Multipolar Structure of Equilibrium Shear Flow Field in Toroidal Plasmas∗

Liu-Xiu He(何柳秀),Ming-Hai Liu(刘明海), and Hua-Kun Cai(蔡华坤)

1School of Physics,Huazhong University of Science and Technology,Wuhan 430074,China

2International Joint Research Laboratory of Magnetic Confinement Fusion and Plasma Physics,Huazhong University of Science and Technology,Wuhan 430074,China

3State Key Laboratory of Advanced Electromagnetic Engineering and Technology,School of Electrical and Electric Engineering,Huazhong University of Science and Technology,Wuhan 430074,China

Abstract The multipolar velocity field structures are investigated by 2D momentum conservation equation with 3D equilibrium sheared flows in the full toroidal system.Numerical results show that the non-existence of radial velocity field in equilibrium surfaces is suitable only for the zero-order term of our 2D simulation.The non-zero-order radial velocity field is still preserved,even when converted to conventional magnetic surface coordinates.The distribution of velocity field vectors of the order of 1,2,and 3 are presented respectively in 2,4,and 6 polar fields with the local vortex structure.The excitation mechanisms of these velocity vortices are the coupling e ff ects of the magneto- fl uid structure patterns and the toroidal e ff ects.These results can help us understand the complexity of core physics,the transverse transport across magnetic field by the radial plasma flow and the formation of velocity vortices.

Key words:equilibrium shear flow,multipolar field,vortex structure

1 Introduction

Plasma rotation on the practical operation of advanced fusion reactors as well as physical properties of plasma research plays a vital role.Velocity field in Tokomak’s plasma is a very important parameter,it not only participates in the traditional balance,macroscopic stability or microscopic instability,and transport process,also appears in various auxiliary heating and the driven electric currents.Based on the advanced magnetic confinement fusion device to run securely and cost-effectively,the most threatening MHD instabilities are the resistive wall modes(RWM)and the neoclassical tearing modes(NTM).The effects of plasma rotation on diversified instabilities have been extensively investigated in different devices,[1−2]such as the stabilization of the driven instability of ion temperature gradient by toroidal rotating shear in the core area,[3]local exchange mode instability and internal kink mode driven by non-uniform density of magnetic surfaces,[4−5]the inhibition of RWM by plasma toroidal rotation gained through tangential neutral beam injection technique.[6−7]Popov’s calculation results also indicated plasma rotation was advantageous in suppresses NTM instability,moreover the different spiral configurations had the different stimulation patterns.[8]

One of the most successful discoveries during fusion study decades is the efficiency ofvelocity shear.Initially this model have been successfully explained the formation of transport barriers during transition of low(L)to high(H)confinement mode at plasma boundary.[9]It is indicated also that the effects of velocity shear played a decisive role for suppressing high-energy state of turbulence and transport,meanwhileflow is closely related to plasma rotation itself indicated by the experimental results.The experimental observations by Xu et al.[10−11]showed that plasma rotation and its shear play a dominant role in the formation of radial electric field and its shear.These results manifest intrinsic plasma rotations(including poloidal,toroidal one)and their shear degrees before and after the external driving input are very significant for understanding theflow.

Because the externally-driven implement is exerted mostly in the flat-top discharge operation,two unavoidable questions are involved,one is spontaneous plasma rotation state for Ohmic discharge,and another is how to alter the rotation of plasma after applying the external driving source.Spontaneous rotation is the case where no external momentum source is injected,plasma has a spontaneous rotation in the toroidal direction,which is found in almost all Tokomak’s devices in Ohmic heating,L modes,and H modes.The phenomenon was first identified by the famous Rice calibration.[12−13]

At present,many models have been proposed for the theory for the driving toroidal and poloidal flows,the physical mechanisms include the following forms:pressure gradientmotivation,[14]neoclassicalviscous effect,[15−17]up-down asymmetric excitation,[18]Coriolis drift manner,[19]ion orbit loss model,[20]high-energy particle transport mechanism,[21]and the strong turbulence Reynolds stress[22−23]or residual stress[24−26]etc.One of the most representative mechanisms is the last one proposed by Diamond for driving plasma flow.It is pointed out that the previous model is not appropriate without consideration of the interaction between the turbulence itself and the distribution of plasma flow field.With regard to the driving mechanism of the toroidal flow,the turbulent acceleration is a relatively new conception.[27]Previously,it was believed that the source of the toroidal rotation was input from the outside,and the residual stress was driven by its divergence,later studies had shown that the source of the driving force was also generated inside the plasma,the recent research results show that the residual stress driving theory was the main mechanism of the formation of the toroidal flow.But whether the poloidal flow driven by the Reynolds stress or the toroidal one triggered by the residual stress,the fl uctuation levels caused by the turbulent flow involved plasma itself velocity field,or there are close relationships between them.

