Stationary Rossby Waves and Shocks on the Sverdrup Coordinate

Journal of Oceanography Vol. 51, pp. 207 to 224. 1995 Stationary Rossby Waves and Shocks on the Sverdrup Coordinate ATSUSHI KUBOKAWA Graduate School of Environmental Earth Science, Hokkaido University, Sapporo 060, Japan (Received 21 June 1994; in revised form 6 October 1994; accepted 8 October 1994) Gyre-scale frontal structures, e.g., the Subtropical Front, have been well documented in the oceans. Although the generation mechanism of such a front remains unclear, it may be ascribed to the steepening of nonlinear planetary waves as discussed by Dewar (1992). To discuss the stationary wave characteristics and their shock formation in a two-layer wind-driven gyre, the present paper introduces a new coordinate, referred to as the Sverdrup coordinate here, in which the Sverdrup function is used instead of the longitudinal coordinate, and especially, investigates the possibility of the frontgenesis caused by the Rossby waves emanating from the western boundary region. On the Sverdrup coordinate, since the advection by the Sverdrup flow is in the direction normal to that of the Rossby wave propagation, the system becomes much simpler than that on the (x, y)-coordinate, and the solution to this coordinate does not explicitly depend on the distribution of the Ekman pumping, i.e., we can treat cases, in which the Ekman pumping is a function of both x and y, in a similar way to the case with zonally uniform Ekman pumping. 1. Introduction Wind-driven ocean circulation theories started from Sverdrup (1947) have been giving us an image of smooth ocean circulation. It is, however, well-known that frontal structures with gyre-scale can be found even in the central gyre, e.g., the Subtropical Front (see, e.g., Hasunuma and Yoshida, 1978; Roden, 1980). Takeuchi (1984) carried out a numerical experiment and succeeded to reproduce the mid ocean front accompanied by eastward current called the Subtropical Countercurrent. The mechanism of generating the front, however, is still unclear. Recently, Dewar (1992) discussed the possibility of frontgenesis caused by nonlinear Rossby waves. If we consider a two-layer model shown in Fig. 1 without forcing (w e = 0), the propagation speed of long Rossby waves can be represented as c = βg h f 2 H H h, 1 where g is the reduced gravity (=g ρ/ρ 0 ), f is the Coriolis parameter and β = df/dy. Since the propagation speed depends on the upper layer depth, a sufficiently large amplitude wave can steepen and cause a shock as discussed by Dewar (1987). It is expected that a similar mechanism can work on stationary Rossby waves in a wind-driven gyre and may cause a stationary shock or front, like the Subtropical Front. Dewar (1992) discussed this type of frontgenesis by using the method of characteristics and gave two examples in which the steepening of nonlinear stationary Rossby wave along their characteristics generates a front. He called this type of front spontaneous shock. One of those examples is by the Rossby wave originated from the cross-gyre

208 A. Kubokawa Fig. 1. The two-layer model adopted in the present study. flow (see, e.g., Pedlosky, 1984) and the other is that originated from the outcrop line of the ventilated thermocline. While the Rossby wave from the cross-gyre flow produces only the southwestward current in the upper layer in the subtropical gyre, that from the outcrop line can produce the northeastward current. Therefore, the latter example gives a candidate of the generating mechanism of the Subtropical Front, although the northward inclination of surface density contour, which is necessary for the Rossby wave to produce a front in a ventilated thermocline model, cannot be observed in Takeuchi s experiment. In addition to the above two examples presented by Dewar (1992), the western boundary can also be the source of Rossby wave, because it is an open boundary. The frontgenesis caused by such Rossby waves will be discussed in the present paper. The simplest equation for studying the thermocline structure is the so-called Planetary Geostrophic Thermocline Equation (PGTE) derived by Dewar (1987). Since this equation includes only the first-order-derivatives, the equation can easily be transformed to any coordinate systems. The present paper proposes the use of the coordinate in which the Sverdrup function is used instead of the longitudinal coordinate. This coordinate system is referred to as the Sverdrup coordinate here. On this coordinate system, since the advection by the Sverdrup flow is in the direction normal to that of the Rossby wave propagation, the equation for the steady state is almost the same as that for free (long) Rossby waves, and the solution does not depend on the distribution of Ekman pumping. Although the time-dependent problem can also be well described by this coordinate, we discuss only the steady solutions in the present article as a first step. After discussing the steady solution on this coordinate in the next section, we study, using this coordinate, the problem of spontaneous shocks or mid ocean frontgenesis, including the cases treated by Dewar (1992), and clarify the conditions for those caused by the Rossby waves emanating from the western boundary. Although we treat the problems only for zonally uniform distribution of Ekman pumping in Sections 2 and 3, the same technique can be applied to more

