## Abstract

The relation between lens translation speed and potential thickness anomaly in the second layer is investigated in a 2½-layer *β*-plane primitive model in the case of southward movement of lens-shaped eddies. The trapped dipole in the second layer under the lens generated by the initial westward movement of the lens drives the first-layer lens southward. The subsequent southward translation has an almost steady stage before the lens succumbs to baroclinic instability, and the lens in the first layer and a negative potential thickness anomaly in the second layer remain coupled. It is shown that the translation speed is related to the potential thickness anomaly and that the *β* effect in the first and second layers plays an important role in sustaining the steady translation. To examine this relation and discuss the dynamic balance of the steadily translating eddy, an analytic solution in a 2½-layer *f*-plane model is derived, assuming that the relative vorticity in the second layer is negligible. The solution shows the following: 1) If the translation speed is not zero, the area integral of the potential thickness anomaly in the second layer is constant irrespective of the translation speed. 2) The lens speed is related to the area average of the potential thickness anomaly in the second layer. 3) For steady translation, the area average of the potential thickness anomaly must be larger than a certain value. On the *β* plane, Rossby wave radiation, leakage of the potential thickness anomaly, and meridional displacement of the vortex structure lead to a transformation of the potential vorticity anomaly, and the constraints of the *f* plane are thus difficult to hold. From the comparison between the *f*-plane theory and the numerical experiments, however, these constraints are found to be almost satisfied on the *β* plane if the effect of the relative vorticity is included in the *f*-plane theory. This suggests that the increasing of the potential thickness anomaly due to the second-layer *β* during the southward translation balances the leakage of the potential thickness anomaly from the lens region, with the result that the constraints obtained by the *f*-plane theory hold.

## 1. Introduction

Recent observations have revealed the presence of anticyclonic interior submesoscale eddies such as Mediterranean eddies (meddies). Meddies move southward at speeds near 2 cm s^{−1} (Armi et al. 1989; Richardson et al. 1989), and many attempts have been devoted to explain their meridional motion.

It is well known that anticyclones (cyclones) move southwestward (northwestward) under the influence of the variation of the Coriolis parameter with latitude (beta effect) in the Northern Hemisphere (McWilliams and Flierl 1979; Cushman-Roisin et al. 1990; Chassignet and Cushman-Roisin 1991). Monopolar vortices in a one-layer model are advected by secondary eddies developed when vortices evolve on a *β* plane, the resulting vortex structure is called a beta gyre (Fiorino and Elsberry 1989; Smith and Ulrich 1990; Sutyrin and Flierl 1994).

Although the beta gyre provides a possible mechanism for the meridional translation of eddies, it does not work in lens-shaped eddies whose thickness vanishes along the outer edge since the strong nonlinearity retards Rossby wave dispersion (Flierl 1984). Flierl (1984) investigated the behavior of a warm core ring using a two-layer model in which the first-layer has a finite volume. The second layer is assumed to be deep so that the second-layer dynamics are quasigeostrophic and are forced by the motion of the warm pool. His model implies that warm core rings will radiate energy in the form of Rossby waves in the deep water. The radiated wave field carries energy away from the eddy and exerts a form drag on the warm pool. This form drag can only be balanced by a net Coriolis force due to the southward motion of the warm pool.

Chassignet and Cushman-Roisin (1991) studied the transition regime between a finite-depth system and the reduced-gravity approximation in detail with a primitive-equation, isopycnic-coordinate, two-layer numerical model whose first-to-second-layer depth ratio was varied. A series of experiments were performed with a Gaussian distribution of the first-layer thickness for cyclones, anticyclones (eddies whose thickness does not vanish along their rim), and lenses. The anticyclonic and cyclonic eddies tended to translate southwestward and northwestward, respectively. They tended to move westward faster and southward (or northward) slower as the first-to-second-layer depth ratio decreases. On the other hand, the lens is unstable to a baroclinic instability when the second layer is not very deep, and they did not find appreciable meridional translation of lenses, such as predicted by Flierl (1984).

Morel and McWilliams (1997) investigated the beta effect on the evolution of intrathermocline vortices, such as anticyclonic meddies, in a quasigeostrophic numerical model with fine vertical resolution. They classified the isolated vortices into two patterns depending on their potential vorticity (PV) anomaly structure. One of them was called an S vortex with a PV anomaly dominated by vortex stretching. An S vortex whose main PV anomaly is embedded between opposite-sign PV poles develops a tilt in its vertical axis; it leads to the formation of a tilted dipole. A cyclonic circulation develops below the main anticyclonic vortex and they propel each other, namely the hetonic interaction (Hogg and Stommel 1985). The mean displacement is southward or southwestward with complicated paths. These studies indicate that not only the barotropic structure of the beta gyre but also the baroclinic structure of potential vorticity such as an S vortex can cause meridional movement of vortices. It appears that the southward translation speed due to the beta gyre is insufficient to explain the propagation of meddies (McWilliams 1985).

