A stochastic Lagrangian scheme for parametrizing cloud microand macro-physics in the Unified Model

Water in the atmosphere, in all its forms (vapour, liquid and ice), plays a crucial role in our weather and climate. Extreme events of heavy rainfall lead to flooding. Water vapour is a potent greenhouse gas that traps outgoing longwave radiation and thus affects the Earth’s climate. Reliable weather forecast and climate prediction are critical to minimising the impact of hazardous weather and climate changes. However, despite decades of intensive research effort, the modelling of atmospheric moist processes is still a challenge and remains a key error in weather and climate models. This project will tackle this problem by developing a parametrization scheme that is radically different from the current standard approach.

Water vapour is being carried by fluid motions in the atmosphere. As temperature changes along the way, clouds may form by condensation and may eventually lead to precipitation as rain or snow. The latent heat released from cloud formation can in turn affect the atmospheric flow. This complicated interaction between fluid dynamics and phase change occurs over an extremely wide range of spatial scales. For example, condensation of cloud droplets takes place at the micrometers, the length scale of deep moist convection is between 1–10 km while large-scale global motions typically extend over 10^4 km. A numerical weather or climate model solves a discretised version of the differential equations governing the atmospheric motions on a grid. Typical horizontal grid size ranges from about 10 km in weather models to 100 km in climate models. Clearly, many important moist processes happening below the size of a grid box cannot be directly simulated. However, we can account for these unresolved subgrid-scale processes by including their average effects on the resolved processes in the model. This technique is known as parametrization (Jakob and Miller, 2002). Traditional cloud parametrizations often employ some form of turbulence closure in an empirical or somewhat ad hoc manner. Our objective is to develop a scheme in which the statistical effects of unresolved moist processes are rigorously built upon the underlying cloud physics. To this end, we shall take the stochastic Lagrangian approach.

Figure 1 (a) A snapshot from a Monte Carlo simulation of an ensemble of moist parcels advected by a cellular flow. Colour indicates relative humidity (RH). (b) RH field obtained from the Monte Carlo data by averaging over parcels locally in space. We consider this to be the “truth”. (c) RH field obtained from a coarse-resolution solution of the partial differential equation describing the same system in (a) and (b). Large discrepancy from (b) can be seen. (d) Same as (c) but with the stochastic Lagrangian parametrization implemented resulting in good agreement with (b). (Tsang and Vallis, 2018)

The Project

Since Einstein’s work on Brownian motion, stochastic method has developed into a rigorous mathematical tool that found application in numerous areas. In atmospheric sciences, stochastic Lagrangian models—which describe the trajectories of air parcels advected by a prescribed random velocity—have long been used to study turbulent dispersion in the atmosphere (Thomson and Wilson, 2012). We propose to introduce this approach into cloud parametrization. Specifically, we imagine there is an ensemble of moist air parcels inside a model grid box. These parcels are advected by a velocity which consists of two parts: a deterministic component resolved by the grid and an unresolved turbulent component modelled by a stochastic process. The water content of each parcel varies due to condensation and evaporation as the parcel moves. This system can be described by a set of stochastic differential equations from which the statistical properties of the ensemble can be derived and incorporated into a parametrization. Previous studies have demonstrated the feasibility of this method in different idealised setups (Field et al., 2014; Furtado et al., 2015; Tsang & Vallis, 2018). Figure,1 shows this parametrization successfully produces a more realistic global distribution of humidity in a cellular flow mimicking the Hadley cell. Further examples and a movie showing the parametrization in action are available at http://www1.maths.leeds.ac.uk/~amtyt/research.

We plan to develop this method from its mathematical foundations through to its implementation and testing in the Met Office Unified Model in three phases:

+ We first lay the theoretical foundation by extending our previous work to deal with multiple moisture and thermodynamic variables. This will result in a flexible mathematical framework to construct stochastic Lagrangian models for different atmospheric conditions.

