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Generalized point-vortex model for the motion of a dipole-vortex on the β-plane
Fluid Dynamics Research 23 (1998) 113–124
Generalized point-vortex model for the motion of a dipole-vortex on the -plane B.K. Shivamoggi ∗ , G.J.F. van Heijst Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands Received 11 October 1996; revised 29 September 1997; accepted 6 December 1997
Abstract In this paper, several re nements of the point-vortex model to describe the motion of a dipole vortex on the -plane are undertaken. These include the e ects of nite , squeezing and stretching of the dipole vortices due to bottom topography, c 1998 The Japan Society and e ects of the secondary vorticity eld associated with the advection of the ambient uid. of Fluid Mechanics Incorporated and Elsevier Science B.V. All rights reserved.
1. Introduction Coherent structures, known as dipole vortices, (or modons) have been used to model large-scale
uid motions in planetary atmospheres (Pedlosky, 1979) and low-frequency disturbances in magnetized plasmas (Hasegawa and Mima, 1978). The dipole vortex is self-propelled and has a net linear momentum which is associated with a steady translation in the direction de ned by the dipole axis. Kono and Yamagata (1977) and Zabusky and McWilliams (1982) considered the modulated pointvortex pair on the -plane to model the motion of a modon (here the circulation of a point vortex varies as a function of its location on the -plane). The early stages of the motion of a modon interacting with a large-scale, small- planetary vorticity variation can be adequately described by a point vortex model. (However, in the later stages, the Rossby wave radiation e ects become important and they cannot be described by this model1.) Hobson (1991) investigated the possible motions exhibited by this model to predict qualitative kinematical features of modons in non-uniform motion. This was done by constructing the phase-plane portraits for the point-vortex system. ∗
Corresponding author. Permanent address: University of Central Florida, Orlando, FL 32816, USA. Besides, real oceanic and atmospheric vortex motions are subject to lateral divergence due to the existence of strati cation. The latter e ect is neglected despite the fact that planetary eddies have lateral scales comparable to the Rossby radius. Therefore, the present work provides a framework to understand only the role of planetary e ect rather than to describe the overall features of real planetary dipoles.
1
c 1998 The Japan Society of Fluid Mechanics Incorporated and 0169-5983/98/$19.00 Elsevier Science B.V. All rights reserved. PII S 0 1 6 9 - 5 9 8 3 ( 9 7 ) 0 0 0 5 6 - 7
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B.K. Shivamoggi, G.J.F. van Heijst / Fluid Dynamics Research 23 (1998) 113–124
In this work, several re nements of the point-vortex model are undertaken. First, the e ects of nite on the eastward and westward motions of the vortex pair system are discussed. Next, the alternating squeezing and stretching of the dipole vortices due to a bottom topography is incorporated into the point-vortex model; the latter then predicts that the squeezed dipoles translate more slowly than the stretched ones, as observed in the laboratory experiments (Velasco Fuentes and van Heijst, 1994). Finally, the secondary vorticity eld associated with the advection of the ambient uid (which is believed to play a major role in the modon break-up) is incorporated into the point-vortex model in a suitable manner. One of the e ects of the secondary vorticity eld predicted by this generalized point-vortex model is a monotonically decreasing amplitude for the wobbling motion of an eastward moving modon. 2. The point-vortex model The point-vortex model assumes that each vortex moves with the velocity induced by all other vortices. The point vortices are further modulated, i.e., the circulation of a point-vortex is varied as a function of position in space, in accordance with the requirement of conservation of potential vorticity. Consider a pair of point-vortices separated by a nite distance roughly equal to that between the centers of the monopolar vortices in the modon. We have for the velocity eld of the ith-point vortex having circulation i and situated at (xi ; yi ),
j yi − yj dxi =− ; dt 2 r 2
j xi − xj dyi = ; dt 2 r 2
i 6= j;
(1)
where i; j = 1; 2. Since the velocity of each point-vortex is always perpendicular to the line joining the two point vortices, the distance r between the two point vortices is constant: r = [(x2 − x1 )2 + (y2 − y1 )2 ]1=2 = const;
(2)
The two point vortices therefore stay a xed distance apart and form a coherent system. Let us introduce the centroid coordinates (Zabusky and McWilliams, 1982), =
x1 + x2 ; 2
=
y1 + y2 ; 2
tan = −
x1 − x2 : y1 − y2
(3)
Here, is the elevation of the center of this system above the x-axis, the distance moved by the center of this system along the horizontal axis from its initial position, and the angle between the horizontal axis and the line perpendicular to the line joining the two point vortices, called the axis of the system (see Fig. 1). The vortex system moves along this axis. We then obtain from Eqs. (1)–(3), 1 d = ( 1 − 2 ) cos ; dt 4r
(4)
1 d = ( 1 − 2 ) sin ; dt 4r
(5)
B.K. Shivamoggi, G.J.F. van Heijst / Fluid Dynamics Research 23 (1998) 113–124
115
Fig. 1. Point-vortex pair.
