## 1 Introduction
Historically, [PERSON] [14] argued that the CSA was physically wrong and that it did not in fact converge to a system of equations originating from the Navier-Stokes equations. It was postulated that the vorticity field is correctly diffused; however, the vorticity advection is incorrectly computed, even in the limit of infinitely many vortices. Later, [PERSON] [15] proposed a solution for the CSA based on vortex blob splitting, which proved to be convergent and effective. The title \"corrected core-spreading method\" originated from the correction suggested for the CSA by [PERSON] [15]; however, no attempt to include turbulence modelling was incorporated by him. As well as many Eulerian methods, which require the use of sufficiently refined meshes, Lagrangian vortex methods need a great number of vortex blobs to handle higher-resolution turbulent vortical structures, which rapidly increases the computational time cost. In order to overcome that crucial point, several algorithms have been developed over the years to reduce the computational time cost associated with vortex methods; most of them are based on multipole expansions [16; 17], and the more recent algorithms use parallel implementation with CUDA technologies [18]. The vortex method has been also applied to build a neural network description of Lagrangian vortices and their interaction dynamics, aiming to reconstruct the high-resolution Eulerian flow field in a physically precise manner [19]. In this scenario, our research group has examined a spatial averaging model of turbulence in high-Reynolds-number flows for Lagrangian large-eddy simulation, including a roughness model [7; 8]. The numerical approach incorporated viscous effects through the random walk diffusion scheme proposed by [PERSON] [12] blended with LES theory [5] and the rough wall model [7]. The present paper contributes to a new Lagrangian vortex method based on a scheme that combines a blob splitting algorithm [15] with LES theory [5] and a roughness model [7]. The second-order velocity structure function model of the filtered field is then used to compute the local kinetic energy spectrum [20]. The corrected core-spreading method with turbulence modelling is necessary to control the consistency error of the CSA, maintaining the maximum blob radius below a stipulated value. The roughness model is inspired by the key idea to include a new momentum source term because of additional inertial effects; the numerical strategy simulates the momentum injection into the boundary layer, which is effectively reflected, changing both the viscous core and the strength of the nascent vortex blobs [7]. The new algorithm is then applied to analyze the dynamic interactions of aircraft wake vortices in the vicinity of a rough ground plane at two Reynolds number values, namely Re = 7650 and 75,000. The combined effects of crosswind and roughness are investigated and compared with the configuration of a smooth ground plane. Comparisons with experimental data published by [PERSON] and [PERSON] [21] using Re = 7650 are also presented. The available experimental data are scarce and did not include combined effects of surface roughness and crosswind, which motivated the present paper to contribute to the literature. The problem of the dissipation of aircraft vortex wakes has been the focus of extensive studies aiming to predict the potential hazard of wake vortices and their lifetime in the ground effect [22; 23; 24; 25; 26]. The air always flows from a region of high pressure to a region of low pressure, and in the case of the aircraft wing, air flows from below the wing to the upper surface of the wing. Physically, the air rotates as a consequence of trailing vortices created by a wing because of pressure differences, producing the lift of the wing. The literature classifies the circular motion that the air takes as a \"vortex\". The term \"wingtip vortex\" relates to the vortex being generated at the wingtip.