# A Tsunami Ball Approach to Storm Surge and Inundation: Application to Hurricane Katrina, 2005

Authors: Steven N. Ward
Journal: International Journal of Geophysics
 (2009)

---

## Abstract

Most analyses of storm surge and inundation solve equations of continuity and momentum on fixed finite-difference/finite-element meshes. I develop a completely new approach that uses a momentum equation to accelerate bits or balls of water over variable depth topography. The thickness of the water column at any point equals the volume density of balls there. In addition to being more intuitive than traditional methods, the tsunami ball approach has several advantages. (a) By tracking water balls of fixed volume, the continuity equation is satisfied automatically and the advection term in the momentum equation becomes unnecessary. (b) The procedure is meshless in the finite-difference/finite-element sense. (c) Tsunami balls care little if they find themselves in the ocean or inundating land. (d) Tsunami ball calculations of storm surge can be done on a laptop computer. I demonstrate and calibrate the method by simulating storm surge and inundation around New Orleans, Louisiana caused by Hurricane Katrina in 2005 and by comparing model predictions with field observations. To illustrate the flexibility of the tsunami ball technique, I run two “What If” hurricane scenarios—Katrina over Savannah, Georgia and Katrina over Cape Cod, Massachusetts.


---

## Body

## 1. Introduction

Traditionally, storm surge  and inundation have been modeled using finite-difference/finite-element approaches on fixed meshes. A few forefront computer codes of this type (e.g., ADCIRC; seehttp://adcirc.org/) have been under group development for decades and are true marvels of complexity.
Arguably, however, the downside of finite-difference/finite-element approaches may be complexity itself. Researchers interested in trying their hand at storm surge face a steep learning curve to operate ADCIRC-like codes. Moreover, most large-scale finite-difference/finite-element programs are geared specifically for multiprocessor supercomputers. The need for supercomputing to run traditional storm surge codes may form an insurmountable roadblock for scientists who wish
to contribute to, or better understand the subject. Might there be a simpler, more intuitive alternative to storm surge than finite difference/finite elements? Could such an approach be scaled down to laptop computer scale?This article addresses these questions by developing a new approach to storm surge and inundation modeling. The concept springs from Ward and Day [1] who modeled wave runup and inundation using
“tsunami balls.” Simply, tsunami balls are bits of water accelerated over a 3D surface. The volume density of balls at any point equals the thickness of the water column there. In storm surge applications, the forces
that accelerate tsunami balls derive from surface wind drag, air pressure, gravity, corilois, and bottom friction. As framed earlier, the motivation of this work is not to compete with numerical codes like ADCIRC, but to provide an alternative; an
alternative aimed at researchers who wish to learn from 100 km scale storm surge simulations run on their laptop computers.


## 2. Storm Surge Basics

Letx^ and y^ directions be east and  north in the horizontal
plane and z^ be up. At vector position x=(x,y), let still water depth to
seafloor be h(x) measured positive downward and the perturbation of the surface about
the still level be ζ(x,t) measured positive upward (Figure 1). Most storm surge calculations
solve depth-averaged continuity and momentum equations for variation in water
column thickness H(x,t)=h(x)+ζ(x,t) and mean horizontal water
velocity v(x,t) at fixed mesh nodes: (1)∂H(x,t)∂t=−∇•[v(x,t)H(x,t)],(2)∂v(x,t)∂t=−v(x,t)•∇v(x,t)−g∇ζ(x,t)−ρ−1∇P(x,t)+ρaCa|Va(x,t)|Va(x,t)ρH(x,t)−Cd|v(x,t)|v(x,t)H(x,t)+f(v(x,t)×z^). In (2), g is the acceleration of
gravity, ρ(1000 kg/m3) and ρa (1.2 kg/m3) are the
densities of water and air, and f  is the corilois parameter f=2 (7.2921×105s−1)sin(θ),
with θbeing latitude. Also, Ca and Cd are
surface and bottom drag coefficients. Equations (1) and (2) are solved given
prescribed surface air pressure P(x,t)  and surface wind velocity Va(x,t). Be aware that shallow water equations (1) and (2) assume that
surface deformations have wavelengths much longer than water depth h(x). If for any reason, the forcing
functions on the right-hand side develop short-wavelength components, then this
assumption breaks down and solutions to (1) and (2)
become unstable.Figure 1
Geometry for storm surge calculations. Still water
depthh(x) is measured positive downward. The perturbation of
the water surface about still water ζ(x,t) is measured positive
upward.


## 3. Tsunami Ball Approach

Replacing the
established finite-difference/finite-element approach, I intend to employ a
momentum equation to accelerateN tsunami balls around on a tabletop of varying topography. The tsunami balls are
treated as point masses and the thickness of the water column at any location
is calculated from the volume density of tsunami balls near that location: (3)H(x,t)=∑j=1NVjA(x,xj(t)). In (3), Vj is the fixed water volume of the jth ball, xj(t) is the ball's location
at time t, and A(x,xj(t)) is an averaging area.
For instance, A(x,xj(t)) might span a circle
of radius R about x, then (4)A(x,xj(t))=πR2:|x−xj(t)|<R=∞:|x−xj(t)|>R.The primary
advantages  of a tsunami ball approach to storm surge are four.(1)
Because I track water balls
of fixed volume, the continuity equation (1) is satisfied automatically. In
locations where the volume density (3) of balls grows, the water column
thickness increases. In locations where the volume density drops, the thickness
of the water column falls.(2)
The tsunami ball procedure is
meshless. Meshless applications offer a huge simplification over finite-difference/finite-element methods in that one can download a coastal DEM and start the
calculation immediately without having to worry about mesh density, node locations,
and so forth.(3)
Tsunami balls care little if
they find themselves in the oceanh(x)>0 or if they have blown onto land
h(x)<0 (Figure 1). Such indifference is made-to-order for inundation
applications because it obviates the need for special “dry cell” and
“wet cell” behaviors.(4)
When programmed
efficiently, large-scale (several 100 km) tsunami ball simulations of storm surge can be
implemented on a laptop computer.The primary
disadvantage to the tsunami ball approach is granularity. Questions like “How
many balls are needed?;” or, “How should averaging areaA(x,xj(t)) be
selected?” are always close at hand. Smoothness is a particular concern in
evaluating the surface gradient term  ∇ζ(x,t) in (2).


### 3.1. Advection

As was
mentioned, basic storm surge calculations solve (1) and (2) at fixed mesh nodes x. A portion of the change in water
velocity at x as described by (2)
represents real accelerations of the water in the vicinity. Another part of the
velocity change accounts for the fact that the water particles at x at time t are not the same ones that
will be there at t+dt. If the new particles have different velocities than the
old ones, x experiences an apparent
(or advected) acceleration. Here, because I intend to track specific bits of
water, the advection term v(x,t)•∇v(x,t)
in (2) is not needed. In the
tsunami ball approach, quantities like (3), evaluated at fixed locations
automatically, account for the fact that different tsunami balls contribute to
the count at different times.


## 3.1. Advection

As was
mentioned, basic storm surge calculations solve (1) and (2) at fixed mesh nodes x. A portion of the change in water
velocity at x as described by (2)
represents real accelerations of the water in the vicinity. Another part of the
velocity change accounts for the fact that the water particles at x at time t are not the same ones that
will be there at t+dt. If the new particles have different velocities than the
old ones, x experiences an apparent
(or advected) acceleration. Here, because I intend to track specific bits of
water, the advection term v(x,t)•∇v(x,t)
in (2) is not needed. In the
tsunami ball approach, quantities like (3), evaluated at fixed locations
automatically, account for the fact that different tsunami balls contribute to
the count at different times.


## 4. Application to Hurricanes

This article simulates storm
surge due to hurricanes. The primary goal is to introduce and demonstrate the
utility of the tsunami ball approach and not to reproduce the details of any
specific situation. Accordingly I use “off the shelf” prescriptions
of hurricane driving forces and do not dwell deeply on any
choice of parameter.


### 4.1. Surface Pressure and Wind Velocity

Ifx−xc=RR^ is the vector from the hurricane center to
position x, the Holland wind model [2] gives generic surface
pressure and zonal (tangential) wind velocity as
(Figure 2) (5)P(R)=Pc−(Pn−Pc)e−(Rm/R)B,(6)Va(R)={[Rρa∂P(R)∂R+R2f24]1/2−Rf2}(z^×R^) with the B in (5) taken such that (6) gives a peak wind
speed Vm at radius R=Rm, (7)B≈2.7183Vm2ρaP(Rm)−Pc.  To evaluate (5)–(7) one needs to specify as a function of
time:Figure 2
(Top) Holland's [2] relation between storm
pressure (5) (purple) and zonal wind velocity (6) (red). (Bottom)
Zonal wind speeds are nearly proportional to the square root of
pressure gradient.(a)
the storm's central positionxc;(b)
the storm's central pressurePc;(c)
the storm's peak wind speedVm;(d)
the storm's radius of maximum velocityRm;(e)
the background air pressurePn.I augment the
purely zonal wind model (6) in two ways.(a) As a consequence of friction, the direction of the
encircling wind (6) is rotated inward by positive angle β(R) (8)V˜a(R)=|Va(R)|[cosβ(R)(z^×R^)−sinβ(R)R^]. Following Martino
et al. [3], I take this inward
angle to be (9)β(R)=25°;R>1.2Rm,β(R)=10°+75°(RRm−1);Rm<R<1.2Rm,β(R)=10°(RRm);R<Rm.(b) Also following Martino et al. [3], I include in the
forcing wind an additional component in the direction of hurricane storm
velocity Vhur(t) as (10)V⌢a(R)=V˜a(R)+RmRRm2+R2Vhur.  The inverse barometer effect (see
Section 4.4), together with the inward (8) and along track wind components (10),
builds and pulls along a pile of water near the hurricane's low-pressure eye.
This water eye or “eye-pile” is larger in diameter than the pressure
eye and increases in height as it moves into coastal shallows and washes onto
land when the storm crosses a shoreline. Water in the eye-pile, together with
the water-driven ashore by the zonal winds, comprises the two components of
hurricane storm surge.