1 d = ( 1 + 2 ): dt 2r 2 Now, the conservation of potential vorticity gives
i + y = const; i = 1; 2; R2 where R is a reference radius of each vortex. Thus,
i = 0i + R2 (y0i − yi );
i = 1; 2:
(6)
(7)
(8)
Let us non-dimensionalize ; and t, according to ˆ = ; r
ˆ = ; r
tˆ =
t (2r 2 = 0 )
(9)
and drop the hats. Eqs. (4)–(6), then become
1−a d = − (cos − cos 0 ) cos ; dt 2
(10)
1−a d = − (cos − cos 0 ) sin ; dt 2
(11)
d = (1 + a) − 4( − 0 ); dt where we have put
01 = 0 ; =
02 = a 0 ;
2
R r : 2 0
Thus, an initially symmetric dipole corresponds to a = −1.
(12)
(13)
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B.K. Shivamoggi, G.J.F. van Heijst / Fluid Dynamics Research 23 (1998) 113–124
Fig. 2. Phase-plane portrait.
Let the initial conditions associated with Eqs. (10)–(12) be t=0 :
= 0 ;
dx = 1 + a: dt
(14)
Eqs. (10)–(12) enable one to describe the motion of the vortex pair in terms of the position of the centroid (; ) of the pair and the direction of motion given by . Eqs. (11) and (12) may be combined to give d2 + 2(1 − a) sin = 42 (cos − cos 0 ) sin : dt 2
(15)
If is small, then the right-hand side of Eq. (15) can be neglected, and we then obtain the simple pendulum equation: d2 + 2(1 − a) sin = 0 dt 2
(16)
along with the initial conditions (14). Hobson (1991) studied the qualitative behavior of the motion of the two-point-vortex system by considering the xed points of Eq. (16) and their stability (which depends only on the parameters a; and 0 ). The xed points correspond to uniform motions in which the elevation and inclination of the vortex system are constant while it propagates horizontally. The stability of the xed points determines the stability of these uniform motions with respect to perturbations in the elevation and inclination of the vortex system. The xed points are at = 0 and (the phase-plane portrait for Eq. (16) is shown in Fig. 2). The xed point at = 0 corresponds to a center which represents stable uniform motions, so that nearby orbits remain close to this xed point for all time. The xed point at = is a saddle point which
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117
represents unstable motions, so that nearby orbits are de ected into nite-amplitude orbits (near the separatrices). The nonuniform motions corresponding to orbits near the center at = 0 are small-amplitude periodic oscillations of and . The system moves up and down while turning side to side slightly and propagating horizontally – wobbling. On the other hand, the nonuniform motions corresponding to orbits near the saddle point at = are nite-amplitude monotonic variations of and . Thus, the point-vortex model predicts that eastward moving modons will be stable in the path sense and show wobbling motions while westward moving modons will be unstable in the path sense and show nite-amplitude monotonically developing motions.
3. E ects of a nite The foregoing considerations were made under the restriction of small . We will now relax this restriction and consider the e ects of nite on the motion of the present vortex system. Consider rst the eastward propagation (i.e., small). Expanding in powers of , Eq. (15) now becomes d2 1 + [2(1 − a) − 42 (1 − cos 0 )] = [2(1 − a) − 42 (4 − cos 0 )] 3 : 2 dt 6
(17)
Eq. (17) shows that, for the linear problem, the e ect of nite (and hence ) is to cause reduction in the frequency ! of the wobbling motion ! = [2(1 − a) − 42 (1 − cos 0 )]1=2 ; ! may even become imaginary for a large enough (see Fig. 3).
Fig. 3. Frequency-dependence on of an eastward-moving dipole.
(18)
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B.K. Shivamoggi, G.J.F. van Heijst / Fluid Dynamics Research 23 (1998) 113–124
Fig. 4. (a) E ect of nite- on the nonlinear motion. (b) Phase-plane portrait for the hard-spring case. (c) Phase-plane portrait for the soft-spring case.
For the nonlinear problem, on the other hand, Eq. (17) shows that the e ect of nite is to change the behavior from soft-spring type to hard-spring type 2 (see Fig. 4). This represents the nonlinear stabilization due to nite of the direction of motion for the eastward propagation. 2
For a soft (hard) spring, the frequency decreases (increases) with amplitude.
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Fig. 5. Nonlinear stabilization in the path sense for westward propagation.
