• 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.

  • View in gallery

    Maclaurin spheroid with an equatorial radius equal to twice the polar radius. (For the earth, the relative difference between both radii is of the order of Ω2a/g ≈ 0.003.) Top: The larger (smaller) arrow represents the gravitational attraction (centrifugal force) at a point of the surface, where the local vertical is indicated by a dashed line. Bottom: Detail. Thick (thin) arrows represent the horizontal (vertical) components of the gravitational attraction and the centrifugal force.

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    Trajectory of a particle in the frictionless surface of a weakly rotating planet (Ω2a/g ≪ 1). To cross the equator it must be |V/Ωa| > sin2ϑ0 (=0.25 in these examples).

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    Classification of all possible trajectories of a particle in a rotating sphere. Those with negative angular momentum A can be obtained from those with A > 0 by means of the transformation (13).

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    Projection of three different 20° × 20° uniform grids in (x, y) into the sphere. The labels −1, 0, and 1 correspond to a choice of coordinate y, which is linear in the Mercator variable [tanh−1(sinϑ) = tanh−1(sinϑ0) + y/a coshμ0], the latitude (ϑ = ϑ0 + y/a), or its sine (sinϑ = sinϑ0 + y/a cosϑ0) respectively. The small circle indicates the reference latitude ϑ0 (the reference longitude is arbitrary).

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    Large amplitude inertial oscillation. The total time equals one period of the exact calculation (thick solid line). The thin solid line represents the traditional β-plane approximation, whereas the dashed (dot–dashed) lines correspond to correct first-order approximations in Mercator (spherical) coordinates.

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    As in Fig. 5 but for a larger energy.

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    As in Fig. 6 but for the position as a function of time.

  • View in gallery

    As in Fig. 6 but for the velocity as a function of time. Notice that the classical β-plane approximation overestimates the zonal velocity.

  • View in gallery

    As in Fig. 6 but for the orientation of the horizontal velocity as a function of time. Notice that the particle spends less time when it is moving eastward (90° > ξ > −90°) than otherwise (−90° > ξ > −270°); hence, a mean westward drift.

  • View in gallery

    Root-mean-square distance to the first cycle exact solution of Fig. 5 for linear approximations defined by the parameter φ2 in (28), (29). The circles at φ2/tanϑ0 = −1, 0, 1 correspond to the quadratic approximations (30), for the projections with labels −1, 0, and 1 in Fig. 4. The result for the classical β-plane approximation and both improvements proposed by Verkley (1990) are denoted by “;as,” “x,” and “t,” respectively.

  • View in gallery

    As in Fig. 10 but as a function of time. (The curves correspond to root-mean-square distances to the exact solution for every period T; instantaneous distances show oscillations that obscure the meaning of the figure.) The labels C, V1, V2, N1e, and N1o correspond to the linear approximations given by the classical β plane, both systems proposed by Verkley, and the equations derived here for φ2 = 0 and φ2 = ;d1 tanϑ0, respectively. The curve labeled N2 corresponds to the quadratic approximation for φ2 = 0.

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“Inertial” Oscillations and the β-Plane Approximation(s)

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  • 1 Centro de Investigación Científica y de Educación Superior de Ensenada, Ensenada, México
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Abstract

The general solution for the motion of a particle in the frictionless surface of a rotating planet is reviewed and a physical explanation of asymptotic solutions is provided. In the rotating frame at low energies there is a well-known quasi-circular oscillation superimposed to a weak westward drift; the latter is shown to be due to three different contributions, in the relative proportion of 1:tan2ϑ0:− tan2ϑ0, where ϑ0 is the mean latitude. The first contribution is due to the “β effect,” that is, the variation of the Coriolis parameter with latitude. The other two contributions are geometric effects, due to the tendency to move along a great circle and the change of the distance to the rotation axis with the latitude. The mean zonal velocity is produced by the first two contributions, and therefore is underestimated by the classical β-plane approximation [by a factor of (1 +tan2 ϑ0)−1 = cos2ϑ0] because of the lack of geometric effects in such a system. Correct first-order approximations are derived and found to belong to a one-parameter family, whose optimum element is obtained.

