ESTABLISHMENT OF THE OCEAN DYNAMIC SYSTEM WITH FOUR SUB-SYSTEMS AND THE DERIVATION OF THEIR GOVERNING EQUATION SETS*

YUAN Ye-li, QIAO Fang-li, YIN Xun-qiang, HAN Lei, LU Ming

The First Institute of Oceanography, State Oceanic Administration of China, Qingdao 266061, China,

E-mail: yuanyl@fio.org.cn

ESTABLISHMENT OF THE OCEAN DYNAMIC SYSTEM WITH FOUR SUB-SYSTEMS AND THE DERIVATION OF THEIR GOVERNING EQUATION SETS*

YUAN Ye-li, QIAO Fang-li, YIN Xun-qiang, HAN Lei, LU Ming

The First Institute of Oceanography, State Oceanic Administration of China, Qingdao 266061, China,

E-mail: yuanyl@fio.org.cn

Based on their differences in physical characteristics and time-space scales, the ocean motions have been divided into four types in the present paper: turbulence, wave-like motion, eddy-like motion and circulation. Applying the three-fold Reynolds averages to the governing equations with Boussinesq approximation, with the averages defined on the former three sub-systems, we derive the governing equation sets of the four sub-systems and refer to their sum as “the ocean dynamic system”. In these equation sets, the interactions among different motions appear in two forms: the first one includes advection transport and shear instability generation of larger scale motions, and the second one is the mixing induced by smaller scale motions in the form of transport flux residue. The governing equation sets are the basis of analytical/numerical descriptions of various ocean processes.

ocean dynamic system, four sub-systems, advection transport, shear instability, transport flux residue

Introduction

There are two kinds of research that requires more attention, namely, the developments in ocean energetics[1-4]and coupled numerical modeling[5-8]. The former attempts to study the interactions among various ocean motions from the view point of energy balance and then enhance our understanding of ocean motions as a whole (see Fig.1 for more detail). The latter tries to enhance our capability in understanding ocean dynamic system through physical-mathematical descriptions of the interactions among various ocean motions. In these two kinds of research, even though people recognized that the consideration of ocean motions as a whole is of significant importance, to our knowledge, no one studied the ocean dynamic system based on the principles of system sciences. This paper is the continuation of our previous work[9], with the main objectives of establishing a physical description of the ocean dynamic system with four sub-systems and deriving their governing equation sets.

Ocean mixing is an important research field, which needs to consider various ocean motions as a complete system. For the ocean circulation at large scale, the motions involved are not only the turbulence at sub-small scale but also the waves at small and sub-meso scales and the eddies at meso scale, so we need to consider mixing effects of all the motions at relative smaller scales. Therefore, the derivation of the governing equation sets for the ocean dynamic system is important, which helps us to describe the two forms of interactions among the four sub-systems analytically.

In Section 1, based on their difference in physical characteristics and time-space scales of four kinds of motions, we propose partition principles of the ocean dynamic system into four sub-systems, and design a decomposition scheme with a group of Reynolds averages. The decomposition operation is the basic method for deriving the governing equation sets of the ocean dynamic system. In Section 2, we apply theoperation on the governing equations of ocean motions in earth based coordinate system under the Boussinesq approximation, and the governing equation sets of the four sub-system of turbulence, waves, eddies and circulation, are derived respectively, the sum of which is called the governing equation set for the ocean dynamic system. In Section 3, we select the ocean mixing as an example, to sketch the application prospect of the derived equation set in the study of ocean motions as a whole.

Fig.1 Air-sea interactions and the energy flow among different types of motion

Fig.2 Relationship of time and space scales of the motions in ocean dynamical system (quasi-dispersion relation)

1. Partition principle and decomposition operation

1.1 Partition of ocean dynamical system into four sub

systems

In our previous work[9], we suggested that the quasi-dispersion relation, which shows the physical characteristics and time-space scales of different ocean motions, can be taken as the scientific basis for subsystem partition. By carefully inspecting, the relation shown in Fig.2, we noticed that the ocean motions can be divided into the following four groups.

1.1.1 Ocean turbulence motion with sub-small scale

This type of motion, in fact, includes all the forms of motion in 10–Nm-100m scale grade, where N＞0. The sub-small scale ocean motions which are controlled by strong non-linearity can be regarded as isotropic or near-isotropic statistically.

