## 1 Introduction  
In fluid simulations dealing with water, smoke, fire, etc., studies have been continuously attempted to express turbulent flow by restoring details lost by numerical dissipation: Particle-based fluids techniques such as MPM (Material Point Method) [1, 2], FLIP (Fluid-Implicit Particle) [3, 4, 5], APIC (Affine Particle-In-Cell) [6, 7], Polynomial PIC [8], and SPH (Smoothed Particle Hydrodynamics) [9, 10, 11] are useful in creating splashes and expressing free surfaces in detail, so they have been widely used in computer graphics in recent years. [PERSON] et al. proposed a particle-grid hybrid framework for synthesizing turbulent flow [12]. [PERSON] et al. proposed a technique to express fluid turbulence more efficiently by upsampling the grid structure [13]. [PERSON] et al. proposed a technique to express turbulent smoke by upsampling the vortex particles method to a higher resolution grid [14]. [PERSON] et al. expressed vorticity using vortex filaments [15]. [PERSON] and [PERSON] expressed more accurate vortex filaments by improving the resampling scheme proposed in the above technique [16].  
Physically-based fluid simulations often deal with turbulent flows expressed in fluid-solid interactions or large fluid bodies. Key issues in expressing turbulent fluids are the generation and preservation of small-scale details: Methods have been proposed to solve this problem based on physics [17, 18, 19, 20, 21] or noise and texture [22, 23, 24]. Depending on the fluid materials corresponding to these issues, the expression technique should be modeleddifferently. For example, when expressing water, the motion of the main body and the turbulent flow are important, but the secondary effects (e.g., splash, foam, bubble, mist, etc.) resulting from it are also important. However, most approaches have a strong linking structure in which secondary effects are expressed only by the underlying water simulation. Since it is difficult to independently control the motion of the foam effects created in the strong linking structure, even the motion of the underlying water needs to be modified in order to control the motion of the foam particles in detail. The Navier-Stokes equation is often used to simulate the splash or foam effects with complicated motion in wild water flows. However, in the process of calculating the foam particles, it is necessary to check all the water particles, so a huge computational cost and resources are required [25]. In addition, since it is bound to the movement of water particles, it is difficult to properly express intrinsic patterns or styles expressed in foam. Recently, [PERSON] et al. expressed foam effects efficiently using 2D projective space and expressed intrinsic patterns of foam by proposing a clamping function based on curvature [26; 27]. This method tried to express intrinsic patterns by controlling foam generation like a curvature-based mask map. However, since it does not affect the motion, it does not dynamically enhance the motion of the foam particles.  
In the existing methods, it is difficult to express wave patterns or dynamic motion of foam because the foam particles are advected while being strongly bound to the underlying water simulation. In addition, it is difficult to intuitively control the foam effects because the user has to modify the underlying water to control the secondary effects. In this paper, not only foam particles are expressed based on the motion of the underlying water, but also lively foam effects can be expressed by adding the angular momentum of water particles to the foam advection process.