Chapter 2
Sawtooth Model
The goal of this thesis was to design a model-based controller for the sawtooth period in ITER. Therefore, a control-oriented model had to be developed in order to simulate the behaviour of this instability and estimate its period under different conditions. A model for this purpose is not required to describe precisely every detail of the plasma phenomena happening in a tokamak, and it is not even advisable to do so since that requires large amounts of computational power. This fact is even more relevant if a controller is to be designed with this model, because the respective code will have to be run over a large number of cycles. Additionally if the code includes too much details about the phenomena involved it becomes very hard to understand the causes for the observed output. Besides, the only information needed from the code to perform control is the sawtooth period, or in other words, the instants when each crash happen. This means that the main requisite to a control-oriented model is to be as fast, and simple as possible, while keeping a reasonable level of accuracy on the description of the response of the sawtooth period to the actuators.
A computational code exactly with this goal was developed by Witvoet for the TEXTOR tokamak [6], which we used as a starting point. It was adapted for ITER conditions, adding the necessary features including the ICRH actuator and anαparticle population. However the triggering criteria of the sawtooth crash implemented for TEXTOR is not applicable to ITER conditions. Therefore, we adopted the criteria used in another code developed by Lennholm [33]. He developed a much simpler code that although is not control-oriented, includes a crash triggering criteria that is more suited to plasmas with large populations of fast particles. This chapter gives a brief overview of these two codes or models and in the end summarizes the changes made to each one to obtain ours which we named extended Witvoet model.
2.1 Witvoet model
Witvoet made an heuristic model of the sawtooth period that takes into account the effects of the driven current given by the application of an Electron Cyclotron Current Drive (ECCD) source. A good descrip- tion of this model can be found in [29], however, a brief one will be given here.
The model uses only the magnetic diffusion equation to derive the evolution of the magnetic field
during each sawtooth cycle, keeping the electron and ion temperature constant during the full simulation.
At the same time, it is constantly using Porcelli’s criteria [7] which predicts from the plasma quantities, when the internal kink mode becomes unstable and a crash is triggered. The magnetic field configuration after a crash is determined using Kadomtsev’s full-reconnection model, taking into account the plasma conditions before the crash [34]. This implies that there is a certain memory inherent to the model, and the current sawtooth period will depend on previous cycles.
The model was originally adapted to TEXTOR conditions and some parameters tuned in order to obtain sawtooth periods comparable to the ones observed experimentally.
As already explained, the evolution of the simulation can be divided in three main events which are sequentially repeated:
1. Magnetic diffusion
2. Porcelli’s triggering criteria 3. Kadomtsev’s reconnection model
The way each of these events was implemented will be addressed next.
2.1.1 Magnetic field diffusion
When at the ramp phase the code evolves according to the diffusion of the magnetic field, which is governed by the following equation:
∂B
∂t =−∇× η
µ0∇×B−ηJCD
, (2.1)
whereBis the magnetic field,µ0the plasma permeability,JCD is the driven current by an ECCD source, andηSpthe Spitzer resistivity:
ηSp= 1.65×10−9ln ΛZeffTe−3/2, (2.2) where the Coulomb logarithmln Λ = 17and the effective ion chargeZeff= 1.7. For the electron temper- ature a parametrized profile is assumed given by the expression:
Te=T0
1 +ρ2qa−4/3
, (2.3)
whereρ=r/a,T0is the central temperature andqa the safety factor at the wall. Since the temperature profile is assumed constant during the whole simulation so will be the resistivity profile.
The code assumes toroidal and poloidal axis-symmetry which is to say that the tokamak is essentially a cylinder. Such approximation can only be considered accurate to some extension, especially valid for tokamaks with large aspect ratio ε=a/R. In these conditions the toroidal field is considered constant along the minor radius and equation (2.1) becomes an equation for the poloidal field which is only
function of the minor radius:
∂Bθ
∂t = ∂
∂r η
µ0r
Bθ+r∂Bθ
∂r
−ηJCD
. (2.4)
The boundary conditions are imposed by the Ampere’s law, which states that at the plasma centre Bθ(r= 0, t) = 0, and at the wallBθ(r=a, t) = µ2πa0Ip, whereIpis the total plasma current.
After a sawtooth crash the current profile is flattened inside the mixing radius and the magnetic field assumes a new shape that then evolves towards an equilibrium state which for a sawtoothing plasma is never reached. This process evolves with a time scale given by the current diffusion time. An example of the evolution of the current profile simulated with Witvoet’s code can be seen in figure 2.1.
time after crash(ms) 0.15 0.48 3.20 14.90
0.0 0.2 0.4 0.6 0.8 1.0
0 50 100 150 200 250 300 350
ρ j(kA/m2 )
Figure 2.1: Current profile during a full sawtooth cycle simulated with Witvoet model. A discontinuity emerges at the mixing radius after the crash due to the Kadomtsev full reconnection model discussed in section 2.1.3. It can be seen that this discontinuity fades quickly due to the diffusion of the magnetic field.
2.1.2 Crash triggering
The sawtooth crash is triggered by the onset of anm= 1mode, whose dynamics is influenced by many factors. In [7] Porcelli et al. summarised all the effects that influence the stability of this mode in a list of statements based on its potential energy,δW. A more convenient way to express this quantity is by its normalized version:
δWˆ = 4R0
s1ξr2r1B2δW, (2.5)
whereR0is the major radius of the tokamak,r1 the minor radius of theq = 1surface,Bthe magnetic field andξthe radial displacement of the magnetic axis.
