Angelo Rubino, Peter Brandt and Katrin Hessner- Analytical Solutions for Circular Eddies of the...

8/3/2019 Angelo Rubino, Peter Brandt and Katrin Hessner- Analytical Solutions for Circular Eddies of the Reduced-Gravity, Shallow-Water Equations

1/4

MAY 1998 999N O T E S A N D C O R R E S P O N D E N C E

1998 American Meteorological Society

Analytical Solutions for Circular Eddies of the Reduced-Gravity,Shallow-Water Equations

ANGELO RUBINO, PETER BRANDT, AND KATRIN HESSNER

Institut fu r Meer eskunde, Universitat Hamburg, H amburg, Ge rmany

4 March 1997 and 14 July 1997

ABSTRACT

New analytical, circular eddy solutions of the nonlinear, reduced-gravity, shallow-water equations in a rotatingsystem are presented. While previous analytical solutions were limited to the description of pulsons, which areoscillating, frontal, warm-core eddies with paraboloidic shape and linear velocity components, the new solutionsdescribe more general radial structures of eddy shape and azimuthal velocity. In particular, the new solutions,which contain as a subset the circular pulson solution, also allow for the description of circular, frontal, warm-core eddies with small azimuthal velocities at their periphery and/or with motionless cores, which are frequently

observed characteristics of warm-core eddies in the World Ocean.

1. Introduction

Frontal, warm-core eddies are mesoscale flow fea-tures associated with isolated water masses of anoma-lous water. Thus their properties differ significantly fromthose of the surrounding ambient water, from which theyare separated by a closed frontal line at the sea surface.Such eddies are observed very frequently in the worldocean, the most famous example being perhaps thewarm-core rings released by the meanders of the GulfStream (Cheney et al. 1976; Evans et al. 1984; Joyce1984). The influence of frontal, warm-core eddies onthe larger-scale ocean circulation involves the transferof energy and physical, chemical, and biological prop-erties across frontal zones and their impact on mixing(Olson 1991). This relevance of frontal, warm-core ed-dies explain why, in the last two decades, many ana-lytical, numerical, experimental, and observational in-vestigations have been carried out to elucidate differentaspects of their dynamics (e.g., Saunders 1971; Olsonet al. 1985; Pavia and Cushman-Roisin 1988; Rogers1989; Pavia and Lopez 1994; Matsuura 1995; Cushman-Roisin and Merchant-Both 1995).

In the study of frontal, warm-core eddies, the reduced-

gravity, shallow-water equations have been widely used(Csanady 1979; Nof 1983; Killworth 1983; Cushman-Roisin et al. 1985; Cushman-Roisin 1987; Arai et al.1994). Although these equations exclude relevant oce-anic processes such as baroclinic instabilities, they rep-

Corresponding author address: Dr. Angelo Rubino, Institut furMeereskunde , Universitat Hamburg, Troplowitzstr. 7, D-22529 Ha m-burg, Germany.E-mail: [email protected]

resent an efficient tool for the description of many char-acteristics of eddy dynamics. Moreover, an exact ana-lytical solution of these equations has been found, whichdescribes the temporal and spatial evolution of a specialclass of frontal, warm-core eddies, the so-called pulsons,characterized by a paraboloidic shape and by horizontalvelocities that are linear functions of the horizontal co-ordinates (Thacker 1981; Cushman-Roisin 1987; Rogers1989). Actually, observed frontal, warm-core eddies of-ten differ remarkably from the pulson either in their

velocity field or in their shape (Richardson et al. 1973;Andrews and Scully-Power 1976; Evans et al. 1985 andreferences therein). In the present paper we propose anextension of the circular pulson solution that can be usedto describe more general radial structures of eddy shapeand azimuthal velocity. The paper is organized as fol-lows: In section 2 the reduction of the problem to a setof series of ordinary differential equations in time ispresented. The general analytical solutions of the prob-lem are given in section 3, and, in section 4, these so-lutions are discussed and conclusions are presented.

