Deterministic Chaos of N Stochastic Waves in Two Dimensions

Kinematic exponential Fourier (KEF) structures, dynamic exponential (DEF) Fourier structures, and KEF-DEF structures with time-dependent structural coefficients are developed to examine kinematic and dynamic problems for a deterministic chaos of N stochastic waves in the two-dimensional theory of the Newtonian flows with harmonic velocity. The Dirichlet problems are formulated for kinematic and dynamics systems of the vorticity, continuity, Helmholtz, Lamb-Helmholtz, and Bernoulli equations in the upper and lower domains for stochastic waves vanishing at infinity. Development of the novel method of solving partial differential equations through decomposition in invariant structures is resumed by using experimental and theoretical computation in MapleTM. This computational method generalizes the analytical methods of separation of variables and undetermined coefficients. Exact solutions for the deterministic chaos of upper and lower cumulative flows are revealed by experimental computing, proved by theoretical computing, and justified by the system of Navier-Stokes PDEs. Various scenarios of a developed wave chaos are modeled by 3N parameters and 2N boundary functions, which exhibit stochastic behavior.


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 ( ) (1) 0, ∇ ⋅ = v (2) where ( ) 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, ( ) are the gradient and the Laplacian in the 2d Cartesian coordinate system ( ) , 0, x z = x of the three-dimensional (3d) space with unit vectors ( ) , , i j k , respectively, and t is time.By a flow vorticity ( ) of the velocity field , ∇ × = v ω (3) Equation ( 1) may be written into the Lamb-Pozrikidis form [1] [2] 1 0, 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 , φ = ∇ + ∇ × v ψ (6) and the vortex force Equation ( 4) is reduced to the Lamb-Helmholtz PDE [5] 0 e e b ∇ + ∇ × = h (8) for a scalar Bernoulli potential and a vector Helmholtz potential, respectively, ω v , 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.Reduction of (1) to (8) means the potential-vortical duality of the Navier-Stokes PDE since writing equation (8) as shows that a virtual force s n of (1) may be represented both in the potential form s e b = −∇ n and the vortical form s e = ∇ × n h .The exponential Fourier eigenfunctions obtained by the classical method of separation of variables of the 2d Laplace equation in [1] and [4] were primarily used for a linear part of the kinematic problem for free-surface waves of the theory of the ideal fluid with 0 ν = in [6].This analytical method was recently developed into the computational method of solving PDEs by decomposition into invariant structures.Topological flows away from boundaries were computed by the Boussinesq-Rayleigh-Taylor structures in [3].Spatiotemporal cascades of exposed and hidden perturbations of the Couette flow were modeled by the trigonometric Taylor structures and the trigonometric-hyperbolic structures, respectively, in [7].Dual perturbations of the Poiseuille-Hagen flow were treated by the invariant trigonometric, hyperbolic, and elliptic structures in [8].Exact solutions for the conservative interaction of N internal waves were recently obtained by experimental and theoretical computing with kinematic Fourier (KF) structures with space-dependent structural coefficients and exponential kinematic Fourier (KEF) structures, dynamic exponential Fourier (DEF) structures, and KEF-DEF structures with constant structural coefficients in [5].
To examine linear and nonlinear parts of kinematic and dynamic problems for 2d stochastic waves in the theory of Newtonian flows with harmonic velocity, the KEF structures, the DEF structures, and the KEF-DEF structures with time-dependent structural coefficients are developed in the current paper.The structure of this paper is as follows.The KEF structures are used to compute theoretical solutions of the kinematic problems for the velocity components and the dual potentials of the velocity field in Section 2. The KEF and KEF-DEF structures are employed for theoretical computation of the dynamic problems for the Helmholtz and Bernoulli potentials, the kinetic energy, and the total pressure in Section 3. Verification of the experimental and theoretical solutions is also provided in Section 3. Various scenarios of a developed wave chaos are treated in Section 4. A summary of main results is given in Section 5.

Kinematic Problems for Internal Waves
The following solutions and admissible boundary conditions for the kinematic problems of Section 2 in the KEF and DEF structures with time-dependent coefficients were primarily computed via experimental programming techniques, which use lists of equations and expressions of Maple™ in the virtual environment of a global variable Eqs with 25 procedures of 600 code lines in total.

