Deep ocean influence on upper ocean baroclinic instability

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 106, NO. Cll, PAGES 26,863-26,877, NOVEMBER 15, 2001 Deep ocean influence on upper ocean baroclinic instability M. J. Olascoaga Departamento de Oceanografia Fisica, CICESE, Ensenada, M6xico Abstract. Reduced gravity models, namely, those having an active layer of fluid floating on top of a motionless one, have been largely used to study the upper ocean. This approximation is formally valid when the total ocean depth tends to infinity. The effect of a finite ocean depth on the upper ocean baroclinic instability is examined here using a quasi-geostrophic three-layer model and comparing the results with those obtained using a reduced gravity quasi-geostrophic two-layer model or 2.5-layer model. The basic state is a zonal current with uniform velocities within each layer. The ratio s between the sum of the upper two layers mean depths and the lower layer mean depth is the relevant new parameter of the problem. Even for very small values of s, important differences between the 2.5-layer (s = 0) and the three-layer ( > 0) model are found. As increases, the region of Arnold stable states decreases. For certain basic states, new normal mode instability branches are found, whose growth rates increase with s. An asymptotic expansion in s is made in order to shed some light on the transition regime between both models. This allows one to interpret the new instabilities as a consequence of the resonant interplay between the stable modes in the 2.5-layer model and a short Rossby wave in the deep layer. The growth rates of the new instability branches are O(s 13) and O(s 12) and cannot be neglected even for reasonably small values of s. 1. Introduction Motivated by the observed enhancement of the variability toward the surface of the ocean, reduced gravity models have been used to study the problem of baroclinic instability in the upper layer of the ocean (i.e., above the thermocline) by several authors [Fukamachi et al., 1995; Young and Chen, 1995; Beton-V era and Ripa, 1997; Ripa, 1999; Olascoaga and Ripa, 1999; Ripa, 2000]. Such models consist of an active layer of fluid floating on top of a quiescent one which is assumed to be infinitely deep. In principle, the reduced gravity approximation seems to be a reasonable one since the total depth of the ocean is much larger than that of the upper thermocline layer. However, this is not always the case: a thick (but finite) abyssalayer may have an important influence on the stability of the upper ocean. For instance, the stability of a surface front was studied with both a reduced gravity one-layer model (or, for short, 1.5-layer model) and a two-layer model by 1 Now at Rosenstiel School of Marine and Atmospheric Science, Applied Marine Physics, University of Miami, Miami, Florida, USA. Copyright 2001 by the American Geophysical Union. Paper number 2000JC J C $ ,863 Killworth and Stern [1982] and Killworth et al. [19841, respectively. The two-layer front is always unstable and the growth rate of the most unstable perturbation is generally much larger than the one in the 1.5-layer front. Similarly, the stability characteristics of oceanic (lens-like) eddies has been shown to change substantially when a finite lower layer is present [Chassignet and Cushman-Roisin, 1991; Ripa, 1992a]. None of these previous works allowed for baroclinic instability in the active layer. Benilov [1995], in turn, studied the stability of baroclinic flows confined to a thin layer using different models: a three-layer model, a mixed model with a thin continuously stratified layer over a thick homogeneous layer, and a continuously stratified model with the flow localized in a thin upper layer. However, Benilov [1995] did not study the validity of the reduced gravity approximation in the context of upper ocean baroclinic instability; such an examination is the object of this work. Quasi-geostrophic three-layer models have been extensively used to investigate the stability of a baroclinic zonal current. The basic state can be characterized by the following nondimensional parameters: (1) s, which is the ratio of the buoyancy jumps in the first and second interfaces; (2), which is the ratio of the slope of the second and first interfaces; (3) b, which measures the importance of the planetary effect relative to the shear between the two upper layers; and (4) e, which is