Benilov (2000) examined the problem formulated by Flierl (1984) assuming *βR*_{d}/*f* ≪ *H*_{1}/*H*_{0} where *H*_{0} is the total depth and *H*_{1} is the thickness of the lens. He observed that the meridional translation speed grows linearly with time and overtakes the zonal translation speed in the numerical solution. He showed that the westward movement of the lens due to the *β* effect in the first layer is essential for southward translation.

In this paper, we shall consider a 2½-layer model and explore in detail the effect of the second layer on the translation of eddies in the first layer and the dynamic balance of eddies with strong southward propagation tendency such as the S vortex. In order to study the translation of eddies, a series of experiments are performed with a 2½-layer *β*-plane model in which the second-layer depth and *β* are varied. The model is initialized with a lens with zero potential vorticity as a first-layer eddy whose thickness vanishes at its periphery. This type of configuration is similar to that used in previous works (Flierl 1984; Chassignet and Cushman-Roisin 1991; Benilov 2000) but differs from them in the following point. Although the lens in a two-layer ocean is unstable when the lens thickness is comparable to the total depth (Chassignet and Cushman-Roisin 1991), the baroclinic instability in the 2½-layer model is weakened because of the small relative vorticity in the second layer. As will be shown, the lens has a stably propagating stage in this configuration, in contrast to that in Chassignet and Cushman-Roisin (1991).

In the next section, we outline the model configuration. In section 3, results of the numerical experiments with varying *β* and second-layer depth are presented. Some relations between the negative potential thickness anomaly and the lens translation speed will be shown. Three particular experiments in which *β* is set to zero are shown to make clear its importance for southward translation of the lens. In section 4, to investigate the dynamic balance of such a steadily translating lens, we derive an analytic solution under the assumption that both the *β* effect and the relative vorticity in the second layer are negligible. This solution shows that the translation speed is related to the area average of potential thickness (reciprocal of the potential vorticity) anomaly under the lens in the second layer (anomaly from the potential thickness in the resting case without the lens), and the area integral of the potential thickness anomaly is constant, independent of the translation speed. In the numerical experiments, however, the relative vorticity is important in some degree as well. We estimate and parameterize the effect of the relative vorticity. We compare the results of numerical experiments with the modified theory and obtain good agreement between them. Section 5 gives a summary and discussion.

## 2. Model configuration

We use a 2½-layer *β*-plane primitive model for numerical calculation. The flux-corrected transport (FCT) algorithm is applied to the first layer where the thickness outside the isolated lens vanishes. This algorithm is necessary to avoid the thickness becoming negative or the sharp density boundary diffusing. The original Bleck and Boudra (1986) reduced-gravity isopycnal model is used in Nof and Gorder (1999). The computational domain is chosen to span 600 km × 600 km horizontally with a grid spacing of 2.5 km. Comparing this grid spacing with solutions on finer grids, we found that it adequately resolves the vortex evolution. The north and south edges of the model ocean are slip boundaries; for the meridional boundaries, cyclic boundary conditions are used. The Coriolis parameter in a typical case is *f*_{0} = 1.0 × 10^{−4} s^{−1} with *β* = 2 × 10^{−11} m^{−1} s^{−1}. Density differences between layers are (*ρ*_{2} − *ρ*_{1})/*ρ*_{2} = (*ρ*_{3} − *ρ*_{2})/*ρ*_{2} = 1.529 052 × 10^{−3}, where subscripts 1, 2, and 3 represent the first, second, and third layers, respectively. The Boussinesq approximation has been made.

Schematic initial conditions are shown in Fig. 1. Here *h*_{1} and *h*_{2} are the thicknesses of the first and second layers, respectively, *H*_{2} is the ambient second-layer depth, and *η*_{2} = *h*_{1} + *h*_{2} − *H*_{2}. As the initial condition, an axisymmetric lens that is in gradient-wind balance is used. The potential vorticity of the lens is set to zero for simplicity. As a typical case, the depth of the interface between the second and third layers is initialized to 2000 m and the second layer is at rest initially. Therefore, the initial conditions of the lens in the first layer satisfy

where *r* is the distance from the center of the lens, *g*^{′}_{1} = *g*(*ρ*_{2} − *ρ*_{1})/*ρ*_{2}, and *υ*_{1θ} is the azimuthal velocity. Here we neglect the *β* effect and assume axisymmetry (∂/∂*θ* = 0). From these conditions, the lens structure at *t* = 0 will be determined as follows:

where *H*_{1} is the thickness at the center of the lens and *r*_{0} is the lens radius. The value for *H*_{1} is chosen to be 600 m; as a result, the lens radius is 84.9 km and the Rossby deformation radius, based on *H*_{1}, *g*^{′}_{1}*H*_{1}/*f*_{0}, is 30 km.