### 4.2. Surface Wind DragCa

Many
formulations exist for fixing the wind drag coefficientCa as a
function of wind speed. I employ a standard linear  relation from Garratt [4]: (11)Ca(Va)=(0.75+0.067Va)×10−3, where Va is the wind
speed in m/s, 10 m above the sea surface. In this article, Va will be
the absolute value of (10).


### 4.3. Bottom ResistanceCd

For tsunami balls in the ocean, I select a bottom drag coefficientCd = 0.001.
When tsunami balls blow onto land, presumably they encounter more resistance to
flow from vegetation and structures. I increase Cd for those balls
to 0.005.


### 4.4. “Fly Apart” Surge—Limits onCa and Cd

When viewed from
a particle perspective, certain storm surge relationships emerge that might not
be apparent in a fixed node perspective. Consider a stationary hurricane with
purely zonal winds in which the ocean has reached a steady state with bits of
water smoothly orbiting the hurricane pressure eye at distanceR  with constant
speed v. From a particle viewpoint, to orbit stablely, water bits must
continually experience an inward centripetal acceleration of (12)Acent=v2R. If inward accelerations fall
below (12), the water particles composing the eye-pile will “fly
apart.” Steady-state water velocity will be attained when  the surface and
seafloor drag accelerations in (2) balance, that is, (13)ρaCaVa2ρ=Cdv2orv2=ρaCaVa2ρCd. The
steady-state inward accelerations are −g∇ζ(R)−ρ−1∇P(R), so critical condition (12)  becomes (14)−g∇ζ(R)−ρ−1∇P(R)=ρaCaVa2ρCdR.  The
hurricane pressure eye accompanies a water surface high and an air pressure
low, so −g∇ζ(R) is directed outward and −ρ−1∇P(R) is directed inward. Elementary notions ignore
the orbital motions of the water and simply strike a balance in these forces to
generate the inverted barometer concept:
(15a)−g∇ζ(R)=ρ−1∇P(R)or(15b)ζ(R)=(ρg)−1[Pn-P(R)],
where
the height of the eye-pile is proportional to the pressure low. In deep water,
for a pressure difference [Pn−Pc] = 100 mbar (104 Pa), the inverted barometer
effect (15b) draws up an eye-pile of about 1 m height—a rather minor contributor
to surge. Shoaling of the tracking water as the storm runs ashore, however,
amplifies the eye-pile height several times.Returning to stability condition (14) and
dropping the  −g∇ζ(R)
term
for the moment (16)−∂P(R)∂R=ρaCaVa2CdR. From
(6), air speeds are approximately (17)Va(R)=[Rρa∂P(R)∂R]1/2whence∂P(R)∂R=ρaVa2(R)R. Under
these assumptions, stability condition (16) reduces to a remarkably simple form: (18)CaCd<1,
for
a stable state to exist Ca<Cd. If  Ca>Cd,
then  centrifugal forces will eventually cause the eye-pile to fly apart. If Ca is 0.001 as I assume, Garratt's [4] formula (11) for Cd violates
condition (18) in winds of just 10 m/s.Of the several assumptions leading to (18), the
strongest is that of steady state. Numerical tests using Garratt's [4] formula and Cd = 0.001
(0.004) indicate that in 100 m of water, it takes 11.7 hours (5.8 hours) of
sustained 50 m/s winds to accelerate the fluid to 90% of its limited speed of
3.5 m/s (1.7 m/s). Time to steady state increases in proportion to the water
depth, so it is unlikely that a travelling storm will remain over a given
location long enough to accelerate the water below to steady state speed (13).
Still, the concept of water orbiting a hurricane eye with pressure and topographic
forces summing to balance centripetal forces is at least as useful as the
inverted barometer concept that ignores zonal water velocities all
together.


## 4.1. Surface Pressure and Wind Velocity

Ifx−xc=RR^ is the vector from the hurricane center to
position x, the Holland wind model [2] gives generic surface
pressure and zonal (tangential) wind velocity as
(Figure 2) (5)P(R)=Pc−(Pn−Pc)e−(Rm/R)B,(6)Va(R)={[Rρa∂P(R)∂R+R2f24]1/2−Rf2}(z^×R^) with the B in (5) taken such that (6) gives a peak wind
speed Vm at radius R=Rm, (7)B≈2.7183Vm2ρaP(Rm)−Pc.  To evaluate (5)–(7) one needs to specify as a function of
time:Figure 2
(Top) Holland's [2] relation between storm
pressure (5) (purple) and zonal wind velocity (6) (red). (Bottom)
Zonal wind speeds are nearly proportional to the square root of
pressure gradient.(a)
the storm's central positionxc;(b)
the storm's central pressurePc;(c)
the storm's peak wind speedVm;(d)
the storm's radius of maximum velocityRm;(e)
the background air pressurePn.I augment the
purely zonal wind model (6) in two ways.(a) As a consequence of friction, the direction of the
encircling wind (6) is rotated inward by positive angle β(R) (8)V˜a(R)=|Va(R)|[cosβ(R)(z^×R^)−sinβ(R)R^]. Following Martino
et al. [3], I take this inward
angle to be (9)β(R)=25°;R>1.2Rm,β(R)=10°+75°(RRm−1);Rm<R<1.2Rm,β(R)=10°(RRm);R<Rm.(b) Also following Martino et al. [3], I include in the
forcing wind an additional component in the direction of hurricane storm
velocity Vhur(t) as (10)V⌢a(R)=V˜a(R)+RmRRm2+R2Vhur.  The inverse barometer effect (see
Section 4.4), together with the inward (8) and along track wind components (10),
builds and pulls along a pile of water near the hurricane's low-pressure eye.
This water eye or “eye-pile” is larger in diameter than the pressure
eye and increases in height as it moves into coastal shallows and washes onto
land when the storm crosses a shoreline. Water in the eye-pile, together with
the water-driven ashore by the zonal winds, comprises the two components of
hurricane storm surge.


## 4.2. Surface Wind DragCa

Many
formulations exist for fixing the wind drag coefficientCa as a
function of wind speed. I employ a standard linear  relation from Garratt [4]: (11)Ca(Va)=(0.75+0.067Va)×10−3, where Va is the wind
speed in m/s, 10 m above the sea surface. In this article, Va will be
the absolute value of (10).


## 4.3. Bottom ResistanceCd

For tsunami balls in the ocean, I select a bottom drag coefficientCd = 0.001.
When tsunami balls blow onto land, presumably they encounter more resistance to
flow from vegetation and structures. I increase Cd for those balls
to 0.005.


## 4.4. “Fly Apart” Surge—Limits onCa and Cd

When viewed from
a particle perspective, certain storm surge relationships emerge that might not
be apparent in a fixed node perspective. Consider a stationary hurricane with
purely zonal winds in which the ocean has reached a steady state with bits of
water smoothly orbiting the hurricane pressure eye at distanceR  with constant
speed v. From a particle viewpoint, to orbit stablely, water bits must
continually experience an inward centripetal acceleration of (12)Acent=v2R. If inward accelerations fall
below (12), the water particles composing the eye-pile will “fly
apart.” Steady-state water velocity will be attained when  the surface and
seafloor drag accelerations in (2) balance, that is, (13)ρaCaVa2ρ=Cdv2orv2=ρaCaVa2ρCd. The
steady-state inward accelerations are −g∇ζ(R)−ρ−1∇P(R), so critical condition (12)  becomes (14)−g∇ζ(R)−ρ−1∇P(R)=ρaCaVa2ρCdR.  The
hurricane pressure eye accompanies a water surface high and an air pressure
low, so −g∇ζ(R) is directed outward and −ρ−1∇P(R) is directed inward. Elementary notions ignore
the orbital motions of the water and simply strike a balance in these forces to
generate the inverted barometer concept:
(15a)−g∇ζ(R)=ρ−1∇P(R)or(15b)ζ(R)=(ρg)−1[Pn-P(R)],
where
the height of the eye-pile is proportional to the pressure low. In deep water,
for a pressure difference [Pn−Pc] = 100 mbar (104 Pa), the inverted barometer
effect (15b) draws up an eye-pile of about 1 m height—a rather minor contributor
to surge. Shoaling of the tracking water as the storm runs ashore, however,
amplifies the eye-pile height several times.Returning to stability condition (14) and
dropping the  −g∇ζ(R)
term
for the moment (16)−∂P(R)∂R=ρaCaVa2CdR. From
(6), air speeds are approximately (17)Va(R)=[Rρa∂P(R)∂R]1/2whence∂P(R)∂R=ρaVa2(R)R. Under
these assumptions, stability condition (16) reduces to a remarkably simple form: (18)CaCd<1,
for
a stable state to exist Ca<Cd. If  Ca>Cd,
then  centrifugal forces will eventually cause the eye-pile to fly apart. If Ca is 0.001 as I assume, Garratt's [4] formula (11) for Cd violates
condition (18) in winds of just 10 m/s.Of the several assumptions leading to (18), the
strongest is that of steady state. Numerical tests using Garratt's [4] formula and Cd = 0.001
(0.004) indicate that in 100 m of water, it takes 11.7 hours (5.8 hours) of
sustained 50 m/s winds to accelerate the fluid to 90% of its limited speed of
3.5 m/s (1.7 m/s). Time to steady state increases in proportion to the water
depth, so it is unlikely that a travelling storm will remain over a given
location long enough to accelerate the water below to steady state speed (13).
Still, the concept of water orbiting a hurricane eye with pressure and topographic
forces summing to balance centripetal forces is at least as useful as the
inverted barometer concept that ignores zonal water velocities all
together.