2 26,864 OLASCOAGA ET AL.' DEEP OCEAN INFLUENCE ON UPPER OCEAN BAROCLINIC INSTABILITY Table 1. Summary of Baroclinic StabilityInstability Studies with Three Layers a Reference s v b e n Type Davey [1977] 1 any any 2 any spectral Ikeda [1983] > any spectral $meed [1988] _> I any 0 _> I any spectral Benilov [1995] any any O(e) << 1 0(514) spectral Vanneste [1995] I any any 2 any formal Pichevin [1998] any any 0 any any spectral Olascoagand Ripa [1999] any any any 0 any spectral and formal This work any any any << I any spectral and formal a Parameters are as follows: s is the ratio of the buoyancy jumps in the first and second interfaces; v is the ratio of the slope of the second and first interfaces; b measures the importance of the planetary? effect relative to the shear between the two upper layers; e is the depth ratio of the upper two layers relative to the lower layer; and n is the wave number nondimensionalized by the internal deformation radius of the system. the depth ratio of the upper two layers relative to the lower layer. (There could be a fifth basic state parameter, the ratio of the depths of the upper two layers, which is not varied in this paper and is taken equal to 1.) The perturbation is characterized by n, which is the wave number nondimensionalized by the internal deformation radius of the system. Linear (normal modes) baroclinic instability with these models has been studied by several authors [Davey, 1977; Ikeda, 1983; Smeed, 1988; Benilov, 1995; Pichevin, 1998]. Table 1 summarizes these previous studies in the space defined by the above characteristic parameters. Davey [1977] considered the case of equal depth layers (s = 2) and equal buoyancy jumps across the interfaces (s - 1), finding that the range of unstable wavenumbers increases as the vertical curvature of the current increases. Ikeda [1983] studied a system in which the buoyancy jump across the first interface is greater than that across the second one (s > 1), the depth of the second and third layer are equal and twice the depth of the first one (s = 32), and there is no relative motion between the two lower layers (F = 0). That work focused the attention on the effect of the stratification on the stability properties of the lower ocean as well as on the effects of the hori- nonlinear saturation bounds on unstable basic states were obtained by Purer and Vanneste [1996] for the particular case of equal buoyancy jumps (s - 1) and equal layer depths (s- 2). Finally, Benilov [1995] worked on the plane and considered the limit s -- 0 assuming that b = O(s). In the work by Benilov [1995] the similarities between the three-layer and the classical twolayer model of Phillips were particularly considered. In Olascoagand Ripa [1999], here after referred to as OR99, the baroclinic instability properties of a zonal current using a 2.5-layer model was studied. Impor- tant differences with the classical Phillips (two-layer) model [Phillips, 1954] were found. In Phillips' model the effect always stabilizes, while in the 2.5-layer model it may either strengthen or weaken the stability of the basic current. The maximal instability for the Phillips' model is achieved for a perturbation whose wavelength is of the order of the internal deformation radius, as in the case of the 2.5-layer model it is achieved for a perturbation with an intermediate wavelength between the external and internal deformation radius. These instability characteristics are found to be almost the same as the ones obtained with a con- tinuously stratified reduced gravity model, e.g., Eady's zontal velocity shear and bottom slope. Smeed [1988] discussed different values of the velocity and buoyancy jumps across the interfaces and reference layer depths. That work concentrated on the case in which the buoyancy jump across the upper interface is greater than that across the lower one (s _ 1), and the lower layer is thinner than the total depth of the upper two layers ( > 1). An asymptotic expansion in s - and s- as s - and s -, respectively, was also performed by classical problem [Eady, 1949] with a free boundary [Fukamachi et al., 1995; Beton-V era and Ripa, 1997; Ripa, 2000]. The similarity between continuously stratified and layered models has been extended to 0 cases by Ripa [2001]. The goal of the present paper is to discuss the effects of the interior ocean on the baroclinic instability of the upper thermocline layer. We analyze the extent to which the stability properties of the 2.5-layer 5'meed [1988]. Pichevin [1998], in turn, characterized model discussed in OR99 vary with respect to those the instabilities as a result of the resonant interaction corresponding to a problem with a finite third layer. of free topographic Rossby waves; the results compared well with the exact normal modes solutions. Among all those normal modes studies the planetary effect was included only by Davey [1977]. In addition, Arnold sta- Particular attention is therefore paid to the limit - 0 in order to investigate the transition regime between an infinite and a finite depth third layer on the instability. The rest of the paper has the following organization. bility was established by Vanneste [1995], and rigorous In section 2, the three-layer model is presented. Formal

3 OLASCOAGA ET AL.' DEEP OCEAN INFLUENCE ON UPPER OCEAN BAROCLINIC INSTABILITY 26,865 ( H Dq Dt = 0 Dq21Dt= O g' Dq31Dt= O Figure 1. Sketch of the Three-layer model, g and g} are the buoyancy jumps across the upper and lower interfaces, respectively, (top graph). The model equations are conservation of quasi-geostrophic potential vorticities in each layer. Vertical profile of the baroclinic zonal flow in the basic state (bottom left graph) and the corresponding slopes of the interfaces (bottom right graph), x (V) is the eastward (northward) coordinate and f0 is the reference Coriolis parameter. and spectral stability properties are addressed in section 3 and 4, respectively. In section 5, a small c asymptotic expansion is carried out, and a resonant mechanism is invoked to interpret the instabilities. A summary and conclusions are presented in section 6, and Appendices A-D are reserved for some mathematical details. 2. Three-Layer Model Consider three layers of constant density and inviscid fluid with a horizontal rigid surface and flat bottom, as sketched in the top graph of Figure 1. The reference layer depths of the three layers are chosen as H1 - H2 _ 1H H3-6-1H. (1) Let x: = (x,y) denote the horizontal position with Cartesian coordinates x and y, which point eastward and northward, respectively. The governing equations are those of conservation of quasi-geostrophic potential vorticities qi(x,t) in the upper (i = 1), intermediate (i: 2), and lower (i: 3) layers, namely, Dqi Dt :0, (2) DDr '- Ot +,. 7 )j x V is the material derivative, t is the time,, is the vertical unit vector, and V '- (Ox, Oy). The stream functions )i(x, t) and potential vorticities are related through 3 qi '-- V 2 )i -I- Z Mij ½j -+- fo -+- V, j--1 fo and are the reference value and gradient of the Coriolis parameter, respectively. Here the matrix M is given by M '- 2L s s 0 es 1 _ 21_ 8 (3) L '- vig H - g--- Ifol ' g > 0, (4) g is the buoyancy jump across the ith and (i + 1)th layer. The top and the bottom boundaries are considered rigid and horizontal. The horizontal domain D is an infinite (or periodic) zonal channel of width W with the usual boundary conditions of no normal flow and conservation of Kelvin circulations at the solid boundaries ODJ of the channel (y = O, W) in each layer, i.e.,

