Seasonal weather forecasting — the motivations — the

Seasonal weather forecasting: - the motivations - the capabilities - the atmospheric physics involved photo credit: J. D. Mc. Alpine photo credit: pixomar /freedigitalphotos. net J. D. Mc. Alpine ATMS 611, autumn 2010 Class project Graphic from www. cpc. noaa. gov photo credit: Michelle Meiklejohn /freedigitalphotos. net

“Them weatherfolks can’t tell me if it’s going to rain tomorrow or not!” • Motivations - our sample case: Tacoma Umbrellas, inc. - economic planning, weather derivatives, and actuaries • Capabilities - limits of weather forecasting - chaos theory applied to computational modeling - probability/statistics vs. deterministic forecasting (why a turbulence-guru is interested in this topic) • Useful high-frequency climate modes and the physics involved - ENSO and ocean-atmosphere coupling - Arctic Oscillation and cryosphere-atmosphere coupling • Conclusion - Buy or Sell? : what do today’s indicators say

Our motivation: limit risk Tacoma Umbrella, LLC • Dry season may lead to losses • Looking into weather derivatives to limit risk of loss • Assess climatology AND seasonal forecasts photo credit: admin /freedigitalphotos. net

Results of climate/ demand study Rainfall (inches) Departure from norm Probab. of scen. 28 -30% 0. 01 30 -25% 0. 02 0. 03 32 -20% 0. 03 0. 06 34 -15% 0. 16 36 -10% 0. 11 0. 27 38 -5% 0. 13 0. 4 40 0% 0. 2 0. 6 42 5% 0. 15 0. 75 44 10% 0. 08 0. 83 46 15% 0. 06 0. 89 48 20% 0. 04 0. 93 50 25% 0. 03 0. 96 52 30% 0. 02 0. 98 54 35% 0. 01 0. 99 56 40% 0. 01 1 Cumulative probabiliy

Results of climate/ demand study rainfall % of normal probability of % rainfall units sold profit $ return on investment -30% 0. 01 20000 -16. 67% -25% 0. 02 20500 -11275 0. 03 21500 -5375 -4. 35% -15% 0. 1 22500 1125 0. 11 23000 6900 5. 36% -5% 0. 13 24500 18375 13. 76% 0% 0. 2 26000 18. 87% 5% 0. 15 28000 56000 39. 22% 10% 0. 08 32500 82875 53. 68% 15% 0. 06 36500 102200 60. 87% 20% 0. 04 38500 113575 64. 84% 25% 0. 03 40000 140000 77. 78% 30% 0. 02 41000 149650 82. 02% 35% 0. 01 42000 161700 88. 51% 40% 0. 01 42500 180625 100. 00% • Expected profit: $44, 200 • 65% chance of a return < expected • ~10% chance of loss 0. 88% -10% Expected return on investment: 29% -9. 40% -20% •

Weather derivative: Linear put-option purchase Payoff function: Where L$=D(K-L), the maximum payout D is the “tick”: the amount of money per unit = $/(% precip below normal) example: $2500/1% K is the “strike”: the threshold value of payout = 3. 5% above normal X is the actual value that occurs = ? Look at climate indicators to guess what % of precip we can expect this year

