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We present a detailed guide to advanced collisionless fluid models thatincorporate kinetic effects into the fluid framework, and that are much closerto the collisionless kinetic description than traditional magnetohydrodynamics.Such fluid models are directly applicable to modeling turbulent evolution of avast array of astrophysical plasmas, such as the solar corona and the solarwind, the interstellar medium, as well as accretion disks and galaxy clusters.The text can be viewed as a detailed guide to Landau fluid models and it isdivided into two parts. Part 1 is dedicated to fluid models that are obtainedby closing the fluid hierarchy with simple (non Landau fluid) closures. Part 2is dedicated to Landau fluid closures. Here in Part 1, we discuss the CGL fluidmodel in great detail, together with fluid models that contain dispersiveeffects introduced by the Hall term and by the finite Larmor radius (FLR)corrections to the pressure tensor. We consider dispersive effects introducedby the non-gyrotropic heat flux vectors. We investigate the parallel andoblique firehose instability, and show that the non-gyrotropic heat fluxstrongly influences the maximum growth rate of these instabilities.Furthermore, we discuss fluid models that contain evolution equations for thegyrotropic heat flux fluctuations and that are closed at the 4th-moment levelby prescribing a specific form for the distribution function. For thebi-Maxwellian distribution, such a closure is known as the "normal" closure. Wealso discuss a fluid closure for the bi-kappa distribution. Finally, byconsidering one-dimensional Maxwellian fluid closures at higher-order moments,we show that such fluid models are always unstable. The last possible nonLandau fluid closure is therefore the "normal" closure, and beyond the4th-order moment, Landau fluid closures are required.

An introductory guide to fluid models with anisotropic temperatures Part 1 -- CGL description and collisionless fluid hierarchy