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  <front>
    <journal-meta><journal-id journal-id-type="publisher">NHESS</journal-id><journal-title-group>
    <journal-title>Natural Hazards and Earth System Sciences</journal-title>
    <abbrev-journal-title abbrev-type="publisher">NHESS</abbrev-journal-title><abbrev-journal-title abbrev-type="nlm-ta">Nat. Hazards Earth Syst. Sci.</abbrev-journal-title>
  </journal-title-group><issn pub-type="epub">1684-9981</issn><publisher>
    <publisher-name>Copernicus Publications</publisher-name>
    <publisher-loc>Göttingen, Germany</publisher-loc>
  </publisher></journal-meta>
    <article-meta>
      <article-id pub-id-type="doi">10.5194/nhess-19-1265-2019</article-id><title-group><article-title>Numerical simulations of the 2004 Indian Ocean <?xmltex \hack{\break}?> tsunami deposits' thicknesses and emplacements</article-title><alt-title>Numerical simulations of the 2004 Indian Ocean tsunami deposits' thicknesses</alt-title>
      </title-group><?xmltex \runningtitle{Numerical simulations of the 2004~Indian Ocean tsunami deposits' thicknesses}?><?xmltex \runningauthor{Syamsidik et al.}?>
      <contrib-group>
        <contrib contrib-type="author" corresp="yes" rid="aff1 aff2">
          <name><surname>Syamsidik</surname><given-names/></name>
          <email>syamsidik@tdmrc.org</email><email>syamsidik@unsyiah.ac.id</email>
        <ext-link>https://orcid.org/0000-0002-0124-5822</ext-link></contrib>
        <contrib contrib-type="author" corresp="no" rid="aff1">
          <name><surname>Al'ala</surname><given-names>Musa</given-names></name>
          
        </contrib>
        <contrib contrib-type="author" corresp="no" rid="aff3">
          <name><surname>Fritz</surname><given-names>Hermann M.</given-names></name>
          
        <ext-link>https://orcid.org/0000-0002-6798-5401</ext-link></contrib>
        <contrib contrib-type="author" corresp="no" rid="aff1 aff4">
          <name><surname>Fahmi</surname><given-names>Mirza</given-names></name>
          
        </contrib>
        <contrib contrib-type="author" corresp="no" rid="aff1">
          <name><surname>Hafli</surname><given-names>Teuku Mudi</given-names></name>
          
        </contrib>
        <aff id="aff1"><label>1</label><institution>Tsunami and Disaster Mitigation Research Center (TDMRC), Universitas Syiah Kuala, <?xmltex \hack{\break}?> Jl. Prof. Dr. Ibrahim Hasan, Gp. Pie, Banda Aceh, 23233, Indonesia</institution>
        </aff>
        <aff id="aff2"><label>2</label><institution>Civil Engineering Department, Faculty of Engineering, Universitas Syiah Kuala, <?xmltex \hack{\break}?> Jl. Syeh Abdurrauf No. 7, Banda Aceh, 23111, Indonesia</institution>
        </aff>
        <aff id="aff3"><label>3</label><institution>School of Civil Engineering and Environmental Engineering, Georgia Institute of Technology, <?xmltex \hack{\break}?> 790 Atlantic Drive, Atlanta, GA 30332, USA</institution>
        </aff>
        <aff id="aff4"><label>4</label><institution>Politeknik Negeri Lhokseumawe, Jl. Banda Aceh-Medan, km.280.3 Buketrata, Lhokseumawe, 24301, Indonesia</institution>
        </aff>
      </contrib-group>
      <author-notes><corresp id="corr1">Syamsidik (syamsidik@tdmrc.org, syamsidik@unsyiah.ac.id)</corresp></author-notes><pub-date><day>27</day><month>June</month><year>2019</year></pub-date>
      
      <volume>19</volume>
      <issue>6</issue>
      <fpage>1265</fpage><lpage>1280</lpage>
      <history>
        <date date-type="received"><day>16</day><month>November</month><year>2018</year></date>
           <date date-type="rev-request"><day>26</day><month>November</month><year>2018</year></date>
           <date date-type="rev-recd"><day>4</day><month>May</month><year>2019</year></date>
           <date date-type="accepted"><day>1</day><month>June</month><year>2019</year></date>
      </history>
      <permissions>
        <copyright-statement>Copyright: © 2019 </copyright-statement>
        <copyright-year>2019</copyright-year>
      <license license-type="open-access"><license-p>This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit <ext-link ext-link-type="uri" xlink:href="https://creativecommons.org/licenses/by/4.0/">https://creativecommons.org/licenses/by/4.0/</ext-link></license-p></license></permissions><self-uri xlink:href="https://nhess.copernicus.org/articles/.html">This article is available from https://nhess.copernicus.org/articles/.html</self-uri><self-uri xlink:href="https://nhess.copernicus.org/articles/.pdf">The full text article is available as a PDF file from https://nhess.copernicus.org/articles/.pdf</self-uri>
      <abstract><title>Abstract</title>
    <p id="d1e144">After more than a decade of recurring tsunamis,
identification of tsunami deposits, a part of hazard characterization, still
remains a challenging task that is not fully understood. The lack of sufficient
monitoring equipment and rare tsunami frequency are among the primary
obstacles that limit our fundamental understanding of sediment transport
mechanisms during a tsunami. The use of numerical simulations to study
tsunami-induced sediment transport was rare in Indonesia until the
2004 Indian Ocean tsunami. This study aims to couple two hydrodynamic numerical
models in order to reproduce tsunami-induced sediment deposits, i.e., their
locations and thicknesses. Numerical simulations were performed using the
Cornell Multi-grid Coupled Tsunami (COMCOT) model and Delft3D. This study
reconstructed tsunami wave propagation from its source using COMCOT, which
was later combined with Delft3D to map the location of the tsunami deposits
and calculate their thicknesses. Two-dimensional horizontal (2-DH) models
were used as part of both simulation packages. Four sediment transport
formulae were used in the simulations, namely van Rijn 1993, Engelund–Hansen 1967,
Meyer-Peter–Mueller (MPM) 1948, and Soulsby 1997. Lhoong, in the Aceh
Besar District, located approximately 60 km southwest of Banda Aceh, was
selected as the study area. Field data collected in 2015 and 2016 validated
the forward modeling techniques adopted in this study. However, agreements
between numerical simulations and field observations were more robust using
data collected in 2005, i.e., just months after the tsunami (Jaffe et al.,
2006). We conducted pit (trench) tests at select locations to obtain tsunami
deposit thickness and grain size distributions. The resulting numerical
simulations are useful when estimating the locations and the thicknesses of
the tsunami deposits. The agreement between the field data and the numerical
simulations is reasonable despite a trend that overestimates the field observations.</p>
  </abstract>
    </article-meta>
  </front>
<body>
      

