ABSTRACT

Mass, angular momentum, and energy budgets are examined in an analytical model of vortex merging relevant to midlatitude mesoscale eddies. The vortices are baroclinic and cyclogeostrophic. The fluid surrounding them is assumed to remain quiescent. It is shown that due to this surrounding fluid, angular momentum is conserved when expressed in both the inertial and rotating frames of reference.

Lens-shaped solid-body vortices can conserve mass, angular momentum, and energy when they merge. If an upper-layer of thickness H₁ is included in the model, the merged vortex must have either less energy or mass than the sum of the original two vortices.

A more complex model of the vortex azimuthal structure is then considered, which includes a constant vorticity shell surrounding the solid-body core. If the shell is large compared to the core, the mass, angular momentum, and energy can all be conserved in the merged vortex. However, if the shell is small, the merged vortex must have less energy or mass than in the solid-body case.

1. Introduction

Like-sign vortices merge in rotating tanks (Nof and Simon 1987 ; Griffiths and Hopfinger 1987 ) and in the ocean (Cresswell 1982 ; Tokos et al. 1994 ). Merging has been numerically modeled in a wide range of settings (Melander et al. 1988 ; Verron and Valcke 1994 ; Carton and Bertrand 1994 ; Valcke and Verron 1997 ). Presumably, the characteristics of the merged vortex are set by those of its parents and the relevant conservation laws. But what properties are conserved in vortex merging? Gill and Griffiths (1981) demonstrated that, if potential vorticity and mass are conserved by two merging anticyclones, the final state contains more energy than the initial state. This “energy paradox” is at odds with the known spontaneous nature of vortex merging (Cushman-Roisin 1989 ). Several attempts to resolve this paradox have questioned the assumptions of potential vorticity or mass conservation (Nof 1988 ; Cushman-Roisin 1989 ; Pavia and Cushman-Roisin 1990 ).

Once vortices come into contact, merging takes place in two stages: fusion and axisymmetrization. During fusion, the vortices rapidly exchange fluid and homogenize into a central, elliptical vortex. In the subsequent, relatively slow axisymmetrization stage, the elongated vortex becomes S-shaped and sheds part of its mass into spiral-shaped filaments while its core becomes circular. This inviscid process is particularly dramatic in potential-vorticity-conserving numerical models (Melander et al. 1988 ; Pavia and Cushman-Roisin 1988 ) and has been observed in tank experiments (Griffiths and Hopfinger 1987 ). Cushman-Roisin (1989) proposed that filamentation plays a fundamental role in the merging, acting to relax mass conservation in the framework of the Gill and Griffiths (1981) merging model.¹ He also introduced an additional constraint, conservation of the absolute angular momentum (the angular momentum formulated in the inertial reference frame). He argued that this angular momentum must be conserved in the final vortex if the merging is complete, a condition not met by the solutions of Gill and Griffiths (1981) (Pavia and Cushman-Roisin 1990 ).

Cushman-Roisin’s (1989) model assumes that the potential vorticity of each fluid parcel is conserved. He concludes that a significant fraction of the initial mass must be shed into filaments. However, when Nof and Simon (1987) allowed anticyclonic lenses to merge in a rotating tank, the initial vortex mass was conserved in the merged vortex (i.e., filamentation was negligible). Furthermore, the central depth of the vortices rapidly decreased as they fused. This depth must increase to conserve potential vorticity, suggesting that potential vorticity is not conserved during fusion (Nof and Simon 1987 ). Nof (1988) argued that highly viscous, turbulent mixing occurs along the interface of the vortices, which can cause an O(1) alteration in the fluid parcel’s potential vorticity (Nof 1986 ). This hypothesis suggests that the energy paradox can be resolved by relaxing the a priori assumption of potential vorticity conservation. However, Pavia and Cushman-Roisin (1990) examined solutions to the merging model that conserve mass and energy in the final vortex; they argued that angular momentum was not in general conserved, rendering these solutions physically unacceptable.

