Jie Gao, Rick Luettich University of North Carolina at Chapel Hill Jason Fleming

Development and Initial Evaluation of A Generalized Asymmetric Tropical Cyclone Vortex Model in ADCIRC

Jie Gao, Rick LuettichUniversity of North Carolina at Chapel Hill

Jason FlemingSeahorse Coastal Consulting

ADCIRC Users Group Meeting, USACE Vicksburg, MS, April 29, 2013

Introduction

NW NE

SW SE

Storm forward motion

Schematic cross section of a hurricane wind field

Goal: use tropical cyclone storm parameters provided in NHC ATCF Best Track or Forecast Advisories to generate dynamically consistent pressure and wind fields for storm surge predictions.

Storm Parameterslon, lat of center of eye VmaxR64 , R50 , R34 in 4 storm quadrantsPc - ATCF BT only

Pn , Rmax - estimated separately

Holland (1980) Wind Model (ADCIRC: NWS=8)

𝑃 (𝑟 )=𝑃𝑐+ (𝑃𝑛−𝑃𝑐 )𝑒−𝐴 /𝑟 𝐵

Hyperbolic hurricane pressure profile (Schloemer 1954):

(1)

Substitute into the gradient wind equations, the vortex wind velocity is:

(2)

Hurricane Isabel of 2003 showing the circular, symmetric eye associated with annular hurricanes at Sept 13 2013 1710Z.

The Holland Wind Model produces a symmetric hurricane vortex with the spatially constant Rmax.

𝑅𝑚𝑎𝑥

𝑉 𝑔 (𝑟 )=√𝑉𝑚𝑎𝑥2 𝑒(1− ( 𝑅𝑚𝑎𝑥 /𝑟 )𝐵 ) (𝑅𝑚𝑎𝑥 /𝑟 )𝐵+( 𝑟𝑓2 )

2

−( 𝑟𝑓2 )

Substitute Eqs. (3) & (4) into Eqs. (1) & (2) - final Holland equations:

𝑃 (𝑟 )=𝑃𝑐+ (𝑃𝑛−𝑃𝑐 )𝑒−(𝑅𝑚𝑎𝑥 /𝑟 ) 𝐵

(6)

(5)

Derive relationships for scaling parameters A, B

• Assume V=Vmax @ r =Rmax (dV/dr = 0 @ r =Rmax)• Assume << Vmax = cyclostrophic wind balance @ r =Rmax • Setting dV/dr = 0 from Eq. (2) after dropping Corriolis terms:

𝐵=𝑉𝑚𝑎𝑥2 𝜌𝑒 /(𝑃𝑛−𝑃𝑐 ) (4)

(3)

Holland (1980) Wind Model (ADCIRC: NWS=8)

𝑉 𝑔 (𝑟 )=√𝑉𝑚𝑎𝑥2 𝑒(1− ( 𝑅𝑚𝑎𝑥 /𝑟 )𝐵 ) (𝑅𝑚𝑎𝑥 /𝑟 )𝐵+( 𝑟𝑓2 )

2

−( 𝑟𝑓2 )

Substitute Eqs. (3) & (4) into Eqs. (1) & (2) - final Holland equations:

𝑃 (𝑟 )=𝑃𝑐+ (𝑃𝑛−𝑃𝑐 )𝑒−(𝑅𝑚𝑎𝑥 /𝑟 ) 𝐵

(6)

(5)

Derive relationships for scaling parameters A, B

• Assume V=Vmax @ r =Rmax (dV/dr = 0 @ r =Rmax)• Assume << Vmax = cyclostrophic wind balance @ r =Rmax • Setting dV/dr = 0 from Eq. (2) after dropping Corriolis terms:

𝐵=𝑉𝑚𝑎𝑥2 𝜌𝑒 /(𝑃𝑛−𝑃𝑐 ) (4)

(3)

Holland (1980) Wind Model (ADCIRC: NWS=8)

Holland “B”

• Holland Equations (4), (5), (6)

• Use either R64 , R50 , or R34 distance to strongest wind isotach (64kt, 50kt, 34kt) to solve for a different Rmax in each storm quadrant (NE, NW, SW, SE)

Asymmetric Holland Wind Model (ADCIRC: NWS=19)

Hurricane Bob of 1991 was extremely asymmetrical, having uneven distribution of the wind radii at Aug 19 1991 1226Z.

The asymmetrical characteristic of a hurricane can be addressed in AHW with spatially varying Rmax.