<sec id="Ch1.S1" sec-type="intro">
  <label>1</label><title>Introduction</title>
      <p id="d1e156">In recent decades, sediment transport dynamics due to extremely long waves,
such as tsunamis, has increasingly caught the interest of researchers,
especially from countries in the direct path of these phenomena. Studies of
tsunami impact on coastal morphology, involving a comparison with actual
field data, are rare for previous tsunami events. This limits our ability to
understand the physical processes occurring during erosion and sediment
deposition. The 2004 Indian Ocean tsunami, triggered by a magnitude <inline-formula><mml:math id="M1" display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mi mathvariant="normal">w</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn mathvariant="normal">9.1</mml:mn></mml:mrow></mml:math></inline-formula>
earthquake, dramatically changed coastal areas around Aceh of Indonesia
(Borrero et al., 2006). Erosion processes and its effects were quite
profound and severe along most of Aceh's west coast (Syamsidik et al.,
2015a). Severe erosion associated with the 2004 tsunami actually created a
new island (Al'ala et al., 2015). Tsunami waves transported<?pagebreak page1266?> large volumes of
sediment of several hundred meters at certain locations, resulting in a
large-scale coastline recession. Large shear stresses generated by the long
waves overpowered the critical shear stress of bedload materials throughout
the coastal zone. However, full investigation into the physical processes of
longwave shear stresses during the 2004 tsunami did not occur. Owing to the
lack of sufficient coastal monitoring equipment in tsunami-prone areas and
the rare occurrence of tsunamis, insights into sediment transport mechanisms
during tsunami events remain poorly understood. Overland flow velocities
extracted from spontaneous eyewitness videos recorded during the 2004 Indian
Ocean tsunami are limited to a few inland locations in Banda Aceh (Fritz et
al., 2006). The coupling of tsunami hydrodynamics and sediment transport
warrants future studies in order to improve our fundamental understanding of
tsunami processes. Before the 2004 tsunami, paleo-tsunami studies were quite
rare in this region (Andrade et al., 2014).</p>
      <p id="d1e174">Two major factors, i.e., episodic land subsidence due to tectonic movement
and massive sediment transport during tsunami propagation, play various
roles in modifying coastal morphology (Moore et al., 2006). Several studies
conducted in this region include the investigation of tsunami deposits at
Khao Lak in Thailand (Dawson and Stewart, 2007; Jankaew et al., 2008) and
Meulaboh in the Aceh Province (Monecke et al., 2008). Another study reported
the presence of tsunami sediment deposits in a coastal cave, which is less
than 5 km from our study area in Lhoong of Aceh Besar (Rubin et al., 2017).
Both studies revealed that information from tsunami deposits is capable of
improving our understanding of the physical processes involved during the
transport of sediment. Paris et al. (2007) performed their study around Lhok
Nga of Aceh Besar, which is about 20 km to the east of our study areas.
Their study exhibited the influence of local topography on the sediment
thicknesses found in the area. The thickest deposits were found in a low topography
situation, and steep slopes gave varied results in spatial distribution
of the tsunami deposit.</p>
      <p id="d1e177">Previous studies on tsunami deposits have inspired tsunami mitigation
efforts in the USA and in Indonesia (Dunbar et al., 2008; Rubin et al., 2017).
Data on tsunami deposits would provide more scientific evidence for past
tsunamis for a significant period of record, such as in the case of the coastal
cave in Aceh that preserved a very long record of past tsunamis in the
region (Rubin et al., 2017). A set of tsunami deposit data could help us to
estimate the recurrence period of the tsunami from the subduction zone
(Fujiwara, 2007; Minoura et al., 2001). Finding more tsunami deposits would
strengthen the validation process and help find some ways to estimate the sources of tsunamis through inverse mechanisms (Buckley et al., 2012; Moore et al., 2011).</p>
      <p id="d1e180">The recurrence rate of tsunamis throughout this region, information critical
for hazard assessment, is inferable from events recorded in the tsunami
deposits. Sediment transport due to tsunami waves has certain characteristic
properties and hydrodynamic regimes during both the tsunami runup and
backwash. During transport, material that is mobile depends on grain size
and density, which vary greatly because of the erratic nature of tsunami
sediment transport. Very few studies investigated the energy involved during
sediment transport produced by the 2004 tsunami in Aceh. Furthermore, only a
few of these published studies confirmed the thickness of the sediment
deposits after the tsunami via geological surveys. This is the first study
in northern Sumatra which applies numerical simulations in order to estimate
the locations of tsunami deposits prior to the field surveys. Earlier
investigations concentrated on forward tsunami modeling, such as in Kuala
Meurisi (Apotsos et al., 2011a), Lhok-Nga (Gusman et al., 2012; Li et al.,
2012), and Ulee Lheue Bay (Syamsidik et al., 2017). This research aims to
couple two hydrodynamic numerical models in order to reproduce the
spatiality of the 2004 tsunami sediment deposits, i.e., their locations and
thicknesses. We analyze the possible sources of large shear stresses
produced by the tsunami which transported sediment grains of various sizes.
Several studies observed that backwash processes, associated with tsunami
waves, deposited large volumes of sediments offshore (Jiang et al., 2015). Sediment
properties, such as grain size, correlate poorly with the thicknesses of
tsunami sediment deposits (Cheng and Weiss, 2013). Another poorly
characterized physical process related to the tsunami is the location and
extent of inland sediment deposition, which differs from the extent of
tsunami inundation (Goto et al., 2012).</p>
      <p id="d1e184">We combine both numerical simulations and field surveys to investigate
sediment transport induced by the tsunami at Lhoong of Aceh Besar. We also
conducted topographic surveys and sediment deposit pit tests in the
vicinity. Studies rarely focus on changes along the shoreline due to the
impacts of tsunami waves and subsequent erosion and deposition. Morphologic
changes to the shoreline and sediment thickness are attributable to the
amount of energy involved in tsunami propagation and inundation. This study
estimates the energy required to transport sediments via a tsunami wave. The
trajectory of the 2004 Indian Ocean tsunami was numerically simulated using
the Cornell Multi-grid Coupled Tsunami (COMCOT) model, from its rupture area
to the shoreline, and we applied Delft3D-FLOW to simulate sediment transport
processes by inland flooding in the study area. The combination of these two
models provides an ideal re-creation of the 2004 tsunami scenario, since
COMCOT does not include sediment transport, and the original version of the
Delft3D model does not have the option of multi-fault scenarios to generate
tsunami waves (Syamsidik and Istiyanto, 2013). The numerical model is
described briefly in Sect. 3, and the model results are presented in
Sect. 4. Field observations are also described in Sect. 4. The
observations and model results are discussed in Sect. 5, and conclusions
are presented in Sect. 6.</p>
</sec>
<?pagebreak page1267?><sec id="Ch1.S2">
  <label>2</label><title>Study area</title>
      <p id="d1e195">To carry out the objectives of this study, we selected an area where human
intervention has remained minimal during the decade following the 2004 tsunami
in order to obtain well-preserved tsunami deposits. The study area
is located at Lhoong in the Aceh Besar District, Indonesia (Fig. 1). The
2004 Indian Ocean tsunami destroyed a wide swath of coastal area, including
the Aceh Besar District. The study area is situated on the west coast of
Sumatra, approximately 40 km south of Banda Aceh, the city most devastated by
the tsunami. Several prior studies have been conducted in the general study
area (Jaffe et al., 2006; Rubin et al., 2017). Our study area is situated in
a tropical area where the rainy season occurs about 4–5 months in a year, with a high precipitation rate. After more than 10 years, the 2004 Indian Ocean
tsunami deposits in this area have experienced some natural processes despite
the selection of the study area as the most sediment-preserved
area in the Aceh Besar District. Szczucinski (2012) also argues that tsunami inundation
less than 3 m would unlikely preserve the sediment deposit years after
the tsunami event. Coastal mountains, representing
extension of the Bukit Barisan Mountains, surround the study area. We
selected four specific areas for detailed sediment transport investigations.
The selected areas are shown in red boxes in Fig. 1, namely, Birek, Pasie
Janeng, Jantang, and Saney. Among these locations, only Jantang has a wide
coastal plain, and the other three sites have more rugged terrain. Birek and
Pasie Janeng have V-shaped coastal areas characterized by steep hill slopes
100 to 300 m from their coastlines. Saney headland features a narrow plain
sandwiched between coastline both to the north and south. To the north of
Saney, there is an area which could be classified as plain coast,
about 3 km long. However, this area has undergone heavy anthropogenic
intervention after the tsunami as it was formerly a paddy field and settlement area.
Therefore, we determine the four study sites as the most tsunami-deposit-conserved area.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F1"><?xmltex \currentcnt{1}?><label>Figure 1</label><caption><p id="d1e200">Field survey sites at Lhoong, Aceh Besar, Sumatra, approximately
40 km south of Banda Aceh, Indonesia.</p></caption>
        <?xmltex \igopts{width=199.169291pt}?><graphic xlink:href="https://nhess.copernicus.org/articles/19/1265/2019/nhess-19-1265-2019-f01.png"/>

      </fig>

      <p id="d1e209">The 2004 tsunami heavily impacted the Lhoong region, leaving a number of
areas conserved after the tsunami because of their relative isolation from
major human activities. After more than a decade, it is difficult to find a
suitable location to investigate tsunami deposits because many areas have
been rebuilt, modifying the deposits during reconstruction processes.
Birek, Pasie Janeng, and Saney are the sites that conserve the best sediment
deposits because of the absence of anthropogenic influences that might
disturb the tsunami deposits. Furthermore, researchers conducted a field
survey of Jantang 3 months after the tsunami (Peters and Jaffe, 2010; Jaffe
et al., 2006). Aerial photographs of the study area are shown in Fig. 2.
Birek is surrounded by a range of hills covered by dense forests that regrew
rapidly after the 2004 tsunami. The characteristics of Pasie Janeng are
similar to Birek. Jantang (Fig. 2c and d), however, has a wider flat
area that allowed the tsunami to propagate farther inland compared with
Birek and Pasie Janeng. Since the 2004 tsunami, however, Jantang has
recovered, new settlements have been established, and it is actively used
for agricultural purposes.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F2" specific-use="star"><?xmltex \currentcnt{2}?><label>Figure 2</label><caption><p id="d1e215">Aerial drone photos of the study areas at Birek <bold>(a, b)</bold>, Pasie
Janeng <bold>(c, d)</bold>, Jantang <bold>(e, f)</bold>, and Saney <bold>(g, h)</bold>
captured in October 2017.</p></caption>
        <?xmltex \igopts{width=341.433071pt}?><graphic xlink:href="https://nhess.copernicus.org/articles/19/1265/2019/nhess-19-1265-2019-f02.jpg"/>

      </fig>

</sec>
<sec id="Ch1.S3">
  <label>3</label><title>Methods</title>
<sec id="Ch1.S3.SS1">
  <label>3.1</label><title>Numerical simulations</title>
      <?pagebreak page1269?><p id="d1e251">During the simulation process, we combined two numerical models. COMCOT
simulated the formation and propagation of tsunami waves. A series of
laboratory experiments and prior tsunami events have validated the use of
the COMCOT model (Liu et al., 1995). COMCOT was successfully implemented to
investigate historical tsunami events, such as the 1992 Flores Island
tsunami (Liu et al., 1995) and the 2004 Indian Ocean tsunami (Wang and Liu,
2006, 2007; Syamsidik et al., 2015b).
Using the incident tsunami wave heights generated by COMCOT, Delft3D-FLOW
models sediment transport from the ocean to the inundation zone, as well as
the reverse flow. Both models coupled the hydrodynamic and morphodynamic
models. The hydrodynamic model has been validated comparatively with
numerical and laboratory data following several tsunami benchmark standards
(Apotsos et al., 2011a). For the hydrodynamic model, Delft3D uses a finite
difference scheme on a three-dimensional grid with nonlinear shallow
water equations (NSWEs). The linear shallow water equations used in COMCOT
in a spherical coordinate system are as follows:
<?xmltex \hack{\newpage}?><?xmltex \hack{\vspace*{-6mm}}?>