The westward propagation speed of a lens in a 1½-layer model is proportional to *β* (Nof 1981; Killworth 1983) and as Chassignet and Cushman-Roisin (1991) indicated, the propagation of the vortex is expected to depend on the first-to-second-layer depth ratio. Thus *β* and the ambient second-layer depth (*H*_{2}) are varied to investigate the dependence on lens propagation. The initial anticyclonic lens in the first layer implies a deepening of the interface between the first and second layers, hence the potential thickness (reciprocal of the potential vorticity) in the second layer is small under the lens.

We define the potential thickness anomaly in the second layer, Γ_{A}, by

where Γ_{2} is the potential thickness in the second layer:

Here, PV_{2} is the potential vorticity in the second layer.

The depths *H*_{1} = 600 m and *H*_{2} = 2000 m chosen for the standard run may be somewhat too large for direct application to the real ocean. We use these values, however, because we focus our attention only on the dynamics, without requiring that the results apply directly to specific ocean eddies, and we carry out experiments over a wide range of parameters. If we nondimensionalize the equations by using *f*^{−1}_{0}, *g*^{′}_{1}*H*_{1}/*f*_{0}, and *H*_{1} as the timescale, horizontal length scale, and vertical length scale, respectively, with *g*^{′}_{1} = *g*^{′}_{2}, then the resultant equation set contains only two external parameters, *β*∗ = *β**g*^{′}_{1}*H*_{1}/*f*^{2}_{0} and *H*_{1}/*H*_{2}. Therefore, the reduction of *g*^{′}_{1}*H*_{1}/*f*_{0} (or *H*_{1}) has the same physical meaning as the reduction of *β.* As will be shown in section 3b, even when *β* is reduced by one-half, the solution behavior is qualitatively unchanged.

## 3. Results of experiments

### a. Standard case

Figure 2 illustrates the time evolution of the potential vorticity in the second layer [hereinafter PV_{2}; see (2.2)] and the trajectory of the lens (left column), the interface displacement between second and third layers *η*_{2}, and velocity vectors in the second layer (right column) with *H*_{2} = 2000 m and *β* = 2 × 10^{−11} m^{−1} s^{−1}. There is a region in which PV_{2} contours close. The location of the lens is depicted by the bold dotted line, which represents the 10-m depth contour of the lens. Solid dots represent the location of the maximum thickness of the first layer (left column). We define this position as the center of the lens in our study. The lens begins its westward translation under the *β* effect, as Nof (1981) and Killworth (1983) indicated. Then below the lens in the second layer, the western part is squeezed and the eastern part is stretched; consequently, the generated anticyclonic and cyclonic vortices form a dipole (Fig. 2 on day 10), as Cushman-Roisin et al. (1990) suggested. This is similar to what was found in Morel and McWilliams (1997): namely the vertical tilting of the vortex core. The first-layer lens translation will be determined by the-self-propelling westward translation and the pressure distribution due to the displacement of the second density interface, *η*_{2}. It is noticeable that the lens has a strong southward drift tendency rather than a westward one, similar to that in Morel and McWilliams (1997). The translation is slow initially and gradually evolves toward a steady speed translation. In a longer calculation with a latitudinal extent 1800 km, the lens succumbs to baroclinic instability after about day 160 (Fig. 3). The growing of the translation speed in the early stage is the same as what was found in Benilov (2000). The near-steady translation in the later stage has not been observed in previous works, and though the lens is unstable in last stage, our analysis in section 4 will be focused on this near-steady state.

The high PV_{2} (negative potential thickness anomaly) fluid is always found to be located below the lens, shifted eastward and leaked out from the lens region little by little. The amount of the negative potential thickness anomaly fluid under the lens will be reduced as it translates southward. As is shown in Fig. 4, however, the southward translation speed does not decrease throughout the calculation. Figure 5 shows the lens profile in the initial state and on day 50. It is shown that the shape and velocity profile of the lens change little, except near its periphery. This will be important for analysis of the translation properties in section 4.

### b. The effect of the magnitude of β

Next we investigate the effect of *β.* For this purpose we change the value of *β* with *H*_{2} = 2000 m with *β* = 1 × 10^{−11} and 4 × 10^{−11} m^{−1} s^{−1}. The increase of *β* physically corresponds to an increase of the lens size.

The parameter dependence of the lens propagation is shown in Fig. 6a, which displays the trajectories of the position of the maximum thickness of the lens for three different values of *β.* As *β* increases, the translation speed increases. Benilov (2000) showed that this speed increases linearly with *β* in the first layer. In the early stage, the speed seems to be proportional to *β* (Fig. 4), as in Benilov (2000), but in a later stage it is rather proportional to the square root of *β* (Fig. 6b). This suggests that the dynamic balance in the latter stage is different from the early one.