4 26,866 OLASCOAGA ET AL.: DEEP OCEAN INFLUENCE ON UPPER OCEAN BAROCLINIC INSTABILITY i,x i,ydx - const. In this paper we consider a basic state defined by a baroclinic current which implies that the basic state potential vorticities Qi take the form Qi(Y) -- - Y. MijUj y q- fo - Q iy q- fo, j--1 the Ui are constant and different in each layer and The nondimensional parameters v and b are given by v'- U8 s b'- Us' U := U1 -U2 is the velocity jump across the upper interface. Notice that is the ratio of the slopes of the interfaces in the basic state; U3 will be chosen as nil without loss of generality (see bottom graph of Figure 1). The parameters s,, and b are the same as those of the 2.5-layer model we discussed in OR99 and qualitatively the same as those used by Ripa [1999, 2000, 2001] in a three-dimensional model. The parameter e, which is nil in OR99 calculation, is typically small for the ocean (see Benilov [1995] for an account on possiblexamples of ocean currents with small e). 3. Stability Properties Inferred From Conservation Laws The evolution of the system (2) is constrained by the existence of an infinite set of integrals of motion, namely, quantities which are preserved by the dynamics. These invariants being energy zonal momentum œ - - H qi idxdy, i--1 YM --1H Dqiydxdy, 2 i=1 and a family of potential vorticity related invariants known as Casimirs Fi( stants. ], J J C -- y ai i q- Fi(qi)dxdy i--1 j=l ) are arbitrary functions and a i are con- Choosing a Casimir such that (œ + ½- c 2M) = 0 for any constant c, the pseudoenergy œp[sqi] :: (A-5)(œ + ½) and pseudomomentum.a p[( qi] :---- (A- ( )(.A q- C) are exactly quadratic invariants in the departure ( qi J from the basic state. Here A and 5 refer to the total and first variations of a functional, respectively. Arnold's theorems provide sufficient conditions for stability [e.g., Holm et al., 1985; Ripa, 1992b] and rely upon the sign definiteness of the linear combination of the pseudoenergy and pseudomomentum, i.e., l kd Hi(e )ie qiq- Ui-oz 5q )dxd p-- O. p -- 2 i----1 Q'i Arnold's first theorem states that the basic flow is stable only if a value of c exists such that the quantity œp -c 2Mp is positive definite, which implies (c - Ui)Q'i > 0 (i - 1, 2, 3). (6) Such a condition is satisfied for values of the parameters b,, and e such that b 5' b 2+e 2e (b,vs) e 1- < v< - b> _J -<v<l- b<-2 Notice that this condition is independent of the stratification parameter s. Condition (6) also guarantees nonlinear (Lyapunov) stability in a quadratic norm, which is a combination of the energy and enstrophy of the perturbation. Furthermore, rigorous nonlinear saturation bounds on unstable basic states can be obtained using Shepherd's method [Shepherd, 1988; Pater and Vanneste, 1996]. If (6) is not satisfied, linear (normal modes) instability is usually examined. In this case, a shortwave cutoff for the unstable modes can be identified. That is, if n '- Vk L (7) is the nondimensional wave number of the perturbation, with k and 1 the zonal and meridional wave numbers, respectively, then there exists ns(b,, s, e) such that for n > ns, all modes are stable (see section 4). Arnold's second theorem states that the basic flow will be stable if there exists an c such that œp- c 2Mp is negative definite, which implies that (oz- Ui)Q'i < 0 (i - 1, 2, 3) (8) nmin > ns, (9) nmi n -- 7rLW is the minimum wave number in the zonal channel. The basic state is thus stable if (6) is satisfied or if (8) and (9) are satisfied. Moreover, these conditions are not only sufficient but also necessary ones for stability [Vanneste, 1995]. That is, if neither of Arnold's conditions is satisfied, the basic state is unstable either by linear instability (if condition (9) is violated) or by explosive resonant interaction (ERI). The latter process refers to the existence of a resonant triad of waves whose

5 OLASCOAGA ET AL.' DEEP OCEAN INFLUENCE ON UPPER OCEAN BAROCLINIC INSTABILITY 26, layer 3-layer ( --> 0) '5 V V Figure 2. Stabilityinstability regions on the (,b) plane, b :- fil2us;, := (U2- U3)sU -, Us := U1 - U2 is the velocity jump at the upper interface; L is a length scale proportional to the internal deformation radius; s '- gig ; nmin is the minimal nondimensional wave number in the zonal channel; and ns(b,, s,e) is the shortwave cutoff for instability. The dark shaded (Arnold 1) and the white (Arnold 2) regions are proved nonlinearly stable according to Arnold's first (equation(3.1)) and second ((3.3) and (3.4)) theorems, respectively. The light shaded (explosive resonant interaction (ERI)) regions are unstable due to explosive resonant interaction, even though they are linearly stable according to (3.10). components grow simultaneously by extracting energy from the basic flow [Cairns, 1979; Craik and Adam, 1979]. This occurs for (a- Ui)Q i sign indefinite, i.e., for (b.,.z) e {z, < b < 2-2.., > O) U{2-2, < b < -2) < < -2,. > (lo) Of particular relevance is the fact that for nmin > ns, basic states defined by (10) are nonlinearly unstable due to ERI even though they are linearly stable (see section 4). The occurrence of ERI for nmin < ns in this region is unimportant because the basic states are already unstable from linear theory. Figure 2 shows the stability region according to Arnold's first and second theorems in the (y, b) plane for the 2.5-layer model (OR99) and the three-layer model in the limit e - 0. Notice that as e - 0, the region Arnold's first theorem is valid (labeled "Arnold 1" in Figure 2) does not reduce to that of the 2.5-layer model. That reduction would require that both H35 k35q3-0 and H3(U3- a)6q Q - O, even though H3 -, that is, the lower layer must be both of infinite depth and quiescent. Unstable regions due to ERI, only for the cases in which the basic state is linearly stable, are also displayed in Figure 2. It is remarkable that even in the limit - 0, there are spectrally stable regions which are unstable due to ERI. In the 2.5-layer case, however, ERI can be present only in the region of linearly unstable states (not shown in Figure 2). That in the 2.5-layer model there is no possibility for nonlinear instability the basic state is linearly stable is a con- sequence of the fact that spectral and Arnold's stability conditions coincide in that model (OR99). 5 3-layer (e = 0.1 & s = 1) 3-layer (e = 3 & s = 1) 0 Arnold 2 (Kmi n > KS) Arnold 2 (Kmi n > KS) -2õ V V Figure 3. Stabilityinstability regions the (y, b) plane, as in Figure (2), for finite values of '- HH3. Notice that the scales are different in each graph.

