Doctoral dissertation

Genuinely third-order advection scheme for atmospheric flows: the Multidimensional Positive Definite Advection Transport Algorithm approach



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Maciej Waruszewski

Hanna Pawłowska

Wydział Fizyki, Uniwersytet Warszawski



Numerical modelling of the atmosphere is crucially important for the state-of-the-art weather forecasting and climate prediction. In weather, climate, and chemistry-transport models, advective transport of moist air, chemical species, or pollutants has to be represented with high accuracy. Furthermore, advection schemes need to respect the fundamental physical principles of transport, such as conservation, monotonicity, compatibility with mass continuity, and correlations between tracers.
Numerical errors and behavioural properties of an advection scheme are dependent on its order of accuracy. In atmospheric modelling, second-order accurate schemes are common, and a trend towards high-order (i.e. third-order or higher) accurate algorithms can be observed. However, combining high-order accuracy with robustness and physical realisability is far from trivial.
To achieve high-order accuracy without compromising advantageous numerical properties, this thesis proposes a third-order accurate advection scheme based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA). MPDATA-based solvers have a rich history of successful applications in geo- and astrophysics. The standard MPDATA advection scheme is second-order accurate, sign-preserving (optionally nonoscillatory), and fully multidimensional. Remarkably, in simulations of turbulent flows, MPDATA can provide an implicit subgrid-scale turbulence model.
This thesis extends MPDATA to third-order accuracy for temporally or spatially varying flows, while preserving its beneficial characteristics. This is accomplished by deriving the leading truncation error of the standard second-order MPDATA, performing the Cauchy-Kowalevski procedure to express it in a spatial form and compensating its discrete representation—much in the same way as the standard MPDATA corrects the firstorder accurate upwind scheme. The procedure of deriving the spatial form of the truncation error was automated using a computer algebra system. This enables various options in
MPDATA to be included straightforwardly in the third-order scheme, thereby minimising the implementation effort in existing code bases. Following the spirit of MPDATA, the error is compensated using the upwind scheme resulting in a sign-preserving algorithm, and the entire scheme can be formulated using only two upwind passes. Established MPDATA enhancements, such as formulation in generalised curvilinear coordinates, the nonoscillatory option or the infinite-gauge variant, carry over to the fully third-order accurate scheme.
The novel scheme was implemented in libmpdata++, which is an open-source library of MPDATA-based solvers. Highlights of the library are presented, particularly stressing its design based on object-oriented programming and modern software development practices. Benefits of the adopted design choices for implementing the fully third-order accurate scheme are discussed.
A manufactured 3D analytic solution is used to verify the theoretical development and its numerical implementation. Global tracer-transport benchmarks facilitate comparison of the fully third-order accurate MPDATA to other schemes popular in computational meteorology, while also demonstrating its benefits for chemistry-transport models fundamental to air quality monitoring, forecasting and control.
Advantages of the fully third-order-accurate MPDATA for fluid dynamics applications are illustrated by simulations of a double shear layer, a convective boundary layer, and an idealised supercell storm. The double shear layer simulations quantify the increased accuracy of the new scheme in an overall lower-order accurate flow solver. Simulations of the convective boundary layer reveal its implicit subgrid-scale turbulence model. Characteristics of the scheme in simulations with parametrised cloud microphysics are explored in the idealised supercell storm benchmark.
Using the genuinely third-order accurate MPDATA consistently improved simulation results in a variety of test cases relevant to atmospheric modelling. The improvement was especially significant for the tracer transport benchmarks; therefore, the novel scheme can be generally recommended for tracer transport applications. In simulations of turbulent flows, the fully third-order accurate MPDATA revealed an implicit subgrid-scale model with beneficial characteristics. An increased complexity of the new scheme is offset by the availability of its open-source implementation.