## 5. Computational Elements

Like every
numerical method, tsunami balls utilize certain techniques to stabilize and to
speed the calculation. Moreover, because a primary intent is to keep storm
surge calculations to a laptop computer scale, additional shortcuts become
necessary. Below I list some “tricks of the trade.”


### 5.1. Grid Not Mesh

I have touted an advantage of tsunami balls as
being meshless in the finite-difference/finite-element sense. Still, it is
convenient to interpolate certain quantities to a grid. A grid differs from a
mesh in that the former is a simple rectangular frame and its selection does
not influence the calculation greatly. Interpolation to a grid involves the
weighted density of tsunami balls in the area. Equation  (3) for instance, evaluates
the water column thickness H(x,t) at fixed grid point x. Interpolating flow velocity to grid
point x might involve (19)v(x,t)=1H(x,t)∑j=1N(VjA(x,xj(t)))vj(t), where xj(t) and vj(t)
are the jth ball's location and velocity at time t. Interpolating quantities
associated with N balls to M grid locations by direct summations like (19) is
computationally costly however. In practice, I smooth and interpolate quantities
simultaneously (see Section 5.5 in what follows).Grid-interpolated quantitiesH(x,t) and  ζ(x,t) appear in the variant of momentum equation used to accelerate the
tsunami balls: (20)∂vj(t)∂t=−g∇ζ(x,t)−ρ−1∇P(xj(t),t)+ρaCa|Va(xj(t),t)|Va(xj(t),t)ρH(x,t)+Cd|vj(t)|vj(t)H(x,t)+f(vj(t)×z^)P(xj(t),t)
and Va(xj(t),t) are evaluated
explicitly at xj(t) from
(5) and (10). Quantities H(x,t) and ζ(x,t)
are smoothed interpolated values read from the
grid point nearest to xj(t).


### 5.2. One Way Gravity

Equations (1) and (2) arise
from shallow water wave theory. Not surprisingly, they generate waves in
addition to flows. By and large, oscillatory
waves contribute less to storm surge than do longer period flows. Also, compared
to flows, waves vary much more rapidly in time and space. Resolving wave
actions requires far finer steps in time and space than does capturing flow
actions. The presence of waves also contributes to numerical instabilities if
the long wave assumption inherent in  (1) and (2) becomes violated. In
view of this, I invoke “one way” gravity to damp out waves as quickly
as possible. One way gravity selects the value of  g  in momentum equation (20)
at each location and time based on the current velocity vj(t) of the tsunami ball being accelerated and the
gradient ofthe surface
slope −∇ζ(x,t)
at the nearest grid location. Specifically, if
(21a)−∇ζ(x,t)•vj(t)<0;theng=9.8m/s,(21b)−∇ζ(x,t)•vj(t)>0,|∇ζ(x,t)|<10−5;theng=9.8m/s,(21c)−∇ζ(x,t)•vj(t)>0,|∇ζ(x,t)|>10−5;theng=0m/s.
In words, if acceleration −g∇ζ(x,t) opposes ball velocity vj(t) (21a) or if the surface slope  ∇ζ(x,t)  is small (21b), gravity acts in full force. If acceleration −g∇ζ(x,t) reinforces velocity vj(t) and the surface slope is not small (21c), gravity does not act at all. How does
one way gravity suppress waves? Think of a playground swing—by fully
opposing the upswing (21a) and by not accelerating most of the downswing (21c),
one way gravity effectively damps the oscillation. One way gravity curbs wave
actions while hardly affecting long-maintained, low-slope flows and surges associated
with wind drag and air pressure.What surge
slopes do we expect from wind drag and air pressure? For pressure alone, (15a) and (15b)
give
(22)∇ζ(x,t)=∇P(x,t)gρ~[104Pa/100km]9.8×103~10−5.  From (2), a static force balance predicts surge slopes due
to winds alone of (23)∇ζ(x,t)=ρaCa|Va(x,t)|Va(x,t)gρH(x,t).  Using (11) under winds of 30 m/s or 50 m/s, storm surge
slopes would be (24)∇ζ(x,t)=(3.04×10−4)mH(x,t)or(1.25×10−3)mH(x,t).  In 50–100 m of water, both (22) and (24) predict pressure
or wind-driven storm surge slopes to be about |∇ζ(x,t)|~10−5.
In shallow water (10 m), wind-driven storm surge slopes might reach 10−4 or 1 m in 10 km. Water slopes larger than 10−4–10−5 likely associate with wave actions and should
be extinguished. This line of reasoning motivated the parameter choice in (21a), (21b), and (21c).


### 5.3. Inner and Outer Loops

One wishes to evaluate (20) and
update tsunami ball positions at time intervals such that ΔT times the ball velocity is less than spacing
of base topography, otherwise, the balls might skip over obstructions. On the
other hand, updating ocean surface ζ(x,t) on N grid points from M balls (see
Section 5.5) takes considerable computational effort, so I want to do this less
frequently. Accordingly, this calculation incorporates inner and outer time
loops. In the inner loop, tsunami ball positions are updated using (20) with
small ΔT,
but with the water surface ζ(x,tfix) fixed at recent time
tfix. Only in the outer loop is the surface shape re-evaluated to a
new tfix. The calculation passes 20 or 30 inner loops for each pass
of the outer loop.


### 5.4. Balls at Domain Boundaries

During the calculation,
accelerations (20) may push tsunami balls off of the computational grid. I
replace out-of-bounds balls at the closest in-bounds location and then zero the
ball's velocity. Near domain walls there will be artifacts, so I try to keep
the walls as far as possible from the region of interest.


### 5.5. Rectangular Smoothing

Interpolating quantities to the
grid must be done efficiently to hold this calculation to laptop scale. Summing
quantities overN balls at M grid locations directly by (3) or (19) will not be
tenable when N and M each count many 100,000. Instead, I depend on a two-step
smoothing/interpolation that reduces the number of numerical operations from
being proportional to N×M to being proportional to only N. First, cycle through
all M ball locations xj(t)
and assign that ball's value (volume, velocity, etc.) to the nearest grid point.
Certain grid points may have zero value while others may have the summed values
of several balls. Second, apply a rectangular smoothing to all grid rows and
then to all grid columns. The qth smoothed value in a grid row might be (25)G¯q=ΣqNorm;Σq=∑r=q−wq+WGr,  where Gr are raw values, W is the half width of
smoothing and Norm is some normalization. The q+1th smoothed value is (26)G¯q+1=Σq+1Norm;Σq+1=Σq+Gq+w+1−Gq−w. You can see that each value (26) derives from the previous
value (25) by only one addition, one subtraction, and one division. The number
of numerical operations in rectangular smoothing is proportional to the number
of grid points N only, and independent of both the number of balls M and the
smoothing half width W.  In most cases, I
apply rectangular smoothing of half width W twice to get triangle smoothing of
half width 2W. Proper selection of half width W takes trial and error. I desire
a smooth solution, but not so smooth as to lose resolution or to drag the water
surface artificially far onto land. Because all of the tsunami balls retain a
fixed water volume and no balls leave the grid, ocean volume and mean surface
height ought to be conserved in smoothing. I select the normalization in (25)
and (26) to make this so.


## 5.1. Grid Not Mesh

I have touted an advantage of tsunami balls as
being meshless in the finite-difference/finite-element sense. Still, it is
convenient to interpolate certain quantities to a grid. A grid differs from a
mesh in that the former is a simple rectangular frame and its selection does
not influence the calculation greatly. Interpolation to a grid involves the
weighted density of tsunami balls in the area. Equation  (3) for instance, evaluates
the water column thickness H(x,t) at fixed grid point x. Interpolating flow velocity to grid
point x might involve (19)v(x,t)=1H(x,t)∑j=1N(VjA(x,xj(t)))vj(t), where xj(t) and vj(t)
are the jth ball's location and velocity at time t. Interpolating quantities
associated with N balls to M grid locations by direct summations like (19) is
computationally costly however. In practice, I smooth and interpolate quantities
simultaneously (see Section 5.5 in what follows).Grid-interpolated quantitiesH(x,t) and  ζ(x,t) appear in the variant of momentum equation used to accelerate the
tsunami balls: (20)∂vj(t)∂t=−g∇ζ(x,t)−ρ−1∇P(xj(t),t)+ρaCa|Va(xj(t),t)|Va(xj(t),t)ρH(x,t)+Cd|vj(t)|vj(t)H(x,t)+f(vj(t)×z^)P(xj(t),t)
and Va(xj(t),t) are evaluated
explicitly at xj(t) from
(5) and (10). Quantities H(x,t) and ζ(x,t)
are smoothed interpolated values read from the
grid point nearest to xj(t).


