Three-dimensional quasi-geostrophic vortex equilibria with m fold symmetry

Transcription

1 This raft was prepare using the LaTeX style file belonging to the Journal of Flui Mechanics 1 Three-imensional quasi-geostrophic vortex equilibria with m fol symmetry Jean N. Reinau Mathematical Institute, University of St Anrews, North Haugh, St Anrews, KY16 9SS, UK (Receive xx; revise xx; accepte xx) We investigate arrays of m three-imensional, unit Burger number, quasi-geostrophic vortices in mutual equilibrium whose centrois lie on a horizontal circular ring; or m + 1 vortices where the aitional vortex lies on the vertical central axis passing through the centre of the array. We first analyse the linear stability of circular point vortex arrays. Three istinct categories of vortex arrays are consiere. In the first category, the m ientical point vortices are equally space on a circular ring an no vortex is locate on the vertical central axis. In the other two categories, a central vortex is ae. The latter two categories iffer by the sign of the central vortex. We next turn our attention to finite volume vortices for the same three categories. The vortices consist in finite volumes of uniform potential vorticity an the equilibrium vortex arrays have an (impose) m fol symmetry. For simplicity all vortices have the same volume an the same potential vorticity, in absolute value. For such finite volume vortex arrays, we etermine families of equilibria which are spanne by the ratio of a istance separating the vortices an the array centre to the vortices mean raius. We etermine numerically the shape of the equilibria for m = 2 up to m = 7, for each three categories, an we aress their linear stability. For the m vortex circular arrays, all configurations with m 6 are unstable. Point vortex arrays are linearly stable for m < 6. Finite volume vortices may however be sensitive to instabilities eforming the vortices for m < 6 if the ratio of the istance separating the vortices to their mean raius is smaller than a threshol epening on m. Aing a vortex on the central axis moifies the overall stability properties of the vortex arrays. For m = 2, a central vortex tens to estabilise the vortex array unless the central vortex has opposite sign an is intense. For m > 2, the unstable regime can be obtaine if the strength of the central vortex is larger in magnitue than a threshol epening on the number of vortices. This is true whether the central vortex has the same sign as or the opposite sign to the peripheral vortices. A moerate strength like-signe central vortex tens however to stabilise the vortex array when locate near the plane containing the array. On the contrary, most of the vortex arrays with an opposite-signe central vortex are unstable. 1. Introuction Arrays of vortices in mutual equilibrium have arguably been of theoretical interest since the earliest works on vortex motion. Thomson (1883) first escribe the stability of m two-imensional ientical point vortices equally space on a horizontal circular ring for m = 2 up to m = 7. Such a configuration is referre to as a circular m-vortex array. There is a very large boy of literature eicate to the stuy of vortex equilibria, in particular for two-imensional vortices. Morikawa & Swenson (1971) stuie the effect of a central point vortex on the stability of an array of m point vortices for two-imensional vortices as aress for corresponence: jean.reinau@st-anrews.ac.uk

2 2 J. N. Reinau well as for single-layer quasi-geostrophic shallow water vortices. The two-imensional m- vortex problem was also revisite in epth by Kurakin & Yuovich (2002). The stability of point vortex multipoles has also been to focus of numerous other stuies incluing Aref (2009) in two imensions an Kizner (2011, 2014) for a two-layer flow. Thomson s seminal work also inspire further stuies, incluing Dritschel (1985) where the point vortex configurations were generalise to arrays of two-imensional finite area patches of uniform vorticity. Other configurations of finite area vortex equilibria with m fol symmetries have been sought for two-imensional vortices by Burbea (1982); Wu et al. (1984); Crowy (2002, 2003); Kizner & Khvoles (2004a,b); Xue et al. (2017) an for geophysical vortices by Kizner et al. (2007); Shteinbuch-Friman et al. (2015); Kizner et al. (2017); Shteinbuch-Friman et al. (2017); Reinau et al. (2017) to name but a few stuies. Observations by the Juno spacecraft have recently reveale the presence of persistent polygonal arrays of cyclonic vortices on Jupiter s poles, see Ariani et al. (2018), motivating further stuies of vortex equilibria in a three-imensional, rapily rotating an stratifie environment. Large scale oceanic an atmospheric motions are also strongly influence by the backgroun planetary rotation an the backgroun stable stratification of the flui. The Quasi-Geostrophic (QG) moel is the simplest ynamical moel that takes these effects into account. It is asymptotically erive from the full equations of motion when both rotation an stratification effects ominate the flow evolution. In this framework, the flow can be fully escribe by the (slow) evolution of a materially conserve scalar quantity: the potential vorticity, see Vallis (2006). Vortices aboun in the oceans an the atmosphere, see Ebbesmeyer et al. (1986); Chelton et al. (2011); Peterson et al. (2013); Zhang et al. (2014) an many other stuies. These vortices can be efine as contiguous regions of potential vorticity. The main objective of the present paper is to stuy, for the first time, equilibria for m (m 2) three-imensional, unit Burger number, finite volume vortices of uniform potential vorticity locate on a circular ring, within the QG approximation, in a rapily rotating, continuously stratifie flui.we also examine the effect of the aition of a vortex on the vertical axis passing through the centre of the ring, a problem often referre in the literature as the m + 1-vortex problem, see Sokolovskiy & Verron (2008). Pairs of three-imensional, continuously stratifie quasi-geostrophic co-rotating vortices in mutual equilibrium were first analyse in Reinau & Dritschel (2002) while pairs to counter-rotating vortices were iscusse in Reinau & Dritschel (2009). Aitionally, a special class of three-vortex equilibria is iscusse by Reinau & Carton (2015). Configurations of three-imensional QG vortices arrange in a nearly regular pattern can be the result of the estabilisation of a torus of potential vorticity as shown by Reinau & Dritschel (2018b). In this paper we show that unstable finite volume vortex equilibria can be foun in some part of the parameter space for all the values of m consiere. For the m-vortex problem however, point vortices arrays are linearly stable for m < 6. Nevertheless, finite volume vortices can be sensitive to moes of instability eforming the vortices for m < 6 if the vortices are close enough to each other. For m = 2, the aition of a central like-signe vortex estabilises the system. For m > 2, aing a weak central like-signe vortex tens to stabilise the vortex array, in particular if the central aitional vortex is locate at a small enough height from the other vortices. An intense like-signe central vortex may however inuce instability. The 2+1-vortex system with an opposite-signe central vortex is also unstable unless the central vortex is intense. For m > 3, aing an opposite-signe vortex on the central axis may estabilise the vortex arrays, an linearly stable solutions are foun in fewer, if any, parts of the parameter space.