### c. The effect of the second-layer depth

The lens evolution is examined as the second-layer depth is varied with *β* = 2 × 10^{−11} m^{−1} s^{−1} with *H*_{2} = 1, 2, 3, 4, 5, 10, 20, 30, 40, and 100 (× 10^{3} m). In the second layer, the potential thickness anomaly inside the lens region relative to the ambient potential thickness, (*h*_{2} − *H*_{2})/*H*_{2}, increases in inverse proportion to the second-layer depth. Although some of the values are unrealistically large, those *H*_{2} values are used for studying the case of a small relative potential thickness anomaly.

Figure 7a shows trajectories of the position of the maximum thickness of the lens for five different values of *H*_{2}. Figure 7b shows that the translation speed increases with 1/*H*_{2}. The lens translation has two tendencies:

as the second-layer depth increases, the southward translation speed decreases, and

the translation direction varies from south-southwestward to southward as

*H*_{2}increases.

One can interpret the tendency 1 by the following argument. As *H*_{2} increases, the relative vorticity becomes dominant in comparison with the stretching effect. In such a case, the shrinking of the fluid column in the second layer at the western side of the lens and the stretching at the eastern side of the lens balance the generation of the relative vorticity rather than the displacement of *η*_{2}. Therefore, the zonal pressure gradient, which drives the lens, and the induced translation speed decrease. We will discuss this in section 4 again.

An example demonstrating tendency 2 is shown in Fig. 8 (*H*_{2} = 10 000 m). In this case, the zonal component of the translation often becomes eastward. The translation direction should depend on the distribution of the high PV_{2} fluid since the location of the cyclonic circulation is consistent with the location of the high PV_{2} fluid. In comparison with Fig. 2, it is found that the leaking position of the high PV_{2} fluid shifts northward. This induces the negative *η*_{2} region to shift northeastward, and the second-layer pressure gradient (−*g*^{′}_{2}∇*η*_{2}) has a northward component (on day 40, day 80, and day 120). This pressure gradient could drive the lens eastward. Then why will this difference according to the value of *H*_{2} happen? The distribution of the leaked high PV_{2} fluid would be determined by the balance between the westward Rossby wave propagation speed and the southward lens translation speed. The deformation radius increases with increasing *H*_{2} as we mentioned before, so that the phase speed of the first baroclinic planetary Rossby wave increases with increasing *H*_{2}. On the other hand, as *H*_{2} increases, the translation speed decreases as mentioned for tendency 1. A combination of these two effects, that is, the fast westward propagation of leaked high PV_{2} fluid and the slow lens translation, is probably what causes the leaking point to shift northward.

### d. The importance of the potential thickness anomaly

The above discussion makes clear that the region in which PV_{2} contours close is important for the southward translation of the lens. If *H*_{2} increases further, the potential thickness of the second layer will be dominated by planetary *β.* Consequently, the area of this closed region decreases and the translation of the lens becomes gradual westward (Fig. 9). An example is shown in Fig. 10 (*H*_{2} = 100 000 m). The region of closed PV_{2} contours does not exist. The lens translates westward with the radiating planetary Rossby wave in the second layer (Flierl 1984).

Further, we show in Fig. 11 the case of second-layer thickness 2000 m everywhere including the area under the lens as the initial condition. The second-layer potential thickness contours are now not distorted and become lines of constant *y*; that is, the potential thickness anomaly does not exist at all. Since the motion is initially at rest in the second layer, there will be an adjustment toward geostrophic balance: this generates anticyclonic vorticity in the second layer. The lens translates westward, accompanied by this anticyclonic vortex. Since this vortex becomes dominant rather than the dipole, southward translation is not induced. The small southward drift results from the beta gyre mechanism, which acts on the second-layer vortex (see section 1).

### e. Translation speed and potential thickness anomaly

It was found in the previous sections that the region in which contours of potential thickness in the second layer close is essential for the southward drift of the first-layer lens. Now we show some relations between the negative potential thickness anomaly and the southward translation speed. Results shown in this section will be discussed in section 4.

Now we define the area integral of potential thickness anomaly in the second-layer *Q*_{E} as

where the suffix *E* denotes the numerical experiments and ∫ ∫_{2} means the integration over the high-PV_{2} (low Γ_{2}) region defined by

since the region *h*_{1} < 10 m spreads out. Figure 12 shows the relation between the translation speed *c*_{E} normalized by

and the area integral of potential thickness anomaly in the second layer nondimensionalized by the mass of the lens and *f*:

where *ɛ* = *R*^{2}_{d}/*r*^{2}_{0} is the ratio of the deformation radius (*R*_{d} = *g*^{′}_{2}*H*_{2}/*f*_{0}) to the lens radius and ∫ ∫_{1} means the integration over the region defined by

Here *c*_{E} is calculated with the location of the maximum thickness of the lens every 2 days. The reason why we divide *c*_{E} by 2*a**g*^{′}_{2}*r*_{0}/*f*_{0}(1 + *ɛ*) will be clarified in section 4 [see (4.25) and (4.26)].

