Conservative Interaction of N Internal Waves in Three Dimensions

The Navier-Stokes system of equations is reduced to a system of the vorticity, continuity, Helm-holtz, and Lamb-Helmholtz equations. The periodic Dirichlet problems are formulated for internal waves vanishing at infinity in the upper and lower domains. Stationary kinematic Fourier (SKF) structures, stationary exponential kinematic Fourier (SKEF) structures, stationary dynamic exponential (SDEF) Fourier structures, and SKEF-SDEF structures of three spatial variables and time are constructed in the current paper to treat kinematic and dynamic problems of the three-dimensional theory of the Newtonian flows with harmonic velocity. Two exact solutions for conservative interaction of N internal waves in three dimensions are developed by the method of decomposition in invariant structures and implemented through experimental and theoretical programming in Maple™. Main results are summarized in a global existence theorem for the strong solutions. The SKEF, SDEF, and SKEF-SDEF structures of the cumulative flows are visualized by two-parametric surface plots for six fluid-dynamic variables.


Introduction
The three-dimensional (3d) 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 ( ) (1) 0, ∇ ⋅ = v (2) where ( ) are the gradient and the Laplacian in the Cartesian coordinate system ( ) , , x y z = x of the 3d space with unit vectors ( ) , , i j k ; t is time; ( ) , , u v 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.By a flow vorticity ( ) of the velocity field , ∇ × = v ω (3) 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 , t d p p p ρ − = − ⋅ g x (5) where 0 p is a reference pressure, a kinetic energy per unit mass 2 e k = ⋅ v v , the 3d Helmholtz decomposition [4] of the velocity field , φ = ∇ + ∇ × v ψ (6) , ∇ ⋅ =0 ψ (7) and the vortex force , ∇ ⋅ =0 a (9) Equation ( 4) is reduced to the Lamb-Helmholtz PDE [5] e b ∇ × ∇ + = 0 e h (10) for a scalar Bernoulli potential and a vector Helmholtz potential ( ) are vector potentials of v and × ω v , re- spectively.The Lamb-Helmholtz PDE (10) sets a dynamic balance between potential and vortical forces of the Navier-Stokes PDE (1), which are separated completely.Reduction of (1) to (10) means the potential-vortical duality of the Navier-Stokes PDE for free flows [3] since writing Equation (10) as (13) shows that a virtual force s n of (1) may be represented both in the potential form .For instance, the potential-vortical duality of ( 1)-( 2) results in formation of the wave-vortex structures in surface waves [6]- [8].

, , fe ge he =
The exponential Fourier eigenfunctions were calculated by separation of variables of the 3d Laplace equation, for instance, see [1] and [4], and applied for a linear part of the kinematic problem for free-surface waves in the theory of the ideal fluid with 0 ν = in [9].The analytical method of separation of variables was recently gene- ralized into the computational method of solving PDEs by decomposition into invariant structures.The Boussinesq-Rayleigh-Taylor structures were used to compute topological flows away from boundaries in [3].The trigonometric Taylor structures and the trigonometric-hyperbolic structures were developed in [10] to model spatiotemporal cascades of exposed and hidden perturbations of the Couette flow.In [11], the invariant trigonometric, hyperbolic, and elliptic structures were utilized to treat dual perturbations of the Poiseuille-Hagen flow.The zigzag hyperbolic structures were studied to derive the exact solution for interaction of two pulsatory waves of the Korteweg-de Vries equation in [12].In two dimensions, the stationary kinematic Fourier (SKF) structures with space-dependent structural coefficients, the stationary exponential kinematic Fourier (SKEF) structures, the stationary dynamic exponential Fourier (SDEF) structures, and SKEF-SDEF structures with constant structural coefficients were developed to obtain the exact solutions of the Navier-Stokes system of PDEs for conservative interaction of N internal waves by the experimental and theoretical programming [5].
In the current paper, the SKF, SKEF, SDEF, and SKEF-SDEF structures are extended in three dimensions to examine kinematic and dynamic problems for internal conservative waves in the theory of Newtonian flows with harmonic velocity.The structure of this paper is as follows.The SKF structures are used to compute theoretical solutions for the velocity components in Section 2. Theoretical solutions for the kinematic potentials of the velocity field and the dynamic potentials of the Navier-Stokes PDE are obtained in Sections 3 and 4, respectively, through the SKEF structures.The SDEF structures are constructed in Section 5.In Section 6, the SDEF and SKEF-SDEF structures are used for theoretical computation of the kinetic energy and the dynamic pressure.Decomposition of harmonic variables in a SKEF structural basis is tackled in Section 7. Verification of the experimental and theoretical solutions by the Navier-Stokes system of PDEs and the existence theorem are provided in Section 8. Discussion of significant outcomes and visualization of the developed structures are given in Section 9, which is followed by a summary of main results in Section 10.

