The Application of the Generalized Differential Formulation of the First Law of Thermodynamics for Evidence of the Tidal Mechanism of Maintenance of the Energy and Viscous-Thermal Dissipative Turbulent Structure of the Mesoscale Oceanic Eddies

The practical significance of the established generalized differential formulation of the first law of thermodynamics (formulated for the rotational coordinate system) is evaluated (for the first time and for the mesoscale oceanic eddies) by deriving the general (viscous-compressible-thermal) and partial (incompressible, viscous-thermal) local conditions of the tidal maintenance of the quasi-stationary energy and dissipative turbulent structure of the mesoscale eddy located inside of the individual fluid region τ of the thermally heterogeneous viscous (compressible and incompressible, respectively) heat-conducting stratified fluid over the two-dimensional bottom topography ( ) h x characterized by the horizontal coordinate x along a horizontal axis X. Based on the derived partial (incompressible) local condition (of the tidal maintenance of the quasi-stationary energy and viscous-thermal dissipative turbulent structure of the mesoscale eddy) and using the calculated vertical distributions of the mean viscous dissipation rate per unit mass ( ) , dis v z ε and the mean thermal dissipation rate per unit mass ( ) , dis t z ε in four regions near the observed mesoscale (periodically topographically trapped by nearly two-dimensional bottom topography) eddy located near the northern region of the Yamato Rise in the Japan Sea, the combined analysis of the energy structure of the eddy and the viscous-thermal dissipative structure of turbulence is presented. The convincing evidence is presented of the tidal mechanHow to cite this paper: Simonenko, S.V. and Lobanov, V.B. (2018) The Application of the Generalized Differential Formulation of the First Law of Thermodynamics for Evidence of the Tidal Mechanism of Maintenance of the Energy and Viscous-Thermal Dissipative Turbulent Structure of the Mesoscale Oceanic Eddies. Journal of Modern Physics, 9, 357-386. https://doi.org/10.4236/jmp.2018.93026 Received: December 22, 2017 Accepted: February 4, 2018 Published: February 7, 2018 Copyright © 2018 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access S. V. Simonenko, V. B. Lobanov DOI: 10.4236/jmp.2018.93026 358 Journal of Modern Physics ism of maintenance of the eddy energy and viscous-thermal dissipative structure of turbulence (produced by the breaking internal gravity waves generated by the eddy) in three regions near the Yamato Rise subjected to the observed mesoscale eddy near the northern region of the Yamato Rise of the Japan Sea.

the observed mesoscale (periodically topographically trapped by nearly two-dimensional bottom topography) eddy located near the northern region of the Yamato Rise in the Japan Sea, the combined analysis of the energy structure of the eddy and the viscous-thermal dissipative structure of turbulence is presented.The convincing evidence is presented of the tidal mechan-