In the case of *H*_{2} = 2000 m, the quantity represented by (3.4) tends to decrease as the nondimensionalized translation speed increases. In cases of *H*_{2} ≥ 3000 m, however, the nondimensionalized area integral of potential thickness anomaly stays at unity. While *Q*_{E} increases because of the *β* effect in southward drift (see Fig. 15 in the next section), *Q*_{E} decreases as the potential thickness anomaly fluid leaves the lens region. These tendencies preserve the unity value [(3.4)] and will be obtained theoretically in section 4 in an *f*-plane geostrophic model.

The area average of potential thickness anomaly in the second layer (Γ_{AE}) is

where *S*_{2E} is the area of the high-PV_{2} region defined by (3.2). Figure 13 shows the relation between *c*_{E} normalized by (3.3) and the area average of potential thickness anomaly in the second layer nondimensionalized by the average thickness of the lens and *f*:

where *S*_{1E} is the area of the region defined by (3.5). Although the plotted circles are somewhat scattered for large translation speed, all circles are almost along a line drawn in Fig. 13. This line is a theoretical one that will be derived in section 4. Figure 13 suggests that the nondimensionalized translation speed is related to the area average of potential thickness anomaly in the second layer, and results shown here are fitted to a line irrespective of the value of *H*_{2}.

### f. The role of β

In this section, we refer to three cases of the numerical experiment investigating roles of *β* in the first and second layers, respectively. In the work of Benilov (2000), the second-layer *β* is neglected, and the pressure gradient in the second layer driving the lens southward is caused only by the westward migration tendency of the lens induced by the first-layer *β* effect.

To investigate whether *β* in the second layer is needed for southward translation, we carry out an experiment in which *β* is set to zero only in the second layer. Figure 14 shows the result with *H*_{2} = 2000 m. After day 30 the translation direction changes gradually to westward. Therefore, *β* is needed in the second layer to sustain the quasi-steady southward translation of the lens. Figure 15 shows the time series of *Q*_{E}/(*M*/*f*). The value of *Q*_{E}/(*M*/*f*) decreases from unity. Comparison with the standard case suggests that the change in the translation direction is caused by the decrease of *Q*_{E}; this decrease of *Q*_{E} is possibly caused by leakage of the anomalous potential thickness fluid. As described in section 3d, the lens translates westward by the self-driving mechanism if there is no potential thickness anomaly in the second layer.

Furthermore, we carry out a case in which *β* is switched off in both layers on day 30 and another case in which *β* is switched off only in the first layer on day 30. In the case with *H*_{2} = 2000 m, the lens succumbs to baroclinic instability after *β* in the first layer is turned off, and so we show cases with *H*_{2} = 10 000 m. Figure 16 shows that the translation speed gradually decreases after *β* is set to zero in both layers. The numerical dissipation decreases the intercentroid distance (the horizontal distance between the two centers) between the potential thickness anomaly and the lens. Subsequently, the motion in the second layer, which is driving the lens, decreases. On the other hand, Fig. 17 shows that, after *β* is switched off only in the first layer, the lens translation immediately ceases. This suggests that *β* in the second layer blocks the southward translation of the lens or the potential thickness anomaly in the second layer. If the lens translates southward when only the first layer *β* is zero, the motion and the energy in the second layer increase during the southward translation while the dynamic energy of the first-layer lens is unchanged. This is physically unreasonable.

Hence *β* in the first layer has two important roles. First, *β*-induced westward translation of the lens keeps the center of the lens separate from the center of the potential thickness anomaly against the dissipation. Second, *β* in the first layer is needed to transfer energy from the first layer to the second layer and drive the potential thickness anomaly southward. On the other hand, *β* in the second layer maintains the value *Q*_{E}/(*M*/*f*) unity against the dissipation and the leakage of the potential thickness anomaly and sustains the quasi-steady southward translation.

## 4. Theory

### a. The f-plane geostrophic model

In section 3e, it is suggested that the translation speed is related to the potential thickness anomaly. In order to examine this relation, we derive in this section an analytic solution for a steadily drifting eddy in a 2½-layer *f*-plane model, neglecting the relative vorticity (geostrophic approximation) as a first step, although *β* and relative vorticity terms should be important in the numerical model. These terms are discussed in the next section.

