Nonlinear Interaction of N Conservative Waves in Two Dimensions

Kinematic Fourier (KF) structures, exponential kinematic Fourier (KEF) structures, dynamic exponential (DEF) Fourier structures, and KEF-DEF structures with constant and space-dependent structural coefficients are developed in the current paper to treat kinematic and dynamic problems for nonlinear interaction of N conservative waves in the two-dimensional theory of the Newtonian flows with harmonic velocity. The computational method of solving partial differential equations (PDEs) by decomposition in invariant structures, which continues the analytical methods of undetermined coefficients and separation of variables, is extended by using an experimental and theoretical computation in MapleTM. For internal waves vanishing at infinity, the Dirichlet problem is formulated for kinematic and dynamics systems of the vorticity, continuity, Helmholtz, Lamb-Helmholtz, and Bernoulli equations in the upper and lower domains. Exact solutions for upper and lower cumulative flows are discovered by the experimental computing, proved by the theoretical computing, and verified by the system of Navier-Stokes PDEs. The KEF and KEF-DEF structures of the cumulative flows are visualized by instantaneous surface plots with isocurves. Modeling of a deterministic wave chaos by aperiodic flows in the KEF, DEF, and KEF-DEF structures with 5N parameters is considered.


Introduction
The two-dimensional (2d) Navier-Stokes system of partial differential equations (PDEs) for a Newtonian fluid with a constant density ρ and a constant kinematic viscosity ν in a gravity field g is ( ) where ( ) , 0, u w = v is a vector field of the flow velocity, ( ) is a vector field of the gravitational acceleration, t p is a scalar field of the total pressure, ( ) Equation ( 1) may be written into the Lamb-Pozrikidis form [1] [2] 1 , 2 which sets a dynamic balance of inertial, potential, vortical, and viscous forces, respectively.Using a dynamic pressure per unit mass [3] 0 , where 0 p is a reference pressure, a kinetic energy per unit mass 2, e k = ⋅ v v the 2d Helmholtz decomposition [4] of the velocity field and the vortex force Equation ( 4 respectively.The Lamb-Helmholtz PDE (8) means a dynamic balance between potential and vortical forces of the Navier-Stokes PDE (1), which are separated completely.
A linear part of the kinematic problem for free-surface waves of the theory of the ideal fluid with 0 ν = im- plies the exponential Fourier eigenfunctions [5], which are obtained by the classical method of separation of variables of the 2d Laplace Equation in [4] and [1].This analytical method was recently developed into the computational method of solving PDEs by decomposition into invariant structures.In [3], the Boussinesq-Rayleigh-Taylor structures were developed for topological flows away from boundaries.The trigonometric Taylor structures and the trigonometric-hyperbolic structures [6] were used to describe spatiotemporal cascades of exposed and hidden perturbations of the Couette flow, respectively.In [7], the theory of the invariant trigonometric, hyperbolic, and elliptic structures was constructed and applied for modeling dual perturbations of the Poiseuille-Hagen flow.
To treat linear and nonlinear parts of kinematic and dynamic problems for 2d internal waves in the theory of Newtonian flows with harmonic velocity, kinematic Fourier (KF) structures, exponential kinematic Fourier (KEF) structures, dynamic exponential Fourier (DEF) structures, and KEF-DEF structures with constant structural coefficients are developed in the current paper.The structure of this paper is as follows.In section 2, the kinematic problems for velocity components and dual potentials of the velocity field are formulated in upper and lower domains and treated in the KF and KEF structures.To compute and explore Jacobian determinants (JDs) of the velocity field, the DEF structure is also constructed in this section.In section 3, the dynamic problems for the Bernoulli potential and the total pressure are formulated and computed in the KF, KEF, and KEF-DEF structures.The Navier-Stokes system of PDEs is employed for verification of experimental and theoretical solutions for cumulative upper and lower flows in this section, as well.Visualization and discussion of the developed structures and fluid-dynamic variables is given in section 4, which is followed by a summary of main results in Section 5.

Kinematic Problems for Conservative Flows
The following solutions and admissible boundary conditions for the kinematic problems of section 2 in the KF and DEF structures were primarily computed experimentally in Maple™ by programming with lists of equations and expressions in the virtual environment of a global variable Eqs with 29 procedures of 670 code lines.

Formulation of Theoretical Kinematic Problems for Velocity Components
Theoretical kinematic problems for harmonic velocity components ( ) ( ) k of a Newtonian fluid are given by vanishing the - y component of the vorticity Equation (3) and the continuity Equation ( 2), respectively, To consider nonlinear interaction of N internal, conservative waves with a harmonic velocity field, the cumulative flow is decomposed into a superposition of local flows such that the local vorticity and continuity equations are 0, 0, ) and a vanishing condition as z → −∞ 0.
Thus, an effect of surface waves on the internal waves is described by the Dirichlet conditions ( 16) and (18).Here, a structural notation ( ) ( )  for the upper and lower flows, respectively.Thus, the - x and -z components of the velocity field of the cumulative flows are expanded in the KF structures with constant structural coefficients ( ) ( ) Fw ca Gw sa and the velocity components vanish as z → ±∞ 0, 0, for the upper and lower cumulative flows, respectively.