2. Reduction of the problem to a system of

ordinary differential equationsAssuming circular symmetry, the nonlinear, reduced-

gravity, shallow-water equations for a rotating systemexpressed in polar coordinates are

r

f 0, (1)r rt r r

2 hr r f g 0, (2)r

t r r r


2/4

1000 VOLUME 28J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y

h h hr r 0. (3)

t r r

Here h is the thickness of the upper layer, f is the (con-stant) Coriolis parameter, g is the reduced gravity, t isthe time, and

and r represent the azimuthal and

radial velocity components of the upper layer along the and r coordinates. Positive r are directed from thecoordinate origin, located at the eddy center, toward theeddy periphery, and positive

are directed counter-

clockwise.Solutions of (1)(3) are searched that represent cir-

cular, frontal, warm-core eddies with the following ve-locity field and shape:

n

2i1 L r (4) ii1

Kr (5)r

2n1

2i

h A r , (6)i

i0

where the coefficients Li , K, and Ai are functions of timeonly and n 1 is the order of the system. Inserting(4)(6) into (1)(3) yields the following series of or-dinary differential and algebraic equations:

dLi 2iKL f K 0, i 1, , n (7)i i1

dt

i

dK 2 L L K f L 2igA 0, i 1, , 2n 1 (8)i1 j ij1 i1 i idt j1

dAi 2(i 1)KA 0, i 0, , 2n 1, (9)i

dt

where i1 is the Kronecker delta and L i 0 for i n.For n 1, this set of series yields the four ordinary

differential equations that describe the circular pulson:

dL1 2KL f K 0 (10)1

dt

dK2 2 L K f L 2gA 0 (11)1 1 1

dt

dA0 2KA 0 (12)0

dtdA1

4KA 0. (13)1dt

From (10)(13) the coefficients L1 , K, A0 , and A1 canbe determined (Cushman-Roisin 1987). This system ofequations has four degrees of freedom. Every successiveorder m introduces in the system three additional or-dinary differential equations [(7) with i m and (9)

with i 2m 2 and i 2m 1] and two algebraicequations [(8) with i 2m 2 and i 2m 1]. Thus,due to these two algebraic equations, only one degreeof freedom is added to the system for every successiveorder. Note that Eq. (8) for i 1 represent the set ofconditions relating Li (1 i m) and Ai (2 i 2m 1). It can be shown that, if these conditions are sat-isfied initially, then they will be satisfied for every time.

Before presenting the general solutions of (7)(9), wehave to note that the description of a circular eddy whosefluid circulates inside of a surface frontal line requires

that the following conditions are satisfied: (i) h(r) 0for r R, where R 0 is the time-dependent eddy

surface radius and (ii) h(r) 0 for r R. It can beshown that, if conditions (i) and (ii) are satisfied initially,then they will be satisfied for every time. Thus the so-

lutions of (10)(13) are valid at all times for r

R.

3. General solutions

The general solutions of (7)(9) are

f LiL ,i i1 i2 [1 sin(ft )]

i 1, , n, (14)

f cos(ft )K , (15)

2 [1 sin(ft )]

AiA ,i i1[1 sin(ft )]

i 0, , 2n 1, (16)

where , , and A0 are arbitrary constants of integration.The range of applicability of these parameters is 0 1, 0 2, and A0 0. The last condition isrequired for the description of eddies whose fluid cir-culates inside of a surface frontal line. The coefficientsAi and Li are related to each other by


3/4

MAY 1998 1001N O T E S A N D C O R R E S P O N D E N C E

FIG. 1. Interface depth h (a), azimuthal velocity component

(b),and radial velocity component r (c) of a circular, frontal, warm-coreeddy for the times t 0, t ()T, t ()T, and t ()T (T 2/f) as function of the radius r. The following values for the pa-rameters of (17)(20) have been used: n 4, f 7 105 s1, g

