simulation of tsunami propagation.pdf

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Coastal Engineering Journal, Vol. 57, No. 4 (2015) 1550016 (30 pages)

c World Scientific Publishing Company and Japan Society of Civil EngineersDOI:10.1142/S0578563415500163

Simulation of Tsunami Propagation

Using Adaptive Cartesian Grids

Qiuhua Liang,,, Jingming Hou, and Reza Amouzgar,

State Key Laboratory of Hydrology-Water Resources

and Hydraulic Engineering, Hohai University,

Nanjing 210098, P. R. ChinaSchool of Civil Engineering and Geosciences, Newcastle University,

Newcastle Upon Tyne, NE1 7RU, [email protected]@ncl.ac.uk

[email protected]

Received 24 July 2014Accepted 28 July 2015Published 16 September 2015

This paper presents a 2D model for predicting tsunami propagation on dynamically adap-tive grids. In this model, a finite volume Godunov-type scheme is implemented to solvethe 2D nonlinear shallow water equations on adaptive grids. The simplified adaptive gridachieves automatic adaptation through increasing or reducing the subdivision level of abackground cell according to certain criteria defined by tsunami wave features. The gridsystem is straightforward to implement and no data structure is needed to store grid infor-mation. The present model is validated by applying it to simulate three laboratory-scaletest cases of tsunami propagation over uneven beds and finally used to reproduce the 2011

Tohoku tsunami in Japan. The model results confirm the models capability in predictingtsunami wave propagation in a reliable and efficient way.

Keywords: Tsunami Modeling; adaptive grid; shallow water equations; finite volumemethod; Godunov-type scheme.

Corresponding author.

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http://dx.doi.org/10.1142/S0578563415500163http://dx.doi.org/10.1142/S0578563415500163


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Q. Liang, J. Hou & R. Amouzgar

1. Introduction

In recent years, tremendous research efforts have been made to develop accurate

and robust numerical models for tsunamis, for example, the COMCOT (Cornell

Multi-grid Coupled Tsunami Model), MOST (Method of Splitting Tsunami) model,TUNAMI (Tohoku Universitys Numerical Analysis Model for Investigation), Geo-

Claw, TSUNAMI3D, THETIS, among others. COMCOT, TUNAMI and GeoClaw

solve the 2D nonlinear shallow water equations (NSWEs) to represent the tsunami

propagation and runup, while the TSUNAMI3D and THETIS models solve 3D

NavierStokes (NS) equations for incompressible fluid flows with free surface and

interfacial boundaries described based on the concept of the fractional volume of

fluid (VOF) method. They have been widely used to model tsunami events [Wang

and Liu,2006;Wei et al.,2008;Oishi et al.,2015;Tang et al.,2012;Imamura,1996;

George and LeVeque,2008;Arcos and LeVeque,2014;Horrillo et al.,2013;Abadie

et al.,2012].The wave fronts of a tsunami usually present as bores/surges rapidly moving

inland, which demands special treatment for accurate simulations. A general way is

to use fine computational meshes with resolution ranging from several meters to tens

of meters. But tsunami propagation can take place over an entire ocean and mesh

refinement over the entire domain may lead to unaffordable computational cost. For

instance, if 10m 10m cells are adopted for a domain of 100km 100km, 100million computational cells will be produced, which are computationally inhibited

for most of the existing tsunami models. So it is desirable to intelligently refine

the grid locally near to the wave fronts instead of the whole domain. This can be

achieved using adaptive mesh refinement (AMR), which has been an active researchtopic in CFD for over 30 years [Berger and Oliger,1984;Yiu et al.,1996;Lee et al.,

2011].

Over recent decades, two grid adaption approaches have been widely used to

carry out the AMR on Cartesian grids. They are respectively known as block adap-

tion and hierarchical grid adaption [Popinet,2011;Liang,2012]. The former method

uses a group of coexisting grids of different resolutions to achieve grid adaption

[Berger and Oliger, 1984; Berger and LeVeque, 1998]. It has been implemented in

CeoClaw and applied byGeorge and LeVeque[2008], Watanabe et al. [2012],Arcos

and LeVeque [2014] to model tsunami propagation. A hierarchical grid generally

employs a quadtree data structure to store grid information such as neighbors and

subdivision levels. The information is updated by searching the data tree during

grid adaptation. Hierarchical grids perform AMR on an individual grid cell rather

than a block of grid cells as used in the block method. Therefore, it gives more

flexibility to track the flow features and is favored by numerous researchers, e.g. Yiu

et al. [1996], Rogers et al. [2001], Popinet[2003], Liang and Borthwick[2009], Lee

et al.[2011].Popinet[2011,2012] used adaptive quadtree grids in his tsunami model.

Despite its greater flexibility, quadtree grids suffer from large storage requirement

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Simulation of Tsunami Propagation Using Adaptive Cartesian Grids

and additional computational overhead for searching the data tree [Liang, 2012].

To improve the efficiency of adaptive quadtree grids, Ji et al. [2010] developed a

new cell-based structured AMR data structure, which requires less storage than the

conventional quadtree. Liang[2012] devised a simplified block-based adaptive grid

system which just imposes subdivision levels on coarse background cells. There aretwo main advantages for the adaptive mesh method proposed in Liang[2012] com-

paring with the aforementioned conventional approaches. Firstly, no complicated

procedure is required for generating an initial grid apart from simple allocation of

specific subdivision level to each of the coarse cells on the background grid. Secondly,

the neighbors of an arbitrary cell are fully determined by simple algebraic relation-

ships and thus no data structure is demanded. Grid adaption is straightforward to

achieve by altering the subdivision level of a cell to follow certain flow conditions.