Consider next the westward propagation (i.e., ( − ) small). Expanding in powers of ; Eq. (15) now becomes 1 d 2 ˆ − [2(1 − a) + 42 (1 − cos 0 )] ˆ = − [2(1 − a) + 42 (4 − cos 0 )] ˆ3 ; 2 dt 6 where
(19)
ˆ ≡ − : Eq. (19) has the following exact solution: s
ˆ = −
√ 2p sec h( pt + C); q
(20)
where 1 q ≡ [2(1 − a) + 42 (4 − cos 0 )]: 6 Eq. (20), which represents a solitary-wave, again describes the nonlinear stabilization due to nite of the direction of motion for the westward propagation (see Fig. 5). It may be noted, however, that this result may be an artifact of the truncated (to cubic order) nonlinear model (19) and the fully nonlinear version may behave di erently! p ≡ 2(1 − a) + 42 (1 − cos 0 );
4. E ect of bottom topography The latitudinal variation of the Coriolis parameter on a rotating sphere can be simulated in the laboratory by a sloping bottom in a rotating uid. As the dipole vortices move on a sloping bottom,
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they undergo alternating squeezing and stretching during their motion across the equilibrium depth level. This leads to an oscillation of the dipole size and a consequent asymmetry in the motion of the dipole vortex below and above the equilibrium line because the squeezed dipole is observed to translate more slowly than the stretched one (Velasco Fuentes and van Heijst, 1994). It is of interest to note that this result is predicted by the point-vortex model, once the model incorporates the fact that the vortices undergo alternating squeezing and stretching as they move back and forth across the equilibrium depth level. This can be modeled by assuming that the distance r between the two point vortices is not constant, as arranged previously in (2), but changes according to dr = kr0 sin ; dt
(21)
where k is a positive constant and r0 is a reference length. Eq. (21) expresses the fact that the dipole vortex undergoes squeezing as it moves toward the shallower side of the equilibrium line and stretching as it moves toward the deeper side. Eqs. (10)–(12) then become
d 1 1 − a = − (cos − cos 0 ) cos ; dt r 2
(22)
d 1 1 − a = − (cos − cos 0 ) sin ; dt r 2
(23)
d 1 = [(1 + a) − 4( − 0 )]: dt r 2
(24)
Eqs. (23) and (24) may be combined to give, when is assumed to be small, to lowest order in ; for the symmetric dipole (i.e., a = −1),
d d2 + 2k + 4 sin = 0: 2 dt dt
(25)
Eq. (25), in conjunction with Eq. (23), then shows that the translation of the squeezed dipole is retarded while the stretched dipole is speeded up (see Fig. 6). 5. E ect of the secondary vorticity eld The point-vortex model is incapable of describing the break-up of the modon because the distance between the two point vortices is a constant. The secondary vorticity eld associated with the advection of the ambient uid for a modon is believed to play a major role in the modon break-up process (Velasco Fuentes and van Heijst, 1994). We will now try to incorporate some features of this secondary vorticity eld into the point-vortex model and examine the consequences. The advection of the ambient uid leads to an anisotropic secondary vorticity eld. For monopole cyclonic vortices, the advecting uid develops a vorticity de cit on the east side and a vorticity excess on the west side of the vortex. This produces a secondary dipolar vorticity eld with its axis oriented toward the northwest. As a result, the whole vortex system moves toward the northwest. Further, the secondary vorticity eld becomes stronger with increasing . Numerical simulations
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121
Fig. 6. Retardation (acceleration) of squeezed (stretched) dipoles.
by Carnevale et al. (1988) and laboratory experiments by Carnevale et al. (1991) con rmed the northwest drift of cyclonic vortices. Application of the above result for dipole vortices would then indicate that the velocity eld induced by the secondary vorticity eld is so as to (1) slow down an eastward moving dipole vortex while separating its two monopolar vortices away, (2) speed up a westward moving dipole vortex while bringing its two monopolar vortices closer (Fig. 7). Laboratory experiments of Velasco Fuentes and van Heijst (1994) con rmed the presence of a secondary vorticity eld around a dipole vortex in the form of a small ring of oppositely signed vorticity around the dipole vortex for the eastward motion. Numerical simulations by Carnevale et al. (1988) on the eastward motion of a modon on a plane of constant slope showed that for small slopes the modon holds itself together during the motion, but for large slopes (which correspond to large ) the modon breaks up. The laboratory experiments of Velasco Fuentes and van Heijst (1994) further showed that, for large , an eastward moving dipole vortex breaks up into two monopolar vortices; (for small , the dipole vortex tries to adjust itself to the eastward motion by radiating away Rossby waves). On the other hand, numerical simulations by Velasco Fuentes and van Heijst (1994) showed that westward moving dipole vortices speed up while becoming more and more compact 3 . 3
These di erences between the eastward traveling and westward traveling dipole vortices show up also in the head-on collision of an eastward traveling dipole vortex with a westward traveling vortex of equal strength and size: Numerical and laboratory experiments of Velasco Fuentes and van Heijst (1995) showed that an eastward traveling dipole vortex exchanges uid with the surroundings while the westward traveling dipole vortex preserves its mass in this process.
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Fig. 7. The secondary vorticity eld due to the advection of ambient uid.
