## 1. Introduction

**Φ, where**

**∇****is the horizontal gradient operator,**

**∇***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

*f*

*θ.*

*θ*=

*θ*

_{0}. Three environmental parameters are needed for the approximate description of such a system (Ripa 1997, hereafter R97):

*f*

_{0}

*β*/

*τ*

_{0}+

*f*

^{2}

_{0}

^{2},

*f*

_{0}/(

*τ*

_{0}

*β*) =

*R*

^{2}, and

*f*

_{0}

*τ*

_{0}/

*β*= tan

^{2}

*θ*

_{0}. [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*=

*f*

_{0}+

*βy*/(1 −

*τ*

_{0}

*y*), in the notation of the present paper. Variables

*a*and

*τ*

_{0}used in R97 correspond to

*R*and

*τ*

_{0}

*R*in this paper, whereas

*φ*

_{2}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 *f*-plane approximation (*f* = *f*_{0}) 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 *c*: that caused by the intrinsic rotation *c*_{i} (discussed by Nycander) and that due the inertial oscillations *c*_{0} (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 *τ*_{0} = 0) and thus it gives incorrect results, except in the equatorial *β* plane (*θ*_{0} = 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**u*〉*ρ* (≪*R*), the drift is the orbital one *c*_{o}, and the temporal mean of the zonal velocity is *u**c*_{o}(1 + *f*_{0}*τ*_{0}/*β*) ≡ *c*_{o} sec^{2}*θ*_{0}; the *β*-plane model predicts correctly the former (due to the fortuitous cancellation of large errors) but subestimates the latter by a factor of cos^{2}*θ*_{0} (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 cos^{2}*θ*_{0} 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 *τ*_{0}, 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 *θ*_{0} 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.

*θ*≈

*θ*

_{0}, it is useful to change to rescaled spherical coordinates:

*x*

*λ*

*λ*

_{0}

*R*

*θ*

_{0}

*y*

*θ*

*θ*

_{0}

*R,*

*θ*∼

*θ*

_{0}. The squared arc element is exactly given by d

**r**

^{2}=

*γ*

^{2}

*dx*

^{2}+

*dy*

^{2}, where

*γ*= cos

*θ*sec

*θ*

_{0}. Since

*γ*= 1 −

*τ*

_{0}

*y*+

*O*(

*y*

^{2}/

*R*

^{2}) and

*f*=

*f*

_{0}+

*βy*+

*O*(

*y*

^{2}

*R*

^{−2}), non-Cartesian terms are of the same order as the

*β*term as

*y*/

*R*→ 0.

*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

*x*

*y*

*r*

*ϕ,*

*ϕ*

*d*

**r**| =

*γ̃*

*dr*

^{2}+

*r*

^{2}

*dϕ*

^{2}

*γ̃*

*dx*′

^{2}+

*dy*′

^{2}

*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.

*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*∗ = (

*δ*Ω)

*t*and

*θ*∗ =

*θ*

_{0}(as in Fig. 2). In the second case, and in any coordinates, the velocity in the new system is

^{1}

**u**

**u**

*Rδ*

*θ*

*λ̂**f*′

**ẑ**×

**u**′, where

*f*

*δ*

*θ*

**Φ′ between the equatorward centrifugal force and the poleward gravitational one, discussed in the introduction, where**

**∇****Φ′ will be called the “geoforce.” If the frame is subrotating, |Ω +**

**∇***δ*Ω| < Ω (suprarotating, |Ω +

*δ*Ω| > Ω), then the geoforce is poleward (equatorward). The parameters for the terrestrial (inertial) frame are recovered for

*δ*Ω = 0 (

*δ*Ω = −Ω), which in particular makes Φ′ ≡ 0 (

*f*′ ≡ 0). In order to write

*f*′ and Φ′ in terms of (

*x*′,

*y*′), we use sin

*θ*= (2

*γ̃*

*θ*

_{0}+

*γ̃y*′

*R*

^{−1}cos

*θ*

_{0}, from (A.5).

*c*=

*Rδ*Ω cos

*θ*

_{0}. 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.