Introduction
It is well known that the problem of turbulence is "the last great unsolved problem of classical physics" [1], the solution of which has the practical significance for humankind.Based on the assumption of the local thermodynamic equilibrium [2], De Groot and Mazur [3], and Gyarmati [4] defined the macroscopic kinetic energy per unit mass k ε as the sum of the macroscopic translational kinetic energy per unit mass t ε and the macroscopic internal rotational kinetic energy per unit mass r ε .We derived [5] the formula for the macroscopic ki- netic energy per unit mass k ε generalizing the classical expression k t r ε ε ε = + [3] [4] by taking into account the shear component of the macroscopic continuum motion related with the rate of strain tensor ij e [1] [5].The macroscopic kinetic energy per unit mass k ε is presented [5] as the sum of the macroscopic translational kinetic energy per unit mass t ε [3] [4] [5] [6] and three Galilean invariants: the classical macroscopic internal rotational kinetic energy per unit mass r ε [3] [4], the established [5] macroscopic non-equilibrium internal shear kinetic energy per unit mass s ε and the established [5] macroscopic non-equilibrium internal kinetic energy of a shear-rotational coupling per unit mass , coup s r ε with a small correction res ε .The generalized formula [5] for the macroscopic kinetic energy per unit mass k ε was the basis of the non-equilibrium statistical thermohydrodynamic theory [5] [7] [8] [9] [10] [11] of the three-dimensional isotropic homogeneous small-scale dissipative turbulence.The physical correctness of the non-equilibrium statistical thermohydrodynamic theory was demonstrated [5] [7]- [13] for laboratory and oceanic three-dimensional isotropic homogeneous small-scale dissipative stratified turbulence in the wide range of the energy-containing length scales from the inner Kolmogorov length scale [14] to the length scales proportional to the Ozmidov length scale [5].
The classical Gibbs' differential formulation [2] [15] of the first law of thermodynamics was generalized [7] [8] [9] [16]- [21] (for the small [7] and for the finite continuum regions τ considered in the Galilean frame of reference) by taking into account (along with the classical [2]  of the internal thermal energy U τ ) the infinitesimal increment dK τ of the macroscopic kinetic energy K τ (which contains (for the for the small conti- nuum region τ [5]  [7]) the classical macroscopic translational kinetic energy [3] [4], the classical macroscopic internal rotational kinetic energy [3] [4], the established [5] macroscopic non-equilibrium internal shear kinetic energy and the established [5] macroscopic non-equilibrium internal kinetic energy of a shear-rotational coupling), the infinitesimal increment d τ π of the gravitational potential energy τ π , the generalized expression for the infinitesimal work , np A τ δ ∂ [7] done by the non-potential terrestrial stress forces (characterized by general symmetric stress tensor T [4]) acting on the boundary surface τ ∂ of the continuum region τ , the infinitesimal increment dG (which is not pre- sented in the generalized differential formulation [7] of the first law of thermodynamics for the small continuum region τ ) of energy due to the combined cosmic and terrestrial non-stationary energy gravitational influence dG on the continuum region τ .We founded the generalized thermohydrogravidynamic model [9] [16] [17] [18] of the earthquake focal region based on the generalized differential formulation [9] [16]- [21] of the first law of thermodynamics and using the generalized expression for the infinitesimal work [20]) together with the generalized ex- [18] for the instantaneous macroscopic kinetic energy K τ of the small macroscopic individual continuum region τ .We founded [9] [21] also the generalized differential formulation of the first law of thermodynamics for the deformed one-component individual finite continuum region τ (considered in the rotational coordinate system K related with the rotating Earth) subjected to the non-stationary Newtonian terrestrial gravitational field, the tidal, Coriolis and centrifugal forces, and non-potential terrestrial stress forces (characterized by general symmetric stress tensor T [4]) acting on the boundary surface τ ∂ of the individual finite continuum region τ .It was pointed out [21] that the generalized differential formulation of the first law of thermodynamics [9] [16] [17] [18] [19] [20] (for the Galilean frame of reference) is preferable (with respect to the derived generalized differential formulation of the first law of thermodynamics [9] [21] formulated for the rotational coordinate system) for consideration of the regional and global seismotectonic activity of the Earth since it gives the possibility to not consider the variable (in time and space) tidal, Coriolis and centrifugal forces acting on the individual finite continuum region τ of the Earth.However, in this article we shall consid- er (for the first time) the established [9] [21] generalized differential formulation of the first law of thermodynamics (formulated for the rotational coordinate system K related with the rotating Earth) for analysis of the energy and dissipative structure of the mesoscale eddy observed [22] in the northwestern part of the Japan Sea near the Yamato Rise.The aim of this article is to bring out the practical significance of the established generalized differential formulation of the first law of thermodynamics [9] [21] (formulated for the rotational coordi-nate system K related with the rotating Earth) for foundation of the tidal mechanism (related with cosmic non-stationary gravitational field of the Moon) of maintenance of the quasi-stationary energy and dissipative turbulent (not isotropic and not homogeneous) structure of the mesoscale oceanic eddies (especially, located near the Yamato Rise of the Japan Sea [22]).To do this, in Section 2 we present the equivalent generalized differential formulations (11) and (17) of the first law of thermodynamics [9] [21] for the deformed one-component individual finite continuum region τ (considered in the rotational coordinate sys- tem K related with rotating Earth) subjected to the non-stationary Newtonian terrestrial gravitational field, the tidal forces (related with the cosmic non-stationary gravitational field), the Coriolis and centrifugal forces, and the non-potential terrestrial stress forces acting on the boundary surface τ ∂ of the individual finite continuum region τ .Based on the established [9] [21] genera- lized differential formulation (17) of the first law of thermodynamics and the related evolution Equation ( 18) for the total mechanical energy ( ) In Section 5, we present also the calculated mean (for all stations in each considered region in the vicinity of the mesoscale eddy) vertical distributions ( ) dis z ε (of the mean viscous-thermal dissipation rates per unit mass ) characterizing the vertical viscous-thermal dissipa-tive structure of turbulence in four regions in the vicinity of the mesoscale eddy.
Based on the partial (incompressible) local condition (30), in Section 6, we present the combined analysis of the energy and viscous-thermal dissipative structure of turbulence in the mesoscale (periodically topographically trapped [22] [25] [26]) eddy located near the northern region of the Yamato Rise in the Japan Sea.In Section 7, we present the summary of main results and conclusion.