## 5.2. One Way Gravity

Equations (1) and (2) arise
from shallow water wave theory. Not surprisingly, they generate waves in
addition to flows. By and large, oscillatory
waves contribute less to storm surge than do longer period flows. Also, compared
to flows, waves vary much more rapidly in time and space. Resolving wave
actions requires far finer steps in time and space than does capturing flow
actions. The presence of waves also contributes to numerical instabilities if
the long wave assumption inherent in  (1) and (2) becomes violated. In
view of this, I invoke “one way” gravity to damp out waves as quickly
as possible. One way gravity selects the value of  g  in momentum equation (20)
at each location and time based on the current velocity vj(t) of the tsunami ball being accelerated and the
gradient ofthe surface
slope −∇ζ(x,t)
at the nearest grid location. Specifically, if
(21a)−∇ζ(x,t)•vj(t)<0;theng=9.8m/s,(21b)−∇ζ(x,t)•vj(t)>0,|∇ζ(x,t)|<10−5;theng=9.8m/s,(21c)−∇ζ(x,t)•vj(t)>0,|∇ζ(x,t)|>10−5;theng=0m/s.
In words, if acceleration −g∇ζ(x,t) opposes ball velocity vj(t) (21a) or if the surface slope  ∇ζ(x,t)  is small (21b), gravity acts in full force. If acceleration −g∇ζ(x,t) reinforces velocity vj(t) and the surface slope is not small (21c), gravity does not act at all. How does
one way gravity suppress waves? Think of a playground swing—by fully
opposing the upswing (21a) and by not accelerating most of the downswing (21c),
one way gravity effectively damps the oscillation. One way gravity curbs wave
actions while hardly affecting long-maintained, low-slope flows and surges associated
with wind drag and air pressure.What surge
slopes do we expect from wind drag and air pressure? For pressure alone, (15a) and (15b)
give
(22)∇ζ(x,t)=∇P(x,t)gρ~[104Pa/100km]9.8×103~10−5.  From (2), a static force balance predicts surge slopes due
to winds alone of (23)∇ζ(x,t)=ρaCa|Va(x,t)|Va(x,t)gρH(x,t).  Using (11) under winds of 30 m/s or 50 m/s, storm surge
slopes would be (24)∇ζ(x,t)=(3.04×10−4)mH(x,t)or(1.25×10−3)mH(x,t).  In 50–100 m of water, both (22) and (24) predict pressure
or wind-driven storm surge slopes to be about |∇ζ(x,t)|~10−5.
In shallow water (10 m), wind-driven storm surge slopes might reach 10−4 or 1 m in 10 km. Water slopes larger than 10−4–10−5 likely associate with wave actions and should
be extinguished. This line of reasoning motivated the parameter choice in (21a), (21b), and (21c).


## 5.3. Inner and Outer Loops

One wishes to evaluate (20) and
update tsunami ball positions at time intervals such that ΔT times the ball velocity is less than spacing
of base topography, otherwise, the balls might skip over obstructions. On the
other hand, updating ocean surface ζ(x,t) on N grid points from M balls (see
Section 5.5) takes considerable computational effort, so I want to do this less
frequently. Accordingly, this calculation incorporates inner and outer time
loops. In the inner loop, tsunami ball positions are updated using (20) with
small ΔT,
but with the water surface ζ(x,tfix) fixed at recent time
tfix. Only in the outer loop is the surface shape re-evaluated to a
new tfix. The calculation passes 20 or 30 inner loops for each pass
of the outer loop.


## 5.4. Balls at Domain Boundaries

During the calculation,
accelerations (20) may push tsunami balls off of the computational grid. I
replace out-of-bounds balls at the closest in-bounds location and then zero the
ball's velocity. Near domain walls there will be artifacts, so I try to keep
the walls as far as possible from the region of interest.


## 5.5. Rectangular Smoothing

Interpolating quantities to the
grid must be done efficiently to hold this calculation to laptop scale. Summing
quantities overN balls at M grid locations directly by (3) or (19) will not be
tenable when N and M each count many 100,000. Instead, I depend on a two-step
smoothing/interpolation that reduces the number of numerical operations from
being proportional to N×M to being proportional to only N. First, cycle through
all M ball locations xj(t)
and assign that ball's value (volume, velocity, etc.) to the nearest grid point.
Certain grid points may have zero value while others may have the summed values
of several balls. Second, apply a rectangular smoothing to all grid rows and
then to all grid columns. The qth smoothed value in a grid row might be (25)G¯q=ΣqNorm;Σq=∑r=q−wq+WGr,  where Gr are raw values, W is the half width of
smoothing and Norm is some normalization. The q+1th smoothed value is (26)G¯q+1=Σq+1Norm;Σq+1=Σq+Gq+w+1−Gq−w. You can see that each value (26) derives from the previous
value (25) by only one addition, one subtraction, and one division. The number
of numerical operations in rectangular smoothing is proportional to the number
of grid points N only, and independent of both the number of balls M and the
smoothing half width W.  In most cases, I
apply rectangular smoothing of half width W twice to get triangle smoothing of
half width 2W. Proper selection of half width W takes trial and error. I desire
a smooth solution, but not so smooth as to lose resolution or to drag the water
surface artificially far onto land. Because all of the tsunami balls retain a
fixed water volume and no balls leave the grid, ocean volume and mean surface
height ought to be conserved in smoothing. I select the normalization in (25)
and (26) to make this so.


## 6. Hurricane Katrina at New Orleans

### 6.1. Hurricane Katrina Parameters

I have previously listed the
hurricane parameters that need to be specified:(a)
storm's central positionxc;(b)
storm's central pressurePc;(c)
storm's peak wind speedVm;(d)
storm's radius of maximum velocityRm;(e)
the background air pressurePn; The Weather
Underground [5]  website provided Hurricane Katrina storm track
information (a)–(c) (see Table 1). I fixed the radius of maximum velocity at
25 km and  background air pressure at Pn = 1008 mbar.Table 1
Katrina Track,
Peak Wind Speed (miles/hour, 1 mph = 0.447 m/s) and Central Pressure (mbar) from Weather Underground [5]. Simulations here spanned 22 GMT 8/28 to
08 GMT 8/30.
DateTimeLatLonVm(mph)Pc(mbar)08/2318 GMT23.175.135100808/2400 GMT23.475.735100708/2406 GMT23.876.235100708/2412 GMT24.576.540100608/2418 GMT25.476.945100308/2500 GMT26.077.750100008/2506 GMT26.178.46099708/2512 GMT26.279.06599408/2518 GMT26.279.67098808/2600 GMT25.980.38098308/2606 GMT25.481.37598708/2612 GMT25.182.08597908/2618 GMT24.982.610096808/2700 GMT24.683.310595908/2706 GMT24.484.011095008/2712 GMT24.484.711594208/2718 GMT24.585.311594808/2800 GMT24.885.911594108/2806 GMT25.286.714593008/2812 GMT25.787.716590908/2818 GMT26.388.617590208/2900 GMT27.289.216090508/2906 GMT28.289.614591308/2912 GMT29.589.612592308/2918 GMT31.189.69094808/3000 GMT32.689.16096108/3006 GMT34.188.64597808/3012 GMT35.688.03598508/3018 GMT37.087.03599008/3100 GMT38.685.33599408/3106 GMT40.182.930996


### 6.2. Hurricane Katrina Simulation-Setup

The Katrina at New Orleans
simulations ran under a 2300 × 1600 topographic grid 170 m square with minimum
and maximum water depths restricted to 3 m and 400 m, respectively. Spaced about
570 m apart, 250 000 tsunami balls were distributed over all wet locations. The
water volume assigned to each ball equaled the ball spacing squared multiplied
by the water depth at the initial location. (Actually, tsunami balls might be
better described as tsunami columns.) To keep the water surface from spreading
far onto land as a consequence of smoothing, the smoothing half width dimensionW, varied from 50 km at locations far from land, to about 2 km at locations
close to, or onshore.The simulation began at 22 GMT
8/28/05 when the Katrina was well south of the grid and ended at 08 GMT 8/30/05,
after the storm moved far north of the grid. I updated tsunami ball location
and velocity everyΔT=10s
in the inner loop and recalculated the ocean surface every 5 minutes in the
outer loop. A movie frame was constructed at 15-minute intervals. Primary
outputs of the simulation include grid-interpolated flow velocity, current flow
depth and peak flow depth versus space and time both off and
onshore.


