Long Rossby Wave Basin-Crossing Time and the Resonance of Low-Frequency Basin Modes

2652 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 32

FRANCËOIS PRIMEAU* Canadian Centre for Climate Modelling and Analysis, Meteorological Service of Canada, University of Victoria, Victoria, British Columbia, Canada

(Manuscript received 12 December 2000, in ®nal form 5 March 2002)

ABSTRACT The ability of long-wave low-frequency basin modes to be resonantly excited depends on the ef®ciency with which energy ¯uxed onto the western boundary can be transmitted back to the eastern boundary. This ef®ciency is greatly reduced for basins in which the long Rossby wave basin-crossing time is latitude dependent. In the singular case where the basin-crossing time is independent of latitude, the amplitude of resonantly excited long-wave basin modes grows without bound except for the effects of friction. The speed of long Rossby waves is independent of latitude for quasigeostrophic dynamics, and the rectangular basin geometry often used for theoretical studies of the wind-driven ocean circulation is such a singular case for quasigeostrophic dynamics. For more realistic basin geometries, where only a fraction of the energy incident on the western boundary can be transmitted back to the eastern boundary, the modes have a ®nite decay rate that in the limit of weak friction is independent of the choice of frictional parameters. Explicit eigenmode computations for a basin geometry similar to the North Paci®c but closed along the equator yield basin modes suf®ciently weakly damped that they could be resonantly excited.

1. Introduction also show that for the case of a square ocean basin these modes can be resonantly excited by weak atmospheric In recent articles, LaCasce (2000) and Cessi and Pri- forcing to produce strong decadal ¯uctuations in the meau (2001) have demonstrated that there exists, in the depth of the thermocline. physically relevant case of weak friction, a set of weakly For quasigeostrophic (QG) dynamics in a square ba- damped low-frequency basin modes. The basin modes sin the crossing time, given by the width of the basin consist of a westward propagating long Rossby wave divided by the long Rossby wave phase speed, is in- excited by boundary pressure ¯uctuations at the eastern dependent of latitude. Real ocean basins, however, are boundary and a uniform pressure adjustment that en- not square, and their width depends on latitude. Fur- forces mass conservation in the basin. This uniform thermore, real ocean basins have large north±south ex- pressure adjustment is determined by imposing an in- tent so that the speed of long Rossby waves propagating tegral constraint for mass conservation in the basin. Fur- in them also depends on latitude contrary to the qua- thermore, it is this pressure adjustment that produces sigeostrophic approximation. Since the basin-crossing the pressure ¯uctuations at the eastern boundary. At the time is actually a function of latitude, it is natural to western boundary a frictional boundary layer damps out inquire how the modes identi®ed by Cessi and Primeau part of the mode's energy so that the resulting modes (2001) are affected. In a follow-up article, Cessi and are weakly damped. The period of the gravest mode is Louazel (2001) studied a shallow-water model and given by the time for long Rossby waves to cross the found that because of the variations of the long Rossby basin. For midlatitude ®rst baroclinic waves, this time wave phase speed in the shallow-water model, the decay is on the order of a decade. Cessi and Primeau (2001) rate of the basin modes remained ®nite as the friction parameter was decreased. Here we will investigate the role played by the basin shape in both a quasigeostrophic * Current af®liation: Department of Earth System Science, Uni- versity of California, Irvine, Irvine, California. setting and in a shallow water setting. To this end, we begin by revisiting in section 2 the quasigeostrophic case previously considered by Cessi Corresponding author address: FrancËois Primeau, Canadian Centre and Primeau (2001). The difference is that here we allow for Climate Modelling and Analysis, Meteorological Service of Can- ada, University of Victoria, P. O. Box 1700, Victoria, BC V8W 2Y2, the width of the basin to vary with latitude. After having Canada. gained an understanding of the quasigeostrophic case E-mail: [email protected] with latitude-dependent crossing time, it is straightfor-

᭧ 2002 American Meteorological Society SEPTEMBER 2002 PRIMEAU 2653

ward to extend the results to planetary scales via the (x, y) ϭ Lx(x*, y*), (9) shallow-water equations in spherical coordinates (section 3). The conclusions are presented in section 4. t ϭ tt0 *, (10)

Ϫ1 ␺ ϭ ␶ 0(␤␳) ␺*, (11) 2. The quasigeostrophic case where time has been nondimensionalized by the time

The linear evolution of the transport streamfunction for a long Rossby wave to cross a basin of width Lx, forced by wind stress ␶ is governed by L t ϭ x . (12) ␺ 0 ␤R2ץ 1 ץ ␤ 2␺ Ϫ ␺ ϩٌ x The resulting nondimensional equation after droppingץ tR΂΃2ץ the asterisks is 11 22 ϭ curl␶ ϩ ␬ٌٌ␺ Ϫ ␺ , (1) 222 (␺ Ϫ ␺) ϩ ␺ ϭ G(x, y, t) ϩ ␦ٌ (⑀ٌ ␺ Ϫ ␺) (13 ٌ⑀) 2 ␳ ΂΃R tx where in which G(x, y, t) is the wind forcing. The nondimensional boundary conditions are ͙gЈH R ϵ , baroclinic deformation radius (2) ␺ ϭ ␺ (t), (14) f 0 2 (١␺ ´ n ϭ 0, or ⑀␦ٌ ␺ ϭ 0, (15␦⑀ and ␬ ϵ large-scale eddy diffusivity. (3) together with the integral constraint for mass conservation, For geostrophic ¯ow, the kinematic boundary condition of no ¯ow normal to the basin walls requires that there ␺(x, y, t) dx dy ϭ 0. (16) be no pressure gradient along the basin boundary. In a ͵ simply connected domain, the resulting boundary con- D dition is that There are two small parameters