*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

*x*′ ∼

*x*−

*x*∗ −

*τ*

_{0}(

*x*−

*x*∗)

*y*and

*y*′ ∼

*y*−

*y*∗ + ½

*τ*

_{0}(

*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 −

*τ*

_{0}

*y*+

*O*(

*y*

^{2}

*R*

^{−2}), whereas

*γ̃*

*r*

^{2}

*R*

^{−2}+

*O*(

*r*

^{4}

*R*

^{−4}).

*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 +

*τ*

_{0}

*y*′)

**x̂**′ +

*τ*

_{0}

*x*′

**ŷ**′ and

**∇***y*∼ −

*τ*

_{0}

*x*′

**x̂**′ +

**ŷ**′, respectively: directions in both frames differ when

*τ*

_{0}

*x*′ ≠ 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

*τ*

_{0}

*X*′ =

*O*(

*R*

^{−1}) ≪ 1, this transformation is just a rotation, which can be appreciated more clearly in polar coordinates. Writing

*α*∼

*ϕ*+

*τ*

_{0}

*X*′, whose time derivative gives

*ω*

*σ*

*τ*

_{0}

*U*

*ω*

*α̇,*

*σ*

*ϕ̇*

*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

_{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

**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}− Ω

**ê**×

**x**

_{I}is the velocity measured by a terrestrial observer. Notice that the energy can also be written as

**ẋ**

^{2}

_{I}

**ê**×

**x**

_{I})

^{2}〉 − Ω

_{e}, where the term ½(Ω

**ê**×

**x**

_{I})

^{2}is the potential Φ mentioned in the introduction and the term −Ω

_{e}arises from the transformation from the inertial to the terrestrial frame (White 1989).

Approximations of _{e}, and **x**_{1} = *R*(cos*θ* cos(*λ* + Ω*t*), cos*θ* sin(*λ* + Ω*t*), sin*θ*), 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.

*x*momentum in the

*f*plane,

_{e}/(

*R*cos

*θ*

_{0}) −

*R*Ω cos

*θ*

_{0}will be used instead of

_{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

*x*∗,

*y*∗) = (

*ct,*0). Notice that the energy in the frame rotating with velocity Ω +

*δ*Ω is

*f*

_{0}

*cY*′ is the leading term of Φ′ from (2.5).

*ω*and that observed following the object

*σ,*the expressions for

*r, ϕ, α*), for a particle, and (

*ρ, κ*), for the center of mass, are functions of time. Neither

*σ*=

*ϕ̇*

*ω*=

*α̇*

*ṙ*and

*ρ̇*

*τ*

_{0}

*c*is

*O*(

*R*

^{−2}) and is not included at this level of approximation]. Similarly, the kinetic energy of the relative motion is ½〈

*ṙ*

^{2}+

*r*

^{2}(

*σ*+

*τ*

_{0}

*U*)

^{2}〉 in case 1 and ½〈

*ṙ*

^{2}+

*r*

^{2}

*ω*

^{2}〉 in case 2. The expressions for

*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

*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

*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.

*brute force,*as follows. Spherical coordinates in the inertial frame are

*λ*

_{I}(

*t*) = Λ(

*t*) + Ω

*t*+

*F*

_{1}(

*λ*′,

*θ*′, Θ(

*t*)) and

*θ*(

*t*) =

*F*

_{2}(

*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

*F*

_{1}and

*F*

_{2}are defined in Eq. (A.2). The coordinates of an arbitrary particle in the disk are

*λ*′ =

*ϕ*

_{0}+

*ϕ*(

*t*) and

*θ*′, where

*ϕ*

_{0}is a label. The average on the disk then takes the form

*R*

^{2}〈

*θ̇*

^{2}

*λ̇*

^{2}

_{I}

^{2}) cos

^{2}

*θ*〉, but this is a quite cumbersome calculation. Instead, there is a much simpler and physically meaningful way to obtain

*ω*

_{I}(

*t*) made up of three contributions (see Fig. 4): 1) an angular velocity component

*π*/2; and 3) the intrinsic rotation

*ϕ̇*

*ω*is different from both the rotation rate

*ϕ̇*

*ω*

_{z}

*ω*

*σ*

*UR*

^{−1}

*σ*=

*ϕ̇.*

*ω*and

*σ*corresponds to the approximate formula (2.9).