Porcelli stated that the mode is meta-stable when this quantity becomes less than the normalized ion Larmor radiusρˆi=ρi/r1, and at the same time the ion diamagnetic frequencyω∗ibecomes less than the product of the mode’s growth rate by a scaling factorc∗γ. Since the mode’s growth rate depends directly ons1, the magnetic shear at theq= 1surface, this condition can be written in the form of a critical shear
condition:
s1> scrit (2.6)
where scrit is a critical value for the magnetic shear that depends on many factors, but for the sake of simplicity is assumed constant, and equal to0.133. If however, this second condition is not verified and the mode’s potential energy becomes negative and less than −0.5ω∗iτA, then the mode gets unstable and the crash will be triggered independently of the shear at theq= 1surface. For TEXTOR conditions however, the mode is always destabilized before this happens.
In summary, a crash will be triggered if one of the following criteria is verified
δW <ˆ −0.5ω∗iτA, (2.7)
−0.5ω∗iτA< δW <ˆˆ ρicρ and s1> scrit. (2.8) The work developed by Witvoet focuses only on a specific case of the kink mode, for which the left side of (2.8) is normally verified and the relevant test condition for the triggering of a crash therefore iss1> scrit.
2.1.3 Kadomtsev full reconnection model
Since the actual time scale of a crash is very short when compared to the entire cycle period and we are only interested in this last quantity, in Witvoet model is assumed that the crash happens infinitely fast, which is to say that the magnetic field configuration is changed instantly after the crash.
The shape of the magnetic field after the crash is determined from the one before, with the Kadomt- sev’s full reconnection model. According to this description, all the magnetic surfaces up to a point called the mixing radius reconnect, while conserving a quantity called the helical flux. This quantity is defined as
Ψ∗(r) = Z r
0
(Bθ(x)−xBφ/R0)dx, (2.9)
and the principle of flux conservation essentially states that only surfaces with the same helical flux reconnect. A direct implication of this statement is that the cross sectional area between the surfaces must also conserve like is depicted in figure 2.2.
Figure 2.2: Illustration of two reconnecting surfaces before (left) and after (right) the crash, showing the conserved area in yellow.
As a consequence if the inner and outer surfaces have minor radiusr1andr2respectively, and the post
reconnection surface has minor radiusrc, these must satisfyr2c =r22−r21andΨc∗(rc) = Ψ∗(r1) = Ψ∗(r2).
This rule is applied from the plasma centre to the mixing radius generating a new helical flux profile, whilst outside this radius the helical flux is kept unchanged. From this profile the magnetic field can easily be obtained by inverting equation (2.9). An inherent consequence of this model is the occurrence of a discontinuity on the poloidal field at the mixing radius, shortly after the crash. This yields an infinite current density that then diffuses out very quickly, as can be seen in figure 2.1. As a result also the safety factor after the crash has a discontinuity at the mixing radius as can be seen in the simulation result of figure 2.3. This picture also shows the quick diffusion around the mixing radius (ρ≈0.3) compared to
time after crash(ms) 0.1 1.3 6.8 14.9
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 1.00
1.05 1.10 1.15
ρ
q
Figure 2.3: Safety factor profiles during a full sawtooth cycle simulated with Witvoet model. Theq-profile after the crash is minimum atr = 0and equal to 1, therefore according to this model the radius of the q= 1surface is zero after the crash and then is constantly growing.
the rest of the profile.
With this rearrangement of the field the safety factor always becomes larger than 1 everywhere except atr= 0. Hence, as depicted in figure 2.3 we haverq=1 = 0ands1= 0shortly after the crash and so we conclude that, according to the condition (2.6) the kink mode is always stable after the crash.
2.1.4 Witvoet Model input and output
In Witvoet’s work the sawtooth period is controlled using the ECCD. This actuator drives additional current to the plasma in a very localized way and thus can increase or decrease the magnetic shear locally by driving either co or counter current. The crash can this way be artificially triggered by varying the deposition location of the ECCD relatively to theq= 1surface, and by changing the amount of total current drive.
The model receives as input the total plasma currentIp, the ECCD launcher mirror orientation angles and the total current driven with this actuator ICD. The deposition location is calculated taking into account the tokamak’s geometry, the plasma equilibrium conditions like the Shafranov shift and the mirror’s orientation. The driven current is modelled as a Gaussian distribution over the minor radius around the deposition location, with a certain width, which for TEXTOR is assumed to be 3.5% of the minor radius.
The instant when each crash happens is stored and the sawtooth period is calculated from it. This period is passed out as the output of the code. This means that the output is only updated after each crash, staying stationary during the rest of the cycle. Consequently, there will be a delay between the output of the code and the corresponding input used, since it is necessary to wait until a new crash is triggered to determine the sawtooth period. Hence, this delay is well determined: it is equal to the output itself and so, it can be compensated.
2.1.5 Implementation
Witvoet model was implemented in C-code and structured specifically to be used as an S-function on the MatlabR SimulinkR environment. The radial profile of the poloidal field is discretized in200points, using a denser distribution near the current mixing radius which provides a better description of the large diffusion happening there. As a consequence, the discretization grid is reset after each crash, and the temperature and diffusion profiles updated accordingly.
From the control point of view, the model is seen as a black box that receives the ECCD launcher mirror orientation and the total driven current, and outputs the sawtooth period. This approach allows to easily build and test all different types of controllers, by keeping all the complex physical phenomenon hidden in a diagram block.