                <disp-formula specific-use="align" content-type="numbered"><mml:math id="M2" display="block"><mml:mtable displaystyle="true"><mml:mlabeledtr id="Ch1.E1"><mml:mtd><mml:mtext>1</mml:mtext></mml:mtd><mml:mtd><mml:mstyle class="stylechange" displaystyle="true"/></mml:mtd><mml:mtd><mml:mrow><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mo>∂</mml:mo><mml:mi mathvariant="italic">η</mml:mi></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>+</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mn mathvariant="normal">1</mml:mn><mml:mrow><mml:mi>R</mml:mi><mml:mi>cos⁡</mml:mi><mml:mi mathvariant="italic">φ</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mfenced close="}" open="{"><mml:mrow><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mo>∂</mml:mo><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi mathvariant="italic">ψ</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>+</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mo>∂</mml:mo><mml:mrow><mml:mo>∂</mml:mo><mml:mi mathvariant="italic">φ</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>(</mml:mo><mml:mi>cos⁡</mml:mi><mml:mi mathvariant="italic">φ</mml:mi><mml:mi>Q</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mo>∂</mml:mo><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mlabeledtr><mml:mlabeledtr id="Ch1.E2"><mml:mtd><mml:mtext>2</mml:mtext></mml:mtd><mml:mtd><mml:mstyle displaystyle="true" class="stylechange"/></mml:mtd><mml:mtd><mml:mrow><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mo>∂</mml:mo><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>+</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mi>g</mml:mi><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi>R</mml:mi><mml:mi>cos⁡</mml:mi><mml:mi mathvariant="italic">φ</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mo>∂</mml:mo><mml:mi mathvariant="italic">η</mml:mi></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi mathvariant="italic">ψ</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>-</mml:mo><mml:mi>f</mml:mi><mml:mi>Q</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">0</mml:mn><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mlabeledtr><mml:mlabeledtr id="Ch1.E3"><mml:mtd><mml:mtext>3</mml:mtext></mml:mtd><mml:mtd><mml:mstyle displaystyle="true" class="stylechange"/></mml:mtd><mml:mtd><mml:mrow><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mo>∂</mml:mo><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>+</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mi>g</mml:mi><mml:mi>h</mml:mi></mml:mrow><mml:mi>R</mml:mi></mml:mfrac></mml:mstyle><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mo>∂</mml:mo><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi mathvariant="italic">φ</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>+</mml:mo><mml:mi>f</mml:mi><mml:mi>P</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">0</mml:mn><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mlabeledtr></mml:mtable></mml:math></disp-formula>

            The equations governing nonlinear shallow water behavior in COMCOT for a
spherical coordinate system can be expressed as follows:
            <disp-formula id="Ch1.E4" content-type="numbered"><label>4</label><mml:math id="M3" display="block"><mml:mrow><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mo>∂</mml:mo><mml:mi mathvariant="italic">η</mml:mi></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>+</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mn mathvariant="normal">1</mml:mn><mml:mrow><mml:mi>R</mml:mi><mml:mi>cos⁡</mml:mi><mml:mi mathvariant="normal">∅</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mfenced open="{" close="}"><mml:mrow><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mo>∂</mml:mo><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi mathvariant="italic">ψ</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>+</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mo>∂</mml:mo><mml:mrow><mml:mo>∂</mml:mo><mml:mi mathvariant="normal">∅</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>(</mml:mo><mml:mi>cos⁡</mml:mi><mml:mi mathvariant="normal">∅</mml:mi><mml:mi>Q</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mo>∂</mml:mo><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>
          <?xmltex \hack{\newpage}?><?xmltex \hack{\vspace*{-6mm}}?>
            <disp-formula id="Ch1.E5" content-type="numbered"><label>5</label><mml:math id="M4" display="block"><mml:mtable columnspacing="1em" class="aligned" rowspacing="0.2ex" displaystyle="true" columnalign="right left"><mml:mtr><mml:mtd><mml:mrow><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mo>∂</mml:mo><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:mfrac></mml:mstyle></mml:mrow></mml:mtd><mml:mtd><mml:mrow><mml:mstyle class="stylechange" displaystyle="true"/><mml:mo>+</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mn mathvariant="normal">1</mml:mn><mml:mrow><mml:mi>R</mml:mi><mml:mi>cos⁡</mml:mi><mml:mi mathvariant="normal">∅</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mo>∂</mml:mo><mml:mrow><mml:mo>∂</mml:mo><mml:mi mathvariant="italic">ψ</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mfenced open="{" close="}"><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:msup><mml:mi>P</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msup></mml:mrow><mml:mi>H</mml:mi></mml:mfrac></mml:mstyle></mml:mfenced><mml:mo>+</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mn mathvariant="normal">1</mml:mn><mml:mi>R</mml:mi></mml:mfrac></mml:mstyle><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mo>∂</mml:mo><mml:mrow><mml:mo>∂</mml:mo><mml:mi mathvariant="normal">∅</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mfenced close="}" open="{"><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mi>P</mml:mi><mml:mi>Q</mml:mi></mml:mrow><mml:mi>H</mml:mi></mml:mfrac></mml:mstyle></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mstyle displaystyle="true" class="stylechange"/></mml:mtd><mml:mtd><mml:mrow><mml:mstyle displaystyle="true" class="stylechange"/><mml:mo>+</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mi>g</mml:mi><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>R</mml:mi><mml:mi>cos⁡</mml:mi><mml:mi mathvariant="normal">∅</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mo>∂</mml:mo><mml:mi mathvariant="italic">η</mml:mi></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi mathvariant="italic">ψ</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>-</mml:mo><mml:mi>f</mml:mi><mml:mi>Q</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn mathvariant="normal">0</mml:mn><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>

            <disp-formula id="Ch1.E6" content-type="numbered"><label>6</label><mml:math id="M5" display="block"><mml:mtable rowspacing="0.2ex" columnspacing="1em" class="aligned" displaystyle="true" columnalign="right left"><mml:mtr><mml:mtd><mml:mrow><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mo>∂</mml:mo><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:mfrac></mml:mstyle></mml:mrow></mml:mtd><mml:mtd><mml:mrow><mml:mstyle class="stylechange" displaystyle="true"/><mml:mo>+</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mn mathvariant="normal">1</mml:mn><mml:mrow><mml:mi>R</mml:mi><mml:mi>cos⁡</mml:mi><mml:mi mathvariant="normal">∅</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mo>∂</mml:mo><mml:mrow><mml:mo>∂</mml:mo><mml:mi mathvariant="italic">ψ</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mfenced close="}" open="{"><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mi>P</mml:mi><mml:mi>Q</mml:mi></mml:mrow><mml:mi>H</mml:mi></mml:mfrac></mml:mstyle></mml:mfenced><mml:mo>+</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mn mathvariant="normal">1</mml:mn><mml:mi>R</mml:mi></mml:mfrac></mml:mstyle><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mo>∂</mml:mo><mml:mrow><mml:mo>∂</mml:mo><mml:mi mathvariant="normal">∅</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mfenced open="{" close="}"><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:msup><mml:mi>Q</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msup></mml:mrow><mml:mi>H</mml:mi></mml:mfrac></mml:mstyle></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mstyle displaystyle="true" class="stylechange"/></mml:mtd><mml:mtd><mml:mrow><mml:mstyle class="stylechange" displaystyle="true"/><mml:mo>+</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mi>g</mml:mi><mml:mi>H</mml:mi></mml:mrow><mml:mi>R</mml:mi></mml:mfrac></mml:mstyle><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mo>∂</mml:mo><mml:mi mathvariant="italic">η</mml:mi></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi mathvariant="normal">∅</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>+</mml:mo><mml:mi>f</mml:mi><mml:mi>P</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn mathvariant="normal">0</mml:mn><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>

                <disp-formula specific-use="align" content-type="numbered"><mml:math id="M6" display="block"><mml:mtable displaystyle="true"><mml:mlabeledtr id="Ch1.E7"><mml:mtd><mml:mtext>7</mml:mtext></mml:mtd><mml:mtd><mml:mstyle displaystyle="true" class="stylechange"/></mml:mtd><mml:mtd><mml:mrow><mml:mstyle displaystyle="true" class="stylechange"/><mml:mi>f</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">2</mml:mn><mml:mi mathvariant="normal">Ω</mml:mi><mml:mi>sin⁡</mml:mi><mml:mi mathvariant="italic">φ</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mlabeledtr><mml:mlabeledtr id="Ch1.E8"><mml:mtd><mml:mtext>8</mml:mtext></mml:mtd><mml:mtd><mml:mstyle displaystyle="true" class="stylechange"/></mml:mtd><mml:mtd><mml:mrow><mml:mstyle class="stylechange" displaystyle="true"/><mml:msub><mml:mi>F</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mi>g</mml:mi><mml:msup><mml:mi>n</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mi>H</mml:mi><mml:mrow><mml:mn mathvariant="normal">7</mml:mn><mml:mo>/</mml:mo><mml:mn mathvariant="normal">3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac></mml:mstyle><mml:mi>P</mml:mi><mml:msup><mml:mfenced close=")" open="("><mml:mrow><mml:msup><mml:mi>P</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mi>Q</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msup></mml:mrow></mml:mfenced><mml:mrow><mml:mn mathvariant="normal">1</mml:mn><mml:mo>/</mml:mo><mml:mn mathvariant="normal">2</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mlabeledtr></mml:mtable></mml:math></disp-formula>

