Inertial Instability of Arbitrarily Meandering Currents Governed by the Eccentrically Cyclogeostrophic Equation

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1 Jornal of Oceanography, Vol. 59, pp. 163 to 17, 3 Inertial Instability of Arbitrarily Meandering Crrents Governed by the Eccentrically Cyclogeostrophic Eqation HIDEO KAWAI* Shibagahara, Kse, Joyo, Kyoto Pref. 61-1, Japan (Received 11 Jne 1; in revised form 9 Jly ; accepted 5 Agst ) Using natral coordinates, we have derived a criterion for the inertial instability of arbitrarily meandering crrents. Sch crrents, governed by the eccentrically cyclogeostrophic eqation, are adopted as the basic crrent field for the parcel method. We assme that any virtal displacement which is given to a water parcel moving in the basic field has no inflence on this field. From the conservation of mechanical energy for a virtal displacement we derive an inertial instability freqency ω m = [( f + /r)z].5 for the eccentrically cyclogeostrophic crrent, where f is the Coriolis parameter, the velocity (always positive), r the radis of crvatre of a streamline (negative for an anticyclonic meander), and Z the vertical component of absolte vorticity. If ω m is negative, the eccentrically cyclogeostrophic crrent becomes nstable. Althogh the conventional, centrifgal instability criterion, derived from the conservation of anglar momentm in a circlarly symmetric crrent field, has a certain meaning for a monopolar vortex, it contains a radial shear vorticity that is difficlt to se in arbitrarily meandering crrents. The new criterion ω m contains a lateral shear vorticity that is applicable to arbitrarily meandering crrents. Examining instabilities of concentric rings with radii of 5 1 km, we consider reasons why the anticyclonic spersolid rotation has been very mch less freqently observed than the cyclonic spersolid rotation, despite a prediction of some common stability and a rapid change in radial velocity gradient for the former. Classifying eccentric streamlines into the large and small crvatre-gradient types, we point ot that the largegradient crvatre in anticyclonic rings is apt to be nstable. Keywords: Inertial instability, eccentrically cyclogeostrophic crrent, natral coordinates, parcel method, virtal deflection, instability freqency, ring, sbsolid rotation, spersolid rotation, zero potential vorticity. 1. Introdction While the inertial instability of circlarly symmetric flows has been treated since Rayleigh (1916), that of arbitrarily meandering crrents, having no symmetry with respect to any point or any line, has received little attention, as far as the athor is aware. To derive the instability freqency for sch arbitrarily meandering crrents, we apply the parcel method to the crrents that are governed by the eccentrically cyclogeostrophic eqation, expressed in terms of natral coordinates. In the parcel method we assme that any virtal displacement of a water parcel has no inflence on the basic crrent field arond the parcel. The basic crrent field means a steady crrent field that is to be examined to see whether it is stable or not with respect to the virtal displacement. Applying the parcel method to the basic crrent field * address: kawaihid@d.dion.ne.jp Copyright The Oceanographic Society of Japan. of meandering crrents, governed by the eccentrically cyclogeostrophic eqation, we find a relation between a virtal displacement of a water parcel across streamlines of the basic field and a virtal deflection-angle of the corse of the parcel, made to the streamline. From the conservation of mechanical energy for the virtal displacement we derive an inertial instability freqency for the eccentrically cyclogeostrophic crrent in terms of natral coordinates. When this instability freqency is imaginary, we can regard the basic crrent field as nstable. Using the inertial instability criterion we examine velocity strctre in anticyclonic and cyclonic rings with radii of 5 1 km.. Natral Coordinates We se two right-hand coordinate systems. One is the sal coordinate system (x, y, z): positive x and y are eastward and northward, respectively, on a horizontal plane; positive z is pward along a vertical line. The other is a natral coordinate system (ξ,, z): positive ξ is down- 163

