It has been shown that two-fluid effects are important in many cases such as stability of a high beta plasma ¹, relaxation of a flowing high beta plasma ^2,3 and equilibrium of a flowing plasma ⁴. In particular it was found that the two-fluid effect can be significant depending on the effective size of gradient of physical quantities, the beta value and how close the flow speed is to the ion diamagnetic drift speed ⁴. Unfortunately because of the complexity of the analysis, so far the density has been assumed uniform and attention has been paid mainly to the effect of flow. The formalism to analyze a flowing two-fluid equilibrium with non-uniform density was developed in Ref.5. Recently we have found more accurate way to treat the entropies for the electron and ion fluids. The equation for the axisymmetric equilibrium can be expressed using second order differential equations for the stream functions of the generalized vorticities of the electron and ion fluids plus an algebraic equation for the density. These simultaneous equations have six arbitrary functions for the stream functions of poloidal flows, the specific entropies and the generalized enthalpies for the electron and ion fluids. In this paper we will present mainly an improved formalism for flowing two-fluid equilibrium with non-uniform density and show preliminary examples of 1D equilibria.

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