A matter of degree, here or there: Changing extreme daily rainfalls, Zarcero, Costa Rica. Peter Waylen and Marvin Quesada Universidad de Costa Rica, San Ramón

Flooding – Nosara, Costa Rica, October 5, 2018 http://www.ticotimes.net/2018/10/05/live-heavy-rains-lead-to-mass-floods-in-the-pacific

Four Americans and their guide drown during a rafting accident on the Naranjo River, Costa Rica – October 22, 2018. http://time.com/5430802/costa-rica-rafting-accident/

Zarcero

Monthly Precipitation Regimes Gainesville and Zarcero

Frequency Daily Probability Magnitude Daily Mean Fifteen-day Moving Averages

Definition of Variables Prendergrass, A.G., 2018 Science, June 8th, vol. 360 p. 1072-3

Precipitation Climatology

Precipitation Climatology North Atlantic Anticyclone H Northeast Trades

Precipitation Climatology Cross- Equatorial Westerlies 10 Inter-tropical Convergence Zone

Dry Season – November-April N.E. Trades ITCZ

Pre-Veranillos – May -June Trades ITCZ Cross- Equatorial Westerlies

Veranillos– July-August N.E. Trades ITCZ Cross- Equatorial Westerlies

Post-Veranillos – September-October N.E. Trades ITCZ Cross- Equatorial Westerlies

Dry Season – November-April N.E. Trades ITCZ

Macro-scale Drivers of Precipitation Variability HIGH LOW Pacific (ENSO) Central American Cordillera Atlantic (AMO)

(AMO)

El Niño - Southern Oscillation (ENSO) 13 Warm Phase (El Niño) Incomplete/missing rainfall records Source: FSU COAPS Japanese Met . Agency 13 Cold Phase (La Niña)

Boring (but essential) Bits Annual number of extremes (Poisson)   𝑝 𝑥=𝑛 = 𝑒 −Λ Λ n 𝑛! Λ = Mean number of occurrences per year Modified for seasonal occurrence (Non-homogeneous Poisson)   𝑝 𝑡 𝑥=𝑛 = 𝑒 − 𝜆 𝑡 𝜆 𝑡 𝑛 𝑛! 𝜆 𝑡 = Mean number of occurrences per year up to time t Magnitude above critical level (Exponential) 𝐹 𝑋≤𝑥 =1− (𝑒− 𝛼 𝑥 ) 𝛼=Average size of extremes above selected threshold

INCREASING THE THRESHOLD THAT DEFINES EXTREME Threshold = 60mm 60 80 100 120 140 160 180 200 Pcrit=60mm, # Events = 100, in 50yrs Λ = 100/50 = 2.0 Events per yr. Mean = 30mm over threshold (0r 90mm)

INCREASING THE THRESHOLD THAT DEFINES EXTREME Threshold = 60mm Threshold = 80mm 60 80 100 120 140 160 180 200 60 80 100 120 140 160 180 200 Pcrit=60mm, # Events = 100, in 50yrs Λ = 100/50 = 2.0 Events per yr. Mean = 30mm over threshold (0r 90mm) Pcrit=80mm,

INCREASING THE THRESHOLD THAT DEFINES EXTREME Threshold = 60mm Threshold = 80mm 60 80 100 120 140 160 180 200 60 80 100 120 140 160 180 200 Pcrit=60mm, # Events = 100, in 50yrs Λ = 100/50 = 2.0 Events per yr. Mean = 30mm over threshold (0r 90mm) Pcrit=80mm, # Events = 50, in 50yrs Λ = 50/50 = 1.0 Events per yr.

INCREASING THE THRESHOLD THAT DEFINES EXTREME Threshold = 60mm Threshold = 80mm 60 80 100 120 140 160 180 200 60 80 100 120 140 160 180 200 Pcrit=60mm, # Events = 100, in 50yrs Λ = 100/50 = 2.0 Events per yr. Mean = 30mm over threshold (0r 90mm) Pcrit=80mm, # Events = 50, in 50yrs Λ = 50/50 = 1.0 Events per yr. Mean = 30mm over threshold (or 110mm)

