Vol. 14 No.2 CHIN. 1. OCEANOL. LIMNOL. 1996

ON MULTIPLE SOLUTIONS TO A LOW~PECTRUM MODELOF OCEANIC CuRRENT DRIVEN BYTHE BOUNDARY FORCE

-THE BIMODALITY OF THE KUROSHIOSOUTH OF JAPAN

DoNG Chang-mingCiIl§ tm), CHEN Shui-ming( !*7.Ktm )ZHANG Qing-hua( 5l.E.ec~), YUAN Ye-li(1lit.fr. )

(First Institute of Oceanography, SOA, Qingdao 26(i)()3)

Received Mar. 12, 1995; revision accepted Aug. 20; 1995

AlNract

This low-spectrum model study on the multiple solutions to a nonlinear quasi.geostrophic oceanic cur-

rent equation shows that- they depend on the oombination of Ro, Re, A and e, that the bimodality of the

Kuroshio depends strongly on the nonlinear effect represented by Ro and A, and that its occurrence

probability is reduced by the dissipation represented by Re and e. The stability of solutions is discussed in

detail with Hurwitz's theory.

Key words: multiple solutions, low-spectrum model, stability, Hurwitz's theory

INIRODUCTION

It is well known that a forced nonlinear dissipative system has multiple solutions.For oceanic flow, especially western boundary current, due to its high speed and strongshearing, the nonlinear effect is obvious. Lots of observational data and numerical experi-ments show the bimodality (straight and meandering paths) of the Kuroshio south ofJapan (Dong & Zhang, 1993; Masuda, 1982; Yasuda & Yoon, 1985; Yoon, 1987). Nu-merical models are not always available to reveal its general characteristics because of theircomplexity. As known, it is very difficult to acquire directly a general analytical solutionto a strongly nonlinear system. In the present paper, Galerkin's high -truncated spectrummethod is applied to the quasi-geostrophical nonlinear vortex equation and thelow-spectrum solution obtained is used to discuss the bimodality of the Kuroshio southof Japan.

G. Veronis (1963) applied the spectrum method to a quasi-geostrophical wind-driven current equation to obtain its multiple solutions. Ji Z. et al. (1990), in whosepaper the external force is wind stress and solid boundary conditions are applied, cor-rected some of Veronis's inappropriate assumptions and further discussed the problem.

However, due to the speciality of the Kuroshio bimodality, the boundary conditioncannot be solid or streamline. In the present paper, it is with the non-zero boundarycondition that the multiple equilibrium of the Kuroshio's path south of Japan is discussed.

MODEL

The nonlinear vortex equation controlling barotropic quasi-geostrophic oceanic currentis nondimensioned as:

• This work was supporded by NSFC (No. 4~70252)

114 CHINFSE JOURNAL OF OCEANOLOGY AND LIMNOLOGY Vo1.14

L(tjJ)= -aa ('V~)+Ro' J(tjJ,v~)+ aatjJ - R1 'VtjJ+8~=0 (1)t . x e

_ u _ f3L3 _ 'Yhere Ro - f3L2 ,the Rossby Number, Re- AH ,the Reynolds Number, and 8- f3L '

in which u is characteristic velocity, L horizontal characteristic length, AH horizontal vortexdissipative coefficient, 'Y bottom friction coefficient, and f=i+ f3y the Coriolis parameter.

The study region is: (x, y)E {xIO~x ~n} U {YIO~y~n}, which includes: inlet: {x=O,

Yo- ; ~y~Yo+ ~ }, and outlet: {x=n, Yl- ~ ~Y~Yl+ .~ }, where a is the width

of inlet and outlet, as shown in Fig. 1. We select the symmetrical inlet and outletcondition in order to simplify the solving process.

The velocities at inlet and outlet are supposed

where A. is selected to ensure that the velocityreduces quickly to zero laterally away from the in-let and outlet. So Eq .. (2) defines the velocity con-dition in the solid parts of the boundaries.