The key to develop a consistent approximation, with the right conservation laws, is to redefine three geometric coefficients by means of an expansion in a meridional coordinate, up to a fixed order. Making the expansion directly in the equations of motion, as done by other authors, leads to undesirable consequences for the conservation laws. This is true not only for particle dynamics but also for fields, as illustrated with the shallow-water equations. Correct approximations developed for this system are found to have the same integrals of motion as the exact one (angular momentum, energy, volume, and the potential vorticity of any fluid element). In the quasigeostrophic approximation, the geometric corrections cancel out in the potential vorticity law, and therefore the classical β plane gives the right prognostic equation, even though the (diagnostic) momentum equations are incorrect.

Corresponding author address: Dr. Pedro Ripa, CICESE, Km 107 Carretera Tijuana-Ensenada, (22800) Ensenada, BC, Mexico.

Email: ripa@cicese.mx

Abstract

The general solution for the motion of a particle in the frictionless surface of a rotating planet is reviewed and a physical explanation of asymptotic solutions is provided. In the rotating frame at low energies there is a well-known quasi-circular oscillation superimposed to a weak westward drift; the latter is shown to be due to three different contributions, in the relative proportion of 1:tan2ϑ0:− tan2ϑ0, where ϑ0 is the mean latitude. The first contribution is due to the “β effect,” that is, the variation of the Coriolis parameter with latitude. The other two contributions are geometric effects, due to the tendency to move along a great circle and the change of the distance to the rotation axis with the latitude. The mean zonal velocity is produced by the first two contributions, and therefore is underestimated by the classical β-plane approximation [by a factor of (1 +tan2 ϑ0)−1 = cos2ϑ0] because of the lack of geometric effects in such a system. Correct first-order approximations are derived and found to belong to a one-parameter family, whose optimum element is obtained.

The key to develop a consistent approximation, with the right conservation laws, is to redefine three geometric coefficients by means of an expansion in a meridional coordinate, up to a fixed order. Making the expansion directly in the equations of motion, as done by other authors, leads to undesirable consequences for the conservation laws. This is true not only for particle dynamics but also for fields, as illustrated with the shallow-water equations. Correct approximations developed for this system are found to have the same integrals of motion as the exact one (angular momentum, energy, volume, and the potential vorticity of any fluid element). In the quasigeostrophic approximation, the geometric corrections cancel out in the potential vorticity law, and therefore the classical β plane gives the right prognostic equation, even though the (diagnostic) momentum equations are incorrect.

Corresponding author address: Dr. Pedro Ripa, CICESE, Km 107 Carretera Tijuana-Ensenada, (22800) Ensenada, BC, Mexico.

Email: ripa@cicese.mx

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.

The centrifugal force (per unit mass) can be derived from the potential −(½)r2cos2ϑΩ2, 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 ϕG must be such that
i1520-0485-27-5-633-e1
(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 ϕG|surface =(½)a2cos2ϑΩ2, 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 ϕG|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
fϑ
which is twice the vertical component of the planet’s angular velocity.

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 A of the angular momentum in the direction of that axis is a constant of motion, whereas the other two components of the angular momentum are not conserved. The energy E is also conserved. For continuous systems, like the shallow-water equations, there are additional integrals of motion.

Instead of working with the full equations in the spherical geometry represented by the longitude and latitude (λ, ϑ), it is sometimes preferred to use an approximation, valid in some vicinity of a point (λ0, ϑ0). 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 = f0 or f = f0 + βy, respectively, where
i1520-0485-27-5-633-e3
However, the classical β-plane approximation is not a consistent one in the sense that the term βy is of the same order as the corrections to the planar geometry (except for ϑ0 = 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.
One way of maintaining the conservation laws with any approximation is to use the Lagrangian formalism, in which integrals of motion are related to symmetries of the Lagrangian function L(j(t), qj(t), t) (where {qj(t)} is any set of generalized coordinates and {j(t)} their time derivatives), which is used to write the equations of motion by the Euler formula
i1520-0485-27-5-633-e4
Now, if the longitude q = λ and the time t do not appear in L, then these equations are such that the angular momentum and the energy are conserved; namely,
i1520-0485-27-5-633-e5a
and
i1520-0485-27-5-633-e5b
In section 3 new approximations are derived by expanding geometric and Coriolis terms in a meridional coordinate y. The expansion is made in the Lagrangian L, not in the equations of motion, and in such a way that the symmetries of L are preserved. In this form the conservation laws of the original system are not lost, as shown in (5a).

