Slab model for the ion mode and the barotropic equation

Florin Cadarache and Magurele

Abstract:This part introduces a simple model for the nonlinear ion mode and shows that it is possible to reduce it to the barotropic equation.

The slab model of the ion mode instability

We consider cylindrical geometry with circular magnetic surfaces. Locally the model can be reduced to a slab geometry with $\left( x,y\right)$ cartezian coordinates replacing respectively the radius and poloidal angle coordinates $(r,\theta )$ . At equilibrium the plasma parameters are constant on the magnetic surfaces. The effects of the toroidicity and of the particle drifts are not included instead the nonlinearity related to the ion polarization drift is fully retained. The plasma model consists of the continuity equations and the equations of motion for electrons and ions: MATH MATH where $\alpha =e,i$ . The friction forces , which are important for the parallel electron momentum balance, vanish for infinite plasma conductivity, which we will assume. The collisional viscosity will be neglected as well. However we will need to include it later when we will consider the balance of the forces contributing to the poloidal rotation. The electron and ion temperatures are considered constant along the magnetic lines . The equilibrium quantities are perturbed by the wave potential $\phi$ : . A sheared poloidal plasma rotation is included, and we later will make explicit the corresponding part in the potential, $\phi _{0}$ .

The momentum conservation equations are used to determine the perpendicular velocities of the electrons and ions. The parallel momentum conservation equation for electrons, in the absence of dissipation or drifts gives the adiabatic distribution of the density fluctuation. The velocities are introduced in the continuity equations to find the dynamical equations for the density and electric potential.

From the equations of motion for the ions the velocities are obtained in the form: MATH where the ion diamagnetic velocity is MATH The versor of the magnetic field is ; the versors along the transversal coordinate axis $\left( x,y\right)$ will be noted . The ion-polarization velocity is: MATH Using the notation the perpendicular ion velocity can be written MATH

The equation for the velocity of the electrons is MATH MATH

We assume neutrality $n_{e}=n_{i}=n$ and introduce the expressions of the velocities in the continuity equations for ions and for electrons.

The ion continuity equation is MATH

The electron continuity equation is MATH The equations are substracted , to obtain MATH From the last term in the left we get: MATH Including the similar term for the electrons, we obtain MATH or: MATH From the continuity equation for electrons MATH we obtain MATH where the seed poloidal velocity is . The parallel momenttum balance gives the parallel electron velocity MATH In the absence of friction and of particle drifts the electron response is adiabatic MATH and the potential is determined from Eq.(eq2). To develop separately the ion-polarization drift term, we introduce the notation: MATH where we make explicit the electric potential $\phi _{0}$ associated to the initial plasma poloidal rotation, . MATH We have MATH or MATH with the relations MATH MATH After very simple calculations we obtain: MATH and MATH It will be useful to calculate the derivaties of these quantities MATH and MATH

The quantity denoted by takes the form MATH

The mode evolution in a fixed plasma rotation profile

We will assume that the mode evolves initially without perturbing the equilibrium profiles, in particular the seed poloidal rotation. This allows us to simplify the expressions above, taking: MATH Then the first lines in the formulas Eqs.(ir), (didr), (didy) are zero. Let us consider in the expression of the part $W_{0}$ which does not contain the fluctuating density $\widetilde{n}$ . Writting MATH we have MATH and MATH Replacing the perturbed velocity by the perturbed potential, writting all terms and summing, we get: MATH Collecting all what we have at this moment the ion continuity equation (eq2) becomes: MATH

In the above equation (which is exact) we shall make the following approximations:

neglect the term containing the parallel electron velocity, assuming infinite electric conductivity;
neglect the term which contains $dn_{0}/dx$ since it is in the ratio $k:1/L_{n}$ with the other terms, and we consider
neglect $\widetilde{W}$ ; these terms are in the ratio with the terms which are retained;
neglect of the terms in the first lines of the expressions for $I_{x}$ and $dI_{x}/dx$ , $dI_{y}/dy$ . (These are the terms in the curly brakets, the last line). As explained above, we assume that the mode evolves in a background of fixed rotation profile, .

The resulting equation is MATH and, replace the adiabatic form of the density perturbation MATH We define MATH and obtain MATH which is the barotropic equation.

Nondimensional form of the equation

We consider that the ion mode extends in the spatial () direction over a length . A typical value for the sheared poloidal rotation is noted $U_{0}$ . We make the replacements MATH MATH MATH MATH such that from now on $y,t,\phi$ and are nondimensional quantities. We also change the radial coordinate into a dimensionless variable MATH and rewrite the equation MATH The coefficients are MATH MATH

For an order of magnitude, $\varepsilon$ is the ratio of the diamagnetic electron velocity to the rotation velocity $U_{0}$ multiplied by the ratio of the density gradient length to the length of the spatial domain. This quantity, $\varepsilon$ is in general smaller than unity.

The quantity $\beta ^{\prime }$ is the ratio of the ion cyclotron frequency to the inverse of the time required to cross the spatial domain with the typical flow velocity. Since the later involves macroscopic quantities this ratio can be large. It is multiplied by the ratio of the spatial length to the density gradient length (these quantities can be comparable and the ratio not too different of unity).

We change the notations eliminating the primes. The equation becomes MATH

This is the barotropic equation, known in the physics of the atmosphere.