Stationary Rossby Waves and Shocks on the Sverdrup Coordinate 209 complicated distributions. Section 4 presents an example and briefly discusses what difference will occur in a case in which the zero Ekman pumping line is not longitudinal. The results are summarized in Section 5. 2. Stationary Rossby Waves on the Sverdrup Coordinate 2.1 Governing equation and Sverdrup coordinate If we adopt the geostrophic balance and mass conservation, the equation governing the twolayer system with planetary scale, driven by the Ekman pumping, w e, shown in Fig. 1, can be reduced to h t + 1 fh J φ,h βg h h f 2 H H h x = β f 2 H ( H h) φ x, ( 2) where x and y are the longitudinal and meridional coordinate, t the time, f the Coriolis parameter, β the y-derivative of f, g the reduced gravity, H the total depth, h the thickness of the upper layer and φ is the Sverdrup function, φ = f 2 x w e dx. 3 β x e Here x e is the longitude of the eastern boundary of the basin. Equation (2) is called Planetary Geostrophic Thermocline Equation (PGTE) by Dewar (1987), and its derivation can be found in that paper. The nondimensional versions of these equations can be written as h t + 1 f J( φ,h) β f h( 1 h) h 2 x = β f ( 1 h) φ 2 x, ( 2 ) φ = f 2 x w e dx 3 β x e where we have used the horizontal scale, L, which is the meridional extent of the gyre, vertical scale H and the time scale, T = f 0 L 2 /g H. The Sverdrup function, the Coriolis parameter and β- parameter are normalized by g H 2, f and f /L, respectively, where f is the Coriolis parameter at the intergyre boundary whose coordinate is y = 0, and we use the β-plane approximation, i.e., f = 1 + βy, in nondimensional form. Using the potential thickness of the lower layer, p = (1 h)/f, in Eq. (2 ) instead of h, we get p t + 1 f J φ 1 ( 1 fp 2 )2, p = 0. 4 If we set φ = 0, Eq. (4) becomes the equation governing the temporal evolution of the

210 A. Kubokawa nonlinear Rossby wave, p t β f ( 1 fp)p p x = 0. ( 5) Since the equation is nonlinear and non-dispersive, the solution will steepen as the wave propagates, and eventually, it will be multi-valued. Although we do not know the actual conservation law of the planetary fluid, we can derive the lowest order conservation equation directly from Eq. (5), and we may simply assume that p t x β f 1 2 p2 f 3 p3 = 0, 6 δ[ p] x s t δ β f 1 2 p2 f 3 p3 = 0, 7 is satisfied at the shock front, where x s is the position of the shock front (discontinuous surface of p) and δ [A] = lim δ 0 {A(x s +δ ) A(x s δ )}. If there is a small diffusion term, (Kp x ) x where K is a diffusion coefficient, on the right hand side of Eq. (5), Eq. (7) is strictly correct at the limit of K 0. This assumption was used by Dewar (1987). On the other hand, for the steady state with non-zero φ, using the relations, = φ J φ, A J A, B x = φ x y y A y φ, A B φ y φ A y φ B φ, ( 8) we can simplify Eq. (4) on a coordinate, in which φ is used instead of x, as p y p β( 1 fp)p φ = 0. ( 9) We refer this (φ,y)-coordinate to as the Sverdrup coordinate. When y and φ are replaced by t and x, Eq. (9) is the same as Eq. (5) except for the factor f in the characteristic speed. On the Sverdrup coordinate, since Rossby waves are simply advected in y direction in the subtropical (and +y direction in subarctic) gyre, the analogy between Eq. (5) and Eq. (9) occurs. Note that the equation does not depend on the distribution of Ekman