| (t). (4) 2 ␺ boundaryϭ ␺ 0 R ␬ ⑀ ϵ , ␦ ϵ . (17) The scalar function, ␺ (t), gives the time evolution of 2 0 ΂΃Lxx␤LR the boundary pressure. It is determined as part of the solution by requiring that the total mass be conserved: The parameter ␦ is the ratio of the basin crossing timescale to the damping timescale for long Rossby waves and the parameter ⑀ measures the relative importance of inertia to vortex stretching. ␺(x, y, t) dx dy ϭ 0. (5) ͵ For midlatitude scalings, area Ϫ11 Ϫ1 Ϫ1 Because of the fourth-order derivative in the friction ␤ ϭ 2 ϫ 10 m s , (18) 6 term, an additional boundary condition is required, for Lx ϭ 8 ϫ 10 m, (19) which one can choose either no-slip or free-slip boundary conditions: R ϭ 3 ϫ 104 m, (20) (١␺ ´ n ϭ 0, (no slip) (6) ␬ ϭ 1000 m2 sϪ1 . (21 2␺ ϭ 0, (free slip). (7) Typical values for the small parameters areٌ In order to allow the basin crossing time for long ⑀ ϭ 2.4 ϫ 10Ϫ5 , ␦ ϭ 6.9 ϫ 10Ϫ3 (22) Rossby waves to be a function of latitude within the quasigeostrophic context, we consider a basin whose so that width is a function of latitude: ⑀ K ␦ K 1. (23)

D ϭ {(x, y)|0 Յ y Յ YNW, X (y) Յ x Յ X E(y)}. (8)

Here XW(y) and XE(y) are the positions of the eastern b. Numerical method and western basin boundaries as a function of y, and YN is the maximum north±south extent of the basin. We will be looking for eigenmode solutions of the form a. Nondimensionalization ␺ ϭ eϪi␻t␾(x, y). (24) It is useful to recast the problem in nondimensional Substitution into the governing equation yields the fol- form by introducing the following scales: lowing eigen problem: 2654 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 32

222 Ϫi␻[⑀ٌ ␾ Ϫ ␾] ϩ ␾x ϭ ␦ٌ (⑀ٌ ␾ Ϫ ␾), (25) frictional term or the inertial term depending on which is larger. In contrast, the integral constraint (16) can be which can be written as a matrix eigenvalue problem, satis®ed even when both ⑀ and ␦ vanish. Ϫi␻B␾ ϭ A␾, (26) For basin-scale modes, the scaling given by (23) im- by discretizing the equations. For the solutions pre- plies that both friction and inertia can be neglected away sented in the following sections, the discretization ap- from thin boundary layers. To O(␦), the interior problem proximates the solution using the ®nite element method reduces to on a unstructured mesh made of triangles. The domain ␺ ץ ␺ץ triangulation is done using the two-dimensional mesh ϪϩII ϭ0  (x  . (28ץ tץ -generator ``Triangle'' available from Netlib (see ac knowledgments). The basis functions are ®rst-order tent  ␺ [X (y), t] ϭ ␺ (t) functions. Second-order tent functions were used as well IE 0 to check the convergence of the solutions. Again, because of the scaling ⑀ K ␦ K 1, the eastward propagating short Rossby waves are damped out before c. Long-wave approximation with weak friction they can leave the frictional western boundary layer. Consequently, the integrand in the integral constraint In order to make the long-wave approximation for can be well approximated by the interior long-wave so- the solution of modes in a closed basin, it is important lution ␺I everywhere except in the frictional boundary that short Rossby waves generated by re¯ection at the layers. Only in the frictional boundary layers are the western boundary be damped out suf®ciently fast that corrections to ␺I of order one, but those corrections of they will not propagate out of the western boundary order one are con®ned to a boundary layer of thickness layer and spoil the long-wave approximation in the basin ␦ so that their contribution in the total integral will only interior. Rossby waves re¯ected from the western be of order ␦: boundary must have an eastward group velocity. Con- sequently, their wavelengths are shorter than the Rossby ␺ dx dy ϭ ␺ ϩ ␺ dx dy deformation radius. In terms of our nondimensional pa- ͵͵ ͵͵ Ib rameters, eastward propagating Rossby waves have wavelengths ␭ Յ ⑀1/2 so that their damping time is ϭ ␺ ϩ O(␦). (29) ͵͵ I ␺⑀ tsw ϳ O Յ . (27) Thus the interior solution must satisfy the integral con- ␦2␺ٌ␦ ΂΃ straint Thus for the long-wave approximation to be valid, it is YXNE(y) necessary that ⑀ K ␦ as is the case for scaling (23). The dy dx{␺ } ϭ 0. (30) scaling ⑀ K ␦ K 1 implies that short Rossby waves ͵͵ I 0 XW(y) with eastward group velocity will be damped quickly by friction, while westward propagating long Rossby waves will be damped slowly with a long timescale, tlw 1) ENERGY CONSIDERATIONS ϳ O(1/␦). It is important to distinguish the limit ⑀ K ␦ → 0 The energy equation for the interior problem is ob- from the alternate limit ␦ K ⑀ → 0. The former leads tained by multiplying equation (28) by ␺I,toget (F ϭ 0, (31´١ to the long-wave basin modes whose properties we focus E ϩ on in the present paper. The later does not. It leads to t 2 the small Rossby radius limit of the inviscid basin modes in which the energy density is given by␺I /2 and the discussed by Flierl (1977). energy ¯ux is given by Note that if either ⑀ or ␦ vanish, the fourth-order ␺2 derivatives in the governing equation (13) vanish so that F ϭϪx I . (32) 2 one cannot impose the boundary condition on the tan- gential velocity. This is why we have multiplied the no- Note that the energy ¯ux is everywhere westward. For slip or free-slip boundary condition by ⑀␦ in (15). If a resonant mode to be established, some of the energy both ⑀ and ␦ vanish, then the second-order derivatives ¯uxed onto the western boundary must be returned to in the governing equation also vanish. In this case, the the eastern boundary where it can radiate back into the governing equation reduces to a ®rst-order hyperbolic interiorÐotherwise the mode will be completely equation, Ϫ␺t ϩ ␺x ϭ G(x, y, t) and one can only satisfy damped out after one long-wave basin-crossing time. the no normal-¯ow boundary condition (14) at the east- As is well known, for inviscid basin modes (␦ ϭ 0, ⑀ ern boundary from which the characteristics emanate. ± 0), the energy is re¯ected back to the eastern bound- In order to satisfy the no-normal ¯ow boundary con- ary by short Rossby waves with eastward group veloc- dition at the western wall, one must retain either the ity. For the long-wave basin modes satisfying scaling SEPTEMBER 2002 PRIMEAU 2655

