  Recently, a new singularity formation scenario for the 3D axi-symmetric Euler
equation and the 2D inviscid Boussinesq system has been proposed by Hu and Luo
based on extensive numerical simulations [15, 16]. As the firrst step to
understand the scenario, models with simplified sign-definite Biot-Savart law
and forcing have recently been studied in [7, 6, 8, 12, 14, 18]. In this paper,
we aim to bring back one of the complications encountered in the original
equation - the sign changing kernel in the Biot-Savart law. This makes analysis
harder, as there are two competing terms in the fluid velocity integral whose
balance determines the regularity properties of the solution. The equation we
study here is based on the CKY model introduced in [6]. We prove that finite
time blow up persists in a certain range of parameters.
