Copyright © 2017 by Roland Stull. Practical Meteorology: An Algebra-based Survey of Atmospheric Science. v1.02b 20 NUMERICAL WEATHER PREDICTION (NWP) Contents Most weather forecasts are made by computer, 20.1. Scientific Basis of Forecasting 746 and some of these forecasts are further enhanced 20.1.1. The Equations of Motion 746 by humans. Computers can keep track of myriad 20.1.2. Approximate Solutions 749 complex nonlinear interactions among winds, tem- 20.1.3. Dynamics, Physics and Numerics 749 20.1.4. Models 751 perature, and moisture at thousands of locations and altitudes around the world — an impossible 20.2. Grid Points 752 task for humans. Also, data observation, collection, 20.2.1. Nested and Variable Grids 752 20.2.2. Staggered Grids 753 analysis, display and dissemination are mostly au- tomated. 20.3. Finite-Difference Equations 754 20.3.1. Notation 754 Fig. 20.1 shows an automated forecast. Produced 20.3.2. Approximations to Spatial Gradients 754 by computer, this meteogram (graph of weather vs. 20.3.3. Grid Computation Rules 756 time for one location) is easier for non-meteorologists 20.3.4. Time Differencing 757 to interpret than weather maps. But to produce such 20.3.5. Discretized Equations of Motion 758 forecasts, the equations describing the atmosphere 20.4. Numerical Errors & Instability 759 must first be solved. 20.4.1. Round-off Error 759 20.4.2. Truncation Error 760 20.4.3. Numerical Instability 760 20.5. The Numerical Forecast Process 762 20.5.1. Balanced Mass and Flow Fields 763 20.5.2. Data Assimilation and Analysis 765 (a) 20.5.3. Forecast 768 20.5.4. Case Study: 22-25 Feb 1994 768 20.5.5. Post-processing 770 (b) 20.6. Nonlinear Dynamics And Chaos 773 20.6.1. Predictability 773 20.6.2. Lorenz Strange Attractor 773 (c) 20.6.3. Ensemble Forecasts 776 20.6.4. Probabilistic Forecasts 777 20.7. Forecast Quality & Verification 777 (d) 20.7.1. Continuous Variables 777 20.7.2. Binary / Categorical Events 780 20.7.3. Probabilistic Forecasts 782 (e) 20.7.4. Cost / Loss Decision Models 785 20.8. Review 786 20.9. Homework Exercises 787 (f) 20.9.1. Broaden Knowledge & Comprehension 787 20.9.2. Apply 788 Local Time 00 06 12 18 00 06 12 20.9.3. Evaluate & Analyze 790 31 Oct | 1 Nov 2015 | 2 Nov 20.9.4. Synthesize 791 Figure 20.1 Two-day weather forecast for Jackson, Mississippi USA, plotted “Practical Meteorology: An Algebra-based Survey as a meteogram (time series), based on initial conditions valid of Atmospheric Science” by Roland Stull is licensed under a Creative Commons Attribution-NonCom- at 12 UTC on 31 Oct 2015. (a) Temperature & dew-point (°F), mercial-ShareAlike 4.0 International License. View this license at (b) winds, (c) humidity, precipitation, cloud-cover, (d) rainfall http://creativecommons.org/licenses/by-nc-sa/4.0/ . This work is amounts, (e) thunderstorm likelihood, (f) probability of precipi- available at https://www.eoas.ubc.ca/books/Practical_Meteorology/ tation > 0.25 inch. Produced by US NWS. 745 746 CHAPTER 20 • NUMERICAL WEATHER PREDICTION (NWP) INFO • Alternative Vertical Coordinate 20.1. SCIENTIFIC BASIS OF FORECASTING Eqs. (20.1-20.7) use z as a vertical coordinate, where z is height above mean sea level. But local terrain ele- vations can be higher than sea level. The atmosphere 20.1.1. The Equations of Motion does not exist underground; thus, it makes no sense Numerical weather forecasts are made by solv- to solve the meteorological equations of motion at ing Eulerian equations for U, V, W, T, r , ρ and P. heights below ground level. T From the Forces & Winds chapter are forecast To avoid this problem, define aterrain-follow - ing coordinate σ (sigma). One definition forσ is equations for the three wind components (U, V, W) (modified from eqs. 10.23a & b, and eq. 10.59): based on the hydrostatic pressure Pref(z) at any height z relative to the hydrostatic pressure difference be- ∆U ∆U ∆U ∆U tween the earth’s surface (Pref bottom) and a fixed pres- =− U −V −W (20.1) sure (Pref top) representing the top of the atmosphere: ∆t ∆x ∆y ∆z Pzref ()− Pref top 1 ∆P ∆(FUzturb ) σ= − · ·+ fVc − PPref bottom − ref top ρ ∆x ∆ z Pref bottom varies in the horizontal due to terrain eleva- tion (see Fig. 20.A) and varies in space and time due ∆V ∆V ∆V ∆V =− U −V −W (20.2) to changing surface weather patterns (high- and