The primary e ect of the secondary vorticity eld is to cause a northwest drift of the cyclonic vortex and a southwest drift of the anti-cyclonic vortex. This can be modeled by modifying Eq. (1) as follows:
j yi − yj dyi
j xi − xj dxi =− − i ; = − i ; i 6= j; (26) 2 dt 2 r dt 2 r 2 where i ¿0 for i = 1; 2 and 1 ¡0; 2 ¿0. It may be noted that the parameters i and i characterizing the secondary vorticity eld actually depend on the vortex strengths i , but for the sake of simplicity, we will ignore this aspect and treat i and i as constants. Eqs. (10)–(12) then become
1 d 1 1 − a = − (cos − cos 0 ) cos − (1 + 2 ); dt r 2 2
(27)
d 1 1 − a 1 = − (cos − cos 0 ) sin − (1 + 2 ); dt r 2 2
(28)
d 1 1 = [(1 + a) − 4( − 0 )] − [(2 − 1 ) cos − (2 − 1 ) sin ] tan ; dt r 2 r where we have non-dimensionalized i and i , according to i i ˆi = ; ˆi = ; i = 1; 2
0 =2r0
0 =2r0
(29)
(30)
and dropped the hats. The distance r between the two point vortices is no longer constant, but changes according to dr = −(2 − 1 ) sin + (2 − 1 ) cos : (31) dt Eqs. (28), (29) and (31) may be combined to give, when is assumed to be small, to lowest order in ; for the eastward moving symmetric dipole (i.e., a = −1), d d3 + [4 − 2(2 − 1 )2 ] − 12(2 − 1 ) = −6(22 − 21 ): dt 3 dt
(32)
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Fig. 8. E ect of the secondary vorticity eld on the wobbling motion of the eastward-moving dipole.
Let us look for a solution of the form (t) = A(t) exp[±i[4 − 2(2 − 1 )2 ]1=2 t]
(33)
where A(t) is a slowly varying function of t. Substituting Eq. (33), and neglecting the second- and higher-order derivatives of A; Eq. (32) gives 2[4 − 2(2 − 1 )2 ] from which,
dA + 12(2 − 1 )A = 0 dt
(34)
6(2 − 1 ) A(t) = A0 exp − t : 4 − 2(2 − 1 )2
(35)
Eq. (35) shows that the e ect of the secondary vorticity eld is to produce a monotonically decaying oscillation in , i.e., a wobbling with monotonically decreasing amplitude (see Fig. 8). Physically, this is due to the reduction in the coherence of the eastward moving dipole vortex caused by the secondary vorticity eld. This coherence reduction may eventually lead to the break-up of the eastward-moving dipole. 6. Discussion In this work, we have undertaken several re nements of the point-vortex model. We have considered the e ects of nite on the eastward and westward motions of a vortex-pair system and found them to cause a nonlinear stabilization of the direction of motion of the vortex-pair. Thus, for an
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eastward moving dipole vortex, the e ect of nite is to change the behavior from soft-spring type to hard-spring type while, for a westward moving dipole vortex, the e ect of nite is to limit the monotonically developing motions for small enough amplitudes. Next, we have incorporated in the point-vortex model the alternating squeezing and stretching of the dipole vortices due to a bottom topography. This led to a prediction that the squeezed dipoles translate more slowly than the stretched ones, as observed in the laboratory experiments (Velasco Fuentes and van Heijst, 1995). Finally, we have incorporated into the point-vortex model the secondary vorticity eld associated with the advection of the ambient uid (which is believed to play a major role in the modon break-up) in a suitable manner. One of the e ects of the secondary vorticity eld predicted by this generalized point-vortex model is a reduction in the coherence of the eastward-moving dipole vortex and the consequent monotonically decreasing amplitude for the wobbling motion of dipole vortex. Acknowledgements This work was carried out when BKS held a visiting research professorship at Eindhoven University of Technology under the auspices of the J.M. Burgerscentrum. Our thanks are due to Dr. Oscar Velasco Fuentes for his help with the numerical computation. We are also thankful to the referees for their helpful comments that improved the presentation in this paper. References Carnevale, G.F., Kloosterziel, R.C., van Heijst, G.J.F., 1991. J. Fluid Mech. 233, 119. Carnevale, G.F., Vallis, G.K., Purini, R., Briscolini, M., 1988. Geophys. Astrophys. Fluid Dyn. 41, 45. Hasegawa, A., Mima, K., 1978. Phys. Fluids 21, 87. Hobson, D.D., 1991. Phys. Fluids A 3, 3027. Kono, J., Yamagata, T., 1977. Proc. Oceanogr. Soc. Japan 36, 83. Pedlosky, J., 1979. Geophysical Fluid Dynamics. Springer, Berlin. Velasco Fuentes, O.U., van Heijst, G.J.F., 1994. J. Fluid Mech. 259, 79. Velasco Fuentes, O.U., van Heijst, G.J.F., 1995. Phys. Fluids 7, 2735. Zabusky, N.J., McWilliams, J.C., 1982. Phys. Fluids 25, 2175.
