The change of the position of the Sun and the Moon with respect to the Sea is not the only effect of Earth’s rotation (Laplace 1778).

## 1. Introduction

The rotation of the earth produces a change of its shape, manifested by a flattening at the poles. This is illustrated in Fig. 1 (top) for a homogenous axial-symmetric planet at a fast rotation rate. Notice that the gravitational force is noncentral due to the attraction of the equatorial bulge. Even though the nonradial component of this force points toward the equator, its *horizontal* component is directed toward the nearest pole, due to the angle between the local vertical (dashed) and radial (dotted) directions. From the point of view of an inertial observer, and in the absence of other horizontal forces (like friction), this poleward horizontal force is responsible for accelerated motions, usually round a pole. From the point of view of an observer rotating with the earth, on the other hand, two other forces must be added to Newton’s equation: Coriolis and centrifugal; the last one is exactly balanced (by definition of horizontal!) by the poleward force due to gravitational attraction (see bottom of Fig. 1), and therefore one is left with just the Coriolis force. In the absence of other horizontal forces, the Coriolis one is responsible for the so-called inertial oscillations, which are no more than the accelerated motions seen by the inertial observer, from which the earth’s rotation has been added, as a change of coordinates.

*r*

^{2}cos

^{2}

*ϑ*Ω

^{2}, where

*r*is the distance to the center,

*ϑ*is the (geocentric) latitude, and Ω is the rotation rate of the earth. Consequently, at the surface the gravitational potential

*ϕ*

_{G}must be such that(or equal to an arbitrary constant) so as to achieve the balance indicated at the bottom of Fig. 1 (see Durran 1993). Notice that it is not necessary to know the gravitational potential away from the surface. Earth’s eccentricity is so small (see caption in Fig. 1) that it is convenient to approximate its surface by that of a sphere,

*r*=

*a,*but obviously keeping the balance (1), that is, including a gravitational potential

*ϕ*

_{G}|

_{surface}=(½)

*a*

^{2}cos

^{2}

*ϑ*Ω

^{2}, which yields the poleward horizontal force (Gill 1982). As mentioned above, this force cancels the horizontal component of the centrifugal force, leaving just the Coriolis force in the equations of motion used by a rotating observer. If

*ϕ*

_{G}|

_{surface}were not included, a rotating observer would have to use both the Coriolis

*and*the centrifugal force, and his/her conclusions would be quite different indeed (for instance, an object deposited with no impulse at a frictionless horizontal surface would go toward the equator). In the approximation of taking the surface as that of a sphere, rotation effects are parameterized by the so-called Coriolis parameter

*f*

*ϑ*

In section 2 is described the motion of a particle in such a system, in the absence of other horizontal forces (Whipple 1917; for more recent literature see Cushman-Roisin 1982; Stommel and Moore 1989). (The results are generalized in the appendix for a finite eccentricity Maclaurin spheroid, such as that of Fig. 1.) The gravitational force is noncentral but directed towards the earth’s axis. Consequently, the component

*λ*,

*ϑ*), it is sometimes preferred to use an approximation, valid in some vicinity of a point (

*λ*

_{0},

*ϑ*

_{0}). The most common approximations are the

*f*-plane and the

*β*-plane, which use a Cartesian (planar) geometry with coordinates (

*x, y*) and make

*f*=

*f*

_{0}or

*f*=

*f*

_{0}+

*β*

*y,*respectively, whereHowever, the classical

*β*-plane approximation is not a consistent one in the sense that the term

*β*

*y*is of the same order as the corrections to the planar geometry (except for

*ϑ*

_{0}= 0). Other

*β*-plane approximations have been derived by means of a perturbation expansion made directly in the equations of motion (e.g., Verkley 1990; Müller 1995), but they do not satisfactorily conserve exactly the integrals of motion of the original system, as explained below.

*q̇*

_{j}(

*t*),

*q*

_{j}(

*t*),

*t*) (where {

*q*

_{j}(

*t*)} is any set of generalized coordinates and {

*q̇*

_{j}(

*t*)} their time derivatives), which is used to write the equations of motion by the Euler formulaNow, if the longitude

*q*=

*λ*and the time

*t*do not appear in

*y.*The expansion is made in the Lagrangian

The procedure to derive a correct approximation starts by defining coordinates (*x, y*) in some region of the sphere that includes the reference point (*λ*_{0}, *ϑ*_{0}). Phillips (1973) and Verkley (1990) choose coordinates such that the axis *x* = 0 is the meridian *λ* = *λ*_{0}, whereas *y* = 0 is another great circle, tangent to the zonal displacement (*λ*_{0} + d*λ*, *ϑ*_{0}). [Phillips makes a stereographic projection from the antipode (*λ*_{0} + *π*, −*ϑ*_{0}), whereas Verkley uses spherical coordinates such that *y* = 0 is their equator.] With these choices of variables, the corrections to the geometric coefficients are *O*(*x*^{2}/*a*^{2}, *y*^{2}/*a*^{2}), and thus the classical *β*-plane equations are correct up to order **x**/*a.* Notice, however, that with these coordinates *λ* and *ϑ* are functions of both *x* and *y,* and thus the zonal symmetry related to the conservation of the angular momentum *ϑ* is only a function of *y*; there is not a unique choice for this function, though, and therefore the problem of finding the optimum one will also be addressed.

The idea of making approximations directly in the Lagrangian or the Hamiltonian, instead of in the equations of motion, can also be implemented for systems with infinite degrees of freedom; for example, in geophysical fluid dynamics it has been successfully applied by Salmon (1983) and Shutts (1989). The use of the new *β* plane, developed here, in geophysical fluid dynamics is exemplified in section 4, with the shallow-water equations. The main conclusions are presented in section 5.

