Gyrofluid vortex interaction

Low-frequency turbulence in magnetised plasmas is intrisically influenced by gyroscale effects across ion Larmor orbits. Here we show that fundamental vortex interactions like merging and co-advection in gyrofluid plasmas are essentially modified under the influence of gyroinduced vortex spiraling. For identical initial vorticity, the fate of co-rotating eddies is decided between accelerated merging or explosion by the asymmetry of initial density distributions. Structures in warm gyrofluid turbulence are characterised by gyrospinning enhanced filamentation into thin vorticity sheets.


Introduction
Vortices can be regarded as the basic constituents of turbulence. Vortex motion and interactions govern nonlinear structure formation, flow and convective transport properties in a variety of fluids. A particular case of interest are quasitwo-dimensional fluids, which are characterised by the possibility for formation of coherent structures and large scale (zonal) flows, a dual cascade, and the ideal conservation of enstrophy in addition to energy [1,2]. Merging and filamentation of vortices are fundamental processes that underly these properties. Examples for quasi-2D fluids encompass stratified and rotating fluids, atmospheric and oceanic flows, or the cross-field dynamics of magnetised plasmas [3].
In magnetised space, laboratory or fusion plasmas, the fluidlike convection perpendicular to a magnetic field B is governed by drifts, which describe the mean motion on top of fast charged particle gyration. In particular, the magnetic confinement of fusion plasmas is crucially determined through turbulent transport generated by drift-type instabilities, and their suppression by (turbulence driven) zonal or equilibrium flows [4,5]. Turbulent convection in magnetised plasmas is dominated by the electric drift velocity i ) they effectively experience a ring-averaged electric potential f i rather than the potential at the gyrocenter (for low-frequency perturbations with w w  i ), and contribute to a quasi-neutral spatial distribution with approximately equal electron and ion particle densities on average over the gyroorbit. The framework for a description of magnetised plasmas governed by drift motion with FLR effects is efficiently given by gyrokinetic models evolving a 5D distribution function in phase-space [6][7][8], or by gyrofluid models of appropriate respective fluid moments in 3D space [9][10][11][12]. In the following, the effects of finite Larmor orbits on fundamental vortex interactions are analysed within an isothermal gyrofluid model [13]. It is found that spiraling of vorticity induced by FLR effects in the presence of spatial asymmetry significantly alters the merger process and generates fine structured vorticity sheets. Specific initial conditions result either in strongly accelerated merging or in vortex explosion, with consequences for inverse turbulent cascade properties and zonal flows.

Gyrofluid model for basic vortex interactions
Here we employ a basic local ('delta-f') gyrofluid model, derived from an energetically consistent gyrofluid electromagnetic model [14] in the limit of 2D isothermal dynamics in a straight and constant magnetic field. Details on the normalisation of the model equations and their numerical implementation can be found in [15]. Gyrofluid models do not dynamically evolve the particle densities ( ) N x t , s of electrons and ions ( Î s e i , ), but rather their gyrocenter densities ( ) n t x, s . For small density fluctuation amplitudes both are connected through polarisation [10,13,16,17] of gyroorbits by f: can be assumed at ion gyro scales. The small electron mass also leads to negligible polarisation, so that particle and gyrocenter densities » N n e e coincide. The electron and ion particle densities are connected by strict quasi-neutrality to º N N i e at scales much larger than the Debye length.
In the absence of driving, damping or parallel coupling the evolution equation for the gyrocenter densities in a homogeneous magnetic field then reflects incompressible mass conservation, and can (for normalised B=1) in 2D be written as is added to the right hand side of equation (2).
The continuity equation (2) is closed by the quasineutrality condition N i = N e , which by the gyrodensity relation in equation (1) gives the polarisation equation: The gyro-operators can be written in the Padé approximate forms G = +  [10]. These expressions are useful for analytical considerations. Although strictly valid for  b 1, the Padé forms give numerically nearly indistiguishable results compared to the original Bessel forms for all here presented simulations. , which for normalised B = 1 describes the deviation between electron and ion gyrocenter densities in relation to a scalar E × B vorticity f W = 2 . Subtracting the ion from the electron gyrocenter density equation, equation (2) in this cold limit transforms into the classical 2D Euler equation in vorticity representation:

