A Stochastic Lagrangian Basis for a Probabilistic Parameterization of Moisture Condensation in Eulerian Models

TL;DR

This paper introduces a probabilistic parameterization method for moisture condensation in coarse-resolution Eulerian models, using stochastic Lagrangian moments to accurately represent subgrid-scale saturation effects efficiently.

Contribution

It presents a novel approach that derives probability distributions from stochastic Lagrangian models to improve moisture condensation parameterization in Eulerian frameworks.

Findings

The method accurately mimics Lagrangian moisture transport in idealized tests.

It provides a computationally efficient way to incorporate subgrid-scale moisture variability.

The approach is theoretically grounded using the Fokker-Planck equation.

Abstract

In this paper we describe the construction of an efficient probabilistic parameterization that could be used in a coarse-resolution numerical model in which the variation of moisture is not properly resolved. An Eulerian model using a coarse-grained field on a grid cannot properly resolve regions of saturation---in which condensation occurs---that are smaller than the grid boxes. Thus, in the absence of a parameterization scheme, either the grid box must become saturated or condensation will be underestimated. On the other hand, in a stochastic Lagrangian model of moisture transport, trajectories of parcels tagged with humidity variables are tracked and small-scale moisture variability can be retained; however, explicitly implementing such a scheme in a global model would be computationally prohibitive. One way to introduce subgrid-scale saturation into an Eulerian model is to assume the humidity within a grid box has a probability distribution. To close the problem, this distribution is conventionally determined by relating the required subgrid-scale properties of the flow to the grid-scale properties using a turbulence closure. Here, instead, we determine an assumed probability distribution by using the statistical moments from a stochastic Lagrangian version of the system. The stochastic system is governed by a Fokker--Planck equation and we use that, rather than explicitly following the moisture parcels, to determine the parameters of the assumed distribution. We are thus able to parameterize subgrid-scale condensation in an Eulerian model in a computationally efficient and theoretically well-founded way. In two idealized advection--condensation problems we show that a coarse Eulerian model with the subgrid parameterization is well able to mimic its Lagrangian counterpart.