Effect of electron and ion mobility on edge biasing in tokamak plasmas

We present an improved model for the study of edge biasing in a tokamak plasma that incorporates electron and ion mobility contributions. The non-ambipolar nature of the drifts due to the electron/ion mobility terms influences the space charge separation due to edge biasing and affects plasma dynamics in the edge and SOL regions in a significant manner. In contrast to earlier studies, the present model enables simulation studies at higher biasing voltages. The inclusion of mobility enhances/decreases the effect of negative/positive biasing. The radial profiles of plasma density, electron temperature, radial electric field, and its shear for positive as well as negative biasing are investigated as a function of mobility.


Introduction
The boundary region of a tokamak is highly turbulent mainly due to the presence of drift and interchange instabilities [1][2][3][4][5], that leads to several deleterious effects in tokamak plasmas such as anomalous transport, confinement degradation, intense heat and particle loads on the plasma-facing components, etc [6][7][8][9].Mitigation/reduction of these effects is essential for a safe and controlled operation of the tokamaks.In the past, various approaches have been considered-one of them being edge biasing.
The edge biasing imposes an external electric field in the radial direction, which reduces the plasma turbulence by turbulence decorrelation mainly via ⃗ E × ⃗ B velocity shear that causes higher zonal flows and reduces the anomalous plasma transport in the edge and SOL regions [10][11][12][13].Several authors have studied the effect of biasing in the boundary region (edge and SOL regions) experimentally [9,[14][15][16][17][18][19][20][21][22][23][24][25][26][27] as well as theoretically/numerically [4,7,[28][29][30][31].Theoretical/numerical studies of edge biasing are attempted in [30,31] using a finite electron temperature gradient.In these studies, it was found that the high positive edge biasing leads to a high charge imbalance [30,31] that increases the radial electric field and its shear such that it quenches the turbulence almost entirely in the edge and SOL regions whereas in experiments such as J-TEXT [24], ISTTOK [16,22], TCABR [26], Aditya-U [32] and various other tokamaks [9,12,19,33,34] the turbulence exists at high biasing voltages.The high electric field shear in simulations [30,31] thus places an upper limit on the magnitude of the high positive biasing voltage.This indicates that the model in [30,31] is inadequate to reproduce turbulence for the higher edge biasing voltages that are used in many tokamak experiments.
To incorporate the high edge biasing in the simulation other mechanisms that are not considered in [30,31] such as ion orbit loss [35], neutral friction [36], and mobility [37] that may prevent the quenching of the turbulence are required.The ion orbit loss can decrease the floating potential created by electron loss into the limiter/divertor plates, therefore, it reduces parallel (∥) current that effectively increases the resistivity.The neutral friction also increases the resistivity inherently in the parallel direction.The mobility in the perpendicular direction (⊥) increases ⃗ ∇ ⊥ • ⃗ J ⊥ , therefore ∇ ∥ J ∥ decreases as ⃗ ∇ • ⃗ J = 0; here ⃗ J indicates current density.This effect also indirectly increases the parallel resistivity.As the interchange plasma instability increases by the resistivity, the plasma turbulence can be increased even in the presence of high voltage biasing.Our model is based on the fluid description and fully ionized plasma therefore ion-orbit loss and neutral friction can not be taken into account but mobility may be considered.In our present extended model, we find that the non-ambipolar drift arising from the mobility of the electrons and ions serves the purpose.The non-ambipolar nature arises from the fact that the mobility-induced drifts of the electrons and ions are in opposite directions in contrast to the ⃗ E × ⃗ B drift that moves the electrons and ions together.The diamagnetic drift and the ion polarization drift also have a non-ambipolar nature as they are charge-dependent.However, these two drifts are already included in the earlier model [30,31].This indicates that another drift due to the mobility of ions and electrons must be included for the electrode biasing [30,31] studies.In this work, we have included the effect of the mobility of electrons and ions and studied its impact on positive as well as negative biasing cases as a function of mobility.
The extended model equations have been simulated on the BOUT++ framework [38].The simulation results show that the inclusion of mobility of electrons and ions reduces the space charge (charge imbalance) for positive edge biasing that modifies the radial electric field (E x ) and its shear (E ′ x ) in the edge and SOL regions.To the best of our knowledge, the inclusion of electron and ion mobility has received very little attention [37] in the context of edge biasing as well as plasma turbulence studies.Our work highlights the importance of including this contribution for more realistic modeling of the tokamak edge dynamics in the presence of external biasing.
The rest of the paper is organized as follows.The derivation of the model equations in the presence of electron and ion mobility is described in section 2. The numerical simulation details are given in section 3. The main simulation results are presented in section 4 and finally, all the results have been summarized in section 5.