Tokamak’s magnetic field type is two-dimensional(2D)spiral structure based on one-dimensional(1D)plasma balance model.[28]In the theory study of 3D simulation,just considering the toroidal magnetic field with a corrected factor of the toroidal distribution(equivalent ripple effect),the macro fine structure of the flow field are rarely mentioned,and the toroidal rotation is mainly considered where the poloidal rotation is thought to be damped by the neoclassical effect,[29]or the poloidal flow is also reserved in other simulations.[30−32]In our previous investigation based on 2D simulation,[33]it is elucidated that the magnetic configuration of toroidal plasma should be 3D structure,and it is nature that radial plasma flow occurs due to the existence of radial magnetic field.

In the existing fusion theoretical and experimental study,with the exception of electric field,the radial component of physical quantity seems to be a rare territory.After considering the coherent radial flow field,velocity field distribution(VFD)under certain condition can possess multifarious spiral structures.Vortices exist widely in various kinds of plasmas,and there are an affluent of structural patterns in di ff erent systems,such as the “cat eyes”structure and drift-Alfven vortex structures in the edge region of fusion plasma,[34−35]The purpose of this paper is to explore the multipolar structure of self-consistent flow after taking the intrinsic radial comportment of velocity into account.Past research focus one system on the peculiar vortex structure,but multi-structure of vortices coexist simultaneously in the same system is rarely reported,especially for the hexagon and octupole field structures with local triangle vortexes.

2 Physical Model

For Ohmic discharge,the velocity profile of plasma spontaneous rotation has no local strong shear,and the biggest Mach number normalized by ion Alfven velocity is of the order of 0.01,so the segregation of pressure,poloidal current to magnetic surfaces is insignificant,the effect offlow field on plasma balance can be seen as a small amount.The algorithms of plasma static equilibrium are described in detail in Ref.[33].Here considering self-consistent flow field,thermodynamic equilibrium momentum conservation equation with flow is used without impurity and viscosity:

where mi,meis the mass of ion and electron,are their velocity,

is the partial pressure of ion and electron.Ti,Teare their temperature,τeiis the collision time of electrons and ions.

Ignoring the inertia of electrons:

where p=pi+peis the total pressure,andis the total electric current densidy respectively.Generalized Ohm’s Law can be obtained from Eqs.(1)and(2):

The steady state of Eq.(3)is expressed as:

With the property of variable fields:

The coordinate components in the symmetric toroidal geometry of Eq.(5)are listed below:

where r,θ,φ is the minor radial,azimuthal,and toroidal angle of coordinates respectively.ur,uθ,uφ,jr,jθ,jφ,and Br,Bθ,Bφare the corresponding components of ion velocity,electric current density and magnetic field sequentially,ρ is the mass density of hydrogen.To solve the problem on toroidal coordinate system(r,θ,φ),the effect offlow field is taken as a small order,pressure(p)and plasma current at this scenario are the composite functions of the magnetic flux surfaces sand the flow functions.It is assumed that the profiles of uφare the function of the magnetic flux surfaces,and the values depend on the experimental measurement based on a tangential x-ray imaging crystal spectrometer at the core and radiation diagnostic at the inside boundary:

where k1and k2are the distributional coefficients determined by experimental observations and ψ(r,θ)is the 2D magnetic flux surfaces of static state of the Grad-Shafranove quilibrium,[33]which is a Fourier expansion,and similar to the expression of the work of Paulo Rodrigues and Felix I.Parra.[36]A nested magnetic surface is considered in this work excluding the current effects of the auxiliary coils for controlling plasma horizontal and vertical displacement.Br,Bθare completely controlled by the distribution of plasma discharge current.The vertical velocity is decided by Eq.(4)without plasma resistance:

where E and B are electric field,the total magnetic field respectively.It is very difficult to solve directly the equation because the coupling of electric filed and flow.In order to describe succinctly the vertical velocity,it could be written implicitly as another expression:[37]

where R is the major radius and f is the 2D stream distributions,which are generally adopted as the functions of magnetic flux surfaces.The selection principle is based on the fact that electrons are the carrying current body,and the ionic component contribution in the total current is a few percent.Under such conditions,the vertical velocity distributions satisfy automatically the incompressible conditions in the toroidal geometry of Eq.(9).And the first two terms of the left side of Eq.(10)is zero,at which point poloidal currents are separated from the magnetic flux surfaces to a minimum values,and this is referred to as equilibrium flow field.Reforming the analogous distribution type to magnetic flux surfaces,[33]they could be written as follows:

where the zero-order term refers to the pure radial part(independent of the azimuthal angle)in a Fourier expansion,the zero-order quantity of the radial component of flow field is grimly zeroare the amplitude of the radial and azimuthal VFD for the corresponding order number(m)respectively.These magnitudes are functions of radial positions,which are also the determinants of vortex structural features.If returning to 1D model,it is noticed that the average radial component of velocity field along the azimuthal direction is also zero,and the circumferential average of the azimuthal component of velocity field is exactly the traditional poloidal velocity fieldbecause of the average value of non-zero-order velocity field is null.So the refined structures of velocity fields disappear voluntarily on 1-D equilibrium configuration.