3 QG vortex equilibria 3 The paper is organise as follows. Section 2 escribes the mathematical moel an the numerical tools use in the stuy. The main results for point vortices are iscusse in 3 while the results for finite volume vortices are presente in 4. Conclusions are presente in Mathematical setup We consier an aiabatic, invisci, three-imensional, horizontally an vertically unboune, continuously stratifie, rapily rotating flui. For simplicity we assume that the backgroun rotation is uniform so that the Coriolis frequency f is constant. We also assume that the buoyancy frequency N, efine by N 2 ρ/z uner the Boussinesq approximation, is constant so that the stratification is linear with epth. Here g is the gravitational acceleration, ρ 0 is the mean ensity an ρ(z) is the basic state ensity. The Boussinesq approximation assumes that the ensity variations are small compare to the mean ensity. This assumption is vali for the oceans. For convenience we rescale the physical vertical coorinate by the ratio N/f. Typically N/f 1 in most parts of the oceans, see Dijkstra (2008). In this vertically stretche reference frame the equations become inepenent of N an f, hence our results are vali for all values of N/f. We efine the Froue number F r = U/(NH), where U is a characteristic scale of horizontal velocity an H is a characteristic vertical length scale, an the Rossby number Ro = U/(f L), where L is a characteristic horizontal length scale. For rapi backgroun rotation Ro 1 an strong stratification F r 2 Ro, the Boussinesq equations can be asymptotically expane in terms of small Ro an F r to obtain the quasi-geostrophic (QG) moel (see Vallis, 2006 for etails). The equations rea = gρ 1 0 with q = 2 ψ x ψ y ψ z 2, (2.1) Dq Dt = q t ψ q y x + ψ q = 0, (2.2) x y where q efine by equation (2.1) is the QG potential vorticity anomaly, hereinafter referre to as PV for simplicity, ψ is the streamfunction an D/Dt stans for the material erivative. Equation (2.2) states that PV is materially conserve for an aiabatic, invisci flui. It shoul be note that the full ynamics is controlle by the PV, q, a single scalar quantity. Equation (2.1) can formally be inverte using the Green s function for the three-imensional Laplacian G(x; x 1 ) = 4π x x, (2.3) which by construction provies the streamfunction, evaluate at x = (x, y, z), inuce by a point vortex of unit intensity locate at x = (x, y, z ). Differentiating explicitly the Green s function with respect to x an y respectively gives the velocity v = ψ/ x an u = ψ/ y inuce by the point vortex. Secon orer erivatives provie the velocity graients which are use to stuy the linear stability of the point vortex arrays, see appenix A for etails. For finite core vortices, the inversion of equation (2.1) leas to volume integrals. The flui omain is represente by horizontal layers of equal thickness z in which the Green s function (an its erivatives) can be integrate analytically. The remaining (horizontal) surface integrals are converte to contour integrals over the