Model equations
The model equations related to edge biasing had been discussed in [30,31], where it was found that the high positive biasing voltages led to a high radial electric field and its shear that greatly reduced the turbulence in the edge and SOL regions.It was found that these simulations did not allow the application of very high biasing voltages at the electrode [30,31] as at higher voltages the plasma turbulence was seen to almost disappear.In reality, the experimental edge biasing has been performed for significantly higher biasing voltages where the plasma turbulence is still present.To simulate the biasing at higher voltages, in this work, we have modified the existing model equations by incorporating the mobility of electrons and ions as indicated in the introduction.
The high positive biasing voltage creates a large electron current into the electrode mainly due to Λ b ≫ ϕ/T e − Λ that drives a charge imbalance near the biasing region.Here, Λ and Λ b denote the floating and biasing potentials, respectively.The imbalance of the charge generates a large radial electric field E x which creates a large E ′ x that is responsible for the suppression of the turbulence.To prevent this unrealistic suppression of the turbulence one needs to reduce the large E ′ x at high positive biasing.To do so the extra charge imbalance should be reduced by a non-ambipolar mechanism of electron/ ion transport so that it mainly balances eΛ b ∼ −eΛ e,non (E x ) + eΛ i,non (E x ).Here Λ e,non (E x ) and Λ i,non (E x ) represent the nonambipolar potential of electron and ion as a function of E x ; e is the electronic charge.The dominant ⃗ E × ⃗ B drift is ambipolar, while other drifts such as diamagnetic and polarization are non-ambipolar but weak.Hence, these are insufficient to reduce the space charge and consequent high E x .This leads to considering another drift that is directly related to the electric field so that ⃗ v µ ∝ ⃗ E ⊥ , which can be a simple drift velocity driven by ⃗ E ⊥ .This leads to ⃗ v µi,e = µ i,e ⃗ E ⊥ ; µ i and µ e denote the mobility of ions and electrons.The ion mobility can be given by Einstein's formula µ i = D µ /T i0 , where D µ and T i0 represent the particle diffusion coefficient and equilibrium ion temperature, respectively.The relation between the mobility of ions and electrons can be given by µ e = µ i exp(Λ), where Λ is the floating potential of the material probe inserted into the plasma (biasing electrode).µ e = µ i exp(Λ) has been obtained assuming electron response for the potential fluctuations as Boltzmann.After including the drift velocity due to the mobility of electrons and ions, the drift-related terms will be added to the existing model equations [30].These drifts can be written as and Here

ion diamagnetic, and electron diamagnetic drift velocity, respectively with
/en e B 0 : e, c, B 0 are electronic charge, speed of light, and the magnitude of the toroidal magnetic field, respectively.
These drifts have been used to derive model equations related to the electron continuity, quasi-neutrality, and electron energy equations.These equations in the normalized units are 5 T e n µ ne (5) where µ ni and µ ne are newly added terms related to Here, Boltzmann distribution has been used for the electrons near the electrode which can be written mathematically as It is to be noted that the first terms of the equations ( 6) and ( 7) are fluctuating terms and the last term has a second derivative of ϕ, therefore, these terms are smaller compared to the second term inside the bracket as the second term is related to the radial gradient of the equilibrium quantities.The second term of equations ( 6) and ( 7) has a dependency on the biasing voltage with its appropriate sign.
For positive biasing ∂ϕ/∂x is positive and for negative biasing, it is negative (more discussion will be given in section 4).Thus positive biasing decreases the electron density and negative biasing increases the electron density after the inclusion of µ ni and µ ne in the model equations.The basic assumptions in the derivation of the model equations ( 3)-( 5) and the definition of all the symbols are given in [30,31].The effect of ionorbit loss [35] and neutral friction [36] has not been considered in our simulation due to fluid and fully ionized description of the plasma.The normalization has been used as n ← n/n 0 , T e ← T e /T e0 , ϕ ← eϕ/T e0 (x, y) ← (x, y)/ρ s , and t ← tΩ s .It is to be noted that x and y correspond to the radial and poloidal directions.