## 2. Particle on a rotating sphere

*in an inertial frame*(½)

**u**

^{2}

_{I}

**u**

_{I}=

*a*[cos

*ϑ*(

*λ̇*

*ϑ̇*

*λ*+ Ω

*t.*Thus, the equations of motion are derived from the least action principle

*δ*[∫(

**u**

^{2}

_{I}

*ϕ*

_{G}) d

*t*]/

*δ*

*q*

_{j}= 0, where (

**u**

^{2}

_{I}

*ϕ*

_{G}) is the Lagrangian in the

*inertial*frame but

*q*

_{j}= {

*λ*,

*ϑ*} are coordinates in the

*rotating*frame. This is essentially the procedure used by Pierre Simon de Laplace to derive his tidal equations, before Gustave Coriolis had even been born (Ripa 1995, 1997). The advantage of using a scalar like the Lagrangian (or what now is known as d’Alembert principle, which is what Laplace used) is the freedom in the choice and change of variables, freedom that will be fully exercised in this paper.

*q*

_{j}and

*q̇*

_{j}the Lagrangian

*ϑ*and (

*λ̇*

*ϑ̇*

*λ*in this case) is important in view of conservation law (5a). Use of (4) with the Lagrangian above yields second-order evolution equations. However, it is possible to augment the number of variables in order to get first-order equations. This is done here introducing the horizontal velocity components in the

*rotating frame,*

*u*=

*a*cos

*ϑ*

*λ̇*

*υ*=

*a*

*ϑ̇*

*λ*and

*t*; use of (5a) and (5b) givesas integrals of motion of the system. The angular momentum and energy in the inertial frame are

_{I}=

*a*cos

*ϑ*,

*u*

_{I}≡

_{I}= (

*u*

^{2}

_{I}

*υ*

^{2}

_{I}

^{2}

*a*

^{2}cos

^{2}

*ϑ*)/2 ≡

*V*=

*u*

^{2}+

*υ*

^{2}

*u, υ*) plane is a circle

*u*

*t*

*iυ*

*t*

*V*

*e*

^{iξ(t)}

*f*is given by (2). Notice that (8) is satisfied, whereis the instantaneous angular velocity in the (

*u, υ*) plane.

There are two types of geometrical terms in the evolution equations (9), namely, those proportional to sec*ϑ* and to tan*ϑ*. The first one simply shows that the angular velocity *λ̇**u* decreases with the distance *a* cos*ϑ* to the planet’s axis. In order to understand the meaning of the second geometric coefficient, it is better to temporarily make *f* = 0 (i.e., Ω = 0). In the absence of rotation effects the motion is uniform and along a great circle (as predicted by Galileo in 1632) which is not given by (*u, υ*) = const because the coordinates (*λ*, *ϑ*) are not planar. The term proportional to *a*^{−1}*u* tan*ϑ* in (9c,d) vanishes for motion along a meridian (*u* = 0) or along the equator (*ϑ* = 0), which are great circles. On the other hand, if at a certain time the motion is in the direction of other parallel, with *ϑ* ≠ 0, then that term “deflects” the particle toward the equator; this is true for both signs of *u* and of *ϑ*. There is not a real change of horizontal direction on the sphere (hence the quotation marks) but on the representation of the trajectory in the (*λ*, *ϑ*) plane. Therefore, this is different from the real deflection produced by the Coriolis term, proportional to *f,* which, in particular, does depend on the direction of motion, as shown next.

As an example, Fig. 2 shows trajectories on the sphere, corresponding to an initial condition *λ*(0) = *λ*_{0}, *ϑ*(0) = *ϑ*_{0} = 30°, *u*(0) = 0, and *υ*(0) = *V,* for different values of *V*/Ω*a* (the value of *λ*_{0} is of course irrelevant). For small values of *V*/Ω*a* (upper panels) the trajectory is an “inertial circle” in the horizontal surface, with a weak westward drift. Larger values of *V*/Ω*a* allow the particle to cross the equator (lower panels) and even to have a mean eastward drift (lower right panel). Notice that the motion is not seen with constant speed and along a great circle in the inertial frame—or in any frame, for that matter—so the name “inertial oscillations” is not quite appropriate. Only for *V*/Ω*a* → ∞ the particle travels a great circle, as predicted by Galileo, which for these particular initial conditions corresponds to the union of opposite meridians.

### a. Exact solution

*α*and ε are nondimensional measures of angular momentum and speed,For instance, the trajectories of Fig. 2 correspond to solutions with

*α*= cos

^{2}

*ϑ*

_{0}and ε = |

*V*/Ω

*a*|, that is, to the range of parameters 0 <

*α*< 1 and ε > 0.

*α*> 0 (the case

*α*< 0 will be discussed later); it is then easy to show ε > max(0,

*α*− 1). Since the left-hand side of (11a) cannot be negative, it follows that

*α*+ ε

^{2}/4

*ϑ*≤

*α*+ ε

^{2}/4

*α*, 0 ≤

*α*≤ 1. Consequently, there are two possibilitiesThe first one corresponds to motion in only one hemisphere (e.g., upper panels in Fig. 2) and the second one to trajectories with equatorial crossings (lower panels in Fig. 2). The meridional motion

*ϑ*(

*t*) is found to be periodic,

^{1}between the extreme latitudeswhere

*ρ*

*α*

^{2}

*σ*

^{4}/4 + ε

^{2}

*α*

*t*=

*t*(

*ϑ*), by making the change of variables sin

^{2}

*ϑ*=

*ρ*+

*σ*sinΛ(

*t*) or sin

^{2}

*ϑ*= (

*ρ*+

*σ*)sin

^{2}Λ(

*t*) in the first or second case, respectively. The period is then given byNotice that as ε → ∞ it is

*T*→ 2

*π*

*a*/|

**u**| (which corresponds to the particle moving in a great circle, unaffected by Coriolis), whereas for ε → 0 it is

*T*→ 2

*π*/|

*f*

_{0}|.