For convenience, let Yl = Yo= ; ,uo= 1, in or-

der to eliminate the none-zero boundary condition;let tjJosatisfy:

atjJo =v=O atjJo =-u=-uoe-Jl.y-yci (3)ax 'ay

Obviously, ~~ + ~~ =0. From (3), we get:

fy IrE...

tjJo=- E... e-A<Y-fl' dy=- 0 2 e-Jidy2

Fig. 1 Sketch of study region

as:x=O: u=Urft-)J.y-yci, v=O (2)

(4)

Let tjJ= <p+ tjJo.Using (3), Eq. (1) becomes:

L(tjJ)=L(<p)+ F(<p)+L(tjJJ=O (5)

a<p i?tjJo atjJo a~<pwhere F(<p)= - • -- - - • -- (6)

ax al ay axwhich represents the interaction between the boundary force and ip, Eq. (5) yields:

L(<p)+F(<p)= - L(tjJJ

Through some complicated computations,

L(tjJJ'= r- ye-JJ.y- f l'+Q • le-JJ.y- f l'

(7)

(8)

No.2 LOW.,sPECfRUM MODEL STUDY 115

where P=2EA+ ~)? Q= - ~ A3Re > Re

ThenL(<p) +F(<p) = - L(t/JJ

<pIc = 0 (boundary condition)

(9)

(10)

In the following, using Galerkin's high -truncated spectrum method, we expand theEquation (10).. As Galerkin's truncated spectrum method requires that the basic functionsmeet the boundary condition, we expand tp, L(t/JJ and t/Jo as follows:

4

tp= I a;t/J; (11);=1

(12)

4

F( <p) =I F;t/J;i=1

(13)

4

L(t/JJ= IL;t/J;;=1 .

(14)

where t/JI =sinx· sinyt/J, =sinzx - sinyt/J3=sinx' sin2yt/J4=Sin2X· sin2y

are basic functions. It is easily shown that ~ n = - Ant/J n' where An = 2, 5, 5, 8 when n = 1,2, 3, 4, respectively.

Through some complicated calculations for (12), (13) and (14), we haveCI=S=C4=0

. 1 .=l.c3= 2 e A .J relA Irt

Fl=<;~' F2=<;a}> F3=F4=0 (15)

. L}=L2=L4=0

. ( 2P 3Q 2Q).=l.4=-T +7 -7 e A .JreIA!rt

where rt= ; . Substitute (11), (12), (13), (14) and (15) into (10), and project eachterm of (lO) on the basic functions. The spectrum coefficients of Equation (10) can be de-duced as follows:

da} re+s: +(rt<;+-3 )a2+Q1al=O

da, 9Ro~ 2ren Tt - 80 a}~+(rtC3-15 )a}+q2~=0

da, 9Ro~ 2re 4rtrtTt + 80 a}~-15 a4+Q3a3=-5-

(16)

116 CHINESE JOURNAL OF OCEANOLOGY AND LIMNOLOGY Vo1.14

da, ttrJdt - 12 ll:J+q4a4=0

in which qn=( ~~ +1»rJ (n=l, 2, 3, 4)

STEADY SOLUTIONS

For the steady case, (16) yields

(rJ£;+ ~ ')~+qlal=O

-ralll:J+(rJ£;- ~~ )al+qA=O

2n L3rJral~-15 a4+q3ll:J= -5-

(17.1)

(17.2)

(17.3)

(17.4)

9~ .where ~=80 . Ro. According to Eq. (17), we have

1. a, =0, then~=Oa = ~rJq4

3 5(q3q4- :a ) (18)

nL3rJa4 = -(-r--=-'--~'---""')

6\q3q4- 90 .