Formulation of Theoretical Kinematic Problems for the Velocity Field
Theoretical kinematic problems for harmonic velocity components ( ) ( ) Newtonian fluid are given by vanishing the y-component of the vorticity Equa- tion (3) and the continuity Equation ( 2), respectively, To consider a deterministic chaos of N internal, stochastic waves, the cumulative flow is decomposed into a superposition of local flows , , , , , ,   , and a vanishing condition as z → ∞ 0.
A lower cumulative flow is identified by the Dirichlet condition on an upper boundary 0 z = of a lower do- main ( ) , and a vanishing condition as z → ∞ 0.
Thus, an effect of surface waves on the internal waves is described by the Dirichlet conditions (15) and (17 .
Similarly to w, u vanishes as z → ±∞ 0, 0, for the upper and lower cumulative flows, respectively.

Theoretical Solutions for the Velocity Field
Theoretical solutions of kinematic problems (12)-(18) are constructed in the KEF structure ( ) , , p x z t of two spatial variables x, u and time t with a general term n p , which in the structural notation may be written as , , , , exp , where signs "−" and "+" of the exponential term refer to the upper and lower flows, respectively, first letters f and g of structural coefficients while vanishing boundary conditions ( 16) and ( 18) are obviously satisfied.

The DEF Structure and Theoretical Jacobian Determinants of the Velocity Field
Define two KEF structures ( ) , , l x z t and ( ) , , h x z t with general terms n l and m h by using the generalized Einstein notation for summation that is extended for exponents in [5] ( ) Following [5], define structural functions computed by rectangular summation of non-diagonal terms becomes By ( 37) and (39), summation formula for the product of the KEF structures may be written as the DEF structure with time-dependent structural coefficients with the following structural coefficients: Thus, the global JD does not vanish for parallel waves with non-vanishing ( ) ( ) .
So, the global JD does not vanish also for orthogonal waves with non-vanishing .
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 , , , , , , such that the local Helmholtz PDEs are 0, 0, the lower flows Substitution of solutions (59) in AEs (56) and (58) reduces them to identities.Finally, substitution of structural coefficients (59) in the KEF structures (53) (54) and superpositions (50) yields the cumulative kinematic potentials in the KEF structures for the upper and lower cumulative flows, respectively, ( The theoretical solutions in the KEF and DEF structures for the kinematic problems of Section 2 were computed utilizing theoretical programming methods with symbolic general terms in the virtual environment of a global variable Eqt with 21 procedures of 522 Maple code lines in total.The theoretical formulas for velocity components (34), the products of the KEF structures (40) (41), and the kinematic potentials (60) of the upper and lower cumulative flows were justified by the correspondent experimental solutions for N = 1, 3, 10.

Dynamic Problems for Internal Waves
The following solutions for the dynamic problems of Section 3 in the KEF, DEF, and KEF-DEF structures were primarily computed by experimental programming with lists of equations and expressions in the virtual environment of the global variable Eqs with 18 procedures of 470 code lines in total.

Theoretical Solutions for the Dynamic Potentials in the KEF Structures
Equations ( 61) are complemented by the local Lamb-Helmholtz PDEs 0, 0 , , since the cumulative dynamic potentials are again decomposed into the local dynamic potentials , , , , , . N and the lower flows Substitution of solutions (69) in AEs (66) and (68) reduces them to identities.Eventually, substitution of structural coefficients (69) in the KEF structure (64) and superpositions (63) returns the cumulative dynamic potentials in the KEF structures for the upper and lower cumulative flows, respectively,

Theoretical Solutions for the Total Pressure in the KEF-DEF Structures
for the upper and lower cumulative flows, respectively.Substitution of ( 71), (75), and ( 73) in ( 74) yields for the upper and lower cumulative flows, respectively.So, the kinetic energy is obtained in the DEF structures, the dynamic pressure is expressed in the KEF-DEF structures, and the total pressure is computed in the KEF-DEF and polynomial structures.

Harmonic Relationships between the Kinematic and Dynamic Variables
Similar to the invariant trigonometric, hyperbolic, and elliptic structures [8], there are two pairs of independent KEF structures: generating structures with general terms Expressing velocity components (34), kinematic potentials (60), and dynamic potentials (70) (71) through the generating and complementary structures (77) (78) and solving for n As Taking derivatives of (34) and (60) with respect to x, z and solving for n As and n Bs yields differential relationships, which extend algebraic ones (79), In fluid dynamics, relationships (79) (80) mean that a harmonic flow, which is non-uniform in - x or -z directions, produces a complementary flow in z-or x-directions, respectively.
Computing velocity components (34) and dynamic potentials (70) (71) through the generating and complementary structures (77) (78), taking temporal and spatial derivatives, and solving for n be and n he returns , .
Differential relationships (81) mean that spatial derivatives of the dynamic potentials generate temporal rates of a harmonic flow.