            <?xmltex \hack{\newpage}?><?xmltex \hack{\vspace*{-6mm}}?>

                <disp-formula specific-use="align" content-type="numbered"><mml:math id="M7" display="block"><mml:mtable displaystyle="true"><mml:mlabeledtr id="Ch1.E9"><mml:mtd><mml:mtext>9</mml:mtext></mml:mtd><mml:mtd><mml:mstyle displaystyle="true" class="stylechange"/></mml:mtd><mml:mtd><mml:mrow><mml:mstyle displaystyle="true" class="stylechange"/><mml:msub><mml:mi>F</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mi>g</mml:mi><mml:msup><mml:mi>n</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mi>H</mml:mi><mml:mrow><mml:mn mathvariant="normal">7</mml:mn><mml:mo>/</mml:mo><mml:mn mathvariant="normal">3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac></mml:mstyle><mml:mi>Q</mml:mi><mml:msup><mml:mfenced open="(" close=")"><mml:mrow><mml:msup><mml:mi>P</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mi>Q</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msup></mml:mrow></mml:mfenced><mml:mrow><mml:mn mathvariant="normal">1</mml:mn><mml:mo>/</mml:mo><mml:mn mathvariant="normal">2</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mlabeledtr><mml:mlabeledtr id="Ch1.E10"><mml:mtd><mml:mtext>10</mml:mtext></mml:mtd><mml:mtd><mml:mstyle class="stylechange" displaystyle="true"/></mml:mtd><mml:mtd><mml:mrow><mml:mstyle class="stylechange" displaystyle="true"/><mml:mi>H</mml:mi><mml:mo>=</mml:mo><mml:mi mathvariant="italic">η</mml:mi><mml:mo>+</mml:mo><mml:mi>h</mml:mi><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mlabeledtr></mml:mtable></mml:math></disp-formula>

            Meanwhile, for a Cartesian coordinate system, COMCOT applies the following
equations for linear shallow water equations.

                <disp-formula specific-use="align" content-type="numbered"><mml:math id="M8" display="block"><mml:mtable displaystyle="true"><mml:mlabeledtr id="Ch1.E11"><mml:mtd><mml:mtext>11</mml:mtext></mml:mtd><mml:mtd><mml:mstyle class="stylechange" displaystyle="true"/></mml:mtd><mml:mtd><mml:mrow><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mo>∂</mml:mo><mml:mi mathvariant="italic">η</mml:mi></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>+</mml:mo><mml:mfenced close="}" open="{"><mml:mrow><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mo>∂</mml:mo><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>x</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>+</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mo>∂</mml:mo><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:mfrac></mml:mstyle></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mo>∂</mml:mo><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mlabeledtr><mml:mlabeledtr id="Ch1.E12"><mml:mtd><mml:mtext>12</mml:mtext></mml:mtd><mml:mtd><mml:mstyle displaystyle="true" class="stylechange"/></mml:mtd><mml:mtd><mml:mrow><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mo>∂</mml:mo><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>+</mml:mo><mml:mi>g</mml:mi><mml:mi>h</mml:mi><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mo>∂</mml:mo><mml:mi mathvariant="italic">η</mml:mi></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>x</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>-</mml:mo><mml:mi>f</mml:mi><mml:mi>Q</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">0</mml:mn><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mlabeledtr><mml:mlabeledtr id="Ch1.E13"><mml:mtd><mml:mtext>13</mml:mtext></mml:mtd><mml:mtd><mml:mstyle displaystyle="true" class="stylechange"/></mml:mtd><mml:mtd><mml:mrow><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mo>∂</mml:mo><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>+</mml:mo><mml:mi>g</mml:mi><mml:mi>h</mml:mi><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mo>∂</mml:mo><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>+</mml:mo><mml:mi>f</mml:mi><mml:mi>P</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">0</mml:mn><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mlabeledtr></mml:mtable></mml:math></disp-formula>

            For linear shallow water equations, the following formulae were applied in COMCOT.

                <disp-formula specific-use="align" content-type="numbered"><mml:math id="M9" display="block"><mml:mtable displaystyle="true"><mml:mlabeledtr id="Ch1.E14"><mml:mtd><mml:mtext>14</mml:mtext></mml:mtd><mml:mtd><mml:mstyle displaystyle="true" class="stylechange"/></mml:mtd><mml:mtd><mml:mrow><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mo>∂</mml:mo><mml:mi mathvariant="italic">η</mml:mi></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>+</mml:mo><mml:mfenced close="}" open="{"><mml:mrow><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mo>∂</mml:mo><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>x</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>+</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mo>∂</mml:mo><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:mfrac></mml:mstyle></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mo>∂</mml:mo><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mlabeledtr><mml:mlabeledtr id="Ch1.E15"><mml:mtd><mml:mtext>15</mml:mtext></mml:mtd><mml:mtd><mml:mstyle class="stylechange" displaystyle="true"/></mml:mtd><mml:mtd><mml:mrow><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mo>∂</mml:mo><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>+</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mo>∂</mml:mo><mml:mrow><mml:mo>∂</mml:mo><mml:mi>x</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mfenced open="{" close="}"><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:msup><mml:mi>P</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msup></mml:mrow><mml:mi>H</mml:mi></mml:mfrac></mml:mstyle></mml:mfenced><mml:mo>+</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mo>∂</mml:mo><mml:mrow><mml:mo>∂</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mfenced open="{" close="}"><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mi>P</mml:mi><mml:mi>Q</mml:mi></mml:mrow><mml:mi>H</mml:mi></mml:mfrac></mml:mstyle></mml:mfenced><mml:mo>+</mml:mo><mml:mi>g</mml:mi><mml:mi>H</mml:mi><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mo>∂</mml:mo><mml:mi mathvariant="italic">η</mml:mi></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>x</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>+</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn mathvariant="normal">0</mml:mn><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mlabeledtr><mml:mlabeledtr id="Ch1.E16"><mml:mtd><mml:mtext>16</mml:mtext></mml:mtd><mml:mtd><mml:mstyle displaystyle="true" class="stylechange"/></mml:mtd><mml:mtd><mml:mrow><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mo>∂</mml:mo><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>+</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mo>∂</mml:mo><mml:mrow><mml:mo>∂</mml:mo><mml:mi>x</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mfenced close="}" open="{"><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mi>P</mml:mi><mml:mi>Q</mml:mi></mml:mrow><mml:mi>H</mml:mi></mml:mfrac></mml:mstyle></mml:mfenced><mml:mo>+</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mo>∂</mml:mo><mml:mrow><mml:mo>∂</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mfenced close="}" open="{"><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:msup><mml:mi>Q</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msup></mml:mrow><mml:mi>H</mml:mi></mml:mfrac></mml:mstyle></mml:mfenced><mml:mo>+</mml:mo><mml:mi>g</mml:mi><mml:mi>H</mml:mi><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mo>∂</mml:mo><mml:mi mathvariant="italic">η</mml:mi></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>+</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn mathvariant="normal">0</mml:mn><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mlabeledtr></mml:mtable></mml:math></disp-formula>

            where <inline-formula><mml:math id="M10" display="inline"><mml:mi mathvariant="italic">η</mml:mi></mml:math></inline-formula> is the water surface elevation, (<inline-formula><mml:math id="M11" display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula>, <inline-formula><mml:math id="M12" display="inline"><mml:mi>Q</mml:mi></mml:math></inline-formula>) gives the volume flux in the
<inline-formula><mml:math id="M13" display="inline"><mml:mi>X</mml:mi></mml:math></inline-formula> (west–east) direction and in the <inline-formula><mml:math id="M14" display="inline"><mml:mi>Y</mml:mi></mml:math></inline-formula> (north–south) direction, respectively,
(<inline-formula><mml:math id="M15" display="inline"><mml:mi mathvariant="italic">φ</mml:mi></mml:math></inline-formula>, <inline-formula><mml:math id="M16" display="inline"><mml:mi mathvariant="italic">ψ</mml:mi></mml:math></inline-formula>) is the latitude and longitude, <inline-formula><mml:math id="M17" display="inline"><mml:mi>R</mml:mi></mml:math></inline-formula> is the Earth's radius, <inline-formula><mml:math id="M18" display="inline"><mml:mi>g</mml:mi></mml:math></inline-formula> is
gravitational acceleration, and <inline-formula><mml:math id="M19" display="inline"><mml:mi>h</mml:mi></mml:math></inline-formula> is water depth. The <inline-formula><mml:math id="M20" display="inline"><mml:mrow><mml:mo>-</mml:mo><mml:mo>∂</mml:mo><mml:mi>h</mml:mi><mml:mo>/</mml:mo><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:math></inline-formula>
expression reflects the effect of transient seafloor motion,
<inline-formula><mml:math id="M21" display="inline"><mml:mi>f</mml:mi></mml:math></inline-formula> represents the Coriolis force coefficient due to the Earth's rotation,
<inline-formula><mml:math id="M22" display="inline"><mml:mi mathvariant="normal">Ω</mml:mi></mml:math></inline-formula> is the Earth's rotation rate, <inline-formula><mml:math id="M23" display="inline"><mml:mi>H</mml:mi></mml:math></inline-formula> is the total water depth, <inline-formula><mml:math id="M24" display="inline"><mml:mrow><mml:msub><mml:mi>F</mml:mi><mml:mi>x</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> and
<inline-formula><mml:math id="M25" display="inline"><mml:mrow><mml:msub><mml:mi>F</mml:mi><mml:mi>y</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> represent bottom friction in the <inline-formula><mml:math id="M26" display="inline"><mml:mi mathvariant="italic">ψ</mml:mi></mml:math></inline-formula> and <inline-formula><mml:math id="M27" display="inline"><mml:mi mathvariant="italic">φ</mml:mi></mml:math></inline-formula> direction,
respectively, and <inline-formula><mml:math id="M28" display="inline"><mml:mi>n</mml:mi></mml:math></inline-formula> is Manning's roughness coefficient. Since Delft3D-FLOW is
unable to generate tsunamis formed from a multi-fault scenario, we therefore
used COMCOT to propagate tsunami waves from layers 1 to 3. In the nested
layer 4, we ran Delft3D-FLOW with boundary conditions based on water
elevations produced in layer 3. We were able to vary Manning's roughness
coefficient based on actual conditions, in reference to natural channels and
floodplains (Arcement and Schneider, 1989). An open boundary condition for
Delft3D-FLOW was placed about 14.5 km from shoreline. The hydrodynamic
boundary condition was obtained from the COMCOT numerical simulation at layer 3.
Here, the boundary condition was set to follow the water level. The
water level boundary condition is a modified Riemann boundary condition
to which Stelling added the time derivative of the water level and
velocities (Stelling, 1984). This was meant to reduce the reflection process
caused by the eigenfrequency of the simulation.</p>
      <p id="d1e1602">The four simulation domain layers applied to COMCOT simulation all used
spherical coordinates. Figure 3 shows the layers of the simulations used in
COMCOT. Layer 4 was later converted to the modeling domain for use in the
Delft3D simulations. Coordinates, grid size, and types of shallow water
equation (SWE) used in each individual layer are shown in Table 1.
Delft3D-FLOW used the finest COMCOT layer as a domain, with coordinate
transformation from spherical to Cartesian. We combined GEBCO and other
nautical charts to produce improved bathymetry data. For the inland regions,
we combined Shuttle Radar Topography Mission (SRTM) 30 with some cross-topography measurements to improve the topographical data.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F3" specific-use="star"><?xmltex \currentcnt{3}?><label>Figure 3</label><caption><p id="d1e1607">Nested simulation layers incorporated into COMCOT and the boundary condition
location for the Delft3D simulation (a solid blue line) in layer 4.</p></caption>
          <?xmltex \igopts{width=341.433071pt}?><graphic xlink:href="https://nhess.copernicus.org/articles/19/1265/2019/nhess-19-1265-2019-f03.png"/>