In this note, we examine an analytical model of vortex merging similar to those of Gill and Griffiths (1981) and Pavia and Cushman-Roisin (1990) . We examine Nof’s (1988) hypothesis for resolving the energy paradox by relaxing the constraint of potential vorticity conservation. We formulate angular momentum conservation in the inertial reference frame; unlike Cushman-Roisin (1989) and Pavia and Cushman-Roisin (1990) , we include the fluid that surrounds the merging vortices. While this fluid is assumed to remain quiescent, it bears absolute angular momentum, which is altered when the fluid is rearranged during merging. We demonstrate that the absolute angular momentum of the total system, vortices and surrounding fluid, is equal to the background angular momentum (the absolute angular momentum in the absence of vortices), plus the sum of the vortex relative angular momentum (the angular momentum seen in the rotating reference frame). As a consequence, conservation of the total absolute angular momentum is equivalent to conservation of the relative angular momentum of the vortices. We conclude that solutions exist for the merging model that do not present an energy or angular momentum paradox. Using a generalized model of the vortex structure, we present two possible merging scenarios based on the choices of conserved quantities. In the discussion, we highlight the difference between our formulation of angular momentum conservation and that of Cushman-Roisin (1989) and discuss physical mechanisms that can account for reduced mass or energy in the final state.

2. Development of the model

The model has two layers: the upper layer (density ) extends from z = 0 to z = H₁ and the lower layer (density + ) extends from z = H₁ to z = H₂ (Fig. 1 ). The system is rotating at angular speed (Fig. 2 ) and is subject to a potential that includes gravity and centrifugal force (the familiar f-plane approximation). Before merging, there are two identical, azimuthally symmetric vortices in the upper layer, touching at a single point. We assume that the fluid is reorganized into quiescent fluid surrounding a single vortex. We assume that the final vortex has the same velocity structure as its parents; for example, vortices in solid-body rotation merge to produce a vortex also in solid-body rotation (though not necessarily rotating at the same rate). Because we assume the vortices are in contact at time t = 0, we do not address the sensitivity of merging on the initial separation distance of the vortices.

.

M = m_H + m_h + m, (3)

= L + MR²_o + (I_H + I_h + I) (7)

b. Angular momentum of vortices immersed in quiescent fluid

In the absence of vortices, the absolute angular momentum of the quiescent background state would be

In the presence of N vortices (denoted by the subscript = 1, ···, N), the absolute angular momentum of the quiescent surrounding fluid is

E = L gH₁m. (19)

3. Vortex merging

When two identical vortices (denoted by subscripts a and b) merge to produce a single vortex (c) without entraining surrounding fluid, mass conservation requires

M_c = M_a + M_b, (21)

By definition, the background angular momentum _back remains unchanged. Thus, from (13), conservation of angular momentum requires

L_c = L_a + L_b. (22)

From (20), energy conservation E_c = E_a + E_b requires

L_c + gH₁m_,c = L_a + gH₁m_,a + L_b + gH₁m_,b. (23)

m_,c = m_,a + m_,b (24)

We now examine the merging properties of particular vortex structures.

a. Solid-body vortices

A solid-body vortex has the velocity structure (r) = r, where is constant. By imposing = 0 at r = r_o, cyclogeostrophy (1) may be integrated to yield

L = (I_H + I_h + I). (26)

Consider two identical solid-body vortices (a and b) merging to produce vortex c, assumed to be also in solid-body rotation. Given _a, _b, r_a, and r_b, two independent conservation laws determine _c and r_c. This leads to the dilemma faced by Pavia and Cushman-Roisin (1990) : which of the three constraints (mass, angular momentum, and energy conservation) should apply? Following Pavia and Cushman-Roisin, we consider two separate merging scenarios; in each we conserve two of these properties and examine what happens to the third. Physical interpretations of the results will be discussed in the final section. We restrict the development to the particular case of identical merging vortices (_a = _b, r_a = r_b), and reject values of (_a, r_a) that yield a nonpositive central thickness H₁ + h(r) + (r).