*I*

_{z}≪

*I*

_{z}except for huge disks, e.g.,

*I*

_{z}=

*R*

^{2}+

*O*(

*a*

^{2}) and

*I*

_{z}= ½

*a*

^{2}+

*O*(

*a*

^{4}

*R*

^{−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),

^{3}

_{j=1}

*I*

_{j}

*ω*

^{2}

_{j}

_{0}+ Ω

_{1}+ Ω

^{2}

_{2}. The potential energy part of

*ϕ̇*

^{2}

*I*

_{x}Ω

^{2}cos

^{2}Θ + ½

*I*

_{z}Ω

^{2}sin

^{2}Θ ≡ Ω

^{2}

_{2}. Consequently, the Lagrangian (3.1) is simply equal to the expression derived for

^{2}, namely,

_{0}+ Ω

_{1}. Finally, recall that any multiple 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,

*a*/

*R*→ 0. The case of the particle, studied in R97, is recovered setting

*I*= 0.

*ϕ̇*)

_{0}+ Ω

_{1})

*R*

^{2}/

*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

*ϕ̇,*

*U, V, ω*

_{e}

*RV*

*ϕ̇*

_{e},

*ϕ,*Λ, and

*t*in

*d*

*dt*= 0, under a change in the reference longitude gives

*d*

_{e}/

*dt*= 0, and under a change of the origin of time gives

*d*

*dt*= 0. Note that the three terms on the right-hand side of (3.3) are included in the vertical angular velocity

*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

*f*+ 2

*ω*= const, is very similar to conservation of potential vorticity of a fluid element (in nondivergent setting). The first part of

_{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

*U*and not

*U*+

*R*Ω cosΘ, and in the internal kinetic energy appears

*ω*and not

*ω*+ Ω sinΘ.

*U, V,*and

*ω*gives the definitions (3.2) of these variables. The other three equations of motion are

*d*

_{e}/

*dt*= 0, the one obtained from the variation of

*d*

*dt*= 0, which take the form

*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, *d***z**/*dt* = **J** · ∂**z,** in three different interpretations: 1) **z** groups the generalized coordinates [Λ, Θ, *ϕ,* . . . and their conjugate momenta . . . , _{e}, *RV,* *U* and *ω* expressed as functions of (_{e}, Θ, **J** is the canonical 6 × 6 Poisson tensor. 2) **z** = [Θ, *U, V, ω*]; **J**(**z**) is a singular matrix of corank two. The null space of **J** is spanned by the gradients of _{e} and **J** · ∂_{e}/∂**z** = **0** = **J** · ∂**z**). 3) For given values of _{e} and **z** = [*R*Θ, *V*]; **J** is the canonical 2 × 2 Poisson tensor.

*ω*=

*I*− ΩsinΘ and

*U*=

*R*

^{−1}

_{e}secΘ −

*R*

^{−1}

*R*Ω cosΘ; then from energy conservation it follows

^{2}

*R*

^{−2}

*R*

^{−2}

_{e}

*R*

^{−2}

^{2}

*IR*

^{−2}

*I*

^{2}

*t*(

*Θ*) can be obtained by a simple integration;

*U*(

*t*) and

*ω*(

*t*) are then calculated from Θ(

*t*) using conservation of

_{e}; 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}(=

*I*

_{z}/

*I*

_{x}; see Fig. 5) and the integrals of motion

*R*

^{2}Ω

^{2},

*I*Ω, &

_{e}/

*R*

^{2}Ω for a particular orbit. Unlike the case of the particle (Cushman-Roisin 1982; Paldor and Killworth 1988; R97), equator-crossing orbits, which correspond to 2

_{e}

*R*

^{−1}− Ω

*R*)

^{2}+

^{2}/

*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

*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 ½

*R*

^{2}Ω

^{2}cos

^{2}Θ 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.