(23) this is not possible since short Rossby waves are western boundary and the energy transmitted to the east- damped within a thin western boundary-layer. Cessi and ern boundary gives the rate of change of total energy, Primeau (2001) demonstrate that in the long-wave limit the time-dependent boundary pressure allows the energy E ϭϪ(F ITϪ F ) to be transferred back to the eastern boundary. Physi- ͵ t D cally, the boundary pressure adjustment can be thought of as parameterizing the effect of very low-frequency YYNN␺22(␺Ј) ␺ 2 ϭϪ dy 0 ϩϩI dy 0 gravity waves forced by the mass redistributions in- ͵͵22 2 00΂΃ duced by the propagation of long Rossby waves in a closed basin. We will return to this interpretation in YN (␺Ј)2 ϭϪ I Յ 0. section 3 when we consider the shallow-water equations. ͵ 2 Energy considerations allow us to obtain an expres- 0 sion for the energy rate of change in terms of the interior Thus we see that the total energy must decrease unless ␺Ј ϭ 0, that is, unless the wave fronts arrive at the solutions ␺I, without having to consider the details of I the frictional western boundary layer. The incident en- western boundary parallel to the wall. This will be the ergy ¯uxed onto the western boundary layer, F I, and case only if the basin-crossing time for long waves is the energy transmitted to the eastern boundary by the independent of latitude. time-dependent boundary pressure, F T, are given by The re¯ection process at the western boundary splits the energy into two parts. One part is re¯ected into the YN ␺ (X (y), y, t)2 uniform boundary pressure adjustment. The other part F I ϭ dy IW , ͵ 2 that is re¯ected into the short eastward propagating 0 [] Rossby waves is quickly damped out in the frictional Y boundary layer. The energy partitioning is determined N ␺ (t)2 F T ϭ dy 0 . (33) by the meridionally averaged long-wave interior solu- ͵ 2 0 [] tion evaluated along the western boundary. The part of the energy that is re¯ected into the uniform pressure Note that the interior solution ␺ ϭ ␺I does not satisfy the no-normal ¯ow boundary condition at the western adjustment is transmitted to the eastern boundary where boundary; F I can be rewritten in a more useful form by it excites the long Rossby waves. making use of the mass conservation constraint. Dif- The square basin considered by Cessi and Primeau ferentiating (30) with respect to time and using (28) (2001) is a singular case since the basin crossing time gives for long Rossby waves is independent of latitude. This is why the amplitude of the resonating basin modes YYNNbecomes unbounded as the friction parameter ␦ ap- dy{␺ (X (y), y, t)} ϭ ␺ (t) dy, ͵ IW ͵ 0 proaches zero. In general, however, the long-wave ba- 00sin-crossing time will not be independent of latitude and whence, the boundary pressure term is given by the the resonating basin modes will have a ®nite amplitude meridional average of the interior solution evaluated even as the friction parameter approaches zero. along the western boundary 1 YN 2) FREE MODES ␺0(t) ϭ ␺IW(X (y), y, t) dy. (34) YN ͵ Cessi and Louazel (2001) show that it is possible to 0 obtain an estimate of the period and damping time of If we then decompose ␺I(XW(y), y, t) into an average free modes of the planetary geostrophic equations with- plus deviation, out having to deal explicitly with the frictional boundary ␺ (X (y), y, t) ϭ ␺ (t) ϩ ␺Ј(y, t), (35) layers. If we apply their method to the quasigeostrophic IW 0 I equations we can make the connection to the inviscid where␺ЈI (y, t) ϵ ␺(XW(y), y, t) Ϫ ␺0(t), the incident energy results of Cessi and Primeau (2001) more direct. For ¯uxed onto the western boundary can be written as this method, we consider free mode solutions of the ␴t 1 YN form ␺(x, y, t) ϭ e h(x, y). Substitution into (28) and F I ϭ (␺22ϩ (␺Ј))dy a little algebra yields 2 ͵ 0 I 0 ␴(xϪXE(y)ϩt) ␺ ϭ he0 , (36) YN since #0 ␺Ј dy ϭ 0. Note that if the basin-crossing time is independent of latitude, the wave fronts arrive at the where ␴ is yet to be determined and h 0 is an arbitrary western boundary with their lines of constant phase par- amplitude independent of x and y. The conservation of mass constraint (16) yields an equation for ␴, allel to the wall. In that case ␺I(XW(y)) ϭ const and ␺Ј ϭ 0. YN 1 Since there is no dissipation in the interior of the dy (1 Ϫ e␴(XWE(y)ϪX (y))) ϭ 0. (37) ͵ ␴ basin, the difference between the energy incident on the 0 Ά· 2656 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 32

FIG. 1. Basin geometry for various values of the parameter m. For m ϭ 0 the basin geometry is square, for m Ͻ 0 it wider to the north, and for m Ͼ 0 it is narrower to the north.