        </fig>

<?xmltex \floatpos{t}?><table-wrap id="Ch1.T1" specific-use="star"><?xmltex \currentcnt{1}?><label>Table 1</label><caption><p id="d1e1620">Specification of layers applied in the COMCOT simulation.</p></caption><oasis:table frame="topbot"><oasis:tgroup cols="6">
     <oasis:colspec colnum="1" colname="col1" align="left"/>
     <oasis:colspec colnum="2" colname="col2" align="right"/>
     <oasis:colspec colnum="3" colname="col3" align="right"/>
     <oasis:colspec colnum="4" colname="col4" align="right"/>
     <oasis:colspec colnum="5" colname="col5" align="left"/>
     <oasis:colspec colnum="6" colname="col6" align="left"/>
     <oasis:thead>
       <oasis:row>
         <oasis:entry colname="col1">Layer</oasis:entry>
         <oasis:entry colname="col2">Longitude (<inline-formula><mml:math id="M29" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula>)</oasis:entry>
         <oasis:entry colname="col3">Latitude (<inline-formula><mml:math id="M30" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula>)</oasis:entry>
         <oasis:entry colname="col4">Grid</oasis:entry>
         <oasis:entry colname="col5">Type of</oasis:entry>
         <oasis:entry colname="col6">Coordinate</oasis:entry>
       </oasis:row>
       <oasis:row>
         <oasis:entry colname="col1"/>
         <oasis:entry colname="col2"/>
         <oasis:entry colname="col3"/>
         <oasis:entry colname="col4">size</oasis:entry>
         <oasis:entry colname="col5">SWE</oasis:entry>
         <oasis:entry colname="col6">system</oasis:entry>
       </oasis:row>
       <oasis:row rowsep="1">
         <oasis:entry colname="col1"/>
         <oasis:entry colname="col2"/>
         <oasis:entry colname="col3"/>
         <oasis:entry colname="col4">(m)</oasis:entry>
         <oasis:entry colname="col5"/>
         <oasis:entry colname="col6"/>
       </oasis:row>
     </oasis:thead>
     <oasis:tbody>
       <oasis:row>
         <oasis:entry colname="col1">1</oasis:entry>
         <oasis:entry colname="col2">79.2 to 107.6</oasis:entry>
         <oasis:entry colname="col3"><inline-formula><mml:math id="M31" display="inline"><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">13.6</mml:mn></mml:mrow></mml:math></inline-formula> to 8.2</oasis:entry>
         <oasis:entry colname="col4">1856</oasis:entry>
         <oasis:entry colname="col5">Linear</oasis:entry>
         <oasis:entry colname="col6">Spherical</oasis:entry>
       </oasis:row>
       <oasis:row>
         <oasis:entry colname="col1">2</oasis:entry>
         <oasis:entry colname="col2">94.61 to 97.78</oasis:entry>
         <oasis:entry colname="col3">3.41 to 6.29</oasis:entry>
         <oasis:entry colname="col4">618</oasis:entry>
         <oasis:entry colname="col5">Linear</oasis:entry>
         <oasis:entry colname="col6">Spherical</oasis:entry>
       </oasis:row>
       <oasis:row>
         <oasis:entry colname="col1">3</oasis:entry>
         <oasis:entry colname="col2">94.8 to 95.99</oasis:entry>
         <oasis:entry colname="col3">4.903 to 5.89</oasis:entry>
         <oasis:entry colname="col4">124</oasis:entry>
         <oasis:entry colname="col5">Linear</oasis:entry>
         <oasis:entry colname="col6">Spherical</oasis:entry>
       </oasis:row>
       <oasis:row>
         <oasis:entry colname="col1">4</oasis:entry>
         <oasis:entry colname="col2">95.15 to 95.31</oasis:entry>
         <oasis:entry colname="col3">5.25 to 5.33</oasis:entry>
         <oasis:entry colname="col4">18</oasis:entry>
         <oasis:entry colname="col5">Nonlinear</oasis:entry>
         <oasis:entry colname="col6">Cartesian</oasis:entry>
       </oasis:row>
     </oasis:tbody>
   </oasis:tgroup></oasis:table></table-wrap>

