## 1 Introduction
The vortices that increase in diameter and decrease in intensity as the time passes are called \"trailing vortices\" [25; 26]. These extend for a reasonable distance behind the aircraft wing. The contrails observed when an aircraft flies high at cruising altitude, developing velocity of about 850 km/h, are from the engine, not the vortices. Contrails from wingtip vortices are usually seen when an aircraft flies at low speeds (e.g., 240 km/h) and low altitudes (e.g., 70 m) in humid ambient conditions. The most common situation to visualize contrails occurs at the ends of the aircraft flaps when they are extended for an aircraft approaching or leaving a runway. So, wingtip vortices are more intense for a configuration of an aircraft: heavy, slow, and flying in a clean configuration. However, the real interest is in investigating the potential hazard for an aircraft approaching on the same (or a parallel) runway, since vortex size is affected by proximity to the ground too. The effect of strong crosswinds is also a factor that can interfere with vortices; the intensity of a vortex is not directly affected by crosswinds but they contribute to the dissipation of trailing vortices faster than still air [27; 28]. Recently, intending to avoid accidents, [PERSON] et al. [27] assessed various aspects of developing pairwise wake vortex separations for approach and landing at Vienna airport using what the authors called the Wake Vortex Prediction System (WSVS). The simulation study evaluated the probability that wake vortices remain within a defined radius around the follower aircraft when separations are applied. This probability was compared to data collected at five international airports. It was found that for WSVS predictions, wake vortices in ground proximity resided within a distance of 25 m to the follower aircraft in 0.25% of the landings. It is known that wake decay can be improved by introducing artificial perturbations. [PERSON] et al. [28] used a large-eddy simulation to investigate air blowing and suction as active flow control methods on wake vortex decay. Thus, blowing and suction zones were embedded in the ground to generate airflow below the wake vortex tubes. According to the authors, it was found that even with a small blowing velocity, the aircraft wake strength could be reduced. In further work, [PERSON] et al. [29] used a large-eddy simulation to show that under certain conditions of crosswind, the wake vortex deformed sinusoidally and linked with its mirror counterpart, which may cause an imminent risk of the vortex corridor to aircraft landing or taking off on the same or parallel runway. In a recent paper from our research group, [PERSON] et al. [30] utilized a Lagrangian temperature particle method to study buoyance effects on aircraft wake vortices near a ground plane; however, turbulence modelling and roughness effects were not included. Talking about this field of application, the present paper contributes with a new method to investigate the combined effects of surface roughness and crosswind for the problem of vortex rebound followed by continued separation at approximately constant distance from the ground plane. The new technique discretizes the vorticity field using vortex elements. The method of images is chosen to precisely represent the ground plane. The corrected core-spreading method simulates the vorticity diffusion by maintaining vortex elements with small sizes through a deterministic approach using a splitting algorithm. The vorticity diffusion incorporates viscosity effects and includes the local eddy viscosity coefficient computation. The local eddy viscosity coefficient is computed for every vortex element during every time step of a typical numerical simulation. The roughness model is also supported by the local eddy viscosity coefficient computation. The second-order velocity structure function model is adopted to compute the eddy viscosity coefficient. More details of the new technique will be properly discussed in Section 3.