The first-order jump model for the potential temperature or buoyancy variable at the capping inversion atop a convectively mixed layer is reexamined and found to imply existence of an entrainment rate equation which is unreliable. The model is therefore extended here to allow all the negative buoyancy flux of entrainment to occur within the interfacial layer of thickness Δh and to allow realistic thermal structure within the layer. The new model yields a well behaved entrainment rate equation requiring scarcely any closure assumption in the cases of steady-state entrainment with large-scale subsidence, and pseudo-encroachment. For nonsteady entrainment the closure assumption need only be made on d(Δh)/dt in order to obtain the entrainment rates at both the outer and inner edges of the interfacial layer. A particular closure assumption for d(Δh)/dt is tested against five laboratory experiments and found to yield favorable results for both Δh and the mixed-layer thickness if the initial value of Δh... Abstract The first-order jump model for the potential temperature or buoyancy variable at the capping inversion atop a convectively mixed layer is reexamined and found to imply existence of an entrainment rate equation which is unreliable. The model is therefore extended here to allow all the negative buoyancy flux of entrainment to occur within the interfacial layer of thickness Δh and to allow realistic thermal structure within the layer. The new model yields a well behaved entrainment rate equation requiring scarcely any closure assumption in the cases of steady-state entrainment with large-scale subsidence, and pseudo-encroachment. For nonsteady entrainment the closure assumption need only be made on d(Δh)/dt in order to obtain the entrainment rates at both the outer and inner edges of the interfacial layer. A particular closure assumption for d(Δh)/dt is tested against five laboratory experiments and found to yield favorable results for both Δh and the mixed-layer thickness if the initial value of Δh...