Such an efficient grid system may be well-suited for large-scale tsunami simulations

that involve multi-level bathymetric datasets.

This paper aims to introduce an efficient tsunami model based on the afore-

mentioned simplified adaptive grid system. The remainder of the paper is organized

as follows: the governing equations that mathematically describe the propagation

of the tsunami waves and the numerical scheme to solve them are briefly reviewed

in Sec. 2; the adaptive Cartesian grid system is introduced in Sec. 3; the model is

verified against three laboratory tests and applied to a field-scale tsunami event in

Sec. 4; brief discussions and conclusions are drawn up in Sec. 5.

2. Governing Equations and Numerical Scheme

As inJi et al.[2010] andPopinet[2011] the NSWEs derived from the conservation of

mass and momentum are chosen to simulate tsunamis. In a vector form, the NSWEs

can be written as[Liang and Borthwick,2009]

q

t +

f

x+

g

y =S, (1)

q=

qx

qy

, f=

uh

u2h+g(2 2zb)/2uvh

, g=

vh

vuh

v2h+g(2 2zb)/2

, (2)

S= Sb+ Sf=

0

gzb/xgzb/y

+

0

Cfu

u2 +v2

Cfv

u2 +v2

, (3)

where t is the time; x and y represent the Cartesian coordinates; q denotes the

vector of conserved flow variables consisting of ,qx=uh and qy =vhy, i.e. the free

surface water level, unit-width discharges in the x- and y-directions, respectively;

h, u and v are water depth, depth-averaged velocities in the x- and y-directions,

respectively; f and g are the flux vectors in the two Cartesian directions; Sdenotes

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the source vector including slope Sb and friction source terms Sf; zb represents the

bed elevation; Cf is the bed roughness coefficient that is generally calculated by

gn2/h1/3 with n being the Manning coefficient.

On the adaptive grid system, the SWEs are solved numerically using a two-step

MUSCL-Hancock scheme originally developed by van Leer [1984], which consistsof a predictor step and a corrector step and has been widely applied by numerous

researchers (e.g. Zhou et al. [2002]) for shallow water flow modeling. It has been

incorporated in a cell-centered finite volume Godunov-type scheme for nonuniform

grid as presented by Liang [2011, 2012]. In the predictor step, intermediate flow

variables are predicted over half of a time interval. These intermediate variables

are then utilized in the corrector step to update the results to a new time level.

The friction source terms Sfare not evaluated within the MUSCL-Hancock scheme

but independently calculated using a splitting point-implicit method proposed by

Bussing and Murmant[1998]. In order to prevent the reversing flow direction caused

by unrealistic friction force, a limiting step introduced by Liang and Marche[2009]

is taken into account. Since the numerical scheme is overall explicit, the solution

stability is essentially controlled by the CourantFriedrichsLewy (CFL) condition

[Courant et al., 1928], which is used to estimate the time step for next iteration

during a simulation. Open and closed boundaries are implemented, as by Liang and

Borthwick[2009].

3. Dynamically Adaptive Cartesian Grid

The dynamically adaptive Cartesian grid system developed by Liang [2012] isadopted in the current tsunami model to allow effective grid adaption. After the

initial grid is generated, dynamical grid adaption can be easily carried out by spec-

ifying the subdivision level on a coarse background cell according to certain criteria

related to flow conditions. This section reviews the procedure of implementing the

dynamically adaptive Cartesian grid system.

3.1. Grid generation and cell identification

The initial grid is generated by simply following three steps:

A rectangular computational domain is selected and discretized using a coarseuniform Cartesian grid which is called background grid. For example, the domain

in Fig.1(a) is divided into a 2 3 background grid; Each cell on the background grid is checked and refined by giving a specific subdi-

vision level according to certain criterion. For instance, the subdivision levels on

background cells (i, j) and (i, j +1) shown in Fig.1(b) are respectively subdivided

by respectively specifying as level one and two;

To ensure grid quality, the final mesh should satisfy a 2:1 rule, i.e. no cell is allowedto have a neighbor that is more than two times bigger or smaller. As shown in

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(a) (b) (c)

(d)

Fig. 1. (Color online) Structured but nonuniform grid generation: (a) background grid, (b) irregulargrid, (c) regularized grid, (d) cell indexes of regularized grid. Red-bold numbers in this figurerepresent the number of subdivision level.

Fig. 1(b), cell (i, j+ 1) does not meet the 2:1 rule since its western neighbor is

four times bigger. So cell (i, j+ 1) should be regularized and subdivided into levelone [Fig.1(c)].

As illustrated in Fig. 1(d), each cell on such a nonuniform grid can be identified by

four indexes (i, j, is, js), where is and js are numbered form 1 to Ms. Ms denotes

the number of sub-cells in the x- or y-direction at background cell (i, j);Ms = 2lev

with lev representing the subdivision level of (i, j). The size of a background cell

is x y, and subsequently the dimensions of the sub-cells can be computed byxs= x/2

lev and ys= y/2lev.