Velocity Components in the SKF Structures
The following theoretical solutions and admissible boundary conditions of Sections 2-8 were primarily computed in Maple™ using experimental programming with lists of equations and expressions for numerical indices and N = 3 in the virtual environment of a global variable Eqe by 33 developed procedures of 1748 code lines.
Theoretical problems for harmonic velocity components ( ) To consider conservative interaction of N internal waves, the cumulative flows are decomposed into superpositions of local flows , , , , , , , , , , , , such that the local vorticity and continuity equations are 0,  .An upper cumulative flow is specified by a Dirichlet condition, which is periodic in the x-and y-directions, through the two-dimensional (2d) SKF structure on a lower boundary 0 z = of an upper domain ( ) Fw cc w cs Gw sc Rw ss w Q and a vanishing Dirichlet condition in the z-direction A lower cumulative flow is identified by a periodic Dirichlet condition on an upper boundary 0 z = of a lower domain Fw cc w cs Gw sc Rw ss w Q and a vanishing Dirichlet condition in the z-direction Thus, an effect of surface waves on the internal waves is described by the Dirichlet conditions ( 23) and (25).While notations of boundary coefficients , , , Fw Qw Gw Rw coincide in ( 23) and (25) for computational simplicity, their values are different for the upper and lower flows, which model internal waves produced by surface waves in atmosphere and ocean.In Equations ( 23) and (25), a structural notation Xa Yb are initial coordinates for all n.The experimental solutions show that similar to [5], boundary conditions for , Similarly to w, u and v vanish as z → ±∞ 0, 0, for the upper and lower cumulative flows, respectively.Theoretical solutions of ( 14)-( 26) are constructed in the SKF structure ( ) , , , p p x y z t = of three spatial variables x, y, z and time t with a general term n p , which in the structural notation may be written as where first letters f, q, g, r of space-dependent structural coefficients , Application of ( 32)-( 34) to (31) and substitution in (19)-( 22) reduce the four PDEs to three ordinary differential equations (ODEs) and an algebraic equation (AE).For these equations to be satisfied exactly for all independent variables, independent parameters, structural functions, and structural coefficients of the local flows Substitution of (39) in the second ODEs and addition/subtraction of the first ODEs reduces the second ODEs to identities.Substitution of functional relations (39) into the fourth ODEs reduces them to the following system:  , .
Finally, substitutions of (39) and ( 41)-( 43) into (31) and (18) give the following velocity components of the upper and lower cumulative flows, respectively: ( ) ( ) ( ) ( ) where , and the 3d solution (44) is transformed into a 2d solution in the - where return cosine and sine of local front angles n θ with respect to x-axis.In the general case, the local front angles n θ differ from local celerity angles n ϕ , which are defined by ( ) ( )