The Generalized Differential Formulation of the First Law of Thermodynamics for the Rotational Coordinate System Related with the Rotating Earth
Let us consider an individual finite continuum region τ (characterized by the closed continual boundary surface τ ∂ ), which moves in the three-dimensional Euclidean space with respect to rotational Cartesian coordinate system K ( ( )

, K K C
≡ Ω ) related with the rotating Earth (see Figure 1).The rotational Cartesian coordinate system K is centred at the mass center 3 C of the rotating Earth and is determined by the axes 1 2 3 , , X X X (see Figure 1) defined by the unit normal coordinate vectors 1 2 3 , , µ µ µ , respectively.
The local hydrodynamic velocity ( ) is determined by the general equation of continuum movement (for the rotational coordinate system K) [9] [21]: where, d dt v is the total acceleration of the physically infinitesimal continuum arbitrary symmetric stress tensor [4], ρ is the local density of mass distribu- tion, is the local terrestrial gravitational acceleration of the non-stationary gravitational field of the Earth, ter ψ is the non-stationary terrestrial gravitational potential, Ω is the angular velocity vector of the Earth's ro- tation, r is the vector characterizing the physically infinitesimal continuum element.According to the general Equation (1), the moving rotating deforming heat-conducting stratified one-component individual finite continuum region τ is subjected to the terrestrial force 1 div ρ T , the terrestrial non-stationary Newtonian gravitational field characterized by the local terrestrial gravitational acceleration [27] and the classical tidal force tidal F [28], which is related predominantly with the non-stationary gravitational field of the Moon and the Sun.
The pressure tensor = − P T [4] is given by the decomposition [3]: defined by the delta-tensor δ , the thermodynamic pressure p and the viscous-stress tensor Π [3].The differential formulation of the first law of ther- modynamics for the one-component deformed continuum element (physically infinitesimal continuum region) with no chemical reactions [3]: determines the time evolution of the specific (per unit mass) internal thermal energy u by taking into account the specific volume 1 ϑ ρ = , the infinitesimal (differential) change of heat dq (related with the thermal molecular conduc- tivity) across the boundary surface of the continuum element.The infinitesimal change of heat dq is determined by the classical heat equation [3]: which takes into account the density of the heat flux q J [3] due to the thermal molecular conductivity of heat in the considered continuum.
We use the classical de Groot and Mazur expression [3] for the entropy production (per unit mass) pr σ in thermally heterogeneous one-component Newtonian fluid (with no chemical reactions): where T is the absolute temperature.The density of the heat flux q J is deter- mined by the classical Fourier's law where λ is the coefficient (designated [6] as ae) of thermal molecular conductivity of heat [3] [4].The relations ( 5) and ( 6) give the expression for the total S. V.
, 3 where ( ) is the classical [3] [6] thermal dissipation rate per unit mass determined by the coefficient λ of thermal molecular conductivity of heat, the local absolute temperature T, and the local gradient T ∇ of the local temperature field.
Based on the general Equation ( 1), the decomposition (2), the differential formulation (3) and the heat Equation ( 4) [3], we derived [9] [21] the generalized differential formulation of the first law of thermodynamics (for the symmetric stress tensor Т and for the rotational coordinate system ( ) taking into account the classical differential (during the differential time interval , the classical differential change dU τ of the internal thermal energy U τ [2] [4] [15], the differential change dK τ [9]   [21] of the macroscopic kinetic energy K τ : the differential change the generalized [9] [21] differential work done by non-potential terrestrial stress forces acting on the boundary surface τ ∂ of the considered individual continuum region τ , the differential terrestrial energy gravitational influence d ter G [9] [21] on the continuum region τ : due to the non-stationary terrestrial Newtonian gravitational field, and the tid- done by the combined tidal and centrifugal forces acting on the considered individual continuum region τ during the differential time interval dt . Based on relations ( 11), ( 12), ( 13), ( 14), ( 15) and ( 16), we obtained [9] [21] the equivalent generalized differential formulation of the first law of thermodynamics (for rotational coordinate system ( ) ter ter Based on the generalized differential formulation (17) of the first law of thermodynamics, we derived [9] [21] the evolution equation for the total mechanical energy ( ) which will be used in the next Section 3 for formulation of the general (compressible) and partial (incompressible) local conditions of the tidal maintenance of the quasi-stationary energy and dissipative structure of the mesoscale oceanic eddy located over the two-dimensional bottom topography.Based on the evolution Equation ( 18) and the expression (7) for the total kinetic energy dissipation   9)) of the macroscopic kinetic energy inside of the individual region τ .The total power of the reversible compressibility effect (re- lated with the influence of the divergence 0 div ≠ v and the thermodynamic pressure p on the total mechanical energy ( ) [21] by the third term.The total powers of the mechanical energy exchange across the boundary surface τ ∂ (between the individual continuum region τ and its