Description

In addition to the experimentally known external coil currents and limiter position, the foregoing predictive FP analysis assumes knowledge of $p(s)$, the equilibrium pressure profile as a function of normalized toroidal flux, which is known only when the equilibrium problem itself has already been solved. Any scheme to find $p(s)$ from experimental measurements of electron and ion temperatures and densities inevitably involves guessing the spatial to flux transformation, clearly leading to some degree of approximation. We prefer to avoid making any assumptions, however, and instead adopt an interpretive approach to implicitly determine $p(s)$. Here, the PCs of $p(s)$ are iteratively varied in reconstructions to find the best matching FP-derived equilibrium for the considered experimental data, in this case Thomson scattering data of electron temperature and density.

The Thomson scattering diagnostic on W7-AS [49] currently measures electron temperature $T_{\mathrm{e}}$ and density $n_{\mathrm{e}}$ (and thus the electron pressure ${p_{\mathrm{e}}}$) on up to 20 channels at a single time-point during a discharge. The measurement is performed along a horizontal line-of-sight through the magnetic axis in a symmetry plane ($\phi$=0) and is especially useful in that it yields a complete ${p_{\mathrm{e}}}$ profile covering the entire mid-plane diameter of the plasma. A least-squares spline in $R$ is fitted to ${p_{\mathrm{e}}}$, which both smoothes the data and allows continuous evaluation between channels. The $\chi^2$ goodness of fit test is used to select the optimum degree of smoothing and extract the maximum possible information from the data, whereby the relative importance of points in the fit scales inversely with their associated errors.

Our iterative procedure attempts to identify the spatial to flux transformation implicit in ${p_{\mathrm{e}}}(R)$, assuming that ${p_{\mathrm{e}}}$ is constant on flux surfaces. Starting from an initial guess, a number of PCs of the equilibrium pressure profile $p(s)$ is varied to minimize a cost function. This consists of the sum of squared differences between the electron pressure evaluated on the high and low field radii on the same flux surface ( ${R_{\mathrm{in}}}(s)$ and ${R_{\mathrm{out}}}(s)$ respectively), normalized to the global maximum of ${p_{\mathrm{e}}}(R)$, i.e. the quantity:

$$\frac{1}{\mbox{max}({p_{\mathrm{e}}})^{\,2}} \int_0^1 \Big( ... ...in}}}(s)] - {p_{\mathrm{e}}}[{R_{\mathrm{out}}}(s)] \Big)^2 ds$$ (4.1)

subject to the constraint that $p(s)$ monotonically decreases from centre to edge. It is worth emphasizing that this procedure does not hinge on the absolute measured electron pressure profile being true; it merely uses the profile to identify inboard and outboard points of equal flux along the line-of-sight and thus produces correct results as long as the topology of ${p_{\mathrm{e}}}(R)$ is accurate.

Extensive testing of interpretations based solely on the condition of equation 4.1 often resulted in badly behaved interpreted $p(s)$ profiles (e.g. containing multiple turning points or even negative values), despite the fact that the asymmetry was minimized. This indicated the need for further regularization of the shape of $p(R)$, the equilibrium pressure profile as a function of major radius, which was mildly constrained to reflect that of ${p_{\mathrm{e}}}(R)$. This is quite reasonable, since in most experimental scenarios we expect the ion pressure profile not to introduce new features into the pressure profile; thus the shape of $p(R)$ is largely given by ${p_{\mathrm{e}}}(R)$. The extra regularization produced profiles similar to those of standard calculations without undue raising of the asymmetry and also proved helpful in cases with poor input data.

A standard nonlinear least-squares minimization routine E04FCF from the NAG library is used to perform the parameter variation. Note that due to the above normalization, the interpreted fit depends only on the topology of the ${p_{\mathrm{e}}}(R)$ profile and not on its magnitude.

In fact, we are not exclusively limited to minimizing the asymmetry in the ${p_{\mathrm{e}}}$ profile, but could also do this separately for the $n_{\mathrm{e}}$ or $T_{\mathrm{e}}$ profile. We discuss these possibilities later in section 4.3.3 and proceed using ${p_{\mathrm{e}}}$ for now.

The figure of merit for the quality of the equilibrium simulation is taken to be the residual asymmetry in ${p_{\mathrm{e}}}(R)$, i.e. the difference between the ${p_{\mathrm{e}}}(R)$ fit evaluated on inboard and outboard points of equal flux along the Thomson line-of-sight. This quantity vanishes for an equilibrium that is perfectly consistent with the Thomson data. An additional consistency check on the interpreted equilibrium is that the ion pressure, $\pi(R) = p(R) - {p_{\mathrm{e}}}(R)$, be everywhere positive inside the plasma boundary.

The computed kinetic energy content can be optionally forced to match that derived from the diamagnetic flux measurement: this can stabilize the procedure when the input data is of poor quality, since it fixes the volume integral of the equilibrium pressure profile. The procedure generally lends itself well to adaptation, since matching additional profile or global data merely requires adding appropriate residuals to the cost function.

However, mismatches between the theoretical and physical magnetic configurations could falsify both the present scheme and the standard NEMEC calculations. This might arise due to differences between the actual geometry of the magnetic field coils and that used in the code. Taking this into account, however, would be most difficult and is outside the scope of this paper. Consistency checks with additional spatially resolved or global diagnostic data would reveal such discrepancies, if present. The impact of systematic errors such as these are investigated later.

Error bounds on the interpreted pressure profile can be calculated from the $N \times N$ variance-covariance matrix V of the $N$ fitted parameters $\alpha_i, i = 1 \ldots N$ as follows: if $y$ is any parameter of the interpreted equilibrium and $\nabla _{\bar{\alpha}} \, y$ is the gradient of $y$ with respect to the $\alpha_i$, then the standard error in $y$, $\Delta y$, is given by:

$${(\Delta y)}^{\ 2} = \nabla _{\bar{\alpha}}^{T} \,y \cdot \mbox{\bf V} \cdot \nabla _{\bar{\alpha}}^{} \, y$$ (4.2)

This error is specific to the choice of regression model and is useful for identifying the correct choice of model size. In particular, if the error bounds are very large, this indicates that the interpreted $p(s)$ is essentially undetermined and further regularization needs to be applied.