Kinematic Potentials through the SKEF Structures
Theoretical problems for the kinematic potentials ( ) , , χ η ψ = ψ and φ of v are set by seven global Helm- holtz PDEs ( 6)-( 7) since a scalar-vector duality the velocity field admits two descriptions: a scalar description φ = ∇ v for = 0 ψ and a vector description = ∇ × v ψ for 0 φ = .The cumulative kinematic potentials are decomposed into super- positions of local kinematic potentials ( The local kinematic potentials are governed by local Helmholtz PDEs 0, The periodic Dirichlet conditions for ψ of the upper and lower cumulative flows are specified on a lower boundary 0 z = of an upper domain , Computation of derivatives of ( 65) by ( 32)-( 34) and substitution in (57)-( 63) reduce the seven Helmholtz PDEs to four Helmholtz ODEs and three Helmholtz AEs.For these equations to be satisfied exactly for all independent variables, independent parameters, structural functions, boundary coefficients, and structural coefficients of the upper and lower flows x, y, z, t all coefficients of the kinematic structural functions must vanish.Vanishing four coefficients of seven equations yields 28 equations in total for the upper flows and 28 equations for the lower flows.For ( ) , , , 16 equations are separated into four systems of four equations each with respect to four groups of the SKF structural coefficients  , for the upper and lower flows, respectively, ( where first and second ODEs are produced by ( 57)-(58), third AEs are generated by (59), and fourth ODEs are created by (60).
For n φ , 12 equations are separated into four systems of three equations each with respect to four SKF struc- tural coefficients  for the upper and lower flows, respectively, ( ) ( ) ( ) where first and second AEs are produced by ( 61 Substitution of (74) in the second ODEs of (66)-(69) and addition/subtraction of the first ODEs reduces the second ODEs to identities.Substitution of (74) into the fourth ODEs reduces them to the following system: Solving the first AEs of separated systems (70)-( 73) gives structural coefficients Substitution of (76) in the second AEs and third ODEs of ( 70)-( 73) reduces them to identities.Construct solutions of the first ODEs of (66)-( 69) and (75) in the SE structures with the following general terms for the upper and lower flows, respectively, Substitution of ( 77) and ( 78) into (75) reduces them to identities.Finally, substitutions of (74), ( 76)-( 78) into (65) and (56) give the following kinematic potentials in the SKEF structures for the upper and lower cumulative flows, respectively, where structural coefficients are given by (78) and Since the cumulative dynamic potentials are decomposed into superpositions of local dynamic potentials the local dynamic potentials are governed by three definitions and three Lamb-Helmholtz PDEs Application of (93) to ( 87) and (79) gives the following Helmholtz potentials in the SKEF structures for the upper and lower cumulative flows, respectively, ( where structural coefficients are Computation of derivatives of (97) and ( 91) by ( 94)-( 96) and ( 32)-(34), respectively, and substitution in (88)-(90) reduce the three Lamb-Helmholtz PDEs to two Lamb-Helmholtz AEs and one Lamb-Helmholtz ODE.For these equations to be satisfied exactly for all independent variables, independent parameters, structural functions, boundary coefficients, and structural coefficients of the upper and lower flows x, y, z, t Substitution of (106) in the second AEs and third ODEs of ( 101 f t vanishes for the SKEF structures because of the vanishing Dirichlet conditions (24) and (26).Using the scalar-vector duality, the Lamb-Helmholtz PDE (10) where ψ = ψ ψ .Integration of (108) returns dual formulas for the global and local Bernoulli potentials , be t since an integration constant again vanishes for the SKEF structures in agreement with (24) and (26).Therefore, computation of be by (110), (79), and (93) also results in (105)-(106).

Stationary Dynamic Exponential Fourier (SDEF) Structures
By the generalized Einstein notation for summation that is extended for exponents in [5], define two SKEF structures ( ) , , , l x y z t l = and ( ) , , , h x y z t h = with general terms n l and m h for the upper and lower cumulative flows, respectively, (    , , ,  p x y z t p lh = = by one-dimensional summation of diagonal terms yields for the upper and lower flows, respectively, ( where structural coefficients are , ,

Gssp SsCs Rsdp SsSd
Rssp SsSs r r z Conversion of ( 115)-( 116) by triangular summation of non-diagonal terms and addition of ( 113)-( 114) yields summation formulas for the product of the SKEF structures written through the SDEF structures for the upper and lower flows, respectively, (

