By Hartmut Zohm

This publication bridges the distance among normal plasma physics lectures and the true international difficulties in MHD balance. for you to help the certainty of thoughts and their implication, it refers to actual international difficulties comparable to toroidal mode coupling or nonlinear evolution in a conceptual and phenomenological method. distinctive mathematical therapy will contain classical linear balance research and an summary of more moderen suggestions comparable to the ballooning formalism.

The e-book is predicated on lectures that the writer has given to grasp and PhD scholars in Fusion Plasma Physics. Due its robust hyperlink to experimental ends up in MHD instabilities, the booklet is usually of use to senior researchers within the box, i.e. experimental physicists and engineers in fusion reactor science.

The quantity is geared up in 3 components. It begins with an creation to the MHD equations, a piece on toroidal equilibrium (tokamak and stellarator), and on linear balance research. ranging from there, the perfect MHD balance of the tokamak configuration may be taken care of within the moment half that is subdivided into present pushed and strain pushed MHD. This comprises many examples near to experimental effects for vital MHD instabilities resembling kinks and their transformation to RWMs, infernal modes, peeling modes, ballooning modes and their relation to ELMs. ultimately the insurance is done by means of a bankruptcy on resistive balance explaining reconnection and island formation. back, examples from contemporary tokamak MHD reminiscent of sawteeth, CTMs, NTMs and their relation to disruptions are generally mentioned.

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60) Ψvac,1 = ????0 Ip r ???????? ???????? r 2 r which is an inhomogeneous diﬀerential equation for Ψvac,1 . 57) gives the two conditions, B????,plasma (a, ????) = B????,vac (a, ????) and Br (a, ????) = 0. The latter means that the circle at r = a is a ﬂux surface, that is Ψvac,1 (a) = 0. 64) 2 2 a Inserting these into Eq. 62) gives the ﬂux function for r ≥ a. 8 Addition of a vertical ﬁeld Bv to the ﬁeld Bp,plasma generated by the plasma current alone provides force balance and leads to the Shafranov shift as can be seen from the resulting ﬁeld Bp,tot .

12 shows the corresponding ﬁeld lines resulting from such a current distribution in the ???? − ???? plane for ???? = 2, M = 4. 12 Map of ﬁeld lines generated by helical windings according to Eq. 87) plotted in the ????, ????-plane (???? = 2, M = 4). 37 38 2 MHD Equilibria in Fusion Plasmas The conﬁguration of a ‘classical’ stellarator consists of an axisymmetric toroidal ﬁeld and helical windings as discussed earlier. Depending on ????, the q-proﬁles12) can vary appreciably. For example, the shear in an ???? = 2 stellarator is much smaller than that in a tokamak, whereas an ???? = 3 stellarator has a shear comparable to that of a tokamak, but the q-proﬁle is reversed, that is q decreases from the centre to the edge.

6 introduces the coordinates we will use for the calculation of general axisymmetric equilibria, which are cylindrical coordinates oriented such that the azimuthal angle ???? represents the ignorable toroidal coordinate, whereas the poloidal plane is decomposed in R and Z. Following the discussion in the previous section, we can freely chose the integration surface to calculate the poloidal ﬂuxes and we use a circular surface oriented such that its surface normal points in Z-direction. 23) 5) To prove this, use Gauss’ theorem for the sum of the integrals of two arbitrarily chosen surfaces bounded by the same curve C.