𝑅𝑚𝑎𝑥 (𝜃 )

𝑉 𝑔 (𝑟 )=√𝑉𝑚𝑎𝑥2 𝑒(1− ( 𝑅𝑚𝑎𝑥 /𝑟 )𝐵 ) (𝑅𝑚𝑎𝑥 /𝑟 )𝐵+( 𝑟𝑓2 )

2

−( 𝑟𝑓2 )

Substitute Eqs. (3) & (4) into Eqs. (1) & (2) - final Holland equations:

𝑃 (𝑟 )=𝑃𝑐+ (𝑃𝑛−𝑃𝑐 )𝑒−(𝑅𝑚𝑎𝑥 /𝑟 ) 𝐵

(6)

(5)

Derive relationships for scaling parameters A, B

• Assume V=Vmax @ r =Rmax (dV/dr = 0 @ r =Rmax)• Assume << Vmax = cyclostrophic wind balance @ r =Rmax • Setting dV/dr = 0 from Eq. (2) after dropping Corriolis terms:

𝐵=𝑉𝑚𝑎𝑥2 𝜌𝑒 /(𝑃𝑛−𝑃𝑐 ) (4)

(3)

Holland (1980) Wind Model (ADCIRC: NWS=8)

Holland “B”

e.g., Vg=64 ktr=R64

Solve forRmax

• Holland Equations (4), (5), (6)

• Use either R64 , R50 , or R34 distance to strongest wind isotach (64kt, 50kt, 34kt) to solve for a different Rmax in each storm quadrant (NE, NW, SW, SE)

• Interpolate Rmax around storm Rmax (θ)

Asymmetric Holland Wind Model (ADCIRC: NWS=19)

Hurricane Bob of 1991 was extremely asymmetrical, having uneven distribution of the wind radii at Aug 19 1991 1226Z.

The asymmetrical characteristic of a hurricane can be addressed in AHW with spatially varying Rmax.

𝑅𝑚𝑎𝑥 (𝜃 )

• Holland Equations (4), (5), (6)

• Use either R64 , R50 , or R34 distance to strongest wind isotach (64kt, 50kt, 34kt) to solve for a different Rmax in each storm quadrant (NE, NW, SW, SE)

• Interpolate Rmax around storm Rmax (θ)

Asymmetric Holland Wind Model (ADCIRC: NWS=19)

Hurricane Bob of 1991 was extremely asymmetrical, having uneven distribution of the wind radii at Aug 19 1991 1226Z.

The asymmetrical characteristic of a hurricane can be addressed in AHW with spatially varying Rmax.

𝑅𝑚𝑎𝑥 (𝜃 )

𝑉 𝑔 (𝑟 ,𝜃 )=√𝑉𝑚𝑎𝑥2 𝑒(1− (𝑅𝑚𝑎𝑥 /𝑟 )𝐵) (𝑅𝑚𝑎𝑥 /𝑟 )𝐵+( 𝑟𝑓2 )

2

−( 𝑟𝑓2 )𝑃 (𝑟 ,𝜃 )=𝑃𝑐+(𝑃𝑛−𝑃𝑐 )𝑒−( 𝑅𝑚𝑎𝑥 /𝑟 )𝐵

(6)

(5)

Problems with Asymmetric Holland Model (AHM) • Inconsistency between Rmax ,Vmax and full gradient wind velocity, Vg , Eq. (6)

when Rmax ≮≮ Vmax• In some cases unable to compute Rmax • B is constant in space Eq. (4)

• Only uses single (strongest) isotach in each quadrant

Asymmetric Holland Wind Model (ADCIRC: NWS=19)

Generalized Asymmetric Wind Model (GAM)

• Start again with the initial pressure and gradient wind equations from Holland (1980)

• Do not assume cyclostropic balance dVg/dr = 0 @ r =Rmax

𝜓=1+𝑉𝑚𝑎𝑥❑ 𝑅𝑚𝑎𝑥 𝑓 /𝐵 (𝑉𝑚𝑎𝑥

2 +𝑉𝑚𝑎𝑥𝑅𝑚𝑎𝑥 𝑓 )

𝐵=(𝑉𝑚𝑎𝑥2 +𝑉𝑚𝑎𝑥𝑅𝑚𝑎𝑥 𝑓 )𝜌𝑒𝜓 /𝜓 (𝑃𝑛−𝑃𝑐 )

𝑉 𝑔 (𝑟 ,𝜃 )=√ (𝑉𝑚𝑎𝑥2 +𝑉𝑚𝑎𝑥𝑅𝑚𝑎𝑥 𝑓 )𝑒𝜓 (1− (𝑅𝑚𝑎𝑥 /𝑟 )𝐵) (𝑅𝑚𝑎𝑥 /𝑟 )𝐵+( 𝑟𝑓2 )

2

−( 𝑟𝑓2 )

