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Using Applied Mathematics to Assess Hurricane Risk

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### Using Applied Mathematics to Assess Hurricane Risk

Kerry Emanuel

Program in Atmospheres, Oceans, and Climate

Massachusetts Institute of Technology

Limitations of a strictly actuarial approach

- >50% of all normalized damage caused by top 8 events, all category 3, 4 and 5
- >90% of all damage caused by storms of category 3 and greater
- Category 3,4 and 5 events are only 13% of total landfalling events; only 30 since 1870
- Landfalling storm statistics are inadequate for assessing hurricane risk

Additional Problem:Nonstationarity of climate

Atlantic Sea Surface Temperatures and Storm Max Power Dissipation

(Smoothed with a 1-3-4-3-1 filter)

Years included: 1870-2006

Power Dissipation Index (PDI)

Scaled Temperature

Data Sources: NOAA/TPC, UKMO/HADSST1

Risk Assessment by Direct Numerical Simulation of DissipationHurricanes:The Problem

- The hurricane eyewall is a region of strong frontogenesis, attaining scales of ~ 1 km or less
- At the same time, the storm’s circulation extends to ~1000 km and is embedded in much larger scale flows

Global Models do not simulate the storms that cause destruction

Modeled

Histograms of Tropical Cyclone Intensity as Simulated by a Global Model with 50 km grid point spacing. (Courtesy Isaac Held, GFDL)

Observed

Category 3

Numerical convergence in an axisymmetric, nonhydrostatic model (Rotunno and Emanuel, 1987)

How to deal with this? model (Rotunno and Emanuel, 1987)

- Option 1: Brute force and obstinacy

Multiply nested grids model (Rotunno and Emanuel, 1987)

How to deal with this? model (Rotunno and Emanuel, 1987)

- Option 1: Brute force and obstinacy
- Option 2: Applied math and modest resources

Time-dependent, axisymmetric model phrased in R space model (Rotunno and Emanuel, 1987)

- Hydrostatic and gradient balance above PBL
- Moist adiabatic lapse rates on M surfaces above PBL
- Boundary layer quasi-equilibrium convection
- Deformation-based radial diffusion
- Coupled to simple 1-D ocean model
- Environmental wind shear effects parameterized

Angular Momentum Distribution model (Rotunno and Emanuel, 1987)

How Can We Use This Model to Help Assess Hurricane Risk in Current and Future Climates?

Risk Assessment Approach: Current and Future Climates?

Step 1: Seed each ocean basin with a very large number of weak, randomly located cyclones

Step 2: Cyclones are assumed to move with the large scale atmospheric flow in which they are embedded, plus a correction for beta drift

Step 3: Run the CHIPS model for each cyclone, and note how many achieve at least tropical storm strength

Step 4: Using the small fraction of surviving events, determine storm statistics

Details: Emanuel et al., Bull. Amer. Meteor. Soc, 2008

Synthetic Track Current and Future Climates?Generation:Generation of Synthetic Wind Time Series

- Postulate that TCs move with vertically averaged environmental flow plus a “beta drift” correction
- Approximate “vertically averaged” by weighted mean of 850 and 250 hPa flow

Synthetic wind time series Current and Future Climates?

- Monthly mean, variances and co-variances from re-analysis or global climate model data
- Synthetic time series constrained to have the correct monthly mean, variance, co-variances and an power series

Risk Assessment Approach: Current and Future Climates?

Step 1: Seed each ocean basin with a very large number of weak, randomly located cyclones

Step 2: Cyclones are assumed to move with the large scale atmospheric flow in which they are embedded, plus a correction for beta drift

Step 3: Run the CHIPS model for each cyclone, and note how many achieve at least tropical storm strength

Step 4: Using the small fraction of surviving events, determine storm statistics

Details: Emanuel et al., Bull. Amer. Meteor. Soc, 2008

Comparison of Random Seeding Genesis Locations with Observations

Example: 50 Synthetic Tracks Observations

Cumulative Distribution of Storm Lifetime Peak Wind Speed, with Sample of 1755Synthetic Tracks

95% confidence bounds

Return Periods with Sample of

Coupling large hurricane event sets to surge models (with Ning Lin)

- Couple synthetic tropical cyclone events (Emanuel et al., BAMS, 2008) to surge models
- SLOSH
- ADCIRC (fine mesh)
- ADCIRC (coarse mesh)

- Generate probability distributions of surge at desired locations

ADCIRC Mesh CNRM model

Application to Other Climates CNRM model

Summary: CNRM model

- History too short and inaccurate to deduce real risk from tropical cyclones
- Global (and most regional) models are far too coarse to simulate reasonably intense tropical cyclones
- Globally and regionally simulated tropical cyclones are not coupled to the ocean

- We have developed a technique for downscaling global models or reanalysis data sets, using a very high resolution, coupled TC model phrased in angular momentum coordinates
- Model shows skill in capturing spatial and seasonal variability of TCs, has an excellent intensity spectrum, and captures well known climate phenomena such as ENSO and the effects of warming over the past few decades

Spare Slides or reanalysis data sets, using a very high resolution, coupled TC model phrased in angular momentum coordinates

- Ocean Component or reanalysis data sets, using a very high resolution, coupled TC model phrased in angular momentum coordinates:
((Schade, L.R., 1997: A physical interpreatation of SST-feedback.