Numerical simulation
We have used the Neumann and periodic boundary conditions for the x and y directions, respectively for n, T e , and ∇ 2 ⊥ ϕ.To solve ϕ from ∇ 2 ⊥ ϕ, the Dirichlet boundary condition has been considered at x = 0 [ϕ = −T e log(n)] and x = L x [ϕ = ΛT e ] while the periodic boundary condition has been used in the y-direction.It is to be noted that the Laplacian solver has been used for solving ϕ from ∇ 2 ⊥ ϕ. 128 × 128 grid resolution has been considered for x and y directions.The size of the simulation zone in x and y directions is L x = 5.12 cm and L y = 10.24 cm, respectively.We have used normalized input parameters; Λ = 3.0, D = 2 × 10 −3 , ν = 0.03, κ e = 4 × 10 −3 , g = 1.0 × 10 −3 , σ 0 = 2.0 × 10 −4 , σ 0b = 6.67 × 10 −5 , χ 0 = 6.0 × 10 −4 [1].D, ν, and κ e are used mainly for numerical stability purposes as these remove the nonphysical large k y modes.These input parameters have been calculated using Aditya Tokamak parameters; n 0 = 5 × 10 18 m −3 , T e0 = 16 eV, R = 100 cm, a = 0.25 cm, and B = 1 Tesla at the last closed flux surface (LCFS).All the simulations have been performed on the Antya HPC of IPR.The BOUT++ framework [38] has been used to simulate the above model equations ( 3)-( 5).The radial width of the biasing electrode and its radial position are 13.8ρ s and 42ρ s , respectively [30] in the simulation.The biasing potential propagates on the magnetic flux surfaces as explained in [30] where it is shown that the shear flows generated due to edge biasing effectively control the turbulence in both linear and non-linear regimes.The same model has been used in this work.The biasing electrode has a Gaussian shape in the radial direction and is continuous in the poloidal direction (as indicated in the relevant figures with the shaded regions).The I-V characteristic of the electrode follows the typical Langmuir probe characteristic which shows that the current drawn in the positive biasing case is higher compared to the negative biasing case which is responsible for the strong charge imbalance in the case of positive biasing [30] as discussed earlier.This shows that a strong effect of mobility can be seen in the case of the positive biasing compared to the negative biasing.After the mobility-related drift velocity correction to the model equations, the code can run up to Λ b = +160 volt, unlike the previous simulation where biasing of the electrode was not possible beyond +64 volt [30,31].In this work, we have varied biasing voltage from Λ b = −160 to +160 volts.Also, edge biasing has been performed with different mobility cases of electrons and ions.For this purpose, the different mobility cases have been obtained by changing D µ using Einstein's formula (µ = D µ /T i0 ) in the simulation.Furthermore, D µ can be estimated using It is to be noted that [39,40] have used the same formula to calculate the diffusion coefficients for their simulation.We have used the same formulae to calculate the diffusion coefficients for the calculation of mobility.q, ρ e , ν ei , ρ s , and Ω i denote the safety factor, electron gyroradius, electron-ion collision frequency, ion gyro-radius, and ion gyro-frequency, respectively.This formula estimates D µ ∼ 10 −3 for the above input parameters for Aditya Tokamak.In this work, we have used three different values of D µ (5 × 10 −4 , 2 × 10 −3 , and 4 × 10 −3 ) to show the dependence of the results on the mobility.