2) CONSERVING ANGULAR MOMENTUM AND ENERGY

b. Vortices with finite-shear edges

In the solid-body model, vortices have an infinite-shear edge separating vortex fluid from surrounding upper-layer fluid of identical density. However, momentum diffusion would rapidly smear the edge vorticity into a shell surrounding the solid-body core. Within this shell, (r) decreases with increasing r. If the shell’s width is allowed to adjust during merging, its inclusion in the model adds a degree of freedom to the solutions. Consider vortices with the velocity structure:

Vortices of this structure may be centrifugally unstable. If a fluid parcel orbiting a cyclogeostrophic vortex is infinitesmally perturbed, it experiences a restoring force proportional to

F = _r(r + r²)² (32)

, r

There is no guarantee that the set of nonlinear conservation laws can be simultaneously satisfied by a unique combination of _c, r_i,c, and r_o,c. Paralleling the solid-body case, we examine two merging scenarios: one in which mass is conserved a priori and the other in which energy is conserved. In both scenarios, we assume that the mass M_i of the solid-body core is conserved:

This constraint demands that fluid parcels retain the sign of their vorticity; that is, the core of the merged vortex is the fused cores of the initial vortices, and mass changes in the energy-conserving scenario are associated solely with shell fluid.⁴

1) CONSERVING TOTAL MASS AND ANGULAR MOMENTUM

The energy loss is nearly independent of _a/. Energy is lost for small r_i,a/R_d, but less so than in the solid-body counterpart (Fig. 3 ). For r_i,a/R_d 8, energy is conserved within 0.1% for all values of _a/.

Figure 7 shows how the characteristics of the merged vortex vary as a function of shell width r_o,a/r_i,a, for the limiting case of a zero potential vorticity anticyclonic core (_a/ = 1). For nearly irrotational shells (r_o,a r_i,a), the merged vortex has a core period nearly double that of the original vortices (_a/_c 2) and inner and outer radii larger by (2). The dependence of the solutions on r_i,a/R_d increases as r_o,a r_i,a (the solid-body limit), and energy is lost (particularly for small r_i,a/R_d).

2) CONSERVING ANGULAR MOMENTUM AND ENERGY

Mass is conserved in the limit r_i,a R_d. Smaller vortices lose mass when they merge, nearly independently of _a/. For r_i,a = R_d, 7%–8% of the mass is lost in the merging.

Figure 9 shows the characteristics of the merged vortex as a function of r_o,a/r_i,a, for _a/ = 1. For vortices with nearly irrotational shells, mass is conserved. The mass loss exceeds 10% for shell radius smaller than r_o,a/R_d 10.

Although we assume that the surrounding fluid remains quiescent, when the total depth H₂ is finite, fluid columns underlying the vortices may be compressed or stretched in the merging. A column of quiescent fluid beneath a vortex bears potential vorticity 2/(H₂ H₁ h), where h is the downward displacement of the interface. If during merging h increases by h, the underlying column must gain anticyclonic vorticity ,

In their examination of lens-shaped vortex merging, Pavia and Cushman-Roisin (1990) conserved the mass and energy of the initial vortices, and showed that the absolute angular momentum of the vortices was not conserved. However, there is no “angular momentum paradox” inherent in their result, due to the role of the surrounding fluid in the total angular momentum budget. A simple thought experiment can serve to highlight this role: consider a pair of cylinders (density , mass m_a, radius r_a in solid-body rotation at relative rate _a) touching at the center of a tank that rotates at . The cylinders are surrounded by fluid of density . Later, they have merged to produce a single cylinder of the same density with mass m_c = 2m_a, radius r_c, and rotation rate _c. It is centered on the contact point of the parent vortices. Conservation of mass requires r_c = (2)r_a, and angular momentum conservation for the entire system requires