*t*) =

*θ*

_{0}(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

*ω*=

*ϕ̇*

*θ*

_{0}. This equation can be written as (Ω +

*I*

_{y}

*ω*

_{y}sin

*θ*

_{0}−

*I*

_{z}

*ω*

_{z}cos

*θ*

_{0}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 −∂

*I*

_{z}

_{e}is constant). There are two values for

*U*) for each

*ϕ̇*

*I*

_{z}≪ 2

*I*

_{x}(or

*a*

^{2}≪ 4

*R*

^{2}). For this pure precession solution, it follows from (B.4) 〈

*u*〉 ∼

*U*+

*τ*

_{0}〈

*x*′

*υ*′〉 ∼

*U*sec

^{2}

*θ*

_{0}. Since the center of the disk is at a fixed latitude, the drift velocity

*c*=

*R*cos

*θ*

_{0}

*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*^{−2}*I* = 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 2*π*/|*f*_{0}| corresponding to the latitude *θ*_{0}. 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**U**V**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**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}. Table 1 shows the lowest-order term of *c,* *U**u*〉*ρ* ∼ |*V*(0)/*f*_{0}| and *ω**t*) − *θ*_{0} 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 *τ*_{0} terms appears in the transformation back to (*x, y*) coordinates.

### a. Approximate solution in spherical coordinates

*X, Y*) the scaled spherical coordinates of the center of the disk, the Lagrangian (3.4) takes the form

*Y,*

*Ẋ,*

*Ẏ,*

*ϕ̇*

*U,*

*V,*

*ω*

*Ẋ*

*ẎV*

*ϕ̇*

*Y*the trigonometric functions in the definitions of

*U, V, ω*) gives

*X, Y, ϕ*) gives the same equations of motion (3.6), where now

*ϕ, X,*and

*t*in

*Iβ*has been subtracted from the definitions of

*τ*

_{0}

*βY*

^{3}, 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*=

*f*

_{0}+

*βY*/(1 −

*τ*

_{0}

*Y*) +

*τ*

_{0}

*Ẋ*(as used in R97 for the particle).]

*R*

^{−1}. The lowest-order solution, an

*f*-plane inertial oscillation with constant intrinsic rotation,

*V̇*

*I*∼ ½

*a*

^{2}. Second, since

*U*

*τ*

_{0}

*Y*)

*Ẋ*

*c*−

*τ*

_{0}

*YẊ*

*c*∼

*U*

*τ*

_{0}

*Y*

_{0}

*Ẋ*

_{0}

*c*is a factor cos

^{2}

*θ*

_{0}smaller than

*u*〉

*U*

*u*〉

*c*in the term due to the intrinsic rotation.

*σ*and

*ω,*namely,

*σ*∼

*ω*−

*τ*

_{0}

*U,*to next order rotation functions are given by

*σ*

_{1}/

*ω*

_{1}= 1 + 2 tan

^{2}

*θ*

_{0}; in the case of Fig. 6, it is 1 + 2 tan

^{2}

*θ*

_{0}= 7.

The next-order center of mass solution (*X*_{1}, *Y*_{1}) 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

*X, Y*) → (

*X*′,

*Y*′) in the Lagrangian (3.8), using

*α̇*

*ϕ̇*

*τ*

_{0}

*Ẋ*′ from (2.9), and neglecting terms of

*o*(

*R*

^{−1}), we obtain

*Y*

*Ẋ*

*Ẏ*

*α̇*

*U*

*V*

*ω*

*Ẋ*

*Ẏ*

*V*

*α̇*

*τ*

_{0}is an exact time derivative and was subtracted from the definition of

*O*(

*R*

^{−2}) and

*c*

*O*(

*R*

^{−2}), whereas

*τ*

_{0}

*O*(

*R*

^{−1}) because it is lacking the term

*τ*

_{0}

_{CM}. However, from the equations of motion it follows that

*d*

_{CM}/

*dt*=

*O*(

*R*

^{−1}) and therefore

*d*

*dt*=

*O*(

*R*

^{−2}). It is not surprising that

*τ*

_{0}

*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*∼ ½

*τ*

_{0}

*X*

^{2}) instead of a latitude circle (

*Y*= 0); even though the errors are

*O*(

*R*

^{−2}), they would not be uniformly so with time.