2 stream along a streamline; along an orthogonal to streamlines; z the same as the above. ξ and are not crvilinear bt metric coordinates on a horizontal plane. The relation between nit vectors (i, j) along the (x, y)-axes and (i ξ, i ) along the (ξ, )-axes becomes i ξ icosθ + jsinθ, i isinθ + jcosθ, i ξ, (1) where θ is an angle of i ξ made to i, is the horizontal crrent velocity vector, and (the scalar of ) becomes positive in any case. The horizontal gradient operator H and the directional differentiation along the natral coordinate axes ( / ξ, / ) are given by H i / x + j / y = i ξ / ξ + i /, H + k / z () / ξ i ξ = cosθ / x + sinθ / y / i = sinθ / x + cosθ / y, (3) where k is the nit vector along the z-axis. Definition (1), () and (3) easily lead to the following relations: = i / x + j / y + k / z = i ξ / ξ + i / + k / z i ξ = (i / x + j / y + k / z) (icosθ + jsinθ) = i ξ θ/ z + k θ/ ξ i = i θ/ z + k θ/ i ξ = (i / x + j / y + k / z) (icosθ + jsinθ) = θ/ i = θ/ ξ = (i ξ ) = i ξ i ξ = i ξ θ/ z + i / z + k( θ/ ξ / ) (4) = (i ξ ) = i ξ + i ξ = / ξ + θ/. The radis of crvatre of a streamline is defined by 1/r θ/ ξ, (5) where r is measred from a point on the streamline toward the center of crvatre of the streamline (Fig. 1(b)). This measring direction is opposite to the measring direction of R in polar coordinates (Fig. 1(a)). Since r defined by (5) becomes positive (negative) for a cyclonically (anticyclonically) meandering streamline, the direction of r coincides with the positive (negative) -axis. Fig. 1. Difference in kinematic notations between horizontal polar coordinates (a) and natral coordinates (b). + signs in front of R, ξ and indicate the positive sense of each coordinate axis. Here and hereafter the cyclonic or anticyclonic deflection means that in the Northern Hemisphere. By the definition (3), the directional differentiation of sm, prodct and qotient of two or more fnctions, composite fnction, etc., are obtained nder the same rle as in the sal differentiation, even if θ is dependent on space coordinates and time. When considering the directional differentiation of the second order, however, the difference between the sal directional differentiation and the extended one becomes distinct (Kawai, 1957). 3. Eqations of Motion of an Unsteady State Althogh we regard the basic crrent field for the parcel method as steady, we need eqations that govern the motion of a virtally displaced parcel. This section is devoted to expressing eqations of motion of an nsteady state in terms of the natral coordinates. Assming hydrostatic eqilibrim of the basic crrent field, we can redce p/ρ at z in a top or pper layer with niform density to p/ρ = g (ς z), (6) where p is the presser, ρ the density, g the acceleration of gravity, and ς an elevation of the sea-srface of the basic field above a reference level. The eqation of motion for horizontal inviscid flow becomes t H Hp + + kz + =, ( 7) ρ where Z is the vertical component of absolte vorticity. From (4) and (5), 164 H. Kawai