### 6.3. Hurricane Katrina Simulation-Results

Figure3 shows five
frames from the Katrina simulation. (All of the simulations presented in this article
are accompanied by Quicktime movie animations. See Table 2. Please view the
Quicktime movie version of Figure 3
at
http://es.ucsc.edu/~ward/k-at-no.mov) Panel 3(a) sets
the stage with geographic information of the New Orleans region. In the
Mississippi Delta, levees along the Mississippi River comprise the local high
ground. We will see that these linear levees act as dams for surge incoming
from either side. Panel 3(a) also presents the surge color scale used throughout
this article—colors red, orange, and yellow being positive, and blue and
violet being negative.Table 2
Quicktime movie links to the storm surge examples
presented in this article.
Katrina at New Orleans:http://es.ucsc.edu/~ward/k-at-no.movhttp://es.ucsc.edu/~ward/k-at-no-close.movhttp://es.ucsc.edu/~ward/k-at-no-peak.movKatrina at Savannah:http://es.ucsc.edu/~ward/k-at-sav.movhttp://es.ucsc.edu/~ward/k-at-sav-peak.movKatrina at Cape Cod:http://es.ucsc.edu/~ward/k-at-cod.movhttp://es.ucsc.edu/~ward/k-at-cod-peak.movTsunami Ball storm surge
calculation for Hurricane Katrina at New Orleans. Panels (b)-(f) show conditions at
3 hour intervals beginning at 06 GMT 8/29/06. Storm center and central
pressure in mbar are marked by white circle. Bold arrows plot wind direction.
Thin lines plot water flow direction. Red and blue colored areas show
significant deviations from sea level at 1 m intervals. Tsunami Ball storm surge
calculation for Hurricane Katrina at New Orleans. (Panel (e) shows peak surge for the
entire storm event occurs along the Mississippi State Coast. (Panel (f) shows peak
surge migrates eastward toward Alabama.In Panel 3(b) (06 GMT 8/29), 8 hours
after the calculation began, Katrina moves onto the computational grid at the
south. The hurricane eye-pile is 2 m high and 60 km across. In all of the
simulations, the wind-driven component of surge is most noticeable where winds
blow nearly perpendicular to the coastline. Offshore winds of ~35 m/s have
begun to draw down the sea surface west of the Mississippi River Delta. On the
north-south trending coasts on the east side of the Delta, directly onshore
winds of 30 m/s have begun to send surge overland.In Panel 3(c) (09 GMT 8/29), Katrina
has progressed to within about 30 km of first landfall. Water in the eye-pile-driven under the storm, shoals to nearly 5 m. North of the storm, 10 km of
overland flooding reaches  the Mississippi River and begins to pond against
its eastern levee. Northwest of the storm, offshore winds increasing to 50 m/s
draw down the sea level by 4  m.In Panel (d) (12 GMT 8/29), Katrina
has moved 20 km inland and crossed to the east side of the   river. The eye-pile
component of surge now pushing onto land cannot keep pace with the storm and
breaks off from the pressure eye itself. Driven by 40 m/s southwest winds,
water from the eye-pile runs over 20 km of flat land and dams to 8 m against the
River's western levee. Further north, flood water penetrates north and east of
the New Orleans town site.In Panel (e) (15 GMT 8/29), Katrina
has crossed the Mississippi State Coast and lies 20 km inland. Near the
pressure eye, a quick wind shift after the storm passes causes a redirection
of surge. Westerly winds now blow flood waters on the east side of the river
back toward the sea from the Delta lands. About this time, surges at New
Orleans town site peak at 3-4 m. Squarely onshore southwest and south winds of
50 m/s now force the highest flows of the entire storm event against the Mississippi
State Coast. The model registers 11 m of surge 20–40 km east ofKatrina's
track.In Panel  (f) (18 GMT 8/29), Katrina
has crossed the upper edge of the topographic grid and continues to travel
north and weaken. Maintained westerly winds >20 m/s cause the peak of the
surge to migrate eastward along the Mississippi State   Coast toward Alabama for
several hours. Westerly winds persist in ponding water on the west side of the
Mississippi River in the southern Delta. Several meters of surge remain there
until the end of the simulation (08 GMT 8/30).Figure4 contours peak storm
surge from 1 to 7 m. (Please view the Quicktime movie of this Figure at
http://es.ucsc.edu/~ward/k-at-no-peak.mov)
The “dots-on-a-string” map Katrina's position at 1-hour intervals. In
the south, note the residual trail from the eye-pile. The eye-pile component of
surge increases in height from 2 m well offshore to over 5 m as its tracking
water shoals near  the Delta. Highest predicted surges for the entire storm
(>9 m) occur along the lower Mississippi River and the Mississippi State
Coast. At New Orleans, this simulation predicts 3-4 m of flooding. (At the ~200 m
scale of the base topography, I do not expect to reproduce a block by block
flooding history in New Orleans. Still, if given higher-resolution topography,
I do not see why tsunami balls could not be used for that purpose.) Wind-driven
surge, banked against the mainland coast, extends 60 km in the shallow waters
offshore and as far east as the Florida panhandle. Surge banked  up against the
mainland covers the numerous islands of Southern Mississippi and Alabama. The
10−4 slope of the banked up wind blown surge is consistent with that
predicted by (24) in shallow water.Figure 4
Peak storm surge in 1 m increments starting at 1 m. Note
the trail left by the eye pile and how its height grows toward
shore. Areas in the lower Mississippi Delta and the Coast of
Mississippi State fare worst in this simulation. Surge banks up
against the mainland and extends 60 km out into the Gulf of Mexico.Evident in Figure4 is the strong
asymmetry in peak surge relative to the storm track. Counter clockwise rotation
of the zonal winds expose coasts on the right-hand side of the storm to more
extensive surge than equally distant locations on the left hand side. The zone
of maximum  surge extends for 100 km to the right side of the track. From Figure
2, 100 km corresponds to the radius of major storm pressure gradient and the
distance at which zonal winds fall to 1/2 their peak strength.Figure5 contains three large-scale frames from the Katrina simulation. (Please view the Quicktime movie of
Figure 5 at http://es.ucsc.edu/~ward/k-at-no-close.mov)
At large scale, it is easier to compare wind and water flow directions. As
discussed  in Section 4.4, the time required for flows to respond to changes in
wind direction increases in proportion to water depth. In shallow water, wind
and flow directions track fairly closely, and peak flow speeds reach
about 2 m/s. At that speed, overland flows take 2-3 hours to attain maximum
inundation
distance of 20 km.  Offshore, this
simulation predicts instantaneous surge to be rather lumpy and bumpy over a
several km scale (see Figure 5(c) especially). It is difficult to say for certain
if this irregularity attributes to a physical process (e.g., vestiges of wave
actions) or is an artifact of granularity and smoothing.Expanded view of storm surge calculation
from Hurricane Katrina at New Orleans at 3 hour intervals
beginning at 11 GMT. The small yellow or white circles are
locations of field-measured storm surge.
(a)(b)(c)


### 6.4. Comparison with Storm Surge Observations

How well does this tsunami ball
simulation using  off-the-shelf hurricane parameters predict Katrina's actual
surge? The best comparison would be made with time histories of sea level
measured at many offshore locations. Multiple measurements make it possible to
assess variability of surge in time and space (like the lumps and bumps
mentioned earlier) and to average noisy information. Also, offshore measurements
suffer less from the vulgarities of overland fluid flow than do onshore measurements.
A few permanently moored buoys did survive the storm, but these locate at some
distance from the action. I do not know of any satellite radar measurements of
sea height during the storm, but possibly some exist. Lacking open water surge
data, I fall back to information from post-Katrina surveys that measured peak
storm surge height onshore. I find a survey by Fritz et al. [6, 7] to be most complete.Like tsunami
runup data, peak storm surge data has several weaknesses. Firstly, peak surge,
being a point measure in space and time, can vary dramatically over short
distances. Secondly, peak surge, being an extreme measure not a mean  measure,
incorporates considerable happenstance. Imagine two waves momentarily  interfering
constructively here, but not over there. Thirdly, peak surge is the sum of peak
flow depth plus topographic elevation. If a point at 5 m elevation experiences a
flow depth of 3 m, then surge is listed as 8 m. Discrepancies in observed versus
model surge heights may be due to differences in flow depth at that point,
differences in model versus observed elevation at that point, or both.
(Offshore measurements avoid this added difficulty because in water, peak flow
depth equals surge height.) At a given latitude and longitude, elevations
extracted from the base topography used in these calculations can differ by
several meters from those reported by Fritz
et al. [6, 7].In
appreciation of the problems inherent in onshore measurements of peak surge,
the comparison proceeds as follows.(1)
Given latitude, longitude, and elevation of anobservation location, search the base
topography file within a 500 m radius and select a reference location that has an elevation as close as possible to
the stated one. Elevations at the reference locations mostly matched observed
elevations within 50 cm, but there were cases where no high ground could be
found and reference location fell as much as 2 m lower than the observation location.(2)
Peak surge height (elevation plus peak flow depth) at
acomparison location anywhere within
500 m of the reference location was extracted and compared to field data.
Selecting the largest surge anywhere near reference location helps account for
happenstance.(3)
Lastly, recall that the smoothing process outlined in
Section4.4 ignores topographic elevation. Near the coast, even a 2 km smoothing
dimension can artificially lap up water high onto otherwise dry land. I
restricted the comparison location to be no more than 2 m higher than the
reference location.The top numbers in Figure6 lists observed
peak surge heights in decimeters at 40 locations selected from the Fritz et al. [6, 7] survey. The
highest observed surges (>6 m) are found within a 100 km wide band east of Katrina's
track. Like the predicted pattern in Figure 4, surge waters backed against the
Mississippi State Coast and extended at least 20 km offshore to engulf the
barrier islands in several meters of water. The lower numbers in Figure 6 list
computed peak surge heights at the comparison locations, selected following the
procedure above. Given all of the weaknesses and caveats regarding peak surge,
a first look suggests that the trends of the two datasets, if not the values
themselves, agree fairly well.Figure 6
Comparison of computed peak storm surge height
(bottom number) with field values (top number) measured by
Fritz et al. [6, 7]. Red-colored dots are locations where
observed surge exceed 5 meters. The little numbers within the
circles are location identifiers used in Figure 7.To quantify “eye
ball”  agreements of Figure6, Figure 7 plots observed and modeled peak surge
heights, peak flow depths, and elevations for the 40 selected data of Figure 6.
Observed and predicted peak surge values (Figure 7(a)) show good correlation—85% of the peak surge values fall within a
factor of two of the observed values (red-dashed lines). with 40% of predictions
being too large and 60% being too small. Considerable scatter exists, but net
bias is low: mean of observations = 5.01 m; mean of predictions = 4.96 m.  It is true to say there are noticeable outliers.
Some outliers seem to be isolated, others show systematic differences. For
instance, the model predicts higher surges for the three lower Delta sites
(number 28, number 32, number 15) than observed. Southernmost location number 15 is one of the worst
outliers, discrepant by over 6 m. These three sites lie close to the storm
track where surge is dominated by the eye-pile, not so much the zonal wind. The
large misfit there might suggest that the modeled eye-pile was exaggerated.
Possibly too, the selected value for onland bottom friction (Cd =  0.005)
was low. Higher friction might keep more of the surge from reaching these far
inland sites.Comparison of observed and modeled peak surge height
(a), peak flow depth (b), and elevation of observation point
(c). The little numbers in the circles are location identifiers
of Figure6. There is considerable scatter in both the observations
and in the calculations.
(a)(b)(c)Investigating the origin for
systematic differences in observed versus computed Katrina surge is a topic for
follow-on research. Seeing that the hurricane pressure and wind fields were
generic, the water surface drag formulation came off-the-shelf, and no tidal
corrections were included, there are plenty of avenues for refinement. In fact,
the only adjustments that I made were in the bottom friction coefficient and
smoothing half-width. Although better fits to data could be made, the thrust of
this article is to demonstrate the approach and not necessarily to reproduce
any specific situation. From the Katrina at New Orleans example, I am convinced
that a tsunami ball approach can capture most of the important physical aspects
associated with storm surge and that it can provide a viable alternative to
traditional methods.


