Relation of plasma displacement with experimental measurements

In experiments, the beam emission spectroscopy (BES) and microwave reflectometer can measure electron density fluctuation. The electron cyclotron emission (ECE) radiometer can measure electron temperature fluctuation. In the ideal MHD theory, we assume that $n_{e 0} = n_{i 0}$, $n_{e 1} = n_{i 1}$, and the mass density is given approximately by $\rho_m = n_i m_i$. Then the linearized continuity equation, $\rho_1 = -\ensuremath{\boldsymbol{\xi}}_1 \cdot \nabla \rho_0 - \rho_0 \nabla \cdot \ensuremath{\boldsymbol{\xi}}_1$, is written as

$$n_{e 1} = -\ensuremath{\boldsymbol{\xi}} \cdot \nabla n_{e 0} - n_{e 0} \nabla \cdot \ensuremath{\boldsymbol{\xi}},$$ (307)

which gives relationship between the density fluctuation and plasma displacement. Similarly, in the ideal MHD theory, we assume that $T_{e 0} = T_{i 0}$, $T_{e 1} = T_{i 1}$, and $p = n_e T_e + n_i T_i = 2 n_e T_e$. Then the linearized equation of state, $p_1 = -\ensuremath{\boldsymbol{\xi}} \cdot \nabla p_0 - \gamma p_0 \nabla \cdot \ensuremath{\boldsymbol{\xi}}$, is written as

$$n_0 T_1 + n_1 T_0 = -\ensuremath{\boldsymbol{\xi}} \cdot (T_0 \na... ... + n_0 \nabla T_0) - \gamma n_0 T_0 \nabla \cdot \ensuremath{\boldsymbol{\xi}},$$ (308)

Using Eq. () to eliminate $n_1$, Eq. () is written as

$$\Longrightarrow n_0 T_1 -\ensuremath{\boldsymbol{\xi}} \cdot T_0 ... ...dot n_0 \nabla T_0 - \gamma n_0 T_0 \nabla \cdot \ensuremath{\boldsymbol{\xi}},$$ (309)

$$\Longrightarrow T_1 = -\ensuremath{\boldsymbol{\xi}} \cdot \nabla T_0 - (\gamma - 1) T_0 \nabla \cdot \ensuremath{\boldsymbol{\xi}},$$ (310)

$$\Longrightarrow \frac{T_1}{T_0} = -\ensuremath{\boldsymbol{\xi}} ... ...rac{\nabla T_0}{T_0} - (\gamma - 1) \nabla \cdot \ensuremath{\boldsymbol{\xi}},$$ (311)

The continuity equation is written as

$$\rho_1 = - \rho_0' \xi_{\psi} - p_0 \nabla \cdot \ensuremath{\boldsymbol{\xi}},$$ (312)

Neglecting the compressible term, the above equation is written as

$$\rho_1 = - \rho_0' \xi_{\psi} .$$ (313)

Using $\rho \approx n_i m_i$ and $n_i = n_e$, the above equation is written as

$$n_{e 1} = - n_{e 0}' \xi_{\psi} .$$ (314)

Equation (314) relates the radial displacement obtained from a eigenvalue code with the density perturbation $n_{e 1}$, which can be measured by the reflectometer in experiments.

We know that the radial plasma displacement $\xi_{\psi}$ is related to the perturbed thermal pressure through the relation:

$$p_1 = - p_0' \xi_{\psi} - \gamma p_0 \nabla \cdot \ensuremath{\boldsymbol{\xi}},$$ (315)

Neglecting the compressible term, the above equation is written as

$$p_1 = - p_0' \xi_{\psi}$$ (316)