3 Z = f + k = f +/r /, (8) where f is the Coriolis parameter. From (1), = ( iξ) t t = i ξ + i t θ. 9 t ( ) From (9), (8) and (6), scalar prodcts of (7) with i ξ and i become d 1 p d + g = + ς = ( 1 ρ ξ ξ ) dθ p d f + f g + 1 = + θ + ς ρ =, ( 11 ) respectively, where the material differentiation is defined by d/ / t + / ξ. (1) 4. Acceleration de to Virtal Displacement We regard the state of a field of the eccentrically cyclogeostrophic crrent as basic (Section 1). Neglecting / t and θ/ t contained in (1) and (11), respectively, we get eqations that govern the eccentrically cyclogeostrophic crrent, ξ b b f + r + gς b b = ( 13 ) g + ς =, ( 14) where the sbscript b indicates the basic state. Adopting the parcel method, we conventionally assme that any virtal displacement of a water parcel has no inflence on the basic crrent field arond the parcel. Using (1), (13), (11), (5) and (14), we derive the ξ- and -components of the eqation of motion for the parcel as follows: d b = g ς b = ξ ξ ( 15) θ b g f fb f = ς t r = + r. r ( 16 ) b Fig.. Geometry of virtal displacement. P is the initial position occpied by a water parcel. P d is the final position after a virtal displacement v = θ v ξ, where ξ = t means a displacement of the parcel along a basic streamline ξ for a minte time interval t. ξ and ξ d are basic streamlines passing throgh P and P d, respectively. θ v is a virtal deflection-angle made to ξ at P. C and C d are crvatre centers of streamlines, ξ at P and ξ d at P d, respectively. r and r d are radii of crvatre of streamlines, ξ at P and ξ d at P d, respectively. Althogh (16) does not contain any time derivative of v, from (19) below we shall find the term θ/ t in (16) eqivalent to d v /. Let a parcel, initially located at P, move with a crrent velocity along a basic streamline ξ that passes throgh P (Fig. ). Add to the moving parcel a virtal, minte, cross-stream displacement v for a minte time interval t. Denoting a virtal deflectionangle of the corse of the parcel, made to ξ, by θ v, we express v in terms of θ v as follows: v = θ v ξ = θ v t, (17) where ξ = t is a displacement of the parcel along the streamline ξ for t. From (17), d v / = θ v, (18) becase t is an independent variable, which can be sfficiently minte. From (18), 19 d v d θv d θ, = = ( ) becase θ v is an arbitrary variable independent of spatial coordinates. In the last term of (19), the sbscript has been removed from, in order that (19) may hold not only at P bt also at any point. Before sbstitting the Inertial Instability of Arbitrarily Meandering Crrents 165

4 last term of (19) by (16), sbscripts on the rightmost side of (16) shold be changed as below, becase the acceleration of (19) is indced by the virtal displacement, namely, by the vales of the qantities on the rightmost side of (16) at the displaced position P d. Sbstitte by c to denote a qantity conserved for a virtal displacement. Sbstitte the sbscript b by d to indicate qantities at P d. Sbstitte r by r d from the assmption of no inflence on the basic field by any virtal displacement of a parcel. Ths, sbstitting the last term of (19) by (16), which has been changed as mentioned above, we get The material differentiation of (5) becomes de = d g d d + ς g = + ς ξ =, ( 6) becase of relations, ς/ t = for the basic field and (1). This represents the conservation of mechanical energy of a parcel for the virtal displacement. From (6) with (5), ths c = + g (ς ς c ). (7) d v c θ d = f d c = ( )+ t r d. ( ) Expansion of ς c in powers of ξ and v abot ς, by sing (13) and (14), yield 5. Redcing Qantity Vales at Virtally Displaced Position This section is devoted to redcing vales of qantity at P d in () to those at P. 5.1 Qantities depending on basic crrent field at new position P d Vales of d, d and 1/r d at P d redce to their vales and gradients at P as follows: = + O + ξ ξ ξ, d v v + ( ) ( 1) d = + + v O v ξ ξ, + ( ) ( ) = + v O v 3 rd r r + ξ r + ( ξ, ). ( ) ξ Here O( ξ, v ) denotes a sm of negligible terms with the sqared ( ξ, v and v ξ) or frther higher (n ) powers of infinitesimal qantities ( ξ, v ) n, omitting some factors of finite qantities, for simplicity. 5. Qantities depending on property conserved by a parcel for virtal displacement Mechanical energy per nit mass of an incompressible water parcel is given by Sbstittion of (6) into (4) yields E = / + p/ρ + gz. (4) E = / + gς. (5) ς ς g( ς ςc)= g ξ g v O v ξ, = + ( ) ξ + f + v O ξ v 8,. r Sbstittion of (8) into (7) yields + ( ) ( ) c = + ξ + f + v O ξ, v, 9 r which leads to c ξ = 1+ + f + r v + ( ) ( ) 1 / = + ξ + f + v O ξ, v. r + O( ξ, ) + ( ) ( 3) 6. Inertial Instability Criterion Sbtracting (3) from (1), in consideration of (8), we get f ( d c ) = f Z v + O( ξ, v ). (31) Sbtracting (9) from (), in consideration of (8), we get d c = Z v + O( ξ, v ). (3) Mltiplying (3) by (3), we get ( d c )/r d = Z v /r + O( ξ, v ). (33) v 166 H. Kawai