Generalized point-vortex model for the motion of a dipole-vortex on the -plane B.K. Shivamoggi ∗ , G.J.F. van Heijst Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands Received 11 October 1996; revised 29 September 1997; accepted 6 December 1997
Abstract In this paper, several re nements of the point-vortex model to describe the motion of a dipole vortex on the -plane are undertaken. These include the e ects of nite , squeezing and stretching of the dipole vortices due to bottom topography, c 1998 The Japan Society and e ects of the secondary vorticity eld associated with the advection of the ambient uid. of Fluid Mechanics Incorporated and Elsevier Science B.V. All rights reserved.
1. Introduction Coherent structures, known as dipole vortices, (or modons) have been used to model large-scale
uid motions in planetary atmospheres (Pedlosky, 1979) and low-frequency disturbances in magnetized plasmas (Hasegawa and Mima, 1978). The dipole vortex is self-propelled and has a net linear momentum which is associated with a steady translation in the direction de ned by the dipole axis. Kono and Yamagata (1977) and Zabusky and McWilliams (1982) considered the modulated pointvortex pair on the -plane to model the motion of a modon (here the circulation of a point vortex varies as a function of its location on the -plane). The early stages of the motion of a modon interacting with a large-scale, small- planetary vorticity variation can be adequately described by a point vortex model. (However, in the later stages, the Rossby wave radiation e ects become important and they cannot be described by this model1.) Hobson (1991) investigated the possible motions exhibited by this model to predict qualitative kinematical features of modons in non-uniform motion. This was done by constructing the phase-plane portraits for the point-vortex system. ∗
Corresponding author. Permanent address: University of Central Florida, Orlando, FL 32816, USA. Besides, real oceanic and atmospheric vortex motions are subject to lateral divergence due to the existence of strati cation. The latter e ect is neglected despite the fact that planetary eddies have lateral scales comparable to the Rossby radius. Therefore, the present work provides a framework to understand only the role of planetary e ect rather than to describe the overall features of real planetary dipoles.
1
c 1998 The Japan Society of Fluid Mechanics Incorporated and 0169-5983/98/$19.00 Elsevier Science B.V. All rights reserved. PII S 0 1 6 9 - 5 9 8 3 ( 9 7 ) 0 0 0 5 6 - 7
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B.K. Shivamoggi, G.J.F. van Heijst / Fluid Dynamics Research 23 (1998) 113–124
In this work, several re nements of the point-vortex model are undertaken. First, the e ects of nite on the eastward and westward motions of the vortex pair system are discussed. Next, the alternating squeezing and stretching of the dipole vortices due to a bottom topography is incorporated into the point-vortex model; the latter then predicts that the squeezed dipoles translate more slowly than the stretched ones, as observed in the laboratory experiments (Velasco Fuentes and van Heijst, 1994). Finally, the secondary vorticity eld associated with the advection of the ambient uid (which is believed to play a major role in the modon break-up) is incorporated into the point-vortex model in a suitable manner. One of the e ects of the secondary vorticity eld predicted by this generalized point-vortex model is a monotonically decreasing amplitude for the wobbling motion of an eastward moving modon. 2. The point-vortex model The point-vortex model assumes that each vortex moves with the velocity induced by all other vortices. The point vortices are further modulated, i.e., the circulation of a point-vortex is varied as a function of position in space, in accordance with the requirement of conservation of potential vorticity. Consider a pair of point-vortices separated by a nite distance roughly equal to that between the centers of the monopolar vortices in the modon. We have for the velocity eld of the ith-point vortex having circulation i and situated at (xi ; yi ),
j yi − yj dxi =− ; dt 2 r 2
j xi − xj dyi = ; dt 2 r 2
i 6= j;
(1)
where i; j = 1; 2. Since the velocity of each point-vortex is always perpendicular to the line joining the two point vortices, the distance r between the two point vortices is constant: r = [(x2 − x1 )2 + (y2 − y1 )2 ]1=2 = const;
(2)
The two point vortices therefore stay a xed distance apart and form a coherent system. Let us introduce the centroid coordinates (Zabusky and McWilliams, 1982), =
x1 + x2 ; 2
=
y1 + y2 ; 2
tan = −
x1 − x2 : y1 − y2
(3)
Here, is the elevation of the center of this system above the x-axis, the distance moved by the center of this system along the horizontal axis from its initial position, and the angle between the horizontal axis and the line perpendicular to the line joining the two point vortices, called the axis of the system (see Fig. 1). The vortex system moves along this axis. We then obtain from Eqs. (1)–(3), 1 d = ( 1 − 2 ) cos ; dt 4r
(4)
1 d = ( 1 − 2 ) sin ; dt 4r
(5)
B.K. Shivamoggi, G.J.F. van Heijst / Fluid Dynamics Research 23 (1998) 113–124
115
Fig. 1. Point-vortex pair.