𝑃 (𝑟 ,𝜃 )=𝑃𝑐+(𝑃𝑛−𝑃𝑐 )𝑒−𝜓 (𝑅𝑚𝑎𝑥 / 𝑟 )𝐵

(10)

(9)

(8)

(7)

Comparison of Wind Formulations

𝑃 (𝑟 ,𝜃 )=𝑃𝑐+(𝑃𝑛−𝑃𝑐 )𝑒− ( 𝑅𝑚𝑎𝑥 /𝑟 )𝐵(5)

𝑉 𝑔 (𝑟 ,𝜃 )=√𝑉𝑚𝑎𝑥2 𝑒(1− (𝑅𝑚𝑎𝑥 /𝑟 )𝐵) (𝑅𝑚𝑎𝑥 /𝑟 )𝐵+( 𝑟𝑓2 )

2

−( 𝑟𝑓2 ) (6)

𝐵=𝑉𝑚𝑎𝑥2 𝜌𝑒 /(𝑃𝑛− 𝑃𝑐 ) (4)

𝑃 (𝑟 ,𝜃 )=𝑃𝑐+(𝑃𝑛−𝑃𝑐 )𝑒−𝜓 (𝑅𝑚𝑎𝑥 / 𝑟 )𝐵 (7)

𝑉 𝑔 (𝑟 ,𝜃 )=√ (𝑉𝑚𝑎𝑥2 +𝑉𝑚𝑎𝑥𝑅𝑚𝑎𝑥 𝑓 )𝑒𝜓 (1− (𝑅𝑚𝑎𝑥 /𝑟 )𝐵) (𝑅𝑚𝑎𝑥 /𝑟 )𝐵+( 𝑟𝑓2 )

2

−( 𝑟𝑓2 )(8)

𝜓 (𝜃)=1+𝑉𝑚𝑎𝑥❑ 𝑅𝑚𝑎𝑥 𝑓 /𝐵 (𝑉𝑚𝑎𝑥

2 +𝑉𝑚𝑎𝑥 𝑅𝑚𝑎𝑥 𝑓 )

𝐵(𝜃)=(𝑉𝑚𝑎𝑥2 +𝑉𝑚𝑎𝑥 𝑅𝑚𝑎𝑥 𝑓 )𝜌𝑒𝜓 /𝜓 (𝑃𝑛− 𝑃𝑐 ) (9)

(10)

AHM NWS=19

GAM

Holland B

Wind

Pressure

• Re-calculate B and ψ using the latest Rmax in each quadrant

B , ψ

• Use brute-force marching to solve for Rmax in each quadrant

Rmax

• Guess initial values without considering coriolis force

B0 , ψ0

converge ?

• Spatially interpolate Rmax, B, and ψ at each ADCIRC node.

Spatial Interpolation

• Procedures

Implementation of GAM

• Calculate dynamic wind and pressure fields

Vg(r, θ), P(r, θ)

• Storm center locations, Vmax, Pn, Pc, multiple Isotachs and their radii, etc

Inputs

ASWIP ADCIRC

Fort.22

Output

Weighted Composite Wind Field

NE

Isot 34

Isot 50

Isot 64

Composite

SE SW NW

Weighted Composite Wind Field

NE

Isot 34

Isot 50

Isot 64

Composite

SE SW NW

Weighted Composite Wind Field

NE

Isot 34

Isot 50

Isot 64

Composite

SE SW NW

AHM (NWS = 19) only uses the highest isotach in each quadrant to generate its wind/pressure field.

GAM uses a linear weighting of parameter sets computed from all available isotachs in each quadrant.

Weighted Composite Wind Field in GAM

R64 R50 R34

Weighted Composite Wind Field

NE

Isot 34

Isot 50

Isot 64

Composite

SE SW NW

Comparison of Spatial Wind Fields (Strong Wind)

GAMAHM NWS=19

Comparison of Spatial Wind Fields (Weak Wind)

GAMAHM NWS=19

• A new “Generalized Asymmetric vortex Model" is implemented in ASWIP / ADCIRC, that solves the full gradient wind equation, and utilizes all available isotachs to generate composite wind fields.

• The new formulation allows the model to (i) faithfully represent weaker and larger storms and (ii) to exactly fit multiple wind isotachs that are typically specified in each storm quadrant in either forecast or best track input.

• Because GAM is still a parametric model, it lacks complexity when compared to re-analysis H*Wind. Does best available job of representing available ATCF BT / forecast parameters.

Conclusions

NWS = 19 GAM H*Wind

Thank you!

NWS = 19 NWS = 20 H*Wind

Comparison of Spatial Wind Fields (Strong Wind) Cont.

AHM NWS=19 GAM

Comparison of Spatial Wind Fields (Weak Wind) Cont.AHM

NWS=19 GAM