ATM S 582 A Sp 21: Advanced Numerical Modeling Of Geophysical Flows
ATM S 582 A Sp 21: Advanced Numerical Modeling Of Geophysical Flows
ATM S 582
Professor Dale Durran
502 ATG Building
Email Me
Our goal: Let's try to see the forest and the trees.
The purpose of the course is to obtain a deeper understanding of the basic numerical techniques that form the foundation for the computer models commonly used to simulate geophysical flows, particulary weather and climate. The class builds on material covered in ATMS 581/AMATH 586, although some familiarity with numerical methods for the solution of partial differential equations is generally adequate preparation.
We will look closely at behavior of some representative numerical methods in the simplest possible contexts. At the same time, we will strive to develop an overview of the pros and cons of these methods for the simulation atmospheric flows.
Textbook: Durran, D.R., 2010: Numerical Methods for Fluid Dynamics: With Applications in Geophysics. 2nd Ed. Springer-Verlag.
The book is available by chapters for free as pdfs via a subscription through the UW library. That subscription also entitles students to buy $25 paperback copies. To access these priviliges, connect here via a UW computer, and click 'Read Online'.
Lecture Format: Live remote via Zoom MW 9:00-10:20. Lectures will be available as Zoom recordings.
Office Hours: Tues 1:00-2:00 and by appointment (don't hesitate to ask!)
Grading: The grade will be based on a short project and short bi-weekly homework assignments, one of which will be identified as a take-home midterm that must be done independently. You may work with other students on the other homeworks. The project will ideally be something related to your research.
Course Outline
Differential-Difference Equations for the Scalar Transport
- Numerical Dissipation and Numerical Dispersion (pp. 100-110)
- Figures
Further Considerations in Finite Difference Approximations
- Phase speed as a function of numerical resolution for compact schemes (Fig. 3.10)
- Staggered meshes (pp. 153-157)
- Discrete dispersion relation for systems of equations (pp. 167-169)
- Stability constraints in simulations of multi-dimensional waves (pp. 157-159)
-
Splitting into fractional steps (pp. 169-176)
- Swirl test
- Effect of Strang splitting (from Skamarock, 2006, MWR, p. 2243)
-
Optional background reading on the stability of systems (pp. 148-151)
Finte-Volume Methods
- Example of how fine scale features are produced in a scalar tracer field by flow deformation
- Conservation laws and conservation form (pp. 203-206, 211-213)
- Monotone, TVD and monotonicity preserving methods (pp. 213-220)
- Flux-corrected transport (pp. 221-226)
- Flux-limiter methods (pp. 226-235)
- Subcell Polynomial Reconstruction (pp. 235-243
-
Preserving Smooth Extrema (pp. 253-255)
- Need to grow extrema
- Swirl test
- Mass consistent split: PPM no limiting, 50x50 mesh
- PPM with global and selective limiting, 50x50 mesh
- PPM with global and selective limiting, 100x100 mesh
- As above with initial cube distribution on 50x50 mesh
- Convergence tests (selective)
- Convergence tests (Table 5.1)
- Postive-Definite Advection Schemes (pp. 271-273)
Spectral and Pseudo-Spectral Models
- Basics of series expansion methods (pp. 281-293)
- Conservation and the Galerkin approximation (pp. 298-299)
- Aliasing error (pp. 178-179)
- Improving efficiency with the transform method (pp. 292-298)
- The pseudo-spectral method (pp. 299-303)
- Spherical Harmonic Functions (pp. 303-308)
- Elimination of the Pole Problem (pp. 308-310)
- Finite-element method, linear elements (pp. 320-323)
- Quadratic elements (pp. 327-336)
The Cube Sphere
The Discontinuous Galerkin and Spectral Element Methods
- h-p methods
- Discontinuous Galerkin method (pp. 341-350)
Semi-Lagrangian Methods -- or maybe Boundary Conditions depending on student preferences. The following is for semi-Lagrangian methods.
-
The scalar advection equation (pp. 357-360)
- Backward trajectory
- CFL condition (pp. 98-100)
- 250 hPa winds forecast for March 14, 2015; Todays loop
- Cascade interpolation (pp. 362-365)
- Finite volume integrations with large time steps (pp. 369-372)
- Forcing in the Lagrangian frame (pp. 372--377)
- Comparison with the method of characteristics (pp. 377-379)
Course Summary:
Date | Details | Due |
---|---|---|