Weather derivative: Linear put-option purchase Precip. % depart. Prob. of % depart. Profit $ % return Put pay out Profit $ with put % return with put -30% 0. 01 -20000 -16. 67% 31825 5412. 5 4. 51% -25% 0. 02 -11275 -9. 40% 27075 9387. 5 7. 83% -20% 0. 03 -5375 -4. 35% 22325 10537. 5 8. 52% -15% 0. 1 1125 0. 88% 17575 12287. 5 9. 67% -10% 0. 11 6900 5. 36% 12825 13312. 5 10. 34% -5% 0. 13 18375 13. 76% 8075 20037. 5 15. 01% 0% 0. 2 26000 18. 87% 3325 22912. 5 16. 63% 5% 0. 15 56000 39. 22% 0 49587. 5 34. 73% 10% 0. 08 82875 53. 68% 0 76462. 5 49. 53% 15% 0. 06 102200 60. 87% 0 95787. 5 57. 05% 20% 0. 04 113575 64. 84% 0 107162. 5 61. 17% 25% 0. 03 140000 77. 78% 0 133587. 5 74. 22% 30% 0. 02 149650 82. 02% 0 143237. 5 78. 51% 35% 0. 01 161700 88. 51% 0 155287. 5 85. 00% 40% 0. 01 180625 100. 00% 0 174212. 5 96. 45% Example Case: worse case return > inflation D (tick): $950 / % precip departure K (strike): ~3. 5% $6400: Expected payout Put option cost: $6400+fees

Motivations: overview Imagine the economic advantages of an accurate seasonal forecast!!!! Weather Derivatives: - 50 billion $ industry photo credit: graur razvan ionut /freedigitalphotos. net Commodity pricing: -Impacts a huge portion of the economy photo credit: pixomar /freedigitalphotos. net photo credit: francesco marino /freedigitalphotos. net Other items: -Emergency management planning -Utility planning and management -Product demand mapping photo credit: dan /freedigitalphotos. net

Seasonal forecasting capabilities Weather Forecast Seasonal Forecast TIMESCALE - Hours -Months PHYSICS -Eqs. of motion -Parameterizations -Statistical/ Dynamical -Energy balances - Ensembles ACCURACY 3 -10 days 2 -6 months PRODUCTS -Temps, precip, winds: -Anomolies of temp, precip:

Weather Forecasting Limitations “ the atmosphere exhibits no periodicities of the kind that enable one to predict the weather in the same way one predicts the tides” Charney (1951) 1) Initial Conditions a) 6 -hr forecasted field b) objective analysis c) data assimilation 2) Smoothing and filtering - time-step limits based on grid size and highest frequency waves 3) Forward integration photo credit: Graeme Weatherston /freedigitalphotos. net Why have forecasts improved? a) Increased computing power c) Better data assimilation methods From Kalnay (2003) b) improved physics parameterizations c) Increased data volume and accuracy

Weather Forecasting Limitations Lorenz’s fundamental theorem of predictability: Unstable systems have a finite limit of predictability, and conversely, stable systems are infinitely predictable (since they are either periodic or stationary) [Kalnay, 2003] From Kalnay (2003) From: http: //www. stsci. edu/~lbradley/seminar/attractors. html • The non-linear primitive equation set has a finite limit of predictability • Even with a perfect set of equations and near identical initial conditions a total loss in forecast skill after 2 weeks (in best case)

Weather Forecasting Limitations • Longer term forecasting abilities: - ensembles - interannual predictability • Potential Predictability: Total variance – weather noise • Regions with high potential predictability: Tropics, Arctic, Deserts • Global predictability using teleconnections Precip/ Temp correlations to oscillation indices • Major oscillations: ENSO and AO www. rap. ucar. edu/weather

El-Nino Southern Oscillation (ENSO) • El Nino: weakening of Walker circulation • La Nina: strong Walker circulation • Two modes are analogous to strange attractors in non -linear chaotic system • Strongly impacts on seasonal averages in tropics and extratropics Walker Circulation • Non-periodic: random transfer between states

El-Nino Southern Oscillation (ENSO) feedback mechanisms Increasing instability and convection Increased Moisture flux convergence West Pacific Low strengthens Easterly trade wind response increases Increasing heat and moisture flux to boundary layer East Pacific High strengthens East Pacific Ocean Upwelling -La Nina: strong cycle, El Nino: weak cycle What processes slow/strengthen the non-linear feedback cycle? 1) Rossby/Kelvin oceanic waves (delayed oscillator mechanism) 2) Mid-latitude systems (seasonal cycle mechanism) 3) Large tropical disturbances / Hurricanes OVERALL: High-frequency ‘weather noise’ determines lifetime of feedback