+ Next, we shall implement and test the scheme in simplified dynamical models. An idealised version of the Met Office Unified Model and the idealised general circulation model ISCA developed at the University of Exeter (execlim.github.io/IscaWebsite) will be used.

+ Finally, we shall test the scheme in the Met Office Global Atmosphere configuration for weather cases and climatologies.

Potential for High Impact Outcome

The major impact of this PhD project will be a wholly new approach to parametrizing subgrid-scale clouds. It has the potential to drastically improve the prediction of cloud properties in the Unified Model and ultimately increases the confidence of our weather forecast. Furthermore, the stochastic framework introduced here is very general and can readily be adapted for other problems, for example, the parametrization of mixdown time for atmospheric chemical reactions. We expect the output from this project to lead to advance in other areas of atmospheric sciences. Previous work on stochastic Lagrangian models almost exclusively focuses on non-buoyant, non-reactive particles. We are making major theoretical contributions to this area by developing models involving condensation and other moist processes. We therefore anticipate this project to produce high quality publications in atmospheric sciences journals as well as fluid dynamics journals.

Training and Research Environment

The student will be based in the School of Earth and Environment at the University of Leeds and work under the supervision of Dr. Yue-Kin Tsang in the Department of Applied Mathematics and Dr. Jochen Voss in the Department of Statistics. The multidisciplinary supervising team also includes Dr. Kalli Furtado and Prof. Paul Field in the Atmospheric Processes and Parametrizations team at the Met Office and Prof. Geoff Vallis at the University of Exeter. Experts in the Met Office Atmospheric Dispersion and Air Quality team will also informally provide further guidance. To facilitate effective interaction, the student is expected to spend periods of time (roughly 1–2 months per year) working at the Met Office and the University of Exeter.

During the project, the student will develop theoretical skills in stochastic methods. At the University of Exeter, the student will learn to use and modify ISCA for the purpose of this project. At the Met Office, the student will familiarise with the source code of the Unified Model—the current operational model for weather forecast. In addition to skills specialised for the project, the student is also expected to acquire a broad range of analytical and computational skills through the wide variety of modules offered to PhD students at the University of Leeds.

The University of Leeds traditionally has strong fluid dynamics research. Its reputation is further enhanced by the recent establishment of the interdisciplinary Leeds Institute for Fluid Dynamics. The student will be a member of the Astrophysical and Geophysical Fluid Dynamics group. The group holds a biweekly AGFD meeting where group members, who have a wide range of interest, discuss their research and foster a strong sense of research community.

Student Profile

The student should have a strong interest in atmospheric sciences and numerical modelling, a strong background in applied mathematics or physics and some knowledge of programming and scientific computing.

References and Further Reading

C. Jakob and M. Miller (2002), Parameterization of Physical Processes: Clouds.
Encyclopedia of Atmospheric Sciences, 1st ed. J. A. Curry and J. A. Pyle, Eds., Elsevier, 1692–1698

D. J. Thomson and J. D. Wilson (2012)}, History of Lagrangian Stochastic Models for Turbulent Dispersion, Lagrangian Modeling of the Atmosphere, Volume 200, J. Lin, D. Brunner, C. Gerbig, A. Stohl, A. Luhar and P. Webley, Eds., American Geophysical Union, 19–36

P. R. Field, A. A. Hill, K. Furtado and A. Korolev (2014), Mixed-phase clouds in a turbulent environment. Part 2: Analytic treatment, Q. J. R. Meteorol. Soc., 140, 870–880

K. Furtado, P. R. Field, I. A. Boutle, C. J. Morcrette and J. M. Wilkinson (2015), A Physically Based Subgrid Parameterization for the Production and Maintenance of Mixed-Phase Clouds in a General Circulation Model, J. Atmos. Sci.,73, 279–291

Y.-K. Tsang and G. K. Vallis (2018), A Stochastic Lagrangian Basis for a Probabilistic Parameterization of Moisture Condensation in Eulerian Models, J. Atmos. Sci., 75, 3925–3941