This expression is analogous to the formulation of angular momentum conservation of Cushman-Roisin (1989) and Pavia and Cushman-Roisin (1990) . With = 0, we are essentially considering coalescing fixed-height disks on a rotating table. To an observer in the rotating frame, the rotation rate of the resulting disk is more positive (for positive ) than it would be in the absence of rotation, due to Coriolis deflection applying a net torque on the coalescing particles (cf. Feynman et al. 1987 , p. 19-8). However, if = , (35) simplifies to

Our model demonstrates that solid-body vortices can conserve mass, angular momentum, and energy in the lens-shaped limit H₁ = 0. For nonzero H₁ either energy or mass conservation must be relaxed, leading to the two merging scenarios discussed in this paper.⁵ In the first scenario, if mass is conserved, the final vortex has less total energy than its parents. These solutions are thus “energetically allowable” (Dewar and Killworth 1990 ) in the sense that an external energy source is not required, analogous to the classical Rossby adjustment problem that has a steady-state solution with one-third of the initial energy (cf. Gill 1982 , pp. 191–203), the remaining energy having been removed via Poincaré wave radiation. Turbulent mixing during merging can also lower the energy state of the merged vortex (Nof 1986 ).

In the second scenario, if energy is conserved, the final vortex has less mass than its parents. Cushman-Roisin (1989) proposed that merging vortices eject fluid in narrow filaments as they become axisymmetric (Griffiths and Hopfinger 1987 ; Melander et al. 1988 ). Because these filaments bear negligible energy and relative angular momentum but a significant fraction of the mass, they are modeled implicitly by relaxing mass conservation, as was done in the second scenario (Pavia and Cushman-Roisin 1990 ).

In summary, vortices can conserve mass, angular momentum, and energy when they merge in two physically important limits of vortex structure: lens-shaped anticyclones and Rankine-like eddies. All three properties cannot be simultaneously conserved over the full range of parameters considered here; in general, mass or energy must be lost, presumably due to Poincaré wave radiation, turbulent dissipation, or filamentation. Future laboratory experiments and field observations could determine which scenario best describes the merging of oceanic vortices.

Acknowledgments. The authors would like to thank B. Cushman-Roisin, J. Verron, E. Hopfinger, Gert Jan van Heijst, B. Le Cann, J. Paillet, Y. Morel, and X. Carton for valuable conversations on the subjects of vortex dynamics and merging. Support for P. Flament and R. Lumpkin was provided by Office of Naval Research Contract N000149710147. R. C. Kloosterziel and P. Flament received support from National Science Foundation Grant OCE 97-30843. R. Lumpkin would like to thank IFREMER for hosting during a French Minister of Foreign Affairs postdoctoral fellowship.

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Fig. 1. Side view of a vortex and surrounding quiescent fluid in the 2-layer model. For a vortex in solid-body rotation, the surface and interface displacements are parabolic, as sketched here.

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Fig. 2. Top view of the merging model. The center of rotation of the system is at the .

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Fig. 3. Top: ratio of final rotational frequency _c to initial frequency _a as a function of _a/, for solid-body vortices, which conserve mass and angular momentum when they merge. The label for each curve indicates the value of r_a/R_d. The heavy line gives the solution for a lens-shaped vortex (R_d = H₁ = 0). Some curves end before reaching _a = ; at greater values of _a, the vortex has negative thickness H₁ + h + at its center and is thus not geometrically valid. Middle: ratio of final radius r_c to initial radius r_a as a function of _a/. Bottom: ratio of final energy E_c to initial energy 2E_a. Except in the lens-shaped case, the merged vortex has less energy.

