Analytical Approach for Solving the Internal Waves Problems Involving the Tidal Force

The mathematical model for describing internal waves of the ocean is derived from the assumption of ideal fluid; i.e., the fluid is incompressible and inviscid. These internal waves are generated through the interaction between the tidal currents and the basic topography of the fluid. Basically the mathematical model of the internal wave problem of the ocean is a system of nonlinear partial differential equations (PDEs). In this paper, the analytical approach used to solve nonlinear PDE is the Homotopy Analysis Method (HAM). HAM can be applied to determine the resolution of almost any internal wave problem involving tidal forces. The use of HAM in the solution to basic fluid equations is efficient and simple, since it involves only modest calculations using the common integral.


Introduction
Internal waves are gravitational waves that exist on two layers of fluid having different densities.Internal waves are formed due to a meeting among layers of seawater that have different densities of generating forces coming from wind, tide, or even movement of ships.The density difference causes the seawater to become layered where water with a larger density will be below that with a smaller density.This condition stimulates the formation of boundary of the two layers fluid (interfaces) where in case of external disturbance (by the existing generating force), an interlayer wave occurs without affecting the waves on the surface.Generating internal waves requires a large force, for instance, generated by the interaction of strong tidal currents, fluid coating, and lower topography.Research on internal waves at sea has previously been applied to various applications and ranges, for example, to detect the strength of offshore oil platform pylons [1] and to measure how the impact of internal waves can affect Chlorophila [2].In addition, this wave can also affect the marine habitat that is the spatial distribution of Planktothrix rubescens [3].
Internal waves of the ocean can be modeled in terms of mathematical equations using the ideal fluids assumptions (incompressible and inviscid) of mass conservation laws and the law of momentary vapor.Internal waves are generated through the interaction between the tidal flow and the topography in a nonuniform fluid layer by solving the Navier-Stokes equation in Boussinesq approximation.Basically the mathematical representation of the internal waves of the ocean is a system of nonlinear partial differential equations (PDEs) [4].In many cases, nonlinear PDE systems are very difficult to be resolved analytically.Thus an analytic approach can provide a solution which is almost needed.
The analytical approach for solving the nonlinear PDEs was first introduced by Liao in 1992, i.e., the homotopy analysis method (HAM).HAM excellence lies in the selection mechanism of initial values and auxiliary parameters so as to extend the convergence region [5].Earlier version of HAM methods has been applied for various nonlinear problem solving such as the Klein-Gordon equation [6], El Nino Southern Oscillation [7], Huxley [8], Zakharov-Kuznetsov equation [9], and one species growth model in the polluted environment [10].In this article, we review the internal wave issues in the sea that involve tidal forces using the HAM method.The completion of almost this method will be compared to the numerical settlement of error calculations and graphical visualization of the settlement.
The equation used in this study is the Navier-Stokes equation with Boussinesq approximation, in which it is assumed that the internal waves are generated by the interaction between pairs of currents with two-dimensional topography in a nonuniform fluid layer.In this model,  is the density,  0 a reference density,  pressure, and  and  velocity, respectively, in the horizontal and vertical directions where  is time,  is the fluid depth,  is the constant of gravity,  represents the mean of fluid depth, ] is the kinematic viscosity, and   is the tidal force given by In ( 2),  is tidal excursion and it was found that the value of A was less than 10% variation in the measured quantities of this range; the data presented are for A = 20 m;  is the tidal frequency.Another parameter is the height caused by the change in horizontal directional pressure formulated by  = −(/), where  = 2.5 m/s and the Coriolis parameter  = 2Ωsin() depending on the angular velocity earth rotation Ω.

Analysis Method
In this part we illustrate the concept of homotopy method.Suppose that a nonlinear equation is given in the form as below: where N is a nonlinear derivative operator, (, ) is an unknown function,  and  are independent variables, and L is defined as linear operator which satisfies Let  0 (, ) be the initial approach of solving (3);  ∈ [0, 1] is an embedding parameter, ℎ is auxiliary parameter, and () is an additional function.In the frame of the homotopy method, we first construct such a continuous variation (or deformation) (, ; ) that as  increases from 0 to 1, (, ; ) varies from the initial approach  0 (, ) to the solution (, ) of (3).Such kind of continuous variation (or mapping) is governed by the so-called zero-order deformation equation At  = 0, the zero-order deformation equation ( 5) becomes such that  (, ; 0) =  0 (, ) .
Thus, as  increases from 0 to 1, the solution (, ; ) varies continuously from the initial approach  0 (, ) to the exact solution (, ).So, (5) defines a homotopy of function (, ; ) :  0 (, ) ∼ (, ).Such kind of continuous variation is called deformation in topology, and this is the reason why we call (5) the zero-order deformation equation.By using the Taylor expansion from (, ; ) to , the following is obtained where Suppose that given the initial value of  0 (, ), the linear operator L and the auxiliary parameters ℎ are not equal to zero and the auxiliary function () is chosen so that (10) is from (, ; ) convergent at  = 1.Hence, we may assume the following series solution: (, ) =  (, ; 1) =  0 (, ) According to (10), ( 5) can be rewritten as follows: such that By deriving (14) as much as  times with respect to , then the following is obtained: such that where and
Now, the solution of the  th -order deformation equation (24) for m ≥ 1 becomes where According to (10) and ( 18 For simplification, then select ℎ 1 = ℎ 2 = ℎ 3 .Further, the boundary conditions used in the solution of (1) are a polynomial determined by the settlement of HAM.The solution of ( 1) is numerically determined with the aid of a symbolic computing program.The resulting numerical settlement will be compared to the almost-resultant settlement with the HAM.The parameters used for the evaluation need the inclusion of the tidal force parameter   =  0  2 sin , where  is the tidal excursion; in this case  should be less than the channel width (A = 20 m)  0 = 1000 kg/m 3 ; ] = 0.01 m 2 /s is kinematic viscosity, tidal frequency ( =  M2 = 1.4052 × 10 −4 rad/s), and earth's rotational angle velocity Ω = 7.29 × 10 −5 and  = /3 as the constant geostrophic current velocity.Furthermore there is also a constant of gravity  = 9.8 m/s 2 .
In the HAM application, the completion of high-order deformation is determined by (26).The completion of the high-order deformation obtained is the basis of determining the completion of the series.The result of series completion is a function that depends on the values of  and .In this section, the completion of the obtained series is evaluated at a certain  and  value to determine the completion of the HAM.Nearly obtained solutions compared to their Note that (31) contains the auxiliary parameter ℎ.To obtain an appropriate range for ℎ, we consider the ℎ-curves.
Based on Figure 1 we get the value of ℎ = −1.