Impurity transport and the Parallel momen- tum balance equation (PMBE) modication

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Boundary plasma concepts

Magnetic con nement in a tokamak does not prevent 100% of the plasma particles from escaping. Particles escape both in the direction parallel and perpendicular to the magnetic eld. While the radial escape of the plasma is rather slow, the parallel one is very e cient – particles escape with their thermal velocity, which in the case of the fusion plasma often creates heat ow su cient for damaging the plasma facing components. Another e ect of plasma surface interaction (PSI) is the contamination of plasma by impurity particles, sputtered from the material surface.
In order to limit both e ects: components damaging and plasma contamination in fusion devices, limiter and divertor con gurations were introduced.

Limiter and divertor con gurations

In the limiter and divertor con gurations the area inside the chamber becomes split in the con ned area (core) and boundary layer (SOL, scrape-o -layer). The magnetic surface between these two is called Last Closed Flux Surface (LCFS) or separatrix.
SOL consists of the so-called open magnetic eld lines. In reality these magnetic eld lines are still closed, but they intersect the material element, so the particles trajectories on these lines are not closed.

Limiter Con guration

The limiter con guration is a tokamak chamber con guration with a material ele-ment, physically limiting the plasma. Example of the limiter con guration on the ASDEX Upgrade tokamak is shown in Fig 2.1, where the plasma is limited by the HFS (High Field Side) heatshield [2].
In the case of limiter con guration there is no strong separation between open and closed magnetic surfaces. This leads to two major drawbacks: direct contact of Plasma density outside of the LCFS in the limiter con guration decreases mono-tonically. Most of the particles reach the limiting structure by fast transport parallel to the magnetic eld before they can reach the wall by the slower radial transport. The width of the SOL is determined by the balance of the parallel and radial trans-port [22].

Divertor con guration

A more advanced plasma limiting con guration is called a divertor. In this con g-uration, separation of the closed and the open eld lines is done through speci c shaping of the con ning magnetic eld.
In order to separate the con ned region additional currents are introduced creating a point (or a number of points) in which the poloidal magnetic eld is equal to zero (X-point). In this con guration the core plasma is con ned without direct proximity to the material surfaces.
The separatrix contacts the special areas of the magnetic chamber, called divertor targets; the points of this contact are called strike points. They become the most intense Plasma Surface Interaction (PSI) area.


The most studied type of the divertor con guration has one X-point in the bottom of the device. This con guration is called Lower Single Null (LSN) and is intensively studied on many tokamaks including ASDEX Upgrade.
An example of the ASDEX Upgrade LSN magnetic con guration is shown on the right of Fig 2.1.
In Fig 2.1 detailed designations of all the elements of the magnetic geometry and chamber structure of ASDEX Upgrade are given. To discuss the tokamak SOL it is important to de ne the equatorial midplane (midplane in the Fig 2.1) – a horizontal plane in the poloidal cross section at the height of the magnetic axis. The outer midplane (point of = 0 in Fig 2.1) is often taken as a reference « upstream point » in the discussion of power and particle transport.
The region of open eld lines is called the SOL (orange region in the Fig 2.1), apart from the region of the open eld lines between the two legs of the separatrix. This region (marked green in the Fig 2.1) is called the private ux region (PFR).
It is common to call the region of the SOL, located poloidally above the X-point the upstream region (main SOL), and the region below the X-point is usually called the divertor region.
For ITER a LSN divertor con guration is planned due to the simplicity of this con guration. More complicated magnetic con gurations (double null con guration, snow ake divertor, etc.) are less studied. Another reason to keep the ITER divertor con guration simple is the need to protect all additional magnetic structure from neutron uxes, which is simpler in the LSN con guration.

Sheath in the tokamak SOL

At the plasma-surface interface in front of the divertor target speci c arrangement of the electric potential occurs. The closest to the target region of this formation is called the sheath. The structure of this layer is shown in Fig. 2.2 . It is important to discuss this layer since it sets the ion parallel velocity in the SOL to the ion sound speed, cs = emi i . This condition should be satis ed in the vicinity of the divertor targets.
Due to the much lower mass of the electron in comparison with ions they travel to the surface faster. This causes a negative charge of the surface and a potential builds up, which slows down the electrons and speeds up the ions. The plasma quasineutrality condition in this layer is violated and the width of this layer is equal to the Debye radius. The thickness of the sheath layer is de ned by the scale on which it is possible to violate plasma quasineutrality condition. For the sheath layer the Bohm criterion exist; velocity of plasma ions on the sheath entrance should be larger than the local sound speed.
Due to the presence of the large toroidal magnetic eld in the tokamak plasma the incidence angle of the plasma particles to the target can be very far from the normal direction. In this case a so called magnetic pre-sheath layer is formed. This layer is quasineutral but it has the same condition at the entrance: ion velocity should be larger or equal than the sound speed.
Since the bulk velocity in general cannot be higher than the sound velocity, in the simple approximation the plasma velocity in the SOL is assumed to be equal to the sound speed. This boundary condition for ion velocity is applied on both divertor targets.