For the mixing in the ocean circulation at large scale, we give up the previous way in which the sum of all the perturbations at smaller scales are ambiguously called as the ocean turbulence.

1.1.2 Ocean wave-like motion with small and submeso scale

In the ocean, this type of motion includes ocean surface waves in 101m-102m scale grade and internal waves in 103m-104m scale grade. The ocean wave-like motion has much larger phase speed than that of the water mass, and therefore it is controlled by dynamic gravity balance. This type of motion is weakly nonlinear in general, so the linearization and the perturbation method are efficient approaches to reveal their physical essence.

1.1.3 Ocean eddy-like motion with meso-scale In the real ocean, the motion in 104m-105m grade includes not only meso-scale eddies and frontal waves with vertical axis but also the secondary circulation and the multi-core structures of the Kuroshio and the Gulf Stream with horizontal axis. The oceaneddy-like motion differs from the wave-like one. Its speed of pattern movement (phase speed) is not so much different from that of water mass, and therefore it is controlled by static gravity balance. This kind of motion is weakly nonlinear in general, so we can use the linearization and the perturbation method to reveal their physical essence.

1.1.4 Ocean circulation motion at large scale

In the real ocean, motions of this form include ocean circulations and ocean waves that are controlled by static gravity balance in vertical and by geotropic balance in horizontal. They are the ocean motions in 106m or bigger scale grade. Due to the fact that the speeds of the motions in geotropic balance are extremely low, they also show some degree of nonlinearity.

Due to the fact that the four types of motions cover all the ocean motions in physical characteristics and time-space scale, we can divide the whole ocean motion into four sub-systems and refer to the sum of them as the ocean dynamic system.

1.2 Decomposition approach of the original governing equations

1.2.1 Original governing equations for ocean motions

The governing equations of ocean motions in the earth based coordinate system with the Boussinesq approximation[10]include motion equations and boundary conditions as follows.

(1) Motion equations

here ｛u,p,ρ,T,s｝ represent velocity, pressure, density, temperature and salinity, and ｛ν0,κ0,D0, Cp0｝ are the coefficients of molecular viscosity, thermal conductivity, diffusion coefficient of salinity and the isobaric specific heat respectively. Q is the equivalent temperature source ofQ1

are the boundary conditions at the sea surface and bottom, n represents the normal vector,ΔS≡are the fluxes of momentum, salinity and heat

where iirepresents the coordinate orientations of the earth bas ed coordinate systems. P≡P/ρ is theequivalent fresh water flux at sea surface,PAis the air stress at sea surface, PHis the waterstress at ocean bottom, and QAand QHare the heat input fluxes at sea surface and bottom, respectively.

1.2.2 Design of the threefold Reynolds averages

Due to their differences in physical characteristics and time-space scale, we define a group of Reynolds averages for the first three types of motion as follows:

(1) First-fold Reynolds average

For the ocean turbulence motion, the first-fold Reynolds average is defined and marked by the symbolso we have,

are the sums of motions except ocean turbulence.

(2) Second-fold Reynolds average

For the wave-like ocean motion, the second-fold Reynolds average is defined and marked by symbolso we have,

are the sums of the motions beside ocean turbulence and wave-like ocean motions.

(3) Third-fold Reynolds average

For the eddy-like ocean motion, the third-fold Reynolds average is defined and marked by symbolso we have,

are the perturbations relative to eddy-like ocean motion with averaged values of zero, and

are the ocean circulation motion at large scale.

2. Governing equations for the ocean turbulence sub-system

Applying the first-fold Reynolds average to the original governing Eqs.(1)-(13), we can obtain equations for the ocean turbulence motion sub-system and the sum of wave-like ocean motions, eddy-like ocean motions, and circulation.

2.1 Governing equations for the sum of wave-like motions, eddy-like motions, and circulation

2.1.1 Motion equations

3. Governing equations for the ocean wave-like motion sub-system

Applying the second-fold Reynolds average to the governing Eqs.(20)-(32) for the sum of ocean wave-like motions, ocean eddy-likemotions, and circulation, we can obtain equations for the ocean wavelike motion sub-system and the sum of eddy-like motion and circulation.

3.1 Governing equations for the sum of ocean eddylike motion and circulation

3.1.1 Motion equations

3.2 Governing equations for the ocean wave-like motion sub-system

3.2.1 Motion equations

4. Governing equations for the ocean eddy-like motion and circulation sub-systems

Applying the third-fold Reynolds average to the governing Eqs.(46)-(58) for the sum of ocean eddylike motion and circulation, we can obtain equations for the ocean eddy-like motion and circulation subsystems.