## 6.1. Hurricane Katrina Parameters

I have previously listed the
hurricane parameters that need to be specified:(a)
storm's central positionxc;(b)
storm's central pressurePc;(c)
storm's peak wind speedVm;(d)
storm's radius of maximum velocityRm;(e)
the background air pressurePn; The Weather
Underground [5]  website provided Hurricane Katrina storm track
information (a)–(c) (see Table 1). I fixed the radius of maximum velocity at
25 km and  background air pressure at Pn = 1008 mbar.Table 1
Katrina Track,
Peak Wind Speed (miles/hour, 1 mph = 0.447 m/s) and Central Pressure (mbar) from Weather Underground [5]. Simulations here spanned 22 GMT 8/28 to
08 GMT 8/30.
DateTimeLatLonVm(mph)Pc(mbar)08/2318 GMT23.175.135100808/2400 GMT23.475.735100708/2406 GMT23.876.235100708/2412 GMT24.576.540100608/2418 GMT25.476.945100308/2500 GMT26.077.750100008/2506 GMT26.178.46099708/2512 GMT26.279.06599408/2518 GMT26.279.67098808/2600 GMT25.980.38098308/2606 GMT25.481.37598708/2612 GMT25.182.08597908/2618 GMT24.982.610096808/2700 GMT24.683.310595908/2706 GMT24.484.011095008/2712 GMT24.484.711594208/2718 GMT24.585.311594808/2800 GMT24.885.911594108/2806 GMT25.286.714593008/2812 GMT25.787.716590908/2818 GMT26.388.617590208/2900 GMT27.289.216090508/2906 GMT28.289.614591308/2912 GMT29.589.612592308/2918 GMT31.189.69094808/3000 GMT32.689.16096108/3006 GMT34.188.64597808/3012 GMT35.688.03598508/3018 GMT37.087.03599008/3100 GMT38.685.33599408/3106 GMT40.182.930996


## 6.2. Hurricane Katrina Simulation-Setup

The Katrina at New Orleans
simulations ran under a 2300 × 1600 topographic grid 170 m square with minimum
and maximum water depths restricted to 3 m and 400 m, respectively. Spaced about
570 m apart, 250 000 tsunami balls were distributed over all wet locations. The
water volume assigned to each ball equaled the ball spacing squared multiplied
by the water depth at the initial location. (Actually, tsunami balls might be
better described as tsunami columns.) To keep the water surface from spreading
far onto land as a consequence of smoothing, the smoothing half width dimensionW, varied from 50 km at locations far from land, to about 2 km at locations
close to, or onshore.The simulation began at 22 GMT
8/28/05 when the Katrina was well south of the grid and ended at 08 GMT 8/30/05,
after the storm moved far north of the grid. I updated tsunami ball location
and velocity everyΔT=10s
in the inner loop and recalculated the ocean surface every 5 minutes in the
outer loop. A movie frame was constructed at 15-minute intervals. Primary
outputs of the simulation include grid-interpolated flow velocity, current flow
depth and peak flow depth versus space and time both off and
onshore.


## 6.3. Hurricane Katrina Simulation-Results

Figure3 shows five
frames from the Katrina simulation. (All of the simulations presented in this article
are accompanied by Quicktime movie animations. See Table 2. Please view the
Quicktime movie version of Figure 3
at
http://es.ucsc.edu/~ward/k-at-no.mov) Panel 3(a) sets
the stage with geographic information of the New Orleans region. In the
Mississippi Delta, levees along the Mississippi River comprise the local high
ground. We will see that these linear levees act as dams for surge incoming
from either side. Panel 3(a) also presents the surge color scale used throughout
this article—colors red, orange, and yellow being positive, and blue and
violet being negative.Table 2
Quicktime movie links to the storm surge examples
presented in this article.
Katrina at New Orleans:http://es.ucsc.edu/~ward/k-at-no.movhttp://es.ucsc.edu/~ward/k-at-no-close.movhttp://es.ucsc.edu/~ward/k-at-no-peak.movKatrina at Savannah:http://es.ucsc.edu/~ward/k-at-sav.movhttp://es.ucsc.edu/~ward/k-at-sav-peak.movKatrina at Cape Cod:http://es.ucsc.edu/~ward/k-at-cod.movhttp://es.ucsc.edu/~ward/k-at-cod-peak.movTsunami Ball storm surge
calculation for Hurricane Katrina at New Orleans. Panels (b)-(f) show conditions at
3 hour intervals beginning at 06 GMT 8/29/06. Storm center and central
pressure in mbar are marked by white circle. Bold arrows plot wind direction.
Thin lines plot water flow direction. Red and blue colored areas show
significant deviations from sea level at 1 m intervals. Tsunami Ball storm surge
calculation for Hurricane Katrina at New Orleans. (Panel (e) shows peak surge for the
entire storm event occurs along the Mississippi State Coast. (Panel (f) shows peak
surge migrates eastward toward Alabama.In Panel 3(b) (06 GMT 8/29), 8 hours
after the calculation began, Katrina moves onto the computational grid at the
south. The hurricane eye-pile is 2 m high and 60 km across. In all of the
simulations, the wind-driven component of surge is most noticeable where winds
blow nearly perpendicular to the coastline. Offshore winds of ~35 m/s have
begun to draw down the sea surface west of the Mississippi River Delta. On the
north-south trending coasts on the east side of the Delta, directly onshore
winds of 30 m/s have begun to send surge overland.In Panel 3(c) (09 GMT 8/29), Katrina
has progressed to within about 30 km of first landfall. Water in the eye-pile-driven under the storm, shoals to nearly 5 m. North of the storm, 10 km of
overland flooding reaches  the Mississippi River and begins to pond against
its eastern levee. Northwest of the storm, offshore winds increasing to 50 m/s
draw down the sea level by 4  m.In Panel (d) (12 GMT 8/29), Katrina
has moved 20 km inland and crossed to the east side of the   river. The eye-pile
component of surge now pushing onto land cannot keep pace with the storm and
breaks off from the pressure eye itself. Driven by 40 m/s southwest winds,
water from the eye-pile runs over 20 km of flat land and dams to 8 m against the
River's western levee. Further north, flood water penetrates north and east of
the New Orleans town site.In Panel (e) (15 GMT 8/29), Katrina
has crossed the Mississippi State Coast and lies 20 km inland. Near the
pressure eye, a quick wind shift after the storm passes causes a redirection
of surge. Westerly winds now blow flood waters on the east side of the river
back toward the sea from the Delta lands. About this time, surges at New
Orleans town site peak at 3-4 m. Squarely onshore southwest and south winds of
50 m/s now force the highest flows of the entire storm event against the Mississippi
State Coast. The model registers 11 m of surge 20–40 km east ofKatrina's
track.In Panel  (f) (18 GMT 8/29), Katrina
has crossed the upper edge of the topographic grid and continues to travel
north and weaken. Maintained westerly winds >20 m/s cause the peak of the
surge to migrate eastward along the Mississippi State   Coast toward Alabama for
several hours. Westerly winds persist in ponding water on the west side of the
Mississippi River in the southern Delta. Several meters of surge remain there
until the end of the simulation (08 GMT 8/30).Figure4 contours peak storm
surge from 1 to 7 m. (Please view the Quicktime movie of this Figure at
http://es.ucsc.edu/~ward/k-at-no-peak.mov)
The “dots-on-a-string” map Katrina's position at 1-hour intervals. In
the south, note the residual trail from the eye-pile. The eye-pile component of
surge increases in height from 2 m well offshore to over 5 m as its tracking
water shoals near  the Delta. Highest predicted surges for the entire storm
(>9 m) occur along the lower Mississippi River and the Mississippi State
Coast. At New Orleans, this simulation predicts 3-4 m of flooding. (At the ~200 m
scale of the base topography, I do not expect to reproduce a block by block
flooding history in New Orleans. Still, if given higher-resolution topography,
I do not see why tsunami balls could not be used for that purpose.) Wind-driven
surge, banked against the mainland coast, extends 60 km in the shallow waters
offshore and as far east as the Florida panhandle. Surge banked  up against the
mainland covers the numerous islands of Southern Mississippi and Alabama. The
10−4 slope of the banked up wind blown surge is consistent with that
predicted by (24) in shallow water.Figure 4
Peak storm surge in 1 m increments starting at 1 m. Note
the trail left by the eye pile and how its height grows toward
shore. Areas in the lower Mississippi Delta and the Coast of
Mississippi State fare worst in this simulation. Surge banks up
against the mainland and extends 60 km out into the Gulf of Mexico.Evident in Figure4 is the strong
asymmetry in peak surge relative to the storm track. Counter clockwise rotation
of the zonal winds expose coasts on the right-hand side of the storm to more
extensive surge than equally distant locations on the left hand side. The zone
of maximum  surge extends for 100 km to the right side of the track. From Figure
2, 100 km corresponds to the radius of major storm pressure gradient and the
distance at which zonal winds fall to 1/2 their peak strength.Figure5 contains three large-scale frames from the Katrina simulation. (Please view the Quicktime movie of
Figure 5 at http://es.ucsc.edu/~ward/k-at-no-close.mov)
At large scale, it is easier to compare wind and water flow directions. As
discussed  in Section 4.4, the time required for flows to respond to changes in
wind direction increases in proportion to water depth. In shallow water, wind
and flow directions track fairly closely, and peak flow speeds reach
about 2 m/s. At that speed, overland flows take 2-3 hours to attain maximum
inundation
distance of 20 km.  Offshore, this
simulation predicts instantaneous surge to be rather lumpy and bumpy over a
several km scale (see Figure 5(c) especially). It is difficult to say for certain
if this irregularity attributes to a physical process (e.g., vestiges of wave
actions) or is an artifact of granularity and smoothing.Expanded view of storm surge calculation
from Hurricane Katrina at New Orleans at 3 hour intervals
beginning at 11 GMT. The small yellow or white circles are
locations of field-measured storm surge.
(a)(b)(c)