      <p id="d1e1820">To simulate the rupture area caused by the <inline-formula><mml:math id="M32" display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mi mathvariant="normal">w</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn mathvariant="normal">9.1</mml:mn></mml:mrow></mml:math></inline-formula> earthquake on
26 December 2004, we used a model proposed by Piatanesi and Lorito (2007)
and Romano (2009). The rupture that generated the initial waves around the
source area is shown in Fig. 3. We divided the rupture area into eight
segments as described in Piatanesi and Lorito (2007). This model simulated a
multi-fault scenario and was validated by Syamsidik et al. (2015b). Using
NSWEs for layers 1 to 3 provides insignificant differences to the waves
generated around the offshore area. On the other hand, applying nonlinear SWEs for
layers 1 and 3 will reduce computational time without reducing the
accuracy of the simulation results.</p>
      <p id="d1e1838">We initiated sediment transport using a grain diameter switch for each of
the three sites. By producing shear stress, we calculated a uniform
Manning's roughness coefficient of 0.02 for the seafloor and a coefficient
of 0.04 for overland areas combined with a water level fluctuation boundary
condition from COMCOT simulations. We used Delft3D to simulate sediment
transport processes caused by tsunami waves. We modeled this at the
innermost layer of the simulation domain (layer 4 in the COMCOT model).
Delft3D uses NSWEs combined with finite difference methods. The NSWEs in the
model apply the equations of conservation mass, momentum, and energy flux
and have the capability to simulate rapidly varying flow conditions (Stelling
and Duijmeijer, 2003). In its basic module, Delft3D computes sediment
transport using bedload and suspended load characteristics as described in
van Rijn (1993, 2007). Changes in topography caused by erosion and
sedimentation processes are updated at each time step of the simulation
(Lesser et al., 2004). A more detailed explanation of the sediment
transport formulas used in Delft3D can be found in Apotsos et al. (2011a).
One study proved that van Rijn (1993) formulas provided the best results
compared to tsunami-induced sediment transport experiments in a wave flume
(Li and Huang, 2013). Notwithstanding the reviews, we incorporated more
sediment transport formulas in our numerical simulation, such as
Engelund–Hansen 1967, Meyer-Peter–Mueller (MPM) 1948, and Soulsby 1997.
Recently, researchers have used Delft3D to simulate and test extreme events
that triggered morphological changes, such as in the case of the Indian
Ocean tsunami (Syamsidik et al., 2017; Gelfenbaum et al., 2007; Apotsos et
al., 2011a, b, c). Previously, Delft3D was used and validated based on
conditions less extreme than a tsunami, such as wind–wave-driven<?pagebreak page1270?> sediment
transport (Broekema et al., 2016; Lesser et al., 2004; van Rijn et al., 2011).</p>
</sec>
<sec id="Ch1.S3.SS2">
  <label>3.2</label><title>Field measurements</title>
      <p id="d1e1849">The results of the numerical simulations present estimations of the
locations of the tsunami deposits, as well as their respective thicknesses.
To assess the validity of our results, we conducted a series of surveys at
three locations, i.e., Birek, Pasie Janeng, and Saney. We conducted these
surveys in October 2015, January 2016, and April 2016 for Birek, Saney, and
Pasie Janeng, respectively. Surveys were conducted during different periods
of the year because of the intensive labor and time needed for each survey.
At each location, we performed manually excavated pit tests.</p>
      <p id="d1e1852">To define the tsunami deposit, we followed the USGS procedure as described
in Peters and Jaffe (2010). The same procedure was also applied during a
tsunami deposit survey in Jantang conducted by Peters and Jaffe (2010). The
identification of tsunami deposits relies on 10 criteria. Tsunami deposits
are easily identifiable by their sharpness and erosional basal contact with
original soil below or under the tsunami deposit. In this study, tsunami
sediments were easily identified because of the characteristics of adjacent
sediments, which have high organic contents from surface runoff processes.
Tsunami sediments form thin layers, ranging between 1 and 30 cm thick in
most cases (Srinivasalu et al., 2009). Beaches near the study area are
composed of sandy-quartz materials. Patches of coral reefs exist around
Birek, Saney, and Pasie Janeng. Therefore, it was difficult to distinguish
between tsunami deposits and sediments produced by littoral transport, also
highlighted in Jaffe et al. (2003). The presence of sea shells was
identified through a microscopic observation of the sediment material. This
could distinguish the tsunami-induced deposit from original topsoil material
or other surface runoff process. Other criteria of the tsunami deposit were
advised by Jaffe et al. (2003) and Peters and Jaffe (2010) who put
the methods of the tsunami-induced sediment transport investigation into one
practical guideline. We followed the steps of the guideline exactly and
clarify it using some microscopic observations. We spent a significant
period (from 0.5 to 2.0 h) for each sample to carefully identify the
tsunami deposit and distinguish it from other sources of sediment. In total,
we collected 14 samples of the tsunami deposits and 22 locations of sampling
performed by Jaffe et al. (2006). Rip-up clasts were also found in the
layers of the tsunami deposit layers, indicative of the energy that
transported the material, such as in the case of a<?pagebreak page1271?> tsunami. Therefore, the
existence of rip-up clasts further confirms the presence of tsunami deposits
(Morton et al., 2007). Costa et al. (2012, 2015) proposed that the shape of the
zircons in the sediment deposit could be used to interpret the number of waves
and tsunami runup or backwash processes. Costa et al. (2015) stated that
euhedral zircon could be associated with backwash processes. Meanwhile, rounded
zircons could be attributed to deposition that occurred during tsunami runup processes. Detailed observation is absent in our present study.</p>
</sec>
</sec>
<sec id="Ch1.S4">
  <label>4</label><title>Tsunami runup and tsunami deposits</title>
      <p id="d1e1864">We analyzed the impacts that the 2004 tsunami had on coastal morphology
using field surveys and satellite images. Tsunami-inundated areas were
observed using satellite images to measure changes in coastal morphology in
the vicinity of Lhoong, Aceh Besar. We identified tsunami deposits during a
field survey in October 2015. Surveys started with coastal topography
measurements and then continued by location of potential deposits based on
topography and surrounding conditions.</p>
<sec id="Ch1.S4.SS1">
  <label>4.1</label><title>Tsunami inundation area</title>
      <p id="d1e1874">We analyzed the inundation area caused by the tsunami using satellite images
captured in February 2005 using DigitalGlobe. After the 2004 tsunami
propagated around the rupture source, waves moved inland, reaching the
foothills approximately 200 m from the coastline at Birek and approximately
1 km inland at Jantang. The extent of inundation is shown in Fig. 4 by
comparing the boundaries from numerical simulations and satellite images.
The inundation limits generated by numerical simulation and those shown in
the digitized images are generally consistent. On the basis of the
inundation zones, we selected three locations to perform pit tests to sample
the tsunami sediment deposits. For Jantang, we refer to descriptions in
Jaffe et al. (2006).</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F4"><?xmltex \currentcnt{4}?><label>Figure 4</label><caption><p id="d1e1879">The 2004 tsunami inundation limit at Lhoong based on the numerical
simulation (dashed black lines) and satellite image digitization (dashed red lines).</p></caption>
          <?xmltex \igopts{width=199.169291pt}?><graphic xlink:href="https://nhess.copernicus.org/articles/19/1265/2019/nhess-19-1265-2019-f04.png"/>

        </fig>

</sec>
<sec id="Ch1.S4.SS2">
  <label>4.2</label><title>Coastal topography</title>
      <p id="d1e1896">Surrounded by hills, the topography conditions in the Birek coastal area are
classified as V-shaped, indicating that the elevation of the center region
is lower than its surroundings. At Birek, a small creek channels surface
runoff to the sea. Figure 5 shows the topography of Birek and cross
profiles of the two transects, i.e., Cross 1 and Cross 2. The locations of
the pit tests for the tsunami deposit sample collected are marked as B01–B03 in Fig. 5.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F5" specific-use="star"><?xmltex \currentcnt{5}?><label>Figure 5</label><caption><p id="d1e1901">The topographic profiles at Birek and the locations of the pit tests
(B01–B03), Pasie Janeng (P01–P04), and Saney (S01–S05).</p></caption>
          <?xmltex \igopts{width=398.338583pt}?><graphic xlink:href="https://nhess.copernicus.org/articles/19/1265/2019/nhess-19-1265-2019-f05.png"/>

        </fig>

      <p id="d1e1910">Pasie Janeng has a limited plain area, stretching approximately 100 m from
the shoreline. Steep slopes characterize the inundation zone further inland
at Pasie Janeng (Fig. 5b). Given this topography, we estimate that surface
runoff from rainfall, flowing from the hillside to the beach, significantly
affects the current thickness of tsunami deposits. We assume that hydrologic
runoff processes are the main mechanism that erodes the tsunami deposits.
At Saney headland (Fig. 5c), topography is flat, but the study area is
sandwiched by the coastline. We conducted pit tests to investigate tsunami
deposits at the Point S01 to Point S05 locations.</p>
</sec>
<sec id="Ch1.S4.SS3">
  <label>4.3</label><title>Sediment characteristics</title>
      <p id="d1e1921">Tsunami deposits were identified by the sharp, clear presence of a sandy
layer between new and old topsoil. The presence of seashells also
strengthened the indication of tsunami deposits (Jaffe and Peters, 2010).
Sediment deposition from the tsunami waves almost reaches the foothills. The
energy produced by the earthquake was capable of generating tsunami waves
that deposited sediments up to 6.3 m in elevation above sea level. The
thicknesses of the tsunami deposits correlate with distance from the
shoreline, becoming thinner the farther inland. At sampling location B01, we
did not find tsunami deposits. At location B02, tsunami deposit thickness
was approximately 29 cm. The location of Point B02 is approximately 320 m
from the coastline. At sampling location B03, we identified a thinner
tsunami deposit, approximately 7.5 cm thick. This location is approximately
400 m from the coastline (Fig. 5). Pit dimensions at these two locations
were approximately 1.8 m in length, 0.9 m in width, and 1.50 m in depth.</p>
      <p id="d1e1924">The profiles of the pit tests at Birek, Pasie Janeng, and Saney are shown in
Fig. 6. All of the profiles have similar trends in thicknesses of tsunami deposits.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F6" specific-use="star"><?xmltex \currentcnt{6}?><label>Figure 6</label><caption><p id="d1e1929">Profiles of tsunami deposit samples B02 and B03 at Birek, PJ01 and
PJ03 at Pasie Janeng, and S01 and S02 at Saney.</p></caption>
          <?xmltex \igopts{width=497.923228pt}?><graphic xlink:href="https://nhess.copernicus.org/articles/19/1265/2019/nhess-19-1265-2019-f06.png"/>

        </fig>

      <?pagebreak page1272?><p id="d1e1939"><?xmltex \hack{\newpage}?>The majority of the sediment deposits at several locations in Lhoong contain
sand, i.e., <inline-formula><mml:math id="M33" display="inline"><mml:mrow><mml:mo>&gt;</mml:mo><mml:mn mathvariant="normal">70</mml:mn></mml:mrow></mml:math></inline-formula> % sand (Birek <inline-formula><mml:math id="M34" display="inline"><mml:mo>=</mml:mo></mml:math></inline-formula> 82.15 %, Pasie Janeng <inline-formula><mml:math id="M35" display="inline"><mml:mo>=</mml:mo></mml:math></inline-formula> 78.85 %,
and Saney <inline-formula><mml:math id="M36" display="inline"><mml:mo>=</mml:mo></mml:math></inline-formula> 71.88 %). Waves carried the majority of the sand
from the ocean bed, which mixed with deposits from each locality. Each
sampling location contains less than 25 % coarse grains (Birek <inline-formula><mml:math id="M37" display="inline"><mml:mo>=</mml:mo></mml:math></inline-formula> 6.3 %,
Pasie Janeng <inline-formula><mml:math id="M38" display="inline"><mml:mo>=</mml:mo></mml:math></inline-formula> 17.29 %, and Saney <inline-formula><mml:math id="M39" display="inline"><mml:mo>=</mml:mo></mml:math></inline-formula> 20.46 %), which is
produced by local erosion. The tsunami flow depth drove the sediment
transport process, which resulted in the characteristic sandy tsunami
deposits. Additionally, we found rip-up clasts in the tsunami deposits at
Birek and Pasie Janeng.</p>
</sec>
<sec id="Ch1.S4.SS4">
  <label>4.4</label><title>Model hindcast</title>
      <p id="d1e2005">To simulate both hydrodynamic and morphodynamic processes during the 2004
Indian Ocean tsunami, COMCOT and Delft3D were run simultaneously. Transient
waves produced by COMCOT successfully drove the sediment transport
simulation performed by Delft3D. Tsunami wave heights, shear stresses, and
the sediment transport process produced by Delft3D-FLOW are useful when
trying to explain the sediment transport mechanism during the runup of
tsunami waves. Here, we present the results of the sediment transport
process simulations in cumulative sedimentation and erosion thicknesses.
Extreme changes in coastal morphology from plains to steep slopes
significantly contributed to the wide<?pagebreak page1273?> distribution of tsunami deposits
throughout the area affected by the tsunami.</p>
<sec id="Ch1.S4.SS4.SSS1">
  <label>4.4.1</label><title>Tsunami wave height and sedimentation</title>
      <p id="d1e2015">Tsunami occurrence induced tsunami waves, inundating areas far inland. These
results came first from COMCOT, which produced the tsunami wave height that
was used as a parameter for Delft3D simulations. The modeled tsunami
consisted of two major waves: (1) the first wave, which peaked at a height of
5.23 m, and (2) the second wave, which peaked at a height of approximately 5.61 m (Fig. 7).</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F7"><?xmltex \currentcnt{7}?><label>Figure 7</label><caption><p id="d1e2020">Delft3D-FLOW boundary flow input from COMCOT results.</p></caption>
            <?xmltex \igopts{width=236.157874pt}?><graphic xlink:href="https://nhess.copernicus.org/articles/19/1265/2019/nhess-19-1265-2019-f07.png"/>