Simulation results
This section presents the simulation results of electrode biasing using the effect of mobility of ions and electrons.Here we will present some common basic features of the edge biasing such as the various radial profiles (plasma density, temperature, potential, radial electric field and its shear, particle flux), fluctuations in density, and k y spectra as a function of mobility of electrons and ions.It is found that these features qualitatively are similar to the experimental results obtained in [9,15,18,22,[24][25][26][27]41].The time series of the plasma density (⟨n⟩ x,y , where ⟨⟩ x,y indicates radial and poloidal averages) has been shown in figure 1 using D µ = 2 × 10 −3 for Λ b = +160, +64, −64, and −160 volts by blue, orange, green, and red colored lines, respectively.From figure 1, it has been observed that ⟨n⟩ x,y attains the saturated value after t ∼ 7 × 10 4 /Ω s for all biasing voltages.It is found that the magnitude of ⟨n⟩ x,y is lower for Λ b = +160 volt compared to Λ b = +64 volt, and the magnitude of ⟨n⟩ x,y for Λ b = −160 volt is higher than Λ b = −64 volt (figure 1).These phenomena can be explained in terms of µ ne .In the case of positive biasing, µ ne is positive in the electron density continuity equation (equation ( 3)) that acts as a sink term, therefore, ⟨n⟩ x,y decreases.The opposite is true for the negative biasing.
Here, we have analyzed the impact of mobility on radial profiles of ⟨n⟩ y,t , ⟨T e ⟩ y,t , and ⟨ϕ⟩ y,t .Here, ⟨ f ⟩ y,t indicates the poloidal and longtime averages of f.These profiles for Λ b = +64 volt with D µ = 5 × 10 −4 , 2 × 10 −3 , 4 × 10 −3 , and w/o mobility are shown in figures 2(a)-(c).Figures 2(d)-(f ) shows the same but in the case of Λ b = −64 volt.In the case of Λ b = +64 volt biasing, the high D µ decreases the magnitude of ⟨n⟩ y,t , ⟨T e ⟩ y,t , ⟨ϕ⟩ y,t compared to w/o mobility as µ ne acts as a sink in the density continuity equation.Different behavior of ⟨n⟩ y,t , ⟨T e ⟩ y,t , and ⟨ϕ⟩ y,t has been observed for the Λ b = −64 volt edge biasing.Here, the high mobility increases ⟨n⟩ y,t , ⟨T e ⟩ y,t , and ⟨ϕ⟩ y,t as the mobility term in the case of negative biasing acts as a source term in the density continuity equations, therefore, after time integration the magnitude of plasma density increases than the w/o biasing in the edge region.The same phenomenon is true for other negative biasing voltages (not shown here).x is attributed to the enhancement in the space charge via the non-ambipolar mechanism as E x has a negative sign in the edge region for the negative biasing.The higher magnitude of the mobility corresponds to the higher drift velocity which indicates that the space charge and consequent E x and E ′ x increase with the increase in the magnitude of mobility that can be seen in figures 3(c) and (d).
From the cases of positive and negative edge biasing, observations show that the effect of the mobility of electrons and ions modifies the space charge.This leads to the modification in the different plasma parameters that may affect the density fluctuations.To show these effects a snapshot of the plasma density in x where radially outward motion has been reduced.This may give rise to the increase of the plasma confinement by the effect of mobility in the case of negative biasing.A time series of density at coordinate x, y = 64ρ s , 128ρ s for w/o and w/ mobility cases using Λ b = +64 & − 64 volts has been shown in figures 5(A), (C) and (B), (D), respectively.For the case of positive edge biasing, it can be seen from figure 5(A) that the fluctuations of density have almost disappeared for a time interval of ∼4000/Ω s which indicates the stabilization of the turbulence for this duration of time.After the inclusion of the mobility in the model equations, the temporal variations in the density have been observed at all instants of time which can be seen in figure 5(B) although the amplitude of the fluctuations has been reduced in this case.It can be seen from figure 5(D) that the amplitude of the density fluctuations is smaller but their numbers increase (they appear more frequent) for w/ mobility compared to w/o mobility.
The effect of the mobility of electrons and ions also modifies the cross-field transport by the modification in E x and E ′ x .The radial profiles of the particle flux (Γ) for w/ and w/o mobility have been shown in figure 6 using Λ b = +64 volt which is indicated by the blue-colored solid and black-colored dashed lines.It has been observed from figure 6 that the effect of mobility increases Γ in the edge region and decreases in the SOL region [30].This is also a reason behind the reduction in the plasma density and electron temperature for the positive edge biasing (figures 2(a) and (c)).To see the effect of mobility on k y -PSD (power spectral density), we have plotted it for D µ = 5 × 10 −4 , 2 × 10 −3 , 4 × 10 −3 , and w/o mobility cases at x = 38ρ s using Λ b = +64 volt which is shown in figure 7 and indicated by blue, orange, and green-colored solid and black-colored dashed lines, respectively.It is to be noted that x = 38ρ s has been chosen as the reversal of flow takes place at this point.A small increase in the magnitude of k y -PSD has been observed after the inclusion of the mobility of electrons and ions (figure 7).Still, an almost negligible shifting of the k y -PSD toward lower k y modes has been observed and modes peak around 5-8 k y .The shifting of the k y -PSD may be arising due to the ⃗ E × ⃗ B shear.Since, the reduction in the radial electric field shear may lead to an increment in the scale length of the turbulence or scale of coherent structures such as blobs.A recent experimental paper [27] reports that the increase in shearing rate via edge biasing decreases the blob size.This phenomenon may be attributed to the Doppler shifting of the turbulence but the detailed analysis is out of the scope of this paper.