Total Pressures through the SKEF-SDEF Structures
The kinetic energy per unit mass of the upper and lower cumulative flows, ( ) ( ) is computed as a superposition of three products of the SKEF structures for the velocity components that are converted from the functional form (44) to the structural form (92) for the upper and lower cumulative flows, respectively, as where the structural coefficients for the upper and lower flows, respectively, structural parameters n C and n S are given by (47).Application of the product rules (117)-(118) for transformation of the SKEF structures into the SDEF structures to the cumulative kinetic energy of the upper and lower flows, respectively, yields where the structural coefficients are ( 1 Here, structural parameters of energy pulsations Since the velocity components have a unique presentation both for the vector and scalar descriptions of the kinematic and dynamic potentials, the kinetic energies also have a single description.
Substitution of the dual formula (109) for the Bernoulli potential be into the Bernoulli Equation (11) in the vector description returns the same expression for the dynamic pressure as the Bernoulli Equation in the scalar description ( p x y z t be x y z t k x y z t = − (125) Thus, the kinetic energies, the dynamic pressures, and the total pressures have a unique presentation both in the vector and scalar descriptions.
Substitution of the dynamic pressure in the hydrostatic Equation ( 5) yields the total pressure for the upper and lower cumulative flows, respectively, ( where structural coefficients are given by ( 106) and (123).When 0 n σ = , (126) is reduced to the 2d solution in the - x z plane [5] ( In ( 127)-( 128

Decomposition of Harmonic Variables in the SKEF Structural Basis
Similar to two independent SKEF structures in two dimensions [5], there are four independent SKEF structures in three dimensions  Fa Qa Ga Ra are structural coefficients.Computation of the spatial and temporal derivatives of the general terms of (129) by ( 93)-(96) yields for the upper and lower flows, respectively, , , Thus, there are three couples of symmetric isosurfaces with the same absolute values of the scalar products of gradients.In two dimensions, the non-orthogonal isosurfaces are reduced to orthogonal isolines [5].
For velocity components (44), the non-orthogonal harmonic SKEF structural basis is Aw Bw Cw are harmonic functions, the local and cumulative velocity components are also harmonic both for the upper and lower flows, in agreement with 0, 0, 0, 0, 0, 0.
Taking the temporal derivatives of (44) by ( 93) and reducing to the SKEF structures using (47) give For kinematic potentials (79) and dynamic potentials (97) and (105), the non-orthogonal harmonic SKEF Since the SKEF structural basis is harmonic, there are 11 pairs of local and global functions ( ) , n be be , which are harmonic together with all their spatial and temporal derivatives.

Experimental and Theoretical Verification by the System of Navier-Stokes PDEs
The classical proofs of existence theorems for series solutions of PDEs, see existence theorems of [3] and references therein, include three following steps: 1) to derive formal solutions, 2) to show that PDEs are satisfied, and 3) to find conditions of convergence.For the structural solutions of this paper, the first step is implemented in Sections 2-7, the second step is the point of this section, and the third step is not required since the structural solutions are exact and decompositions in the invariant structures are truncated.The Navier-Stokes PDEs (1) in the scalar notation become Theoretical computation of the directional derivatives of ( 146)-( 148) by ( 120), ( 94)-(96), and (117)-(118) as a superposition of three products of the SKEF structures, which is reduced to the SDEF structure, yields for the upper and lower cumulative flows, respectively, (   In the vector description, the pressure force of (146)-( 148) is reduced by (5)   tions.Invariance of the SKEF and SDEF structures with respect to differentiation is shown.The nonlinear algebra of the SKEF structures is developed.The non-orthogonal SKEF structural basis for harmonic functions is constructed and decompositions of the fluid-dynamic variables in this basis are obtained.The conservative system of N internal waves is neutrally stable with respect to M wave perturbations, and it is not affected by viscous dissipation.The computational method of decomposition in invariant structures continues the analytical methods of separation of variables, undetermined coefficients, and series expansions [3] [5] [10]- [12].In the current paper, the method of decomposition in invariant structures is extended into three dimensions.By this method, the vorticity and continuity PDEs are reduced to 12 homogeneous ODEs of first order and four linear AEs, the Helmholtz PDEs to 12 inhomogeneous ODEs of first order and 16 linear AEs, the Lamb-Helmholtz PDEs to four inhomogeneous ODEs of first order and eight linear AEs, and the Bernoulli equation is reduced to a linear equation in the SKEF and SDEF structures both for the upper and lower flows.To summarize, the system of four Navier-Stokes PDEs is reduced to the linear system of 57 equations, including 28 ODEs and 29 AEs.
Experimental discovery and theoretical proof of the exact solutions are implemented through experimental programming in Maple™ with lists of equations and expressions for numerical indices and numeric N = 3 by 33 developed procedures of 1748 code lines and theoretical programming with symbolic general terms, symbolic indices, and code-generated names of structural variables for symbolic N by 33 developed procedures of 1938 code lines.The developed procedures allow for theoretical derivations in the environment of novel COMP MATH data structures-the SKEF, SDEF, and SKEF-SDEF structures that are extended in this paper into three dimensions.
Newtonian fluid are given by vanishing the -, x -, y z components of the vorticity Equation (3) and the continuity Equation (2), respectively,