The procedure to derive a correct approximation starts by defining coordinates (x, y) in some region of the sphere that includes the reference point (λ0, ϑ0). Phillips (1973) and Verkley (1990) choose coordinates such that the axis x = 0 is the meridian λ = λ0, whereas y = 0 is another great circle, tangent to the zonal displacement (λ0 + dλ, ϑ0). [Phillips makes a stereographic projection from the antipode (λ0 + π, −ϑ0), 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(x2/a2, y2/a2), 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 A is not explicit (in fact, this integral of motion is lost for these approximations). To avoid this problem, the present paper adopts a transformation in which 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

An appropriate Lagrangian is obtained as the difference between the kinetic energy in an inertial frame (½)u2I and the potential energy of the poleward force (1); namely,
i1520-0485-27-5-633-eq1
Notice that uI = a[cosϑ(λ̇ + Ω), ϑ̇] is the horizontal velocity in the inertial frame because the longitude in such a frame is simply λ + Ωt. Thus, the equations of motion are derived from the least action principle δ[∫(u2I/2 − ϕG) dt]/δqj = 0, where (u2I/2 − ϕG) is the Lagrangian in the inertial frame but qj = {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.
In this and following equations, the argument of L(. . .) indicates the qj and j the Lagrangian L is a function of, for example, in this case ϑ and (λ̇, ϑ̇). The explicit absence of a variable (like 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 cosϑ λ̇ and υ = a ϑ̇, so that the new L is
i1520-0485-27-5-633-e6
It can be checked that both Lagrangians yield equivalent systems.
This Lagrangian is independent of λ and t; use of (5a) and (5b) gives
i1520-0485-27-5-633-e7
as integrals of motion of the system. The angular momentum and energy in the inertial frame are AI = a cosϑ, uIA, and EI = (u2I + υ2I + Ω2a2cos2ϑ)/2 ≡ E + ΩA, respectively. Notice that the (conserved) energy in the rotating frame is purely kinetic; that is, the speed V = u2 + υ2 is constant and thus the orbit in the (u, υ) plane is a circle
uttVeiξ(t)
The evolution equations, obtained using (6) in (4), are
i1520-0485-27-5-633-e9
where f is given by (2). Notice that (8) is satisfied, where
i1520-0485-27-5-633-e10
is 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 secϑ and to tanϑ. The first one simply shows that the angular velocity λ̇ corresponding to a zonal velocity u decreases with the distance a cosϑ 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−1u tanϑ 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) = λ0, ϑ(0) = ϑ0 = 30°, u(0) = 0, and υ(0) = V, for different values of Va (the value of λ0 is of course irrelevant). For small values of Va (upper panels) the trajectory is an “inertial circle” in the horizontal surface, with a weak westward drift. Larger values of Va 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 Va → ∞ 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

The meridional motion of a general solution of (9) can be found using A = const and E = const in (7), from which is obtained
i1520-0485-27-5-633-e11a
where α and ε are nondimensional measures of angular momentum and speed,
i1520-0485-27-5-633-e11b
For instance, the trajectories of Fig. 2 correspond to solutions with α = cos2ϑ0 and ε = |Va|, that is, to the range of parameters 0 < α < 1 and ε > 0.
Assume for the moment that 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 + ε2/4 − ε/2 ≤ cosϑα + ε2/4 + ε/2. Moreover, the lower bound is always in the interval [0, 1] whereas the upper bound belongs to that interval only if ε ≤ 1 −α, 0 ≤ α ≤ 1. Consequently, there are two possibilities
i1520-0485-27-5-633-e12
The 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 latitudes
i1520-0485-27-5-633-eq2
where
ρα2σε4/4 + ε2α
Equation (11a) is easily integrated as t = t(ϑ), by making the change of variables sin2ϑ = ρ + σ sinΛ(t) or sin2ϑ = (ρ + σ)sin2Λ(t) in the first or second case, respectively. The period is then given by
i1520-0485-27-5-633-eq4
Notice that as ε → ∞ it is T → 2πa/|u| (which corresponds to the particle moving in a great circle, unaffected by Coriolis), whereas for ε → 0 it is T → 2π/|f0|.
To complete the solution, the longitudinal motion is obtained from the law of angular momentum conservation (7a) and Eq. (9a), which yield
i1520-0485-27-5-633-eq5
Since ϑ(t) is periodic, it follows that u(t) is also periodic, whereas λ(t) has two parts: a secular one, 〈α sec2ϑ − 1〉Ωt, and a periodic one, λ0 + Ωα t0 (sec2ϑ − 〈sec2ϑ〉) dt′, 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 = λ0 + Ωt; 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 = cos2ϑ0), 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 = Ωa(α − 1)]. All these possibilities are depicted in Fig. 3; see also a summary in Fig. 7 and Table 1 of Cushman-Roisin (1982).

