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Asymptotic solution of liquid water content equation at equilibrium state of radiation fog
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    The liquid water content equation at a state of equilibrium for radiation fog was asymptotically solved with the singular perturbation technique. This report presents the detailed solution procedures, sensitivity testing of related parameters and discussions. The asymptotic solution describes the vertical structure of liquid water content which explicitly reveals the balance relationship between various factors in a steady radiation fog. It shows that cooling produces liquid water, which is regulated by droplet gravitational settling, while turbulence consumes the produced liquid water.The asymptotic solution also reveals that there exists a surface turbulence sub-layer, or fog boundary layer (FBL) within a radiation fog. The characteristic FBL depth is thinner for weaker turbulence and a larger cooling rate, whereas, if turbulence increases or the cooling rate decreases the FBL will develop from the ground. Through the FBL, a critical turbulence exchange coefficient for radiation fog was identified. This parameter is most interesting because it defines the upper bound of turbulence that a steady radiation fog can withstand. The deeper a fog is, the stronger a turbulence intensity it can endure, which explains why turbulence hinders or disperses a shallow fog near the ground but promotes a deeper fog at higher altitudes. Furthermore, the conditions for maintaining the state of equilibrium in a radiation fog can be characterized by either the critical turbulence exchange coefficient or the characteristic FBL depth. If turbulence decreases to below the critical turbulence intensity fog will form, while if turbulence exceeds the critical turbulence intensity or if the characteristic FBL depth dominates most of the fog bank, the state of equilibrium will be broken, leading to the dissipation of the radiation fog. The breakdown of the equilibrium condition can be applied to forecasts as a criterion to diagnose the formation or dissipation of radiation fog.

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