3 Numerical Results

J-TEXT device is a medium size Tokamak with a circular cross section,which the major radius R0=1.05 m,the minor radius a=0.27 m.The experimental data of#1032849 is employed to demonstrate plasma simulation.The main parameters in Ohmic discharge are Ip=160 kA,Bt=1.6 T,plasma density,electron and ion temperature are 3.0×1019m−3,0.85 keV,0.45 keV at the core of plasma respectively.As a calculation example,the proportion of ions in plasma current is assumed to be 2.5%.

Fig.1 The experimental toroidal rotational Mach number.

Figure 1 is the experimental measurement of toroidal rotational velocity distribution described by Eq.(13)whose value normalized by ion Alfven velocity,it can be seen that the maximum Mach number is 0.12 at plasma core. The separations of pressure,electric current to magnetic surfaces are[–1.3e–03,0.9e–03],[8.5e–9,3.6e–5]respectively,means that pressure deviation maximum is parts per thousand,the one of electric current is much smaller,parts per million,taking the effect of the equilibrium flow field on plasma balance into account.Thus the effect offlow on the plasma equilibrium for Ohmic discharge can be negligible in the range of experimental measurement errors.[38]FD.

Fig.2 The zero-order of V

The vector diagrams of the zero-order VFD is shown in Fig.2,where the blue star is the device center,the magenta one denotes the Shafranov displacement at plasma core(new magnetic axes).It is not only the contour map of the 2D azimuthal VFD,also is the average value along the whole rotation angles,and is as well as the equivalent of the traditional poloidal VFD.Its value increases firstly and then decreases with the increase of the radius,the maximum value is 1.39 km/saround the radial position of 0.44 a.The overall nature is laminar structurewith excellent symmetry.

The first-order of vertical VFD on the magnetic surfaces is given in Fig.3,where the back ground color is the corresponding potential function. In general,it is indicated that there is the dipole structure of the weak asymmetry of the low and high magnetic field sides due to Shafranov shifts.There is vortex pair composed of two vortexes with opposite rotation direction at the radial position of 0.5 a.Another distinctive feature in Fig.3 is that the amplitudes of radial,azimuthal component of the VFD are roughly equivalent on the majority of profiles,but on the inside edge,the value of the former is much smaller than the latter,so the streamline renders the closed structure.

The vector distribution of the second-order velocity field on magnetic surfaces is illustrated in Fig.4 whose background color denoting the corresponding potential functions,and the four-fold polar field with a weak asymmetry structures are presented.The amplitude of radial velocity at first increases and then decreases with the radial positions,the azimuthal one has double distributional characteristics with the first increases and then decreases over the profiles and its maximum is 5 m/s around 0.7 a.There are two pairs of vortex structure rotating in the opposite direction at the radial position 0.47 a,and each vortex is characterized by a local center of vorticity.

Fig.3 The first order of VFD on magnetic surfaces.

Fig.4 The second-order of VFD on magnetic surfaces.

Fig.5 The three-order of VFD on magnetic surfaces.

A map of the vector distribution of the third-order velocity field on magnetic surfaces is plotted in Fig.5,where the background colors indicate the corresponding potential functions.The overall presentation of the weak asymmetric structure of the six-fold filed is presented.The magnitude of the radial velocity distribution decreases monotonically with the increase of the radius.For the amplitude of the azimuthal,one firstly decreases,then increases and decreases again with the increase of the radius,and its maximum is−0.1 m/s at the core.In the radial position of 0.44 a with a reversal amplitude of azimutal velocity,there are three-swirl structures with the opposite rotation directions,which characterized by triangular vortices.