Even though for low energies ε ≪ 1, the description in the rotating frame is preferred; the general solution is simpler in the inertial frame where, for instance, the equations of motion are invariant under a change of sign of longitude. In terms of the coordinates of the rotating frame, this symmetry expresses that the map
λtλtt,ϑtϑt
transforms solutions into solutions.2 For the integrals of motion, this transformation corresponds to
i1520-0485-27-5-633-eq6
or αα, ε ε2 + 4α. This covariance can be used to obtain the solutions with 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

Assume an infinitesimal energy in the rotating frame ε → 0, which implies an angular momentum such that O(ε) < α < 1 + O(ε), and expand (λ, ϑ, u, υ) as a0 + a1 + a2 + · · ·, where an = On) [recall that V = O(ε) and thus u0 = υ0 = 0]. To lowest order from (9) it is found
i1520-0485-27-5-633-e14
where R := V/f0; 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〉 = aϑ̈〉 = 0, which follows from ϑ(t) being periodic. From (9d) it is then found: 0 = 〈2Ωa u sinϑ + u2tanϑ〉 ∼ af0u2〉 + βa2u1 ϑ1〉 + 〈u21〉tanϑ0, and therefore
i1520-0485-27-5-633-e15
There are two different contributions, with the same sign, to this mean “Lagrangian” velocity: −(½)βR2, due to the variation of the Coriolis parameter 2Ω sinϑ with latitude (known as the β effect), and −(½)βR2tan2ϑ0, due to the geometric term proportional to (u/a)tanϑ. (Notice that for tan2ϑ0 > 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| of the circular orbit in the (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.
The mean zonal displacement is not given by 〈ut 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 is actually smaller. From (9a) it is found: 〈u2〉 = a cosϑ0λ̇2〉 − a sinϑ0λ̇1ϑ1〉, and therefore
i1520-0485-27-5-633-e16
This last geometric effect is of the same magnitude and opposite in sign to that due to the term proportional (u/a)tanϑ (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°.
The complete O2) contribution is
i1520-0485-27-5-633-e17
where 〈u2〉 is given by (15). Energy conservation at O3) requires u1(t)u2(t) + υ1(t)υ2(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 − f0tβVf−20 sec2ϑ0(1 − cos f0t) + O2), which is consistent with (10). More precisely, ξ̇(t) = −f0[1 + βf−20 sec2ϑ0u] + O2), and thus |ξ̇| is smaller when 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 (λ0, ϑ0) 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

This corresponds to making the change of variables λ = λ0 + secϑ0x/a, ϑ = ϑ0 + y/a, simplifying the equations of motion to a Cartesian geometry, but making a first-order expansion in y of the Coriolis parameter:
u,υ,u̇,υ̇f0βyυ,u
where f0 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 is
i1520-0485-27-5-633-eq7
whereas an expansion of (7a) gives
i1520-0485-27-5-633-e19
In other words, for τ0 ≠ 0 the classical β-plane approximation gives the correct angular momentum conservation law only up to O(y1), something which already is true for the f-plane approximation.
In order to assess the accuracy of this approximation, consider the problem of inertial oscillations in the limit of infinitesimal energy, studied at the end of section 2b. This system of equations gives, to lowest order in ε, the inertial circle of (14); namely, x = R(1 − cosf0t) and y = R sinf0t (this is also true for the f-plane approximation). However, to second order in ε, from υ̇ = 0 it follows
i1520-0485-27-5-633-eq8
instead of (15). This result is incorrect, the drift velocity is too small by a factor cos2ϑ0. However, the errors of not considering the geometric effects compensate each other for the calculation of the drift in λ̇ instead of u (see Table 1); namely, from λ = λ0 + secϑ0(x/a) it follows 〈u2〉 = a cosϑ0λ̇2〉 and therefore
i1520-0485-27-5-633-eq9
which is the exact result given in (16)!