6 26,868 OLASCOAGA ET AL.: DEEP OCEAN INFLUENCE ON UPPER OCEAN BAROCLINIC INSTABILITY s=0.1 &s=l 2.5 t' C =C C + =C R C_=C R b=l V i i i i I b-- -1 o o v Figure 4. Stabilityinstability regions on the (, n) plane for the three-layer model on the fi plane (b 0). The shaded region is linearly unstable, as the white region is linearly stable. Solid, dashed and dot-dashed curves are such that c+ = c_, c+ = cr, and c_ = cr, respectively, cñ are the perturbation phase speeds in the 2.5-layer model and cn is the phase speed of a short Rossby wave. These curves correspond to instabilities due to double resonance. The solid curves delimit the instability region in the 2.5-layer model. Vanneste [1995] presents a detailed analysis of conditions (6) and (8) and (9) as well as the conditions for _n-2l2 the existence of ERI but for the particular case e = 2 E (rl + 1) r + 4srl and s - 1. The results for different finite values of e are shown in Figure 3. The domain of Arnold stable states, by the first theorem, diminishes as e increases. For e < 2s- 1 the region which is unstable due to ERI increases as e decreases, independently of s. For e > 2s -1 persion relation this region depends on both e and s. More precisely, the minimum area for ERI in the (, b) plane corresponds to emin -- $--1[$ (14- S 2)x2112 _ 1. Conse- quently, for e > emin the ERI area increases as e increases, as for 2s -1 < e emin this area increases as e decreases. 4. Spectral Instability Spectral instability is studied by assuming infinitesimal perturbations of the form 5qi - c i ½ik(x-ct) sin ly. This results in the eigenvalue problem 3 Hijc j - O, (11) j--1 H -- diag[(ui- c)q'i] 4- œ, (M- n2l-2[) - namely, E '= 4s (r- 1) + rz z 4s r rr 4st 2se 2rse 4 r._ 1 + n2, p '- r2 +s0-1, and r := 2 (n 2 + es). Nontrivial solutions of (11) imply the dis- (c - c ) (c - c+)(c - c_) - -esn-2(c - U2)P(c), (12) c - cñ denotes the dispersion relation of the 2.5- layer model (see Appendix A), c - - k (= -busin 2) is the phase speed of a short Rossby wave, and ].,--18½ IT] (C- Vl) 4- (1 + «b) Us]. No simple expression for c can be obtained for an arbitrary basic state [e.g., Davey, 1977; $meed, 1988]. The cubic polynomial (12) has one real and a pair of complex conjugate roots for certain values of (s,, b, e, n) there exist growing perturbations.

7 OLASCOAGA ET AL.: DEEP OCEAN INFLUENCE ON UPPER OCEAN BAROCLINIC INSTABILITY 26,869 s=1, b=-i & v= , s=1, b=-i & v= s= =0.1 ' , 4 CR Im(c+) Re Re(c_) Im(c_) Im(c_): Im(c+) Figure 5. Imaginary part of the perturbation phase speed for the three-layer model for two sets of values of the basic state parameters (top graphs). Eigenvalues in the 2.5-layer model (½+) and the phase speed of a short Rossby wave (½R)(bottom graph). Notice that the instability peaks corresponds to the values of for which there is coincidence between any of the curves ½+, ½_, and cr. s=1 &b=-i 0.4,,,, 0.3 v=o œ=0 œ=0.1 œ=1 œ=2 0 I i I I I I I 0.8- V = ',, x. x _ ß \ \ ' \ \. - \ \ E 0.4- x x ill K Figure 6, Perturbation growth r tes for two slices of he bottom graph of Figure 4 t v - 0 nd v - 3. Different v ]ues of s re lso displayed in order o ppreci te he effec of the finite depth lower l yer on he instability problem.