Stationary Rossby Waves and Shocks on the Sverdrup Coordinate 211 pumping, w e, although the line of w e = 0 must be longitudinal for φ to be a single valued function of x. The y-distribution of w e changes just the φ coordinate of the western boundary, φ w (y). When w e < 0, i.e., in the subtropical gyre, φ w is positive, while φ w < 0 for positive w e. If we consider the Ekman pumping, w e = β f w 2 0 sin πy, ( 10) as an example, we get The φ-coordinate of the western boundary becomes φ = ( x x e )w 0 sin πy. ( 11) φ w ( y) = x e w 0 sin πy, 12 and the eastern boundary is at φ = 0. The mapping from the Sverdrup coordinate to the (x,y)- coordinate can be done by x = x e + φ w 0 sin πy ( 13) which is obtained from Eq. (11). This coordinate system is shown in Fig. 2 with the Rossby wave characteristics described below. The intergyre boundary shrinks into a point on the Sverdrup coordinate, because there is no meridional component of the Sverdrup flow. The asymmetry in the shape of domain between the subtropical and subarctic gyres is explained in the next subsection. The present example is the simplest one, and we will discuss a more complicated case in which w e is a function of both x and y in Section 4. 2.2 Stationary Rossby wave Equation (9) can easily be solved by the method of characteristics. The characteristics originated from (φ 0,y 0 ) are obtained by integrating the characteristic velocity, β(1 fp)p, with respect to y, φ c = φ 0 ( f f 0 )p + 1 2 f 2 2 ( f 0 )p 2, ( 14) where f 0 = 1 + βy 0. On the characteristic curve, p is constant, and h(φ,y) is obtained from h( φ c, y) = 1 f( y)p( φ 0, y 0 ). ( 15) This solution does not depend on the distribution of w e as mentioned above. To illustrate the behavior of the solution, we consider the purely wind-driven gyre in which

212 A. Kubokawa Fig. 2. The characteristic curves of stationary Rossby wave when the lower layer is at rest: (a) on the Sverdrup coordinate, and (b) on the (x,y)-coordinate. x e w 0 = 0.15, β = 0.7 and h e = 0.3. the lower layer is at rest. In this case, the Sverdrup equation becomes Therefore, β h f x = f h w e. ( 16) h = h 2 e + 2φ, ( 17) where h e is a constant representing the thickness of the upper layer at the eastern boundary. The Rossby wave characteristics in a purely wind-driven gyre is shown in Fig. 2, in which h on the boundary is set to satisfy Eq. (17), and the parameters are x e w 0 = 0.15 and h e = 0.3. Since the upper layer depth, h, vanishes at φ = (1/2)h e2, the region with φ smaller than (1/2)h 2 e is omitted. In this case, h(φ,y) automatically satisfies Eq. (17) everywhere, i.e., it is only the function of φ. Therefore, we may consider that Eq. (17) represents the basic state without the Rossby wave, and may regard the characteristics shown in Fig. 2 as those of linear waves. From this point of view, we may define the Rossby wave amplitude as the deviation of p from (1 h 2 e + 2φ )/f. In order to obtain a solution with shock formation, we need to give a finite amplitude disturbance at the boundary. All the characteristics on the Sverdrup coordinate run from upper left to lower right as shown in Fig. 2(b) and the distribution of the characteristic curves is much simpler than that on the (x,y)- plane (Fig. 2(b)). The propagation direction of information is northward in the subarctic gyre and