surroundings) are described [9] [21]  by the fourth, fifth and sixth terms.The total power of the terrestrial energy gravitational influence (owing to of the non-stationary terrestrial gravitational field) on the individual continuum region τ is described [9] [21] by the seventh term.The total power of the energy influence (on the individual continuum region τ ) of the centrifugal force is described [9] [21] by the eighth term.The total power of the energy gravitational influence of the tidal force tidal F (due to the cosmic non-stationary gravitation) on the individual continuum region τ is described [9] [21] by the ninth term.Taking into account that the Coriolis force is perpendicular to the hydrodynamic velocity v , the total power of the energy influence of the Coriolis force is vanished [9] [21].The classical [6] thermal dissipation rate per unit mass  9)).To deduce the general (compressible) and partial (incompressible) local conditions of the tidal maintenance of the quasi-stationary energy and dissipative turbulent structure of the mesoscale eddy over the two-dimensional bottom topography ( ) h x , we take into account in the following analysis the first, second and ninth terms on the right hand side of the evolution Equation ( 18) by disregarding the total powers of the mechanical energy exchange across the boundary surface τ ∂ (between the individual continuum region τ and its surroundings) due to the compressibility, pressure and viscous effects (related with the third, fourth, fifth and sixth terms), by disregarding the total power of the terrestrial energy gravitational influence on the individual continuum region τ (related with the seventh term) owing to the time variations of the non-stationary gravitational potential ter ψ of the Earth, and by disregarding the total power (related with the eighth term) of the energy influence (on the individual continuum region τ ) of the centrifugal force.We make these simpli- fied assumptions to found convincingly the predominant tidal mechanism (related mainly with the ninth term of the evolution Equation ( 18)) of maintenance of the quasi-stationary energy and viscous-thermal dissipative turbulent structure of the mesoscale oceanic eddies (especially, located near the Yamato Rise of the Japan Sea [22]).
Let us consider the ninth term on the right hand side of the evolution Equation (18).According to the internal tide generation models [23] for the two-dimensional bottom topography ( ) where F is the force generating the barotropic (surface) tide, i F is the force generating the baroclinic (internal) tide related with the generation of internal tidal waves by the interaction of the barotropic tide with the bottom topography.
Taking into account the decomposition (19), the ninth term on the right hand side of the evolution Equation ( 18) represents the total mechanical energy production per unit time (in the individual macroscopic region τ ) related with the energy power tidal W of the tidal force tidal F : where is the total barotropic kinetic energy production per unit time (in the individual macroscopic region τ ) related with the barotropic tidal force F , ( ) ( ) is the total baroclinic mechanical energy production per unit time bc W (in the individual macroscopic region τ ) related with the baroclinic tidal force i F .
According to the statistical analysis of the temperature variations (based on the empirical orthogonal functions [29]) at different depths throughout the water column near the shelf boundary of the Japan Sea, the baroclinic (internal) tide of the semidiurnal time period 12.4 hr T = is the predominant component of the internal tide in the Japan Sea.
According to the internal tide generation models [23] describing the generation of the internal semidiurnal tide by the barotropic tide over the two-dimensional bottom topography (determined by the bottom depth ( ) as a function of the horizontal coordinate x along a horizontal axis X), the total barotropic kinetic energy production per unit time ( 21) is related with the barotropic tide characterized by the following barotropic velocities (along the horizontal axis X and the vertical axis Z, respectively): ( ) ( ) where 1 u is the horizontal barotropic velocity component along the horizontal axis X, 1 w is the vertical barotropic velocity component along the vertical axis Z, i is the imaginary unity, t is the time, 2π T ω = is the circular frequency of the barotropic semidiurnal tide related with the semidiurnal time period 12.4 hr T = , ( ) Q h x is the maximal horizontal barotropic velocity of the barotropic flow along the horizontal axis X. Owing to the absence of the vertical velocity shear and the vanished divergence ( ]) of the barotropic velocity field ( ) (given by the components (23) [23]), the barotropic tide (inside of an abitrary individual region τ of the Newtonian continuum) is characterized by the vanished (equal to zero) total rate of the viscous dissipation, the vanished total rate of the viscous-compressible dissipation and the vanished total rate of the thermal dissipation of the macroscopic kinetic energy (owing to the constant density of the barotropic tide [23]).Consequently, the total rates of the viscous dissipation, the viscous-compressible dissipation of the macroscopic kinetic energy, and the thermal dissipation of the macroscopic mechanical energy are related mainly with the baroclinic (internal) tide [23].
According to the internal tide generation models [23], the baroclinic tidal force i F (generating the internal tide) is characterized by the following single vertical real component i F : ( ) ( ) ( ) where the stability frequency

