## 1 Introduction
[16] developed a hybrid decomposition technique to decompose the multi-scale structures of a semi-cylinder wake using a 2D orthogonal wavelet first and then further analyzed the decomposed structures with POD to extract distinctive flow patterns from the dominating POD modes of intermediate-scale and small-scale structures. In recent years, a dimensional reduction algorithm, developed by [PERSON] and coworkers [17], called dynamic mode decomposition(DMD), has emerged as a popular alternative to the modal decomposition techniques discussed earlier. DMD decomposes the time series of flow field data into spatial modes with dynamic time histories that exhibit decaying or growing amplitudes with fixed oscillatory frequencies. Unlike POD, DMD modes are not necessarily orthogonal in the spatial domain, although their individual time dynamics might. The spatial-temporal modal decomposition with individual temporal characteristics has given DMD a unique advantage over other decomposition techniques, leading to its widespread application in both fundamental and applied fluid dynamics research [18; 19; 20; 21; 22]. These advantages, alongside its inherent challenges, make DMD the primary decomposition technique employed in this study. The DMD decomposition is closely related to Koopman analysis and has led to numerous extensions and applications across fluid dynamics and other fields like video surveillance, epidemiology, neurobiology, and financial engineering. Since its first deployment in fluid dynamics research, DMD has evolved into various extensions, generalizations, and improvements. These efforts have resulted in an extensive list of published research in the field of fluid dynamics, and for brevity, we will limit our discussions to the works that relate very closely to ours, that is, those involving primarily ground vehicles. However, interested readers are directed to the recently published review paper by [PERSON] [23], which provides an in-depth discussion on the practical aspects of DMD and its variants and their effectiveness as quantitative tools for analyzing complex fluid flows. The dynamics of turbulent flow play a crucial role in determining the characteristics of fluid flows around ground vehicles. Ground vehicles experience three aerodynamic forces (drag, lift, side) and three moments (rolling, pitching, and yawing). Dynamic turbulent structures and their interactions in the wake region, far wake region, and boundary layer near the car body significantly contribute to the six aerodynamic component forces. Therefore, modal decomposition techniques, including dynamic mode decomposition (DMD), are valuable tools for enhancing fuel economy and vehicle maneuverability. They achieve this by identifying dynamic body force features, designing passive and active control systems, and constructing reduced-order models (ROMs). DMD's ability to compute spatiotemporal modes, isolate individual components in the time domain, and facilitate future predictions makes it highly promising for automotive applications. For instance, [PERSON] et al. [24; 25] used DMD to extract flow structures responsible for lift forces with an oscillation frequency of 5 Hz around a realistic car body, revealing that high-energy fluctuations were concentrated on the trunk, spare tire pan, and underbody. Similarly, [PERSON] et al. [26] employed sparse promoting dynamic mode decomposition to capture an asymmetric structure with \(St_{W}=0.22\) in the flow regime of a 47\({}^{\circ}\) [PERSON] body, designing a closed-loop controller to symmetrize the mode. Here, \(f\), \(U\), and \(W\) represent the dimensional frequency, reference velocity, and vehicle width, respectively, with the Strouhal number based on the width defined as \(St_{W}\equiv fW/U\). Note that, in this paper, we will be using the height of the [PERSON] body as the characteristic length, and the corresponding Strouhal number will be denoted simply as \(St\). In another study, [PERSON] et al.