5 Sbstitting (31) and (33) into (), and neglecting higher order terms, we redce () to d v = f + Z r v ωmv = ( 34) ω m = ( f + /r)z. (35) In (34) and (35), the sbscript has been removed from, r and Z, in order that they may hold not only at P bt also at any point. For concentric streamlines, we write r d = r v, which leads to v v = = + + O( v ). ( 36) r r r r r d 1 Taking the same procedres as mentioned before (34) except for sing (36) instead of (3), we obtain (34) and (35), too. Whether adopting the eccentrically cyclogeostrophic flow or the exactly cyclogeostrophic flow as the basic field, we apparently get the same form of ω m as was given by (35), bt expressions of Z are essentially different from each other, as will be shown between (41) and (4), below. If ω m >, (34) represents a simple harmonic motion with a freqency ω m. If all of the relative vorticities in (35) vanish, ω m redces to f, the freqency of the inertial oscillation. If ω m <, d v / increases with increasing v, which means the inertial instability. According to common meteorological textbooks, the inertial instability freqency for a zonal geostrophic wind ω z is given by ω z = f ( f / y) = f Z, (37) becase ( f / y) is eqivalent to Z for a zonal jet stream by (8). The sign of ω z given by (37) depends on the sign of Z only, bt the sign of ω m given by (35) depends on the sign of ( f + /r)z. A postscript to Rayleigh (1916) states, It may have been observed that according to what has been said above the stability of flid motion in cylindrical strata reqires only that the sqare of the circlation increase otwards. This, Rayleigh s stability criterion, is formlated as d(vr) /dr >, (38) according to Kloosterziel and van Heijst (1991, hereafter KH), where R is the radis, which is measred otward from the center of circlar streamlines to a point nder consideration, positive for both anticyclonic and cyclonic vortices, and V is the crrent velocity, which is positive (negative) for an anticyclonic (a cyclonic) vortex (Fig. 1(a)). Since R is always positive, division of (38) by R 3 leads to (V/R) (V/R + dv/dr) >. (39) Assming a circlarly symmetric basic state for the se of horizontal polar coordinates, several athors (e.g., KH; Carnevale et al., 1997; Potylitsin and Peltier, 1998; Orlandi and Carnevale, 1999) mentioned a criterion for the inertial instability in the f-plane, which sed to be called the centrifgal instability. Among them KH obtained a stability criterion d( fr / + VR) /dr >, (4) by applying a parcel method to the basic field of circlarly symmetric flows, governed by the exactly cyclogeostrophic eqation. Since R is positive, division of (4) by R 3 leads to ( f + V/R) ( f + V/R + dv/dr) >. (41) From (35) with (8), the stability criterion obtained in the present stdy becomes ω m = ( f + /r) ( f + /r / ) >. (4) This criterion (4) looks similar to that of (41) or (39), bt there is an essential difference between them, as will be mentioned in the next paragraph. In (39) and (41), R is always positive, directed otward in horizontal polar coordinates, bt V is positive (negative) for a cyclonic (an anticyclonic) vortex (Fig. 1(a)). By contrast, in (4), is always positive, bt r is positive (negative) for a cyclonically (an anticyclonically) meandering streamline, becase the center of crvatre of the streamline is on the left (right) of the streamline, facing downstream (Fig. 1(b)). Becase the difference between (4) and (41) is whether the basic crrent field is circlarly symmetric or not, this difference appears in the expression of shear vorticities