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Fig. 4. Top: ratio of final rotational frequency _c to initial frequency _c as a function of _a/, for solid-body vortices, which conserve angular momentum and energy when they merge. The label for each curve indicates the value of r_a/R_d. The heavy line gives the solution for a lens-shaped vortex (R_d = H₁ = 0). Some curves end before reaching _a = ; at greater values of _a, the vortex has negative thickness H₁ + h + at its center and is thus not geometrically valid. Middle: ratio of final radius r_c to initial radius r_a as a function of _a/. Bottom: ratio of final mass M_c to initial mass 2M_a. Except in the lens-shaped case, the merged vortex has less mass.

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Fig. 5. Top: azimuthal velocity (r) vs radial distance for a vortex with a constant vorticity shell surrounding a solid body core. For this vortex, = . Bottom: sea surface displacement vs radial distance, calculated by integrating the cyclogeostrophic relation (1).

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Fig. 6. Top: ratio of final rotational frequency _c to initial frequency _a as a function of _a/, for finite-shear vortices, which conserve total mass and angular momentum when they merge (r_o,a = 10r_i,a). The label for each curve indicates the value of r_i,a/R_d. Some curves end before reaching _a = ; at greater values of _a, the vortex would have negative thickness H₁ + h + at its center and is thus not geometrically valid. Middle, upper: ratio of final core radius r_i,c to initial core radius r_i,a. Middle, lower: ratio of final shell radius r_o,c to initial shell radius r_o,a. Bottom: ratio of final energy E_c to initial energy 2E_a.

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Fig. 7. Top: ratio of final rotational frequency _c to initial frequency _a as a function of r_o,a/r_i,a, for finite-shear anticyclonic vortices, which conserve total mass and angular momentum when they merge, at the limit _a/ = 1. The label for each curve indicates the value of r_i,a/R_d. Middle, upper: ratio of final core radius r_i,c to initial core radius r_i,a. Middle, lower: ratio of final shell radius r_o,c to initial shell radius r_o,a. Bottom: ratio of final energy E_c to initial energy 2E_a.

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Fig. 8. Top: ratio of final rotational frequency _c to initial frequency _a as a function of _a/, for finite-shear vortices, which conserve angular momentum and energy when they merge (r_o,a = 10r_i,a). The label for each curve indicates the value of r_i,a/R_d. Some curves end before reaching _a = ; at greater values of _a, the vortex would have negative thickness H₁ + h + at its center and is thus not geometrically valid. Middle, upper: ratio of final core radius r_i,c to initial core radius r_i,a. Middle, lower: ratio of final shell radius r_o,c to initial shell radius r_o,a. Bottom: ratio of final mass M_c to initial mass 2M_a.

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Fig. 9. Top: ratio of final rotational frequency _c to initial frequency _a as a function of r_o,a/r_i,a, for finite-shear vortices, which conserve angular momentum and energy when they merge (_a/ = 1). The label for each curve indicates the value of r_i,a/R_d. Middle, upper: ratio of final core radius r_i,c to initial core radius r_i,a. Middle, lower: ratio of final shell radius r_o,c to initial shell radius r_o,a. Bottom: ratio of final mass M_c to initial mass 2M_a.

¹Cushman-Roisin explicitly included an idealized model for the filaments, and concluded that when zero potential vorticity lenses merge, 23% of their mass goes into the filaments while 96% of their energy is retained in the merged vortex.

²Specifically, the vertical component of the angular momentum vector.

³The equivalency of these invariants can also be derived directly from the momentum and continuity equations for a rotating, multilayer fluid (B. Cushman-Roisin, 1999, personal communication).

⁴When 2D vortices of the exactly shielded structure are brought into contact, their cores merge while shell fluid is lost. However, if they have shell vorticity greater than that given by

⁵A third scenario was considered in which mass and energy were conserved. The final state contained more angular momentum than the initial state, leading us to reject this scenario as a physically meaningful description of free vortex merging.

*School of Ocean and Earth Sciences and Technology, University of Hawaii Contribution Number 4897.