( )
N z is defined by the relation [23] [30] ( ) ( ) depending on the local gravity acceleration g, the distribution of the averaged potential density ( ) ρ * [30] as a function of the vertical depth z.The total baroclinic mechanical energy production per unit time bc W (given by the relation ( 22)) is directed to the baroclinic tide due to the interaction of the barotropic (surface) tide with the bottom topography.Based on the decomposition [23]: of the total semidiurnal velocity field as the sum of the barotropic ( 1 v ) and ba- roclinic ( i v ) components, and using the condition , we evaluate the local baroclinic mechanical energy production per unit mass ( ) , P x z (determining by the relation ( 22)): directed to the unit mass of sea water due to the interaction of the barotropic (surface) tide with the two-dimensional bottom topography ( ) depending on the vertical depth (coordinate) z.
To found the general (compressible) and partial (incompressible) local conditions of the tidal maintenance of the quasi-stationary energy and dissipative turbulent structure of the mesoscale eddy over the two-dimensional bottom topography ( ) h x , we shall use the first, second and ninth terms on the right hand side of the evolution equation ( 18), the related relation (27) (for the local baroclinic mechanical energy production per unit mass ( ) , P x z in the relation (22) for the total baroclinic mechanical energy production per unit time bc W in the individual macroscopic region τ ) and the thermal dissipation rate per unit mass , dis t ε given by the relation (10).Assuming the predominance of the first, second and ninth terms on the right hand side of the evolution equation (18), and using the expression (7) for the total kinetic energy dissipation rate per unit mass dis ε , we formulate the following general (compressible) local condition of the tidal maintenance of the quasi-stationary energy and visc-ous-thermal-compressible dissipative turbulent structure of the mesoscale eddy located inside of the individual fluid region τ over the two-dimensional bot- tom topography ( ) Taking into account , , of the tidal maintenance of the quasi-stationary energy and viscous-thermal dissipative turbulent structure of the mesoscale eddy located inside of the individual fluid region τ over the two-dimensional bottom topography ( ) In the next Section 4 we shall present the calculated vertical distributions of the mean viscous dissipation rate per unit mass ( ) of the partial local condition (30)) characterizing the vertical viscous dissipative structure of turbulence in four regions in the vicinity of the mesoscale anticyclonic eddy [22] located just to the north of Yamato Rise in the Japan Sea.