Vortex merging for cold ions
Vortex interactions, in particular merging and co-advection, are in this limit akin to classical fluids, and the gyrocenter densities are just passively advected. For t = 0 i , a vortex that is defined by a localised spatial distribution W( ) x can be initialised by any arbitrary initial density field ( ) n x e with an appropriate choice of : the further evolution of W( ) t x, in time through equation (4) will not depend on the particular initial gyrocenter densities, but only on their difference.
In the following it is shown how finite ion temperature with t > 0 i in magnetised plasmas leads to fundamentally different behaviour of the classical vortex merger and coadvection problems, and thus the resulting spectral properties of fully developed turbulence. In particular, the warm ion vortex merger problem intrinsically depends on the initial gyrocenter density distribution.
For comparability with classical fluid merger problems the vorticity is in the following initialised as a Gaussian In the gyrofluid model this may be obtained by a difference δ between the amplitude a of the electron compared to ion gyrocenter density, in the (x, y) plane perpendicular to B: n n e i e 0 1 0 1 0 0 for both cold and warm plasmas. In the merger problem two vortices with the same amplitude are placed next to each other with an initial density peak distance D 0 .
The Euler equation as a standard model for fluid flow is, like the underlying Newtonian particle motions, invariant under parity transformation . The evolution of flow patterns is thus symmetric with respect to simultaneous point reflection ( x x) and reversal of the . This implies that also for the fluidlike (cold plasma) case t = 0 i a change in sign of the initial x) leads to a spatially anti-symmetric evolution of the vortex merger. . The evolution is a typical example for a 2D fluid vortex merger: vortices orbit each other by mutual advection and after a while (depending on initial separation) develop encircling vorticity and density veils, and coalesce on combined advection-diffusion time scales into a spiraling single vortex [18]. The ensembles with inverted initial vorticity evolve exactly anti-symmetrical: negative vorticity inverts the direction of co-rotation and of the final spiral arms of the merged vortex.

FLR effects on vortex merging
It had already been noted in the seminal work of Knorr et al [9] from 1988 on theory of FLR effects in a guiding centre plasma, that in a (reduced) exemplary simulation of vortex merging with and without FLR effects different behaviour appeared: while for zero Larmor radius the maxima remained separated, a coalescense had been observed for a FLR [9].
In the following it is shown, that such FLR effects on vortex motion and vortex interactions in warm plasmas are generally a result of a breaking of the axial symmetry of the vortex in presence of any spatial asymmetry in the initial gyrocenter density distributions, which leads to FLR induced vortex spiraling.
However, for finite t > 0 i no 'generic' merger scenario like in the fluid case can be constructed: the temporal evolution of vorticity does not only depend on the initial (for example Gaussian) distribution W( ) x , but further on the specific gyrocenter initial density distributions ( ) n x e and ( ) n x i that generate this vorticity. For warm plasmas the densities are not any more passively advected like in the cold case with W = d 0 t from equation (4). The gyrofluid vorticity evolution can be understood more intuitively (compare [19]) in the long wave length limit (  b 1 2 ) when the Padé form of the gyrooperators is Taylor approximated: In the first line terms up to order b 2 are kept, and in the second line up to b. For homogeneous B, the identity has been used.
The first term on the right of equations (7) and (8)  . This can be interpreted as an FLR induced contribution to polarisation by advection of vorticity along isobars of pressure p [19].
The Poisson bracket vanishes when the isocontours of axially symmetric density and vorticity profiles coincide. For a deviation from exact axial symmetry this term gives a significant contribution to the evolution of the (generalised) vorticity. The vortex dynamics for t > 0 i thus depends on the The situation completely changes when the same initial W( ) t x, 0 is obtained by keeping = + a 1 but setting d = -0.01 and thus changing the relative local differences between electron and ion gyrocenter densities. As can be seen by the second term on the right of equation (7), n i determines the FLR effect on polarisation. This means that at the positive Gaussian (quasi-neutral) density perturbation, the ion gyrocenters are not any more shifted outwards (as     same general tendency, with expectedly faster merging for shorter separations.