*U*′,

*V*′,

*ω*) = (

*Ẋ*′,

*Ẏ*′,

*α̇*

*βY*′ come from the dependence of

*f*′ from (2.4) on

*θ,*whereas the term 2

*δ*Ω sin

*θ*

_{0}is

*O*(

*R*

^{−2}) and therefore is not included at this level of approximation. The term −

*f*

_{0}

*c*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).

*R*

^{−1}, to lowest order it follows the inertial oscillation (3.10), but in the variables (

*X*

^{′}

_{0}

*Y*

^{′}

_{0}

*ω*

_{0}). The eigenvalue

*c*is then chosen so that there is no secular drift in (

*X*′,

*Y*′), namely,

*V̇*′

*c*=

*c*

_{i}+

*c*

_{o}with

*c*

_{i}≡ ½

*f*

^{−1}

_{0}

*βI*

*ω*

*c*

_{o}=

*f*

^{−1}

_{0}

*β*

*βρ*

^{2}. Finally (B.5) gives

*U*

*c*+

*τ*

_{0}

*c*

_{i}+

*c*

_{o}sec

^{2}

*θ*

_{0}and

*u*〉

*U*

*τ*

_{0}

*x̃*′

*υ̃*′〉

*U*

*τ*

_{0}

*I*

*ω*

*c*

_{i}+

*c*

_{o}) sec

^{2}

*θ*

_{0}. The bulk results

*c,*

*U*

*u*〉

*O*(

*R*

^{−2}) error is

**Ẋ**′| ∼

*ρ*|

*f*

_{0}+ ½

*βY*′|, 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 −

**(**

**∇***f*

_{0}

*cY*′). 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

*f*

_{0}

*βρ*

^{2}

**ŷ**′, 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

*βI*

*ω*

**ŷ**′. This time-mean Coriolis force is balanced by −

**(**

**∇***f*

_{0}

*cY*′) (the time mean of the acceleration is, of course, zero), giving

*c*= ½

*βI*

*ω*

*βρ*

^{2}. 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 −

*f*

_{0}

*c*

_{o}

**ŷ**. In the case of the disk,

*c*=

*c*

_{o}+

*c*

_{i}might be positive, for sufficiently fast cyclonic inner rotation.

*x, y*) coordinates, are obtained adding an arbitrary shift in

*X*′ and/or an

*f*-plane inertial oscillation with amplitude proportional to

*ρ*

^{2}/

*R.*

### c. The classical *β* plane

*τ*

_{0}= 0 in the Lagrangian (3.8) and therefore has the integrals of motion

*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*(

*dt*=

*O*(

*R*

^{−1}) for the former, whereas

*d*(

*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 *f*_{0}*c* replaced by *f*_{0}*c* + *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 *τ*_{0} 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 tan^{2}*θ*_{0} (=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**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 *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) *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 *I*(*ω* + ½*f*) 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 *θ*_{0}.

*x*′,

*y*′), whose origin moves along the reference latitude with the drift velocity

*c.*The Coriolis parameter is

*f*=

*f*

_{0}+

*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*(cos

*κ,*sin

*κ*) moving in a circle,

*ρ*= const +

*O*(

*R*

^{−2}), with

*κ̇*

*f*

_{0}− ½

*βY*′ +

*O*(

*R*

^{−2}) and

*ω*=

*ω*

*βY*′ +

*O*(

*R*

^{−2}). In order for this solution to be uniformly valid in time,

**(**

**∇***f*

_{0}

*cy*′) of the equatorward centrifugal and poleward gravitational forces (present because the frame is rotating, relative to an inertial one, with velocity Ω +

*cR*

^{−1}sec

*θ*

_{0}instead of Ω); see Fig. 7.