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 (34) by ( 26)-( 27) reduces (87) to identity both for the upper and lower cumulative flows.
Temporal derivatives of v in the KEF structures for the upper and lower cumulative flows, respectively, are The directional derivatives of (86) computed by ( 40)-(41) in the DEF structures for the upper and lower cumulative flows, respectively, become By using ( 26)-( 27), components of the gradient of (76) may be written in the KEF-DEF structures for the upper and lower cumulative flows, respectively, as 0, 0, where n R is the Reynolds number.In agreement with [7], ( ) Fd x τ is composed of an invariant structure with even indices of hyperbolic se- cant-tangent temporal modes and trigonometric sine spatial modes (HETO structure) and an invariant structure of the same temporal modes combined with trigonometric cosine spatial modes (HETE structure) Gd x τ is formed of an invariant structure with odd indices of hyperbolic secant-tangent temporal modes and trigonometric sine spatial modes (HOTO structure) and an invariant structure of the same temporal modes composed with trigonometric cosine spatial modes (HOTE structure) , Here, a structural notation and recurrent relations In ( ) and spatial amplitudes These equations guarantee that absolute values of remainders of the structural approximations of (94) do not exceed 2 c d for all , d n x , and τ , where

Conclusions
The computational method of solving PDEs by decomposition in invariant structures, which continues the analytical methods of separation of variables and undetermined coefficients, is generalized in the current paper at the KEF, DEF, and KEF-DEF structures with time-dependent coefficients.This computational method is implemented in the kinematic and dynamic problems for internal waves by 43 procedures with 1070 code lines of the experimental computing in total and 35 procedures with 932 code lines of the theoretical computing in total.These structures with time-dependent structural coefficients are invariant with respect to various differential and algebraic operations.
For internal waves vanishing at infinity in the upper and lower domains, the Dirichlet problems are formulated for the kinematic and dynamic systems of the vorticity, continuity, Helmholtz, Lamb-Helmholtz, and Bernoulli equations.The exact solutions of the Navier-Stokes PDEs for the deterministic chaos of N stochastic waves are revealed experimentally, proved theoretically, and justified by the system of Navier-Stokes PDEs in the class of flows with the harmonic velocity field.The kinematic and dynamic solutions for stochastic waves coincide with the correspondent solutions for conservative waves [5] when stochastic boundary functions are reduced to constants.
Independence of both propagating and interacting internal waves is shown by computation of the Jacobian determinants in the DEF structures.Conditions for existence of parallel and orthogonal waves with time-dependent amplitudes are obtained through the Jacobian determinants, as well.The harmonic relationships between six pairs of the harmonic, fluid-dynamic variables, their temporal derivatives, and their spatial derivatives with respect to x and z are derived both for the upper and lower flows.
The stochastic boundary functions are constructed from the stochastic solutions of the one-dimensional Navier-Stokes equation [7] with hyperbolic temporal modes and trigonometric spatial modes in the HETO, HETE, HOTO, and HOTE structures.Various scenarios of a developed wave chaos are modeled by 3N parameters for internal waves and 2N stochastic boundary functions, which depend on ( )( )  parameters, where K is a number of temporal modes.
vector potentials, η and b are pseu- dovector potentials of v and × An upper cumulative flow is specified by the Dirichlet condition in the KF structure on a lower boundary 0 z = of an upper domain ( ) , x ∈ −∞ ∞ and [ ) 0, z ∈ ∞ (see Figure 1).

Figure 1 .
Figure 1.Configuration of upper and lower domains for stochastic waves.
C and S stand for dynamic structural functions cosine and sine, letter a for arguments n α , m α , letters s and d for sum and difference of arguments n α and m Theoretical dynamic problems in the KF structures for the Helmholtz and Bernoulli potentials of the cumulative flows are set by the Lamb-Helmholtz PDEs (8)-(10) in the vortical presentation with 0

n
Ds for the upper and lower flows, respectively, (

Figure 3 .
Figure 3. Scalar potential (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 Sas , respectively, and a third letter for variable p. .

. Theoretical Solutions for the Kinematic Potentials in the KEF Structures
Application of (26) (27) to (53) (54), substitution in the Helmholtz PDEs (51) (52), collection of the structural functions, and vanishing their coefficients reduce four Helmholtz PDEs to the following system of eight Helmholtz AEs for the upper flows