We shall seek solutions corresponding to a first-layer lens and potential thickness anomaly in the second layer, which translate together at a constant speed *c.* The first-layer lens shape is found not to change much in time (Fig. 5) so that the first-layer thickness can be taken to be steady in a moving coordinate system. The origin of our moving coordinate system is located at the center of the lens. The first-layer thickness satisfies the gradient-wind balance, which is calculated in section 2:

The center of the lens is located at (*x, **y*) = (0, 0); *r*_{0} is the radius and the maximum thickness of the lens is *a**r*^{2}_{0}. With the assumption of *f*-plane, geostrophic, Boussinesq, and rigid-lid conditions, the governing equations for the second layer in a coordinate system moving at unknown speed *c* can be written in the form:

where

the subscripts 1 and 2 mean the first and second layer, respectively, and *g*^{′}_{2} = *g*(*ρ*_{3} − *ρ*_{2})/*ρ*_{2}. Here we assume that the lens moves toward −*y* direction with a speed *c*(>0). Substitution of (4.2) into (4.3) gives the conservation statement for the potential thickness

where *J* is the Jacobian operator. With the aid of (4.1), we obtain

where

and *G*(*) and *F*(*) are arbitrary functions. Equation (4.6) implies that the potential thickness in the second layer, *h*_{2}/*f*_{0}, is axisymmetric with respect to the point (*x, **y*) = (*x*_{c}, 0), which depends on the translation speed of the lens *c.* Since the point *x* = ±*r*_{0} is located on the edge of the lens where *h*_{1} vanishes and closed contours of *h*_{2}/*f*_{0} can only exist under the lens, the maximum radius of the region in which *h*_{2}/*f*_{0} contours close is found to be *R*_{0} = *r*_{0} − *x*_{c}. Outside of this region *h*_{2}/*f*_{0} must be *H*_{2}/*f*_{0}; that is,

since inertial–gravity and planetary Rossby waves are assumed to be forbidden with the *f* plane and geostrophy assumption. Using (4.7) with the condition that *x*_{c} < *r*_{0}, the translation speed can be written as a function of the area of the closed region,

where *S*_{1}(=*π**r*^{2}_{0}) is the area of the lens and *S*_{2}(=*π**R*^{2}_{0}) is the area of the closed region. The equation gives a kinematic relation between the area of the closed region in the second layer and the translation speed.

So far, we have obtained the relation between the translation speed and the area of the region of closed *h*_{2}/*f*_{0} contours, but the relation between the potential thickness value and the translation speed is still unknown. Following Killworth (1983) and Flierl (1984), a relation between *h*_{2} and the translation speed of the first-layer lens in a 2½-layer model from a dynamical point of view is

where

*M* is the lens volume (=∫ ∫_{r0 }*h*_{1 }*dx **dy*) and (*X, **Y*) are the positions of center of mass of the lens [=(∫ ∫_{r0 }*h*_{1}*x **dx **dy*/*M,* ∫ ∫_{r0 }*h*_{1}*y **dx **dy*/*M*)]. These equations imply that the lens will be driven by the pressure gradient due to the interface displacement between the second and third layers and the inertial oscillation. Since we assume steady translation, the inertial oscillation is taken as zero. Since Eq. (4.6) implies that the *h*_{2} is symmetric in the *y* direction, the rhs in (4.11) vanishes and the southward translation speed *c*(=−*Y*_{t} > 0) can be written as

Equation (4.12) implies that, if *h*_{2} is given, *c* is determined. As shown by (4.6) and (4.7), however, *h*_{2} is also a function of *c.* The combination of (4.6) and (4.12) will yield a relation between the translation speed and the potential thickness in the closed region. Substituting (4.1) and (4.6) into (4.12) yields

where *x*′ = *x* − *x*_{c} and

with *R*_{0} = *r*_{0} − *x*_{c}. Further, (4.13) becomes

If *c* ≠ 0, we obtain

where *Q* represents the area integral of potential thickness anomaly,

Note that *Q* is defined to be positive and has the opposite sign of the standard definition of “anomaly” [see (2.1)]. In other words, the area integral of the potential thickness anomaly is independent of the translation speed.

On the other hand, the area of the region of closed *h*_{2}/*f*_{0} contours depends on *c* [see (4.9)]. So we define the area average of potential thickness anomaly as

where *S*_{2}(=∫ ∫_{R0 }*dx*′ *dy* = *π*(*r*_{0} − *x*_{c})^{2}) represents the area of the closed region of potential thickness, and (4.16) can be written as

The solutions of (4.19) are

where *S*_{1}(=*π**r*^{2}_{0}) represents the area of the lens. Here, a solution in which *x*_{c} > *r*_{0} is excluded. Consequently, as the area average of potential thickness anomaly increases, the translation speed *c* increases. Moreover, from (4.20), the condition

must be satisfied for the solution with *c* ≠ 0 to exist. It is a crucial point that *c* is independent of the particular distribution of potential thickness anomaly and depends only on the area average of potential thickness anomaly.

The theoretical results can be summarized as follows.

