SO441 Synoptic Meteorology

1. SO441 Synoptic Meteorology Numerical weather prediction GFS: 23km Δx NAM: 12km Δx NAM: 4km Δx 3-hour precipitation totals ending 12 UTC 04 Sept 2012

2. A bit of important history . . . • What is numerical weather prediction? • An integration forward in time (and space) of fundamental governing equations: 6 equations, 6 unknowns • Equation of state • Navier Stokes (from F=ma) • Continuity equation • Thermodynamic energy equation (from 1st & 2nd laws of thermodynamics) • Why is it so important? • Moved meteorology away from a collection of rules-of-thumb and educated guesses to an analytic science grounded in physics and calculus

3. What, exactly, is a dynamical model? • A set of computer programs/lines of code (usually written in FORTRAN), designed to simulate the real atmosphere • Integrating the governing equations forward in time • Using “finite differencing” techniques to evaluate partial derivatives • What does a dynamical model need to succeed? • A good set of governing equations • Accurate initial and boundary conditions

4. A simple example • The setup: • A cold front has passed through Oklahoma City, OK (KOKC) at 1800 UTC 10 December 2011 • The initial surface temperature (measured at 2 m above the earth’s surface) is known from the instrument at the KOKC observing location (Will Rogers World Airport) • The temperature gradient (change in temperature) behind the cold front has been observed to be fairly uniform • The wind behind the front is blowing steadily from the northwest at 15 kts (7 m s-1). • Based on the information provided, can we quantitatively predict the temperature in Oklahoma City at 0000 UTC 11 December? (adapted from Lackmann 10.2, pg. 250)

5. A simple example • To setup the model, even though this is a simple example, we actually still need to make 4 assumptions about the factors influencing temperature in OKC • Temperature advection (transport of air to a new location) is the dominant factor • Diurnal heating or cooling is not important • Processes relating to clouds and precipitation do not come into play • If these assumptions are made, the governing equation for temperature uses the following: • Which says: “Quantity changes in time at a fixed point because of advection at that point”. • Temperature advection then becomes: • Where u is the east-west wind, v is the north-south wind, and x and y are the distances in east-west and north-south directions, respectively. (Note: vertical advection, with wand z, is conveniently ignored here)

6. A simple example • Let’s simplify the math even further by rotating the coordinate system to force the temperature gradient to look like the following: • KOKC is located at the point (i,j), and we know the temperature at the point (i-1,j) and (i+1,j). We also know the wind is now blowing straight from the west (i.e., a “westerly wind”). Let’s discretize the advection equation: KOKC

7. A simple example • Rearrange the advection equation to solve for the final temperature: • Plugging in the numerical values from the figure, we see that the predicted temperature at 0000 UTC will be about 6°C less than the temperature at 1800 UTC, all because of cold-air advection behind the front:

8. Grid spacing in a model From: http://www.drjack.info/INFO/model_basics.html

9. Parameterizations • For all processes that take place inside a grid box, i.e., they are smaller than the grid spacing of the model, the model cannot “resolve” them explicitly • Processes requiring parameterization: • Planetary boundary layer • Turbulence (energysmaller scales) • Flux of momentum, heat, and water vapor • Land-surface • Water and water vapor cycle • Microphysics (clouds) • Precipitation The effects that model physics parameterizations attempt to simulate are generally unresolvable at grid scales

10. Parameterizations: planetary boundary layer • Turbulent fluxes need to be “transported” from within the planetary boundary layer to outside it • Example: momentum flux in the governing equation for u: • Similar equations exist for other flux quantities: • Heat • Water vapor Example of the “boundary layer”

11. Parameterizations:Land-surface models • Many complex processes to “pass” on to the model: • Evaporation • Transpiration • Infiltration/runoff • Sublimation • Condensation • Note that nearly all have something to do with water!

12. Parameterizations:Cloud microphysics • Imagine a cloud occupies a model 3-d grid box • How many water molecules are there? • What shapes/sizes are those molecules? • What phases of water (gas, liquid, solid) are present? • Is there more condensation/freezing • And thus heat being added to the atmosphere • Or is there more evaporation/melting • And thus heat being removed?

13. Parameterizations:Cumulus parameterization • Clouds come in many shapes and sizes • Most clouds are between 0.5-3 km in diameter • Thus smaller than model grid boxes • To get precipitation in the model, need to parameterize clouds • “Trigger” precipitation when certain thresholds are met: relative humidity above 70-80%, positive w (rising motion), CAPE • Effect is to warm and dry the atmosphere above the surface • Multiple “schemes” for cumulus parameterization: each differs in how it adjusts the atmosphere column in response to precipitation • Betts-Miller-Janic • Arakawa-Schubert • Kain-Fritsch

14. Parameterizations:Cumulus parameterization • Example: how to handle precipitation in a model grid cell • Difference between cloud water for an explicit (a) vs. parameterized (b) precipitation event

15. Data assimilation • What is data assimilation? • A means of combining all available information to construct the best possible estimate of the state of the atmosphere • What data are assimilated? • In-situ surface observations: temp, dew point, pressure, cloud cover, wind speed and direction, current weather, visibility • Like the weather station we have on the field out at the corner of Hospital Point • Ships, buoys • In-situ upper-air observations • Radiosondes, aircraft • Remotely sensed observations • Satellites: clouds, but also temperature, water vapor, and even vertical profiles • Radar: precipitation, air motion • GPS radio occultation Radiosonde network ACARS observations

16. Data assimilation • How does it work? • Observations must be blended together and interpolated to the nearest model grid point (horizontal and vertical) • Not an easy process! • Which data source is most important? • Sensitivity studies (data denial) show that it depends Flow chart for the GFS model Data sources: Radiosondes, ACAR, and surface Sensitivity of the u-wind forecast to the various components

17. Ensemble modeling • Predicting future weather using a suite of several individual forecasts • Idea began with Ed Lorenz at MIT in 1950s: tried to repeat an experiment he had made with an equation. Found that very small rounding differences completely changed the mathematical answer! • This is now known as the “butterfly effect”: an even miniscule difference in the initial state will eventually amplify and result in a different forecast • Lorenz proposed an upper-limit on weather forecasts of 2 weeks • Also found that some situations “degrade” much faster than others • Ensemble weather prediction attempts to show how fast the solutions “degrade”

18. Why are there still errors in NWP today? • Grid spacing • The atmosphere is divided into cells, and the center point (or edge points) is (are) forecasted • Topography, ground cover, etc. vary – sometimes dramatically – inside one grid box (but the forecast for that grid box gives only one unique value) • Equations of motion are non-linear • Changes in one variable feed back to all others • Differential equations are solved by discretizing them in time • Rounding occurs in finite-differencing techniques • This noise, as found by Dr. Lorenz, leads to growing error • Initial conditions • Current atmospheric state is unknown at all points • Parameterizations • Sub-grid processes are estimated