Electron Cyclotron Heating

Electron cyclotron resonance heating is the simplest of the radio frequency heating methods. In contrast to ion cyclotron and lower hybrid heating, there is no evanescent region between the antenna and the plasma,although cut-offs can exist within the plasma. As a result, the antenna can be retracted into a less hostile environment than for the other schemes. Electron cyclotron heating has been made possible by the invention of the gyrotron millimetre wave source. This and related devices have only emerged since the mid-1970s. Gyrotron tubes capable of 0.5-1 MW out- put and with frequencies in the range 100-200 GHz are now under intense development. The frequency required for a reactor would be in the range 100-200 GHz which corresponds to vacuum wavelengths of 1-2 mm. Since $\omega_{ce}\le \omega_{pe}$ only the electrons respond to electron cyclotron waves and only the electrons are heated directly. However, under reactor- like conditions the ions will be heated collisionally by the electrons. As the density is increased, a limit is encountered above which electron cyclotron waves cannot penetrate to the central regions of a tokamak. Propagation is again described below

$n_{\perp}^2 = 1 - \frac{\omega_{pe}^2}{\omega^2} \ \ (\text{O-mode})$

$n_{\perp}^2=\frac{(1-\frac{\omega_{pe}^2}{\omega^2}-\frac{\omega_{ce}}{\omega})(1-\frac{\omega_{pe}^2}{\omega^2}+\frac{\omega_{ce}}{\omega})}{(1-\frac{\omega_{pe}^2}{\omega^2}-\frac{\omega_{ce}^2}{\omega^2})} \ \ \text{(X-mode)}$

Normalised current drive efficiency | `eccdef()`

One of the methods for calculating the normalised current drive efficiency is the eccdef method found below.

Input	Description
$\mathtt{tlocal}$	Local electron temperature (keV)
$\mathtt{epsloc}$	Local inverse aspect ratio
$\mathtt{zlocal}$	Local plasma effective charge
$\mathtt{cosang}$	Cosine of the poloidal angle at which ECCD takes place (+1 outside, -1 inside)
$\mathtt{coulog}$	Local coulomb logarithm for ion-electron collisions

This routine calculates the current drive parameters for a electron cyclotron system, based on the AEA FUS 172 model. It works out the ECCD efficiency using the formula due to Cohen quoted in the ITER Physics Design Guidelines : 1989 (but including division by the Coulomb Logarithm omitted from IPDG89).

We have assumed $\gamma^2$ -1 << 1, where gamma is the relativistic factor. The notation follows that in IPDG89. The answer ECGAM is the normalised efficiency $n_{\text{e}}IR/P$ with $n_{\text{e}}$ the local density in [ $10^{20} / \text{m}^3$ ], I the driven current in [ $\text{MA}$ ], $R$ the major radius in [ $\text{m}$ ], and $P$ the absorbed power in [ $\text{MW}$ ].

$\mathtt{mcsq} = 9.1095\times10^{-31} \frac{c^2}{1 \text{keV}}$

$\mathtt{f} = 16\left(\frac{\mathtt{tlocal}}{\mathtt{mcsq}}\right)^2$

$\mathtt{fp}$ is the derivative of $\mathtt{f}$ with respect to gamma, the relativistic factor, taken equal to $1 + \frac{2T_{\text{e}}}{(m_{\text{e}}c^2)}$

$\mathtt{fp} = 16 \left(\frac{\mathtt{tlocal}}{\mathtt{mcsq}}\right)$

$\mathtt{lam}$ is IPDG89's lambda. legend calculates the Legendre function of order $\alpha$ and argument lam, palpha, and its derivative, palphap. Here alpha satisfies $\alpha(\alpha+1) = \frac{-8}{(1+z_{\text{local}})}$ . $\alpha$ is of the form $(-1/2 + ix)$ , with x a real number and $i = \sqrt{-1}$ .

$\mathtt{lam} = 1.0$

$\mathtt{palpha, palphap} = \text{legend}(\mathtt{zlocal, lam})$

$\mathtt{lams} = \sqrt{\frac{2 \times \mathtt{epsloc}}{1 + \mathtt{epsloc}}}$

$\mathtt{palphas} = \text{legend}(\mathtt{zlocal, lams})$

$\mathtt{hp}$ is the derivative of IPDG89's \mathtt{h} function with respect to $\mathtt{lam}$

$\mathtt{h} = \frac{-4 \times \mathtt{lam}}{\mathtt{zlocal + 5}} \times \frac{1- (\mathtt{lams \times \mathtt{palpha}})}{\mathtt{lam \times \mathtt{palphas}}}$

$\mathtt{hp} = \frac{-4}{\mathtt{zlocal}+5} \times \left(1- \mathtt{lams} \times \frac{\mathtt{palphap}}{\mathtt{palphas}} \right)$

$\mathtt{facm}$ is IPDG89's momentum conserving factor

$\mathtt{facm} = 1.5$

$\mathtt{y} = \frac{\mathtt{mcsq}}{2 \times \mathtt{tlocal}} \times (1+ \mathtt{epsloc} \times \mathtt{cosang})$

We take the negative of the IPDG89 expression to get a positive number

$\mathtt{ecgam} = \left(\frac{-7.8 \times \mathtt{facm \times \sqrt{\frac{1 + \mathtt{epsloc}}{1- \mathtt{epsloc}}}}}{\mathtt{coulog}}\right) \times (\mathtt{h} \times \mathtt{fp} -0.5 \times \mathtt{y} \times \mathtt{f} \times \mathtt{hp})$

Legendre function and its derivative | `legend()`

Input	Description
$\mathtt{zlocal}$	Local plasma effective charge
$\mathtt{arg}$	Argument of Legendre function

The legend() function is a routine that calculates the Legendre function and its derivative. It takes two input parameters: zlocal (local plasma effective charge) and arg (argument of the Legendre function). The function returns two output values: palpha (value of the Legendre function) and palphap (derivative of the Legendre function).

Here is the explanation of the legend() function:

Check if the absolute value of arg is greater than 1.0 + 1.0e-10. If it is, set eh.fdiags[0] to arg and report an error (error code 18).
Set arg2 to the minimum value between arg and 1.0 - 1.0e-10.
Calculate sinsq as 0.5 * (1.0 - arg2).
Calculate xisq as 0.25 * (32.0 * zlocal / (zlocal + 1.0) - 1.0).
Initialize palpha to 1.0, pold to 1.0, pterm to 1.0, palphap to 0.0, and poldp to 0.0.
Start a loop that iterates up to 10000 times.
Check for convergence every 20 iterations:
- If n > 1 and (n % 20) == 1, calculate term1 as 1.0e-10 * max(abs(pold), abs(palpha)) and term2 as 1.0e-10 * max(abs(poldp), abs(palphap)).
- If the absolute difference between pold and palpha is less than term1 and the absolute difference between poldp and palphap is less than term2, return palpha and palphap.
Update pold to palpha and poldp to palphap.
Calculate pterm as pterm * (4.0 * xisq + (2.0 * n - 1.0) ** 2) / (2.0 * n) ** 2 * sinsq.
Update palpha as palpha + pterm.
Update palphap as palphap - n * pterm / (1.0 - arg2).
If the loop completes without returning, report an error (error code 19).

Abramowitz, Milton. "Abramowitz and stegun: Handbook of mathematical functions." US Department of Commerce 10 (1972). ↩

Electron Cyclotron Heating

Normalised current drive efficiency | eccdef()

Legendre function and its derivative | legend()

Normalised current drive efficiency | `eccdef()`

Legendre function and its derivative | `legend()`