In the past years, several observations of AGN and X-ray binaries have suggested the existence of a warm T around 0.5-1 keV and optically thick, tau ~ 10-20, corona covering the inner parts of the accretion disk. These properties are directly derived from spectral fitting in UV to soft-X-rays using Comptonization models. However, whether such a medium can be both in radiative and hydrostatic equilibrium with an accretion disk is still uncertain. We investigate the properties of such warm, optically thick coronae and put constraints on their existence. We solve the radiative transfer equation for grey atmosphere analytically in a pure scattering medium, including local dissipation as an additional heating term in the warm corona. The temperature profile of the warm corona is calculated assuming it is cooled by Compton scattering, with the underlying dissipative disk providing photons to the corona. Our analytic calculations show that a dissipative thick, (tau_{cor} ~ 10-12) corona on the top of a standard accretion disk can reach temperatures of the order of 0.5-1 keV in its upper layers provided that the disk is passive. But, in absence of strong magnetic fields, the requirement of a Compton cooled corona in hydrostatic equilibrium in the vertical direction sets an upper limit on the Thomson optical depth tau_{cor} < 5 . We show this value cannot be exceeded independently of the accretion disk parameters. However, magnetic pressure can extend this result to larger optical depths. Namely, a dissipative corona might have an optical depth up to ~ 20 when the magnetic pressure is 100 times higher that the gas pressure. The observation of warm coronae with Thomson depth larger than ~ 5 puts tights constraints on the physics of the accretion disk/corona systems and requires either strong magnetic fields or vertical outflows to stabilize the system.
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