Figure 1 .
Figure 1.Configuration of upper and lower domains for conservative waves.
and the 3d solution (44) is reduced to the 2d solution in the -x z plane[5] Rs are given boundary coefficients.The experimental solutions show that similar to (28)-(29), the vanishing Dirichlet conditions for ψ and the periodic and vanishing Dirichlet conditions for , , χ η φ are redundant.Construct general terms of the kinematic potentials of the upper and lower flows in the SKF structures by n n fh z cc qh z cs gh z sc rh z ss fe z cc qe z cs ge z sc re z ss fs z cc qs z cs gs z sc rs z ss fp z cc qp z cs gp z sc rp z ss

4 .
, (78)-(81) are reduced to the 2d solution in the x-z plane [5] , (78)-(81) are transformed into a 2d solution in the y-z plane Dynamic Potentials through the SKEF StructuresDefinitions of the dynamic vector potential (the Helmholtz potential) problems for the dynamic scalar potential (the Bernoulli potential) b be ψ = in the vector description are set by three components of the global Lamb-Helmholtz PDEs(10) )-(104) reduces them to identities.The Bernoulli potential (105)-(106) does not depend on boundary coefficients of the kinematic potentials , since ψ of (79) generates the SKEF solution of the homogeneous problem ∇ × = 0 function of time ( ) solutions, construct 16 trigonometric structural functions of the SDEF structure where capital letters C and S stand for cosine and sine, letters s and d for sum and difference of arguments , in the algebraic and trigonometric forms are (126) is converted into the 2d solution in the - y z plane of the SKEF structural basis by (140) and substitution in (144) returns local relationships between the dynamic potentials and the temporal derivatives of the velocity components

Figure 3 .
Figure 3. Kinetic energy (a) and dynamic pressure (b) of the lower cumulative flow.
ss and a second letter to the expanded variable p.General terms of the velocity components of the local flows in the structural notation become , , n Xa , n Yb , n Cx , n Cy , n ρ , n σ , n cc , n cs , n sc , n ss , n Fw , n Qw , n Gw , n Rw , n fh , n qh , n gh , n rh , n fe , n qe , n ge , n re , n fs , n qs , n gs , n rs , n fp , n qp , n gp , n rp fe ge he be depend on the boundary parameters of w and ψ .Construct a general term of the Bernoulli potential in the SKF structure by , , , of the kinematic structural functions must vanish.Vanishing four coefficients of three equations yields 12 equations in total for the upper flows and 12 equations for the lower flows, which are separated into four systems of three equations each with respect to four SKF structural coefficients AEs are produced by (88)-(89) and third ODEs are generated by (90).Solving the first AEs of separated systems (101)-(104) gives the following Bernoulli potentials in the SKEF structures for the upper and lower cumulative flows, respectively, n Xa , n Yb , n Cx , n Cy , n ρ n cs , n sc , n ss , n FF , n QF , n GF , n RF , n FG , n QG , n GG , n RG , n FH , n QH , n GH , n RH , n fB , n qB , n gB , n rB all coefficients , . 2 , .
143)Decomposition of the dynamic potentials in the SKEF structural basis (133), (141) gives for the upper and lower flows, respectively,