1 d = ( 1 + 2 ): dt 2r 2 Now, the conservation of potential vorticity gives
i + y = const; i = 1; 2; R2 where R is a reference radius of each vortex. Thus,
i = 0i + R2 (y0i − yi );
i = 1; 2:
(6)
(7)
(8)
Let us non-dimensionalize ; and t, according to ˆ = ; r
ˆ = ; r
tˆ =
t (2r 2 = 0 )
(9)
and drop the hats. Eqs. (4)–(6), then become
1−a d = − (cos − cos 0 ) cos ; dt 2
(10)
1−a d = − (cos − cos 0 ) sin ; dt 2
(11)
d = (1 + a) − 4( − 0 ); dt where we have put
01 = 0 ; =
02 = a 0 ;
2
R r : 2 0
Thus, an initially symmetric dipole corresponds to a = −1.
(12)
(13)
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B.K. Shivamoggi, G.J.F. van Heijst / Fluid Dynamics Research 23 (1998) 113–124
Fig. 2. Phase-plane portrait.
Let the initial conditions associated with Eqs. (10)–(12) be t=0 :
= 0 ;
dx = 1 + a: dt
(14)
Eqs. (10)–(12) enable one to describe the motion of the vortex pair in terms of the position of the centroid (; ) of the pair and the direction of motion given by . Eqs. (11) and (12) may be combined to give d2 + 2(1 − a) sin = 42 (cos − cos 0 ) sin : dt 2
(15)
If is small, then the right-hand side of Eq. (15) can be neglected, and we then obtain the simple pendulum equation: d2 + 2(1 − a) sin = 0 dt 2
(16)
along with the initial conditions (14). Hobson (1991) studied the qualitative behavior of the motion of the two-point-vortex system by considering the xed points of Eq. (16) and their stability (which depends only on the parameters a; and 0 ). The xed points correspond to uniform motions in which the elevation and inclination of the vortex system are constant while it propagates horizontally. The stability of the xed points determines the stability of these uniform motions with respect to perturbations in the elevation and inclination of the vortex system. The xed points are at = 0 and (the phase-plane portrait for Eq. (16) is shown in Fig. 2). The xed point at = 0 corresponds to a center which represents stable uniform motions, so that nearby orbits remain close to this xed point for all time. The xed point at = is a saddle point which
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117
represents unstable motions, so that nearby orbits are de ected into nite-amplitude orbits (near the separatrices). The nonuniform motions corresponding to orbits near the center at = 0 are small-amplitude periodic oscillations of and . The system moves up and down while turning side to side slightly and propagating horizontally – wobbling. On the other hand, the nonuniform motions corresponding to orbits near the saddle point at = are nite-amplitude monotonic variations of and . Thus, the point-vortex model predicts that eastward moving modons will be stable in the path sense and show wobbling motions while westward moving modons will be unstable in the path sense and show nite-amplitude monotonically developing motions.
3. E ects of a nite The foregoing considerations were made under the restriction of small . We will now relax this restriction and consider the e ects of nite on the motion of the present vortex system. Consider rst the eastward propagation (i.e., small). Expanding in powers of , Eq. (15) now becomes d2 1 + [2(1 − a) − 42 (1 − cos 0 )] = [2(1 − a) − 42 (4 − cos 0 )] 3 : 2 dt 6
(17)
Eq. (17) shows that, for the linear problem, the e ect of nite (and hence ) is to cause reduction in the frequency ! of the wobbling motion ! = [2(1 − a) − 42 (1 − cos 0 )]1=2 ; ! may even become imaginary for a large enough (see Fig. 3).
Fig. 3. Frequency-dependence on of an eastward-moving dipole.
(18)
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Fig. 4. (a) E ect of nite- on the nonlinear motion. (b) Phase-plane portrait for the hard-spring case. (c) Phase-plane portrait for the soft-spring case.
For the nonlinear problem, on the other hand, Eq. (17) shows that the e ect of nite is to change the behavior from soft-spring type to hard-spring type 2 (see Fig. 4). This represents the nonlinear stabilization due to nite of the direction of motion for the eastward propagation. 2
For a soft (hard) spring, the frequency decreases (increases) with amplitude.
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Fig. 5. Nonlinear stabilization in the path sense for westward propagation.