          </fig>

      <p id="d1e2029">On the basis of the COMCOT results, Delft3D-FLOW, at a nearshore observation
point, produced two major tsunami waves: (1) the first wave peaked at 12.2 m,
and (2) the second wave followed 14 min later peaking at 8.9 m, 3.3 m lower
than the first wave. The fluctuation in runup and backwash generated
significant sediment transport in coastal waters and overland flow.</p>
      <p id="d1e2033">Sedimentation processes occurred predominantly during backwash.
Specifically, such a case only occurs on relatively mild slopes. On the
other hand, steep slopes remove sediments during the backwash. Reduced
energy after the second and subsequent waves carried more sediment deposits
but had significantly less energy during the backwash process, therefore
leaving more sediments behind (08:45 LT – local time). As shown in Fig. 8,
backwash produced a sediment deposit that was 0.38 m thick during the second wave.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F8" specific-use="star"><?xmltex \currentcnt{8}?><label>Figure 8</label><caption><p id="d1e2038">Relation between wave height and cumulative sediment on the observation
points (shown in red dots).</p></caption>
            <?xmltex \igopts{width=341.433071pt}?><graphic xlink:href="https://nhess.copernicus.org/articles/19/1265/2019/nhess-19-1265-2019-f08.png"/>

          </fig>

<?xmltex \hack{\newpage}?>
</sec>
<?pagebreak page1274?><sec id="Ch1.S4.SS4.SSS2">
  <label>4.4.2</label><title>Tsunami velocity fields</title>
      <p id="d1e2057">The sediment transport process is highly influenced by the shear process
induced by tsunami currents along the bottom of the bed. The velocity fields
during the tsunami runup process help in explaining the energy involved
during sediment transport processes. Figure 9 shows the velocity fields at
the Birek and Pasie Janeng locations.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F9" specific-use="star"><?xmltex \currentcnt{9}?><label>Figure 9</label><caption><p id="d1e2062">Tsunami wave velocity fields at Birek during wave advance <bold>(a)</bold>
and during wave retreat <bold>(b)</bold> and at Pasie Janeng during wave advance <bold>(c)</bold>
and during wave retreat <bold>(d)</bold>.</p></caption>
            <?xmltex \igopts{width=497.923228pt}?><graphic xlink:href="https://nhess.copernicus.org/articles/19/1265/2019/nhess-19-1265-2019-f09.png"/>

          </fig>

      <p id="d1e2083">As shown in Fig. 9a and b, during the wave's advance, the velocity at
Birek location is lower than that when the wave is retreating. We also found
the same condition at Pasie Janeng (Fig. 9c and d). The maximum value
of shear stress was 83 N m<inline-formula><mml:math id="M40" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">2</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>, recorded 24 min after the first wave
arrived, eroding 0.11 m of the bed layer. High shear stress values
significantly erode land, still depositing sediments 42 min after the first waves arrive.</p>
</sec>
<sec id="Ch1.S4.SS4.SSS3">
  <label>4.4.3</label><title>Tsunami deposit thickness</title>
      <p id="d1e2106">On the basis of the simulations using both Delft3D and field measurements,
we compare the differences in tsunami deposit thicknesses between the four
sample locations. The runup process mostly produces erosion, whereas most
backwash does the opposite. The accumulation sedimentation map gave
relatively reliable information on tsunami deposit locations, similar to the
simulation results found for Lhoong. Figure 10 shows maps of spatial
distribution of cumulative sedimentation and erosion after the tsunami waves
based on the numerical simulation results. Negative values are for erosion,
and positive values are for sedimentation thickness in meters. The following points
compare the results of the numerical simulations and the field data from the
four sample locations.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F10" specific-use="star"><?xmltex \currentcnt{10}?><label>Figure 10</label><caption><p id="d1e2111">Tsunami deposit thickness from numerical models using Delft3D-FLOW at
Birek, Pasie Janeng, Jantang, and Saney.</p></caption>
            <?xmltex \igopts{width=455.244094pt}?><graphic xlink:href="https://nhess.copernicus.org/articles/19/1265/2019/nhess-19-1265-2019-f10.png"/>

          </fig>

</sec>
<sec id="Ch1.S4.SS4.SSSx1" specific-use="unnumbered">
  <title>a. Birek</title>
      <p id="d1e2127">Hydrodynamic process analysis revealed that there is some wave
amplification, along with a decrease in the current velocity. Elevated
topography halted the tsunami runup at the study area. The elevated
topography deflected tsunami waves and reduced their velocity. The reduced
tsunami flow velocity also signifies that the tsunami waves are<?pagebreak page1275?> losing
energy. This explains the presence of a thinner tsunami deposit, which we
found around the foothills compared with areas farther from the hills. The
deflected waves created a larger wave around the hill, but at the same time,
there was a significant decrease in the shear stresses. The lower the value
of the shear stress that is generated by the waves, the more likely it is
for sediment deposition to take place. Figure 11a compares the results of
the numerical simulation and field data at Birek.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F11" specific-use="star"><?xmltex \currentcnt{11}?><label>Figure 11</label><caption><p id="d1e2132">Comparison of tsunami deposit thickness results from the numerical
models using four different sediment transport formulae and the field survey at
Birek, Pasi Janeng, Jantang, and Saney.</p></caption>
            <?xmltex \igopts{width=455.244094pt}?><graphic xlink:href="https://nhess.copernicus.org/articles/19/1265/2019/nhess-19-1265-2019-f11.png"/>