On such a nonuniform grid, it is not necessary to define a data structure to store

neighbor information as the neighbors of an arbitrary cell can be entirely determined

by simple algebraic relationships. For those cells with the same subdivision levels,

the neighbors can be easily obtained as on uniform grids. For cells with neighbors of

different subdivision levels, simple algebraic relationships can be derived to specify

their neighbors. For example, considering the sub-cells of (i, j) in Fig. 1(d), the

western neighbor of (i, j, 1, 1) and (i, j, 1, 2) has a lower subdivision level and is

denoted by (i 1, j, isN, jsN) where the subscript N represents neighbor. Theneighbor sub-cell indices isN and jsNcan be obtained by

isN = 2levN , jsN= Ceiling (js/2), (4)

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where function Ceiling means the smallest integer not less thanjs/2. If the neighbors

are in higher subdivision levels, e.g. the two eastern neighbors of (i, j, 2, 1) can be

denoted by (i+ 1, j, isN, jsN), where

isN= 1, jsN=js 2 1 and js 2. (5)The northern and southern neighbors can be identified in a similar way. Comparing

to other existing adaptive grid systems that need special data structure to store

the neighbor information, the storage requirement for the current adaptive grids is

minimized and only two matrices are required to store the subdivision level of every

background cell and the cell IDs of all sub-cells for flow calculation. Therefore,

cell manipulation during grid adaptation becomes very flexible and efficient [Liang,

2012].

3.2. Adaption indicator

The first step to achieve dynamic grid adaptation is to define an adaptation criterion.

In this work, the adaptation indicator is defined according to water level gradient,

which is a key flow feature indicating the complexity of tsunami hydrodynamics.

At cell ic on the aforementioned nonuniform grid, the adaptation indicator can be

expressed by a dimensionless factor

ic= GicPq(G)

, (6)

where Gic is the averaged gradient of the water level at cell ic and

Gic =

x

2ic

+

y

2ic

, (7)

G is a vector containing the water surface gradients for all of the sub-cells inside

the computational domain, Pq(G) returns the qth percentile of G and q = 1sawhere sa is a user specified coefficient indicating the sensitivity of the adaptation

procedure. For instance, sa = 20% means that 20% of the flow cells are subject to

grid refinement. For coarsening, a critical value of coar must also be prescribed.

Grid coarsening will be carried out if ic is less than coar for all of the sub-cells.

Moreover, the cells defining the wet-dry fronts (i.e. cells have both wet and dry

neighbors) are also refined to accurately capture the moving wet-dry fronts.

3.3. Dynamic grid adaption

For dynamic grid adaptation, if one or more sub-cells of a background cell (i, j)

have values of the adaption indicator ic greater than 1 and the subdivision level

of (i, j) is less than the specified maximum value, (i, j) will be marked for further

subdivision and its subdivision level will be increased to lev + 1. Additionally, any

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cell defining the wet-dry interfaces will be marked for subdivision to the maximum

grid refinement level levm to capture the wet-dry interface. If the values of ic of

all sub-cells in a background cell (i, j) are lower than coarand its subdivision level

is higher than zero, the cell will be coarsened by changing its subdivision level to

lev1. The levels of all newly created cells are then checked to guarantee the 2:1 ruleas explained in Sec. 3.1 to ensure the regularity of the grid. The above procedure of

grid refinement and coarsening are performed at every time step during a simulation.

3.4. Mapping newly created cells

After the grid refinement and coarsening, the flow information will be mapped to

the newly created cells using a simple linear interpolation scheme, as reported by

Liang [2011]. But the values of bed elevation (bathymetry) and friction parameter

are interpolated directly from the original dataset or multiple datasets in order to

represent precisely the domain. In order to reduce the demand of computer memory,the files storing bathymetric data and values of friction parameter are stored offline

in the local drive and read in during grid adaptation as suggested byPopinet[2011].

The flow information in a coarsened cell can be calculated by simply averaging

that of the four child cells. In newly refined cells, the values of bed elevation and

friction parameter are interpolated linearly from the original datasets. In addition,

the approach suggested byLiang et al.[2015] is employed to resolve the contradiction

between the C-property and mass conservation during grid adaption [Liang and

Borthwick,2009;Popinet,2011]. Once the new grid and the flow information in the

newly created cells are obtained, the NSWEs are ready to be solved numerically to

update the flow variables to the next time step by a Godunov-type finite volume

method as described in Sec. 2.

4. Model Applications

The present tsunami model is validated against four test cases in this section, includ-

ing three laboratory-scale and a real tsunami cases, and the results are compared

with measured water levels and maximum runup. The Courant number of 0.5 is

used for all of the simulations.

