# Performance Assessment of NAMI DANCE in Tsunami Evolution and Currents Using a Benchmark Problem
## 2 Materials and Methods  
### The Numerical Model: NAMI DANCE  
Tsunami numerical modeling by NAMI DANCE is based on the solution of a nonlinear form of long wave equations with respect to related initial and boundary conditions [4]. In general, the explicit numerical solution of nonlinear shallow water (NSW) equations is preferred since it consumes reasonable computer time and memory, and also provides the results in acceptable error limits [4]. NAMI DANCE is a numerical model which is able to simulate tsunami evolution, propagation and inundation. It has been developed by the collaboration of Ocean Engineering Research Center, Middle East Technical University, Turkey, and Special Research Bureau for Automation of Marine Researches, Russia [5;6]. NAMI DANCE uses C++ programming language and solves NSW equations using a staggered leapfrog numerical solution procedure. In the theory of long waves, the vertical motion of water particles is not considered since it has a negligible effect on the pressure distribution. Based upon this approximation, using necessary dynamic and kinematic conditions and including the bottom friction terms, the fundamental equations of NAMI DANCE, which are given Equations (1)-(5), are obtained and these equations are discretized by following the staggered leapfrog scheme.  
\[\frac{\partial\eta}{\partial t}+\frac{\partial M}{\partial x}+\frac{\partial N }{\partial y}=0 \tag{1}\]  
\[\frac{\partial M}{\partial t}+\frac{\partial}{\partial x}\left(\frac{M^{2}}{D }\right)+\frac{\partial}{\partial y}\left(\frac{MN}{D}\right)+gD\frac{ \partial\eta}{\partial x}+\frac{gn^{2}}{D^{7/3}}M\sqrt{M^{2}+N^{2}}=0 \tag{2}\]  
\[\frac{\partial N}{\partial t}+\frac{\partial}{\partial x}\left(\frac{MN}{D} \right)+\frac{\partial}{\partial y}\left(\frac{N^{2}}{D}\right)+gD\frac{ \partial\eta}{\partial y}+\frac{gn^{2}}{D^{7/3}}N\sqrt{M^{2}+N^{2}}=0 \tag{3}\]  
\[M=u\left(h+\eta\right)=uD \tag{4}\]  
\[N=v\left(h+\eta\right)=vD \tag{5}\]  
where \(x\) and \(y\) are the horizontal axes; \(t\) is time; \(h\) is undisturbed flow depth; \(\eta\) is the vertical displacement above the undisturbed water surface; \(M\) and \(N\) are the discharge fluxes in \(x\) and \(y\) directions; \(u\) and \(v\) are particle velocities in \(x\) and \(y\) directions, respectively; \(n\) is the Manning's roughness coefficient; \(g\) is the gravitational acceleration, and \(D\) is the total water depth given by \(h+\eta\)[6].  
NAMI DANCE is able to compute (i) tsunami source from either rupture characteristics or predetermined wave form; (ii) propagation; (iii) arrival time; (iv) coastal amplification; (v) inundation (according to accuracy and grid size); (vi) distribution of current velocities and their directions at selected time intervals; (vii) relative damage levels according to drag and impact forces; (viii) time histories of water surface fluctuations; (ix) 3D plots of sea state at selected time intervals from different camera and light positions; and (x) animations of tsunami propagation [7]. NAMI DANCE has been applied to analytical, experimental, and field benchmark problems [8; 9] for validation and verification [10; 11; 12; 13; 14] and also applied to several tsunami events [7; 15; 16; 17; 18].