Predictability of ENSO Onset of El Nino: - sudden collapse of easterlies Onset of La Nina: - sudden acceleration of easterlies • Moisture-flux convergence is key (unpredictable > 2 weeks) • SST distribution is secondary driver (but most predictable) • Once change is evident: a coupled ocean-atmospheric model can derive the rate of change accurately. MEI: weighted average of indices concerning SST, winds, air temp. , cloudiness

Arctic Oscillation Overview *A fundamental internal mode of variability in the atmosphere *May be modulated by various external forcings Positive phase: Jet stream north -warmer, drier midlats -low SLP in Arctic -high SLP in Pacific, Europe Negative phase: Jet stream south -cooler, wetter -Continental Arctic outbreaks -high SLP in Arctic -Low SLP in Pacific, Europe

Arctic Oscillation Overview MAIN EOF Influence of snow-cover on the magnitude of the Arctic oscillation

Snow effects on local mean seasonal climate Heavy snow cover anomaly SR Little or no snow cover SR 1) LR LH & SH SR SR LR 2) LR LH & SH 3) -0. 5 C Atmosphere Cools in anomaly case Atmosphere Warms in anomaly case LH & SH Net Result: Slight to no cooling of column: EXCEPT!!:

Snow effects on local mean seasonal climate Snow anomalies have a greater influence in some regions

Early snow-cover: signal for AO growth

S-Cast seasonal forecasting system

Today’s Indicators: AO www. cpc. noaa. gov

Today’s Indicators: ENSO www. cpc. noaa. gov

Tacoma Umbrella Decisions? ENSO

Tacoma Umbrella Decisions? ENSO http: //www. wrcc. dri. edu/enso/octmar 45. gif

Tacoma Umbrella Decisions? AO • Positive phase illustrated • Negative phase likely • Outbreak of snow more likely • Precipitation similar http: //jisao. washington. edu/wallace/ncar_notes/

Tacoma Umbrella Decisions? Put Precip. % depart. Prob. of % depart. Profit $ % return pay out Expected payout Profit $ with put % return with put -30% 0. 01 -20000 -16. 67% 31825 318. 25 5412. 5 4. 51% -25% 0. 02 -11275 -9. 40% 27075 541. 5 9387. 5 7. 83% -20% 0. 03 -5375 -4. 35% 22325 669. 75 10537. 5 8. 52% -15% 0. 1 1125 0. 88% 17575 1757. 5 12287. 5 9. 67% -10% 0. 11 6900 5. 36% 12825 1410. 75 13312. 5 10. 34% -5% 0. 13 18375 13. 76% 8075 1049. 75 20037. 5 15. 01% 0% 0. 2 26000 18. 87% 3325 665 22912. 5 16. 63% 5% 0. 15 56000 39. 22% 0 0 49587. 5 34. 73% 10% 0. 08 82875 53. 68% 0 0 76462. 5 49. 53% 15% 0. 06 102200 60. 87% 0 0 95787. 5 57. 05% 20% 0. 04 113575 64. 84% 0 0 107162. 5 61. 17% 25% 0. 03 140000 77. 78% 0 0 133587. 5 74. 22% 30% 0. 02 149650 82. 02% 0 0 143237. 5 78. 51% 35% 0. 01 161700 88. 51% 0 0 155287. 5 85. 00% 40% 0. 01 180625 100. 00% 0 0 174212. 5 96. 45%

Conclusions: - Financial impacts of weather - tools to limit risk - great potential of long-term weather forecasting photo credit: Peter Haken /freedigitalphotos. net photo credit: Francesco Marino /freedigitalphotos. net - Predictability - limits of non-linear dynamical system predictability - chaos theory - ENSO and AO - most influential seasonal climate modes - overview of physical concepts - using as a tool to make financial decisions