4.1. Solitary wave runup on a simple beach

The laboratory experiments of a solitary wave on a simple beach were carried out by

Synolakis[1986] and the results can be found in Synolakis et al. [2007]. As sketched

in Fig.2(a), a solitary wave is generated in a channel at x= X1 to propagate over

a simple beach with a slope of 1:19.85. The toe of the slope is at X0 = 9.925m

in this work. X1 = X0 +L/2 and the half wave length L/2 = (1/)arccosh

20

with =

3(H/D)/4 where H and D denote the amplitude of the incipient wave

and the still water depth, respectively [Synolakis et al., 2007]. The incoming wave

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D

H

X

Y

x=X0 x=X1

1:19.85

zb

(a)

(b)

(c)

Fig. 2. Solitary wave runup on a simple beach: (a) definition sketch for the case of a solitary waveon a simple beach[Synolakiset al.,2007], (b) adaptive grid for the wave with H= 0.3 m at:t = 15,(c) t = 25.

is specified at the right-hand side boundary [Nikolos and Delis,2009] as

(t) =Hsech2

3H

4D3C(t T0)

, (8)

u(t) = C

D+, v(t) = 0, (9)

whereT0= 0.0 is the time when the wave crest reaches the domain;C=

g(D+H)

is the wave celerity. Two cases with wave amplitudes ofH= 0.3 m and 0.0185 m are

considered, which represent breaking and nonbreaking waves, respectively. The stillwater depth is specified to be 1.0 m. A Manning coefficient of 0.01 s/m1/3 is used

for the channel of stainless steel bed [Synolakis et al., 2007]. The background cell

resolution is 0.2 m and it will be refined up to 0.1 m according to the grid adaption

criterion withsa= 0.3. Grid coarsening will be undertaken for icless than 0.7. The

grid evolution for the case with the higher wave amplitude is sketched in Figs. 2(b)

and2(c) at t = 15 and 25 where t =t

g/D [Synolakis,1986].

The computed and measured water levels are plotted in Figs. 3 and 4, overall

good agreements with the measured wave profiles are observed for both cases where

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(a) (b)

(c) (d)

Fig. 3. Solitary wave runup on a simple beach: comparison between the computed and measuredwater levels for the wave with amplitude of 0.3 m at (a) t = 15, (b) t = 20, (c) t = 25, (d)t = 30.

H = 0.3 m and 0.0185 m. This demonstrates that the current adaptive grid-based

model is able to reproduce the wave runup process. However, due to the limita-

tion of the shallow water equations in representing 3D fluid patterns and breaking

waves, certain discrepancies between the prediction and measurement are detected,

especially in the area adjacent to the wave fronts. The simulation results may beimproved by including the Boussinesq approximation in shallow water flow model

as shown, for example, byKazolea et al.[2012]. As shown in Figs.3 and 4, the com-

puted water surfaces are almost identical between the simulations on adaptive grid

and uniform grid of 0.1 m resolution, suggesting that the adaptive grid system can

capture the key wave features and produce accurate numerical solutions. Tables 1

and 2 further summarize the performance of the adaptive grid model, in term of

accuracy and efficiency. The computed root mean square errors (RMSE) for the

water level along a section in the x-direction are very similar for the simulations

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(a) (b)

(c) (d)

Fig. 4. Solitary wave runup on a simple beach: comparison between the computed and measuredwater levels for the wave with amplitude of 0.3 m at (a) t = 40, (b) t = 50, (c) t = 60,(d) t = 70.

Table 1. Solitary wave runup on a simple beach: RMSE () and relative computational cost forthe wave with H= 0.0185 m.

t 30 40 50 60 70 Relative CPU time

RMSE() on adaptive grid [mm] 1.72 2.68 2.40 1.40 0.74 1.00RMSE() on uniform grid [mm] 1.71 2.77 2.42 1.43 0.76 2.12

Table 2. Solitary wave runup on a simple beach: RMSE () and relative computational cost forthe wave with H= 0.3 m.

t 15 20 25 30 Relative CPU time

RMSE() on adaptive grid [mm] 5.30 3.11 4.30 1.66 1.00RMSE() on uniform grid [mm] 5.35 3.19 4.34 1.68 2.27

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on adaptive and uniform grids, revealing that the adaptive grid-based model can

achieve similar simulation accuracy to the high-resolution uniform grid. The RMSE

for the adaptive grid is marginally lower than that obtained for the uniform grid.

The propagation of the solitary wave is successfully reproduced by both models.

But the adaptive grid model is computationally more efficient and requires less than50% of computing time on the uniform grid.

4.2. Solitary wave runup over a conical island

A series of experiments to study the runup of tsunami waves on a conical island were

conducted in the US Army Engineer Waterways Experiment Station[Briggs et al.,

1995], and have been widely used to validate SWE models [Titov and Synolakis,

1995, 1998; Bradford and Sanders, 2002; Lynett et al., 2002; Hubbard and Dodd,

2002; Choi et al.

, 2007; Nikolos and Delis, 2009; Zijlema et al.

, 2011; Hou et al.

,2013]. As in [Hou et al.,2013], a 25.92 m27.6 m frictionless computational domainis chosen, in which a conical island with a base diameter of 7.2m, top diameter of

2.2 m and height of 0.625 m, is located at the domain center (Fig.5). Solitary planar

waves are generated by a directional wave-maker at the left boundary and the rest

of the boundaries are closed. The inflow conditions are specified by Eqs. (8) and(9).

Three different waves are considered with H/D = 0.045, 0.091 and 0.181 which are

termed as case A, B and C, respectively, and D is specified to be 0.32 m for all cases.

The simulations are run on a coarse background grid with 130 138 cells with aresolution of 0.2 m, which can be refined up to 0.05 m in certain parts according to

the wave patterns. The sensitivity parametersa and the grid coarsening parametercoar are set to be 0.15 and 0.8, respectively.

