1. SO441 Synoptic Meteorology Numerical weather prediction 3-hour precipitation totals ending 12 UTC today (11 Feb 2015) GFS: 23km ΔxNAM: 12km Δx NAM: 4km Δx

2. Sites http://www.weather.gov/ctp/modelData http://mp1.met.psu.edu/~fxg1/ewall.html

3. 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 1 st & 2 nd 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

4. 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

5. A simple example The setup: – A cold front has passed through Oklahoma City, OK (KOKC) a few hours ago and is now located 75km to the SE, and it is now 1800 UTC – The initial surface temperature at 1800Z (measured at 2 m above the earth’s surface) is known to be 12°C 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 ~1°C every 25km directly behind the cold front (which is oriented SW to NE) with isotherms parallel to the front – 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)

6. Finite Differencing: A simple example To setup the model, even though this is a simple example, we actually still need to make assumptions about the factors influencing temperature in OKC – Horizontal 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 w and z, is conveniently ignored here)

7. 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

8. 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:

9. Grid spacing in a model From: http://www.drjack.info/INFO/model_basics.htmlhttp://www.drjack.info/INFO/model_basics.html In the upper-left figure, the point represent the centers of hypothetical grid boxes. Most models stagger their solutions, solving the wind (u,v,w) at the edges of the boxes and the mass (temp, mixing ratio, height/pressure) at the centers

10. 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

11. 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”

12. 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!

13. 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?

14. 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

15. 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

16. 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

17. 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

18. 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”

19. 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