Surface wind field observations under tropical cyclones, such as retrieved by SAR satellites, allow local-scale and vortex-scale wind field structures to be observed with unprecedented precision. While mean-vortex properties (maximum wind, wind radii) are routinely retrieved operationally, a large part of this high-resolution information, mostly contained in the asymmetric wind field, remains underexploited. The asymmetric surface wind field is an addition of many different contributions, from the synoptic flow imprinting on the vortex to internal wave activity and instabilities in the boundary layer. In SAR imagery, these different contributions result in an entanglement of spiral features, visible in the asymmetric wind field, but their nature and role cannot be directly delineated. This study is a first attempt to separate and qualify these contributions, using a 100m-resolved two hours long LES simulation of a steady-state mature tropical cyclone. The identification, separation and retrieval of spiral patterns was performed using a polar three-dimensional (time, radius, azimuth) spectral analysis. The main components were identified through a manual window filtering of the energy spectrum along time, radius, and azimuth. After extraction, the different identified spiral patterns taken together represent more than 60% of the total asymmetric wind field explained variance, and nearly 70% in the boundary layer. Detailed contributions show that asymmetry at the surface and up to 400m is largely dominated by boundary layer rolls (representing more than 50% of explained variance). The rest of the variability is explained, in part (around 10% of explained variance in the boundary layer), by large-scale outward radiating spiral patterns (wave number smaller than 4) with a constant angular velocity near 0.0015 rad s-1. Unlike the boundary layer rolls, these spiral patterns extend throughout and above the boundary layer and become dominant in the mid troposphere. The propagation properties of these spirals show strong consistency with gravity waves (constant angular velocity, faster than the wind rotation speed). Their nature and relationship with eyewall convection and mean flow properties is investigated through an energy budget. An additional small-scale spiral feature is identified also, aligned with the surface wind field and nearly perpendicular to the dominant A-type rolls, representing a significant 12% of the boundary layer variability, but with an apparent vertical extent above the boundary layer. The remaining spiral patterns are stationary and low-amplitude. While the direct comparison with SAR observations remains to be done, this study proposes a first assessment of the main contributions to the variability of the surface wind field, and to the origin and nature of their dynamics. Dominant local-scale instability rolls in the boundary layer are in fact strongly modulated by different types of spiraling patterns, one identified as fast propagating gravity waves, and the other as smaller-scale spirals aligned with surface winds. Their difference in nature highlights the diversity of dynamics conveyed by the sole measurement of the surface wind field, and the strong potential of SAR-type measurements for tropical cyclone forecasting.