b. Correct approximations

For the reasons explained in the introduction, a change of variables is chosen such that λ = λ(x) and ϑ = ϑ(y); namely,
i1520-0485-27-5-633-e21
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 λ0 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).
With this change of variables the Lagrangian (6) becomes, with no approximations,
i1520-0485-27-5-633-e22
where
i1520-0485-27-5-633-e23
[Notice that in the definition of ψ, a term Ωa cosϑ0 has been subtracted from L, for convenience. This has no consequences whatsoever because any term in L that is an exact time derivative gives no contribution to the Euler equations (4).]
Neither x nor t appear explicitly in L from (22). Consequently, the angular momentum
Auγ1yaaψya
and the energy E = (½)(u2 + υ2) are conserved. (A differs from that in the previous definition by the constant factor a cosϑ0.) The equivalent to Eq. (11a), used to integrate the meridional motion, is
i1520-0485-27-5-633-e25
whose left-hand side is equal to a2ϑ̇2.
The equations of motion, obtained using (4) with (22), are
i1520-0485-27-5-633-e26
with f(y/a) and τ(y/a) defined by
i1520-0485-27-5-633-e27
where 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) ≡ 2Ωsinϑ and τ(η) ≡ tanϑ. 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(ηn+1), then γ1 and γ2 are represented by a polynomial of order n. (The equatorial case, ϑ0 = 0, is a special one and is treated further below.) For instance, the lowest order approximation (f plane) corresponds to making ϑ(η) = ϑ0 + η + O(η2); the geometric coefficients are those of Cartesian coordinates, γ1 = γ2 = 1 and Ωψ(η) = f0η, which with (27) gives f = f0 and τ = 0.

1) First-order approximations

A correct first-order β-plane approximation corresponds to
i1520-0485-27-5-633-eq10
where φ2 is arbitrary, for which the geometric coefficients are
i1520-0485-27-5-633-e28
with τ0 given by (20). The label in the examples of Fig. 4 is the value of φ2/tanϑ0. [The classical β-plane approximation corresponds to making γ1 = γ2 = 1 but ψ′(η) = f0 + βy.]
The equations of motion are (26) and (27), without any further approximation. In particular,
i1520-0485-27-5-633-e29
are such that f(η) = f0 + βy + O(η2) and τ(η) = tan ϑ0 + O(η), ∀ φ2. However, in order to attain the correct angular momentum conservation law (24), f and τ must be kept as in (29); for example, γ−11 should not be approximated as (1 + τ0y/a).
The system (26) with (28)—and thus (29)—is a correct first-order representation of the original problem (9), for any value of the parameter φ2. In particular, it gives the correct result for the inertial oscillations in the infinitesimal energy limit; namely,
i1520-0485-27-5-633-eq10b
which is the correct representation of (14), and
i1520-0485-27-5-633-eq11
which corresponds exactly to (15) and (16). The complete O2) terms are
i1520-0485-27-5-633-eq12
where
i1520-0485-27-5-633-eq13
Notice that the arbitrary parameter φ2 is found only in the result for y2 and that, furthermore, it disappears when calculating the latitude as ϑϑ0 + y1/a + y2/a + (½)φ2y21/a2. These solutions are thus equivalent to the O2) 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 O2).

2) Second-order approximations

Higher order approximations can be developed in a similar way; namely, for
i1520-0485-27-5-633-eq15
where (· · ·) represents the terms already described, it is
i1520-0485-27-5-633-e30
These geometric coefficients now give f(η) = 2Ωsinϑ + O(η3) and τ(η) = tanϑ(η) + O(η2), ∀ φ2, φ3 [although, as pointed out before, the whole expressions (27) must be used]. Values of φ2 and φ3 corresponding to choosing y to be linear in the Mercator coordinate (tanhμ = sinϑ), 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 = 0.2Ωa. The initial latitude is ϑ0 = 60°, and therefore the critical speed to cross the equator is Va = sin2ϑ0 = 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 (φ2 = 0, dot–dash) and Mercator (φ2 = −τ0, 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 = 0.4Ωa, 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, φ2 = 0 gives a better result than φ2 = −τ0 for this finite amplitude comparison. Even though from the point of view of perturbation expansion any value of φ2 in (28), (29) defines a correct approximation, one might wonder whether or not there is an optimum value of φ2 or, more generally, an optimum transformation ϑ(y).