INCREASING THE THRESHOLD THAT DEFINES EXTREME Threshold = 60mm Threshold = 80mm 60 80 100 120 140 160 180 200 Threshold = 150mm 60 80 100 120 140 160 180 200 60 80 100 120 140 160 180 200 Pcrit=60mm, # Events = 100, in 50yrs Λ = 100/50 = 2.0 Events per yr. Mean = 30mm over threshold (0r 90mm) Pcrit=80mm, # Events = 50, in 50yrs Λ = 50/50 = 1.0 Events per yr. Mean = 30mm over threshold (or 110mm) Pcrit=150mm,

INCREASING THE THRESHOLD THAT DEFINES EXTREME Threshold = 60mm Threshold = 80mm 60 80 100 120 140 160 180 200 Threshold = 150mm 60 80 100 120 140 160 180 200 60 80 100 120 140 160 180 200 Pcrit=60mm, # Events = 100, in 50yrs Λ = 100/50 = 2.0 Events per yr. Mean = 30mm over threshold (0r 90mm) Pcrit=80mm, # Events = 50, in 50yrs Λ = 50/50 = 1.0 Events per yr. Mean = 30mm over threshold (or 110mm) Pcrit=150mm, # Events = 5, in 50yrs Λ = 5/50 = 0.1 Events per yr.

INCREASING THE THRESHOLD THAT DEFINES EXTREME Threshold = 60mm Threshold = 80mm 60 80 100 120 140 160 180 200 Threshold = 150mm 60 80 100 120 140 160 180 200 60 80 100 120 140 160 180 200 Pcrit=60mm, # Events = 100, in 50yrs Λ = 100/50 = 2.0 Events per yr. Mean = 30mm over threshold (0r 90mm) Pcrit=80mm, # Events = 50, in 50yrs Λ = 50/50 = 1.0 Events per yr. Mean = 30mm over threshold (or 110mm) Pcrit=150mm, # Events = 5, in 50yrs Λ = 5/50 = 0.1 Events per yr. Mean = 30mm over threshold (or 180mm)

For Modeling Purposes, What is the Initial Critical Level? In absence of “applied” answer, let the data speak for themselves. About 99% of all historic (56 years), non-zero, daily rainfall totals are less than 70mm.

Annual Number of days with Extreme Rainfalls AMO Phase

Annual Average Number of Events – 1.41 AMO ENSO 1950-65 1.33 1.08 Warm 1966-94 1.05 1.40 Neutral 1995- 2017 1.81 1.77 Cold

Number and Timing of Events AMO ENSO Between Drivers

Number and Timing AMO ENSO Ext reme Phases

Changes in Mean Number of Events. Independence of Ocean Phases.   ENSO Phase Period Warm Cold Neutral Sum Rel. Frequ. 1950-1965 4 7 15 0.268 1966-1994 12 20 0.357 1995-2017 5 11 21 0.375 13 30 56 1.000 0.232 0.536 Chi-squared contingency test: Insufficient evidence to reject null hypothesis of independence between frequency of ENSO phases and the phase of the AMO at the 0.05 level. (i.e. they seem to be independent).

Thanks Caroline! Cai, W., Wang, G., Dewitte, B., Wu, L., Santoso, A., Takahashi, K., Yang, Y., Carréric, A. and McPhaden, M.J., 2018. Nature, 564(7735), p.201. Cheng, L., Abraham, J., Hausfather, Z. and Trenberth, K.E., 2019. How fast are the oceans warming?. Science, 363(6423), pp.128-129.

Changing the Critical Level AMO Daily rainfall totals exhibit exponential-like distributions in their upper tail. ENSO

Really Pushing It To The Extreme ! Closed Circles – data used in calibration Open Circles – data used for validation Exponential does not appear to model Extreme Tail Appropriately. Alternatives: Generalized Pareto Gamma Mixed Exponential

Exponential ?

Conlusions: Searching for simple linear-type trends is at best naïve. Although still the largest known cause of interannual variability of climate, ENSO is not the only cause and its effect may be moderated or amplified by other variables. Such extant causes of variability, unless understood, can obscure any trends, yet once enumerated and accounted for, may actually highlight lower frequency changes. The empirical results that emerge concur with the behaviors observed in a variety of simulation studies. The exponential model of magnitudes is inadequate to represent the upper end of the distribution of daily rainfall totals. Particularly interested in pursuing the option of mixed exponentials with Prof. Patra.