Dependence of vortex interactions on initial conditions
It is not a priori clear which vorticity reversal method is physically more relevant. Vortices are in general not seeded, but appear dynamically mostly as a result of the specific effects of instabilities on electron and ion densities. An important mechanism for vorticity generation is the drift wave instability, driven by a nonadiabatic parallel electron response in the presence of a cross-field density gradient [4]. For low collisionality the relation between electron density and potential can however often be regarded as nearly adiabatic, following approximately a Boltzmann relation with f n e . We can also construct a vortex merger problem for such 'adiabatic' vortices. The constraint implies that initially f W =  = ~Ŵn e d 2 2 2 . In particular, setting f = n e in equation (3) for a given f ( , which is readily evaluated in wave number space. The choice is now whether to initialise Gaussian density/potential 'adiabatic blobs' (which yields a shielded vorticity), or 'adiabatic Gaussian' vorticities. The latter can be achieved by inversion of a defined The 'adiabatic blobs' case first develops small satellite vortices which are sheared off through the shielding reversed vorticity rings around the centres. Subsequent collision of the satellites between the vortices rapidly results in an explosive repulsion after formation of a vorticity tangle. The situation is depicted in figure 4.
The 'adiabatic Gaussian' case is shown in figure 5: the development of vorticity is more similar to the fluid-like case (compare figure 1), but again with a pronounced FLR induced filamentary fine structure and accelerated merging. The evolution of peak separation Δ is for both 'adiabatic' cases presented in figure 6. Both cases are again anti-symmetric after reversal of the initial amplitude.
From these examples it is obvious that the gyrofluid merger dynamics strongly depends on the initial vorticity and density distributions, in addition to parameters like initial relative vortex separation as in the fluid case.

Gyrospinning of asymmetric vortices
A unique effect that is here present for all warm ion gyrofluid vortices is FLR induced spinning. A related FLR spin-up has been observed before for the special case of an interchange unstable (magnetic curvature driven) 'blob', which is characterised by the formation of a dipolar potential on top of a monopolar (e.g. Gaussian) density or pressure perturbation [19][20][21]. While interchange 'blobs' can acquire spinning by a range of additional mechanisms [15,22], the FLR spin-up is in the following shown to be a universal phenomenon and essentially a consequence of asymmetry. A single inviscid axially exactly symmetric vortex retains its shape. In Similarly, any radial density gradient from n(r) leads to radial spreading of vorticity. The combined result is the spin-up of spiral arms as in figure 7 with orientation depending on the relative sign of magnetic field B and vorticity W. Two neighbouring vortices mutually induce initial asymmetries similar to elongation, resulting in rapid pre-merging spin-up.
As a side remark, asymmetry can enter not only via noncircular vortex initialisation, but also numerically through a coarse rectangular grid and too close boundary proximity. Grid size and resolution have to be chosen accordingly, that any grid spin-up artefacts evolve much slower than the physical time scales of interest.