The best value of φ2 could be defined as the one that gives the closest representation to some geometric coefficient, like cosϑ or cos2ϑ. However, it seems that the best choice is the value of φ2 that better represents (11a), an equation that determines the latitude range and the period of the meridional motion, for any value of A and E. For the system (26), Eq. (11a) is represented by (25)/(Ωa)2; both left-hand sides are equal to (ϑ̇/Ω)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)secϑ0 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)/(Ωa)2 equals 2sin2ϑ0(;d1)τ0 − φ2) (y/a)4 + O(y/a)5. Consequently, it is
i1520-0485-27-5-633-eq16
The 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 φ2. 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

Let ϑ0 = 0 and take ϑ = y/a for simplicity, which makes γ2 ≡ 1; thus, (26) becomes
i1520-0485-27-5-633-eq16b
The (n+1)th approximation is obtained expanding γ(ϑ)ψ(ϑ) up to ϑ2n. Thus, a first-order approximation is given by γ1=1, f=βy, τ=0, which is the classical equatorial β plane (e.g., see Gill 1982). The next approximation corresponds to γ1=1 − ϑ2/2, f=βy(1 − (2/3)ϑ2)/(1 − (½)ϑ2), and τ = ϑ/(1 − ϑ2/2); notice that the term [f + τ u/a] is represented with an error of O(ϑ4, ϑ3u).

c. Other β-plane approximations

Both Verkley (1990) and Müller (1995) choose λ = λ0 + secϑ0(x/a), ϑ = ϑ0 + 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 are
i1520-0485-27-5-633-e31
instead 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) is
i1520-0485-27-5-633-e32
which, even though it does coincide with (19) up to O(y2, 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.
Verkley (1990) proposed another type of approximation: use of the system (18) but reinterpret coordinates (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 by
i1520-0485-27-5-633-eq17
With this choice of variables (x, y), the metric coefficient and Coriolis parameter are cosy/a = 1 + O(y2) and2Ω sinϑ = f0 + βy + O(x2, y2), and therefore Eqs. (18) represent a consistent first-order approximation. However, in terms of these coordinates the angular momentum conservation law A = const yields (1 − tanϑ0 y/a)uf0y + Ω(x2 sin2ϑ0y2 cos2ϑ0)/a + O(x2u, x3) = 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 tanλ = secϑ0 tan〈t/a and sinϑ = sinϑ0 cos〈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 φ2 = 0. Figure 11 shows the time evolution of the errors made by the classical β-plane approximation (label C), both approximations proposed by Verkley (V1 and V2), the linear approximation proposed here for φ2 = 0 and φ2 = (;d1)tanϑ0 (N1e and N1o), and the quadratic approximation for φ2 = 0(N2). 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 V1, V2, N1e, and N1o 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 V1, V2, N1e, and N1o decrease as V3 as V → 0, whereas C decreases as V2. 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 γ1(y/a), γ2(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 (γ1, γ2, ψ) and keeping everywhere the same expression for them.

Equations (26a,b) give the rate of change of the coordinates (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 by
i1520-0485-27-5-633-eq18
The SWE take the form
i1520-0485-27-5-633-e33
where 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 = 2Ωsinϑ and τ = tanϑ) 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 (φ2 = 0) has γ1 = 1 − τ0y/a (where τ0 = tanϑ0), γ2 = 1, f = [f0 + βy(1 − τ20)]/(1 − τ0y/a), and τ = τ0/(1 − τ0y/a).
Equations (33b,c) are no more than the field representation of (26c,d) with the addition of the pressure gradient force. It can be shown that the right-hand side of (33a) equals −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 that
i1520-0485-27-5-633-e34
Therefore, the volume in a domain D is conserved when its boundary ∂D is a material line.
The local law of angular momentum conservation
i1520-0485-27-5-633-eq19
with
Ahγ1uaψ
follows from (33b) and the definitions (27). Notice that the angular momentum per unit area A equals the particle angular momentum A given in (24), times the depth h.
In order to derive the energy conservation law, it is convenient to rewrite Eqs. (33) in the form
i1520-0485-27-5-633-e35
where
i1520-0485-27-5-633-eq22
is the Bernoulli head and
i1520-0485-27-5-633-e36
is the representation of the absolute vertical vorticity. From (35b,c) it then follows
i1520-0485-27-5-633-e37
where the total energy per unit area equals
i1520-0485-27-5-633-eq23
Potential vorticity conservation
i1520-0485-27-5-633-e38
is found eliminating, by cross derivatives, the gradient of b in (35b,c). Finally, the functionals
i1520-0485-27-5-633-eq42
are 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 A any rigid boundary must be zonal so there is no contribution from the last term, (½)gxh2, in the local conservation law.

All the equations in this section are exact if γ1, γ2 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(η2) approximation, it is tempting to replace (1 − ητ0)−1 by (1 + ητ0) since this is consistent with the order of the approximation, but this replacement usually spoils strict conservation of the vorticity-related integrals of motion.