Consider next the westward propagation (i.e., ( − ) small). Expanding in powers of ; Eq. (15) now becomes 1 d 2 ˆ − [2(1 − a) + 42 (1 − cos 0 )] ˆ = − [2(1 − a) + 42 (4 − cos 0 )] ˆ3 ; 2 dt 6 where
(19)
ˆ ≡ − : Eq. (19) has the following exact solution: s
ˆ = −
√ 2p sec h( pt + C); q
(20)
where 1 q ≡ [2(1 − a) + 42 (4 − cos 0 )]: 6 Eq. (20), which represents a solitary-wave, again describes the nonlinear stabilization due to nite of the direction of motion for the westward propagation (see Fig. 5). It may be noted, however, that this result may be an artifact of the truncated (to cubic order) nonlinear model (19) and the fully nonlinear version may behave di erently! p ≡ 2(1 − a) + 42 (1 − cos 0 );
4. E ect of bottom topography The latitudinal variation of the Coriolis parameter on a rotating sphere can be simulated in the laboratory by a sloping bottom in a rotating uid. As the dipole vortices move on a sloping bottom,
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they undergo alternating squeezing and stretching during their motion across the equilibrium depth level. This leads to an oscillation of the dipole size and a consequent asymmetry in the motion of the dipole vortex below and above the equilibrium line because the squeezed dipole is observed to translate more slowly than the stretched one (Velasco Fuentes and van Heijst, 1994). It is of interest to note that this result is predicted by the point-vortex model, once the model incorporates the fact that the vortices undergo alternating squeezing and stretching as they move back and forth across the equilibrium depth level. This can be modeled by assuming that the distance r between the two point vortices is not constant, as arranged previously in (2), but changes according to dr = kr0 sin ; dt
(21)
where k is a positive constant and r0 is a reference length. Eq. (21) expresses the fact that the dipole vortex undergoes squeezing as it moves toward the shallower side of the equilibrium line and stretching as it moves toward the deeper side. Eqs. (10)–(12) then become
d 1 1 − a = − (cos − cos 0 ) cos ; dt r 2
(22)
d 1 1 − a = − (cos − cos 0 ) sin ; dt r 2
(23)
d 1 = [(1 + a) − 4( − 0 )]: dt r 2
(24)
Eqs. (23) and (24) may be combined to give, when is assumed to be small, to lowest order in ; for the symmetric dipole (i.e., a = −1),
d d2 + 2k + 4 sin = 0: 2 dt dt
(25)
Eq. (25), in conjunction with Eq. (23), then shows that the translation of the squeezed dipole is retarded while the stretched dipole is speeded up (see Fig. 6). 5. E ect of the secondary vorticity eld The point-vortex model is incapable of describing the break-up of the modon because the distance between the two point vortices is a constant. The secondary vorticity eld associated with the advection of the ambient uid for a modon is believed to play a major role in the modon break-up process (Velasco Fuentes and van Heijst, 1994). We will now try to incorporate some features of this secondary vorticity eld into the point-vortex model and examine the consequences. The advection of the ambient uid leads to an anisotropic secondary vorticity eld. For monopole cyclonic vortices, the advecting uid develops a vorticity de cit on the east side and a vorticity excess on the west side of the vortex. This produces a secondary dipolar vorticity eld with its axis oriented toward the northwest. As a result, the whole vortex system moves toward the northwest. Further, the secondary vorticity eld becomes stronger with increasing . Numerical simulations
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121
Fig. 6. Retardation (acceleration) of squeezed (stretched) dipoles.
by Carnevale et al. (1988) and laboratory experiments by Carnevale et al. (1991) con rmed the northwest drift of cyclonic vortices. Application of the above result for dipole vortices would then indicate that the velocity eld induced by the secondary vorticity eld is so as to (1) slow down an eastward moving dipole vortex while separating its two monopolar vortices away, (2) speed up a westward moving dipole vortex while bringing its two monopolar vortices closer (Fig. 7). Laboratory experiments of Velasco Fuentes and van Heijst (1994) con rmed the presence of a secondary vorticity eld around a dipole vortex in the form of a small ring of oppositely signed vorticity around the dipole vortex for the eastward motion. Numerical simulations by Carnevale et al. (1988) on the eastward motion of a modon on a plane of constant slope showed that for small slopes the modon holds itself together during the motion, but for large slopes (which correspond to large ) the modon breaks up. The laboratory experiments of Velasco Fuentes and van Heijst (1994) further showed that, for large , an eastward moving dipole vortex breaks up into two monopolar vortices; (for small , the dipole vortex tries to adjust itself to the eastward motion by radiating away Rossby waves). On the other hand, numerical simulations by Velasco Fuentes and van Heijst (1994) showed that westward moving dipole vortices speed up while becoming more and more compact 3 . 3
These di erences between the eastward traveling and westward traveling dipole vortices show up also in the head-on collision of an eastward traveling dipole vortex with a westward traveling vortex of equal strength and size: Numerical and laboratory experiments of Velasco Fuentes and van Heijst (1995) showed that an eastward traveling dipole vortex exchanges uid with the surroundings while the westward traveling dipole vortex preserves its mass in this process.
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Fig. 7. The secondary vorticity eld due to the advection of ambient uid.