For the special case where the width of the basin is Figure 1 shows the basin domain. For m ϭ 0 we have independent of y; for example, XE(y) Ϫ XW(y) ϭ 1, Eq. a square basin. For m Ͼ 0 we have a basin that narrows (37) reduces to to the north, and for m Ͻ 0 we have a basin that is wider in the north. Note that the maximum crossing- 1 e␴ 0, (38) Ϫ ϭ time for long waves as a function of latitude is given and we recover the purely oscillatory undamped solu- by tions found by Cessi and Primeau (2001), 1 Ϫ 2m, m Ͻ 0 ␴ ϭ i2␲n for n ϭ 1, 2, ´ ´ ´. (39) tmax ϭ (41) Ά1, m Ն 0. Basins with latitude independent widths allow the long- For the choice of basin geometry (40), the y integration wave basin modes to resonate most strongly. To study in (37) can be carried out explicitly, and the integral the effect of basins with varying width, we consider the constraint reduces to an algebraic equation for ␴, special case of straight, but nonparallel, western and eastern boundaries with YN ϭ 1: 11 1 Ϫ (e␴(2mϪ1) Ϫ eϪ␴) ϭ 0. (42) XWE(y) ϭ my, X (y) ϭ 1 Ϫ my, ␴Ά·2m␴ m Ͻ 1/2. (40) For m ± 0, solutions for ␴ are obtained numerically. Figures 2 and 3 show the frequency, Im{␴}, and e-

FIG. 2. Frequency of the gravest mode as a function of m. For m FIG. 3. Damping rate of the gravest mode as a function of m. For ϭ 0, the basin width is independent of y. m ϭ 0, the basin width is independent of y. SEPTEMBER 2002 PRIMEAU 2657

FIG. 4. Period of the three modes (dashed lines) as a function of FIG. 5. Plot of Q factors for the three modes as a function of m. m, as well as the period of the three modes rescaled by the maximum As m → 0, the basin width becomes independent of latitude, and basin-crossing time t (solid curves) as a function of m. max Q → ϱ. folding decay rate, Re{␴}, for the three gravest modes. plitude of resonantly excited long-wave basin modes We ®rst discuss the frequency of the modes as a function depends on the explicit value of the friction parameter. of the basin geometry. For m Ͼ 0, the basin narrows to the north and the maximum basin-crossing time for long waves is independent of m, the frequency remains 3) RESONANCES relatively constant. For m Ͻ 0, where the basin widens Cessi and Primeau (2001) showed that the long-wave to the north, however, the frequency decreases with m. low-frequency basin modes can be resonantly excited. In Fig. 4, a plot of the period of the three gravest modes, Given that the square basin geometry they considered rescaled by the maximum crossing time, shows that the is optimally suited to produce resonances, one must ask rescaled period varies little as m is varied in comparison if resonances are possible in more general basin ge- to the non-rescaled period. This indicates that the period ometries. The ability of resonators to produce spectral of the modes scales as the maximum basin-crossing time peaks can be measured in terms of the ``quality factor'' for long Rossby waves. or Q-factor (e.g., Marion 1970). For our modal solu- If we now consider the damping of the modes (Fig. tions, the Q factor is 3), we ®nd that even though friction was neglected in obtaining the approximate long-wave solutions, the Q ϵ ␻rr/(2␴ ), (43) modes have a ®nite damping rate except for the special 221/2 where ␻r ϭ (␴␴irϪ ) is the frequency at which the case m ϭ 0 (the square basin case). Thus, as long as resonantly excited mode would produce a spectral peak. the friction is suf®ciently small, the damping rate of Real positive Q factors give spectral peaks, with the long-wave modes is to ®rst-order independent of fric- sharpness of the spectral peak increasing with increasing tion. The amplitude of the resonantly excited long-wave Q factors. Usually, only broad spectral peaks are present basin modes does not depend on the explicit value of in geophysical records. For reference, Wunsch (2000) the dissipation parameter. Instead, it is determined by points out that the prominent El NinÄo±Southern Oscil- the shape of the basin. This is a satisfying result since lation index signal has a Q ϳ 1. Thus, if long-wave frictional parametersÐapart from being smallÐare modes have real Q factors, they have the potential of poorly known. being resonantly excited by stochastic atmospheric forc- For the special case of m ϭ 0, the damping rate goes ing. In the real ocean, their ability to resonate would of to zero in the absence of explicit friction. This is due course be reduced by frictional dissipation and mode to the fact that when the long Rossby wave basin-cross- coupling, which would drain energy out of the modes. ing time is independent of latitude, the energy carried In Fig. 5, the Q factors for the three gravest modes westward by the long Rossby-wave component of the are plotted as a function of m for the basin geometry mode is perfectly re¯ected into the parameterized forced considered in Fig. 1. For m ϭ 0, the Q factors go to gravity wave component of the mode. There is not in in®nity because of the absence of explicit friction, point- this case a ®xed fraction of the energy that is trapped ing to the fact that the square basin is an ideal resonating near the western boundary where it can be ef®ciently cavity for long-wave basin modes. For m ± 0 the Q damped out by friction. In these special basins the am- factors are greater than unity, indicating that long-wave 2658 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 32

friction in the form of Rayleigh drag. With Rayleigh drag, only the no normal-¯ow boundary condition is imposed along the basin boundary. We use a ®nite element method with linear elements for an unstructured triangular mesh, as well as quadratic elements to check for convergence. Figure 6 shows a typical mesh. For the two-dimensional computations, we choose ⑀ ϭ 0.001 25 and vary ␦ and m, and seek modal solutions of the form ␺ ϭ e␴h(x, y). Typical eigenmodes are shown in Fig. 7 where the two gravest modes are contoured for a basin which is wider to the north (m ϭ Ϫ0.5, ␦ ϭ 0.01). Lines of constant phase arrive at the western boundary at an angle. Friction in the thin western boundary layer forces the phase to become constant along the wall such that the no normal-¯ow boundary condition can be satis®ed. Note how the lines of constant phase are parallel to the eastern boundary as they prop- FIG. 6. Basin domain showing a typical unstructured triangular agate westward. In contrast, inviscid basin modes for mesh. which inertial effects allow the no-normal ¯ow boundary conditions to be satis®ed have their phase lines per- basin modes can be resonantly excited even when the pendicular to lines of constant planetary vorticity (Flierl basin-crossing time is latitude dependent. 1977), regardless of the basin shape. With ⑀ Յ ␦, short Rossby waves with eastward group velocity cannot propagate far from the western wall before being d. Basin mode computations with inertial and damped out. Thus only westward propagating long frictional effects waves are present in the eastern part of the basin. Note To con®rm the results obtained in the previous sec- also that, at any given time, the amplitude of the mode tion, we carry out explicit two-dimensional eigenmode increases westward. This can be understood mathemat- computations retaining inertial effects (⑀ ± 0) and with ically because the dynamical balance for the long-wave