## 6.4. Comparison with Storm Surge Observations

How well does this tsunami ball
simulation using  off-the-shelf hurricane parameters predict Katrina's actual
surge? The best comparison would be made with time histories of sea level
measured at many offshore locations. Multiple measurements make it possible to
assess variability of surge in time and space (like the lumps and bumps
mentioned earlier) and to average noisy information. Also, offshore measurements
suffer less from the vulgarities of overland fluid flow than do onshore measurements.
A few permanently moored buoys did survive the storm, but these locate at some
distance from the action. I do not know of any satellite radar measurements of
sea height during the storm, but possibly some exist. Lacking open water surge
data, I fall back to information from post-Katrina surveys that measured peak
storm surge height onshore. I find a survey by Fritz et al. [6, 7] to be most complete.Like tsunami
runup data, peak storm surge data has several weaknesses. Firstly, peak surge,
being a point measure in space and time, can vary dramatically over short
distances. Secondly, peak surge, being an extreme measure not a mean  measure,
incorporates considerable happenstance. Imagine two waves momentarily  interfering
constructively here, but not over there. Thirdly, peak surge is the sum of peak
flow depth plus topographic elevation. If a point at 5 m elevation experiences a
flow depth of 3 m, then surge is listed as 8 m. Discrepancies in observed versus
model surge heights may be due to differences in flow depth at that point,
differences in model versus observed elevation at that point, or both.
(Offshore measurements avoid this added difficulty because in water, peak flow
depth equals surge height.) At a given latitude and longitude, elevations
extracted from the base topography used in these calculations can differ by
several meters from those reported by Fritz
et al. [6, 7].In
appreciation of the problems inherent in onshore measurements of peak surge,
the comparison proceeds as follows.(1)
Given latitude, longitude, and elevation of anobservation location, search the base
topography file within a 500 m radius and select a reference location that has an elevation as close as possible to
the stated one. Elevations at the reference locations mostly matched observed
elevations within 50 cm, but there were cases where no high ground could be
found and reference location fell as much as 2 m lower than the observation location.(2)
Peak surge height (elevation plus peak flow depth) at
acomparison location anywhere within
500 m of the reference location was extracted and compared to field data.
Selecting the largest surge anywhere near reference location helps account for
happenstance.(3)
Lastly, recall that the smoothing process outlined in
Section4.4 ignores topographic elevation. Near the coast, even a 2 km smoothing
dimension can artificially lap up water high onto otherwise dry land. I
restricted the comparison location to be no more than 2 m higher than the
reference location.The top numbers in Figure6 lists observed
peak surge heights in decimeters at 40 locations selected from the Fritz et al. [6, 7] survey. The
highest observed surges (>6 m) are found within a 100 km wide band east of Katrina's
track. Like the predicted pattern in Figure 4, surge waters backed against the
Mississippi State Coast and extended at least 20 km offshore to engulf the
barrier islands in several meters of water. The lower numbers in Figure 6 list
computed peak surge heights at the comparison locations, selected following the
procedure above. Given all of the weaknesses and caveats regarding peak surge,
a first look suggests that the trends of the two datasets, if not the values
themselves, agree fairly well.Figure 6
Comparison of computed peak storm surge height
(bottom number) with field values (top number) measured by
Fritz et al. [6, 7]. Red-colored dots are locations where
observed surge exceed 5 meters. The little numbers within the
circles are location identifiers used in Figure 7.To quantify “eye
ball”  agreements of Figure6, Figure 7 plots observed and modeled peak surge
heights, peak flow depths, and elevations for the 40 selected data of Figure 6.
Observed and predicted peak surge values (Figure 7(a)) show good correlation—85% of the peak surge values fall within a
factor of two of the observed values (red-dashed lines). with 40% of predictions
being too large and 60% being too small. Considerable scatter exists, but net
bias is low: mean of observations = 5.01 m; mean of predictions = 4.96 m.  It is true to say there are noticeable outliers.
Some outliers seem to be isolated, others show systematic differences. For
instance, the model predicts higher surges for the three lower Delta sites
(number 28, number 32, number 15) than observed. Southernmost location number 15 is one of the worst
outliers, discrepant by over 6 m. These three sites lie close to the storm
track where surge is dominated by the eye-pile, not so much the zonal wind. The
large misfit there might suggest that the modeled eye-pile was exaggerated.
Possibly too, the selected value for onland bottom friction (Cd =  0.005)
was low. Higher friction might keep more of the surge from reaching these far
inland sites.Comparison of observed and modeled peak surge height
(a), peak flow depth (b), and elevation of observation point
(c). The little numbers in the circles are location identifiers
of Figure6. There is considerable scatter in both the observations
and in the calculations.
(a)(b)(c)Investigating the origin for
systematic differences in observed versus computed Katrina surge is a topic for
follow-on research. Seeing that the hurricane pressure and wind fields were
generic, the water surface drag formulation came off-the-shelf, and no tidal
corrections were included, there are plenty of avenues for refinement. In fact,
the only adjustments that I made were in the bottom friction coefficient and
smoothing half-width. Although better fits to data could be made, the thrust of
this article is to demonstrate the approach and not necessarily to reproduce
any specific situation. From the Katrina at New Orleans example, I am convinced
that a tsunami ball approach can capture most of the important physical aspects
associated with storm surge and that it can provide a viable alternative to
traditional methods.


## 7. Hurricane Katrina What Ifs

Much of the worth in developing a
geophysical simulator lay in transferability. Once a simulator can credibly
reproduce an actual event, other similar, but hypothetical, situations can be
investigated by rolling over previously established model parameters. What if
Katrina had stalled out over the Mississippi Coast for four hours?  What if
Katrina stuck North Carolina instead of New Orleans? Investigating “What If”
scenarios can be especially fruitful on laptop-scale simulators because one can
set up the scenario and find answers in just hours.For the What If cases
investigated in this article, I employ the identical storm history and physical
parameters that were used for Katrina at New Orleans. I will move the starting
point of hurricane track and rotate its azimuth such that hypothetical Katrina
storms strike at interesting locations and angles.


### 7.1. Hurricane Katrina over Savannah

The first
target for hypothetical Hurricane Katrina will be the Georgia State Coast near
Savannah. Unlike New Orleans, the trend of the Georgia/South Carolina Coast is
generally straight with paralleling bathymetric contours. Many low lying
drainages intersect this stretch of coast and penetrate far inland. Savannah
itself locates at 5–15 m elevation 25 km up the Savannah River. How far will
surge be forced up these channels before the storm passes and the winds shift?
Would Savannah be flooded in a Katrina event? Although a What If storm track
running obliquely onto land might expose more coastline to damage, I run this
Katrina ashore nearly at right angle to the coast to make a control case.Figure8 shows two frames from
the Katrina at Savannah simulation 3 hours apart. (Please view the Quicktime movie
of this figure at http://es.ucsc.edu/~ward/k-at-sav.mov) In Panel (a), the storm has been approaching  16 hours and it is now 20 km
offshore. Interestingly, compared with Katrina at New Orleans (Figure 3(c),
the pattern of surge is still compact, nearly circular, and just 1-2 m high. For
most the time of approach, zonal winds have been oblique to the Georgia/South
Carolina Coast, so little surge has built up. In Figure 8(b),
the storm has run inland about 40 km, and the eye-pile has collided with coast
and squashed. The zonal winds now face on the northeast coast and surge has
quickly grown to several meters.Current storm surge,
surface wind, and water velocities for Katrina at Savannah (S). Panels (a) and  (b)
are 3 hours apart.
(a)(b)Figure9 contours peak storm surge
from 1 to 7 m for Katrina at Savannah. (Please view the Quicktime movie of
Figure 9 at http://es.ucsc.edu/~ward/k-at-sav-peak.mov) Figure 9 highlights the
right-hand/left-hand asymmetry in surge distribution relative to the storm
track. Surges greater than 1 m affect 250 km of coastline northeast of the track
yet, just 20 km southwest, positive surge is nonexistent. Like the New Orleans
case, peak surges reach 9 meters and the zone of maximum surge extends for
100 km to the right of the track. These two features seem to be dictated by the
storm parameters itself and not strongly dependent on the details of coastline.
Any surge feature that can be ascribed to storm input, rather than local bathymetry/topography
input, simplifies inundation prediction because it carries over
case to case.Figure 9
Peak storm surge in 1 m intervals starting at 1 m for
Katrina at Savannah. The dots-on-a-string show Katrina's
hypothetical course at 1-hour intervals. Note the very strong
right hand/left hand asymmetry in surge relative to the storm
track.