Stationary Rossby Waves and Shocks on the Sverdrup Coordinate 213 southward in the subtropical gyre. This direction is determined by the sign of φ/ x. Since the intergyre boundary shrinks into a point on the Sverdrup coordinate, if p varies on the intergyre boundary, it will be represented as a discontinuity in p on this coordinate. This, however, does not necessarily means the discontinuity on the (x,y)-coordinate. There is a well-known continuous solution which is discontinuous on the Sverdrup coordinate, called a cross-gyre flow (Pedlosky, 1984) in which the westward Rossby wave speed cancels out the eastward advection. In such a case, many characteristics with different p emanate from the intergyre boundary (φ,y) = (0,0). If we rewrite the characteristic velocity in terms of h, it becomes βh(1 h)/f. Since it is always negative and has the maximum at h = 0.5, the characteristics tend to converge when h < 0.5 and h/ φ < 0 or h > 0.5 and h/ φ > 0, as propagating to the lower latitude. Because p is constant along the characteristics, h in the south is larger than that in the north. Therefore, there is a possibility that the characteristics diverging in a northern region, where h < 0.5 and h/ φ > 0, begin to converge after passing a certain latitude where h is 0.5. When the characteristics converge, the solution may become multi-valued at some latitude. In such a case, we may apply a shock wave condition, which can be derived in a similar manner as that for Eq. (7), δ[ p] φ s y δ β 1 2 p2 f 3 p3 = 0, ( 18 ) where φ s is the φ-coordinate of the shock front and δ [A] = lim δ 0 {A(φ s +δ ) A(φ s δ ) }, or φ s y = β f 1 2 h + h 1 3 h 2 2 + + h + h + h, 18 where subscripts + and denote the value at the eastern and western sides of the shock, respectively. This is the same as that derived by Dewar (1992) on the (x,y)-coordinate. Since the following analysis is essentially the same in the both gyres except for the ventilated thermocline case, we consider here subtropical gyres only, i.e., w e 0. Although we restrict our attention to the case in which φ is a single-valued function of x in the next section, we will discuss the effects of the multi-valued φ in Section 4. 3. Frontgenesis in the Subtropical Gyre The point where two adjacent characteristics intersect is the origin of the front (shock) which extends southward. This point can be detected by differentiating Eq. (14) with respect to p and by setting it zero. The equation becomes φ c = φ 0 p y= ys p ( f s f 0 ) + f 2 2 ( s f 0 )p + p( 1 f 0 p) f 0 p = 0, ( 19 ) where f s = 1 + βy s is f at the latitude for shock to occur. The terms, φ 0 / p and f 0 / p, represent the longitudinal and meridional distribution of p in the source region. Solving this equation, we obtain

214 A. Kubokawa f s = 1 2 p 1± 1 4 p φ 0 p + 1 f 0 p f 0 + p f 0 p 1/2. 20 In this section, we derive necessary conditions for the shock to occur in the subtropical gyre, i.e., for f s to be real and satisfy 1 β < f s < f 0, for several Rossby wave sources. Three origins of the Rossby waves, which can generate mid ocean fronts, are possible; (i) the intergyre boundary, (ii) outcrop line of ventilated thermocline model and (iii) western boundary. Although Dewar (1992) already discussed cases (i) and (ii), I also give a brief description about them here, because they are good examples to demonstrate the usefulness of the Sverdrup coordinate and the shocks that appear in case (iii) are caused by the same dynamics as those in these two cases. 3.1 Intergyre boundary For case (i), p changes in longitudinal direction at y = 0, i.e., φ 0 = y 0 = f 0 / p = φ 0 / p = 0. Substituting these conditions to Eq. (20), we can easily find that the shock forms at the latitude of f s = (1 p)/p = h 0 /(1 h 0 ). This implies that h 0 should be smaller than 0.5 for the front to occur in the subtropical gyre because f s < 1. Applying the other condition, f s > 1 β, we get 1 β 2 β < h 0 < 1 2. 21 For the divergence near the intergyre boundary to have physical meaning, h/ φ should be positive. If h/ φ < 0, the solution in the region adjacent to the intergyre boundary is multi-valued since the characteristics with larger h propagates westward (toward positive φ-direction) faster than that with smaller h. This case yields the solution of arrested front discussed by Dewar (1991), in which the shock crosses the intergyre boundary. An example of the trajectories of characteristics on the Sverdrup coordinate is shown in Fig. 3(a), where we set the thickness of the eastern boundary h e = 0.2, that of the western boundary h w = (0.4 + 2φ w ) 1/2 and x e w 0 = 0.15. In this case, just after leaving the intergyre boundary (y ~ 0), the characteristics tends to diverse, because h < 0.5 and h/ φ > 0. However, h is increasing as propagating southward along the characteristics, and after h exceeds 0.5, the characteristics begins to converge and the steepening occurs. Figure 3(b) shows the characteristics on the (x,y)-plane, and (c) shows the distribution of the upper layer depth. Since h/ x < 0 in this case as shown in Fig. 3(c), the upper layer baroclinic current is southwestward. 3.2 Outcrop line of ventilated thermocline For case (ii), (φ 0,y 0 ) is the coordinate of the outcrop line where h = 0. Therefore, p(φ 0,y 0 ) = 1/f 0. Using the relations φ 0 / p = ( φ 0 / y)( y 0 / p) = f 02 β 1 φ 0 / y, we can rewrite Eq. (20) as f s = f 0 2 1± 1+ 4 β f 0 φ 0 y 1/2. 22 Since the solution with a plus sign is always larger than that with a minus sign, we may consider