*κ̇*

*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*+ 2

*κ̇*

*f*+ 2

*ω*remain constant. Consequently the inner rotation with respect to the orientation of

**0X**′

*ω*−

*κ̇*

*ω*

*f*

_{0}+

*O*(

*R*

^{−2}). The inner rotation with respect to the local north (i.e., that measured by an observer moving with the disk) is

*τ*

_{0}(=

*R*

^{−1}tan

*θ*

_{0}) is ignored by the classical “

*β*-plane” approximation, which therefore incorrectly predicts

*σ*≡

*ω.*

*τ*

_{0}. For instance, the time-mean zonal velocity of the center of mass is not equal to

*c*but, rather, given by

*f*

_{0}

*τ*

_{0}=

*β*tan

^{2}

*θ*

_{0}). Also, the time and space average of the particles zonal velocity is

*β*-plane approximation fortuitously predicts the correct value of

*c*but, because it makes

*τ*

_{0}= 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)*f*_{0}, *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*/*f*_{0} = *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*(*R*^{0}). In the small oscillation case, *ρ* = *O*(*R*^{−1}), the parameters of (3.6) can be approximated as *φ* = *f*_{0} and *χ* = ½*β,* instead of (3.9), and thus the system reduces to *U̇* − *f*_{0}*V* = 0 & *V̇* + *f*_{0}*U* = ½*βI**ω**f*_{0}*c*), 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

*λ, θ*) of an arbitrary point are calculated from (

*λ*′,

*θ*′) defined in Fig. 1 as follows. The array

*R*(cos

*θ*′ cos

*λ*′, cos

*θ*′ sin

*λ*′, sin

*θ*′)

^{T}contains the Cartesian coordinates of the point, with origin at the center of the earth and in the rotated system; the components

*R*(cos

*θ*cos

*λ,*cos

*θ*sin

*λ,*sin

*θ*)

^{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

*λ*′,

*θ*′) → (

*λ, θ*) and (

*λ, θ*) → (

*λ*′,

*θ*′) have the form

*d*

**r**| =

*R*

*dϑ*′

^{2}+ cos

^{2}

*θ*′

*dλ*′

^{2}

*ϕ*

*λ*

*π*

*θ*′ →

*r*such that |

*d*

**r**| =

*γ̃*

*r*)

*dr*

^{2}+

*r*

^{2}

*dϕ*

^{2}

*γ̃*

*r*/2

*R*)

^{2})

^{−1}≡ cos

^{2}(

*π*/4 −

*θ*′/2). Notice that

*θ*

*γ̃*

*r*

*R,*

*θ*

*γ̃*

*x, y*) and (

*x*′,

*y*′) coordinates,

**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*(cos

*θ*′ cos

*λ*′, cos

*θ*′ sin

*λ*′, sin

*θ*′)

^{γ}. From (A.5) it follows 〈

*γ̃*

**X**′ +

**x**′)〉 =

*μ*Γ̃

**X**′

*γ̃*

*μ*

^{γ}, where

*γ̃*

**X**′; eliminating

*μ*it is found 〈

*γ̃*

**X**′ +

**x**′)〉/〈

*γ̃*

**X**′/(2

*γ̃*

**x**′〉 =

**X**′〈

*γ̃*

**x**′〉 = 0, as a naive use of the

**x**′ coordinates might suggest). Finally using 〈 · · · 〉 = ∫∫ ( · · · )

*γ̃*

^{2}

*d*

^{2}

**x**′/∫∫

*γ̃*

^{2}

*d*

^{2}

**x**′ and expanding in

*R*

^{−1}it follows that

## APPENDIX B

### Center of Mass and Relative Motion

*X, Y*) and the relative ones in the stereographic projection (

*x*′,

*y*′), using (2.7) with (

*x*∗,

*y*∗) = (

*X, Y*) it follows that

*X*′ +

*x*′,

*Y*′ +

*y*′), using (2.7) with (

*x*∗,

*y*∗) = (

*ct,*0) and (

*x*′,

*y*′) replaced by (

*X*′ +

*x*′,

*Y*′ +

*y*′), it follows that

*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

*u*∼ (1 −

*τ*

_{0}

*y*)

*ẋ,*

*υ*=

*ẏ,*

*u*′ ∼

*ẋ*′, and

*υ*′ ∼

*ẏ*′,

## APPENDIX C

### Disk’s Oscillations from Integrals of Motion

*θ*

_{0}

*ρR*

^{−1}

*f*

_{a}

*t*

*O*

*ρ*

^{2}

*V*

*ρf*

_{a}

*f*

_{a}

*t*

*O*

*ρ*

^{2}

*f*

_{a}has yet to be determined. Substituting in

*f*

_{a}

*t*to vanish yields

*I*

*ω*

*R*

^{2}Ω and

*I*≪

*R*

^{2}then

^{3}it follows

*f*

_{b}

*f*

_{0}

*f*

_{a}

*f*

_{0}

*ρ,*given by the equations (3.10).

