Isotropic pressure model in the presence of coarse-graining scale Alfvén waves and its consequence for modulational instability

A scalar pressure model of solar wind plasma in the presence of Alfvén waves at the coarse-graining scale is discussed. Using a simple sub-grid-scale model, it is shown that the scalar pressure model can be applied to describe the background pressure, because the off-diagonal components of the sub-grid-scale stress tensor disappear in the presence of one-dimensional, sub-grid scale Alfvén waves. Local equilibrium velocity distribution functions corresponding to the scalar pressure including wave dynamics (apparent temperature) are also presented. The specific heat ratio of the apparent temperature is given by the Wentzel-Kramers-Brillouin approximation of the Alfvén waves. The consequence of the local equilibrium velocity distribution functions in the coarse-graining scale model is discussed by using a nonlinear equation set describing envelope-modulated nonlinear Alfvén waves. It is recognized that without sufficient separation between the scale of waves and the scale of integration, artificial changes of the growth rates appear in the solutions of linear growth rates owing to the finite length of the filter function.