Stationary Rossby Waves and Shocks on the Sverdrup Coordinate 215 Fig. 3. A solution when the shock forms; (a) the characteristic curves on the Sverdrup coordinate, (b) those on the (x,y)-coordinate, and (c) the distribution of the upper layer depth. The source of the Rossby wave forming the shock is the intergyre boundary. The thick rigid lines in (a) and (b) denote the shock trajectories numerically calculated by Eq. (18). β = 0.7, the contour interval in (c) is 0.025, and for other parameters, see text. only the solution with a plus sign. The condition for f s to be real and smaller than f 0 is 0 > φ 0 y β 4 f 0. 23 This is identical to the condition derived by Dewar (1992). The minimum value of f s occurs when φ 0 / y = β/4f 0, and its value is f 0 /2. When f 0 /2 < 1 β, a more rigorous condition is needed for f s to satisfy f s > 1 β. This condition becomes 0 > φ 0 y > β 1 2 1 β 4 f 0 f 0 1 2. 24 An example of the solution is shown in Fig. 4, in which the outcrop latitude is set to be φ 0 = 0.2 ln f,

216 A. Kubokawa and x e w 0 = 0.15. The outcrop line encounters the intergyre boundary at x = x e 0.2β/πw 0, and for x > x e 0.2β/πw 0, the outcrop line lies on y = 0. Panels (a) and (b) show the trajectories of characteristics and (c) shows the upper layer depth. Since h/ φ < 0 along the characteristics, the Rossby wave begins to steepen westward just after leaving the outcrop line and shock forms. In this case, the current direction of upper baroclinic flow along the front is northeastward. 3.3 Western boundary In the above two cases, the sources of the Rossby waves are in the Sverdrup interior region. Although the western boundary layer is excluded in the present model, the potential vorticity distribution there will drastically change the gyre structure. Therefore, it would be an interesting subject to study what condition at the western boundary can produce mid ocean fronts. For this case, φ 0 = φ w (y 0 ). Using the relations, φ 0 / p = ( φ w / y) ( y / p) y= y0, we can write the solution (20) as where f s = 1 ( 2 p 1± 1 2h 0 µ ) ( 25) Fig. 4. Same as Fig. 3, but for the case in which the source of the Rossby wave forming the shock is the outcrop line of the ventilated thermocline. The thick broken line is the outcrop line. β = 0.7, the contour interval in (c) is 0.025, and for other parameters, see text.