4 4 J. N. Reinau (a) y κ 3 = κ κ 2 = κ (b) z κ 4 = κ κ 0 α = κ 0 /κ R = 1 x Vortex 1 κ 1 = κ κ 0 κ 4 = κ κ 2, κ 6 = κ x κ 5 = κ κ 6 = κ κ 3, κ 5 = κ κ 1 = κ Figure 1. Geometry of the point vortex array an efinition of the parameters presente for vortices: (a) view from the top in the x, y plane, (b) sie view in the x, z plane. contours bouning the horizontal cross-sections of the uniform PV vortices using Green s theorem. The numerical metho use to perform the simulation of the evolution of the flow is the purely Lagrangian Contour Surgery algorithm introuce by Dritschel (1988) for two imensional flows an aapte to three-imensional QG flows by Dritschel & Saravanan (1994); Dritschel (2002). Contour Surgery is an extension on Contour Dynamics (see Zabusky, Hugues & Roberts, 1979) which allows one to control the complexity of the vortex bouning contours by topological reconnections. To obtain finite volume vortex equilibria, we use an iterative metho which makes the vortex bouning contours converge to streamlines. The approach is base on a metho evelope by Pierrehumbert (1980) for two-imensional flows an aapte to three-imensional QG flows by Reinau & Dritschel (2002) an further use in Reinau & Dritschel (2009); Reinau & Carton (2015). The metho is presente in Appenix B. The linear stability of the finite volume vortex arrays is aresse by analysing eformation moes of the vortex bouning contours (Reinau & Dritschel, 2002) an is briefly escribe in Appenix C. It inclues a moe representing the isplacement of the full contours, hence the relative isplacement of the vortices. 3. Point vortices We first consier arrays of point vortices. We refer to the vortices lying along the ring as the peripheral vortices. Vortex i carries an intensity or charge of potential vorticity Γ i which has the physical imension of a volume integrate PV, see for example Gryanik (1983). The m peripheral vortices are ientifie by their inex 1 i m. If an aitional central vortex is present, it is ientifie by the inex 0. For convenience, we efine the strength of vortex i by κ i = Γ i /(4π). The point vortex problem has a unique length scale an a unique time scale which can be both chosen arbitrarily. The length scale is set by the raius R = 1 of the ring on which the m peripheral vortices are locate, an the time scale is implicitly efine by taking κ i = κ = 1 for 1 i m, where κ is the common strength of the peripheral vortices. Without loss of generality, the ring of vortices is locate on the horizontal plane z = 0. The location of the m peripheral vortices at t = 0 is (x i, y i, z i ) = (cos θ i, sin θ i, 0) where θ i = 2π(i 1)/m. The central vortex has strength κ 0 an is locate at (0, 0, ). We take 0 without loss of generality. We efine α = κ 0 /κ. The general geometry of the vortex array is shown in figure 1.

5 QG vortex equilibria 5 Figure 2. (a) Maximum growth rate σ max versus the number of vortices m for QG point vortices in a three-imensional, continuously stratifie flui. The results for the m vortex problem are inicate by black +, an the m + 1 vortex problem with = 0 an α = 1 by blue while α = 1 by re. (b) Same but for two-imensional vortices for comparison. We analyse the linear stability of the m-vortex arrays an of the m+1-vortex arrays for = 0 an α = ±1 for 2 m 8. When = 0, all m+1 vortices are co-planar. We analyse the normal moes of perturbation of the horizontal locations of the vortices (x i, y i, 0). We o not consier perturbations of the vertical location of the vortices which coul also moify the istance separating the vortices as no external quasi-geostrophic flow may move the point vortices in the vertical irection. We o not consier perturbations on the strength of the vortices neither. These moes of pertubations have a time epenence proportional to e σt = e σrt (cos σ i t + i sin σ i t), where the real part of σ is the growth rate of the moe, σ r, an its imaginary part, σ i, is its frequency. There are n m = 2m moes for the m-vortex problem or n m = 2m + 2 moes for the m + 1-vortex problem. This approach is the same as the one use in ifferent contexts in Reinau & Carton (2015, 2016) an is further escribe in Appenix A. Figure 2 shows the maximum growth rate σ max = max 1 j nm {σ rj } versus the number of peripheral vortices m. For comparison, similar results for two-imensional vortices are inclue. The results, presente here, are obtaine using the same numerical technique as for the QG computations but aapte to the two-imensional situation. For the twoimensional vortices, the circulation of the peripheral vortices is set to Γ 2D = 2π. When a central vortex is ae, = 0 by construction, an again we use a central vortex of circulation Γ 0 = ±Γ 2D = ±2π. The parameter α is also use to enote the circulation ratio of the central vortex to the peripheral ones. For the m-vortex problem, results shown in figure 2 inicate that the three-imensional QG vortex array is less linearly stable than the equivalent two-imensional one. Inee the two-imensional vortex arrays are linearly stable for m 7, see figure 2(b) an Thomson (1883); Kurakin & Yuovich (2002), whereas the three-imensional QG vortex arrays are only linearly stable for m 5, see figure 2 (a). Recall that the ifference between the two situations lies in the nature of the Green s function which is G 2D (x ; x ) = (1/2π) ln x x for the two-imensional case in contrast with (2.3). Hence the velocity graients are proportional to r 3 in QG, where r = x x is the istance between the source an the evaluation point, compare to r 2 for the two imensional vortices.

