1. Basic Concepts of Numerical Weather Prediction 6 September 2012

2. Thematic Outline of Basic Concepts Representation of the spatial derivatives in the primitive equations Time integration methods Representation of the initial atmospheric state Representation of the horizontal and vertical boundaries Physical process parameterizations

3. Note Before Proceeding The material in this lecture is not meant to be comprehensive. Instead, it is meant to further introduce many of the key topics that we will be covering in the remainder of this course. Each of these topics will be discussed in greater detail as we cover Chapters 3-6.

4. Spatial Derivatives Consider a simplified form of the u-momentum equation: How do you solve for the spatial derivatives in x and y (and to a lesser extent, z)? There are two popular (and many other) means of doing so…

5. Spatial Derivatives Method 1: Grid-Point / Finite Difference The atmosphere is represented as a three- dimensional spatial grid defined on a specified map projection. Grid points are generally evenly spaced (or nearly so). Exceptions: lat-lon grids; adaptive grids Spatial derivatives are solved using Taylor series- derived finite difference approximations.

6. Spatial Derivatives The distance between grid points is selected such that there are a sufficient number of grid points to adequately represent the smallest feature of interest Related concept: truncation error (Section 3.4.1) ~6∆x (~10∆x) points needed to represent a feature (wave) Pitfalls of grid-point methods… Introduce non-physical properties to the model solution Have stability criteria, often necessitating a short time step (one that requires more computations to get a forecast)

7. Spatial Derivatives Method 2: Spectral Methods Involves the forward transformation of the standard dependent variables (u, v, T, etc.) into transform space utilizing Fourier transforms. The resultant Fourier series and Fourier-Legendre functions represent horizontal variability. Temporal and vertical derivatives are handled with conventional methods. The dependent variables are subsequently transformed back into physical space to get interpretable forecast fields.

8. Temporal Derivatives Again, consider a simplified form of the u- momentum equation: How do you solve for the temporal derivative in t and thus integrate the model forward in time? We will discuss means of doing so in Section 3.3.1.

9. Stability Considerations For many temporal differencing schemes, the time step (i.e., the time elapsed between individual forecast times) is constrained. Constraining mechanism: the Courant number… (U = speed of the fastest wave on the grid, ∆t = time step, ∆x = distance between horizontal grid points chosen to adequately resolve the meteorological processes of interest)

10. Stability Considerations For grid-point methods, the Courant number must be less than or equal to 1. This is known as the Courant-Friedrichs-Lewy (CFL) criterion. – This is the stability requirement for the advection terms in an Eulerian (grid-point) framework. Typically, U can be determined and ∆x is specified. We want the largest ∆t satisfying the CFL criterion.

11. Stability Considerations If we set ∆t too large, the CFL criterion will be violated and non-meteorological features will grow exponentially in the solution. If we set ∆t too small, the CFL criterion won’t be violated but the forecast will take longer to complete. In other words, the model will have to solve the primitive equations more times for a specified forecast duration.

12. Stability Considerations Example: assume that the fastest wave propagates along a 100 m s -1 upper-tropospheric jet stream… – In other words, we have filtered out fast acoustic waves. What is the largest ∆t that we can have for convective-scale, mesoscale, and synoptic-scale simulations? Convective-scale (∆x = 4 km): 40 s Mesoscale (∆x = 20 km): 200 s Synoptic-scale (∆x = 50 km): 500 s

13. Stability Considerations If we want a 24-h forecast, we will need to solve our equation set the following number of times… Convective-scale: 2160 Mesoscale: 432 Synoptic-scale: 173 From the above, over an equally-sized domain, a convective-scale simulation will take ~12.5 times as long to complete as a synoptic-scale simulation!

14. Time Step Considerations The importance of a long time step becomes even more apparent when the number of horizontal grid points also enters the equation. Over a 1000 km x 1000 km domain, each simulation has the following number of grid points: Convective-scale: 250 x 250 = 62500 Mesoscale: 50 x 50 = 2500 Synoptic-scale: 20 x 20 = 400

15. Time Step Considerations Thus, a convective-scale simulation over an equally-sized domain will take 12.5 * (12.5 * 12.5) times as long as a synoptic-scale simulation. This works out to ~1950 times longer! Similarly, a convective-scale simulation over an equally- sized domain will take 5 * (5 * 5) times as long as a mesoscale simulation. This works out to 125 times longer! Thus, using as long of a time step as possible is vital to timely and efficient numerical weather prediction.

16. Boundary Conditions Numerical simulations are boundary-value problems. In the horizontal… Global simulations – domain is periodic; not a horizontal boundary value problem Limited area simulations – domain is limited; need information on the boundaries Boundary value data for limited area simulations typically comes from larger-scale observational or forecast data (e.g., from another simulation).

17. Boundary Conditions And in the vertical… Model atmosphere cannot extend upward to infinity Model atmosphere also constrained by the Earth’s surface The upper boundary is typically constrained to the tropopause or lower stratosphere. The interaction between the atmosphere and the lower boundary is typically parameterized.

18. Initial Conditions Numerical simulations are also initial value problems. The process of providing initial value data to a model is known as initialization. Initial conditions are typically provided by a numerical synthesis of available observations. Methods for obtaining initial conditions are covered in Chapter 6.

19. Initial Conditions The quality of a numerical forecast is strongly constrained by the quality of the initial conditions. But, our observing system is limited – the totality of the true state of the atmosphere is unknowable! Thus, much effort is expended upon trying to obtain as accurate of initial conditions as possible.

20. Initial Conditions There are two general types of initializations: Static initialization (cold start) Dynamic initialization (warm / hot start) Static initializations… Interpolate observations at t=0 to the model grid Ensure that initial conditions are appropriately balanced Begin model forward integration Static initializations require a “spin up” period No data are provided on scales smaller (e.g., convective or topographic scale) than that of the available observations

21. Initial Conditions Dynamic initializations utilize some means of model “spin up” to ensure that local circulations are represented at the initial forecast time. Typically accomplished via forecast cycling. A short-term forecast from an earlier model simulation is used as a “first guess” for the desired simulation. Available observations are assimilated, whether at specified times (three-dimensional) or through time (four- dimensional), to improve the “first guess” analysis. The model is then integrated forward in time.

22. Forecast Cycling

23. Physical Parameterizations The “interesting” parts of equations (2.1)-(2.7) are often difficult to incorporate into a numerical model! Relevant processes include… Diabatic heating (such as with deep, moist convection) Turbulence (friction, etc.) Microphysical processes (phase changes) Land-surface feedbacks Solar (longwave) and atmospheric (shortwave) radiation

24. Physical Parameterizations Thus, the impacts of these physical processes upon the atmosphere are parameterized. Why parameterize these processes? We may not know enough about them to explicitly resolve them within the model. The process(es) may occur on unresolvable scales. The physical relationships may be sufficiently complex so as to require excessive resources to represent explicitly.

25. Physical Parameterizations Underlying physical parameterizations: representing a process based upon its known relationship to resolved variables within the model. For example: we know (or think we know, at least) that turbulence is related to vertical wind shear and static stability, both of which a model can resolve. We will discuss the underlying formulations of physical parameterizations in Chapters 4 and 5.