8 26,870 OLASCOAGA ET AL.: DEEP OCEAN INFLUENCE ON UPPER OCEAN BAROCLINIC INSTABILITY s=0.1 &s=l I I I I I I I! Figure?. Stabilityinstability regions on the (, n) plane for the three-layer model on the f plane (b - 0) case. Solid curves delimit the instability region in the 2.5-layer model, and the dashed curves delimit the unstable regions of the asymptotic solutions as e - 0. Figure 4 displays the stabilityinstability regions on the (, n) plane for the three-layer model with b 0 (fi plane). The neutral stability curves of the 2.5-layer model, which corresponds to c_ = c+, are displayed as a solid curve for reference. Thus the region bounded by the solid curves is unstable in the 2.5-layer model. Notice that even for a very small value of e, the differences between the instability regions corresponding to both models are important. The appearance of "strips" of instability emanating from the 2.5-layer model instability region is remarkable. Anticipating some results of section 5, we call the attention to the fact that these "strips" of instability coincide with the c_ - cr and c+ - cr curves (dashed and dot-dashed curves in Figure 4). These curves will be shown to correspond to instabilities due to the resonant interaction between the 2.5-layer model stables modes (c+) and a short Rossby wave (cn) in the limit e - 0. In turn, the points these "strips" coincide with the neutral 2.5-layer model stability curve satisfy c_ - c+ - cn, which corresponds to an instability due to the resonant interplay of these three waves. As an example, Im c and c_, c+, and cr are plotted in Figure 5 as a function of n for fixed values of e, s, b, and. The shorter wave number instability coincides with the 2.5-layer model instability, namely, Im c+ 0 for this range of n (see left graphs). Notice that the other peak of Im c corresponds to the value of n at which c_, c+, and cn collide (see bottom left graph). In the right graphs of Figure 5, we depict a case in which the 2.5- layer model is always stable, i.e., Im c+ - 0 for all n. The two peaks of instability are centered on the value of n at which c_ = cn and c+ = cr, respectively (see bottom right graph). Figure 6 depicts the perturbation growth rate for two slices of the bottom graphs of Figure 4 at - 0 and - 3. Different values of e are displayed in order to show that the growth rates increase as e increases. Even for small values of e the growth rates are of the order of the maximal growth rate in the 2.5-layer model. Notice also that the new instability "strips" widen with increasing values of e, namely, as e increases (for, s, and b fixed), there are more values of n which are linearly unstable (see Figure 6, bottom graph). The stabilityinstability regions on the (, n) plane for the three-layer model with b - 0 (f plane) are shown in Figure 7. The solid curve delimits the instability region in the 2.5-layer model. The most striking difference between the three- and 2.5-layer models, in this case, is that in the former, there is instability at small n for any value of. This instability region increases with s and e, as has been described by Ikeda [1983], Benilov [1995], and Pichevin [1998]. 5. Small s Expansion In order to shed some light on the transition regime between an infinite and finite third layer on the insta-

9 OLASCOAGA ET AL.: DEEP OCEAN INFLUENCE ON UPPER OCEAN BAROCLINIC INSTABILITY 26,871 S= Figure 8. Instability curve (thick solid and dashed curves) due to triple resonance, i.e., c+ - c_ - c t, in the (b, t, n) space and the "tube" of neutral stability ellipses (thin curves) in the 2.5-layer model. bility we take (s, t, b) fixed and consider the asymptotic expansions c - C(o) + c( ) + O( 2' ),?- + + as - 0. Substituting (13) in (12), the values of n are determined such that (12) is balanced [e.g., Nayfeb, 1973]. This process results in a -, and 1, namely, three different expansions will be considered. The coefficien;s of each of ;hese expansions are de;ermined by equating ;erms of ;he same order in as explained below. The f and plane problems behave in a quite differen; way and ;hus are ;reared separa;ely The Plane Case To zeroth order in c the three roots of the right-hand side of (12) are qo)- For most values of (v, b,s) it is n = 1, and c( ) would not be complex unless C(o) = c+ or c_ already is complex. Thus, in this case, there are no new instabilities; there are only corrections of order c to the instabilities of the 2.5-layer model. The most interesting differences with respect to the 2.5-layer model solutions appear for n- andn- because they give rise to the possibility of complex c( ) for real c(0). That is, there are unstable solutions the 2.5-layer model is stable. ß Hence results for these values are analyzed in more detail below Triple resonance. For particular values in the curve (b, t) E I'(s), triple resonance is obtained, i.e., c+ - c_ - c t at - (0). Polynomial (12) requires 1 n- to be balanced. This yields (b, v) F(s) C(o) - ca - ' ½(«) --(S(½R U2)I'6-2 (o)p(c ) { 1, I (1+ivY)} ß The zeroth-order term is always real, and thus the instability depends only on the one-third-order solution. The curve F(s) and the resonance wave number n(0) cannot be obtained explicitly. Instead, at fixed s, there are two possible values of b and tfor each n(0), which are denoted here as b+ and tq: (see Appendix B for the definitions). Figure 8 shows b+(n) and tq:(n) possible for triple resonance with s = 1, which corresponds to a curve in the (b, t, n) space rolling up the "tube" of neutral stability ellipses in the 2.5-layer model. That is, for a given choice of the basic state parameters t, b, and s, triple resonance exists for a perturbation with n lay- ing on the curve. Notice that in Figure 8 the 2.5-layer model is unstable inside the "tube." Figure 9 shows the perturbation growth rates and the corresponding b+(n) and tq: (n). The growth rates exhibit various relative maxima, and, in particular, those corresponding to (b_, t+) are larger than those associ-