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Divertor asymmetries

Inner and outer divertor targets (located correspondingly on the HFS and the LFS of the divertor), Fig 2.1, can feature signi cantly di erent plasma properties on them. Usually the plasma density is higher at the inner (HFS) target, and the temperature at that target and the heat ux, received by it, is lower.
Reasons for the divertor power asymmetries:
Magnetic ux surfaces are more compressed on the LFS compared to the HFS. This is caused by the Shafranov shift – displacement of the center of the mag-netic equilibrium outwards from the center of the tokamak. This introduces stronger gradients of the plasma parameters at the LFS [22]. If the cross eld transport is assumed to be proportional to the gradients strength, this results in even more power being deposited on the outer target.
Ballooning type radial transport [23] is present only on the LFS of the tokamak due to the unfavorable curvature there. This ampli es the amount of the power deposited on the outer target;
ExB drifts through the private ux region can signi cantly redistribute the particles from the outer to the inner divertor target.
In order to explain the asymmetry in the particle distribution, cross eld drifts have to be introduced.

Drifts in the tokamak

Larmor motion

In order to discuss the motion of the particles in the tokamak it is important to understand the motion of the charged particles in a homogeneous magnetic eld. The equation of motion of one charged particle in a magnetic eld is as follows:
Total energy of the motion in the direction parallel to the magnetic eld is also constant, since the only force, acting on the particle, is perpendicular to this direc-tion.
In order to connect the absolute value of the perpendicular velocity with the radius of the circle in the circular motion, one takes into account that the centrifugal force of the circular motion has to be balanced by the Lorentz force:
Summing up the information about the Vjj and V?, the particle motion in a con-stant magnetic eld can be described as the spiral motion – motion with the constant velocity along the magnetic eld and circular motion in the plane perpendicular to the magnetic eld.

Drifts of the particles in the magnetic eld

For any force F, acting on the particle in the homogeneous magnetic eld the drift velocity corresponding to this force can be derived. In order to do so one has to consider the equation of motion for this particle:
The component of the velocity, perpendicular to the magnetic eld can be split in the constant drift velocity and the Larmor motion velocity:
Here Larmor motion contribution cancels the cross product part in the right hand side of the equation of motion (2.7) and the time derivative in the left hand side. For the drift contribution the following equation is obtained:
Making the cross product of this equation with the magnetic eld and taking into account expression [[A B] B] = B2A one obtains the following expression for the drift velocity:

ExB drifts

Due to the presence of an electric eld in the tokamak plasma particles experience drifts, which are called ExB drifts. Velocity of these drifts can be calculated by eq. 2.11 and since the electric force, acting on the charged particle is F = zaeE the ExB Since the drift velocity VExB is independent of the particle charge or mass, the ExB drift does not cause charge separation (because electrons and ions drift in the same direction) and does not cause any plasma current.
The ExB drifts can be split in the two categories: drifts due to the radial electric eld and drifts due to the poloidal electric eld. Resulting drift uxes would be directed poloidally for the radial electric eld and radially for the poloidal electric eld.
According to the recent SOLPS-ITER modeling studies, supported by the the-oretical explanations [4] poloidal drift ux though the private ux region is one of the main contributor in the divertor densities asymmetry, discussed above.
Radial electric elds in the boundary plasmas are usually formed due to the presence of the temperature and density gradients.
Figure 2.4: An example of the complicated pattern of the ion ow in the SOL. ExB drift which moves particles from the outer to the inner target is shown in green. Reproduced from [4]

Table of contents :

1 Introduction 
2 Boundary plasma concepts 
2.1 Limiter and divertor configurations
2.1.1 Limiter Configuration
2.1.2 Divertor configuration
2.2 Sheath in the tokamak SOL
2.3 Divertor asymmetries
2.4 Drifts in the tokamak
2.4.1 Larmor motion
2.4.2 Drifts of the particles in the magneticeld
2.4.3 ExB drifts
2.4.4 rB drifts
3 Transport equations 
3.1 Kinetic approach to the plasma description
3.2 Fluid approach to the plasma description. Moments of the distribution function
3.3 Momentum equations
3.4 Momentum equations closure
4 SOLPS-ITER code 
4.1 Geometry
4.2 Continuity equation
4.3 Parallel momentum balance equation
4.4 Heat balance equation
4.5 Charge conservation equation
4.6 Boundary conditions
4.6.1 The core boundary
4.6.2 The tokamak wall boundary
4.6.3 The divertor targets boundary
5 Impurity transport and the Parallel momen-tum balance equation (PMBE) modication 
5.1 Old form of the PMBE. Limiting assumptions
5.2 Braginskii form of the PMBE
5.3 Thermal and friction force terms for the Braginskii form of the PMBE
5.4 Eect of the thermal and friction force corrections in the Braginskii form of the PMBE
5.5 Derivation of the corrected thermal and friction force terms for the Braginskii form of the PMBE
5.5.1 Collision times
5.5.2 Final form of the friction and thermal force terms
5.5.3 The parallel current modification
5.6 Comparison of the old form of the PMBE with the Braginskii form with corrected terms
6 Modeling results 
6.1 Modeling setup
6.2 Modeling results
7 Impurity transport 98
7.1 SOL velocity structure in the SOLPS-ITER modeling. Main ions reverse
7.2 Impurity ions velocity
7.3 Impurity retention and leakage
7.4 Radiative patterns of Ne and N in ASDEX Upgrade and ITER modeling
8 Conclusion 
8.1 Improvement of the PMBE
8.2 Impurity retention and leakage mechanism
8.3 Nitrogen and Neon retention and leakage
8.4 Nitrogen and Neon radiation patterns in ASDEX Upgrade and ITER
SOLPS modeling
8.5 Outlook


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