### 7.2. Hurricane Katrina at Cape Cod

For the second What If, I run Hurricane
Katrina obliquely over Cape Cod, Massachusetts, and into Massachusetts Bay,
South and East of Boston. Cape Cod, with coasts trending every direction of the
compass, should present more complex flow and surge patterns than the Savannah
case. I expect Cape Cod to block the eye-pile and zonal wind surges coming from
the open ocean. Will interior coasts in Massachusetts Bay be shielded from
surge? Will the eye-pile regenerate quickly enough as the storm passes over
the Bay to attack mainland coasts for a second time?Figure10 shows two frames from
the Katrina at Cape Cod simulation 5 hours apart. (Please view the Quicktime
movie of this Figure at
http://es.ucsc.edu/~ward/k-at-cod.mov)
In Panel (a), the storm has been approaching 9 hours and it has just
reached Nantucket Island. In contrast to the Savannah case (Figure 8(a)), the
surge pattern is complex and spatially extensive. Because much of the
Massachusetts coastline trends at right angles to the zonal winds, 2 m surges
have built against coasts as far as 250 km to the Northwest. Surge grows both
on the eastern coast of the outer cape and on the western shores inside of
Massachusetts Bay. Nantucket Island took a glancing blow, the eye-pile pushed
water to 9 m on its eastern side. In Figure 10(b), winds shift
rapidly south of the storm. Most locations that suffered onshore zonal winds
now experience offshore winds and visa versa. The elbow of Cape Cod gets clobbered,
as do interior coasts near Provincetown at the Cape tip. The eye-pile reforms
to 5 m height in the shallow waters of Massachusetts Bay. In this example, Cape
Cod offers only limited storm surge protection for the
interior shores.Current storm surge, surface wind and water velocities
at 5 hours apart for Katrina at Cape Cod. (Panel (a) 2 hours prior
to landfall at Cape Cod. NI = Nantucket Island. P = Provincetown.
(Panel (b) 2 hours after passing the Cape.
(a)(b)Figure11 contours peak storm
surge from 1 to 7 m for Katrina at Cape Cod. (Please view the Quicktime movie
of Figure 11 at http://es.ucsc.edu/~ward/k-at-cod-peak.mov) Indeed, an oblique storm approach to a
variably angled coast makes  a complex pattern of surge. The left-hand/right
hand asymmetry relative to storm track, seen so clearly at Savannah, gets
disrupted and surge heights very in unexpected ways. For instance, Nantucket
receives 9 m on its east side but only about 2 on the west side just 20 km
away. The complicated surge patterns in Figure 9 highlight the importance that
hurricane path variability has on certain coasts. In these situations, surge
would be difficult to forecast beforehand, so underscoring the value in
developing rapid simulation schemes of the type that tsunami balls are designed
to perform.Figure 11
Peak storm surge in 1 m intervals starting at 1 m for
Katrina at Cape Cod. Dots-on-a-string show Katrina's hypothetical
course at 1 hour intervals. South and East shores of Cape Cod fare
suffer peak surges of about 9 m.


## 7.1. Hurricane Katrina over Savannah

The first
target for hypothetical Hurricane Katrina will be the Georgia State Coast near
Savannah. Unlike New Orleans, the trend of the Georgia/South Carolina Coast is
generally straight with paralleling bathymetric contours. Many low lying
drainages intersect this stretch of coast and penetrate far inland. Savannah
itself locates at 5–15 m elevation 25 km up the Savannah River. How far will
surge be forced up these channels before the storm passes and the winds shift?
Would Savannah be flooded in a Katrina event? Although a What If storm track
running obliquely onto land might expose more coastline to damage, I run this
Katrina ashore nearly at right angle to the coast to make a control case.Figure8 shows two frames from
the Katrina at Savannah simulation 3 hours apart. (Please view the Quicktime movie
of this figure at http://es.ucsc.edu/~ward/k-at-sav.mov) In Panel (a), the storm has been approaching  16 hours and it is now 20 km
offshore. Interestingly, compared with Katrina at New Orleans (Figure 3(c),
the pattern of surge is still compact, nearly circular, and just 1-2 m high. For
most the time of approach, zonal winds have been oblique to the Georgia/South
Carolina Coast, so little surge has built up. In Figure 8(b),
the storm has run inland about 40 km, and the eye-pile has collided with coast
and squashed. The zonal winds now face on the northeast coast and surge has
quickly grown to several meters.Current storm surge,
surface wind, and water velocities for Katrina at Savannah (S). Panels (a) and  (b)
are 3 hours apart.
(a)(b)Figure9 contours peak storm surge
from 1 to 7 m for Katrina at Savannah. (Please view the Quicktime movie of
Figure 9 at http://es.ucsc.edu/~ward/k-at-sav-peak.mov) Figure 9 highlights the
right-hand/left-hand asymmetry in surge distribution relative to the storm
track. Surges greater than 1 m affect 250 km of coastline northeast of the track
yet, just 20 km southwest, positive surge is nonexistent. Like the New Orleans
case, peak surges reach 9 meters and the zone of maximum surge extends for
100 km to the right of the track. These two features seem to be dictated by the
storm parameters itself and not strongly dependent on the details of coastline.
Any surge feature that can be ascribed to storm input, rather than local bathymetry/topography
input, simplifies inundation prediction because it carries over
case to case.Figure 9
Peak storm surge in 1 m intervals starting at 1 m for
Katrina at Savannah. The dots-on-a-string show Katrina's
hypothetical course at 1-hour intervals. Note the very strong
right hand/left hand asymmetry in surge relative to the storm
track.


## 7.2. Hurricane Katrina at Cape Cod

For the second What If, I run Hurricane
Katrina obliquely over Cape Cod, Massachusetts, and into Massachusetts Bay,
South and East of Boston. Cape Cod, with coasts trending every direction of the
compass, should present more complex flow and surge patterns than the Savannah
case. I expect Cape Cod to block the eye-pile and zonal wind surges coming from
the open ocean. Will interior coasts in Massachusetts Bay be shielded from
surge? Will the eye-pile regenerate quickly enough as the storm passes over
the Bay to attack mainland coasts for a second time?Figure10 shows two frames from
the Katrina at Cape Cod simulation 5 hours apart. (Please view the Quicktime
movie of this Figure at
http://es.ucsc.edu/~ward/k-at-cod.mov)
In Panel (a), the storm has been approaching 9 hours and it has just
reached Nantucket Island. In contrast to the Savannah case (Figure 8(a)), the
surge pattern is complex and spatially extensive. Because much of the
Massachusetts coastline trends at right angles to the zonal winds, 2 m surges
have built against coasts as far as 250 km to the Northwest. Surge grows both
on the eastern coast of the outer cape and on the western shores inside of
Massachusetts Bay. Nantucket Island took a glancing blow, the eye-pile pushed
water to 9 m on its eastern side. In Figure 10(b), winds shift
rapidly south of the storm. Most locations that suffered onshore zonal winds
now experience offshore winds and visa versa. The elbow of Cape Cod gets clobbered,
as do interior coasts near Provincetown at the Cape tip. The eye-pile reforms
to 5 m height in the shallow waters of Massachusetts Bay. In this example, Cape
Cod offers only limited storm surge protection for the
interior shores.Current storm surge, surface wind and water velocities
at 5 hours apart for Katrina at Cape Cod. (Panel (a) 2 hours prior
to landfall at Cape Cod. NI = Nantucket Island. P = Provincetown.
(Panel (b) 2 hours after passing the Cape.
(a)(b)Figure11 contours peak storm
surge from 1 to 7 m for Katrina at Cape Cod. (Please view the Quicktime movie
of Figure 11 at http://es.ucsc.edu/~ward/k-at-cod-peak.mov) Indeed, an oblique storm approach to a
variably angled coast makes  a complex pattern of surge. The left-hand/right
hand asymmetry relative to storm track, seen so clearly at Savannah, gets
disrupted and surge heights very in unexpected ways. For instance, Nantucket
receives 9 m on its east side but only about 2 on the west side just 20 km
away. The complicated surge patterns in Figure 9 highlight the importance that
hurricane path variability has on certain coasts. In these situations, surge
would be difficult to forecast beforehand, so underscoring the value in
developing rapid simulation schemes of the type that tsunami balls are designed
to perform.Figure 11
Peak storm surge in 1 m intervals starting at 1 m for
Katrina at Cape Cod. Dots-on-a-string show Katrina's hypothetical
course at 1 hour intervals. South and East shores of Cape Cod fare
suffer peak surges of about 9 m.


## 8. Conclusions

This article introduces a new approach to storm
surge and inundation. It uses a momentum equation to accelerate balls of water
over variable depth topography and it computes surge height from the volume density
of balls. Compared with traditional finite-element/finite-difference
approaches, tsunami balls are more intuitive plus they have the advantages
that the continuity equation is satisfied automatically; the procedure is
meshless; inundation is simply tsunami balls run onto land; and
several hundred km size calculations can be done on a laptop computer.I demonstrate and validate the tsunami ball method
by simulating storm surge and inundation in the New Orleans region from
Hurricane Katrina in 2005. Despite the fact that the storm input parameters were off-the-shelf and that the pressure, wind, and drag formulations were generic,
predicted, and observed storm surge heights along a wide swath of coast compared
well with little mean bias. Storm surge has two components—an element
related to the eye-pile and a component related to the zonal winds. The
eye-pile component of surge is generally symmetric relative to storm track and
confined to about 50 km distance on either side. The zonal wind component can
present a much farther flung and asymmetric distribution depending upon the
orientation of the coastline relative to the wind.In addition to Katrina at New Orleans, I consider
two hypothetical scenarios—Katrina at Savannah and Katrina at Cape Cod.
Although the storm parameters were identical, substantially different surge
height outcomes are possible depending on the shape of coastline and the angle of
attack of the storm. Such variability is difficult-to-predict beforehand and it
highlights the value of rapid surge calculations that tsunami balls provide. I
am convinced that tsunami balls capture the important physical aspects
associated with storm surge and that the method provides a viable alternative
to traditional finite-difference/finite-element approaches.



---

*Source: 324707-2009-05-17.xml*