*ϑ*(

*t*) is periodic, it follows that

*u*(

*t*) is also periodic, whereas

*λ*(

*t*) has two parts: a secular one, 〈

*α*sec

^{2}

*ϑ*− 1〉Ω

*t,*and a periodic one,

*λ*

_{0}+ Ω

*α*

^{t}

_{0}

^{2}

*ϑ*− 〈sec

^{2}

*ϑ*〉) d

*t*′, where 〈. . .〉 indicates a time average within one period.

For *α* = 0 the trajectory goes through one or both poles (depending on the sign of ε − 1) and the zonal motion is given by *λ* = *λ*_{0} + Ω*t*; that is, the longitude in the inertial frame is constant (see Fig. 3). For ε = 0, on the other hand, the particle is at rest in the rotating frame. In addition to solutions like those in Fig. 2 (for which *α* = cos^{2}*ϑ*_{0}), there are trajectories with *α* > 1, ε ≥ *α* − 1, for which the zonal velocity *u* is always positive [e.g., at the equatorial crossing it is *u* = Ω*a*(*α* − 1)]. All these possibilities are depicted in Fig. 3; see also a summary in Fig. 7 and Table 1 of Cushman-Roisin (1982).

*λ*

*t*

*λ*

*t*

*t,*

*ϑ*

*t*

*ϑ*

*t*

^{2}For the integrals of motion, this transformation corresponds to

^{}or

*α*−

*α*, ε

^{2}+ 4

*α*

*α*< 0 from those with

*α*> 0 described above, that is, the left side in terms of the right side in Fig. 3.

### b. Weak energy limit

*O*(ε) <

*α*< 1 +

*O*(ε), and expand (

*λ*,

*ϑ*,

*u,*

*υ*) as

*a*

_{0}+

*a*

_{1}+

*a*

_{2}+ · · ·, where

*a*

_{n}=

*O*(ε

^{n}) [recall that

*V*=

*O*(ε) and thus

*u*

_{0}=

*υ*

_{0}= 0]. To lowest order from (9) it is foundwhere

*R*:=

*V*/

*f*

_{0}; that is, |

*R*| is the radius of the circular motion in (14). The lowest order contribution to the westward drift can be calculated using 〈

*ν̇*

*a*〈

*ϑ̈*

*ϑ*(

*t*) being periodic. From (9d) it is then found: 0 = 〈2Ω

*a*

*u*sin

*ϑ*+

*u*

^{2}tan

*ϑ*〉 ∼

*af*

_{0}〈

*u*

_{2}〉 +

*β*

*a*

^{2}〈

*u*

_{1}

*ϑ*

_{1}〉 + 〈

*u*

^{2}

_{1}

*ϑ*

_{0}, and thereforeThere are two different contributions, with the same sign, to this mean “Lagrangian” velocity: −(½)

*β*

*R*

^{2}, due to the variation of the Coriolis parameter 2Ω sin

*ϑ*with latitude (known as the

*β*effect), and −(½)

*β*

*R*

^{2}tan

^{2}

*ϑ*

_{0}, due to the geometric term proportional to (

*u*/

*a*)tan

*ϑ*. (Notice that for tan

^{2}

*ϑ*

_{0}> 1, i.e., more than 45° away from the equator, the geometric contribution is larger than that due to the

*β*effect.) Both contributions tend to decrease (increase) the angular velocity |

*ξ̇*

*u, υ*) plane when the particle is closer to (farther from) the equator [see (8) and (10)]; as a consequence, the particle spends more time where the zonal velocity is negative than where it is positive, and thus 〈

*u*〉 < 0. Since the orbit is traveled with a constant speed

*V,*another manifestation of these effects is that in each cycle the part of the orbit where

*u*< 0 (the one closer to the equator) is larger than the other part.

*u*〉

*t*because of another geometric effect, which has a sign opposite to the other two discussed above: Even though the orbit is larger where the particle is closer to the equator (and

*u*< 0), the distance to the planet’s axis is greater there, and therefore the angular velocity

*λ̇*

*u*

_{2}〉 =

*a*cos

*ϑ*

_{0}〈

*λ̇*

_{2}〉 −

*a*sin

*ϑ*

_{0}〈

*λ̇*

_{1}

*ϑ*

_{1}〉, and thereforeThis last geometric effect is of the same magnitude and opposite in sign to that due to the term proportional (

*u*/

*a*)tan

*ϑ*(see Table 1). Consequently, the

*net*westward drift happens to coincide with that due to the

*β*effect alone, even though this is smaller than each one of the geometric effects poleward of ±45°.

*O*(ε

^{2}) contribution iswhere 〈

*u*

_{2}〉 is given by (15). Energy conservation at

*O*(ε

^{3}) requires

*u*

_{1}(

*t*)

*u*

_{2}(

*t*) +

*υ*

_{1}(

*t*)

*υ*

_{2}(

*t*) = 0, which is true for the solutions found here. Notice that the velocity is indeed of the form (8), where

*ξ*(

*t*) =

*π*/2 −

*f*

_{0}

*t*−

*β*

*V*

*f*

^{−2}

_{0}

^{2}

*ϑ*

_{0}(1 − cos

*f*

_{0}

*t*) +

*O*(ε

^{2}), which is consistent with (10). More precisely,

*ξ̇*

*t*) = −

*f*

_{0}[1 +

*β*

*f*

^{−2}

_{0}

^{2}

*ϑ*

_{0}

*u*] +

*O*(ε

^{2}), and thus |

*ξ̇*

*u*< 0, which was shown to be the reason behind the westward drift.