Spatial Spectra of Temperature Fluctuations and the Viscous Dissipative Structure of Turbulence in Four Regions near the Mesoscale Eddy
Mesoscale eddies of the Japan Sea are significant factor of oceanic structure and dynamics [31] related with the development of the submesoscale motion, which maintains the strong turbulent mixing [22].The experimental studies [22] [31] suggested that the turbulent mixing in the eddies core and the subsequent transport of trapped waters is the significant mechanism of formation of the large-scale structure of the Japan Sea intermediate waters.Taking into account a large number of the eddies and their long life-time, we pointed out [22] the significance of eddies for the vertical transport of heat, salt, dissolved oxygen and the biogenic elements in the deep layers the Japan Sea.
The coexistence of internal gravity waves with mesoscale eddies was revealed [32] based on satellite synthetic aperture radar (SAR) images in the sea south of the Grand Banks.It was shown (based on the linear theoretical analysis [33]) that the shear instability (related with the variability of the eddy current field) is the dynamical mechanism of internal gravity wave generation.
It was shown (based on the revised estimates [34] of net energy transfers between the internal gravity wave and the mesoscale eddy fields) that the wave-eddy coupling is a significant regional source of internal gravity waves.It was confirmed [35] that the dominant source of energy for the internal wave field in the Gulf Stream area is related with the dissipation of mesoscale eddies due to the generation of internal gravity waves during the mesoscale eddy-internal wave interaction.
The prevalent mechanism of the turbulence generation in the oceanic thermocline was associated [30] previously with the breaking internal gravity waves due to the shear instability.We have the proportionality (of the Richardson number Ri and the stability frequency N) Ri N − (in classical consideration [36] of turbulence generation due to the unstable breaking internal gravity waves), which gives the minimal Ri in the oceanic thermocline in accordance with the proportionality 2 ~N ε [36] for the turbulent kinetic energy production rate ε .Consequently, the mean viscous dissipation rate per unit mass dis ε should be also proportional to T E k of temperature fluctuations was suggested previously [37] for the internal gravity waves in the presence of fine structure of the temperature field.
To study the fine structure of the temperature field related with an anticyclonic eddy, the CTD survey of northwestern part of the Japan Sea was carried out on 25 February-9 March, 2003 in the cruise of R/V Akademik M.A. Lavrentyev [22].Special observations were done crossing an anticyclonic eddy of around 70 km in diameter located just to the north of Yamato Rise (see Figure 2 To analyze the calculated [38] spatial spectra ( ) T E k of the temperature fluctuations we have divided the survey area into four regions: 1) the eddy core (St.33, 34, 39 and 40); 2) the edge of the eddy (St.32, 35, 38 and 41); 3) the region of the frontal zone in the south (St.17, 36 and 37); and 4) the region of the subarctic waters in the north (St.30, 31, 42, 43 and 44).
The calculated [38] spatial spectra ( ) T E k of the temperature fluctuations are well approximated (for four regions and for all stations characterized by different integer numbers i) by the suggested [37] dependences (indicated by the black approximating lines on Figures 3(a)-(d)) for the internal gravity waves (characterized by the small spatial wave numbers k) and for the active overturning turbulence (for large k).
We see on Figure 3(a) that the core of the eddy is characterized by the practically identical spatial spectra  we can assume that the mesoscale anticyclonic eddy (located just to the north of Yamato Rise, see Figure 2(a) and Figure 2(b)) generates the breaking internal gravity waves, which produce the intense small-scale dissipative turbulence and related strong turbulent mixing [22] in the mesoscale eddy characterized by the fine microstructure of the temperature field characterized by the suggested [37] dependences (31) for the calculated [38] spatial spectra ( ) T E k of the temperature fluctuations in the four considered regions.
It was evaluated [39] that the viscous dissipation rate (per unit mass) ) throughout the water column.The experimental study [40] reveals also the similar remarkable coexistence of strong stratification, extremely large turbulent kinetic energy and extremely large viscous dissipation rate ( ) , dis v z ε of the turbulent kinetic energy at a very sharp front between two eddies in the Kuroshio-Oyashio confluence zone.The authors [40] argued that this remarkable coexistence "is likely an extreme example of a process that occurs much more widely in the ocean, potentially playing an important role in its dynamics and energetics".Using the calculated [38]  Based on the Kolmogorov's refined hypothesis [24], we founded [10] that the energy spatial spectrum (of the oceanic turbulence) ( ) ( ) ( ) where ν is the kinematic viscosity, ( ) , dis v ε is the mean viscous dissipation rate per unit mass.Considering the power-law dependences ( ) ( )( ) characterized by the power µ , we obtained [10] from relation (32) the power-law expression for the energy spatial spectrum ( ) The power The power 2 µ = − is related with the anisotropic dissipative turbulence characterized by moderate energetics, which is slightly upper than the energetics of the final viscous stage of decay [7] [10].It was correctly pointed out [41] (based on the numerical evidence) that the energy spatial spectrum is not theoretically consistent with the assumption of weak isotropic turbulence.The energy spatial spectrum (33) corresponds to the spatial spectrum ( ) , T E k µ of the turbulent temperature fluctuations ( p c is the specific heat at the constant pressure) [10]: characterized by the same power-law dependence k µ on the spatial wave- numbers k.Using the obtained (as it is evident from Figures 3(a Using the obtained experimental power 2 µ = − in the energy spatial spec- trum (33), we obtain the coefficient ( ) by substituting the relation (37) into the classical condition [30] ( ) where ( ) is the minimal size [7] [30] of the smallest turbulent eddies.Using the condition [30] [44] ( ) ( ) for the interaction time i τ of turbulence and the stability frequency N, we ob- tain the maximal energy-containing scale max l and the turbulent kinetic energy per unit mass tur b ( ) of the anisotropic dissipative turbulence characterized by the power 2 µ = − in the spectra (33) and (36), respectively.Substituting relations (40) and ( 41 semi-empirical relation (refined [45] by introducing the empirical coefficient ( ) 45] and by using the coefficient ( ) ( ) and equating the relation (44) with the obtained relation ( 43), we obtained [38] the expression for the mean viscous dissipation rate per unit mass ( ) used for the calculation of ( ) , dis v z ε for all stations in the four considered regions (see Figure 4).( ) and under condition   with the breaking internal gravity waves [31] [37] [39] generating the anisotropic intermittent (locally strong) dissipative turbulence [38] of the frontal zone (shown on Figure 2(a)) and the edge region of the observed mesoscale eddy.

