^{*}

The Plasma internal energy is not conserved on a magnetic surface if nonlinear flows are considered. The analysis here presented leads to a complicated equation for the plasma internal energy considering nonlinear flows in the collisional regime, including viscosity and in the low-vorticity approximation. Tokamak equilibrium has been analyzed with the magnetohydrodynamics nonlinear momentum equation in the low vorticity case. A generalized Grad–Shafranov-type equation has been also derived for this case.

The linear treatment of the equilibrium equation in tokamaks was carried out by the Russians and result is the Grad-Shafranov equation [1,2]. The poloidal flux solution of the Grad-Shafranov equation determines the magnetic surface in the case when viscosity and nonlinear convective terms are neglected. In this case isobars and magnetic surface are coincidents [3,4]. However if the preceding terms are not neglected, it is not easy to find out a differential equation for the magnetic surface. In this paper, this problem has been treated and a differential equation for the magnetic surface has been found when vorticity is neglected. In the usual Grad-Shafranov equation the internal energy of the plasma does not appear, but in the present case internal energy appears as a quantity to be determined.

The time independent MHD momentum equation including viscosity and non-linear convective terms is [

where, , and are the velocity, current density, and kinematic viscosity coefficient (assumed constant and isotropic), respectively. Here the anisotropic part of the pressure tensor is also neglected. Using the vorticity

the first and last terms in Equation (1) can be written in a more convenient way as

The temperature can also be assumed to be constant along a magnetic line [

where refers to a gradient in the plane of a magnetic surface, giving

where the integral in Equation (6) is performed with T constant along a magnetic line [

where is the flux function and is 1 for an isotherm process or 5/3 for the adiabatic case [

Now Equation (1) becomes

An auxiliary function can be now be defined as

and the equilibrium equation will be written as

Considering now the low vorticity case, that is, is a perturbation, then the low limit level will be with. Then the previous equation becomes,

As in the linear case, the procedure to derive the GradShafranov equation can be followed obtaining an extended Grad-Shafranov equation

where is the same kind of invariant as in the linear case and the operator is

The internal energy in this extended Grad-Shafranov equation is a function of. Since F is only a function of, and is function of then Equation (12) can be written as

That is

Since for the ideal MHD equilibrium confinement the internal energy and magnetic surfaces are coincident, then

From [

and because of the axissymmetry condition

Similarly from [

Then it leads to new surface invariant

From the [

From the plasma pressure equilibrium equation [

If we put together Equations (19), (21) , and (22) we obtain

Experimental observations show that neutral beam injection and rf heating induces poloidal and toroidal plasma rotations in tokamaks. The analysis of plasma equilibrium performed by several authors [3-11] is much more complicated than those of plasma confinement with no rotation. The Grad–Shafranov equation has to be analyzed coupled with a Bernoulli-type equation and furthermore there are regions where that equation is of hyperbolic type instead of elliptic [

A simplified equilibrium analysis in tokamak has been performed for the nonlinear momentum equation with viscosity in the low vorticity case. Internal energy is not constant now on magnetic surfaces, but our analysis shows that other significant magnetic surface new invariant appears, which are useful to determine equilibrium conditions. An extended Grad-Shafranov (GS) -type equation has been derived in this case. This new equation includes the usual invariant depending on the toroidal magnetic field plus some additional functions such as Internal energy. This extended GS equation is a PDE elliptic type, which could be a little more laborious to calculate than the usual GS equation.