2. a,:j= 0, then

-q,a,~= n

rJ£;+ 3

ll:J= - ~ L£;_ 2n )+ _1_ qAr ~ 15 ra,

·na4= 12q4 ll:J

where Q, satisfies:

~+~~I "",)[(~c,- i~- ;-:;;c, ). (90~,q. - J;, )+ ;;, J~o (20)

As ~ +rJ£;>O, + >0, this equation for a} has two solutions if

M=(rJC3- 21n5

- q,q2 ). ( n2 - ~ )+ ~rJ <0 (21)\ ~ +rJ£; 90rq,q4 ra, 5q,

(19)

No.2 LOW-SPECfRUM MODEL STUDY 117

When M~O, one solution exists. When M <0, three solutions exist. Because M=M(Ro, Re, e, A), we discuss M as follows:

Based on the real Kuroshio south of Japan, we adopt horizontal characteristic lengthL=300 km (which is about the amplitude of the Kuroshio's large meander from north tosouth), and [3=2.0 x 10-11 m-Is-I.

~e represents the horizontal dissipation and s the bottom friction. They inhibit the

occurrence of the multiple solutions. A represents the amplitude of the shearing and theinputted energy from the boundary. When A is large" the shearing increases, but theinputted energy from the boundary decreases. So only the rnid-amplitude of A representsthe strong nonlinear effect, and gives rise to three solutions. Ro represents the nonlinearterm, a large Ro implies strong nonlinear effects.

Fig. 2 shows: when Re decreases and s increases, only one solution exists. Otherwise,three solutions exist.

0.21

<.> ~0.11

3

0.01 I I

200 I 21 41 61 81 lOl

0.31 ~-----------,

0.21

0.11

100 150

R,

Fig. 2 Relationship of Re+ e for multiple solutions(I: single solution, 3: three solutions) (Ro =0.5, A= 100)

0.31 r------------,

Fig. 3 Relationship of A-e for multiple solutions(I: single solution, 3: three solutions) (Re= ,00סס1 Ro =0.5)

Fig. 3 shows how A and e affect the multiple solutions. When e is small or A is rather

60 .-----------,0.20

0.18

J.16

cc .

0.14

1-

0.12 3

0.10 I

0.0 0.2 0.4 0.6

--------0.8 1.0

R.Fig. 4 Relationship of Ro - e for multiple solutions

(I: single solution, 3: three solutions) (Re= 1000, A= 100)

40

R.Fig. 5 Relationship of Ro - Afor multiple solutions

(1: single solution, 3: three solutions) (Re= 100, A=O.I)

118 CHINESE JOURNAL OF OCEANOLOGY AND LIMNOLOGY Vo1.14

small three solutions exist. Otherwise, a single solution exists.From Fig. 4, when s is large, there is one solution. Otherwise, there are three solu-

tions.Fig. 5 shows that when Re= 100, e=O.1, which is mid -dissipative when A increases

and the inputted energy decreases, one solution exists.The combination of Ro, Re, e and A affects the multiple solutions. Large Ro and

mid-amplitude A strengthen the nonlinear effect and give rise to the three solutions, smallRe and large' s reduce the occurrence probability of the three solutions.

Fig. 6 gives the stream functions of three solutions when Re= 114, Ro =0.5, A= 100,e=0.08. What do the three solutions represent? After discussing the stability of the threesolutions, we can answer the question.

II ~2~1~~3zl:::::::::t~41r-I~=' ::t:11=::::::r::-2.-1/,====3~1=:::J41r/:~~1l~~231 ~::::-,"31---r_'141

31

21

11

Fig. 6 Three solutions for Re= 144,Ro=0.5, .l.= 100, 8=0.08(a, b. is the stable solutions, c. the unstable solution)

STABILITY

Now let us discuss the stability of the three solutions. In (16), suppose a,(t)=aj+alt)(i= 1, 2, 3, 4) where a, is the steady solution acquired in the preceding section, and at(t)is the small perturbanee of aj • Linearizing equation (16) yields:

(22)

Let a,(t)= Al\ substitute it into (22), then

.4+bt-2+b2•2+ b.x+b, =04

where bt = <Lq)frTi=l

(23)

No.2 LOW-SPECfRUM MODEL STUDY 119

_I " Jil h., h4 h) })b)-lr~k qi%.qk+90 (q,+qJ+ h; (q)+qJ- 90h2 (ql+qJ- 90h2 YJ

rr h, h) 4b4=(qlqiM4+ 90 qlq2+ h q)q4- 90h qJIYJ

2 2

2nwhere h; = 15 -YJC)+~

(24)

~= 1n"3 +YJ~

~ = r'a.a, (25)

h = r~4 n .