The 3D view of wave propagation over the island for case A at t = 28.2, 33.2

and 36.2s is plotted in Fig. 6.The incipient waves propagate along the domain and

hit the island after about 30.5 s. Part of the wave is refracted around the island as

it propagates toward the lee side. Two trapped waves are created at each side of

the island and then collide at the lee side, generating a runup around t = 36.2 s.

After running down from the island, the waves propagate back toward the island.

The computed wave patterns for the case B and C are very similar to case A,

with slight difference in arriving time and magnitude. The corresponding adapted

grids are plotted in Figs. 6(b), 6(d) and 6(f), confirming that the grid adaptionworks well in tracking wave front by refining and coarsening the relevant cells. To

further demonstrate the performance of the adaptive grid model, the measured time

histories of water level at certain wave gauges as sketched in Fig. 5are compared

with the numerical predictions in Figs. 79 for the case A, B and C, respectively.

The computed results agree satisfactorily with the measured data for all of the

three cases. The leading wave-heights and the arrival times are well-predicted at

most gauges, suggesting the model is capable of reproducing the key features of the

propagating waves with H/D ranging from 0.045 to 0.181. But it should be noted

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Fig. 5. Solitary wave runup over a conical island: model set up and locations of selected water-levelgauges[Hou et al.,2013].

that the steepness of the wave fronts and the secondary depression wave following

the leading wave are not exactly reproduced. Such discrepancies may be due to the

inherent inability of a 2D SWE model in representing fully 3D flow features [Choi

et al., 2007]. Similar results are also predicted by other researchers using different

SWE models, for example [Liu et al.,1995;Lynett et al.,2002;Hubbard and Dodd,

2002;Nikolos and Delis,2009;Hou et al.,2013].The computed maximum runup on the island is recorded and compared with the

measurements in Fig. 10. The variable denotes the angle of the section as shown

in the layout plot (Fig. 5). For all of the three cases, the computed highest and

the second highest runups are detected at the front and the lee sides, respectively.

The computed profiles of the maximum runup have similar trends to the measured

ones, despite certain level of deviation. The computed values are found to be slightly

higher than the measured data at the front side of the island, perhaps because the

friction is not taken into account as suggested by other researchers, for example by

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(a) (b)

(c) (d)

(e) (f)

Fig. 6. Solitary wave runup over a conical island: case A: (a) wave propagation at t = 28.2s, (b)adaptive grid at t= 28.2 s, (c) wave propagation at t = 33.2 s, (d) adaptive grid at t = 33.2 s, (e)wave propagation at t = 36.2 s, (f) adaptive grid at t= 36.2 s.

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20 25 30 35 400.04

0.02

0

0.02

0.04

t(s)

(

m)

measuredcomputed

20 25 30 35 400.04

0.02

0

0.02

0.04

t(s)

(

m)

measuredcomputed

(a) (b)

20 25 30 35 400.04

0.02

0

0.02

0.04

t(s)

(

m)

measuredcomputed

20 25 30 35 400.04

0.02

0

0.02

0.04

t(s)

(

m)

measuredcomputed

(c) (d)

Fig. 7. Solitary wave runup a conical island: case A: comparison between the computed and mea-sured water levels at (a) WG3, (b) WG9, (c) WG16, (d) WG22.

Nikolos and Delis [2009]. While the height of the real waves become steeper as a

result of approaching the shorelines, vertical motions are expected to be more and

more significant, which are not represented in the 2D depth averaging assumption.

The computed results by the adaptive grid model are also compared with those

obtained on the uniform grid with a cell length of 0.1 m. As illustrated in Fig. 8,the time histories of computed water surface on the two different grids agree well

with each other at all of the gauges, demonstrating that the AMR can produce

results with equivalent accuracy without compromising computational efficiency.

The RMSEs computed against measured data for all of the three cases at the five

gauge points are shown in Tables 35, further confirming the accurate simulation

results produced by the adaptive grid model. However, the uniform grid model

requires 1.75, 1.87 and 1.69 times of the runtime, compared with the adaptive grid

code for case A, B and C, respectively.

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20 25 30 35 400.04

0.02

0

0.02

0.04

0.06

t(s)

(

m)

measured

computed on adaptive gridcomputed on uniform grid

20 25 30 35 400.04

0.02

0

0.02

0.04

0.06

t(s)

(

m)

measured


(a) (b)

20 25 30 35 40

0.04

0.02

0

0.02

0.04

0.06

t(s)

(

m)

measured


20 25 30 35 40

0.04

0.02

0

0.02

0.04

0.06

t(s)

(

m)

measured


(c) (d)

Fig. 8. Solitary wave runup a conical island: case B: comparison between the computed and mea-sured water levels at (a) WG3, (b) WG9, (c) WG16, (d) WG22.

