (B.14), that is,

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taking the temporal average it follows that

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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, σ ∼ ω − τ₀U, to next order rotation functions are given by

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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 + τ₀Ẋ′ from (2.9), and neglecting terms of o(R⁻¹), we obtain

LYẊẎα̇UVωẊMẎVα̇AE(3.11)

where

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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′ = E − cM + O(R⁻²), whereas M′ = M − τ₀A + O(R⁻¹) because it is lacking the term τ₀A_CM. However, from the equations of motion it follows that dA_CM/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) = (Ẋ′, Ẏ′, α̇) and

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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 (X^′₀, Y^′₀, ω₀). The eigenvalue c is then chosen so that there is no secular drift in (X′, Y′), namely, V̇′ = 0, which yields c = c_i + c_o with c_i ≡ ½f⁻¹₀βIω and c_o = f⁻¹₀β = −½βρ². Finally (B.5) gives U ∼ c + τ₀ = c_i + c_o sec²θ₀ and 〈u〉 ∼ U + τ₀〈x̃′υ̃′〉 ∼ U + ½τ₀Iω = (c_i + c_o) 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

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in the notation of (2.8). This solution is interesting because the orbit is circular, to the given order. The speed is |Ẋ′| ∼ 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′ẑ × Ẋ′ 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 c_o (<0) and the corresponding part of the geoforce − f₀c_oŷ. In the case of the disk, c = c_o + c_i might be positive, for sufficiently fast cyclonic inner rotation.

The motion translated to terrestrial coordinates, obtained using (B.3) and (2.9),

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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

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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") = (X − ct, 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 V̇′ = 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,

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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: ω − κ̇ = 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

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The parameter τ₀ (=R⁻¹ tanθ₀) is ignored by the classical “β-plane” approximation, which therefore incorrectly predicts σ ≡ ω.

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

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(NB: f₀τ₀ = β tan²θ₀). Also, the time and space average of the particles zonal velocity is

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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 U̇ − f₀V = 0 & V̇ + f₀U = ½βIω (≡f₀c), which has a simple analytical solution.