Thermal and Viscous-Thermal Dissipative Structures of Turbulence in Four Regions near the Mesoscale Eddy
The vertical distributions of the mean thermal dissipation rate per unit mass by the horizontal coordinate x along a horizontal axis X.Based on the derived partial (incompressible) local condition (of the tidal maintenance of the quasi-stationary energy and viscous-thermal dissipative turbulent structure of the mesoscale eddy) and using the calculated vertical distributions of the mean viscous dissipation rate per unit mass [3] [4] [15] infinitesimal change of heat Q δ and the classical [2] [3] [4] [15] infinitesimal change d d U U τ ≡

25 February- 9
deformed finite individual macroscopic region τ of the Newtonian conti- nuum (considered in the rotating coordinate system), we formulate in Section 3 the general and partial (incompressible) local conditions ((29) and (30), respectively) of the tidal maintenance of the quasi-stationary energy and dissipative (viscous-thermal-compressible and viscous-thermal, respectively) turbulent structures of the mesoscale eddy located inside of the individual fluid region τ over the two-dimensional bottom topography ( ) h x characterized by the horizontal coordinate x along the horizontal axis X.To evaluate the partial (incompressible) local condition (30) (formulated based on the internal tide generation model [23] and considering the thermally heterogeneous incompressible viscous Newtonian fluid characterized by the classical [6] thermal dissipation rate per unit mass , dis t ε and the classical [3] [4] [5] [6] [7] [14] [24] local viscous dissipation rate per unit mass , dis v ε ), in Section 4, we present the calculated vertical distributions of the mean viscous dissipation rate per unit mass ( ) , dis v z ε characterizing the vertical viscous dissipative structure of turbulence in four regions in the vicinity of the mesoscale eddy.The vertical distributions of ( ) , dis v z ε are calculated based on parametrization (45) established using the analysis of the CTD measurements [22] for four regions in the vicinity of mesoscale eddy observed in the northwestern part of the Japan Sea near the Yamato Rise on March, 2003 in the cruise of R/V Akademik M.A. Lavrentyev.In Section 5, we present the calculated vertical distributions of the mean thermal dissipation rate per unit mass ( ) , dis t z ε characterizing the vertical thermal dissipative structure of turbulence in four regions in the vicinity of the mesoscale eddy.

Figure 1 .
Figure1.The rotational Cartesian coordinate system K centred at the mass center of the rotating Earth and the Lagrangian coordinate system K ′ related with the mass center C of an individual finite continuum region τ subjected to the combined (terrestrial and cosmic) non-stationary Newtonian gravitation.
is the classical[3]-[8] [14] [24] local viscous dissipation rate per unit mass (in the Newtonian continuum characterized by the local coefficient of molecular kinematic viscosity ν η ρ = [5] [6] [7]) related with the local rate of the strain tensor 1 2 ) is the classical [3] [4] [6] [7] viscous-compressible dissipation rate per unit mass (in the Newtonian continuum characterized by the coefficient of molecular kinematic viscosity ν [5] [6] [7] and the coefficient of molecular volume (second) individual macroscopic region τ of the viscous compressible Newtonian continuum (fluid): S. V. Simonenko, V. B. Lobanov DOI: 10.4236/jmp.2018.93026365 Journal of Modern Physics rate per unit mass dis ε (in thermally heterogeneous three-dimensional shear flow of the viscous compressible Newtonian fluid with no chemical reactions), we shall deduce in the next Section 3 the general (compressible) and partial (incompressible) local conditions of the tidal maintenance of the quasi-stationary energy and dissipative structure of the mesoscale eddy located inside of the individual fluid region τ over the two-dimensional bottom topography ( )hx .We shall use in the Section 6 the partial (incompressible) local condition(30) for the combined analysis of the energy and viscous-thermal dissipative structure of turbulence in four regions of the periodically topographically trapped[22] [25] [26] eddy in the quasi-stationary state near the northern region of the Yamato Rise in the Japan Sea.

3 .
The General (Compressible) and Partial (Incompressible) Local Conditions of the Tidal Maintenance of the Quasi-Stationary Energy and Dissipative Turbulent Structure of the Mesoscale Eddy Located over the Two-Dimensional Bottom Topography To derive the general (compressible) and partial (incompressible) local conditions of the tidal maintenance of the quasi-stationary energy and dissipative turbulent structure of the mesoscale eddy located inside of the individual fluid region τ over the two-dimensional bottom topography ( ) h x (characterized by the horizontal coordinate x along the horizontal axis X), it is necessary to understand the physical nature of various terms on the right hand side of the evolution Equation (18).The first term describes [9] [21] the total power of the irreversible viscous dissipation (in the Newtonian continuum due to the viscous dissipation rate (in a unit of mass) , dis v ε according to the expression (8)) of the macroscopic kinetic energy inside of the individual region τ .The second term describes [9] [21] the total power of the irreversible viscous-compressible dissipation (in the Newtonian continuum due to the viscous-compressible dissipation rate (in a unit of mass) , dis c ε related with the compressibility effects (related with the divergence 0 div ≠ v of the local hydrodynamic velocity v ) according to the expression ( the relation(10)) is not presented in the evolution Equation(18) since the thermal dissipation rate per unit mass , dis t ε characterizes the intermediate dissipation of the macroscopic kinetic energy (owing to the creation of the local heterogeneities of the temperature field), which is converted eventually into the internal heat owing to the viscous dissipation rate per unit mass , dis v ε (given by the relation (8)) and the viscous-compressible dissipation rate per unit mass , dis c ε (given by the relation ( x .The expression (27) for ( ) , P x z leads to the vertical distribution (for each horizontal coordinate x ) of the normalized local baroclinic mechanical energy production per unit mass

2 N) 2 T
, i.e. 2 dis N ε explaining the remarkable coexistence of strong stratification and extremely large viscous dissipation of the turbulent kinetic energy in the breaking internal gravity waves.The classical dependence ( E k k − (defined by the spatial wave number k) of the spatial spectra ( ) (a) and Figure 2(b)).Numbers of some stations (St.) referred in the analysis are indicated on Figure 2(a) and Figure 2(b).