10 26,872 OLASCOAGA ET AL.' DEEP OCEAN INFLUENCE ON UPPER OCEAN BAROCLINIC INSTABILITY 0.1 )U O I I I I x, \ [ I \, x I ;.3 E ,, \t 1.5. ;> Figure 9. Asymptotic growth rate of the perturbation due to triple resonance for s = 0.05 (thin curve) and s = 0.2 (thick curve), for b+ and v_ (top left graph) and for b_ and v+ (top right graph). The possible values of b(n) and v(n), i.e., the solid curve (b+, v_) and the dashed curve (b_,v+), are displayed in the bottom graphs. ated to (b+, v_) for small s. For (b_,v+) the growth rate increases as s decreases, as for (b+, v_) the growth rate increases as s increases. In the limit of weak stratification (s --> 0), the maximum perturbation the three-layer model is about twice as large as that one in the 2.5-layer model. This provides evidence that the inclusion of a finite depth third layer, even for reasonably small values of e, produces extra instabilities that growth rate is nim cus = O(s-zd), which is achieved cannot be neglected (for certain basic states) for n vj+ s and (b_, v+) - (3,- ) +O(vJ) ß There Double resonance. There are three possiis also a relative maximum in this limit of n Im cus = 5 8 O(vJ) for n- vj and (b+, v_)- (,- )s + O(s2). In ble cases of double resonance, i.e., c+ - c_, c+ - ca, and c_ - ca, for which polynomial (12) requires n - both cases, (b_,v+) and (b+,v_), it is n = O(vJ) as s - 0, which corresponds to a wavelength of the order of the largest deformation Rossby radius of the system. This shows that the maximum instability due to triple resonance has a length scale which is much larger than that for maximum instability in the 2.5-layer model, i.e., n = O(s 4) (see OR99). Table 2 compares the maxima growth rates of both the 2.5- and three-layer models in the limit s - 0. Notice, for instance, that for s = 0.1 and e = 0.1 the maximum growth rate in to be balanced. This leads to Table 2. Maximum Growth Rate as s - 0 for the 2.5- and Three-Layer Models Which are Achieved at nmax, Vmax, and bmax Model max max bmax maxn,v,b {n Im cus } 2.5-layer (e- 0) x J I + O(vJ) -1 + O(vJ) 4 4 Three-layer (e- 0) vj + O(s) + 0 (vj) (vj) 1 + O(vq) 22aa 2 q- O(, ) 723 V. J

11 OLASCOAGA ET AL.' DEEP OCEAN INFLUENCE ON UPPER OCEAN BAROCLINIC INSTABILITY 26, C -- =C + =C R C + =O R C_=O R I I Figure 10. Asymptotic growth rate of the perturbation due to double resonance (dashed and dotdashed curves) and triple resonance (crosses) for c = 0.1, s -- 1, b = -1, and the corresponding v(rc) for c+ = cr, c_: cr, and c_ = c+ = c. The exact solution is also displayed (solid curve). The arrow indicates the maximum growth rate with respecto (v, b, re) for s: I in the 2.5-layer model. V - 0}, (see Appendix C for the definitions of the F values). which corresponds [o [he eigensolutions in [he 2.5-layer For G Fj(b, s, re), j: 1, 2, 3, the zeroth-order solution of c is real, and thus the instability depends only on model plus a zero phase speed eigensolu[ion. Unlike the one-half-order solution. plane case, in which the unstable perturbations are all of the order of the internal deformation radius, i.e., Figure 4 shows curves corresponding to each possible case of double resonance on the (v, re) plane, the = O(1), in [he f plane case, [hey can be ei[her long, instability regions are also depicted. Triple resonance in Figure 4 corresponds to the points the curves c+ = cr and c_ = cr collide with that of c+ = c_. Consequently, the "strips" of instability mentioned in section 4 can then be seen as a result of a double resonance due to the matchings c+ = c and c_ - cr. The asymptotic growth rates due to these resonant cases are shown in Figure 10 as a function of (rc) for s = 1 and b The obtained asymptotic results compare well with the exact solution given by (12). For the case c+ - cr, the growth rate increases as p increases, reaching values which are considerably larger than those obtained with the 2.5-layer model. The instabilities are thus not negligible whatsoever, even for very small values of c The f Plane Case To zeroth order in the three roots of polynomial (12) are given by I%---- 0(E14), or short, r - 0(1) Long perturbations. Polynomial (12) re- 1 quires n - in order to be balanced, which yields n(o) = 0 :=. C(o) = { U2, O, 0 } --{ ( y2)} == c(«) 0, 4vs2 _, 7 '- (s + v)2 + v2. Hence, to this order instability arises for any v when That is, sufficiently long perturbations are always unstable, independent of the choice of the basic state (see Figure 7). The asymptotic solution is compared with

12 _ 26,874 OLASCOAGA ET AL.: DEEP OCEAN INFLUENCE ON UPPER OCEAN BAROCLINIC INSTABILITY 0.45 b=o, v =-2, s=1&s=0.1 I I I I Exact solution Asymptotic solution '\ O.3 ' 0.25 E O.O5 I - I Figure 11. Imaginary part of the eigenvalue c (asymptotic and exact solution) as a function of. the exact one in Figure 11. For small the asymptotic solution lies very close to the exact one. Notice that the error for the long perturbations is O( ), as expected. In the limit of s - 0, this long perturbation instability can be identified as of Phillips' type due to the velocity shear between the second and third layers [see section 5.3 for the details] Short perturbations. With n - 1, polynomial (12) can also be balanced, which yields 27 0)- - z2 + qo) - {o, o} - {A+, O, A_ }, A+-2A 1 (-B + ½A ) +B (+C). ) Instability to this order exists for (o) + e ( ) < < (o) + e ) with t ) = 2A 1 (-B ß B 2-4AC). (The expressions for A, A, B, B, and C are given in Appendix D.) The correspondingrowth rates are very small up to the present order, and the region of instability is very narrow for fixed (see Figure 7). Pichevin [1998] provides a different explanation for these short-wavelength and long-wavelength instabili- ties in terms of the resonant interaction between cer- tain nonphysical wave modes, which could be identified with a sort of freely propagating Rossby waves due to the slope of the interfaces. To obtain these modes, one has to use ( ½i to obtain the normal modes equation, instead of ( qi as in (l 1), and seek the solutions of the problem that result from retaining only the diagonal terms of the matrix corresponding to this eigenvalue problem. Pichevin [1998] showed that the coupling of these "½ modes" gives rise to an instability whenever the wave components have opposite sign pseudomomentum. An alternative method consists of using the "q modes" (rather than the "½ modes"), namely, those resulting from zeroing the nondiagonal terms of the ma- trix of the eigenvalue problem (11). See Ripa [2001] for a discussion of both possibilities Phillips-Type Instabilities Certain instabilities obtained in this paper (in the limit -+ 0) can be identified with those in a twolayer model for a convenient choice of the parameters. That is, let the upper and lower reference layer depths be equal to H3 and H3, respectively, the deformation radius be equal to Lx, and velocity shear be equal to Uy. According to this choice, the parameter and the nondimensional wave number change as