4.3. Tsunami wave approaching Monai valley

Monai valley is located in Okushiri island, Japan, and was attacked by a severe

tsunami happened in the Japan Sea in 1993. This event was reproduced in labora-

tory using a 1:400 scaled physical model built in the Research Institute for ElectricPower Industry (CRIEPI) in Abiko, Japan [Liu et al., 2008]. The test involves a

5.488m 3.402 m computational domain over the topography as shown in Fig.11.A 22.5 s incident tsunami wave enters the domain from the left boundary and other

domain boundaries are set to be closed. During the simulation, the domain is covered

by a background grid of 98 61 cells with a resolution of 0.056 m. Fine bathymetrydata of 0.014 m resolution is also available for the entire domain, which allows the

background grid to refine up to 2 levels. The sensitivity parameter sa and the

grid coarsening parameter coar are specified as 0.24 and 0.8. Uniform Manning

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20 25 30 35 400.06

0.04

0.02

0

0.02

0.04

0.06

0.08

0.1

t(s)

(

m)

measured

computed

20 25 30 35 400.06

0.04

0.02

0

0.02

0.04

0.06

0.08

0.1

t(s)

(

m)

measured

computed

(a) (b)

20 25 30 35 400.06

0.04

0.02

0

0.02

0.04

0.06

0.08

0.1

t(s)

(

m)

measuredcomputed

20 25 30 35 400.06

0.04

0.02

0

0.02

0.04

0.06

0.08

0.1

t(s)

(

m)

measuredcomputed

(c) (d)

Fig. 9. Solitary wave runup a conical island: case C: comparison between the computed and mea-sured water levels at (a) WG3, (b) WG9, (c) WG16, (d) WG22.

coefficient of 0.001m1/3s is imposed on the whole domain as suggested by other

researchers [Popinet,2011;Funke et al.,2011]. The model is run for 30 s.

The numerical results at different output times (t = 14, 17 and 20s) are plot-

ted in Fig. 12 in a 3D view, including the adapted grids. Grid adaptation clearly

evolves according to the dynamics of the tsunami wave. The adaptive grids effectivelycapture the complex tsunami wave patterns and the moving shoreline by generat-

ing refined mesh at those regions with steep water surface gradients and near the

wet-dry front. Figure13 demonstrates the comparison between the measured and

predicted time histories of water level at three gauges G5, G6 and G7, which are

located at [4.521m, 1.196 m], [4.521m, 1.696 m] and [4.521m, 2.196 m], respectively.

In the first 10 s, because of the initial water disturbances in the realistic wave tank,

the computed water levels do not agree well with the measurements. The results

are consistent with predictions reported by other researchers [Zhang and Baptista,

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0

1

2

3

4

5

0 60 120 180 240 300 360

maximumr

unu

p(cm)

()

measured computed

(a)

0

3

6

9

12

0 60 120 180 240 300 360

maximumr

unup(cm)

()

measured computed

(b)

0

5

10

15

20

25

0 60 120 180 240 300 360

maximumr

unup(cm)

measured

computed

()

(c)

Fig. 10. Solitary wave runup a conical island: computed and measured maximum runup for wavecase (a) A, (b) B and (c) C.

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Table 3. Solitary wave runup over a conical island: RMSE () and relative computational cost forcase A.

Gauges WG3 WG6 WG9 WG16 WG22 Relative CPU time

RMSE () on adaptive grid [mm] 0.792 1.073 0.9234 1.931 0.097 1.00RMSE () on uniform grid [mm] 0.792 1.074 0.9129 1.923 0.1083 1.75

Table 4. Solitary wave runup over a conical island: RMSE () and relative computational costfor case B.


RMSE() on adaptive grid [mm] 1.701 2.313 2.915 0.234 3.309 1.00RMSE () on uniform grid [mm] 1.704 2.313 2.920 0.267 3.325 1.87

Table 5. Solitary wave runup over a conical island: RMSE () and relative computational costfor case C.


RMSE() on adaptive grid [mm] 1.540 3.117 4.234 1.961 1.936 1.00RMSE () on uniform grid [mm] 1.532 3.103 4.104 1.994 2.030 1.69

Fig. 11. Tsunami wave approaching Monai valley: domain and bathymetry for the laboratory exper-iment (unit : m).

2008; Clain and Clauzon,2010;Nicolsky et al., 2011]. Nevertheless, the amplitude

and the phase of the first wave are satisfactorily reproduced at the gauges after

10 s. As shown in Fig.14,the maximum runup height is computed to be 0.072 m in

the Monai valley, which is equivalent to 28.8 m at the prototype scale and roughly

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(a) (b)

(c) (d)

(e) (f)

Fig. 12. Tsunami wave approaching Monai valley: (a) wave propagation at t = 14 s, (b) adaptivegrid at t = 14 s, (c) wave propagation at t = 17 s, (d) adaptive grid at t = 17 s, (e) wave propagationat t= 20 s, (f) adaptive grid at t= 20s.

congruous with the field measurement of over 30 m [Matsuyama and Tanaka,2001].

As compared in Fig. 13 and Table 6, the numerical predictions obtained on the

adaptive grid are very similar to those predicted on the refined uniform grid at

a resolution of 0.014 m. Yet, in order to achieve a similar solution accuracy, the

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0 5 10 15 20 25 300.02

0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

t(s)

(

m)

measuredcomputed on adaptive grid

computed on uniform grid

0 5 10 15 20 25 300.02

0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

t(s)

(

m)

measuredcomputed on adaptive grid

computed on uniform grid

(a) (b)

0 5 10 15 20 25 300.02

0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

t(s)

(

m)

measuredcomputed on adaptive gridcomputed on uniform grid

(c)

Fig. 13. Tsunami wave approaching Monai valley: time histories of computed and measured waterlevel at: (a) G5, (b) G7, (c) G9.

adaptive grid-based model is found to be about four times more efficient than its

uniform grid counterpart.