4.1 Governing equations for the ocean circulation sub-system

4.1.1 Motion equations

4.2 Governing equations for the ocean eddy-like motion sub-system

4.2.1 Motion equations

5. Discussion and conclusions

5.1 The modulation by larger scale motions through advection transport and shear instability generation

In the governing equations of ocean turbulence motion (33)-(45), the ocean wave-like motion (59)-(71), and the ocean eddy-like motion (85)-(97), one motion is modulated by the motions with larger scales through advection transport andshear instability generation as follows:

For ocean turbulence, the larger scale motions include ocean wave-like motion, ocean eddy-like motion, and circulation, for ocean wave-like motion, the larger scale motions include ocean eddy-like motion and circulation, and for ocean eddy-like motion, the larger scale motion is the ocean circulation only.

5.2 The modulation by smaller scale motions through ocean mixing

In the governing equations for the sums of wavelike ocean motion to circulation (20)-(32), of eddy-like ocean motion to circulation (46)-(58), and of ocean circulation (72)-(84), we have the terms of ocean mixings in the equations of momentum, temperature and salinity.

So the ocean mixings are in the form of transport flux residue. They are different from each other in physical background. For the governing equations of ocean circulation, the ocean mixing is in the form of three terms, which are the flux residues of ocean turbulence, wave- and eddy-like motions, for that of the sum of eddy-like ocean motion and circulation, the ocean mixing is in the form of two terms, which are the flux residues of ocean turbulence and wave-like motion, for that of the sum of wave-like ocean motion tocirculation, the ocean mixing is in the form of only oneterm, which is the flux residue of flux of ocean turbulence.

We can conclude that any sub-system has two effects in the ocean dynamic system: one is the advection transport and shear instability generation through which modulate smaller scale motions, and the other is the ocean mixing in the form of flux residues through which modulate larger scale motions.

5.3 Approach for solving mixing problem in the ocean dynamic system

Here we present an approach for solving the mixing problem in the ocean dynamic system.

5.3.1 Oceanmixing in ocean turbulence motion

The turbulence motion sub-system lies on the lowest layer of the ocean dynamic system, which is a type of sub-small scale motion with strong nonlinearity[11]. We derive its closure model of second-order moments under the presupposition of isotropy, and then we can estimate the turbulence mixing by solving shear instability of three kinds of larger scale motions and surface wave breaking. The ocean surface waves and internal waves should mainly responsible for the former, so we may call the turbulence in theupper ocea n layer as the wave-generated so that the closure the closure model. The turbulence sources include model can be simplified and then estimated analytically. The turbulence mixings shown in the form of residue of transport flux on the sums of wave-like ocean motion to circulation, eddy-like ocean motion to circulation and ocean circulation can be expressed as

5.3.2 Ocean mixing in ocean wave-like motion

The ocean wave-like motion includes ocean surface waves and internal waves controlled by dynamicgravity balance. Due to their weak nonlinearity, we may use the analytical solution of the linearized governing equations to describe the main characteristics of ocean surface waves and internal wav es. The wavelike motion mixing shown in t he form of transport flux residue on the sums of eddy-like motion to circulation and ocean circulation can be expressed as

We can use the unified linear theory of the ocean wave-like motion[12], the statistical theory for the stat ionary and homogeneous ocean waves[13], and the ocean wave numerical model[14-16]to obtain an analytical estimation of the residue of transport flux with second-order accuracy.

5.3.3 Ocean mixing inocean eddy-like motion

The ocean eddy-like motion includes both vertical and horizontal ones, which are controlled by static gravity balance. The weak nonlinearity of the motion brings us about analytical solutions of the gove rning equa tions, and then we may calculate the mixing of the ocean eddy-like motion in the form of transport flux residue on the ocean circulation as follows

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Appendix A

Appendix B

March 2, 2012, Revised March 18, 2012)

* Project supported by the National Natural Science Foundation of China (Grant Nos. 40776020, 41106032) the National Basic Research Program of China (973 Program, Grant Nos. G1999043800, 2006CB403600, 2010CB950404 and 2010CB950300).

Biography: YUAN Ye-li (1938-), Male, Ph. D., Professor

Corresponding auther: YIN Xun-qiang,

E-mail: yinxq@fio.org.cn