The center of the potential thickness anomaly in the second layer is related to the translation speed [(4.7)].

If the translation speed is not zero, the area integral of potential thickness anomaly in the second layer is independent of the translation speed and coincides with the mass of the lens divided by

*f*_{0}[(4.17)].The translation speed is related to the area average of potential thickness anomaly in the second layer [(4.20)].

For steady translation of the lens, the area average of potential thickness anomaly in the second layer must be larger than the area average of the lens thickness divided by

*f*_{0}[(4.21)].

These tendencies of the *f* plane lens qualitatively agree with the *β*-plane numerical experiments discussed in section 3. Although the *β* effect is essential for the southward translation as shown in section 3f, *f*-plane lens characters are retained.

### b. The effect of the relative vorticity

In order to give a quantitative comparison, we have to evaluate the effect of the relative vorticity, which is neglected in the previous geostrophic model. The following discussion is only a correction to the previous geostrophic model and is not a rigorous theory. The following argument is only aimed at comparing the geostrophic theory with numerical experiments.

In order to include the effect of the relative vorticity, we assume the quasigeostrophic approximation [*h*_{1}, *η*_{2} ≪ *H*_{2}, (4.2), and *ζ*_{2} ≪ *f*_{0}]. The potential thickness equation becomes

[cf. with (4.5)] where

If the potential thickness is homogeneous at the value of *H*_{2}/*f*_{0},

This equation means that the displacement of *η*_{2} is smaller than *h*_{1} by an order of *R*^{2}_{d}/*r*^{2}_{0} in comparison with the previous geostrophic theory. Since the closed region is defined by the inside of the outermost closed contour of *g*^{′}_{2}*η*_{2}/*f*_{0} + *cx,* the area *S*_{2} decreases with decreasing *η*_{2}. If we assume that closed contours are still circular, this implies that the *x* coordinate of the center of the closed region (*x*^{′}_{c}) will be larger than *x*_{c} in (4.7) for a given *c.* Therefore,

where ɛ is positive and is of order (*R*_{d}/*r*_{0})^{2}, and (4.24) yields

where *S*^{′}_{2}(=*π*(*r*_{0} − *x*^{′}_{c})^{2}) represents modified *S*_{2}. Hereinafter, “≈” represents only an approximation and does not mean a rigorously derived equation. Although the strict value of ɛ is unknown, we will find this works for the moment:

Figure 18 shows the relation between *c*_{E} normalized by (3.3) and *S*_{1E}/*S*_{2E}. Here *S*_{1E} and *S*_{2E} are areas of the region defined in (3.5) and (3.2), respectively. The line in Fig. 18 shows the relation given by (4.25) and (4.26). Figure 18 implies that the above discussion on the effect of the relative vorticity with ɛ = *R*^{2}_{d}/*r*^{2}_{0} is appropriate. Then, let us consider the modification of the relation between the potential thickness anomaly and the translation speed, (4.20), in line with the above discussion of the effect of the relative vorticity on the area of the closed region. From (4.5), (4.6), (4.22), and (4.24), we may assume that the potential thickness in the second layer is

The radius of the closed region becomes *R*^{′}_{0} − *r*_{0} = *x*^{′}_{c}, and outside of that region the potential thickness must be *H*_{2}/*f*_{0}; that is,

Since

(4.27) can be rewritten as

In terms of *η*_{2}, (4.12) becomes

where *Q*′ represents the area integral of potential thickness anomaly,

where

with *x*″ = *x* − *x*^{′}_{c} and *R*^{′}_{0} = *r*_{0} − *x*^{′}_{c}.

The area average of potential thickness anomaly is

where *S*^{′}_{2} = *π*(*r*_{0} − *x*^{′}_{c})^{2}; this yields

Equation (4.34) implies that, as *H*_{2} becomes deeper, the translation speed becomes slower, which explains qualitatively the results of section 3c. Including the modified parameter 1 + *R*^{2}_{d}/*r*^{2}_{0}, we recover the same results obtained in section 4.

The line in Fig. 12 shows the relation given by (4.32). When *Q*_{E} becomes larger than ∫ ∫_{1 }*h*_{1 }*dx **dy*/*f* due to the *β* effect in southward drift (see Fig. 15), the translation speed *c* increases from (4.33) and (4.34). The region of closed *η*_{2} contours in the moving coordinate becomes smaller as the speed increases. Subsequently, the potential thickness anomaly fluid outside the region of closed *η*_{2} contours leaves the lens region. The leaking of the potential thickness anomaly fluid from the lens region contributes to preserve the value of *Q*_{E}. Then the speed *c* decreases, and the region of closed *η*_{2} contours expands and the leaking out stops. The above discussion shows that the balance on the *f* plane holds well on the *β* plane as well. The line in Fig. 13 shows the relation given by (4.34). While the case of *H*_{2} = 2000 m is slightly detached from the curve given by (4.34), the cases of *H*_{2} ≥ 3000 m coincide with (4.34).