"3 +YJ~

According to Hurwitz's theory, the determinant constituted by the coefficients of thecharacteristic equation of the differential equation can determine the stability of the solu-tion of the linearized equation:

AI= b,

I b, bo I~=

b3 b2 (26)

bl bo 0

.13= . b) b2 hI0 b4 b3

.14 =b4.1)

.1;>0 is the sufficient and neccessary condition for a stable solution (i= 1, 2, 3, 4).Following the preoeding section, we can acquire the multiple solutions. According thethe above criterion, we can check which solution is stable or unstable.

In the example of the preoeding section, there are three solutions. After calculating the.1.-, we find the first two solutions are stable and the third is unstable.

For the Kuroshio south of Japan, there are also two stable paths. Our study regionis similar to the ooeanic one south of Japan, and the parameters we select in the exampleof the preoeding section accords with the real Kuroshio. This means that the two sta-ble solutions (Fig. 6a, b) represent the two kinds of paths of the Kuroshio south ofJapan, and the unstable solution (Fig. 6c) represents the transitional path of the Kuroshio.

120 CIllNESE JOURNAL OF OCEANOLOGY AND LIMNOLOGY Vo1.14

DISCUSSION

With Galerkin's low-spectrum method, we simplify the solving process of thenonlinear equation. Although the exact solution to the original equation cannot be ac-quired by the low-spectrum method, the method can still reveal the general characteristicsof the solutions.

The inlet/outlet flowing condition presents the nonlinear interaction between the bound-ary force and .inner flow represented by F(cp) which is absent when the external force isthe wind stress.

Due to the complication of the problem of bimodality of the Kuroshio, and forconvenience in solving the equation, we select a simplified region which has thesymmetrical inlet/outlet condition. When the inlet/outlet condition is asymmetrical, the prob-lem is too difficult to solve. In fact, according to Dong et al. (1993), the effect of the dif-ferent locations between the inlet and the outlet on the flowing paths is minor.

The present study led to the following conclusions:1. The combination of Ro, Re, I) and A affects the multiple solutions: large Ro and

mid-amplitude A strengthen the nonlinear effect and give rise to the three solutions, smallRe and large I) reduce the occurrence probability of the multiple solutions.

2. With Hurwitz's theory, a criterion can be obtained for checking if a solution is sta-ble or unstable.

3. Within a certain range of Re, Ro, A and 8, three solutions can be obtained, twoof which are stable and represent the bimodality of the Kuroshio south of Japan.

References

Dong Changrning, Zhang Qinghua, 1993. On the Bimodal Characteristics of the Kuroshio South of Japan. PACDNSymposiwn. China Ocean Press. Beijing pp. 341- 350.

Ji Zhongzheng, Zheng Chaozhou, 1990. The multiple solutions to a low-spectrum model of wind -driven ocean currentequation. Scientia Atmos. Sinica 14(4): 395-403. (in Chinese)

Masuda, A., 1982. On the interpretation of the bimodal characteristic of the stable Kuroshio path. Deep Sea Res. 29:471-484.

Yasuda, 1., Yoon, 1., 1985. Dynamics of the Kuroshio large meander- barotropic model 1. J. Ocean Soc. JafX1J1.

41: 259 - 273.Yoon, J., Yasuda, 1., 1987. Dynamics of the Kuroshio large meander: twr+Iayer model. J. P. O. 17: 66-81.Veronis, G., 1963. An analysis of wind-driven ooean circulation with a limited number of Fourier component. J. Atmos.

Sci. 20: 577- 593.