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tokens: section’s findings suggest that turbulence is essential for bringing about condensation in a medium with a non-negligible entropy gradient. More sophisticated numerical simulations that solve complete sets of hydrodynamical and magnetohydrodynamical equations will be needed to quantify with greater precision the relationship between turbulence and the critical t cool / t ff ratio for condensation within media with and without dynamically important magnetic fields. Concurrently, simulations of self-regulating mechanical AGN feedback should be analyzed for correlations between the rate at which ambient gas condenses into cold clouds and the values of t cool / t ff and σ t / σ v in the ambient medium. This paper will leave those investigations for the future and instead turn to the question of global self-regulation. 4. Coupling between Precipitation, Feedback, and Turbulence Previous work on precipitation-regulated feedback has assumed that the condensation threshold depends on either the t cool / t ff ratio alone (e.g., McCourt et al. 2012 ; Sharma et al. 2012b ) or the parameter combination , which accounts for the failure of buoyancy damping to suppress condensation when α K is sufficiently small (Voit et al. 2017 ). 3 3 Gaspari et al. ( 2017 , 2018 ) have recently proposed a different criterion for condensation in a turbulent medium that compares t cool with the timescale for eddy turnover. See the Appendix for a discussion of how that criterion relates to the results of Section 3 . However, an additional dependence on σ t / σ v , such as that shown in Figure 7 , can transform the feedback loop into a converging cycle that settles into a particular equilibrium state. This section explores global convergence to thermal balance in a precipitating medium by dropping the thermal balance requirement and developing a toy model that illustrates how turbulence can couple with precipitation. The model illuminates the conditions under which coupling produces a long-lasting equilibrium state in agreement with observations of both the precipitation threshold and the turbulent velocity dispersion in galaxy cluster cores. 4.1. A Toy Model for Coupling In order to keep the overall model as simple as possible, the dynamical timescales of the feedback system are assumed to be much shorter than its thermal and cosmological timescales, so that the ambient gas remains close to hydrostatic equilibrium. The model further assumes that heat input does not change the slope of the ambient entropy profile, which is constrained to be K ( r ) ∝ r 2/3 in a gravitational potential with t ff ∝ r . As a result, the gas temperature remains constant at , with a specific thermal energy that remains invariant while the gas gains and loses heat energy. Instead of causing the temperature to rise and fall, heat input causes the gas to expand and radiative losses cause the gas to contract, while gravitational work keeps kT steady. In such a constrained system, the t cool / t ff