The numerical results offlow structure show that plasma radial flow naturally occurs if considering the intrinsic radial velocity field.The maximum value of zeroorder poloidal filed is around 0.5 a,while the first-order radial ones)is in the plasma core,the ratio ofversusis about one-third.Another universal distribution feature is that the radial and angular amplitudes of the non-zero-order of each order are of the same order of magnitude within the plasma,even the former is a little larger than the latter at the plasma core,while the former is much smaller than the latter at the inner edge,therefore the flow field presents a closed structure.On the other hand,it is important to emphasize thatandmaintain the same order of magnitude level at the plasma core,in other words the zero-order quantity is insufficient to represent the entire physical quantity,so the VFD should be a 3D structure.It can be reduced a 2D distribution after the radial position of 0.5 a due to the nonzero-order magnitudes are much less than the zero-order one.These distributional signatures of self-consistent flow field can in principle be explained by device’s magnetic field con figurations.Since the first-order quantity is much larger than the other non-zero-order ones,therefore,the angular rotation speed observed in the experiment should be the synthesis of zero order laminar velocity and firstorder vortex pair with the opposite rotation velocities.Because of these velocity vortices are also an integral part of the equilibrium shear flow,however most researches on magnetic reconnection are based on the static equilibrium due to the difficulty in energy principle description for steady-state balance with shear flows.Therefore these findings are bene ficial to understand the complexity of core physics and the correlation to transverse transport across the traditional poloidal magnetic field.[35]It can be deduced that the present conclusions have universal consequences in comprehending the origination of fl uid vortex for the noncircular or D-shaped toroidal plasma.Due to the e ff ects of elliptical and triangular distortion,there should be stronger radial velocity fields at all levels of non-zero-order field.

4 Discussion

In fl uid dynamics models,vortex structures have different excitation mechanisms,such as the Larichev-Reznik model of vortex pair,[39]tripolar Vortex system,[40]Beltrami flow model of the quadrupolar structure,[41]Bernard mechanism of the six-fold field.[42]In order to unify the description of each order flow field,the non-zero-order VFD are mapped to the zero-order balance field whose radial component is zero,which is a return naturally to traditional magnetic fluxes coordinates.[37]Their forms can be written as the following expressions:

where m=1,2,3 is the order number of velocity field structure,and rs,θsare the radius and rotation angle on the magnetic surface coordinates respectively

is the azimuthal angles and Λ(rs)is the corresponding Shafranov shifts.

In the mathematical formalism,they may regard as the combination of Beltrami flow and Taylor vortex[43]with variable coefficients here,which can be depicted as the kind of ABC manifold type with variant coefficients[44]by fitting curves.Its pervasive characteristics of vortex are formed by heteroclinic orbits connecting saddle points of periphery of closed loop.According to the di ff erent values of m,the local structure of vortices appear twin vortex(m=1),two pairs(m=2),or hexagon with local tripolar vortices(m=3).Due to the radial VFD decrease on overall trends with the increase of the radial position(its value at the outermost magnetic surface is almost zero),therefore the distributions of velocity field at all orders present the closed structural property.In particular,the closed quadrupole field structures with reverse vorticity are distinctly different from the one in ordinary fluid mechanics with invariant coefficients.Its structure resembles the perturbed flow field during the formation of magnetic island.[45]It can be discovered by the comprehensive analysis that the triggering mechanism of velocity vortex are the coupling interactions between the eigen velocity structural patterns and the toroidal e ff ects,such as vortex pair of the first-order term is determined by the native velocity structural mode together with the function forms of toroidal e ff ects(m=1+0,2−1,3−2),and the quadrupole field is established by their combinations of the intrinsic structural modes(m=2+0,1+1,3−1),and so on.

It is well known that vortex is not entirely turbulence,while turbulent flow has necessarily multiple temporal and spatial scales of spiral structures,while Beltrami flow is to a great extent seen as one of the relevant turbulence structures.If plasma resistance and magnetic viscous effect is considered,once a closed structure found in local charged particle movement,the existing vortex at all levels are likely to grow up in different degrees,and lead to the formation of local island structures.Despite the present amplitude of each order flow field compared with zero-order quantity is also small in vortex position.Especially for the vortex pair of the first-order and second-order of velocity field and the triangle vortex of the threeorder ones,because their locations are close to each other(0.44 a),among of them will have a strong competition relationship or vortex couple.The closed structure of local velocity fl uid will produce self-consistent electromagnetic field,it is only natural that they will interact with the external field,result in the toroidal field correction,and will further a ff ect to the corresponding current density distribution,lead to plasma balance to adjust the layout,the related work is under way.

5 Conclusion

2D plasma simulation results with 3D’s equilibrium shear flow indicate that there are no radial velocity fields in equilibrium surfaces are suitable only for the zero-order term.The non-zero-order radial velocity fields are still present,even when converted to the conventional magnetic surface coordinates.The distributions of velocity field vectors of the order of 1,2,and 3 are respectively presented in 2,4,and 6 polar fields with the local structure features.These local structures are inherent in nature,and are also an integral part of the balanced shear flow field.The excitation mechanisms of these structural modes are the nonlinear coupling e ff ect of the magnetofl uid distributional signatures and the toroidal e ff ects.These results are facilitative to understand the complexity of core physics,the transverse transport across the traditional poloidal magnetic field by the radial plasma flow and the formation of velocity vortices.