6 6 J. N. Reinau (a) 1.5 (b) (c) y 0.0 y 0 y x x x Figure 3. Trajectories of the point vortices for the unstable vortex problem for (a) γ = 1, (b) γ = 0, an (c) γ = 1. We also observe a strong influence of the presence of a central vortex. For = 0, α = 1 an m > 3 the presence of the like-signe central vortex stabilises the array. The vortex array inee remains linearly stable for m = 6 up to 8. The vortex array with m = 9 is linearly unstable. Similar stabilising effects of a central, like-signe vortex, were first observe for two-imensional geostrophic vortices by Morikawa & Swenson (1971). It shoul be note that the like-signe central vortex has however a estabilising effect for a small number of peripheral vortices m = 2, 3. On the other han, the presence of a co-planar opposite-signe central vortex with α = 1 estabilises the ring, except for the special case m = 3. Overall, these trens are also observe for the two-imensional case, as shown in figure 2(b). We next show the nonlinear evolution of a selection of point vortex arrays. Figure 3 shows the trajectories of the point vortices for 2+1-vortex arrays with α = 1, 0, an 1. All trajectories are shown in the reference frame steaily rotating with the equilibrium. In this reference frame, the eparture of the vortices from their initial position is the result of instability. In the case where α = 1, shown in figure 3(a), the central vortex an each peripheral vortices have equal an opposite strength. The central an one of the peripheral vortices get closer together to form a vortex ipole which moves away from the origin. This vortex ipole has an overall zero strength as the strengths of the vortices compensate. To conserve the angular impulse J = 2π 2 i=0 κ i(x 2 i + y2 i ) = 2πκR2, the other peripheral vortex orbits aroun the origin. For α = 0, the central vortex has zero strength an thus is a passive particle. The remaining two vortices are stable, see figure 2(a). The evolution of the vortex array is shown in figure 3(b). The two peripheral vortices, shown in re an black, remain inee at their initial location. The passive particle however moves away from its initial location which is an hyperbolic critical point. It shoul be note that instabilities have also be foun in shallow water for such egenerate tripoles, in which the central pole is passive, by Kizner (2014), see in particular their figure 5. This is further etaile below when we aress the influence of the parameter α. For α = 1, the vortex array is unstable. The evolution of the vortices is shown in figure 3(c). Here all three vortices have equal strength. The trajectories inicate that one of the peripheral vortices may move towars the origin while the central vortex moves outwar to conserve both linear an angular impulses. Similar behaviours where the central vortex moves towars the ring while one of the peripheral vortices moves towars the centre have been observe for larger values of m as shown below. Results presente in figure 2(a) also inicate that the 3+1-vortex array with α = 1 is unstable. The evolution of the vortices is shown in figure 4. The trajectory of the vortices appear to be chaotic. We also see that all vortices transitorily pass near the origin. For a

7 QG vortex equilibria y x Figure 4. Trajectories of the point vortices for the unstable vortex problem for γ = 1. (a) 1.0 y (b) r1, x t Figure 5. (a) Trajectories of the point vortices for the unstable 6-vortex problem for 0 t 500 (left). The unfille circle inicate the initial position of the vortices. The ashe circle inicate the ring of raius R = 1 where the vortices initially lie. (b) istance r 1,5 = x 1 x 5 between the vortex 1 an vortex 5 vs time (right). larger number of vortices (m > 4) an α = 1, the results (not shown) also inicate that the unstable equilibria lea to a chaotic motion of the vortices. The evolution of the weakly unstable 6-vortex array is shown in figure 5. The weak instability results in a very small oscillatory motion of the vortices. The oscillation is ue to nonlinear effects. The oscillation is better seen by plotting on the evolution of the istance r 1,5 = x 1 x 5 between vortex 1 an vortex 5, initially locate at (1, 0, 0) an ( 1/2, 3/2) respectively. After an exponential growth of the istance r 1,5 from its initial value, nonlinear effects bring the vortices back to their initial positions. This relative motion repeats quasi perioically. The evolution of the unstable 7-vortex array is shown in figure 6. In this case the instability is much stronger an the point vortices have an apparent chaotic motion. It can be note that one of the 7 peripheral vortices moves (temporarily) near the centre of the array inicating that the stable vortex array acts as an attractor. By symmetry, the m + 1-vortex array remains in steay rotation even if 0 an α 1, provie the central vortex is locate on the vertical axis passing through the centre of the ring, which is the rotation axis of the system. Moreover, in this case the system remains in steay rotation for all values of α. The central vortex strength κ 0 an its vertical location moifies the angular velocity of the equilibrium but oes not break

8 8 J. N. Reinau y x Figure 6. (a) Trajectories of the point vortices for the unstable 7-vortex problem for 0 t 10. The unfille circle inicate the initial position of the vortices. (a) (b) α α Figure 7. Contours of maximum growth rates of instability for the vortex problem in a three-imensional, continuously stratifie flui in the place ( α, ) for (a) α < 0 an (b) α > 0. the equilibrium. It also affects its stability properties. We thus examine the stability properties of the m + 1-vortex array in the parameter space (α, ) istinguishing the cases α < 0 an α > 0 for clarity. We first present results for m = 2 in figure 7. The vortex array is unstable for small an α > 0. The growth rate of the instability ecays as is increase as a consequence of the ecrease of the interaction between the central vortex an the peripheral vortices as seen in figure 7(b). The same is true for for 1.5 < α < 0, see figure 7(a). A stronger opposite-sign central vortex is however able to stabilise the vortex array. We recover in particular that the 2+1-vortex array with α = 2 is linearly stable as establishe analytically by Reinau & Carton (2015). Results also inicate that the 2+1-vortex array is unstable for α 0, i.e. for a vanishing strength central vortex. In complete absence of a central vortex, a pair of like-sign point vortices, a 2-boy system, is stable. The instability observe for the 2+1-vortex, a 3-boy system, with α 0 is ue to the fact that the central vortex lies initially at a hyperbolic critical point (a stagnation point separating two trajectories aroun the peripheral vortices). Hence, although the peripheral vortices