Bottom: The left graph coordinates (*x, y*) are fixed to the earth, whereas the right graph coordinates (*x*′, *y*′) = *r*(cos*ϕ,* sin*ϕ*) are defined by the transformation of Fig. 1, following the secular drift of an object: *λ*∗ = *λ*_{0} + *δ*Ω*t* and *θ*∗ = *θ*_{0}. 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 *R*Ω.

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

Bottom: The left graph coordinates (*x, y*) are fixed to the earth, whereas the right graph coordinates (*x*′, *y*′) = *r*(cos*ϕ,* sin*ϕ*) are defined by the transformation of Fig. 1, following the secular drift of an object: *λ*∗ = *λ*_{0} + *δ*Ω*t* and *θ*∗ = *θ*_{0}. 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 *R*Ω.

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

Bottom: The left graph coordinates (*x, y*) are fixed to the earth, whereas the right graph coordinates (*x*′, *y*′) = *r*(cos*ϕ,* sin*ϕ*) are defined by the transformation of Fig. 1, following the secular drift of an object: *λ*∗ = *λ*_{0} + *δ*Ω*t* and *θ*∗ = *θ*_{0}. 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 *R*Ω.

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

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

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

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

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 *ω*_{I} are shown by thick lines.

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

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 *ω*_{I} are shown by thick lines.

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

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 *ω*_{I} are shown by thick lines.

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

Inertia momenta *I*_{n} of the disk (with respect to the earth’s center) as a function of its radius *a.* Notice that *I* = *R*^{2}*I*_{z}/*I*_{x} is very close to ½*a*^{2} (dashed line).

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

Inertia momenta *I*_{n} of the disk (with respect to the earth’s center) as a function of its radius *a.* Notice that *I* = *R*^{2}*I*_{z}/*I*_{x} is very close to ½*a*^{2} (dashed line).

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

Inertia momenta *I*_{n} of the disk (with respect to the earth’s center) as a function of its radius *a.* Notice that *I* = *R*^{2}*I*_{z}/*I*_{x} is very close to ½*a*^{2} (dashed line).

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

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) = 0.1*R*Ω. 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

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) = 0.1*R*Ω. 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

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) = 0.1*R*Ω. 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

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**′ = −*f*_{0}**ẑ** × **Ẋ**′ = −*f*^{2}_{0}**X**′. 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 *f*^{2}_{0}**X**′ (thick arrows) is given by the sum of the excess Coriolis force −*f*′**ẑ** × **Ẋ**′ + *f*^{2}_{0}**X**′ (pointing radially) and the (poleward) sum of the mean Coriolis force due to the internal motion 〈−*f*′**ẑ** × **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

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**′ = −*f*_{0}**ẑ** × **Ẋ**′ = −*f*^{2}_{0}**X**′. 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 *f*^{2}_{0}**X**′ (thick arrows) is given by the sum of the excess Coriolis force −*f*′**ẑ** × **Ẋ**′ + *f*^{2}_{0}**X**′ (pointing radially) and the (poleward) sum of the mean Coriolis force due to the internal motion 〈−*f*′**ẑ** × **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

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**′ = −*f*_{0}**ẑ** × **Ẋ**′ = −*f*^{2}_{0}**X**′. 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 *f*^{2}_{0}**X**′ (thick arrows) is given by the sum of the excess Coriolis force −*f*′**ẑ** × **Ẋ**′ + *f*^{2}_{0}**X**′ (pointing radially) and the (poleward) sum of the mean Coriolis force due to the internal motion 〈−*f*′**ẑ** × **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

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* = −*f*_{0} and thus the formulas for both components of the drift, *c _{i}* and

*c*are similar.

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

^{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 *I*_{x} = *I*_{z} and therefore

^{3}

This includes the possibility of very large *ω**I* = *O*(*ρ*^{a}) and *ω**O*(*ρ*^{b}) with *a* > 0 but *b* > −*a*/2.