low- ∆t ∆x ∆y ∆z pressure centers). The new vertical coordinate σ var- 1 ∆P ∆(FVzturb ) ies from 1 at the earth’s surface to 0 at the top of the − · ·− fU − domain. ρ ∆y c ∆ z The figure below shows how thissigma coor- dinate varies over a mountain. Hybrid coordi- ∆W ∆W ∆W ∆W nates (Fig. 20.5) are ones that are terrain following =−U −−V W (20.3) near the ground, but constant pressure aloft. ∆t ∆x ∆y ∆z If σ is used as a vertical coordinate, then (U, V) 1 ∆ P′ θθvp − ve ∆(FWzturb ) are defined as winds along aσ surface. The vertical − + · g − advection term in eq. (20.1) changes from W·∆U/∆ z to ρ ∆ z T ∆ z · ve σ·∆U /∆σ , where σ· ·∆is Uanalogous/∆σ to a vertical veloci- ty, but in sigma coordinates. Similar changes must be made to most of the terms in the equations of motion, From the Heat Budgets chapter is a forecast equa- which can be numerically solved within the domain of tion for temperature T (modified from eq. 3.51): 0 ≤ σ ≤ 1. ∆T ∆T ∆T ∆T σ =− U −V −W + Γ (20.4) Pref top = 1 kPa d 0.0 ∆t ∆x ∆y ∆z * 0.2 1 ∆zzrad L ∆rcondensing ∆(Fzturb θ) − +−v 0.4 ρ·Cp ∆ z Cp ∆t ∆ z 0.6 From the Water Vapor chapter is a forecast equa- z 0.8 tion (4.44) for total-water mixing ratio r in the air: 70 T 85 85 1. 0 ∆rTT∆r ∆rTT∆r (20.5) bottom =− U −V −W P ref ∆t ∆x ∆y ∆z 101 90 kPa x Figure 20.A. ρ ∆ Pr ∆ FFrzturb()T +−L Vertical cross section through the atmosphere (white) and ρ ∆ z ∆ z earth (brown). White numbers represent surface air pres- d sure at the weather stations shown by the grey dots. For From the Forces & Winds chapter is the continu- the equation above, P = 70 kPa at the mountain top, ref bottom ity equation (10.60) to forecast air density ρ: (20.6) which differs from Pref bottom = 90 kPa in the valley. ∆ρρ∆ ∆ρρ∆ ∆U ∆V ∆W =−U − V − W − ρ + + ∆t ∆x ∆y ∆z ∆x ∆y ∆z Although sigma coordinates avoid the problem of coordinates that go underground, they create prob- lems for advection calculations due to small differenc- For pressure P, use the equation of state (ideal gas es between large terms. law) from Chapter 1 (eq. 1.23): PT=ℜρ··dv (20.7) R. STULL • PRACTICAL METEOROLOGY 747 In these seven equations: f is Coriolis parameter; c INFO • Alternative Horizontal Coord. P’ is the deviation of pressure from its hydrostatic value; θvp and θve are virtual potential temperatures Spherical Coordinates of the air parcel and the surrounding environ- For the Cartesian coordinates used in eqs. (20.1- ment;Tve is virtual temperature of the environment; 20.8), the coordinate axes are straight lines. Howev- |g| = 9.8 m s–2 is the magnitude of gravitational ac- er, on Earth we prefer to define x to follow the Earth’s –1 curvature toward the East, and definey to follow the celeration; Γd = 9.8 K km is the dry adiabatic lapse rate; F* is net radiative flux; L ≈ 2.5x106 J kg–1 is Earth’s curvature toward the North. If U and V are de- z rad v fined as velocities along these spherical coordinates, the latent heat of vaporization; C ≈ 1004 J·kg–1·K–1 p then add the following terms (20.1b - 20.3b) to the right is the specific heat of air at constant pressure; r∆ con- sides of momentum eqs. (20.1 - 20.3), respectively: densing is the increase in liquid-water mixing ratio UV··tan(φ)·UW associated with water vapor that is condensing; ρL ≈ +−− []2··Ω W ·cos()φ (20.1b) –3 R R 1000 kg·m and ρd are the densities of liquid water oo and dry air; Pr is precipitation rate (m s–1) of water U2 ·tan()φ VW· accumulation in a rain gauge at any height z; ℜd = −− (20.2b) –1 –1 RooR 287 J·kg ·K is the gas constant for dry air; and Tv is the virtual temperature. For more details, see the UV22+ chapters cited with eqs. (20.1 - 20.7). + + []2···ΩφU cos( ) (20.3b) Ro Notice the similarities in eqs. (20.1 - 20.6). All where R ≈ 6371 km is the average Earth radius, ϕ is lat- have a tendency term (rate of change with time) o itude, and Ω = 0.7292x10–4 s–1 is Earth’s rotation rate. on the left. All have advection as the first 3 terms The terms containing Ro are called the curvature on the right. Eqs. (20.1 - 20.5) include a turbulence terms. The terms in square brackets are small com- flux divergence term on the right. The other terms ponents of Coriolis force (see the INFO Box “Coriolis describe the special forcings that apply to individ- Force in 3-D” from the Forces & Winds chapter).