9 QG vortex equilibria 9 (a) 3.5 (b) α α (c) 1.75 () α α Figure 8. Contours of maximum growth rates of instability for m ientical vortices on a ring with an opposite-signe central vortex, (α < 0), for QG point vortices in a three-imensional, continuously stratifie flui in the plane ( α, ) with m = 3 (a), 4 (b), 5 (c) an 6 (). locations are stable, the vanishing central vortex is unstable. This explains the trajectory of the central vortex observe in figure 3(b). Similar results are iscusse by Kizner (2014) for QG two-layer an single-layer flows, where the pair of active vortices is shown to be stable but an instability can be observe ue to the possible motion of the passive central vortex. Next, results are presente in figure 8 for α < 0 an in figure 9 for α > 0 an 3 m 6. First, the results show that linearly unstable moes can be foun for all values of m investigate, in particular if α is larger than a threshol which epens on m. Since the influence of the central vortex on the peripheral vortices ecreases with its height, this growth rate of the instability is larger for small. This is true for all cases but for m = 3 an α < 0, as seen in figure 8. We also confirm that an opposite-signe central vortex favours instability as the threshol in α which separates the linearly stable an linearly unstable regions is lower, for a given, when α < 0. Finally we see that for α < 0, increasing m makes the vortex array more unstable. On the other han for α > 0, we see that increasing m first shifts the threshol in α to larger values. For m = 6 there is in fact a secon, weaker, unstable moe which arises from (α, ) = (0, 0) an persists for α > 0 an 0, see figure 10(a). Recall inee that the array of 6 point vortex is

10 10 J. N. Reinau (a) (b) α α (c) () α α Figure 9. Contours of maximum growth rates for n ientical vortices on a ring with an like-signe central vortex for QG point vortices in a three-imensional, continuously stratifie flui in the place ( α, ) for m = 3 (a), 4 (b), 5 (c) an 6 (). The panel for m = 6 offers a close-up for small. Results of m = 6 an larger are shown in figure 10. unstable. Therefore, one expects for m 6 that when α 0, the m + 1-vortex problem to be linearly unstable as well. This moe becomes the ominant moe for m 7, see figure 10. The particular case m = 6 is interesting as in that case we see that a weak central vortex locate in the plane of the ring is able to stabilise the vortex array, while instability is recovere if the weak central vortex is vertically offset by a value of of orer of the ring raius R = Finite volume vortices We next turn our attention to finite volume vortices of uniform potential vorticity. The objective of this part is to provie the generic shapes an characteristics of finite core equilibria for 2 m 7 at highest possible resolution. We again investigate vortex arrays with an without a central vortex. We o not inten to provie at this stage a complete catalogue of the equilibria throughout a large parameter space, which arguably woul be of little interest. There is an aitional length scale in the problem, associate with the size of the vortices. In this problem, the raius of the ring along which the peripheral vortices lie is no longer fixe but is varie. Recall that this istance sets the istance

11 QG vortex equilibria 11 (a) (b) α α (c) 1.75 () α α Figure 10. Contours of maximum growth rates for n ientical vortices on a ring with an like-signe central vortex for QG point vortices in a three-imensional, continuously stratifie flui in the place ( α, ) for m = 6 (a), 7 (b), 8 (c) an 9 (). between the vortices. We are still free to fix a length scale in the problem. Without loss of generality we set the height of the vortices to 1, in the reference frame vertically stretche by N/f. We restrict attention to vortex having a unit mean height-to-with aspect ratio, h v /r v, measure in the stretche reference frame ue to the numerical cost of the computations. Here, h v is the half height of the vortices an r v is their mean horizontal raius. Such vortices have therefore a unit Burger number Bu = (h v /r v ) 2 = 1. The vortices are however pancake-like in physical space. The specific choice of unit heightto-with aspect ratio vortices is however consistent with finings in QG turbulence were it has been shown that vortices have typically a near unit aspect ratio (close to 0.8), see Reinau, Dritschel & Kouella (2003). It shoul be note that this specific choice imposes limitation on the variety of equilibria we investigate. The branches of equilibria stem from vortex arrays of infinitely istant spherical vortices in the vertically stretche reference frame. The vortices volume is V = 4πh v r 2 v/3 = π/6, an their PV is set to q = 4π. This means that the time scale associate with the vortices, inversely proportional to q, an their strength qv is the same in all experiments. The m vortices are each iscretise in the vertical irection by n c = 83 horizontal layers. The vortex bouning contours are iscretise by n p = 4n c noes. It shoul be note that