Figure 2 .Figure 3 .
Figure 2. Distribution of water temperature (˚C) at 150 m depth (a) and along meridional section crossing an anticyclonic eddy (b) in the northwestern Japan Sea on 25 February-9 March, 2003.

(
related with the breaking internal gravity waves of the background internal gravity wave field) is distributed proportionally to ( ) 2 N z (i.e., approximating by the suggested[37] dependences (31)) of the temperature fluctuations, we present below the method for calculation of the vertical distributions of the viscous dissipation rate per unit mass ( ) , dis v z ε for different stations located in the four considered regions.
is related with the three-dimensional isotropic homogeneous non-dissipative turbulence of the inertial subrange characterized by the classical [30] Kolmogorov energy spatial spectrum ( very high (large) turbulent Reynolds numbers.The power 13 7 µ = − is related with the weak anisotropic dissipative turbulence at the final viscous stage of decay [7] [10].The power 3 µ = − is related with the strong (active, over- turning) three-dimensional isotropic homogeneous small-scale dissipative turbulence characterized by the energy spatial spectrum [7] [10]

2 Ẽ
)-(d)) experimental power 2 µ = − in the spatial spectrum (36), we have the theoretical power 2 µ = − in the energy spatial spectrum (33) in accordance with the re- vealed [42] [43] energy spatial spectra ( ) k k − (in the surface layers of the California current system) based on the high-resolution numerical simulation of the mesoscale eddy turbulence related with transition from mesoscale to submesoscale fluid motion.
in spectra(33) and (36)) Kolmogorov relation[45] critical kinetic energy viscous dissipation rate per unit mass[5] [7][10], which characterizes the transition from a chaotic overturning turbulent regime to a wave hydrodynamic regime in an incompressible stratified viscous Newtonian fluid.We obtained[38] the numerical coefficient oceanic turbulence.Substituting the relation(40) into the following refined[45]

2 νN 2 N
, which is consistent with the non-equilibrium statistical thermohydrodynamic theory of the small-scale dissipative turbulence[5] [7] [9] [10].The dimensional coefficient in relation (45) is in agreement also with the previous evaluation [39] of the viscous dissipation rate (per unit mass) ( ) , dis v z ε related with the breaking internal gravity waves of the background internal gravity wave field.48] were used for the calculations of the vertical distributions of the viscous dissipation rate per unit mass , dis v ε shown on Figure 4.The revealed very small variance (for stations 33, 34, 39 and 40) of relatively large maximal values of , dis v ε (on Figure 4(a)) confirms the established intense turbulent mixing [22] in the core of the eddies.It is evident from the Figures 4(a)-(c) that the maximal values of , dis v ε for the core of the eddy, the frontal zone and the edge of the eddy, respectively, are larger than the values of the viscous dissipation rate per unit mass , dis v ε (characterized by the range ] in the eddies off Kuril Islands.It confirms the strong dissipative dynamics and energetics of the considered mesoscale eddy (shown on Figure 2) in the northwestern Japan Sea.The revealed maximal values of the viscous dissipation rate per unit mass , dis v ε and the maximal variance of , dis v ε at the edge of the considered eddy (see Figure 4(b)) and in the frontal zone (see Figure 4(c)) confirm the established [22] significance of the submesoscale motion related

Figure 4 .
Figure 4.The calculated vertical distributions of the viscous dissipation rate per unit mass , dis v ε for stations located in the eddy T ∇ of the absolute temperature T of the sea water) are calculated based on the following relation: j) of the same vertical length d=204.8m containing of 4096 numerical values of the vertical gradient T ∇ with the step equal to 0.05 m. Figure 5 shows the calculated vertical distributions of the mean thermal dissipation rate per unit mass ( ) , dis t z ε for all stations located in the eddy core (Figure 5(a)), at the edge of the eddy (Figure 5(b)), in the frontal zone (Figure 5(c)) and in the subarctic waters (Figure 5(d)).The calculated averaged vertical distributions ( ) , dis t z ε (obtained as the average of the all mean thermal dissipation rates per unit mass ( ) , dis t z ε for each region) are shown by black colour in the eddy core (Figure 5(a)), at the edge of the eddy (Figure 5(b)), in the