FIG. 7. Real and imaginary parts for the gravest mode (top two panels), and of the second gravest mode (bottom two panels), for m ϭϪ0.5 and ␦ ϭ 0.05. Solid contours indicate positive values, and dashed contours indicate negative values. SEPTEMBER 2002 PRIMEAU 2659

FIG. 8. Period of the gravest basin mode as a function of the FIG. 9. Plot of Q factors for the gravest basin mode as a function logarithm of the frictional parameter log10 ␦, and the geometry pa- of the log of the frictional parameter log10 ␦ and the geometry parameter m. The period is nondimensionalized by the basin-crossing rameter m. Parallel eastern and western walls have m ϭ 0. time for long waves in a square basin. basin modes is governed by a ®rst-order hyperbolic remains valid for more general basin geometry. The equation, whose solution must be of the form f(x t). ϩ difference is that, for weak friction, the damping of the Modal solutions must therefore be of the form modes in basins with latitude dependent width is not A(y)e␴(xϩt). From this functional form we can see that controlled by the choice of dissipation parameterization the mode's amplitude must decay in the eastward di- as is the case for the special choice of a square ocean rection if it is decaying in time. Physically, this can be basin. understood by the fact that the decaying mode produces a decaying pressure oscillation at the eastern boundary. The boundary pressure ¯uctuations are then carried 3. Planetary geostrophic case westward by the long Rossby waves. Since dissipation in the basin interior is negligible, the ¯uctuations in the In this section we demonstrate how the understanding western part of the basin having originated at a much gained from the quasigeostrophic case carries over to earlier time than those nearer to the eastern boundary the shallow-water equations. The structure of the long- have a larger amplitude. wave modes is essentially unchanged; one simply has In Fig. 8 the period of the gravest mode is contoured to take into account the variations of the phase speed with latitude in addition to the variations of the basin as a function of the frictional parameter ␦ and the parameter m controlling the basin geometry. For a nar- width when determining the variations of the basin- crossing time with latitude. Furthermore, the shallow- rowing basin with m Ͼ 0, the period of the modes is close to unity (recall that time is nondimensionalized water equations retain the necessary dynamics for mass by the basin-crossing time for long waves in the south- conservation. Consequently, the long-wave basin modes computed using the shallow-water equations will ex- ern part of the basin). For a widening basin with m Ͻ 0, the period increases along with the increase in the plicitly resolve the dynamical process that transmits the maximum width of the basin, in agreement with the energy back to the eastern boundary. long-wave results of the previous section. The linearized shallow water equations, with friction Figure 9 contours the Q factor for the gravest mode in the form of Rayleigh drag, can be written in spherical as a function of the basin geometry m and the friction geometry as follows: parameter ␦. The best resonators occur for m ϭ 0. The ut Ϫ f␷ ϭϪgЈh␭ /(a cos␾) Ϫ ru, (44) ability of the modes to be resonantly excited decreases as m moves away from zero. As the friction parameter ␷ t ϩ fu ϭϪgЈh␾ /a Ϫ r␷, (45) is decreased there is a rapid increase in the Q factor (u ϭ 0. (46 ´ h ϩ H١ ␦ for the case m ϭ 0 in accordance with the long-wave th results presented in the previous section. Note that there The ®rst two equations are the momentum equations, is a wide range of frictional parameters with Q ϳ O(1) and the third is the conservation of mass equation, with even with ®nite friction in basins for which the lines of f ϭ 2⍀ sin␾ the Coriolis parameter, a the radius of the constant phase arrive at the western boundary at a large earth, r the Rayleigh friction parameter, gЈ the reduced angle. Thus, the conclusion of Cessi and Primeau (2001) gravity, and h the displacement of the layer depth about that long-wave basin modes can be resonantly excited the mean depth H. Unlike the QG formulation, the shal- 2660 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 32 low-water equations retain gravity wave motions and conserve mass without the need to impose the integral constraint for mass conservation. For modes with a decadal period, ␻ ϳ 2 ϫ 10Ϫ8 sϪ1, the tendency terms in the momentum equations are small compared with the Coriolis, pressure gradient, and friction terms. We can therefore neglect the tendency terms in the momentum equations. The neglect of these terms ®lters out the free gravity modes from the problem as well as the short Rossby waves:

Ϫf␷ ϭϪgЈh␭ /(a cos␾) Ϫ ru, (47)

ϩfu ϭϪgЈh␾ /a Ϫ r␷. (48)

