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Landfalling atmospheric rivers (ARs) frequently trigger heavy and sometimes prolonged precipitation, especially in regions with favored orographic enhancement. The presence and strength of ARs are often described using the integrated water vapor (IWV) and the integrated vapor transport (IVT). However, the associated precipitation is not directly correlated with these two variables. Instead, the intensity of precipitation is mainly determined by the net convergence of moisture flux and the initial degree of saturation of the air column. In this study, a simple algorithm is proposed for estimating the heavy precipitation attributable to the IVT convergence. Bearing a strong resemblance to the Kuo-Anthes parameterization scheme for cumulus convection, the proposed algorithm calculates the large-scale primary condensation rate (PCR) as a proportion of the IVT convergence, with a reduction to account for the general moistening in the atmosphere. The amount of reduction is determined by the column relative humidity (CRH), which is defined as the ratio of IWV to its saturation counterpart. Our analysis indicates that the diagnosable PCR compares well to the forecast precipitation rate given by a numerical weather prediction model. It is also shown that the PCR in an air column with CRH 0.50 is negligibly small. The usefulness of CRH and PCR as two complements to standard AR analysis is illustrated in three case studies. The potential application of PCR to storm classification is also explored.

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