12 12 J. N. Reinau the numerical cost of the algorithms grows as m 2 n 2 pn 2 c. We focus on equilibria having impose symmetries. First, all m vortices have the same shape. Vortex i, 2 i m, is the image of vortex 1 by a rotation of angle 2(i 1)π/m. Secon, each vortex is symmetric with respect to a vertical plane passing through the vortex centre an the vertical axis of rotation (the z axis). Similar numerical setups an symmetries are impose for the m + 1-vortex problem. We focus on cases where the central vortex lies in the plane containing the ring of peripheral vortices ( = 0). We consier the two cases where the PV of the central vortex is q 0 = ±4π. In these cases, the central vortex has an impose m-fol symmetry, an the number of noes n p iscretising each contour is ajuste to be ivisible by m. Even when reucing the computational loa by taking avantages of the problem symmetries, etermining a single equilibrium state at these resolutions for 7+1=8 vortices can take up to a ay on a single core of a moern processor. This imposes limitations on the number of states one can etermine. For each configuration, an a given value of m, we etermine the family of equilibrium states spanne by the ratio of a istance δ separating the peripheral vortices an the centre of the omain, which is the centre of rotation, to the mean vortex raius r v = 0.5. Branches of solutions are sought until we reach an ening point of the solution branch, namely when the vortices touch. The choice of the istance to be use epens on the configuration as not all istances between a point efine on the vortices an the rotation centre varies monotonously along the solution branches, ue to the vortices eformation. In practice, the istance δ is the istance between the innermost ege of the vortices an the centre of the ring when the peripheral vortices evelop a sharp inner ege forms. On the other han, δ is the istance between the outermost ege an the centre of the ring when the vortices evelop a flat, or slightly convex inner surface. We start the branches of solution from large separation istances. When an equilibrium is reache by the iterative metho, the istance is reuce an the numerical metho is resume for the new separation. It shoul be note that if the raius of the ring tens to infinity, spherical vortices shoul be in mutual equilibrium. The shape of the vortices eparts from a sphere as the vortices are locate closer together. The equilibrium vortices are eforme to be able to steaily withstan the shear (proportional to r 3 ) they inuce on each other. The numerical metho use to etermine the equilibrium states is iterative an is escribe in Appenix B. We etermine the equilibria for the m-vortex problem for 2 m 7. It shoul be note that the equilibria for m = 2 were originally obtaine at lower resolution by Reinau & Dritschel (2002) an at high resolution in Reinau & Dritschel (2018a). We first escribe the shape of the equilibria obtaine numerically an we aress next their linear stability. The metho use for the linear stability analysis is escribe in Appenix C. Figure 11 shows a top view on the vortex bouning contours at the en of the solution branch, where the vortices nearly touch for m = 2 to m = 7. Except for m = 2, the vortices near the ening point of the branch o not exhibit a single sharp inner ege. This is in contrast with the two-imensional equilibria shown by Dritschel (1985) (in particular their figure 2) where a single sharp inner ege forms for m 4. Instea, for m 3 the innermost part of the vortices flattens an remains slightly convex. Vortices touch by their sie at the ening point of the solution branch. Figure 12 shows the maximum growth rate σ max = max{σ r } versus the istance δ between the innermost ege of the vortices an the centre of rotation. Results show that the equilibria are unstable for δ less than a threshol epening on m. Importantly, unstable equilibria are foun for all m for all three configurations. Finite volume vortices can eform an therefore can be sensitive to eformation moes as well as isplacement

13 QG vortex equilibria 13 l l δ Figure 11. Top view on the vortex bouning contours to the m-vortex equilibria at the en of the solution branch where the vortices touch. The grey scale inicates the height of the contour: lighter grey contours are nearer the top. moes. By construction, only the isplacement moes can be capture by the analysis of the stability of systems of point vortices. Moreover, there is a funamental ifference between eformation moes which affect the shape of the vortices an isplacement moes which move the vortices. If a point vortex equilibrium is sensitive to a isplacement moe the point vortex equilibrium is funamentally unstable, i.e. it is unstable for all separation istances between the vortices. The istance influences the magnitue of the growth rate, but oes not change the nature of the stability properties. On the other han, the eformation moes for finite volume vortices can be triggere when the vortices are close enough to each other, as it is seen in figure 12. In practice such eformation moes are associate with the phase-locking of vorticity Rossby waves travelling on the

14 14 J. N. Reinau σ max δ Figure 12. Maximum growth rate σ max of instability vs the inner gap δ for the finite volume m-vortex problem for m = 3 (soli black), 4 (otte black), 5 (ashe-otte black), 6 (soli re), an 7 (otte re). vortex bounaries, see Dritschel (1995) for a iscussion for two-imensional vortices. A moe with non-zero growth rate is observe for m = 7 for all istances an is associate with the isplacement moe. The growth rate of the instability increases as the vortices are closer together. This is relate to the increase of the strain vortices inuce on each other. It shoul however be note that the similar (but much weaker) isplacement instability moe for m = 6 observe in the point vortex calculation is not clearly noticeable for the finite volume vortex equilibria. We conjecture that the eforme finite volume vortices have aapte to the external shear inuce by the other vortices. This has weakene the alreay weak moe, an its growth rate cannot be convincingly istinguishe from backgroun numerical noise. Inee the equilibria are obtaine numerically by an iterative metho, escribe in the appenix B. The iterative metho is stoppe when the correction to the rotation rate of the vortex array is less than a threshol (10 11 in the present stuy). This means that a small resiual unsteainess remains together with the unavoiable small truncation errors inherent to the numerical approach. Hence very weak instabilities may have a growth rate of the orer of the precision of our numerical calculation of the equilibrium. Figure 13 shows the vortex equilibria for the m + 1-vortex problem when = 0 an α = 1. Recall that, except for m = 3, the associate point vortex arrays are unstable. The general shape of the equilibria is qualitatively similar to the shape of the twoimensional equilibria obtaine by Kizner et al. (2007) an the vortex arrays obtaine experimentally by Trieling et al. (2010). It shoul be note that we have not been able to reach an ening point in the case m = 2 espite numerous attempts. Importantly, the algorithm requires that we fix a istance between two points iscretising the vortices uring the iterative proceure to etermine the equilibrium. When the equilibrium is reache, this istance is reuce an the proceure is resume for this new istance. As mentione previously not all istances vary monotonously along the branch of solutions, an we may nee to escribe the solution branch piece by piece, aapting the aequate istance to be fixe. In the case m = 2 we were not able to fin a convenient istance to be fixe that woul allow one to go further along the branch. A new approach may be require in this case. The presence of an opposite-signe vortex in the centre of the structure changes the topology of the streamlines hence the shape of the equilibria. This ifference with the

15 QG vortex equilibria 15 Figure 13. Top view on the vortex bouning contours to the m + 1-vortex equilibria at the en of the solution branch where the vortices touch (except for m = 2 in the top left panel where this state coul not be obtaine numerically). The central vortex an the peripheral vortices have opposite signe PV. The grey scale inicates the height of the contour: lighter grey contours are nearer the top.