(u (49´ hehϩ H١ This approximation while correct for realistic parameter values will break down in the limit of zero friction. Eliminating u and ␷ from (49) to obtain a single equation in h and letting r → 0 gives hץ (hc(␾ץ Ϫϭ0, (50) FIG. 10. Real (top panel) and imaginary (bottom panel) parts of ␭ the gravest eigenmode for the Paci®c basin with a wall along theץ tacos␾ץ equator. Shaded areas indicate negative contours (r ϭ 3 ϫ 10Ϫ7 sϪ1). where gЈH 2⍀ cos␾ c(␾) ϭ ␤ , ␤ ϭ . (51) fa2 cause of this, the wave fronts do not arrive at the western boundary parallel to the basin wall and part of the en- For the basin geometry given by ergy carried by the long Rossby waves becomes trapped near the boundary where it is damped out by friction, D ϭ {(␾, ␭)|⌽Ͻ␾ Ͻ⌽ , ⌳ (␾) Յ ␭ Յ ⌳ (␾)}, SNWEgiving the modes a ®nite decay rate even as friction the long-wave basin-crossing time is given by goes to zero. In the following computations we vary the friction (⌳EW(␾) Ϫ⌳ (␾))a cos␾ t0(␾) ϭ . (53) parameter r and hold the reduced gravity gЈ and H ®xed c(␾) (gЈϭ0.013 m sϪ2, H ϭ 1000 m). Varying gЈH changes To emphasize the importance of the latitude depen- the period and the decay rate of the modes, but does dence of the basin-crossing time over the latitude de- not change the Q factors. Contour plots of the two grav- pendence of the phase speed or the basin width we est modes for the Paci®c basin coastline with a wall consider two cases. For the ®rst we consider an idealized along the equator are given in Figs. 10 and 11 (r ϭ 3 basin geometry in which the basin width decreases to ϫ 10Ϫ7 sϪ1). For our choice of gЈH, the period of the the north in such a way that the basin-crossing time is two gravest modes are 14.2 and 7.0 yr. The gravest mode independent of latitude. Speci®cally, we choose the fol- for the optimally resonating basin is given in Fig. 12 (r lowing basin geometry: ϭ 3 ϫ 10Ϫ7 sϪ1). For our choice of gЈH its period is 17.2 yr. For the North Paci®c basin the phase lines do Ϫc(␾)Tc(␾)T not arrive parallel to the western boundary. Conse- D ϭ (␭, ␾) 00Յ ␭ Յ , 2a cos␾ 2a cos␾ quently, a considerable fraction of the energy ¯uxed ΆΗ onto the western boundary will not be transmitted to 25␲ 80␲ the eastern boundary. In contrast, for the optimal basin Յ ␾ Յ , (54) geometry the phase lines arrive at the western boundary 180 180 · parallel to the wall. Except for the effects of the explicit where T 0 ϭ 15 yr. For this case we will show that the frictional dissipation, all of the energy incident on the decay rate of the modes is determined by the friction western boundary in the basin with optimal shape is in an analogous fashion to the case of a rectangular basin transmitted to the eastern boundary. Figure 13 shows a for quasigeostrophic dynamics. In contrast, the second plot of the Q factors for the two gravest modes in the case we consider uses the realistic North Paci®c basin two different basin geometries. The Q factors for the coastline with a wall along the equator. We chose to modes in the optimal-resonator basin shape tend to in- close the North Paci®c basin in order to avoid having ®nity as the friction tends to zero. In contrast, for the to compute the modes for the entire globe. For this latter North Paci®c geometry, the Q factors tend to a ®nite case, the basin-crossing time is latitude dependent. Be- limit as the friction goes to zero (Q ഠ 0.6 for the gravest SEPTEMBER 2002 PRIMEAU 2661

FIG. 12. Real (top panel) and imaginary (bottom panel) parts of the gravest eigenmode for an optimally resonating basin on the sphere. Dashed contours indicate negative values (r ϭ 3 ϫ 10Ϫ7 sϪ1).

in the real ocean if they are being excited. Of course the present calculations are restricted to a basin closed off at the equatorÐa more accurate calculation would solve the problem for the basin con®guration of the FIG. 11. As in Fig. 10 but for the second gravest eigenmode. entire globe. We have neglected nonlinear effects that might become large in the western boundary layer. Also mode and Q ഠ 1.2 for the second gravest mode). Nev- neglected is the effect of the mean background current. ertheless, the computation for the North Paci®c basin For the linearized reduced-gravity formulation mean shows that even for very irregular basin geometries the currents have no effect on the modes because there is long-wave basin modes have Q factors of order one. an exact cancellation between the Doppler shift and the This suggests that these basin modes might be detectable potential vorticity gradient caused by the thickness gra-

FIG. 13. Plot of the Q factor for the two gravest modes for two different basin geometries as a function of the frictional parameter r. The solid squares are for the North Paci®c basin geometry and the open squares are for the optimally resonating basin geometry. The solid lines are for the gravest mode and the dashed line are for the second gravest mode. 2662 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 32 dients supporting the ¯ow. For more realistic model formulations with several vertical modes this would no longer be the case. Topography would also act to scatter energy between different vertical modes. It would be interesting to investigate such effects in a future study.

4. Shallow water case In the previous section's calculations using the planetary geostrophic equations, we neglected the tendency terms in the momentum equations. This approximation is appropriate for low-frequency modes but it also has the effect of ®ltering out the free gravity waves in general, and Kelvin waves in particular. In the QG formulation, the uniform boundary pressure ¯uctuations are sometimes said to parameterize the effects of Kelvin waves propagating around the boundary. The uniformity of the pressure adjustment along the boundary is loosely attributed to the fact that for the QG formulations the gravity waves propagate around the boundary instantaneously. Our goal here is to retain the tendency terms in the shallow-water equations so that we can investigate what role, if any, is played by Kelvin waves in the boundary pressure adjustment of the long Rossby wave basin modes. To address this question we have performed an additional set of computations with the linearized shallow- water equations in which we retain the time derivative terms in the momentum equations. For simplicity, we formulate the problem in Cartesian coordinates on a ␤ plane in a rectangular basin of dimension Lx ϭ 6000 km and Ly ϭ 4000 km:

utoϪ ( f ϩ ␤y)␷ ϭϪgЈh xϪ ru, (55) FIG. 14. Gravest ``Kelvin'' mode (upper panel) and gravest long- wave Rossby mode (lower panel). The thick solid and dashed lines ␷ toϩ ( f ϩ ␤y)u ϭϪgЈh yϪ r␷, (56) denote the phase (cotidal lines) and the amplitude (co-amplitude h ϩ H(u ϩ ␷ ) ϭ 0. (57) lines). The contour interval for the amplitude lines is 0.1. (upper txy panel) The amplitude is unity along the boundary and decreases to We choose the following numerical values for the pa- zero toward the interior. (lower panel) The amplitude is zero at the Ϫ5 Ϫ1 amphidromic point (X ഠ 3000 km, Y ഠ 3900 km) and increases to rameters H ϭ 1000 m, f o ϭ 9 ϫ 10 s , ␤ ϭ 1.8 ϫ 10Ϫ11 mϪ1 sϪ1, g 0.081 m sϪ2, and r 1 10Ϫ6 unity near the western boundary at (X ഠ 200 km, Y ഠ 3000 km). Јϭ ϭ ϫ The mode in the upper panel consists of one wave traveling coun- Ϫ1 s . With these parameters, free gravity waves propagate terclockwise with its amplitude trapped near the boundary and de- with a speed of 9 m sϪ1, and the Rossby radius of caying toward the center of the basin with an e-folding scale equal deformation ranges from 166 km in the south of the to the local Rossby deformation radius. Note how there is a clear basin to 71 km in the north of the basin. This is larger phase propagation along the boundary for the ``Kelvin'' mode, but that the phase for the long-wave Rossby mode is nearly constant than for the real ocean, but allows for adequate reso- along the boundary. For the Rossby mode, the boundary pressure lution of motions on the scale of the Rossby radius of oscillates in time but is nearly spatially uniform. deformation without an exceedingly ®ne grid interval. Using a ®nite difference approximation with a 20-km resolution, we ®rst seek free mode solutions in the form wave trapped within one Rossby radius of the coast. Its period is roughly given by the time for a Kelvin wave  u(x, y, t) uÃ (x, y) to propagate around the boundary ␷(x, y, t) ϭ ␷Ã (x, y) ei␻t. (58)  (2Lxyϩ 2L )/͙gЈH ഠ 26 days. h(x, y, t) hˆ (x, y) It corresponds to the gravest mode obtained by Rao In Fig. 14 we contrast two very different basin modes (1966) for an inviscid f-plane formulation. In contrast, obtained using the above model formulation. The mode the mode in the lower panel has a period of 566 days. in the upper panel has a period of only 25 days and It is the gravest long Rossby wave basin mode. Its period consists of one counterclockwise propagating gravity is comparable to the basin-crossing time for the long SEPTEMBER 2002 PRIMEAU 2663

2 Rossby wave in the north of the basin (Lx/(␤Rd ) ഠ nously for both the ␤-plane case and the f-plane case. 756 days). Furthermore, the low-frequency forced response on the For the ``Kelvin'' mode in the upper panel, the phase f-plane does not have the typical characteristics asso- propagates around the boundary of the basin in the coun- ciated with free Kelvin wave. There is no phase prop- terclockwise direction. The pressure along the boundary agation with the coast to the right, and the amplitude is clearly a function of both space and time. For the of the response decays away from the boundary with long Rossby wave basin mode, the phase propagates an e-folding scale, that is much larger than one Rossby westward in the interior of the basin but is nearly con- radius of deformation as is the case for Kelvin waves. stant along the boundary. Unlike the ``Kelvin'' mode (Compare the dashed co-amplitude lines in the upper the boundary pressure for the long Rossby wave mode left panel with those in the lower left panel. The contour is nearly independent of space. It oscillates in time syn- interval is the same in both panels). chronously at all points along boundary despite the fact For the low-frequency long Rossby wave basin modes that the model formulation has a ®nite Kelvin wave the boundary pressure ¯uctuations are spatially uniform propagation speed. This is not surprising since the time even when the speed of the gravity waves is ®nite. Thus, for a gravity wave to propagate around the boundary for motions with frequencies that are much lower than (26 days) is much shorter than the period of the mode the time for a Kelvin wave to propagate around the (566 days). This suggests that free Kelvin waves are perimeter of the basin the uniform boundary pressure not the means by which the energy ¯uxed onto the west- adjustment in the QG formulation can not be interpreted ern boundary by the long Rossby wave is returned to as being the manifestation of Kelvin waves propagating the eastern boundary. around the boundary instantaneously. The mechanism To further examine how the energy is transmitted by which the energy of the long Rossby wave modes from the western boundary to the eastern boundary, we is transmitted from the western boundary to the eastern now turn to a forced problem in which we prescribe the boundary is attributed to a low-frequency forced gravity pressure along the western boundary to oscillate in time. wave response. This forcing provides a source of energy at the western The midlatitude long-wave basin modes are different boundary. For low-frequency oscillations it is meant to in this respect from the equatorial basin modes described mimic the pressure perturbations caused by the long by Cane and Sarachik (1977) and Cane and Moore Rossby waves incident on the western boundary. The (1981). For the equatorial modes, the energy carried boundary forcing is imposed by prescribing thickness westward by the Rossby waves is returned eastward by along the western boundary as follows: a propagating Kelvin wave. For the midlatitude long- wave Rossby basin modes the resulting modal frequency h(x ϭ 0, y, t) ϭ ei␻t. (59) is so low that the long Rossby waves do not excite any The response to the prescribed forcing is obtained on free gravity waves. Instead, they produce a forced grav- an f plane as well as on a ␤ plane. The advantage of ity wave response at the modal frequency. This forced the f-plane formulation is that by setting ␤ to zero we large-scale response enforces mass conservation and can eliminate the Rossby waves and thus isolate the carries energy back to the eastern boundary where it mechanism by which the boundary pressure ¯uctuations can radiate again as long Rossby waves. on the western boundary are transmitted to the eastern boundary. 5. Conclusions The solution to two different forcing frequencies are contrasted in Fig. 15. The two upper panels show the If long-wave basin modes are to have a damping time- forced response with ␻ ϭ 2␲/756 days, and the lower scale longer than the basin-crossing time so as to avoid panels show the forced response with ␻ ϭ 2␲/25 days. being damped away, it is necessary for the energy carried The left panels are for the f-plane case and the right westward by the long Rossby waves to be returned to the panels are for the ␤-plane case. For the high-frequency eastern boundary. In the inviscid theory of Rossby basin forcing (lower panels) the response is very similar for modes, this is achieved by the re¯ection of short Rossby both the f-plane and ␤-plane solutions. There is a clear waves at the western boundary. For low frequencies, the counterclockwise phase propagation, and the amplitude resulting inviscid basin modes have small spatial scales, is trapped near the boundary with an e-folding decay and we would expect that they would be quickly damped scale equal to the Rossby radius of deformation. For the out by frictional effects. However, as Cessi and Primeau low-frequency forcing (upper panels), the case with the (2001) have demonstrated, the time-dependent boundary ␤-effect (top right panel) shows long Rossby waves ra- pressure adjustment necessary to conserve mass in the diating from the eastern boundary, indicating that the quasigeostrophic limit allows the energy incident on the forcing imposed on the western boundary has been western boundary to be transmitted back to the eastern transmitted to the eastern boundary. It is important to boundary without being damped out. In the quasigeo- note that the response to the low-frequency forcing does strophic case, the energy is returned to the eastern bound- not have any phase propagation along the boundary. ary by the spatially uniform boundary pressure ¯uctua- The pressure around the boundary oscillates synchro- tions. In the shallow-water equations a similar spatially 2664 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 32