4.4. 2011 Tohoku tsunami propagation simulationOn March 11, 2011, a large earthquake with Mw = 9.0 off the Pacific coast of

Tohoku, Japan with epicenter at 38.1035 N, 142.861 E was reported by Japan Mete-

orological Agency (JMA) at 14:46:18 local time. The earthquake subsequently caused

one of the most destructive tsunamis in human history, engulfed part of the eastern

coast of Japan. Research has been reported for different aspects of this tsunami event

since2011, for example [Fujii et al.,2011;Lay et al.,2011;Wei et al.,2012;Hooper

et al.,2013;Melgar and Bock,2013;Satake et al., 2013], due to rich source of data

availability and important scientific value. In this section, the present adaptive grid

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Fig. 14. (Color online) Tsunami wave approaching Monai valley: computed maximum runup att= 16.9 s around Monai valley. The numbers on the contours denote the elevation of the topography(m); The red line is the computed shoreline.

Table 6. Tsunami wave approaching Monai valley: RMSE () at gauges andrelative computational cost.

Gauges G5 G7 G9 Relative CPU time

RMSE() on adaptive grid [mm] 0.414 0.927 0.416 1.00RMSE () on uniform grid [mm] 0.423 0.881 0.395 3.56

tsunami model is used to reproduce the tsunami wave propagation for this event to

demonstrate its robustness for field-scale application.

On a background uniform grid of 1600 m resolution, dynamic grid adaption canbe carried out up to a resolution of 400 m in the computational domain as shown

in Fig. 15, according to local flow features. The bathymetry data for the 1600 m

and 400 m cells are generated from the available 450 m and 1350 m datasets. A

rectangular fault model assuming static deformation of the Ocean floor is applied

to process the initial fault parameters to generate water surface deformation that

initialized the tsunami [Okada,1985]. The initial fault parameters reported byFujii

et al. [2011] are used and the processed initial water surface level is illustrated

in Fig. 15, where the origin (0, 0) locates at (137.732 E, 31.5939 N). The initial

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Fig. 15. 2011 Tohoku tsunami propagation simulation: locations of wave gauges and initial tsunamiwaves [Fujii et al., 2011] (unit: m).

ocean surface disturbance then generates a series of tsunami waves propagating

outward, which is simulated using the proposed adaptive grid-based model in this

work. During the simulation, constant Manning coefficient of 0.025 is imposed and

sa and coar are specified to be 0.17 and 0.8.

The computed process of tsunami propagation is demonstrated in Fig. 16, from

the elapsing time of earthquake up to 20 min. The initial tsunami wave starts to

propagate radially in the ocean, approaching the east coast of Japan where the

first wave arrives the near-shore gauges after about 20 min. Qualitative comparison

with the animation provided in the supplementary material of Wei et al. [2012]

indicates that the current adaptive grid model correctly predicts the overall tsunamipropagation process.

The change of wave surface is also recorded and compared with field measure-

ments. In this study, measured data is available at 15 gauges as shown in Fig. 15(a),

including 6 nearshore GPS buoys, 6 NOWPHAS (The Nationwide Ocean Wave infor-

mation network for Ports and HArbourS, Japan) wave gauges close to the coast, 2

cabled pressure-gauges and 1 DART buoy (Deep-ocean Assessment and Reporting

of Tsunamis) system in offshore about 500 km away from the epicenter, as detailed

in Table7.

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(a) (b)

(c) (d)

Fig. 16. 2011 Tohoku tsunami propagation simulation: computed wave propagation at: (a) t =5 min, (b)t = 10 min, (c) t= 15 min, (d) t= 20 min (unit: m).

From these field records, the maximum tsunami wave amplitude is approximately

6 m at 804 and 802 (Iwate South), located in the sea with the water depths of 200 m

and 204 m, respectively. Looking at that GPS buoys plots in Figs.17and18, it is

observed that the model reasonably reproduced the first leading wave in most of

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Table 7. 2011 Tohoku tsunami propagation simulation: gauges under consideration.

Gauge name Type Depth (m) N Latitude() E Longitude()

807-Iwate North GPS buoy 125 40.11667 142.06667

804-Iwate Central GPS buoy 200 39.62722 142.18667802-Iwate South GPS buoy 204 39.25861 142.09694803-Miyagi North GPS buoy 160 38.85778 141.89444801-Miyagi Central GPS buoy 144 38.2325 141.68361806-Fukushima GPS buoy 137 36.97139 141.18556613 NOWPHAS wave gauge 50 42.8844 144.410602 NOWPHAS wave gauge 50.7 42.5427 141.441202 NOWPHAS wave gauge 43.8 40.9347 141.415203 NOWPHAS wave gauge 27.7 40.5528 141.546219 NOWPHAS wave gauge 49.5 40.2111 141.862205 NOWPHAS wave gauge 21.3 38.2412 141.047D2148 Deep ocean tsunameter 5660 38.8210 148.6550TM1 Pressure gauge 1600 39.1855 142.767

TM2 Pressure gauge 1000 39.2036 142.444

the gauges, as well as the wave series that arrived within the following 5 h affected

by refraction and reflection. A proper match of the arrival times and amplitudes

proves the correct estimation of the position and initial ocean surface deformation

resulting from the earthquake dynamics. However, the source estimation involves

the major uncertainty of the simulation that may lead to discrepancies between

numerical predictions and field measurements, especially in the wave series after

the leading wave. Tsunami waves propagating toward the shore also recorded by

a number of NOWPHAS (The Nationwide Ocean Wave information network forPorts and HArbourS, Japan) wave gauges deployed in 20 to 50 m deep water mostly

in the nearshore area north of Tohoku and south of Hokkaido (Figs. 17 and 18).