sch as / in (4) and dv/dr in (41). Since is eqivalent to V, the essential difference in shear vorticity between (4) and (41) is the se of a variable in (4) instead of R in (41). Althogh the radial, shear vorticity dv/dr has a certain meaning for a monopolar vortex, it loses applicability to arbitrarily meandering crrents becase R is not an independent variable in the latter crrents. Adopting arbitrarily meandering crrents, governed by the eccentrically cyclogeostrophic eqation, as the basic field, and sing natral coordinates, we shall be able to apply this new criterion widely to varios crrent fields, obtained with advanced crrent measrement technology. Inertial Instability of Arbitrarily Meandering Crrents 167

6 7. Anglar-Velocity Gradient in Concentric Rings 7.1 KH stability criterion for concentric rings Becase r/ = 1 (Fig. 3) for both anticyclonic and cyclonic, concentric rings, (8) redces to Z = f + /r / = f + /r + / r. (43) Sbstittion of (43) into (35) yields ω m = ( f + /r)z = ( f + /r) ( f + /r + / r). (44) Since (44) is the same as the left side of (41) obtained by KH, it may be called the KH stability criterion for concentric rings. According to varios observations of anticyclonic and cyclonic rings by Tomosada (1975), Brown et al. (1983), Mied et al. (1983) and others, rings are more or less circlarly asymmetric, e.g., elliptical or D-shaped. Condcting laboratory experiments on barotropic vortices in a rotating flid, KH caght trianglar anticyclonic vortices on streakline photographs of small tracer particles. Despite the asymmetry of rings, Figs. 3 and 4, Tables 1 and and the analyses in Section 7 are based on schematically circlarized rings for simplicity. 7. Dependence of the sign of ω m on Z The soltion of (14), which is a qadratic eqation with respect to b, yields the azimthal crrent velocity fr 5. 4g ς = 1± 1+ f r r > for cyclones ; r < for anticyclones, 45 ( ) ( ) where the sbscript b in (14) has been omitted. In the same way as in (43), sbstittion of = r into the lateral gradient ς/ of (45) for both cyclonic and anticyclonic rings yields the radial gradient ς/ r. We sed to choose the + sign in front of the sqare root in (45), assming that the crrent decreases to zero when the horizontal pressre gradient vanishes. Althogh this assmption has not generally been accepted, it is valid for rings, becase any ring is characterized by a certain, vertical displacement of the main thermocline from that of srronding water, and hence by a remarkable radial pressre gradient. A vortex withot sch vertical displacement can no longer be classed as a ring. From (45), we can derive algebraically the traditional velocity limit for anticyclones, < cr, (46) where cr is the critical velocity defined by cr fr/. (47) Since vales of cr (Table 1) are mch higher than the vales of velocity measred in anticyclonic rings, (46) is also regarded as actally holding. From (46) with (47), f + /r > (48) for anticyclonic rings. For cyclonic rings r > and >, and hence (48) also holds. From Eqs. (48) and (44), the sign of ω m becomes the same as that of Z for both anticyclonic and cyclonic rings. Fig. 3. Radial profiles of velocity for the sbsolid (broken), solid (solid) and spersolid (chained) rotations in schematic, anticyclonic (a) and cyclonic (b) rings, which have positive and negative shear zones, divided by a maximm azimthal velocity m at r = ±r m (r < for anticyclonic rings). Fig. 4. An example of greater vales of / r of three types of rotations of cyclonic rings than those of anticyclonic rings. Typical vales of / r, which are indicated by the maximm slope of a tangent to velocity crves vs. r, are based on Fig. 5. They are measred by the nit of f = s 1, the Coriolis parameter at N. 168 H. Kawai