FIG. 15. Forced response in which the thickness h is prescribed to oscillate in time along the western boundary: (left panels) The response on an f plane and (right panels) the response on a ␤ plane. (upper panels) The response for a forcing frequency of ␻ ϭ 2␲/756 daysϪ1; (lower panels) The forced response for a forcing frequency of ␻ ϭ 2␲/25 daysϪ1. The thick solid and dashed lines denote contours of phase and amplitude. The contour interval for the amplitude is 0.1. The amplitude is equal to unity along the boundary and decreases to zero toward the interior of the basin in all cases except the upper-right panel where the response decreases to zero at the amphidromic point near the northwest corner of the basin.

uniform boundary pressure ¯uctuation returns the energy tition of energy that is either dissipated in the western to the eastern boundary, even when the speed of Kelvin boundary layer or transmitted back to the eastern boundary waves is ®nite. via a forced gravity wave response. We ®nd in the present study that it is the shape of the Finally, we caution on the use of rectangular ocean basin that determines the ability of the modes to be res- basins for theoretical studies of low-frequency vari- onantly excited. If the basin shape is such that the basin ability using the quasigeostrophic equations. Rectan- crossing time for the long Rossby wave is a function of gular basins are widely used for theoretical studies of latitude, the modes' damping rate will remain ®nite and the wind-driven ocean circulation because of the ease become independent of the friction parameter as the fric- with which the boundary conditions can be applied in tion parameter is decreased. Nevertheless, we ®nd that the analytical and ®nite difference models. For quasigeo- resulting decay rate is suf®ciently small for the modes to strophic dynamics, however, the rectangular basin is sin- have real Q factors so that spectral peaks might be de- gular in the sense that it is an optimal resonator for tectable. In fact, Cessi and Louazel (2001) have demon- long-wave basin modes. For the shallow-water equa- strated that this is indeed the case through time-dependent tions, the optimal basin geometry has its width narrow- simulations of the planetary geostrophic equations. Here ing to the north such that the long-wave basin-crossing we have focused on clarifying how the shape of the basin time becomes independent of latitude. For a basin ge- determines the modes decay rate by controlling the par- ometry similar to the North Paci®c with a wall along SEPTEMBER 2002 PRIMEAU 2665 the equator, we obtain Q factors of 0.6 and 1.2 for the REFERENCES two gravest modes. These estimates would of course Cane, M. A., and E. S. Sarachik, 1977: Forced baroclinic ocean decrease if the modes coupled strongly to other modes motions. II: The linear equatorial bounded case. J. Mar. Res., in the vertical because of topography for example. Nev- 35, 395±432. ertheless, if low-frequency long-wave modes are being ÐÐ, and D. W. Moore, 1981: A note on low-frequency equatorial basin modes. J. Phys. Oceanogr., 11, 1578±1584. excited by atmospheric forcing their Q factors appear Cessi, P., and S. Louazel, 2001: Decadal oceanic response to sto- to be suf®ciently large for spectral peaks to be detectable chastic wind forcing. J. Phys. Oceanogr., 31, 3020±3029. in long records of the thermocline displacement. ÐÐ, and F. Primeau, 2001: Dissipative selection of low-frequency modes in a reduced-gravity basin. J. Phys. Oceanogr., 31, 127± 137. Flierl, G. R., 1977: Simple applications of McWilliam's ``A note on Acknowledgments. Numerous conversations with a consistent quasi-geostrophic model in a multiply connected Paola Cessi, as well as useful suggestions from the two domain.'' Dyn. Atmos. Oceans., 1, 443±453. LaCasce, J. H., 2000: Baroclinic Rossby waves in a square basin. J. anonymous reviewers, are gratefully acknowledged. Phys. Oceanogr., 30, 3161±3178. The author wishes to acknowledge the use Jonathan Marion, J. B., 1970: Classical Dynamics of Particles and Systems. Richard Shewchuk's ``Triangle'' program for two-di- Academic Press, 573 pp. mensional quality mesh generation, available through Rao, D. B., 1966: Free gravitational oscillations in rotating rectangular basins. J. Fluid Mech., 25, 523±555. Netlib. It was used for generating meshes for all the Wunsch, C., 2000: On sharp spectral lines in the climate record and two-dimensional ®nite element computations. the millennial peak. Paleoceanography, 15, 417±423.