Theoretical Solutions for the Velocity Field
Theoretical solutions of kinematic problems (11)-(28) are constructed in the KF structure ( ) , , p x z t of two spatial variables , , x z and time t with a general term , n p which in the structural notation may be written as  36) and (38) reduces these ODEs to algebraic equations (AEs) for structural parameters: , .
Substitution of ( 40) and ( 41) in ( 37) and (39) reduces these ODEs to AEs for admissible values of the structural coefficient n c with the following solutions for the upper and lower flows, respectively: , .
Since the admissible values of n c coincide for Equations ( 37) and (39), ODEs for structural coefficients (36)-(39) are compatible both for the upper and lower flows.

The DEF structure and Theoretical Jacobian Determinants of the Velocity Components
Define two KEF structures ( ) , , l x z t and ( ) , , h x z t with general terms n l and m h by using a generalized Einstein notation for summation, which is extended for exponents, , , n m .
Using general terms (48) and (51), summation formula for the product of the KEF structures is written as the DEF structure with the following structural coefficients: , , , e Global JD (55) then becomes .
Thus, the global JD does not vanish for parallel waves with non-vanishing 2 2 .

Jc
vanishes for orthogonal waves with , .
In this case, global JD (55) is reduced to .
Thus, the global JD does not vanish also for orthogonal waves with non-vanishing , , , , , , such that the local Helmholtz PDEs are 0, 0, ) For Equations ( 67)-( 74) to be satisfied exactly for all variables, parameters, and functions of the upper and lower flows: , , , , and the lower flows .
In fluid dynamics, these connections mean that a non-uniform vertical flow generates a horizontal flow and a non-uniform horizontal flow produces a vertical flow.
Similarly, comparison of solutions for n η and n φ with solutions for n u and n w shows that they are also directly proportional, respectively, for the upper flows , , and the lower flows , .
Finally, comparison of solutions for n η and n φ with spatial derivatives in x of n φ and n η shows that they are proportional to each other, respectively, for the upper flows 1 1 , , and the lower flows 1 1 , .
Connections ( 85)-(90) between solutions in the KEF structures are available since there are only two independent combinations of trigonometric structural functions Thus, local isocurves of n η and n φ remain orthogonal for all times in agreement with the Helmholtz Equa- tions (63)-(64).Similarly, local isocurves of n u and n w remain orthogonal since both for the upper and lower flows 0, in agreement with the local vorticity and continuity Equations ( 14)-(15).
Thus, global isocurves of η and φ also remain orthogonal for all times in agreement with the cumulative Helmholtz Equations ( 60)-(61).Finally, global isocurves of u and w remain orthogonal since both for the upper and lower cumulative flows 0, u w u w x x z z in agreement with the cumulative vorticity and continuity Equations ( 11)-( 12).
It is a straightforward matter to show that for the KEF structure ( ) the Laplacian of n p vanishes.Thus, the KEF is an invariant, harmonic structure both for the upper and lower flows.
Application of ( 96)-( 97) to ( 43)-( 46) shows that n u and n w are conjugate harmonic functions since both for the upper and lower flows, in agreement with vector identity By Equations (13), u and w are also conjugate harmonic functions both for the upper and lower cumulative flows, in agreement with vector identity Similarly, applying (96)-( 97) to ( 81)-(84) shows that n η and n φ are conjugate harmonic functions as both for the upper and lower flows, in agreement with 0 By Equation (62), η and φ are also conjugate har- monic functions both for the upper and lower cumulative flows, in agreement with vector identities 0 φ φ The theoretical solutions in the KEF and DEF structures for the kinematic problems of section 2 were computed theoretically in Maple™ by programming with symbolic general terms in the virtual environment of a global variable Equation with 26 procedures of 591 code lines.The theoretical solutions for velocity components (43)-( 46), the products of the KEF structures (52)-(53), and the kinematic potentials (81)-(84) of the upper and lower cumulative flows were justified by the correspondent experimental solutions for 1,3,10.N =

Dynamic Problems for Conservative Flows
The following solutions for the dynamic problems of section 3 in the KF, DEF, and KEF-DEF structures were primarily computed experimentally by programming with lists of equations and expressions in the virtual environment of the global variable Equations with 19 procedures of 472 code lines.

Theoretical Solutions for the Helmholtz and Bernoulli Potentials in the KEF Structures
Theoretical dynamic problems in the KF structures for the Helmholtz and Bernoulli potentials of the cumu- since the cumulative dynamic potentials are again decomposed into the local dynamic potentials as follows: For Equations ( 110)-( 111) to be satisfied exactly for all , , , , , and the lower flows 0.