7 7.3 Stability of rotations in the main body of rings We call portions of a ring inside and otside of the velocity maximm ( m in Fig. 3) the main body and fringe, respectively. The main body ( / r > ) corresponds to the negative shear zone of anticyclonic rings and the positive shear zone of cyclonic rings. The fringe ( / r < ) corresponds to the positive shear zone of anticyclonic rings and the negative shear zone of cyclonic rings. In order to do the algebraic analyses leading to Fig. 5, we introdce three qantities W, X and Y, and express the radial gradient of anglar velocity by where From (48), (/r)/ r = ( / r /r)/r = W/r, (49) W X Y, X / r, Y /r. (5) Y >.5f. (51) Becase (49) with (5) characterizes radial profiles of azimthal velocity in the main body of rings, we classify rotations into three types as follows. (1) Sbsolid rotation is characterized by decreasing, absolte vales of anglar velocity with increasing r, namely, W/r <, which leads to / r <. Becase W > for anticyclonic rings (r < ), from (49) /r < / r <, namely, Y < X <. Hence, the lower limit of (43) becomes Z > f + /r = f + Y. Becase f + Y > from (51), Z becomes positive, which means that the sbsolid rotation is always stable. Becase W < for cyclonic rings (r > ), from (49) /r > / r >, namely, Y > X >. For both anticyclonic and cyclonic rings, /r > / r, namely, Y > X. () Solid rotation is characterized by a constant anglar velocity with increasing r, namely, W/r =, which leads to / r =. Becase W = for both anticyclonic and cyclonic rings, from (5) Y = X. Hence, (43) becomes Z = f + Y. Becase f + Y > from (51), Z becomes positive, which means that the solid rotation is always stable. (3) Spersolid rotation is characterized by increasing, absolte vales of anglar velocity with increasing r, namely, W/r >, which leads to / r >. Becase W < for anticyclonic rings (r < ), from (49) Table 1. Critical velocity cr = fr/ (cm s 1 ) for a radis of crvatre r (km) of anticyclonic rings. Table. Smmary of inertial instability of rings examined by ω m = ( f + /r) Z. Anticyclonic ring [r <, >, f + /r > ] Negative shear zone (Body) [ / r < ] sbsolid rotation [/r < / r < ] ω m > always stable solid rotation [/r = / r < ] ω m > always stable spersolid rotation [ / r < /r < ] If Z >, ω m > stable If Z =, ω m = netral If Z <, ω m < nstable Positive shear zone (Fringe) [ / r > ] ω m > always stable Cyclonic ring [r >, >, f + /r > f ] Positive shear zone (Body) [ / r > ] ω m > f always extremely stable Negative shear zone (Fringe) [ / r < ] If Z >, ω m > stable If Z =, ω m = netral If Z <, ω m < nstable ω m, KH stability freqency; Z = f + /r + / r, vertical component of absolte vorticity. Inertial Instability of Arbitrarily Meandering Crrents 169

8 Fig. 5. Geometrical analysis of ranges of parameters W (= / r /r) and Z (= f + /r + / r) for spersolid, solid and sbsolid rotations in the main body of anticyclonic and cyclonic rings, illstrated on the X (= / r) Y (= /r) coordinate plane. W indicates the extent to which the spersolid and sbsolid crves are warped (Fig. 4). Positive W- and Z-axes are directed to 45 clockwise from the positive X- and Y-axes, respectively. Each axis is gradated in.5f. / r < /r <, namely, X < Y <. The spersolid rotation becomes stable, netral or nstable, according as Z is positive, zero or negative. Becase W > for cyclonic rings (r > ), from (49) /r > / r >, namely, X > Y >. For both anticyclonic and cyclonic rings, / r > /r, namely, X > Y. 7.4 Stability of the fringe of rings (1) The positive shear zone of anticyclonic rings [ / r >, /r < ] always becomes stable, becase Z > as follows. From (43), Z = ( f + /r) + ( / r /r). Since ( f + /r) > from (48), and since ( / r /r) > from the ineqalities bracketed above, we get Z >. () The negative shear zone of cyclonic rings [ / r <, /r > ] becomes stable, netral or nstable, according as Z is positive, zero or negative. 