Summary and conclusions
In this paper, the impact of the mobility of ions and electrons on the positive as well as negative edge biasing has been studied theoretically/numerically in the edge and SOL regions.A set of model equations has been derived by adding the drift that arises due to the mobility of ions and electrons to the conventional drift velocities such as ⃗ E × ⃗ B, polarization, and diamagnetic.The extended model permits simulation to higher applied biasing voltages than the previous model and thereby may be used to explain the results obtained from high-edge biasing experiments.The simulation results show that mobility reduces/increases the space charge in the case of positive/negative biasing and reduces/increases the radial electric field and its shear.Consequently, a reduction/enhancement in the plasma density and electron temperature has been observed for positive/negative biasing voltages.Simulation results for different cases of mobility show a monotonic decrease in space charge with an increase in mobility for positive biasing and vice-verse is true for negative biasing.Further, simulation results show that the effect of the mobility slightly shifts the turbulence towards lower k y , and it peaks around 5-8 k y for the positive biasing voltages.
The present work is not applicable directly in bigger tokamaks such as ITER or DEMO due to the high temperature in the boundary region.However, many present-day fusion devices use radio frequency (RF) antennae for auxiliary heating purposes.In various experimental devices [43], it has been found that the RF antenna generates rectified DC (direct current) potentials due to the presence of an RF sheath.This potential modifies the turbulence and leads to high heat and particle loads on the antenna.Therefore the present work may be relevant for estimating the heat and particle loads on the antenna and to study the modification of turbulence by the RF sheath-induced DC bias.

Figure 1 .
Figure 1.The evolution of density with time for different biasing voltages for Dµ = 2 × 10 −3 .The quantities are expressed in the normalized unit.⟨ f ⟩x,y represents the radial and poloidal averages of the variable f.

2 .
Subplots (a) & (d), (b) & (e), and (c) & ( f ) denote the radial profiles of the plasma density, plasma potential, and electron temperature, respectively with (w/) and without (w/o) mobility for Λ b = +64 & − 64 volts.⟨ f ⟩y,t represents the long-time and poloidal averages of f.The position of the biasing region and LCFS has been indicated by the shaded region and a vertical dashed line, respectively.

3 .
(a) & (c) and (b) & (d) represent the radial profile of the radial electric field (Ex) and its shear (E ′ x ) for different mobility cases and w/o mobility using Λ b = +64 & − 64 volts.The biasing region and the position of LCFS have been indicated by the shaded region and a vertical dashed line, respectively.It is to be noted that the legends for subplot (a) apply to other subplots, also.The modification in the radial profiles of ⟨n⟩ y,t and ⟨T e ⟩ y,t as shown in figures 2(a), (d) and (b), (e) due to the impact of the mobility can be explained from E ′ x .We have calculated E x and E ′ x using data of potential as shown in figures 2(c) and (e).The radial profiles of E x and E ′ x using Λ b = +64 & − 64 volts for D µ = 5 × 10 −4 , 2 × 10 −3 , 4 × 10 −3 , and w/o mobility have been presented in figures 3(a), (c) and (b), (d) that are indicated by blue, orange, and green-colored solid and

Figure 4 .
Figure 4. Subplots (Aa)-(A d ) & (Ca)-(C d ) and (Ba)-(B d ) & (Da)-(D d ) represent the snapshot of density for w/o and w/ mobility using Λ b = +64 & − 64 volts, respectively at four instants of time with an interval of 4000/Ωs.The position of biasing and LCFS have been indicated by the black and white colored dashed lines, respectively.
x − y plane for w/o and w/ mobility (D µ = 4 × 10 −3 ) using Λ b = +64 & − 64 volts has been shown in figures 4(A a )-(A d ) , (C a )-(C d ) and figures 4(B a )-(B d ), (D a )-(D d ), respectively at a time interval of 4000/Ω s .It can be seen in figures 4(A b ) and (A d ) that there is almost no poloidal variation in the plasma density which shows the stabilization of the turbulence at these instants of time, although poloidal variation can be seen in figures 4(A a ), and (A c ).It is to be noted that these two phenomena represent the quiet and burst phases of the turbulence [42].After the inclusion of the mobility, poloidal variation in density exists for all the instants of time, and the fluctuations in the plasma density have been found in figures 4(B a )-(B d ).This phenomenon indicates the modification in the turbulence by the inclusion of mobility by a decrease of the space charge by the nonambipolar mechanism, which increases the poloidal flows in the case of negative biasing.A reduction in the density fluctuations has been observed in figures 4(D a )-(D d ) for w/ mobility at each instant of time compared w/o mobility in the SOL region and vice-versa is true in the edge region.This shows that the inclusion of the mobility of electrons and ions in the model equations improves the effect of negative biasing by the increase in E ′

Figure 6 .
Figure 6.Particle flux (Γ) for w/ and w/o mobility using Λ b = +64 volt.The biasing region and LCFS are indicated by the shaded region and vertical dashed line, respectively.It is to be noted that these profiles are obtained from the poloidal and long-time averages.

Figure 7 .
Figure 7. ky-power spectral density (PSD) for different mobility cases and w/o mobility cases for Λ b = +64 volt at radial position x = 38ρs.It is to be noted that the variables are expressed in the normalized unit.