          </fig>

      <p id="d1e2141">The coastline is marked by the origin along the <inline-formula><mml:math id="M41" display="inline"><mml:mi>x</mml:mi></mml:math></inline-formula> axis. Data from the three
pit tests show that the numerical results overestimate the field data. It is
important to note that this location has a V-shaped topography, surrounded
by hills. The deposit thickness decreases with increasing distance from the coastline.</p>
</sec>
<sec id="Ch1.S4.SS4.SSSx2" specific-use="unnumbered">
  <title>b. Pasie Janeng</title>
      <p id="d1e2157">Compared with Birek, we found slightly different results at Pasie Janeng
(see Fig. 11b). Areas with mild slopes are shorter than those at Birek,
limiting the area capable of trapping sediments. The steep topography of the
hills surrounding Pasie Janeng funnels rainfall runoff into a small channel
that drains directly to the sea. The path that runoff took eroded a small
amount of the tsunami deposit that could have been deposited there for
almost 12 years. The morphology of the site resembles an embayment, which
potentially receives new sediment from coastal processes, making it
difficult to find tsunami deposits nearshore.</p>
</sec>
<sec id="Ch1.S4.SS4.SSSx3" specific-use="unnumbered">
  <title>c. Jantang</title>
      <p id="d1e2166">Jaffe et al. (2006) examined the thickness of tsunami deposits at Jantang
during a survey conducted in 2005. The wide area, with a mild slope, was
advantageous for capturing sediments during the tsunami event. The
relatively short time period between the tsunami and the survey was
beneficial given that there was little activity in the area (i.e., not until 2009)
just after the 2004 tsunami, concerning rehabilitation and
reconstruction. Transect data based on field surveys<?pagebreak page1276?> were compared to the
numerical simulations. Comparisons between the field data and the numerical
simulation results are shown in Fig. 11c.</p>
      <p id="d1e2169">The results from the numerical simulations and the field survey are in good
agreement. Unlike the data from the other three locations, the results of
the numerical simulations at Jantang underestimate the field data. The
absence of variations in the roughness coefficient slightly affected
sediment transport results. The lack of additional land cover parameters
from corals or roads also affected several of the model-calculated
deposition and erosion zones. Besides these<?pagebreak page1277?> limitations, the simulation
results are in good agreement with the field data.</p>
</sec>
<sec id="Ch1.S4.SS4.SSSx4" specific-use="unnumbered">
  <title>d. Saney</title>
      <p id="d1e2178">Saney is protected by a small hill facing the ocean situated at the edge of
the land, which made it more vulnerable to tsunami wave energy. A settlement
complex was completely swept away during the 2004 tsunami. The tsunami wave
that came from the west struck the edge of the land and was amplified after
encountering the mild slope area. The narrow headland of the site provided
a small place for the tsunami deposit to settle down. Years after the
tsunami, other factors may have eroded the deposit, leaving thinner tsunami
deposit. The results of the numerical simulations overestimated values
compared with the field data (see Fig. 11d). Regardless of the process,
we could still find the tsunami deposit 13 years after the event with the
support of numerical modeling.</p>
</sec>
</sec>
</sec>
<sec id="Ch1.S5">
  <label>5</label><title>Discussion</title>
      <p id="d1e2192">Estimations of the deposit locations and the thickness of tsunami deposits
are key factors for site selection and target fieldwork efforts. Even
13 years after the Indian Ocean tsunami, numerical simulations still provide
useful estimates in order to locate tsunami deposits. Finding suitable
tsunami deposit locations may become increasingly difficult as massive
reconstruction efforts significantly alter the tsunami deposits in certain
areas. The tsunami deposits may ultimately disappear or be difficult to
identify. It is important to discuss land use practices with the local
community in order to target well-preserved field sites. Using a linear
regression method, we found that the van Rijn (1993) formulas gave the best
approximation of the field data. Similar results were found at the other three
study sites, with a smaller correlation (<inline-formula><mml:math id="M42" display="inline"><mml:mrow><mml:msup><mml:mi>r</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msup></mml:mrow></mml:math></inline-formula>). Our findings confirmed
suggestions from laboratory experiments done by Li and Huang (2013).</p>
      <p id="d1e2206">We encountered the following limitations during the course of this study.
First, field measurements could not provide the exact number of tsunami
waves. On the basis of our models, two main waves occurred during the
tsunami in this region. This is corroborated by several eyewitness accounts
from the study area given the absence of instrumental wave recordings. The
small number of waves determined the size of the material contained in the
deposits, such as the rip-up clasts in the tsunami deposit layer. We found
rip-up clasts at both Pasie Janeng and Birek. This confirms that there were
only a few waves, as these differ from storm waves (Morton et
al., 2007). Secondly, there are uncertainties about the model itself. For
sediment transport, we used the two-dimensional horizontal (2-DH) Delft3D that did not account for
turbulence kinetic energy, which may simplify the results. Furthermore, our
simulations used depth-averaged velocity produced by the tsunami waves. This
may, in fact, limit the maximum suspended sediment concentration because of
the exclusion of suspended-sediment-induced density stratification (Jaffe et
al., 2016). Other studies have attempted to run Delft3D with the
3-D model that allows interlayer settling and erosion processes
during tsunami wave propagation (Apotsos et al., 2011b; Gelfenbaum et al.,
2007). Our study area has a complex topography and morphology. All of the
sites, except for Jantang, have headlands and patches of coral reefs. The
complexity of the topography could produce a significant difference between
numerical model results and field data, as observed in the case of the Sanriku
coast in Japan (Goto et al., 2017). The roughness coefficient, which was set
to a value similar to all grids of the sea area, could contribute to the
different thicknesses found between the numerical simulations and the field data.</p>
      <p id="d1e2209">Regardless of the limitations and uncertainties, this study has successfully
demonstrated the effectiveness of simulations in estimating the
locations of tsunami deposits. Although natural processes such as rainfall,
runoff, and aeolian sediment transport have taken place in the study area
for more than a decade, location estimation produced by the numerical
simulations is correct. The methods presented here are applicable to areas
without any human activity or other extreme events, such as the effects of
storms, that could not be considered in the simulations. Discontinuous and
patchy tsunami deposit records may be the result of anthropogenic
disturbances (Chague et al., 2018).</p>
      <p id="d1e2212">Further advances in the field of tsunami-induced sediment transport should
incorporate collaborations between multidisciplinary researchers, as this
may increase the comprehensiveness of the study. The limitations and
uncertainties highlighted in this study remain major challenges in
paleo-tsunami studies (Sugawara et al., 2014). A solid understanding of
paleo-tsunamis, by exhibiting tsunami deposits to coastal communities,
serves to allow us to better communicate the risk of tsunamis to them. We
also acknowledge that the use of high-resolution topography and bathymetry
data could significantly contribute to the reliability of the numerical simulation results.</p>
</sec>
<sec id="Ch1.S6" sec-type="conclusions">
  <label>6</label><title>Conclusions</title>
      <p id="d1e2223">This study has successfully demonstrated the coupling method of COMCOT and
Delft3D-FLOW to provide a better understanding of tsunami-wave-induced
sediment transport and to find the locations of tsunami deposits in a
specific area more than one decade after the tsunami. Large shear stresses
generated by tsunami waves transported sediment farther inland. Field survey
topography analysis also contributed to increased accuracy in defining the
deposit location inside the tsunami inundation zone. Combining both methods
contributed to results that are in good agreement between numerical
simulations and field data. Tsunami deposit surveys were<?pagebreak page1278?> conducted at three
locations between 2015 and 2016, showing that actual tsunami deposit
thicknesses are thinner than what the numerical results show. Surface runoff, soil
consolidation, and aeolian sediment transport could contribute to these
differences. On the other hand, tsunami deposit data, obtained just after
the tsunami, provided a good fit to the numerical results. This study
demonstrates the importance of rapid data collection in the immediate
aftermath of a tsunami. Delft3D results displayed the location of sediment
deposition, and field surveys confirmed the presence of tsunami deposits
physically. Ultimately, the characterization of modern tsunami deposits will
allow us to interpret earlier extreme events in sedimentary records.</p>
</sec>

      
      </body>
    <back><notes notes-type="dataavailability"><title>Data availability</title>

      <p id="d1e2230">Data from this research are not publicly available. Interested
researchers can contact the correspondence author of this article.</p>
  </notes><notes notes-type="authorcontribution"><title>Author contributions</title>

      <p id="d1e2236">S coordinated the research team, conducted fieldwork, and
led the writing process of the article. MA led the team during the fieldwork
and performed numerical simulations. HMF supervised the analysis and writing processes.
MF and TMH contributed to fieldwork and sediment property analysis.</p>
  </notes><notes notes-type="competinginterests"><title>Competing interests</title>

      <p id="d1e2242">The authors declare that they have no conflict of
interest.</p>
  </notes><ack><title>Acknowledgements</title><p id="d1e2248">We are grateful for the support of Lhoong residents Alfaisal and Syahrul Mauluddin in
the field survey. Data collection for this research is supported by PEER
Cycle 5 Grant USAID, and the National Academy of Sciences funded this publication
process through PEER Cycle 5 (no. 5-395), sponsor grant award no. AID-OAAA-A-11-00012
and subgrant no. PGA-2000004893, within the research project
titled “Incorporating Climate Change Induced Sea Level Rise Information into
Coastal Cities' Preparedness toward Coastal Hazards”. The writing of this
paper has been part of the World Class Professor Program 2017 (WCP 2017)
(no. 168.A10/D2/KP/2017), promoted by the Ministry of Research, Technology, and Higher
Education (Kemenristekdikti), and within which Universitas Syiah Kuala, Gadjah Mada
University, and Diponegoro University work in collaboration with Hermann M. Fritz
from the Georgia Institute of Technology, USA.</p></ack><notes notes-type="financialsupport"><title>Financial support</title>

      <p id="d1e2253">This research has been supported by the PEER
Cycle 5 Grant USAID and the National Academy of Sciences (USA) (grant nos. AID-OAAA-A-11-00012 and PGA-2000004893) and the Kemenristekdikti Indonesia
(WCP A 2017 UGM-UNDIP-UNSYIAH, grant no. 168.A10/D2/KP/2017).</p>
  </notes><notes notes-type="reviewstatement"><title>Review statement</title>

      <p id="d1e2259">This paper was edited by Maria Ana Baptista and reviewed by
Pedro Costa and three anonymous referees.</p>
  </notes><ref-list>
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    <!--<article-title-html>Numerical simulations of the 2004 Indian Ocean  tsunami deposits' thicknesses and emplacements</article-title-html>
<abstract-html><p>After more than a decade of recurring tsunamis,
identification of tsunami deposits, a part of hazard characterization, still
remains a challenging task that is not fully understood. The lack of sufficient
monitoring equipment and rare tsunami frequency are among the primary
obstacles that limit our fundamental understanding of sediment transport
mechanisms during a tsunami. The use of numerical simulations to study
tsunami-induced sediment transport was rare in Indonesia until the
2004 Indian Ocean tsunami. This study aims to couple two hydrodynamic numerical
models in order to reproduce tsunami-induced sediment deposits, i.e., their
locations and thicknesses. Numerical simulations were performed using the
Cornell Multi-grid Coupled Tsunami (COMCOT) model and Delft3D. This study
reconstructed tsunami wave propagation from its source using COMCOT, which
was later combined with Delft3D to map the location of the tsunami deposits
and calculate their thicknesses. Two-dimensional horizontal (2-DH) models
were used as part of both simulation packages. Four sediment transport
formulae were used in the simulations, namely van Rijn 1993, Engelund–Hansen 1967,
Meyer-Peter–Mueller (MPM) 1948, and Soulsby 1997. Lhoong, in the Aceh
Besar District, located approximately 60&thinsp;km southwest of Banda Aceh, was
selected as the study area. Field data collected in 2015 and 2016 validated
the forward modeling techniques adopted in this study. However, agreements
between numerical simulations and field observations were more robust using
data collected in 2005, i.e., just months after the tsunami (Jaffe et al.,
2006). We conducted pit (trench) tests at select locations to obtain tsunami
deposit thickness and grain size distributions. The resulting numerical
simulations are useful when estimating the locations and the thicknesses of
the tsunami deposits. The agreement between the field data and the numerical
simulations is reasonable despite a trend that overestimates the field observations.</p></abstract-html>
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