Stationary Rossby Waves and Shocks on the Sverdrup Coordinate 217 4 p µ = 1 1 2h 0 φ w 2 y + βph 0 y p φ =φw 1/2. ( 26) Since f 1 βph is the Rossby wave speed and f 1 φ/ y is the eastward component of the Sverdrup flow, βph 0 φ w / y must be positive for the Rossby wave to emanate from the western boundary. Therefore, the condition for real µ becomes or p y φ =φw 4 p φ w 2 1 2h 0 y + βph 0, ( 27) p y φ =φw > 0, 28 where Eq. (27) corresponds to the case in which µ is smaller than unity, while Eq. (28) corresponds to the case of µ larger than unity. The right hand side of Eq. (27) is always negative as mentioned above. These conditions can be rewritten in terms of h 0 as, or h 0 y 1 h 0 1 2h 0 β 2 + 4 φ w f 0 y, 27 h 0 y < β ( 1 h 0 ). ( 28 ) f 0 The right hand side of Eq. (27 ) is always larger than β(1 h 0 )/f 0. First, we consider the solution (25) with a plus sign. In this case, µ should satisfy 21 ( β)p 1 < 1 2h 0 µ < 1 2h 0. ( 29) Since µ is defined as positive, h 0 should be smaller than 0.5, and then 21 ( β)p 1 < µ < 1. ( 30) 1 2h 0 In this case, condition (27) should be applied, because µ is smaller than unity. An example of the solution is shown in Fig. 5 in which the upper layer depth at the western boundary, h w (=h 0 ), is

218 A. Kubokawa set to be where h w = ( ĥ2 + 2φ w ) 1/2, ( 31) ĥ = h e2 + ( y / l +1) ( h e1 h e2 ), for y > l, h e2, for y < l, and φ w l = h e1 h e1 h e2 y y=0 + h e1 h w y y=0 with h e1 = 0.2, h e2 = 0 and h w / y y=0 = 3. This solution is similar to that for case (ii); the 1, Fig. 5. Same as Fig. 3, but for the case in which the source of the Rossby wave forming the shock is the western boundary and the shock is accompanied by the northeastward current. β = 0.7, the contour interval in (c) is 0.025, and for other parameters, see text.

Stationary Rossby Waves and Shocks on the Sverdrup Coordinate 219 characteristics are converging just after leaving (φ w,y 0 ) and the upper layer current along the front is northeastward as expected from the distribution of the upper layer depth (see Fig. 5(c)). On the other hand, µ in the solution with a minus sign should satisfy 1 21 ( β)p > µ > 1, for h 0 < 0.5, ( 32) 1 2h 0 1 21 ( β)p > µ > 1, for h 0 > 0.5. ( 33) 2h 0 1 When µ < 1 and h 0 < 0.5, the shock with a plus sign will occur at a latitude higher than that of a minus sign. Therefore, 1 21 ( β)p > µ > 1, ( 34) 1 2h 0 gives the condition for both h 0 < 0.5 and h 0 > 0.5. Since µ > 1, the condition (28) should be applied. Fig. 6. Same as Fig. 5, but for the case in which the shock is accompanied by the southwestward current. β = 0.7, the contour interval in (c) is 0.025, and for other parameters, see text.

220 A. Kubokawa Fig. 7. The value of h w / y necessary for the characteristics emanating from the northwestern corner of the gyre to form the shock in a rectangular basin with β = 0.7, as a function of h w, for x e w 0 = 0.075, 0.15 and 0.3: (a) the shock accompanied by the northeastward current and (b) the shock accompanied by the southwestward current. In both cases, h w / y should be larger than these values for shock formation, and the attached numerals denote the value of x e w 0. An example of the solution is shown in Fig. 6, in which h w (y) is set to be the same as Fig. 5 but with h e1 = 0.3, h e2 = 0.5 and h w / y y=0 = 10. This case is similar to case (i), in which the upper baroclinic current is southwestward, and if h 0 < 0.5 as the case shown in Fig. 6, the characteristics diverse near the origin, while those are converging after passing a certain latitude. Although the above two examples show that the western boundary can also produce stationary shocks, the necessary conditions request somewhat unrealistic situation. In the case of Fig. 5, h w / y is positive, i.e., the eastward current in the lower layer at the northwestern corner is stronger than that in the upper layer, while in the case of Fig. 6, h w / y is negative but very large, implying that the current in the lower layer at that corner is westward. Figure 7 shows the minimum h w / y necessary for the characteristics emanating from the northwestern corner to make a shock in a rectangular basin. The minimum h w / y was calculated iteratively by using Eqs. (25), (26) and (14), and we imposed the condition, φ c (y s ) < φ w (y s ), in addition to the necessary ones mentioned above. 4. Zonally Inhomogeneous Ekman Pumping As mentioned in Section 2, the solution (14) is also applicable to the case in which φ is a multi-valued function of x. Although this suggests that the same technique will yield the same conditions for shocks to occur on the Sverdrup coordinate, the basin geometry on the Sverdrup coordinate will be different from the case discussed in Sections 2 and 3. Does this geometry change yield any differences? To address this question, we consider the Ekman pumping represented by w e = β { ay + ( 1 2x / x f 2 e )b( y) }. 35