13 ß OLASCOAGA ET AL.: DEEP OCEAN INFLUENCE ON UPPER OCEAN BAROCLINIC INSTABILITY 26,875 respectively. In order to see this we recall the dispersion relation of the classical two-layer model of Phillips [e.g., Pedlosky, 1987] in these variables, which reads := e(2 + b) - b + 2Ve(1 + b)(e - b). Notice that for b = O( ) it is nñ = O( 14), so there is instability for n = O( x4). In particular, for b = 0 it is n_j_ - 4 and n 4-0. On the other hand, for finite b it is nñ = O (1). The corresponding maximum growth rate is given by max {him cp} - U2 n Fs[(s + 1)2-4](14) b e+l l+n2+ ' which is achieved for e-1 b- (s+1) 2. (15) Polynomial (12), with the above rescaled variables, in the limit s - 0 reduces to n n 2 (c_ U2) {(cn2 + U2b) Ic (l + n2) - U2(n2 - b) ] +5 (c-- U2) (1 q- n2 (c- U2) n2) q- U2b } n (16) Notice that, now one of the roots factorizes out, giving c - U2 (which corresponds to c+ for n - O(vJ) as s -> 0). The problem reduces to find the other two roots for which we use the asymptotic approach of the former subsections. Thus, making an expansion in powers of like (13), one obtains, to zeroth order, the following roots: C(o) - U2 n o)' 1 + n o) The first root corresponds to cn (i.e., the phase speed of a short Rossby wave), as the other one corresponds to c_ for n = O(V ) as s - 0 (which is a Rossby wave riding on a uniform current of intensity U2). The following order for which polynomial (16) is balanced is O(V ) and requires -bn o ) - (n o) - b)(1. + n o)), which implies b- -n 0) and c(«) - '+-U2 (n o)- 1)(n o ) + 1). Consequently, the instability giving rise to this order corresponds to a resonant interaction between two Rossby modes. The growth rate associated to these dispersion relations corresponds exactly to the maximum one for the two-layer model in the limit - 0 (see (14) and (15)). In the f-plane case (b = 0), in turn, it follows at O (x )that C(o)- 0---:',, c(«)- U2 1 (n «) Fn «) + _4) as well as n(0) - 0, which coincides with the dispersion relation of Phillips' problem in the limit e Similar results were obtained by Benilov [1995], who assumed that b- O( ). 6. Summary and Conclusions The effects of the interior ocean on the baroclinic instability of the upper thermocline layer have been examined using a quasi-geostrophic three inviscid layer model on the plane. The upper thermocline layer is represented by the upper two layers, with depth H, and the interior ocean by the deep layer with depth He. The three-layer models have been extensively used to investigate the stability of a baroclinic zonal current. Other authors have discussed the case of large e; in this work, we have concentrated on small values of e (see Table 1). Even for very small values of e important differences between the 2.5-layer model and the three-layer model were found. Thus an asymptotic expansion for small e was performed in order to investigate the transition regime between an infinite and finite third layer on the instability. New branches of instability were found and interpreted as being due to a resonance between a short Rossby wave and the stable modes of the 2.5-layer model. The growth rate of these new instabilities were shown to be of the same order or even larger than the maximal one corresponding to the 2.5-layer model. In a particular limit these instabilities were shown to be identifiable as of Phillips type. Our results demonstrate that the interior ocean may alter substantially the stability properties of the upper thermocline layer for reasonably small values of z and particular values of the parameters (y,b, and s) that define the basic state. This is a remarkable conclusion that should not be ignored when the reduced gravity simplification is used in ocean modeling. Appendix A' Two-and-a-half-layer Model Dispersion Relation The dispersion relation for the 2.5-layer model (0R99) is given by cñ =U - u, 5x + 2 n 2(n 2+4+2s)+4s'.- 2 (2 +,? + (,?- - (, + A2 '- 4 (2 + n2) 2 (. -.+)(. - + ( -4),