102

m s2

, A0 500 m, 0.04, 0, L1 1.492 106

s1 , L2

9.169 1015 m2 s1, L3

1.088 1024 m4 s1, andL4 3.177 10

35 m6 s1 .

i2 2f (1 ) 2igA L L ,i i1 j ij1

4 j1

i 1, , 2n 1, (17)

with Li 0 for i n. Together with conditions (17),

the solutions of (1)(3), which describe circular, frontal,warm-core eddies, are

n 2i1f r r L , (18) i i2 [1 sin(ft )]i1

f cos(ft ) r, (19)r

2 [1 sin(ft )]

2n1 2irh A . (20) i i1[1 sin(ft )]i0

In analogy to the circular pulson solution (Cushman-Roisin 1987), these solutions describe oscillating, cir-cular eddies that alternatively contract and deepen, ex-

pand and shoal during an (exact) inertial period T 2/f.

4. Discussion

The new analytical solutions of the nonlinear, re-duced-gravity, shallow-water equations (1)(3) given by(17)(20) can be used to describe the temporal and spa-tial evolution of frontal, warm-core eddies. While theyrepresent an extension of the circular pulson solution(Cushman-Roisin 1987) to more general radial struc-tures of eddy shape and azimuthal velocity, they retainthe radial velocity structure of the circular pulson. This

implies that the temporal evolution of the eddy centerdepth and radius remains unchanged, independent of theorder of the solution. However, frontal, warm-core ed-dies whose azimuthal velocities even largely differ froma linear function of the radius and/or whose shapes evenlargely differ from a paraboloid exist as analytical so-lutions of (1)(3). One example is depicted in Fig. 1.Here, the depth of the interface separating the eddy fromthe ambient water (Fig. 1a), the azimuthal (Fig. 1b), andthe radial velocity components of the eddy (Fig. 1c) areshown as functions of the radius for four different timesof an inertial period. In this case the maximum steepnessof the interface depth is not located at the eddy peripheryas it would be in the case of the pulson. Accordingly,

the associated azimuthal velocity decreases from itsmaximum, located inside the eddy, toward the peripherywhere it is almost zero (Fig. 1b). Thus, due to thischaracteristic of our solutions, a more realistic descrip-tion of eddy dynamics is possible than is possible usingthe previous circular pulson solution (Cushman-Roisin1987), which predicts unrealistically large azimuthal ve-locities at the eddy periphery. Furthermore, a centralregion (core) of the eddy exists where the interface depthis almost constant (Fig. 1a). Accordingly, the associatedazimuthal velocity is, in this region, almost zero (Fig.

1b). This characteristic of our solutions corresponds toa frequently observed characteristic of frontal warm-core eddies that also cannot be described by the pulsonsolution. Note, however, that the radial velocity is, asin the case of the circular pulson, a linear function ofthe radius (Fig. 1c). The temporal evolution of the eddycan be described as follows: At t 0 its radial velocityis divergent and, consequently, the eddy expands andshoals. At t ()T the eddy is shallowest. At this time

its azimuthal velocity is maximum, while its radial ve-locity vanishes. At t ()T the radial velocity of theeddy is convergent and, consequently, the eddy con-tracts and deepens. At t ()T the eddy is deepest. Atthis time its azimuthal velocity is minimum, while itsradial velocity vanishes again.

The presented solutions can also be used to infer theinternal structure of frontal, warm-core eddies from sur-face measurements. For example, remote sensing tech-niques give the possibility to measure the spatial extentas well as the velocity structure of frontal eddies at the


4/4

1002 VOLUME 28J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y

sea surface. If this information is provided, the coeffi-cient Li can be calculated by fitting the measured ve-locity by a polynomial. Thus, using (17), the coefficientsAi (i 1) can be calculated, which describe the spatialstructure of the eddy. The center depth A0 can thus beobtained by letting the measured and calculated eddysurface radius coincide. In this way, the whole stationarystructure of the eddy can be determined. A sequence ofremote sensing measurements could then be used toestimate the eddy pulsation in time.