## 3. *β*-plane approximations

Instead of longitude and latitude (*λ*, *ϑ*), it is convenient to change to some coordinates (*x, y*), such that (*λ*_{0}, *ϑ*_{0}) corresponds to (*x, y*) = (0, 0) and the geometry is Cartesian in an infinitesimal neighborhood of the origin. Approximations are made by means of expansions in (*x, y*)/*a.* The most common of these types of simplified systems is reviewed next and is shown to be incorrect; then followed by a derivation of correct approximations including a discussion on the optimum one, and a comparison with other approximations, proposed in the literature.

### a. Classical β plane

*λ*=

*λ*

_{0}+ sec

*ϑ*

_{0}

*x*/

*a,*

*ϑ*=

*ϑ*

_{0}+

*y*/

*a,*simplifying the equations of motion to a Cartesian geometry, but making a first-order expansion in

*y*of the Coriolis parameter:

*ẋ*

*u,*

*ẏ*

*υ,*

*u̇,*

*υ̇*

*f*

_{0}

*β*

*y*

*υ,*

*u*

*f*

_{0}and

*β*are given by (3). This approximation is clearly not consistent because geometric coefficients should also be expanded to first order in

*y.*For instance, even though the kinetic energy is conserved (simply because the Coriolis force is perpendicular to the velocity), the other conservation law iswhereas an expansion of (7a) givesIn other words, for

*τ*

_{0}≠ 0 the classical

*β*-plane approximation gives the correct angular momentum conservation law only up to

*O*(

*y*

^{1}), something which already is true for the

*f*-plane approximation.

*x*=

*R*(1 − cos

*f*

_{0}

*t*) and

*y*=

*R*sin

*f*

_{0}

*t*(this is also true for the

*f*-plane approximation). However, to second order in ε, from

*υ̇*〉

^{2}

*ϑ*

_{0}. However, the errors of not considering the geometric effects compensate each other for the calculation of the drift in

*λ̇*

*u*(see Table 1); namely, from

*λ*=

*λ*

_{0}+ sec

*ϑ*

_{0}(

*x*/

*a*) it follows 〈

*u*

_{2}〉 =

*a*cos

*ϑ*

_{0}〈

*λ̇*

_{2}〉 and thereforewhich is the exact result given in (16)!

### b. Correct approximations

*λ*=

*λ*(

*x*) and

*ϑ*=

*ϑ*(

*y*); namely,where φ(

*y*/

*a*) is any invertible function, chosen so that φ(0) = 0 and φ′(0) = 1. Even though (

*x, y*) are curvilinear coordinates, they resemble Cartesian coordinates in the limit

*y*→ 0; the reference longitude

*λ*

_{0}is, of course, arbitrary. Figure 4 illustrates choices of φ(

*y*/

*a*) such that

*y*is linear in the latitude (which is the typical choice), in the Mercator coordinate [which makes small circles in the sphere appear as small circles in (

*x, y*)], or in the sine of the latitude (which makes the area between parallels to be proportional to the difference of their

*y*coordinates).

*ψ*, a term

*ẋ*Ω

*a*cos

*ϑ*

_{0}has been subtracted from

*x*nor

*t*appear explicitly in

*u*

*γ*

_{1}

*y*

*a*

*a*

*ψ*

*y*

*a*

*u*

^{2}+

*υ*

^{2}) are conserved. (

*a*cos

*ϑ*

_{0}.) The equivalent to Eq. (11a), used to integrate the meridional motion, iswhose left-hand side is equal to

*a*

^{2}

*ϑ̇*

^{2}.

*f*(

*y*/

*a*) and

*τ*(

*y*/

*a*) defined bywhere the prime indicates derivative with respect to the argument

*η*(=

*y*/

*a*).

The system (26), (27) is an exact representation of Eqs.(9). In particular, it is *f*(*η*) ≡ 2Ωsin*ϑ* and *τ*(*η*) ≡ tan*ϑ*. Approximations are now made in the expansion of the geometric coefficients (23), not in the equations of motion (26) nor the definition (27). This way the form of the integrals of motion, for example, (24), is preserved. If both φ(*η*) and *ψ*(*η*) are approximated by a Taylor expansion up to *O*(*η*^{n+1}), then *γ*_{1} and *γ*_{2} are represented by a polynomial of order *n.* (The equatorial case, *ϑ*_{0} = 0, is a special one and is treated further below.) For instance, the lowest order approximation (*f* plane) corresponds to making *ϑ*(*η*) = *ϑ*_{0} + *η* + *O*(*η*^{2}); the geometric coefficients are those of Cartesian coordinates, *γ*_{1} = *γ*_{2} = 1 and Ω*ψ*(*η*) = *f*_{0}*η*, which with (27) gives *f* = *f*_{0} and *τ* = 0.

#### 1) First-order approximations

*β*-plane approximation corresponds towhere φ

_{2}is arbitrary, for which the geometric coefficients arewith

*τ*

_{0}given by (20). The label in the examples of Fig. 4 is the value of φ

_{2}/tan

*ϑ*

_{0}. [The classical

*β*-plane approximation corresponds to making

*γ*

_{1}=

*γ*

_{2}= 1 but

*ψ*′(

*η*) =

*f*

_{0}+

*β*

*y.*]

*f*(

*η*) =

*f*

_{0}+

*β*

*y*+

*O*(

*η*

^{2}) and

*τ*(

*η*) = tan

*ϑ*

_{0}+

*O*(

*η*), ∀ φ

_{2}. However, in order to attain the correct angular momentum conservation law (24),

*f*and

*τ*must be kept as in (29); for example,

*γ*

^{−1}

_{1}

*τ*

_{0}

*y*/

*a*).