7.5 Contradiction between observation and prediction by the stability criterion Figre 4 illstrates detailed radial profiles of velocity in the main body of rings. The spersolid crves in Fig. 4 appears to be warping otside the short arrows. Becase W defined by (5) is the difference in slope between / r (the slope of a tangent to the velocity crve) and /r; we can se W as an indicator of the extent to which the spersolid and sbsolid crves are warped. According to a geometrical analysis from Fig. 5, which was prepared from algebraic analyses among X, Y, Z and W (Sbsection 7.3), the pper limit of W for the stable (Z > ) spersolid rotation of anticyclonic rings is f. Althogh this vale is smaller than the pper limit of W for either spersolid or sbsolid rotation of cyclonic rings, it is greater than.5f for the sbsolid rotation of anticyclonic rings. Hence, common observations of anticyclonic rings in the spersolid rotation are predicted, as far as Z is positive. According to Kawai (199, sbsection 4.1), however, hardly any measrement has spported the continos existence of anticyclonic rings with a static core or in spersolid rotation. Most measrements have caght anticyclonic rings in solid or sbsolid rotation. Anticyclonic rings with a static core have scarcely been observed, nor have cyclonic rings withot a static core. These observational reslts of rings contradict the prediction of common observations of anticyclonic rings in spersolid rotation based on the above geometrical examination of W according to Fig. 5. This contradiction can be explained in terms of an observational limitation de to discrete measrements of velocity in connection with the difference in magnitde of W in velocity profiles. According to Fig. 5, the spersolid rotation of anticyclonic rings is stable when Z >, bt becomes nstable when Z <. By contrast, the spersolid rotation of cyclonic rings is extremely stable throghot the pper right qadrant, regardless of rotation types. Hence, W of the spersolid rotation of cyclonic rings occpies wider range than W of the stable, spersolid rotation of anticyclonic rings. Conseqently, even in case of missing velocity measrements arond the velocity maximm m, we have more opportnity to confirm the spersolid crve of cyclonic rings by obser- 17 H. Kawai

9 vations of remaining, remarkably warped crves. However, we have less opportnity to confirm the spersolid crve of anticyclonic rings by observations of remaining, less warped crves (Fig. 4). This is one way to interpret the contradiction between observation and prediction. 8. Crvatre Gradient in Eccentric Rings We can write / = ( / r) ( r/ ). (5) The term r/ at the end of (5) takes varios nmerical vales, which mst be obtained by analyzing detailed crrent measrements with modern technology. Sch vales are still navailable to the athor. First of all, we examine simple streamlines in Fig. 6. According to nmerical vales of r/, streamlines come in three types, as follows. (1) Normal-gradient crvatre type (concentric rings) is defined by r/ = 1. Althogh we have examined concentric rings in Section 7, to compare with eccentric types, this type is exemplified in Fig. 6(a) by the formlas r = r, dr = d, r/ = 1. (53) Here r is the radis of crvatre for a streamline, which passes throgh a reference point nder consideration (marked with in Fig. 6), and r is the radis of crvatre for a streamline, which passes throgh a point that is displaced by from the reference point. The +-axis is directed inward (otward) for cyclonic (anticyclonic) circlations for Fig. 6. Sbstittion of the last eqation of (53) into (5) yields / = / r, which redces (8) to (43). () Large-gradient crvatre type (eccentric rings) is defined by r/ < 1. An example is fond arond the bottom of eccentric circles in Fig. 6(b) with formlas r = r, dr = d < d, r/ =. (54) Sbstittion of the last eqation of (54) into (5) yields / = / r, which redces (8) to Z = f + /r + / r. (55) In the negative shear zone of anticyclonic concentric rings, two negative terms /r and / r in (43) redce vales of Z. Becase the radial shear term / r in (55) is twice / r in (43) for concentric rings, it works to redce Z frther. Hence, (55) shows that