Preprints of the 22nd Conf. on Hurr. Trop. Meteor., Amer. Meteor. Soc., Boston, pgs. 439-440.)

- Mixing by bulk-Richardson number closure
- Mixed-layer current driven by hurricane model surface wind

Couple to Storm Surge Model (SLOSH) or reanalysis data sets, using a very high resolution, coupled TC model phrased in angular momentum coordinates

Courtesy of Ning Lin

Surge map for single event or reanalysis data sets, using a very high resolution, coupled TC model phrased in angular momentum coordinates

Histogram of the SLOSH-model simulated storm surge at the Battery for 7555 synthetic tracks that pass within 200 km of the Battery site.

Surge Return Periods, New York City Battery for 7555 synthetic tracks that pass within 200 km of the Battery site.

Genesis Distributions Battery for 7555 synthetic tracks that pass within 200 km of the Battery site.

Why does frequency decrease? Battery for 7555 synthetic tracks that pass within 200 km of the Battery site.

Critical control parameter in CHIPS:

Entropy difference between boundary layer and middle troposphere increases with temperature at constant relative humidity

Change in Frequency when T held constant in Battery for 7555 synthetic tracks that pass within 200 km of the Battery site.

Probability Density by Storm Lifetime Peak Wind Speed, Explicit and Downscaled Events

Change in Power Dissipation with Global Warming Explicit and Downscaled Events

Analysis of satellite-derived tropical cyclone Explicit and Downscaled Events

lifetime-maximum wind speeds

Trends in global satellite-derived tropical cyclone maximum wind speeds by quantile, from 0.1 to 0.9 in increments of 0.1.

Box plots by year. Trend lines are shown

for the median, 0.75 quantile, and 1.5 times the interquartile range

Elsner, Kossin, and Jagger, Nature, 2008

Bringing Physics and Math to Bear Explicit and Downscaled Events

Physics of Mature Hurricanes Explicit and Downscaled Events

Theoretical Upper Bound on Hurricane Maximum Wind Speed: Explicit and Downscaled EventsPOTENTIAL INTENSITY

Surface temperature

Ratio of exchange coefficients of enthalpy and momentum

Outflow temperature

Air-sea enthalpy disequilibrium

Annual Maximum Potential Intensity (m/s) Explicit and Downscaled Events

Compare two simulations each from 7 IPCC models: Explicit and Downscaled Events

1.Last 20 years of 20th century simulations2. Years 2180-2200 of IPCC Scenario A1b (CO2 stabilized at 720 ppm)IPCC Emissions Scenarios Explicit and Downscaled Events

This study

Basin-Wide Percentage Change in Power Dissipation Explicit and Downscaled Events

Basin-Wide Percentage Change in Storm Frequency Explicit and Downscaled Events

7 Model Consensus Change in Storm Frequency Explicit and Downscaled Events

Tracks of 18 “most dangerous” storms for the Battery, under the current (left) and A1B (right) climate, respectively

Explicit (blue dots) and downscaled (red dots) genesis points for June-October for Control (top) and Global Warming (bottom) experiments using the 14-km resolution NICAM model. Collaborative work with K. Oouchi.

Return levels for Tampa, including three cases : control (black), A1B (blue), A1B with 1.1ro and 1.21 rm (red)

Simulated vs. Observed Power Dissipation Trends, 1980-2006 (black), A1B (blue), A1B with 1.1ro and 1.21

Total Number of (black), A1B (blue), A1B with 1.1ro and 1.21 United States Landfall Events, by Category, 1870-2004

U.S. Hurricane Damage, 1900-2004 (black), A1B (blue), A1B with 1.1ro and 1.21 , Adjusted for Inflation, Wealth, and Population

250 hPa zonal wind modeled as Fourier series in time with random phase:

where T is a time scale corresponding to the period of the lowest frequency wave in the series, N is the total number of waves retained, and is, for each n, a random number between 0 and 1.

Example: random phase:

The time series of other flow components: random phase:

or

where each Fi has a different random phase, and A satisfies

where COV is the symmetric matrix containing the variances and covariances of the flow components.Solved by Cholesky decomposition.

Track: random phase:

Empirically determined constants:

6-hour zonal displacements in region bounded by 10 random phase:o and 30o N latitude, and 80o and 30o W longitude, using only post-1970 hurricane data

Calibration random phase:

Absolute genesis frequency calibrated to globe during the period 1980-2005

Genesis rates random phase:

Western North Pacific

Southern Hemisphere

Eastern North Pacific

Atlantic

North Indian Ocean

Seasonal Cycles AMM)

Atlantic

3000 Tracks within 100 km of Miami AMM)

95% confidence bounds

5000 synthetic storm tracks under current climate. (Red portion of each

track is used in surge analysis.)

SLOSH mesh for the New York area ( portion of eachJelesnianski et al., 1992)

Surge Return Periods for The Battery, New York portion of each

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