_{2}. In particular, it gives the correct result for the inertial oscillations in the infinitesimal energy limit; namely,which is the correct representation of (14), andwhich corresponds exactly to (15) and (16). The complete

*O*(ε

^{2}) terms arewhereNotice that the arbitrary parameter φ

_{2}is found only in the result for

*y*

_{2}and that, furthermore, it disappears when calculating the latitude as

*ϑ*∼

*ϑ*

_{0}+

*y*

_{1}/

*a*+

*y*

_{2}/

*a*+ (½)φ

_{2}

*y*

^{2}

_{1}/

*a*

^{2}. These solutions are thus equivalent to the

*O*(ε

^{2}) ones calculated in spherical coordinates (17). Consequently, any of these first-order approximations, that is, correct up to

*O*(

*y*/

*a*), give the exact result up to

*O*(ε

^{2}).

#### 2) Second-order approximations

*f*(

*η*) = 2Ωsin

*ϑ*+

*O*(

*η*

^{3}) and

*τ*(

*η*) = tan

*ϑ*(

*η*) +

*O*(

*η*

^{2}), ∀ φ

_{2}, φ

_{3}[although, as pointed out before, the whole expressions (27) must be used]. Values of φ

_{2}and φ

_{3}corresponding to choosing

*y*to be linear in the Mercator coordinate (tanh

*μ*= sin

*ϑ*), in the latitude, or in its sine (labels −1, 0, and 1, respectively, in Fig. 4) are given in Table 2.

#### 3) Finite energy comparisons

A comparison between the solution of the exact equations and those of several approximations is presented in Fig. 5. As in Fig. 2, the initial velocity is poleward *u*(0) = 0 and *υ*(0) = *V* = 0.2Ω*a.* The initial latitude is *ϑ*_{0} = 60°, and therefore the critical speed to cross the equator is *V*/Ω*a* = sin^{2}*ϑ*_{0} = 3/4. The thick solid line corresponds to the exact equations (9), the total time being one inertial period *T* calculated in that system, whereas the thin solid line gives the result of the classical *β*-plane approximation (18). The difference between both orbits is noticeable. On the other hand, the correct linear approximation, (26) with (28), (29), in spherical (φ_{2} = 0, dot–dash) and Mercator (φ_{2} = −*τ*_{0}, dashed) coordinates give, as expected, a much better result than the classical *β*-plane system, to the point that it is hard to distinguish those orbits from the exact one.

In order for the differences to stand out, a larger speed, *V* = 0.4Ω*a,* is chosen in Fig. 6 (for the orbits), in Figs. 7 and 8 (for the position and velocity as a function of time), and in Fig. 9 (for the horizontal direction). Clearly, φ_{2} = 0 gives a better result than φ_{2} = −*τ*_{0} for this finite amplitude comparison. Even though from the point of view of perturbation expansion any value of φ_{2} in (28), (29) defines a correct approximation, one might wonder whether or not there is an optimum value of φ_{2} or, more generally, an optimum transformation *ϑ*(*y*).

_{2}could be defined as the one that gives the closest representation to some geometric coefficient, like cos

*ϑ*or cos

^{2}

*ϑ*. However, it seems that the best choice is the value of φ

_{2}that better represents (11a), an equation that determines the latitude range and the period of the meridional motion, for any value of

*a*)

^{2}; both left-hand sides are equal to (

*ϑ̇*

^{2}. Cushman-Roisin (1982) calculated the difference between both sides of (11a) as predicted by the classical

*β*-plane approximation, concluding that for a consistent result

*β*= (2Ω/

*a*)sec

*ϑ*

_{0}was needed instead of (3b). This is an awkward result since the parameter

*β*cannot be redefined a posteriori, which reflects the inadequacy of this system for not treating correctly the geometric effects. Instead, making approximation (28), (29), it can be shown that for the difference between the left-hand sides of (11a) and (25)/(Ω

*a*)

^{2}equals 2sin2

*ϑ*

_{0}(;d1)

*τ*

_{0}− φ

_{2}) (

*y*/

*a*)

^{4}+

*O*(

*y*/

*a*)

^{5}. Consequently, it isThe solid line in Fig. 10 shows the root-mean-square distance from the exact solution of Fig. 5 to that of the order-one approximation, (26) with (29), as a function of φ

_{2}. As expected, the error made by these first-order approximations is much smaller than that of the classical

*β*-plane equations (18), shown with a star, and much larger than the error made by the second-order approximations, (26) with (28) and the coefficients from Table 2, indicated by small circles. The meaning of other points in this figure is explained next.

#### 4) The equatorial case

*ϑ*

_{0}= 0 and take

*ϑ*=

*y*/

*a*for simplicity, which makes

*γ*

_{2}≡ 1; thus, (26) becomesThe (

*n*+1)th approximation is obtained expanding

*γ*(

*ϑ*)

*ψ*(

*ϑ*) up to

*ϑ*

^{2n}. Thus, a first-order approximation is given by

*γ*

_{1}=1,

*f*=

*β*

*y,*

*τ*=0, which is the classical equatorial

*β*plane (e.g., see Gill 1982). The next approximation corresponds to

*γ*

_{1}=1 −

*ϑ*

^{2}/2,

*f*=

*β*

*y*(1 − (2/3)

*ϑ*

^{2})/(1 − (½)

*ϑ*

^{2}), and

*τ*=

*ϑ*/(1 −

*ϑ*

^{2}/2); notice that the term [

*f*+

*τ*

*u*/

*a*] is represented with an error of

*O*(

*ϑ*

^{4},

*ϑ*

^{3}

*u*).

### c. Other *β*-plane approximations

*λ*=

*λ*

_{0}+ sec

*ϑ*

_{0}(

*x*/

*a*),

*ϑ*=

*ϑ*

_{0}+

*y*/

*a,*as in the classical

*β*-plane approximation, and make an expansion in

*y*directly in the shallow-water equations. The equivalent of their equations for the particle motion areinstead of (18). Notice that this system also corresponds to expanding up to quadratic terms in (

*y, u, v*) the new approximate system (26) with (29). The angular-momentum-like conservation law for the system (31) iswhich, even though it does coincide with (19) up to

*O*(

*y*

^{2},

*yu*), is not trivial to figure out. There are more serious problems with the conservation laws for the corresponding approximation of the shallow-water equations, as will be pointed out in section 4.