the large-gradient crvatre in anticyclonic rings is apt to have a negative vale of Z, and conseqently to bring an nstable state. Fig. 6. Examples of the relation between r and for concentric (a) and eccentric (b) rings with straight -axes along vertical diameters of circlar streamlines. +, r >, r > for cyclonic rings;, r <, r < for anticyclonic rings. Arond the bottom of eccentric arcs in Fig. 6(b) the interval between adjacent arcs is a half of that for concentric arcs in Fig. 6(a). This interval becomes longer toward both sides of the bottom center and then pward. When this interval for eccentric arcs is shorter than that for the concentric circles in Fig. 6(a), sch eccentric arcs belong to the large-gradient crvatre type. (3) Small-gradient crvatre type (eccentric rings) is defined by r/ > 1. An example is fond arond the top of eccentric circles in Fig. 6(b) with formlas r = r (/3), dr = (/3) d > d, r/ = /3. (56) Sbstittion of the last eqation of (56) into (5) yields / = (/3) / r, which redces (8) to Z = f + /r + (/3) / r. (57) Becase the radial shear term (/3) / r in (57) is smaller than / r in (43) for concentric rings, it works to relax a decrease in Z. Hence, (57) shows that the small-gradient crvatre in anticyclonic rings is apt to have a positive vale of Z, and conseqently to bring a stable state. Arond the top of eccentric circles in Fig. 6(b) the interval between adjacent arcs is 1.5 times that for concentric arcs in Fig. 6(a). This interval becomes shorter toward both sides of the top center and then downward. When this interval for eccentric arcs is longer than that for the concentric circles in Fig. 6(a), sch eccentric arcs belong to the small-gradient crvatre type. Inertial Instability of Arbitrarily Meandering Crrents 171

10 9. Remark What is called the zero-potential-vorticity assmption is eqivalent to the zero-absolte-vorticity assmption. Althogh sch an assmption has become an important method for analyzing nonlinear jets in rotating flids, the zero-absolte-vorticity assmption brings a netral state, according to inertial instability criteria for any type of crrents, sch as the zonal streams, concentric vortices and arbitrarily meandering crrents discssed in this paper. Becase a netral state is apt to lead to an nstable state owing to flctations, the zero-potentialvorticity jets are apt to become nstable. Acknowledgements The athor expresses his sincere thanks to anonymos referees who made critical bt constrctive comments on early drafts of this paper. References Brown, O. B., D. B. Olson, J. W. Brown and R. H. Evans (1983): Satellite infrared observation of the kinematics of a warmcore ring. Ast. J. Mar. Freshw. Res., 34, Carnevale, G. F., M. Briscolini, R. C. Kloosterziel and G. K. Vallis (1997): Three-dimensionally pertrbed vortex tbes in a rotating flow. J. Flid Mech., 341, Kawai, H. (1957): On the natral coordinate system and its applications to the Kroshio system. Bll. Tohok Regional Fisheries Res. Lab., 1, Kawai, H. (199): A model for the strctre of anticyclonic and cyclonic rings with niform dynamical energy density. Deep-Sea Res., 39, Sppl. 1, S1 S18. Kloosterziel, R. C. and G. J. F. van Heijst (1991): An experimental stdy of nstable barotropic vortices in a rotating flid. J. Flid Mech., 3, 1 4. Mied, R. P., G. J. Lindemann and J. M. Bergin (1983): Azimthal strctre of a cyclonic Glf Stream ring. J. Geophys. Res., 88, C4, Orlandi, P. and G. F. Carnevale (1999): Evoltion of isolated vortices in a rotating flid of finite depth. J. Flid Mech., 381, Potylitsin, P. G. and W. R. Peltier (1998): Stratification effects on the stability of colmnar vorticities on the f-plane. J. Flid Mech., 355, Rayleigh, Lord (1916): On the dynamics of revolving flids. Proc. R. Soc. Lond., A93, Tomosada, A. (1975): Observations of a warm eddy detached from the Kroshio east of Japan. Bll. Tokai Regional Fisheries Res. Lab., 81, (in Japanese with English abstract). 17 H. Kawai