Theoretical Verification by the System of Navier-Stokes PDEs
The system of the Navier-Stokes PDEs (1)-( 2) in the scalar notation becomes Computation of spatial derivatives of ( 43)-( 46) by ( 32 ) By using (32) and (33), components of the gradient of (133) may be written in the KEF-DEF structures for the upper and lower cumulative flows, respectively, as

Conclusions
The analytical methods of undetermined coefficients and separation of variables are extended by the computational method of solving 2d PDEs by decomposition in invariant structures.The method is developed by the experimental computing with lists of equations and expressions and the theoretical computing with symbolic general terms.The experimental computing of the kinematic and dynamic problems is implemented by 48 procedures of 1142 code lines and the theoretical computing by 41 procedures of 996 code lines.
To compute the upper and cumulative flows for nonlinear interaction of N internal waves in the KF structures, the KEF, DEF, and KEF-DEF structures were treated both experimentally and theoretically.These structures with constant and space-dependent structural coefficients are invariant with respect to various differential and algebraic operations.The structures continue the Fourier series for linear and nonlinear problems with solutions vanishing at infinity and model flows of a deterministic wave chaos with the period that approaches infinity.
The exact solutions of the Navier-Stokes PDEs for the nonlinear interaction of N conservative waves are computed in the upper and lower domains by formulating and solving the Dirichlet problem for the vorticity, continuity, Helmholtz, Lamb-Helmholtz, and Bernoulli equations.The conservative waves are not affected by dissipation since they are derived in the class of flows with the harmonic velocity field.The harmonic relationships between fluid-dynamic variables and their spatial derivatives with respect to x both for upper and lower flows are obtained.
) is reduced to the Lamb-Helmholtz PDE 0 where φ and d are scalar potentials, vector potentials, η and b are pseudovector potentials of v and , × v ω 14)-(15) for the local flows are fulfilled, then substitution of superpositions (13) into (11)-(12) and changing order of summation and differentiation yield that Equations (11)-(12) for the cumulative flow are also satisfied.Upper flows are specified by the Dirichlet condition in the KF structure on a lower boundary

Figure 1 .
Figure 1.Configuration of upper and lower domains for internal, conservative waves.gation coordinate, n ρ is a wavenumber,

2 . 4 .
vanishing.So, both propagating and interacting waves are independent for structural coefficients with Theoretical Solutions for the Pseudovector and Scalar Potentials in the KEF StructuresTheoretical kinematic problems for cumulative pseudo-vector potential ( ) 61)since the potential-vortical duality the velocity field admits two presentations: The cumulative kinematic potentials are decomposed into a superposition of local kine- matic potentials by using (81)-(84) both for the upper and lower flows gives 0.
lative flows are set by the Lamb-Helmholtz PDEs (104) are complemented by the local Lamb-Helmholtz PDEs 0, conditions are again redundant because the problem is formulated in the KF structures.Construct a general term of the Bernoulli potential of the local flows in the KF structure with space-dependent coefficients ( ) ( ) .

η
application of (32)-(33), substitution in (105)-(106), and collection of the structural functions reduce two Lamb-Helmholtz PDEs to the following system of the Lamb-Helmholtz AE and ODE for the upper flows vanish.Thus, the Lamb-Helmholtz AE and ODE are reduced to the following two AEs and two ODEs for space-dependent structural coefficients of remainders of structural approximations (110)-(111) vanish, exact solutions of (112)-(115) produce exact solutions of (105)-(106).Solving AEs (112) and (114) for structural coefficients (116)-(117) in ODEs (113) and (115) reduced them to identities.Substitution of structural coefficients (116)-(117) in super positions (108) and the KF structure (109) gives the cumulative Helmholtz and Bernoulli potentials in the KEF structures for the upper cumulative flow to the kinematic potentials (87)-(88), the dynamic potentials and the velocity components are directly proportional both for the upper and lower flows , .89)-(90), the Helmholtz and Bernoulli potentials and derivatives of the Bernoulli and Helmholtz potentials in x are directly proportional to each other both for the upper flows 1 1 ,

3 . 2 .
Theoretical Solutions for the Total Pressure in the KEF-DEF Structures Theoretical dynamic problems in the KEF-DEF structures for the kinetic energy per unit mass , e k the dynamic pressure per unit mass , d p and the total pressure t p of the cumulative flows are formulated by definition ( ) )-(33) immediately reduces (136) to identity.Temporal derivatives of v in the KEF structures for the upper and lower cumulative flows, respectively, are of (134)-(135) computed by (52)-(53) in the DEF structures for the upper and lower cumulative flows, respectively, become

Figure 3 ..
Figure 3. Kinetic energy (left) and dynamic pressure (right) of the lower cumulative flow.
Computation of local JDs for the velocity components of the upper and lower flow, respectively, yields n Cad , n Cas , n Sad and n Sas , respectively, and a third letter for variable .p  (54) Thus, velocity components n u and n w are independent for non-trivial structural coefficients n Fw and n Gw since the local JDs vanish when The boundary conditions for n η and n ϕ and redundant when the problem is formu- lated in the KF structures.Construct general terms of the kinematic potentials of the local flows in the KF structure with space-dependent coefficients

Harmonic Relationships for the Velocity Components and the Kinematic Potentials
n Gw all coefficients of structural functions n ca and n sa must vanish.Thus, two Helmholtz ODEs and two Helmholtz AEs are reduced to the following four AEs and four ODEs with respect to , n fe , n ge , n fp and n fe , n ge ,