*x, y*) as proportional to spherical coordinates chosen so that

*y*= 0 corresponds to their equator. Longitude and latitude in the original system are then given byWith this choice of variables (

*x, y*), the metric coefficient and Coriolis parameter are cos

*y*/

*a*= 1 +

*O*(

*y*

^{2}) and2Ω sin

*ϑ*=

*f*

_{0}+

*β*

*y*+

*O*(

*x*

^{2},

*y*

^{2}), and therefore Eqs. (18) represent a consistent first-order approximation. However, in terms of these coordinates the angular momentum conservation law

*ϑ*

_{0}

*y*/

*a*)

*u*−

*f*

_{0}

*y*+ Ω(

*x*

^{2}sin

^{2}

*ϑ*

_{0}−

*y*

^{2}cos2

*ϑ*

_{0})/

*a*+

*O*(

*x*

^{2}

*u,*

*x*

^{3}) = const, which coincides with the result (19) of (18) only up to

*O*(

*y, u*). Another problem with this approximation is that the coordinates do not reflect the zonal symmetry of the exact system. For instance, for the case of the free particle, the mean drift of the center of oscillation, along

*y*= 0, translates into longitude and latitude given by tan

*λ*= sec

*ϑ*

_{0}tan〈

*ẋ*〉

*t*/

*a*and sin

*ϑ*= sin

*ϑ*

_{0}cos〈

*ẋ*〉

*t*/

*a,*which is a great circle and not a parallel.

The mean error made by these two approximations in the one cycle integration of Fig. 5 are labeled by “Verkley 1” and “Verkley 2” in Fig. 10; these approximations perform better than the classical *β*-plane one, but worse than the new one for φ_{2} = 0. Figure 11 shows the time evolution of the errors made by the classical *β*-plane approximation (label *C*), both approximations proposed by Verkley (*V*1 and *V*2), the linear approximation proposed here for φ_{2} = 0 and φ_{2} = (;d1)tan*ϑ*_{0} (*N*1*e* and *N*1*o*), and the quadratic approximation for φ_{2} = 0(*N*2). They initially perform from worst to best in the order mentioned in the previous sentence. However, the linear approximation (31) of Verkley (1990) and Müller (1995) soon becomes worse than the classical *β*-plane equations. The systems corresponding to the curves *V*1, *V*2, *N*1*e,* and *N*1*o* are *formally* equivalent, in the sense that their difference with the exact one is of the same order in (*y,* **u**); however, this difference is responsible for the errors shown in Figs. 10 and 11. It has been checked that, for a fixed time, the correct linear approximations of the curves *V*1, *V*2, *N*1*e,* and *N*1*o* decrease as *V*^{3} as *V* → 0, whereas *C* decreases as *V*^{2}. Even though Figs. 10 and 11 correspond to particular examples, they are taken as good indicators of the performance of the different approximations at moderate energies.

## 4. Shallow-water equations

The shallow-water equations (SWE) can be derived from a Lagrangian functional, which could be used to construct approximations in the same way as it was done for particle dynamics in section 3. However, the essential results are in the equations of motion (26), valid for any choice of meridional coordinate *y.* The whole geometric structure is contained in the coefficients *γ*_{1}(*y*/*a*), *γ*_{2}(*y*/*a*), and *ψ*(*y*/*a*) from which the coefficients *f*(*y*/*a*) and *τ*(*y*/*a*) are derived using (27). The form of the systems is the same in the exact case or in any approximation, which is done by just expanding the geometric coefficients (*γ*_{1}, *γ*_{2}, *ψ*) and keeping everywhere the same expression for them.

*x, y*) (which are not Cartesian) as a function of the velocity (

*u, υ*). The material time derivative of any field

*ζ*(

*x, y, t*) is then given byThe SWE take the formwhere

*h*(

*x, y, t*) is the depth field,

*f*is the Coriolis parameter (including

*β*effect variations), and

*τ*comes from curvature terms. The exact equations (for which

*f*= 2Ωsin

*ϑ*and

*τ*= tan

*ϑ*) as well as any of the approximations proposed in this paper can be written in this form. The simplest of the first-order

*β*-plane approximations (φ

_{2}= 0) has

*γ*

_{1}= 1 −

*τ*

_{0}

*y*/

*a*(where

*τ*

_{0}= tan

*ϑ*

_{0}),

*γ*

_{2}= 1,

*f*= [

*f*

_{0}+

*β*

*y*(1 −

*τ*

^{2}

_{0}

*τ*

_{0}

*y*/

*a*), and

*τ*=

*τ*

_{0}/(1 −

*τ*

_{0}

*y*/

*a*).

*h*times the divergence of the velocity field. Thus, let

*D*be some arbitrary horizontal domain, whose boundary ∂

*D*is parameterized by (

*x, y*) = (

*X*(

*a, t*),

*Y*(

*a, t*)); from (33a) it follows thatTherefore, the volume in a domain

*D*is conserved when its boundary ∂

*D*is a material line.

*A*

*h*

*γ*

_{1}

*u*

*a*

*ψ*

*A*equals the particle angular momentum

*h.*

*b*in (35b,c). Finally, the functionalsare integrals of motion if the domain

*D*is such that its boundary ∂

*D*is rigid and/or made of a front

*h*= 0; in the case of the total angular momentum

*g*∂

_{x}

*h*

^{2}, in the local conservation law.

