Equivalence of pressure and geometric information

At this juncture, we digress in order to illustrate an interesting point concerning the question of the equivalence of pressure and geometry information over the database. For the tokamak, it has been shown [65] that knowledge of the flux surface geometry is sufficient for determination of the current profile in the non-circular case. Here, the current profile is $J = R\frac{\partial p}{\partial\Psi} + \frac{1}{\mu_0 R} F\frac{\partial F}{\partial\Psi}$, where $p(\Psi)$ is the plasma pressure as a function of the poloidal stream function $\Psi = 1/2\pi$ times the poloidal flux) and $F(\Psi) = \mu_0 I_\mathrm{pol}/2\pi$ is proportional to the poloidal current $I_\mathrm{pol}$.

In our database of 3-D equilibria, we have no separate current profile (the plasma current $I_{\mathrm{pl}}(s) = 0$ for all $s$) and therefore, by analogy, knowledge of the geometry should be equivalent to knowledge of the pressure profile. This may seem surprising, since our analysis thus far (comparing tables 5.1 and 5.2) shows the geometry to be far more robustly recoverable than the pressure profile, and although knowledge of the pressure confers knowledge of the geometry, the reverse would appear to be untrue. However, similarity in the reduction in error of both the pressure profile and the flux surface geometry for a model comprising the baseline vacuum configuration parameters and the idealized magnetic measurements, relative to the error for a model consisting of the baseline alone, would be consistent with such an equivalence. Normalizing to the baseline model error allows us to examine the influence on the equilibrium due only to the plasma. If our assertion holds true, then any improvement in identification of the geometry for a particular model must be accompanied by a similar improvement in the pressure and vice-versa.

Firstly, we choose quantities to describe an ``overall'' recovery quality of both the pressure and geometry. The choice for $p(s)$ is relatively obvious, since we have already defined such a quantity, $\langle \delta_\mathrm{p} \rangle$, in equation 5.1 above. A corresponding number that is representative of the average flux surface geometry recovery is the RMS value of the RMSE for each of the set of skeleton Fourier coefficients (which largely determine the geometry) quoted in table 5.1. We denote this by $\langle \delta_\mathrm{geo} \rangle$, thus:

$$\langle \delta_\mathrm{geo} \rangle = \sqrt{ \frac{1}{n} \sum_{i=1}^n (\mathrm{RMSE}_i)^2},$$ (5.2)

where $n$=23 and the sum is carried out over the set of Fourier coefficients listed in table 5.1. We can now compare the reductions in $\langle \delta_\mathrm{p} \rangle$ and $\langle \delta_\mathrm{geo} \rangle$ from the baseline model to that including the magnetic information.

With just vacuum information, the pressure is undetermined, since it is (as it should be) completely decorrelated with the vacuum parameters due to the way in which the database was generated. The RMS recovery error (which is thus the same as the spread) averaged across the profile is quoted in the caption of figure 5.2, as $\langle \delta_\mathrm{p} \rangle$ = 2.63 kPa. Using 9 PCs of the dense measurements with 1% noise, this drops to 0.34 kPa. The overall geometric error with just the baseline model is $\langle \delta_\mathrm{geo} \rangle$ = 3.05 mm; whereas for the baseline model and 9 PCs of the dense measurements with 1% noise it reduces to 0.31 mm. The comparison is then:

$$\frac{{\langle \delta_\mathrm{p} \rangle}_\mathrm{9PC}}{{\langle ... ...rangle}_\mathrm{baseline}} = \frac{0.31~\mathrm{kPa}}{2.63~\mathrm{kPa}} = 0.12$$ (5.3)

$$\frac{{\langle \delta_\mathrm{geo} \rangle}_\mathrm{9PC}}{{\langl... ...rangle}_\mathrm{baseline}} = \frac{0.031~\mathrm{cm}}{0.305~\mathrm{cm}} = 0.10$$ (5.4)

The improvement in identification, or equivalently the reduction factor of the overall recovery error, of the pressure and flux surface geometry are very similar, considering our ad hoc definitions to gauge the recovery. To directly show the symmetry of geometric and pressure information, we endeavour show that the pressure profile can be recovered well from just the geometric information alone, i.e. the flux surface Fourier coefficients. The solution of this problem on a database of limited size such as ours is not straightforward if we are to maintain statistical rigour, as we shall now show.