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The practical significance of the established generalized differential formula-tion of the first law of thermodynamics (formulated for the rotational coor-dinate 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
of the ther-mally heterogeneous viscous (compressible and incompressible, respective-ly) heat-conducting stratified fluid over the two-dimensional bottom topog-raphy characterized by the horizontal coordinate x along a horizon-tal axis X. Based on the derived partial (incompressible) local condition (of the tidal maintenance of the quasi-stationary energy and viscous-thermal dis-sipative turbulent structure of the mesoscale eddy) and using the calculated vertical distributions of the mean viscous dissipation rate per unit mass
and the mean thermal dissipation rate per unit mass
in four regions near the observed mesoscale (periodically topographically trapped by nearly two-dimensional bottom topography
*h*
*(x)* 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 mechanism of maintenance of the eddy energy and viscous-thermal dissipa-tive 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.

It is well known that the problem of turbulence is “the last great unsolved problem of classical physics” [

The classical Gibbs’ differential formulation [

of the internal thermal energy U τ ) the infinitesimal increment d K τ of the macroscopic kinetic energy K τ (which contains (for the for the small continuum region τ [

distributions of the mean viscous dissipation rate per unit mass

calculated based on parametrization (45) established using the analysis of the CTD measurements [

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 [

Let us consider an individual finite continuum region

The local hydrodynamic velocity

where,

arbitrary symmetric stress tensor [

The pressure tensor

defined by the delta-tensor

determines the time evolution of the specific (per unit mass) internal thermal energy u by taking into account the specific volume

which takes into account the density of the heat flux

We use the classical de Groot and Mazur expression [

where T is the absolute temperature. The density of the heat flux

where

where

is the classical [

tensor

is the classical [

is the classical [

Based on the general equation (1), the decomposition (2), the differential formulation (3) and the heat equation (4) [

taking into account the classical differential (during the differential time interval

the differential change

the generalized [

done by non-potential terrestrial stress forces acting on the boundary surface

due to the non-stationary terrestrial Newtonian gravitational field, and the tidal-centrifugal differential work

done by the combined tidal and centrifugal forces acting on the considered individual continuum region

Based on relations (11), (12), (13), (14), (15) and (16), we obtained [

Based on the generalized differential formulation (17) of the first law of thermodynamics, we derived [

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 rate per unit mass

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

pressure p on the total mechanical energy

continuum region

Let us consider the ninth term on the right hand side of the evolution Equation (18). According to the internal tide generation models [

where

where

is the total barotropic kinetic energy production per unit time (in the individual macroscopic region

is the total baroclinic mechanical energy production per unit time

According to the internal tide generation models [

where

According to the internal tide generation models [

where the stability frequency

depending on the local gravity acceleration g, the distribution of the averaged potential density

of the total semidiurnal velocity field as the sum of the barotropic (

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

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

Mesoscale eddies of the Japan Sea are significant factor of oceanic structure and dynamics [

The coexistence of internal gravity waves with mesoscale eddies was revealed [

It was shown (based on the revised estimates [

The prevalent mechanism of the turbulence generation in the oceanic thermocline was associated [

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 [

To analyze the calculated [

The calculated [

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

we can assume that the mesoscale anticyclonic eddy (located just to the north of Yamato Rise, see

It was evaluated [

reveals also the similar remarkable coexistence of strong stratification, extremely large turbulent kinetic energy and extremely large viscous dissipation rate

Based on the Kolmogorov’s refined hypothesis [

where

The power

for very high (large) turbulent Reynolds numbers. The power

The power

The energy spatial spectrum (33) corresponds to the spatial spectrum

characterized by the same power-law dependence

Using the obtained experimental power

by substituting the relation (37) into the classical condition [

where

for the interaction time

of the anisotropic dissipative turbulence characterized by the power

for the coefficient of turbulent (eddy) viscosity

where

and equating the relation (44) with the obtained relation (43), we obtained [

used for the calculation of

The numerical coefficient

and under condition

The founded parameters

with the breaking internal gravity waves [

The vertical distributions of the mean thermal dissipation rate per unit mass

for various vertical subranges

frontal zone (

We calculate the averaged (based on the all stations in the considered regions (a), (b), (c) and (d)) vertical distributions

where the averaged (based on the all stations in the considered regions (a), (b), (c) and (d)) vertical distributions