Stationary Rossby Waves and Shocks on the Sverdrup Coordinate 221 This yields φ = ( x x e )a( y) + x( 1 x / x e )b( y). ( 36) Since the second term of Eq. (35) vanishes at x = 0, the φ-coordinate of the western boundary becomes φ w ( y) = x e ay, 37 irrespective to the value of b(y), and the eastern boundary is always at φ = 0. When b(y) 0, the sign of φ/ x changes, so that the propagation direction is not determined only by y. The x- coordinate where φ/ x = 0, denoted by x m (y), becomes x m ( y) = x e 2b y, 38 ay + by and the extremum of φ, φ m (y) = φ(x m,y), can be represented as φ m ( y) = x e ( 4b y )2. 39 ay by For x m (y) to be in the basin, a(y) and b(y) should satisfy ay by < 1. ( 40 ) A point on the Sverdrup coordinate can be mapped onto the (x,y)-coordinate by x = x e 2b a + b ± { ( a b)2 4bφ / x e } 1/2. 41 One point on the Sverdrup coordinate corresponds to two points on the (x,y)-plane on the latitude satisfying Eq. (40). Since x φ = m2 { ( a b)2 4bφ / x e } 1/2, 42 the minus (plus) sign of Eq. (41) corresponds to the northward (southward) branch. In Fig. 8, we show, as an example, the distribution of characteristics with

222 A. Kubokawa ay = w 0 sin πy, ( 43) by = w 1 cos πy for y < 1 2 0 for y 1 2, where we set w 1 = w 0 and the other parameters are the same as those in Fig. 2, in which the lower layer is at rest. The condition of Eq. (40) can be rewritten as Fig. 8. Same as Fig. 2, but w e = β { f 2 w 0 sin πy + ( 1 2x / x e )by }, with by = w 1 cos πy for y < 1 2 0 for y 1 2. x e w 1 = 0.15 and the other parameters are the same as those in Fig. 1. The thick broken lines in (a) and (b) denote φ = φ m (y) and x = x m (y), respectively.