All the equations in this section are exact if *γ*_{1}, *γ*_{2} and *ψ* are those of (23). They correspond to a consistent approximation if these geometric coefficients are replaced by an expansion like the first-order one (28) or the second-order one (30). While working with an *O*(*η*^{2}) approximation, it is tempting to replace (1 − *η**τ*_{0})^{−1} by (1 + *η**τ*_{0}) since this is consistent with the order of the approximation, but this replacement usually spoils strict conservation of the vorticity-related integrals of motion.

*γ*

_{1}= 1 −

*τ*

_{0}

*y*/

*a*and

*γ*

_{2}= 1. As a consequence, instead of (34) it is foundwhere

*μ*= exp(−

*τ*

_{0}

*y*/

*a*), which is correct up to

*O*(

*y*). Namely,

*h*(

*μ*+

*O*(

*y*

^{2})) d

*x*d

*y*is the element of volume and the terms between parentheses vanish for a material line, with the particle motion calculated up to

*O*(

*yu*). The angular momentum and energy laws are the equivalent of (32) and (37), respectively, with an error of

*O*(

*y*

^{2}). However, there is no conservation law for potential vorticity (or it is very difficult to find it!).

### a. Quasigeostrophic approximation

*y*are assumed small, of the order of the Rossby number ε. To lowest order, Eqs. (35b,c) give the geostrophic balance

*f*

_{0}[

*u, v*] + [∂

_{y}, − ∂

_{x}]

*gh*= 0. Consequently, the following expansions are usually proposed:which satisfy the

*O*(ε) equations (35). The expansions of the absolute vorticity and the Bernoulli head givewith

*ω*

_{1}=

*β*

*y*+ (∇

^{2}−

*f*

^{2}

_{0}

*gH*)

*ψ*and

*b*

_{2}=

*gh*

_{2}+ (

**∇**

*ψ*)

^{2}, where the nabla operator must be understood as if the geometry were Cartesian, that is, ∇

^{2}

*ψ*= ∂

_{xx}

*ψ*+ ∂

_{yy}

*ψ*and (∇

*ψ*)

^{2}= (∂

_{x}

*ψ*)

^{2}+ (∂

_{y}

*ψ*)

^{2}. Using these expansions in (35) it follows that the

*O*(ε

^{2}) terms areThe variation of the metric coefficients is responsible for the terms on the right-hand side; these terms are not present in a similar development with the classical

*β*-plane approximation. However, eliminating the second-order fields from these equations, it is obtainedin which all metric terms, proportional to

*τ*

_{0}and φ

_{2}, have cancelled out. A similar result was found by Phillips (1973) and Pedlosky (1987) for a stratified fluid. Of course, this equation is but the lowest order term of the potential vorticity conservation law (38).

## 5. Conclusions

Two basic approximations usually made in geophysical fluid dynamics are the *shallow* and *traditional* ones, namely, making *r* = *a* in certain metric terms and neglecting the effect of the horizontal component of earth’s rotation. A controversy on the validity of different forms of these approximations was based on their representation (or lack of) of the energy and angular momentum conservation laws (Veronis 1968; Phillips 1968; Wangsness 1970; Phillips 1970). Two points made by Phillips (1968) are worth quoting: “scale factor [should be] approximated initially [and left untouched thereafter]” and “it is preferable to work with a self-contained system of equations in which further approximations are no longer needed [in order to achieve energy and angular momentum conservation].” A similar stance is taken here with respect to the so-called *β*-plane approximation, which is an attempt to incorporate the joint effects of earth’s rotation and curvature, using a geometry simpler than the spherical one.

The classical *β*-plane approximation is not a consistent one, while some proposed alternative approximations (Verkley 1990; Müller 1995) are not satisfactory in the sense that they do not conserve exactly the integrals of motion of the original system. The rate of change of these integrals may be of a low order, consistent with the perturbation expansion, but nevertheless their nonconservation is a nuisance that can—and thus should—be avoided. The problem of those approximations with the conservation laws comes from the fact that they have been derived by means of a perturbation expansion made directly in the equations of motion. This problem can be easily solved by making the expansion in some geometric coefficients and then leaving them untouched, much as Phillips suggested almost three decades ago.

The new approximation proposed here has a particularly simple form for the integrals of motion, both for a particle on the sphere as well as for the shallow water equations. The problem of the particle on the sphere is not strictly an oceanographic one. However, it is the simplest problem in which Coriolis effects on the earth can be examined. (Since already nearing the end of the century there are still textbooks that “explain” the physical meaning of the Coriolis effects by an analogy with a problem where rotation effects are only apparent, I considered it important to clarify this point in the introduction.) Moreover, the problem of the particle on the sphere is also a benchmark to compare different approximations. The linear approximation developed here has been found to perform better than those of Verkley (1990) and Müller (1995). No check has been done on the goodness of the new equations with fields, except to show that it has the right conservation laws. The differences with the classical *β*-plane approximation cancel out in the prognostic equation of the quasigeostrophic system. Consequently, the importance of the corrections to the classical *β*-plane equations will probably be appreciated in problems where high-frequency oscillations or ageostrophic terms are important. That study goes beyond the scope of the present paper.

Critical reading of the manuscript by Federico Graef, José Ochoa, Benoit Cushman-Roisin, and Peter Müller is sincerely appreciated. Dennis Moore called my attention to the “pioneer” paper of Whipple (1917) and was also kind enough to send me a copy. This work has been supported by CICESE’s institutional funding and by CONACyT (Mexico) under Grant 1799-PT.

## REFERENCES

Chandrasekhar, S., 1969:

*Ellipsoidal Figures of Equilibrium.*Yale University Press.Cushman-Roisin, B., 1982: Motion of a free particle on a beta-plane.

*Geophys. Astrophys. Fluid Dyn.,***22,**85–102.Durran, D. R., 1993: Is the Coriolis force really responsible for the inertial oscillation?