The calculated averaged vertical interpolated distributions of the mean viscous-thermal dissipation rates per unit mass

The partial (incompressible) local condition (30) gives the partial local normalized condition (for the calculated distributions of

between the normalized (on the maximal value) local mean viscous-thermal

dissipation rate per unit mass

The proportionality (49) leads to the corresponding proportionality (for the mean distributions obtained by averaging of the several distributions corresponding to the different stations in each considered region):

between the normalized averaged (for several stations in each considered region) local mean viscous-thermal dissipation rate per unit mass

7(a) and

(50) for the eddy core (

We see the satisfactory numerical fulfilment (shown on

incompressible local condition (30)) for the distributions

the all stations (stations 30, 31, 42, 43 and 44) of the subarctic waters. We see for the

several stations of the subarctic waters (d)) local mean viscous-thermal dissipation rate per unit mass

(incompressible) local condition (30) of the quasi-stationary energy and viscous-thermal dissipative structure of the semidiurnal baroclinic tidal motion of the viscous incompressible heat-conducting stratified vortical viscous fluid (over the two-dimensional bottom topography) in the subarctic waters of the Japan Sea.

We use only St. 35 at the edge of the eddy (

The experimental results (of the acoustic tomography of the large-scale heterogeneities in the ocean [

According to the second (more adequate) point of view (taking into account a possible influence of the viscous-compressible dissipation rate per unit mass

The existence of the internal gravity waves is confirmed (based on our statistical analysis of the temperature fluctuations given in Section 4) by the computed spatial spectra

Thus, the calculated dependences (which have the more distinct differences between the calculated distributions in the range of depth 500 ÷ 1400 m of the eddy core for

Based on the evolution equation (18) (deduced from the established [

To use the formulated partial (incompressible) local condition (30), we have presented the analysis of the CTD observations [

distributions of

of the eddy (

Based on the classical [

(47))

The calculated vertical distributions of the mean thermal dissipation rate per unit mass

mean thermal dissipation rate per unit mass

intermediate maximums near the depth of 2000 m for all stations located in the eddy core (

obtained vertical distributions of

averaged (using the all stations in four considered regions shown on

vertical distributions (shown on Figures 6(a)-(d)) of the averaged viscous-thermal dissipation rates per unit mass

(48)) characterizing the averaged vertical viscous-thermal dissipative structure of turbulence in four regions near the mesoscale eddy (see

obtained distributions of

characterized by the deep intermediate maximums near the depth of 2000 m for the eddy core (

Based on the derived partial (incompressible) local condition (30) (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

vertical distributions of

averaged distributions

and averaging of the several distributions corresponding to the all stations (Sts. 30, 31, 42, 43 and 44) of the subarctic waters. This remarkable correspondence

(between

emphasize a remarkable association between structure (related in the considered case with the structure of the normalized averaged local baroclinic mechanical energy production per unit mass

Taking into account that the calculated normalized local baroclinic mechanical energies production per unit mass

viscous-thermal dissipation rate per unit mass

on the partial incompressible local condition (30)) that the semidiurnal baroclinic internal tide (generated by the semidiurnal barotropic tidal current over the nearly two-dimensional bottom topography [

Avoid the stilted expression, authors thank reviewers for significant remarks and questions taken into account with gratitude for correction of the article. Authors thank Mrs. A.V. Sereda for help in numerical calculations and Mr. I.A. Kuskov for help in graphic presentation of the calculated results. One of us (S. V. S.) thanks with gratitude Jane GAO, Editorial Assistant of JMP for the best editorial assistance.

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