Comparison between the measured and computed water levels at these wave gauges

shows a good agreement in wave amplitudes, phase and periods, indicating that the

current model can produce reliable numerical prediction in the nearshore areas. In

those deep water gauges, the measurement for TM1 and TM2 are reported up to

30 min as shown in Fig.18. The recorded peak is about 5 m after about 18 min of

earthquake. The model successfully reproduces such wave forms that compare well

with the measurements although wave amplitude is slightly underestimated. The

main wave features predicted by the current model also agree with those predictedby an alternative model by Fujii et al. [2011]. Only one tsunameter (D21418) in

eastern deep ocean is considered, located about 500 km east of the epicenter with

the water depth of 5660 m. It measured a peak of 1.64 m approximately 33 min after

the earthquake. As plotted in Fig.18(g), the wave amplitude, form, arrival time and

second depression are well-predicted by the current model. However, the amplitude

of the arrival wave is again somehow underestimated. The reason might lie in the

sub-fault positions and the amplitude of the instantaneous uplift of the sub faults

near to the epicenter does not account for the progressive fault procedure.

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(a) (b)

(c) (d)

(e) (f)

(g) (h)

Fig. 17. 2011 Tohoku tsunami propagation simulation: computed and measured water levelevolution: I.

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(a) (b)

(c) (d)

(e) (f)

(g)

Fig. 18. 2011 Tohoku tsunami propagation simulation: computed and measured water level evolu-tion: II [Fujii et al., 2011].

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Table 8. 2011 Tohoku tsunami propagation simulation: RMSE ().

Gauges 804 801 803 806

RMSE () on adaptive grid [m] 1.441 1.736 1.743 0.726

RMSE () on uniform grid [m] 1.498 1.889 1.635 0.733

To further demonstrate the performance of the adaptive grid model, the results

computed by the model on a uniform gird with the resolution of 400 m are compared

at gauges 804, 801, 803 and 806 (Fig. 17). Predictions on the two grid systems

appears to be very similar, which is confirmed by the calculated RMSE () listed in

Table8,indicating that the key wave features are well captured at high-resolution

by the adaptive grid. The computation on the adaptive grid is 2.87 times as fast as

that on the uniform one.

5. Discussions and Conclusions

A dynamically adaptive grid-based 2D shallow water equation model is presented in

this work for tsunami simulation. The model is validated against three laboratory-

scale benchmark tests and then applied to reproduce a field event. Numerical results

compare satisfactorily with experimental measurements and field data wherever

available, demonstrating the models capability in providing reliable and efficient

predictions of tsunami propagation over complex domains.

The model solves the 2D shallow water equations using a finite volume Godunov-

type scheme implemented on the current adaptive grid system. Simulation resultsshow that the second-order Godunov-type scheme is able to reliably predict the

amplitude and arrival times of the tsunami waves. It should be noted that the

proposed model is designed for simulation of long and non-breaking waves, due to

the restriction of the shallow water equations (the ratio between the wave length

and height should be higher than 20 [Hinkelmann, 2005]). The model performs

well when simulating tsunami propagation in relatively deep oceans where long

wave assumption is valid, as shown from the simulation results for the 2011 Tohoku

tsunami. However, in the nearshore zones where wave dispersion becomes important,

a tsunami model based on the shallow water equations may not be able to provide

satisfactory predictions due to its inherent limitation in resolving wave dispersion.Under such circumstances, a model based on Bousinesq equations may be more

appropriate and further improvement to the current model is needed.

On the current simplified adaptive grid system, dynamic adaptation is achieved

by increasing or reducing the subdivision levels of the coarse background cells accord-

ing to certain criteria reflecting local flow features, e.g. water surface gradient and

wetting-drying condition as used in this work. Since no data structure is actually

required to store grid information, the adopted grid system provides flexibility for

grid manipulation and is straightforward to implement comparing to other adaptive

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grid system, e.g. quadtree grid. As demonstrated, the current adaptive grid-based

model is able to capture the key features of the tsunami waves, e.g. steep wave

fronts and wet-dry interfaces, where localized high-resolution meshes are essential

for accurate predictions. By intelligently allocating computational resources by cre-

ating local refined meshes, the current adaptive grid-based tsunami model is able toprovide computationally efficient simulations without compromising solution accu-

racy, as confirmed by the comparison of the numerical results with those predicted

by a uniform grid model based on similar numerical scheme. Therefore, the present

adaptive grid-based model provides an alternative for tsunami simulations.

Acknowledgment

This work is supported by the National Natural Science Foundation of China (Grant

No. 51379074), the Royal Society under the International Exchanges 2013 NSFC costshare scheme (IE131297) and the Chinese Government through the Recruitment

Program of Global Experts. The authors also give big thanks to Dr Tomohiro

Yasuda and Professor Hajime Mase from the Disaster Prevention Research Institute

at Kyoto University for providing data to support the simulations of the 2011 Tohoku

tsunami.

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