*Bull. Amer. Meteor. Soc.,***74,**2179–2184; 1994: Corrigendum.**75,**p. 261.Gill, A. E., 1982:

*Atmosphere–Ocean Dynamics.*Academic Press, 662 pp.King-Hele, D., 1976: The shape of the earth.

*Science,***192,**1293–1300.Lyttleton, R., 1953:

*The Stability of Rotating Liquid Masses.*Cambridge University Press.Müller, P., 1995: Ertel’s potential vorticity theorem in physical oceanography.

*Rev. Geophys.,***33,**67–97.Pedlosky, J., 1987:

*Geophysical Fluid Dynamics.*Springer-Verlag, 710 pp.Phillips, N. A., 1968: Reply.

*J. Atmos. Sci.,***25,**1155–1157.——, 1970: Reply.

*J. Atmos. Sci.,***27,**506.——, 1973: Principles of large scale numerical weather prediction.

*Dynamic Meteorology,*P. Morel, Ed., Reidel, 3–96.Ripa, P., 1995: Caída libre y la figura de la Tierra.

*Rev. Mex. Fís.,***41,**106–127.——, 1997:

*La Increíble Historia de la Malentendida Fuerza de Coriolis*(The Incredible Story of the Misunderstood Force of Coriolis). Fondo de Cultura Económica, in press.Salmon, R., 1983: Practical use of Hamilton’s principle.

*J. Fluid Mech.,***132,**431–444.Shutts, G., 1989: Planetary semi-geostrophic equations derived from Hamilton’s principle.

*J. Fluid Mech.,***208,**545–573.Stommel, H. M., and D. W. Moore, 1989:

*An Introduction to the Coriolis Force.*Columbia University Press, 297 pp.Verkley, W., 1990: On the beta-plane approximation.

*J. Atmos. Sci.,***47,**2453–2459.Veronis, G., 1968: Comments on Phillips’ proposed simplification of the equations of motion for a shallow rotating atmosphere.

*J. Atmos. Sci.,***25,**1154.Wangsness, R., 1970: Comments on the equations of motion for a shallow rotating atmosphere and the traditional approximation.

*J. Atmos. Sci.,***27,**504–506.Whipple, F., 1917: The motion of a particle on the surface of a smooth rotating globe.

*London Edinburgh Philos. Mag. J. Sci.,***33,**457–471.

# APPENDIX

## Motion on a Spheroid

*r*

*a*

*ϑ*

*u*

*a*

*ϑ*

*λ̇*

*υ*

*a*

^{2}+

*a*′

^{2}

*ϑ̇*

*ϕ*

^{surf}

_{G}

^{2}Ω

^{2}cos

^{2}

*ϑ*from (1) takes the forminstead of (6). Use of (5a) and (5b) shows that angular momentum and energy are conserved and have the same form as before (7) [but recall that

*a*=

*a*(

*ϑ*)]. Finally, using (4) it follows the equations of motionwhereare such that (

*f*+

*a*

^{−1}

*u*

*τ*) = (2Ω +

*λ̇*

*ϑ*+

*ζ*). The equivalent of (11a), an equation which fixes the meridional motion and the period, is

*a*

^{2}

*a*

^{2}

*ϑ̇*

^{2}

*a*

^{−1}

*ϑ*

*a*

*ϑ*

^{2}

*ϕ*

^{surf}

_{G}

*a*

^{2}Ω

^{2}cos

^{2}

*ϑ*. For a

*homogeneous*planet, the axialsymmetric solution is the Maclaurin spheroid

*a*

*ϑ*

*a*

*μ*

^{;d1}

^{2}

*μ*

^{2}

*ϑ*

^{−½}

*a*∗ is the radius of the sphere in the absence of rotation and sin

^{2}

*μ*is the eccentricity (the ratio of the polar to the equatorial radii equals cos

*μ*). The gravitational potential inside the spheroid is given by (Lyttleton 1953; Chandrasekhar 1969; King and Hele 1976)where

*g*∗ is the surface gravitational acceleration for vanishing eccentricity (

*μ*=

*ν*= 0). Finally, the relationship with the rotation rate is obtained by imposing that

*ϕ*

_{G}− (½)Ω

^{2}

*r*

^{2}cos

^{2}

*ϑ*be uniform at the surface

*r*=

*a*(

*ϑ*), which may be shown to giveFor infinitesimal eccentricity, it is

*ν*=

^{1}/

_{5}

*μ*

^{2}+

*O*(

*μ*

^{4}) and (Ω

^{2}

*a*∗/

*g*∗) =

^{2}/

_{5}

*μ*

^{2}+

*O*(

*μ*

^{4}), which is a relationship ingeniously proved by Newton in the

*Principia.*(Ω

^{2}

*a*∗/

*g*∗ has a maximum value of 0.337000, corresponding to an eccentricity of 0.864818.) The ellipsoid and forces in Fig. 1 were calculated with

*μ*=

*π*/3, that is, for an eccentricity of

^{3}/

_{4}.

Relative contributions to the westward drift, in the sense of the mean zonal velocity or the mean logitudinal displacement.

Second and third coefficients in the expansion of the latitude as a function of *y,* for the Mercator, linear, and sine transformations.

^{1}

For ε = 1 − *α*, *α* ϵ [0,1], it is *ϑ*_ = 0, but the motion is not periodic: the particle asymptotically approaches the equator as *t* → ±∞.

^{2}

For instance, for every resting solution *λ* = *λ*_{0} and *ϑ* = *ϑ*_{0}, there is a westward rotating solution *λ* = −*λ*_{0} − 2Ω*t* and *ϑ* = *ϑ*_{0}; in the inertial frame both solutions correspond to a circle of latitude traveled with angular velocity ±Ω.