We study the feedback between heating and cooling of the intra-cluster medium ( ICM ) in cooling flow ( CF ) galaxies and clusters .
We adopt the popular view that the heating is due to an active galactic nucleus ( AGN ) , i.e .
a central black hole accreting mass and launching jets and/or winds .
We propose that the feedback occurs with the entire cool inner region ( r \lesssim 5 - 30 { ~ { } \ > kpc } ) , where the non-linear over-dense blobs of gas with a density contrast \rho / \rho _ { a } \gtrsim 2 cool fast and are removed from the ICM before experiencing the next major AGN heating event .
We term this scenario cold-feedback .
Some of these blobs cool and sink toward the central black hole , while others might form stars and cold molecular clouds .

We derive the conditions under which the dense blobs formed by perturbations might cool to low temperatures ( T < 10 ^ { 4 } ~ { } K ) , and feed the black hole .
The main conditions are found to be : ( 1 ) An over-dense blob must be prevented from reaching an equilibrium position in the ICM : therefore it has to cool fast , and the density profile of the ambient gas should be shallow ; ( 2 ) Non-linear perturbations are required : they might have chiefly formed by previous AGN activity ; ( 3 ) The cooling time of these non-linear perturbations should be short relative to few times the typical interval between successive AGN outbursts .
( 4 ) The blobs should be magnetically disconnected from their surroundings , in order not to be evaporated by thermal conduction .