16 16 J. N. Reinau σ max δ Figure 14. Maximum growth rate σ max of instability vs the inner gap δ for the finite volume m + 1-vortex problem for m = 3 where the central vortex an the peripheral vortices have opposite sign. m vortex problem becomes less noticeable for large m as the vortices remain far from the central vortex even at the ening point of the branch. However, the peripheral vortices ten to be thinner in the raial irection when the opposite-signe central vortex is present. It is however important to stress that these qualitatively similar vortex shapes have funamentally ifferent stability properties. Figure 14 shows the maximum growth rate σ max for the 3+1 vortex problem with = 0 an α = 1. We o not show results for the other values of m as they are all unstable for all istances. Inee these configurations are all unstable to at least isplacement moes as foun for the point vortices. The linear stability of the finite volume vortices confirme this. The case m = 3 is the only one which is linearly stable to isplacement moes. Again, we see that eformation moes can estabilise the equilibrium if the vortices are close enough together. Recall vortices at equilibrium are more eforme as they are closer together. Finally, figure 15 illustrates the equilibria at the ening point of the solution branches for the m+1-vortex problem with = 0 an α = 1. Due to the presence of the central likesigne vortex, the vortex exhibit a sharp inner ege for m 6. Note that for m > 6, the istance between two neighbouring peripheral vortices is less than the istance between these vortices an the central vortex. In these cases, the peripheral vortices interact more strongly with their neighbours on the ring. This explains the change in the shape of the equilibria. The maximum growth rate σ max of the unstable moes versus δ are shown in figure 16. Again, unstable moes are obtaine for δ less than a threshol for all m. The weak unstable moe observe for the point vortex system for = 0 an α = 1 is not clearly capture for the finite volume problem. We conjecture that, as seen for the 6- vortex problem, the eformation of the vortices, hence their capacity to aapt to the external shear weakens the instability. The growth rate of the instability is of orer of the backgroun numerical noise. For the sake of completeness one can also calculate two of the funamental invariants of the equilibria, namely their total energy E an their angular impulse J efine by E = 1 2 V qψv, (4.1)

17 QG vortex equilibria 17 Figure 15. Top view on the vortex bouning contours to the m + 1-vortex equilibria at the en of the solution branch where the vortices touch. The central vortex an the peripheral vortices have same PV. The grey scale inicates the height of the contour: lighter grey contours are nearer the top.

18 18 J. N. Reinau σ max δ Figure 16. Maximum growth rate σ max of instability vs the inner gap δ for the finite volume m + 1-vortex problem where the central vortex an the peripheral vortices have same sign for m = 3 (soli black), 4 (otte black), 5 (ashe-otte black), 6 (soli re) an 7 (otte re). J = 1 q(x 2 + y 2 )v. (4.2) 2 V A summary of the results are presente in figure 17. For the two-vortex problem, the margin of stability has been observe empirically to coincie with the maximum of E an minimum of J versus a istance separating the vortices, see Reinau & Dritschel (2002) an Reinau & Dritschel (2005). The fact that the combine extrema can be associate with the onset of instability has been justifie by Saffman (1992) using Kelvin s variational principle. The conition is a sufficient but not necessary conition. We have not foun a systematic match between the onset of instability an the coincience of extrema for E an J for m > 2. It shoul be note that such a match shoul not be expecte at least for the isplacement moe as its origin is inepenent of the istance separating the vortices. For almost all cases, the rotation rate ω an the energy E increase as the vortices are closer to one another hence their interaction is stronger. On the other han, J ecreases with the istance as J is associate with a volume integral of PV weighte by istance to the system centre square. Only for the case m = 3 an = 0, α = 1 these trens iffer. In this case the peripheral vortices are close to the highly-eforme opposite-signe central vortex. This central vortex ominates the overall rotation (which becomes clockwise) an its contribution to the angular impulse is negative. We next illustrate the nonlinear evolution of unstable equilibria for the three categories of vortex arrays. We start with an example of the 3-vortex problem. The equilibria correspons to a state with an innermost gap δ = 0.16 an the maximum growth rate of instability is σ max = Results obtaine with the Lagrangian Contour Dynamics metho are presente in figure 18. The equivalent equilibrium with point vortices is linearly stable inicating that there is no unstable isplacement moe. The instability is associate with the eformation of the vortices. We o not force the instability but simply let it grow from pseuo-ranom numerical errors. For numerical efficiency the number of noes iscretising the contours of the equilibrium (n p = 4n c ) is reuce while maintaining high accuracy. This is one by renoing the contours which is part of Contour Surgery with stanar setup parameters. This proceure is enough to introuce a small perturbation on the vortex shape. Figure 18 shows the vortices in the