Acknowledgments. This work was partly funded bythe German Space Agency (DARA) under Contract50EE9413. The authors would like to thank StefanoPierini, Dmitri Sein, Juan Heberto Gavino-Rodriguez,and Werner Alpers for helpful discussions.

REFERENCES

Andrews, J. C., and P. Scully-Power, 1976: The structure of an EastAustralian current anticyclonic eddy. J. Phys. Oceanogr., 6, 756765.

Arai, M., and T. Yamagata, 1994: Asymmetric evolution of eddiesin rotating shallow water. CHAOS, 4, 163175.

Cheney, R. E., W. H. Gemmill, and M. K. Shank, 1976: Tracking aGulf Stream ring with SOFAR floats. J. Phys. Oceanogr., 6,741749.

Csanady, G. T., 1979: The birth and death of a warm core ring. J.Geophys. Res., 84, 777780.

Cushman-Roisin, B., 1987: Exact analytical solution for ellipticalvortices of the shallow-water equations. Tellus, 39A, 235244., and S. Merchant-Both, 1995: Elliptical warm core rings in a

two-layer ocean with ambient shear. J. Phys. Oceanogr., 25,20112024., W. H. Heil, and D. Nof, 1985: Oscillations and rotations ofelliptical warm-core rings. J. Geophys. Res., 90, 11 75611 764.

Evans, R. H., K. S. Baker, O. B. Brown, R. C. Smith, S. Hooker, D.Olson, and the Warm-Core-Rings Program Ser vice Office, 1984:Satellite images of warm-core ring 82-B, sea surface temperature

and a chronological record of major physical events affectingring structure. Woods Hole Oceanographic Institution Rep., 25pp. [Available from Woods Hole Oceanogr. Inst., Woods Hole,MA, 02543.], , , and , 1985: Chronology of warm-core ring82B. J. Geophys. Res., 90, 88038811.

Joyce, T. M., 1984: Velocity and hydrographic structure of a warm-core ring. J. Phys. Oceanogr., 14, 936947.

Killworth, P. D., 1983: On the motion of isolated lenses on a beta-plane. J. Phys. Oceanogr., 13, 368376.

Matsuura, T., 1995: The evolution of frontal-geostrophic vortices ina two-layer ocean. J. Phys. Oceanogr., 25, 22982318.

Nof, D., 1983: On the migration of isolated eddies with applicationto Gulf Stream rings. J. Mar. Res., 41, 399425.

Olson, D. B., 1991: Rings in the ocean. Annu. Rev. Earth Planet.Sci., 19, 283311., R. W. Schmitt, M. Kennelly, and T. M. Joyce, 1985: A two-layer diagnostic model of the long-term physical evolution of

warm-core ring 82B. J. Geophys. Res., 90, 88138822.Pavia, E. G., and B. Cushman-Roisin, 1988: Modeling of oceanic

fronts using a particle method. J. Geophys. Res., 93, 35543562., and M. Lopez, 1994: Long-term evolution of elongated warmeddies. J. Phys. Oceanogr., 24, 22012208.

Richardson, P. L., A. E. Strong, and J. A. Knauss, 1973: Gulf Streameddies: Recent observations in the western Sargasso Sea. J. Phys.Oceanogr., 3, 297301.

Rogers, C., 1989: Elliptic warm-core theory: The pulsrodon. Phys.Lett., 138, 267273.

Saunders, P. M., 1971: Anticyclonic eddies formed from shorewardmeanders of the Gulf Stream. Deep-Sea Res., 18, 12071220.

Thacker, W. C., 1981: Some exact solutions to the non-linear shallow-water wave equations. J. Fluid Mech., 107, 499508.

Angelo Rubino, Peter Brandt and Katrin Hessner- Analytical Solutions for Circular Eddies of the...

Documents

Transcript of Angelo Rubino, Peter Brandt and Katrin Hessner- Analytical Solutions for Circular Eddies of the...