The primary e ect of the secondary vorticity eld is to cause a northwest drift of the cyclonic vortex and a southwest drift of the anti-cyclonic vortex. This can be modeled by modifying Eq. (1) as follows:
j yi − yj dyi
j xi − xj dxi =− − i ; = − i ; i 6= j; (26) 2 dt 2 r dt 2 r 2 where i ¿0 for i = 1; 2 and 1 ¡0; 2 ¿0. It may be noted that the parameters i and i characterizing the secondary vorticity eld actually depend on the vortex strengths i , but for the sake of simplicity, we will ignore this aspect and treat i and i as constants. Eqs. (10)–(12) then become
1 d 1 1 − a = − (cos − cos 0 ) cos − (1 + 2 ); dt r 2 2
(27)
d 1 1 − a 1 = − (cos − cos 0 ) sin − (1 + 2 ); dt r 2 2
(28)
d 1 1 = [(1 + a) − 4( − 0 )] − [(2 − 1 ) cos − (2 − 1 ) sin ] tan ; dt r 2 r where we have non-dimensionalized i and i , according to i i ˆi = ; ˆi = ; i = 1; 2
0 =2r0
0 =2r0
(29)
(30)
and dropped the hats. The distance r between the two point vortices is no longer constant, but changes according to dr = −(2 − 1 ) sin + (2 − 1 ) cos : (31) dt Eqs. (28), (29) and (31) may be combined to give, when is assumed to be small, to lowest order in ; for the eastward moving symmetric dipole (i.e., a = −1), d d3 + [4 − 2(2 − 1 )2 ] − 12(2 − 1 ) = −6(22 − 21 ): dt 3 dt
(32)
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Fig. 8. E ect of the secondary vorticity eld on the wobbling motion of the eastward-moving dipole.
Let us look for a solution of the form (t) = A(t) exp[±i[4 − 2(2 − 1 )2 ]1=2 t]
(33)
where A(t) is a slowly varying function of t. Substituting Eq. (33), and neglecting the second- and higher-order derivatives of A; Eq. (32) gives 2[4 − 2(2 − 1 )2 ] from which,
dA + 12(2 − 1 )A = 0 dt
(34)
6(2 − 1 ) A(t) = A0 exp − t : 4 − 2(2 − 1 )2
(35)
Eq. (35) shows that the e ect of the secondary vorticity eld is to produce a monotonically decaying oscillation in , i.e., a wobbling with monotonically decreasing amplitude (see Fig. 8). Physically, this is due to the reduction in the coherence of the eastward moving dipole vortex caused by the secondary vorticity eld. This coherence reduction may eventually lead to the break-up of the eastward-moving dipole. 6. Discussion In this work, we have undertaken several re nements of the point-vortex model. We have considered the e ects of nite on the eastward and westward motions of a vortex-pair system and found them to cause a nonlinear stabilization of the direction of motion of the vortex-pair. Thus, for an
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eastward moving dipole vortex, the e ect of nite is to change the behavior from soft-spring type to hard-spring type while, for a westward moving dipole vortex, the e ect of nite is to limit the monotonically developing motions for small enough amplitudes. Next, we have incorporated in the point-vortex model the alternating squeezing and stretching of the dipole vortices due to a bottom topography. This led to a prediction that the squeezed dipoles translate more slowly than the stretched ones, as observed in the laboratory experiments (Velasco Fuentes and van Heijst, 1995). Finally, we have incorporated into the point-vortex model the secondary vorticity eld associated with the advection of the ambient uid (which is believed to play a major role in the modon break-up) in a suitable manner. One of the e ects of the secondary vorticity eld predicted by this generalized point-vortex model is a reduction in the coherence of the eastward-moving dipole vortex and the consequent monotonically decreasing amplitude for the wobbling motion of dipole vortex. Acknowledgements This work was carried out when BKS held a visiting research professorship at Eindhoven University of Technology under the auspices of the J.M. Burgerscentrum. Our thanks are due to Dr. Oscar Velasco Fuentes for his help with the numerical computation. We are also thankful to the referees for their helpful comments that improved the presentation in this paper. References Carnevale, G.F., Kloosterziel, R.C., van Heijst, G.J.F., 1991. J. Fluid Mech. 233, 119. Carnevale, G.F., Vallis, G.K., Purini, R., Briscolini, M., 1988. Geophys. Astrophys. Fluid Dyn. 41, 45. Hasegawa, A., Mima, K., 1978. Phys. Fluids 21, 87. Hobson, D.D., 1991. Phys. Fluids A 3, 3027. Kono, J., Yamagata, T., 1977. Proc. Oceanogr. Soc. Japan 36, 83. Pedlosky, J., 1979. Geophysical Fluid Dynamics. Springer, Berlin. Velasco Fuentes, O.U., van Heijst, G.J.F., 1994. J. Fluid Mech. 259, 79. Velasco Fuentes, O.U., van Heijst, G.J.F., 1995. Phys. Fluids 7, 2735. Zabusky, N.J., McWilliams, J.C., 1982. Phys. Fluids 25, 2175.