19 QG vortex equilibria 19 ω E J l x l x l x ω l x E l x J l x ω 0.2 E 75 J l x l x l x Figure 17. Global iagnostics for the equilibrium state: rotation rate ω, total energy E, an angular impulse J versus l x the istance between the centroi of the peripheral vortices an the centre of the ring for m = 2 (soli blue), m = 3 (soli black), m = 4 (otte black), m = 5 (ashe-otte black), m = 6 (soli re), m = 7 (otte re) an for the m-vortex problem (top row), m + 1-vortex problem with = 0, α = 1 (mile row) an the m + 1-vortex problem with = 0 an α = 1 (bottom row). reference frame steaily rotating with the equilibrium. Therefore any motion observe in this reference frame represents a eparture from equilibrium. The vortices remain at equilibrium for a long perio of time, while the eformation slowly grows from the low numerical noise. By t = 22, the inner most eges of the vortices have eforme an have forme briges which connect them to the neighbouring vortex to their right (in the irection of rotation of the structure). This eformation is qualitatively similar to the one associate with the merger of a pair of like-signe vortices, see Reinau & Dritschel (2002). As the flow evelops, the merge vortex forms a complex structure which resembles a three-blae propeller. Some PV from the central layers converges to the centre of the structure. To conserve the angular impulse J, some PV from the lower an upper layers of the vortices is ejecte away from the centre of the structure. These will turn into filaments an small scale ebris in the late evolution of the flow. The secon illustration of the nonlinear evolution of the equilibria concerns an unstable vortex equilibrium with = 0 an α = 1. Recall that such a vortex array is sensitive to a isplacement moe. Results obtaine with the Lagrangian Contour

20 20 J. N. Reinau Figure 18. Evolution of the vortex bouning contours for an unstable (σ max = 0.037) array of three vortices in mutual equilibrium for δ = 0.16 at t = 0, 22, 24 an The vortex bouning contours are viewe orthographically at an angle of 45 from the vertical. Dynamics metho are shown in figure 19. Here, the istance δ between the peripheral vortex innermost ege an the centre of the central vortex is δ = 1.08 an the most unstable moe has a growth rate σ max = We first observe that the vortices inee move. Some vortices getting closer together creating a larger gap with their other neighbouring vortex on the ring. The straining fiel changes as the vortices epart from their equilibrium. The vortices thus start to eform. The vortices which have got closer together merge to form larger structures. These structures are not stable an can further interact an/or break back into much eforme seconary structures. It resembles instances of partial merger observe uring the interaction between two vortices, see Reinau & Dritschel (2002). Finally, we consier an example of unstable vortex equilibrium with = 0 an α = 1 with δ = 0.16 an σ max = The equivalent point vortex equilibrium is linearly stable. Results obtaine with the Lagrangian Contour Dynamics metho are presente in figure 20. In this case, the instability is associate with the eformation of the vortices. The sharp inner ege of the peripheral vortices elongates an a filament of PV is she near the ege of at least one vortex. This filament is wrappe aroun the central vortex. This in turn breaks the symmetry of the flow. As the vortices eform, some of the peripheral vortices strongly interact with the central vortex in an asymmetric way. This is followe by a series of partial mergers, an breaking into seconary vortices. In the latter two cases the central vortex remains near the centre of the omain at least

21 QG vortex equilibria 21 Figure 19. Evolution of the vortex bouning contours for an unstable (σ max = 0.12) array of six peripheral vortices an an opposite-signe central vortex (α = 1) with = 0, in mutual equilibrium for δ = 1.08 at t = 0, 12, 16 an 21. The vortex bouning contours are viewe from the top. for the uration of the simulation. This is ue to conservation of the linear an angular impulses, as overall the system must remain anchore to the centre of the omain. 5. Conclusion We have investigate the problem of m- an m+1-vortices in mutual equilibrium in the context of three-imensional, unit Burger number, quasi-geostrophic vortices. We have first shown that these vortices have specific stability properties even if they exhibit overall similar patterns with their known two-imensional counterparts. Notably, in the absence of a central vortex, no more than 5 ientical three-imensional quasi-geostrophic vortices can remain stably locate on a circular ring. Recall that 7 two-imensional vortices can remain in mutual equilibrium for long times. We have also seen that aing an opposite signe vortex on the vertical axis of rotation of the system generally tens to estabilise the vortex array. The opposite tren is observe if the central vortex has the same sign as the peripheral ones, except for m = 2. However, instability can be foun in general if the central vortex is strong enough. Moreover, finite volume vortices can be sensitive to eformation moes when the vortices are close enough to each other. Nonetheless, there exists large parts of the parameter space where such vortex arrays are stable an therefore can persist in time. The existence of such equilibria can explain the formation of patterns in geophysical contexts an on the atmosphere of other planets such as the

22 22 J. N. Reinau Figure 20. Evolution of the vortex bouning contours for an unstable (σ max = 0.076) array of four peripheral vortices an an like-signe central vortex (α = 1) with = 0, in mutual equilibrium for δ = 0.16 at t = 10, 14, 20 an 24. The vortex bouning contours are viewe from the top. polygonal clusters of cyclones observe in the polar regions of the Jovian atmosphere, see Ariani et al. (2018). The paper has focuse for the first time on these equilibria on the simplest three-imensional context relevant to environments subject to rapi backgroun rotation an stable ensity stratification. This research can be extene to further stuies in other contexts with ifferent vertical ensity stratification in particular when the Boussinesq approximation fails to be relevant. Aitionally, the present investigation has restricte attention to unit Burger number vortices, which limits the variety of equilibria investigate. Due to the numerical cost of computing three-imensional finite volume equilibria an aressing their stability, it is impractical to perform an exhaustive stuy of the influence of the Burger number in this context. Such a stuy can however be performe using a two-layer moel an will be consiere in a future work. Appenix A. Linear stability for point vortices This appenix briefly escribes the metho use to aress the linear stability of a system of point vortices. It relies on a straightforwar linearisation of the equations of motions of the vortices. The m peripheral vortices of strength κ i are locate on a ring of raius R at a polar angle {θ i } 1 i m, θ i = i 1 2π. (A 1) m