14 26,876 OLASCOAGA ET AL.' DEEP OCEAN INFLUENCE ON UPPER OCEAN BAROCLINIC INSTABILITY with vñ being defined by (B2). Appendix B' Triple Resonance The set of values (b,v) E F(s) that fulfill the resonance condition c+ - c_ - cn must be chosen from bñ i s ( - 2) x2r- 2 [2 ( + 1) p + s] 4 A1 1 2x 2 4- (2-81) hl (B1) There is instability for n(0) + e (1 ) ( ( (0) -e l) A '- 16n 0)ff 2, = i + - 4AD), B-- 16s 2 [7s(-1 + 2v)(s + 2v) + 2v 5] n(o) D'-s2[v2(4+n o,)-2s2] 2. 1 vñ ( -1- «b)(2- s ) - ( - 1) Lt [ 2- ( -b-2) 2], (B2) 1 1= 2 (1-2V) ( ) + 4V (V- 2) (.2 8) +s2. (, + s) + s (s- 4) a2-- (, - 1){4 (1 + s)(s, 2 + 3) - s (16, + 11s) - 4 (, - 2)[,2 (, _ 2) - 2, + 2s 2] } + 4,s. Notice that (B1) and (B2) gives four possible solutions; however, two of them are spuria ((b+, v+) and (b_, v_)). Appendix C: Double Resonance Double resonance for c+ - c_ can occur for v E Fl(b, s, n) given by (B2), and for c+ = cn and c_ = c it can occur for v E F2,3(b, s, n) given by v - - {[2( - 1) 2 ( + 1) + b(srl + 2, - 2)] respectively, Appendix 4. X 22 } 4(r_1;2 (r+l), A2 '- 4b ( - 1) 2 [m(2- ) + s + b] +b2s(4 + s 2-4 ) +4 ( - 1) 3 [02-1)( - 3) - sb]. D' The f Plane Case To O(e) with c(0) - 0, n 0 ) - -27(v 2 + rs), c(«) = 0, and n(«) -0, polynomial (12) is given by A1 AlC i ) q- BlC(1) q- C1 -- O, '- --$[ 0)(S q- 217) +2n o )(ps + s 2 + 2s + 4p)+ 4ps], B1 '--4n(o)n(1)7 +s [-2s2 + p2 (4 + n o))], (7 '- -2t½(o)n( )t2 (u + s). Acknowledgment. This paper would not have been possible without Pedro Ripa's assistance and corrections. Useful discussions with F. J. Beron-Vera are sincerely appreciated. I also thank J. Sheinbaum and M. G. Brown for commenting on the manuscript. Graduate scholarships were awarded by CONACyT and SNI (M xico). This work has been supported by CICESE's core funding and by CONA- CyT under grant T. References Benilov, E. S., Baroclinic instability of quasi-geostrophic flows localized in a thin layer, J. Fluid. Mech., 288, , Beron-Vera, F. J., and P. Ripa, Free boundary effects on baroclinic instability, J. Fluid Mech., 352, , Cairns, R., The role of negative energy waves in some instabilities of parallel flows, J. Fluid Mech., 92, 1-14, Chassignet, E. P., and B. Cushman-Roisin, On the influence of a lower layer on the propagation of nonlinear oceanic eddies, J. Phys. ceanogr., 21, , Craik, A., and J. Adam, "Explosive" resonant wave interactions in a three layer fluid flow, J. Fluid Mech., 92, 15-33, Davey, M. K., Baroclinic instability in a fluid with three layers, J. Atmos. $ci., 3, , Eady, E. T., Long waves and cyclone waves, Tellus, 1, 33-52, Fukamachi, Y., J. McCreary, and J. Proehl, Instability of density fronts in layer and continuously stratified models, J. Geophys. Res., 100, , Holm, D. D., J. E. Marsden, T. Ratiu, and A. Weinstein, Nonlinear stability of fluid and plasma equilibria, Phys. Rep., 123, 1-116, Ikeda, M., Linear instability of a current flowing along a bottom slope using a three-layer model, J. Phys. Oceanogr., 13, , Killworth, P., and M. Stern, Instabilities on density-driven boundary currents and fronts, Geophys. Astrophys. Fluid Dyn., 22, 1-28, Killworth, P., N. Paldor, and M. Stern, Wave propagation and growth on a surface front in a two-layer geostrophic current, J. Mar. Res., 42, , Nayfeh, A. H., Perturbation Methods, John Wiley, New York, Olascoaga, M. J., and P. Ripa, Baroclinic instability in a two-layer model with a free boundary and fi effect, J. Geophys. Res., 10, 23,357-23,366, Paret, J., and J. Vanneste, Nonlinear saturation of baroclinic instability in a three-layer model, J. Atmos. $ci., 53, , Pedlosky, J., Geophysical Fluid Dynamics, 2nd ed., Springer-Verlag, New York, 1987.

15 OLASCOAGA ET AL.: DEEP OCEAN INFLUENCE ON UPPER OCEAN BAROCLINIC INSTABILITY 26,877 Phillips, N. A., Energy transformations and meridional cir- Smeed, D. A., Baroclinic instability of three-layer flows, part culations associated with simple baroclinic waves in a two 1, Linear stability, J. Fluid Mech., 19J, , level quasi-geostrophic model, Tellus, 6, , Vanneste, J., Explosive resonant interaction of Rossby waves Pichevin, T., Baroclinic instability in a three layer flow: A and stability of multilayer quasi-geostrophic flow, J. Fluid wave approach, Dyn. Atmos. Oceans, 28, , Mech., 291, , Ripa, P., Instability of a solid-body-rotating vortex in a two Young, W., and L. Chen, Baroclinic instability and therlayer model, J. Fluid Mech., 2J2, , 1992a. mohaline gradient alignment in the mixed layer, J. Phys. Ripa, P., Wave energy-momentum and pseudo energy- Oceanogr., 25, , momentum conservation for the layered quasi-geostrophic instability problem, J. Fluid Mech., 235, , 1992b. M. J. Olascoaga, Rosenstiel School of Marine and Atmo- Ripa, P., On the validity of layered models of ocean dynam- spheric Science, Applied Marine Physics, University of Miics and thermodynamics with reduced vertical resolution, ami, 4600 Rickenbacker Causeway, Miami, FL 33149, USA. Dyn. Atmos. Oceans, 29, 1-40, (j olas coaga@rs m as.miami. edu) Ripa, P., Baroclinic instability in a reduced gravity, threedimensional, quasi-geostrophic model, J. Fluid Mech., J03, 1-22, Ripa, P., Waves and resonance in free-boundary baroclinic instability, J. Fluid Mech., J28, , Shepherd, T., Nonlinear saturation of baroclinic instability, part I, The two-layer model, J. Atmos. Sci., J5, , (Received October 5, 2000; revised May 29, 2001; accepted June 7, 2001.)