Stationary Rossby Waves and Shocks on the Sverdrup Coordinate 223 y < y b = π 1 tan 1 w 1. ( 44) w 0 In this case, since φ m is the maximum, φ/ x is positive for x < x m and negative for x > x m. Therefore, the information emanating from the eastern boundary for y < y b propagates southward along the characteristic curve while that for y > y b propagates northward. On the other hand, the information from the western boundary for y > y b propagates northward and that for y < y b propagates southward. If they encounter the boundary, (φ m (y),y), they trace back exactly the same trajectories, and passing the starting points, continue to propagate toward the western boundary. In Fig. 8, the characteristics emanating from the western boundary are denoted by thick lines. The distribution of the characteristics is exactly the same as that in the case with b = 0 (Fig. 2(a)) in the region surrounded by φ = φ w (y) and φ = 0. At the point denoted by P, a characteristic curve contacts tangentially with φ = φ m (y). Only such characteristics can cross the intergyre boundary, and if many characteristics tangentially contact with φ = φ m (y), a cross-gyre flow solution will appear. Although the characteristics will cross each other on the Sverdrup coordinate in such a case, they will not cross on the (x,y) coordinate, because φ is multi-valued to x. In regard to the shock formation, we can say that the conditions obtained in the preceding section for cases (ii) and (iii) are applicable to the present case if the shock does not occur in the region surrounded by φ = φ m, φ = φ w and φ = 0, because the trajectories on the Sverdrup coordinate are not affected by the distribution of w e (x,y). For case (i), however, we can say nothing in the present stage because the solution of cross-gyre flow changes when φ is a multi-valued function of x. Before discussing the condition for shocks, we must study the solution of cross-gyre flow. 5. Summary and Discussion In the present paper, we discussed the stationary Rossby waves and conditions for their shock formation on the Sverdrup coordinate, in which the Sverdrup function is used instead of the longitudinal coordinate. In this coordinate system, the equation for stationary Rossby waves in the wind-driven gyre does not depend on the distribution of the Ekman pumping, and has almost the same form as that with the free Rossby waves. As a result, the behavior of stationary Rossby waves can easily be understood, even when the Sverdrup function is multi-valued to x. The derived necessary conditions for causing the shocks show that there are four possible types of fronts. Two of them which are caused by the Rossby waves from the western boundary are recent findings in this paper. Although these newly found two types of shock need somewhat unrealistic condition, this does not necessarily mean that the western boundary cannot be a source of the mid ocean frontlike structure, since the two-layer model is too crude to represent the realistic ocean density structure. Furthermore, even without the shock, if the amplitude of the Rossby wave at the northwestern corner is sufficiently large, eastward upper layer currents may occur in the mid ocean. In the present model, however, the case with strong eastward mid ocean current was not found when the eastward current in the lower layer at the northwestern corner of the gyre is weaker than that in the upper layer. Very recently, Kubokawa (1994) discussed a two-level model of the subtropical gyre under north-south differential heating, and showed that an eastward jet can occur in the mid ocean when the amplitude of the Rossby wave at the western boundary is sufficiently large. The difference between two-level and two-layer models arises form their ways in treatment of the stratification. Under strong north-south differential heating, the stratification

224 A. Kubokawa in the south is much stronger than that in the north of the Rossby repeller in the two-level model. This difference in the stratification causes the difference in the Rossby wave speed and hence, the characteristics tend to converge around the Rossby repeller even when the linear case. This suggests that the north-south differential heating may also be important in the Sverdrup region but not only in the source region. A case in which the zero Ekman pumping line is not zonal was also briefly discussed. The characteristics of the stationary waves are also well discribed on the Sverdrup coordinate. A significant difference, however, occurs in the treatment of the cross-gyre flow, and we need to study the solution of cross-gyre flow in such a situation, before discussing how the condition for the cross-gyre flow to cause a shock depends on the inclination of the zero Ekman pumping line. In regrad to the other two cases, the conditions obtained with zonally uniform Ekman pumping can be applied as far as the shock occurs at the coordinate where the value of Sverdrup function lies between those at the western boundary and eastern boundary. Acknowledgements This research was partialy done when the author was at the Kyushu University and was completed at the Hokkaido University. The comments by the anonymous reviewers were helpful. The author would also like to thank Dr. McCreary for his helpful comment. References Dewar, W. K. (1987): Planetary shock waves. J. Phys. Oceanogr., 17, 470 482. Dewar, W. K. (1991): Arrested fronts. J. Mar. Res., 49, 21 52. Dewar, W. K. (1992): Spontaneous shocks. J. Phys. Oceanogr., 22, 505 522. Hasunuma, K. and K. Yoshida (1978): Splitting the subtropical gyre in the western North Pacific. J. Oceanogr. Soc. Japan, 34, 160 172. Kubokawa, A. (1994): A two-level model of subtropical gyre and subtropical countercurrent (in preparation). Pedlosky, J. (1984): Cross-gyre ventilation of the subtropical gyre: An internal mode in the ventilated thermocline. J. Phys. Oceanogr., 14, 1172 1178. Roden, G. I. (1980): On the variability of surface temperature front in the western Pacific, as detected by satellite. J. Geophys. Res., 85(C), 2704 2710. Sverdrup, H. U. (1947): Wind-driven current in a baroclinic ocean; with application to the equatorial currents of the eastern Pacific. Proc. Nat. Acad. Sci., 33, 318 326. Takeuchi, K. (1984): Numerical study of the Subtropical Front and the Subtropical Countercurrent. J. Oceanogr. Soc. Japan, 40, 371 381.