In the first-order approximation proposed by Verkley (1990) and Müller (1995), the volume conservation law takes the form
i1520-0485-27-5-633-eq43
instead of (35a) with γ1 = 1 − τ0 y/a and γ2 = 1. As a consequence, instead of (34) it is found
i1520-0485-27-5-633-eq24
where μ = exp(−τ0y/a), which is correct up to O(y). Namely, h(μ + O(y2)) dx dy 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(y2). However, there is no conservation law for potential vorticity (or it is very difficult to find it!).

a. Quasigeostrophic approximation

In the usual quasigeostrophic scaling, time derivatives, both components of velocity, and the meridional displacement y are assumed small, of the order of the Rossby number ε. To lowest order, Eqs. (35b,c) give the geostrophic balance f0[u, v] + [∂y, − ∂x]gh = 0. Consequently, the following expansions are usually proposed:
i1520-0485-27-5-633-eq25
which satisfy the O(ε) equations (35). The expansions of the absolute vorticity and the Bernoulli head give
i1520-0485-27-5-633-eq26
with ω1 = βy + (∇2f20/gH)ψ and b2 = gh2 + (ψ)2, where the nabla operator must be understood as if the geometry were Cartesian, that is, ∇2ψ = ∂xxψ + ∂yyψ and (∇ψ)2 = (∂xψ)2 + (∂yψ)2. Using these expansions in (35) it follows that the O2) terms are
i1520-0485-27-5-633-eq27
The 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 obtained
i1520-0485-27-5-633-eq28
in which all metric terms, proportional to τ0 and φ2, 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.

Acknowledgments

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.

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  • 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.

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APPENDIX

Motion on a Spheroid

The analysis of section 2 need not be restricted to motion in a sphere. Let the surface of an axisymmetric spheroid be defined by
raϑ
The radial and vertical direction are not parallel (see top part of Fig. 1) but have an angle
i1520-0485-27-5-633-eq30
both components of the horizontal velocity are
uaϑλ̇υa2 + a2ϑ̇
and thus the Lagrangian of a particle subject to the potential ϕsurfG = (½)a2Ω2cos2ϑ from (1) takes the form
i1520-0485-27-5-633-eq32
instead 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 motion
i1520-0485-27-5-633-eq33b
where
i1520-0485-27-5-633-eq34
are such that (f + a−1u τ) = (2Ω + λ̇)sin(ϑ + ζ). The equivalent of (11a), an equation which fixes the meridional motion and the period, is
a2a2ϑ̇2Ea−1Aϑaϑ2
The shape of the surface can be arbitrary, as long as the internal mass distribution is such to give a gravitational potential ϕsurfG = (½)a2Ω2 cos2ϑ. For a homogeneous planet, the axialsymmetric solution is the Maclaurin spheroid
aϑaμ;d12μ2ϑ−½
where a∗ is the radius of the sphere in the absence of rotation and sin2μ is the eccentricity (the ratio of the polar to the equatorial radii equals cosμ). The gravitational potential inside the spheroid is given by (Lyttleton 1953; Chandrasekhar 1969; King and Hele 1976)
i1520-0485-27-5-633-eq37b
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 ϕG − (½)Ω2r2 cos2ϑ be uniform at the surface r = a(ϑ), which may be shown to give
i1520-0485-27-5-633-eq38b
For infinitesimal eccentricity, it is ν = 1/5μ2 + O(μ4) and (Ω2a∗/g∗) = 2/5μ2 + O(μ4), which is a relationship ingeniously proved by Newton in the Principia.2a∗/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.
Fig. 1.
Fig. 1.

Maclaurin spheroid with an equatorial radius equal to twice the polar radius. (For the earth, the relative difference between both radii is of the order of Ω2a/g ≈ 0.003.) Top: The larger (smaller) arrow represents the gravitational attraction (centrifugal force) at a point of the surface, where the local vertical is indicated by a dashed line. Bottom: Detail. Thick (thin) arrows represent the horizontal (vertical) components of the gravitational attraction and the centrifugal force.

Citation: Journal of Physical Oceanography 27, 5; 10.1175/1520-0485(1997)027<0633:IOATPA>2.0.CO;2

Fig. 2.
Fig. 2.

Trajectory of a particle in the frictionless surface of a weakly rotating planet (Ω2a/g ≪ 1). To cross the equator it must be |V/Ωa| > sin2ϑ0 (=0.25 in these examples).

