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Electromagnetic effects play a key role in tokamak edge turbulence. It has been suggested that the density limit and the L to H mode transition may both be due to an interplay between electromagnetic effects, diamagnetic flows and collisionality. (See, e.g., Ref [1].) The present paper discusses the results of scrape-off layer (SOL) non-linear 3D fluid turbulence simulations including finite beta effects in the shear-less limit. These simulations were carried out using the GBS code [2], which evolves the drift-reduced Braginskii equations for a collisional plasma with cold ions in circular (s-α) geometry with a toroidal limiter in the high-field side midplane. The GBS code has been used to study turbulence in linear devices and in a simple magnetized torus configuration [2, 3]. We have recently adapted the code for tokamak edge geometry, and introduced s-α curvature operators as well as magnetic shear, finite aspect ratio, and finite beta effects. The objective of our work is to describe the phase-space relevant to the tokamak SOL turbulence. In this paper, in particular, the role played by finite beta effects upon the characteristic lengths of the profile gradients, turbulence saturation levels, and other basic turbulence properties, is assessed in the context of fully global non-linear turbulence simulations. The non-linear steady-state turbulent plasma profiles are obtained as the result of a balance between plasma density and heat sources, turbulent fluctuations, and parallel losses at the limiter plates. The turbulence drive is a priori unknown and there is no separation between fluctuations and background profiles. Linear analysis of the fluid equations has been carried out for SOL relevant parameters. In the presence of finite beta effects, we recover three instabilities: drift waves, resistive ballooning modes, and ideal ballooning modes. The onset of ideal ballooning modes is known to correspond to the instability threshold α_MHD = q^2 β R/Lp ~ 1. In the non-linear simulations, however, we observe the onset of catastrophic transport well below the ideal limit. The saturated states in this regime are characterized by large transport due to global ideal modes. These modes are linearly subdominant but non-linearly dominant due to the underlying turbulence saturation mechanism.