Vorticity filamentation in vortex co-advection and turbulence
The complementary problem to merging of co-rotating 2D vortices is the straight co-advection of counter-rotating vortices. In figure 8 it is shown that FLR spin-up again significantly alters this type of vortex interaction: in a warm gyrofluid (bottom row) the vorticity filamentation slows the In combination, merging and co-advection determine the interactions and cascade in a turbulent sea of 2D vortices. We find that in decaying turbulence initialised with a random distribution of density and vorticity fluctuations analogously to equations (7) and (8), the FLR induced spinning also leads to enhanced vorticity filamentation. Self-sustained drift-wave turbulence in inhomogeneous magnetised plasmas is in 2D effectively represented by the Hasegawa-Wakatani model [23]: the turbulent drive is maintained by a dissipative coupling term f -( ) d n e added to the right hand side of equation (2) for electrons, which emulates parallel electron dynamics for a single parallel wave number by the parameter d. A comparison of the vorticity structure between t = 0 i and 1 in a saturated drift wave turbulent state for d=0.01 is shown in figure 9, and demonstrates the persistence of FLR induced vorticity filamentation in fully developed turbulence.  Vorticity thinning has been identified as a possible explanation for the inverse energy cascade of 2D turbulence in general fluids [24,25] and for drift wave turbulence [26], and is here shown to be strongly enhanced by FLR spin-up in a gyrofluid. The amplitude of vorticity is for t = 1 i increased over the whole spectral range, while density fluctuation amplitudes are enhanced on intermediate (rk 1 i ) scales. This can be directly seen by the larger extension of the predominant density fluctuations in the bottom right panel of figure 9.
In 3D warm gyrofluid computations of drift wave turbulence we find that the vorticity sheets are much less pronounced but still discernible. The parallel connection of fluctuations along the magnetic field lines in presence of radial zonal flow [28] and magnetic shear [27] distorts but not completely suppresses the filamentary spin-up. Results on FLR effects in 3D toroidal edge turbulence are going to be presented in a future publication. The present work has focussed on identifying and explaining the FLR induced spiral-arm spin-up in asymmetric vortices and its effects on basic vortex interactions.

Conclusions
In summary, spiraling of asymmetric vortices by gyroorbit effects has been identified as a novel FLR effect in magnetised plasmas. The spin-up of spiral arms in single vortices was understood as an effect of density asymmetries on the governing drift velocities through polarisation of gyroorbits. This FLR induced spiraling was shown to strongly impact all vortex interactions in quasi-2D magnetised plasma dynamics, with examples ranging from dual vortex merging, via co-advection, to fully developed (many-vortex) turbulence. The nature and morphology of drift wave turbulence, which is of overall importance in magnetised fusion plasmas, is essentially changed. Gyrofluid and gyrokinetic simulations are able to consistently account for such FLR effects on vorticity filamentation by sufficient spatial resolution. 3D gyrokinetic and gyrofluid codes are routinely used to numerically study turbulent transport in core, edge and SOL fusion plasmas. The natural focus of interest in such simulations is usually mostly on specific instabilities, nonlinear phenomena like zonal flow emergence, transport scalings or transport reduction, and identification of comparable observables mostly of statistical nature for experimental validation. Simulations of immediate fusion relevance are thus usually operated with as much physics included as is computationally affordable, but at the same time for efficiency restricted to the most coarse grid possible while maintaining convergence.
Then again, it is an essential feature of physics to seek to reduce phenomena down to the most fundamental aspects for better understanding. Vortices may be regarded as the 'fundamental particles' of turbulence. The interaction of vortices underlies all nonlinear turbulent transport dynamics and is therefore at the core of understanding turbulent transport in fusion plasmas. While the single spinning and dual interactions of vortices are of course simplifications which are not directly observable in any fusion plasma, the combined effects on turbulent transport and spectral properties will always enter into any sophisticated modelling, although they may be well disguised by other often dominant effects such as toroidicity, shearing, collisionality, magnetic flutter, temperature fluctuations, and many more. On a marginal note, actually this little essay has had its foundations in present efforts to develop sophisticated 3D full-f 6-moment electromagnetic toroidal edge/SOL gyrofluid turbulence codes. In basic convergence tests for one of our evolving codes, grid refinement towards high resolutions has in some situations produced vorticity filamentations similar to those shown in figure 9. As such fine structure has never been reported in any so far published work (known to the author) on gyrokinetic or gyrofluid turbulence, the first idea was that we had encountered some numerical artefact, perhaps having to do with possible problems regarding some specific numerical handling of FLR effects. After numerical issues had been ruled out (partly by cross-verification between our two gyrofluid codes 'FELTOR' and 'TOEFL' using similar models but fundamentally different numerical methods), simple test cases were devised, and the essence of the present study has emerged.
How now actually the newly identified FLR effects affect particle and energy confinement in more complete and complex simulation scenarios is an important question, which is beyond the scope of this basic presentation. The identification and evaluation of such FLR effects in the broader context of fusion edge turbulent transport, and possible impacts on (zonal or mean) flow generation, is going to be a subject of work to come.