Citation: Journal of Physical Oceanography 27, 5; 10.1175/1520-0485(1997)027<0633:IOATPA>2.0.CO;2

Fig. 3.
Fig. 3.

Classification of all possible trajectories of a particle in a rotating sphere. Those with negative angular momentum A can be obtained from those with A > 0 by means of the transformation (13).

Citation: Journal of Physical Oceanography 27, 5; 10.1175/1520-0485(1997)027<0633:IOATPA>2.0.CO;2

Fig. 4.
Fig. 4.

Projection of three different 20° × 20° uniform grids in (x, y) into the sphere. The labels −1, 0, and 1 correspond to a choice of coordinate y, which is linear in the Mercator variable [tanh−1(sinϑ) = tanh−1(sinϑ0) + y/a coshμ0], the latitude (ϑ = ϑ0 + y/a), or its sine (sinϑ = sinϑ0 + y/a cosϑ0) respectively. The small circle indicates the reference latitude ϑ0 (the reference longitude is arbitrary).

Citation: Journal of Physical Oceanography 27, 5; 10.1175/1520-0485(1997)027<0633:IOATPA>2.0.CO;2

Fig. 5.
Fig. 5.

Large amplitude inertial oscillation. The total time equals one period of the exact calculation (thick solid line). The thin solid line represents the traditional β-plane approximation, whereas the dashed (dot–dashed) lines correspond to correct first-order approximations in Mercator (spherical) coordinates.

Citation: Journal of Physical Oceanography 27, 5; 10.1175/1520-0485(1997)027<0633:IOATPA>2.0.CO;2

Fig. 6.
Fig. 6.

As in Fig. 5 but for a larger energy.

Citation: Journal of Physical Oceanography 27, 5; 10.1175/1520-0485(1997)027<0633:IOATPA>2.0.CO;2

Fig. 7.
Fig. 7.

As in Fig. 6 but for the position as a function of time.

Citation: Journal of Physical Oceanography 27, 5; 10.1175/1520-0485(1997)027<0633:IOATPA>2.0.CO;2

Fig. 8.
Fig. 8.

As in Fig. 6 but for the velocity as a function of time. Notice that the classical β-plane approximation overestimates the zonal velocity.

Citation: Journal of Physical Oceanography 27, 5; 10.1175/1520-0485(1997)027<0633:IOATPA>2.0.CO;2

Fig. 9.
Fig. 9.

As in Fig. 6 but for the orientation of the horizontal velocity as a function of time. Notice that the particle spends less time when it is moving eastward (90° > ξ > −90°) than otherwise (−90° > ξ > −270°); hence, a mean westward drift.

Citation: Journal of Physical Oceanography 27, 5; 10.1175/1520-0485(1997)027<0633:IOATPA>2.0.CO;2

Fig. 10.
Fig. 10.

Root-mean-square distance to the first cycle exact solution of Fig. 5 for linear approximations defined by the parameter φ2 in (28), (29). The circles at φ2/tanϑ0 = −1, 0, 1 correspond to the quadratic approximations (30), for the projections with labels −1, 0, and 1 in Fig. 4. The result for the classical β-plane approximation and both improvements proposed by Verkley (1990) are denoted by “;as,” “x,” and “t,” respectively.

Citation: Journal of Physical Oceanography 27, 5; 10.1175/1520-0485(1997)027<0633:IOATPA>2.0.CO;2

Fig. 11.
Fig. 11.

As in Fig. 10 but as a function of time. (The curves correspond to root-mean-square distances to the exact solution for every period T; instantaneous distances show oscillations that obscure the meaning of the figure.) The labels C, V1, V2, N1e, and N1o correspond to the linear approximations given by the classical β plane, both systems proposed by Verkley, and the equations derived here for φ2 = 0 and φ2 = ;d1 tanϑ0, respectively. The curve labeled N2 corresponds to the quadratic approximation for φ2 = 0.

Citation: Journal of Physical Oceanography 27, 5; 10.1175/1520-0485(1997)027<0633:IOATPA>2.0.CO;2

Table 1.

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

Table 1.
Table 2.

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

Table 2.
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 = λ0 and ϑ = ϑ0, there is a westward rotating solution λ = −λ0 − 2Ωt and ϑ = ϑ0; in the inertial frame both solutions correspond to a circle of latitude traveled with angular velocity ±Ω.

3

 In the inertial frame the transformation implies AA and EI EI; Fig. 3 is left–right symmetric in those variables.

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