## 5. Discussion and conclusions

The numerical experiments described in this paper have shown that a lenslike eddy on the 2½-layer *β* plane always translates southward, trapping the potential thickness anomaly fluid in the second layer whose center shifts eastward from the center of the lens. While the vortex has irregular drift in Morel and McWilliams (1997) and the translation speed grows linearly with time in Benilov (2000), our numerical experiments show that the lens translates southward almost steadily at a later stage before the lens succumbs to instability. We have carried out a series of experiments and found that the translation speed depends on *β* and the second-layer depth: as *β* increases or the second-layer depth decreases, the translation speed increases. If the second layer is deep, the relative vorticity becomes dominant so that the displacement *η*_{2} becomes small, and the southward translation speed decreases. Without the potential thickness anomaly, the lens moves westward, showing that the existence of the potential thickness anomaly is essential for the southward drift. It is shown that the translation speed is related to the potential thickness anomaly and the *β* effect in the first and second layers plays an important role in sustaining the steady translation.

To discuss this steadily southward translation of the lens, we derive an analytic solution in a 2½-layer *f*-plane model with geostrophic approximation. We show three important results.

If the translation speed is not zero, the area integral of potential thickness anomaly in the second layer is constant and independent of the translation speed.

The translation speed is related to the area average of potential thickness anomaly in the second layer.

For steady translation, the area average of potential thickness anomaly must be larger than that in a state in which the interface between the second and third layers is flat.

Relation 1 is the main result, and relations 2 and 3 will be derived from 1 and the relation between the speed and the area of the closed region. Relation 2 physically implies that the translation speed increases as the asymmetric structure of potential thickness anomaly becomes dominant rather than axisymmetric.

On the *β* plane, Rossby wave radiation, leakage of the potential thickness anomaly, and meridional displacement of the vortex structure lead to a transformation of the potential vorticity anomaly, and the constraints of the *f* plane are thus difficult to hold. From the comparison between the *f*-plane theory and the numerical experiments, however, these constraints are found to be almost satisfied on the *β* plane if the effect of the relative vorticity is included in the *f*-plane theory. This suggests that the increasing of the potential thickness anomaly due to the second layer *β* during the southward translation balances the leakage of the potential thickness anomaly from the lens region.

To investigate the roles of *β* described above, three experiments have been carried out. In the experiment in which *β* is set to zero only in the second layer, the lens translates southward initially. Subsequently, following to the tendency of westward translation of the lens, the translation changes the direction from southward to westward. If *β* in the second layer is zero, the potential thickness anomaly cannot preserve its value against the leakage and dissipation. Thus the lens is considered not to be able to maintain the southward translation. On the other hand, from experiments in which *β* is switched off in both layers, it is suggested that the first-layer *β* is needed to preserve the distance between the centers of the lens and the potential thickness anomaly in the second layer in the presence of numerical dissipation. In the experiment in which *β* is set to zero only in the first layer, the lens is stopped immediately since energy does not transfer from the first to the second layer. Therefore, both first and second layer *β* are necessary for quasi-steady southward translation of the lens observed in the experiments.

It can be shown that our configuration of lenses represents a situation of a middle-layer eddy, as follows. Consider a five-layer model where the lens is located in the third layer. The top and bottom layer depths are assumed to be infinite. If we cut at middle depth of the lens horizontally, and suppose vertical symmetry, this situation becomes consistent with our configuration.

We have related the translation speed to the potential thickness anomaly. The next step is to investigate the translation speed as a function of *β* and *R*_{d}. We find the dependence of the translation speed on *β* and 1/*H*_{2} (Figs. 6b and 7b). The reason why the translation speed does not exceed a certain value is still unclear in the present study. It seems that the value of potential thickness anomaly increases due to *β* in the second layer and the lens speed continues to increase. In experiments, however, after the lens speed increases to a certain value, the speed keeps its value. It can be considered that, as the translation speed increases, the area of the region in which the potential thickness anomaly in the second layer is negative shrinks, causing leakage of the potential thickness anomaly. The above discussion suggests that the tendency of decreasing closed region area as the translation speed increases become dominant if the *c*_{E} reaches a certain value. To clarify this mechanism, however, we probably need to consider the full set of nonlinear equations in the *β* plane, which is beyond the scope of the present study.

## Acknowledgments

The authors especially thank Stephen Van Gorder for permission to use his numerical model. They also thank the anonymous reviewers for their valuable comments. Graphical output was created with the GFD-DENNOU library.

## REFERENCES

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## Footnotes

*Corresponding author address:* Yori Ito, Graduate School of Environmental Earth Science, Hokkaido University, Sapporo 060-0810, Japan. Email: yori@ees.hokudai.ac.jp