Modal Interpretation for the Ekman Destabilization of Inviscidly Stable Baroclinic Flow in the Phillips Model

830 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 40

GORDON E. SWATERS Applied Mathematics Institute, and Department of Mathematical and Statistical Sciences, and Institute for Geophysical Research, University of Alberta, Edmonton, Alberta, Canada

(Manuscript received 9 July 2009, in ﬁnal form 29 October 2009)

ABSTRACT

Ekman boundary layers can lead to the destabilization of baroclinic ﬂow in the Phillips model that, in the absence of dissipation, is nonlinearly stable in the sense of Liapunov. It is shown that the Ekman-induced instability of inviscidly stable baroclinic ﬂow in the Phillips model occurs if and only if the kinematic phase velocity associated with the dissipation lies outside the interval bounded by the greatest and least neutrally stable Rossby wave phase velocities. Thus, Ekman-induced destabilization does not correspond to a coalescence of the barotropic and baroclinic Rossby modes as in classical inviscid baroclinic instability. The differing modal mechanisms between the two instability processes is the reason why subcritical baroclinic shears in the classical theory can be destabilized by an Ekman layer, even in the zero dissipation limit of the theory.

1. Introduction (inviscid) model equations. In particular, KM09 have extended Romea’s (1977) work and showed that Ekman The role of dissipation in the transition to instability destabilization within the Phillips model can occur for in baroclinic quasigeostrophic flow can be counterintu- baroclinic shears that are inviscidly (i.e., in the absence of itive (Klein and Pedlosky 1992). It is natural to assume dissipation) nonlinearly stable in the sense of Liapunov. that dissipation acts to reduce the growth rates of baro- It is important to appreciate that the discontinuous clinic flows that are inviscidly unstable and for flows that behavior of the zero dissipation limit of the marginal sta- are inviscidly stable and that dissipation will lead to the bility boundary when an Ekman layer is present cannot decay in the perturbation amplitudes over time (in the be dismissed as akin to the well-known property that unforced initial-value problem). However, it has been solutions to the Orr–Sommerfeld equation need not known since Holopainen (1961) and within the context necessarily reduce to solutions to the Rayleigh stability of the Phillips model (Romea 1977) that subcritical baro- equation in the infinite Reynolds number limit (Drazin clinic shears in the linear inviscid stability theory can be and Reid 1981). As pointed out by KM09, the infinite destabilized by the presence of an Ekman boundary layer Reynolds number limit of the Orr–Sommerfeld equa- and that this destabilization occurs even in the zero tion is singular in the sense that the order of the differ- dissipation limit for the frictional theory; that is, there ential equation changes from fourth order to second is a range of subcritical baroclinic shears (in the linear order so that the mathematical properties of the allowed inviscid theory) that are destabilized by the presence solutions cannot be expected to depend continuously as of an Ekman boundary layer no matter how small the the Reynolds number increases without limit. This is not Ekman number is. Recent work by Krechetnikov and the case for the Phillips model with Ekman layers be- Marsden (2007, 2009, hereafter KM09) has described cause the zero dissipation limit is not singular. this counterintuitive dissipative destabilization within the However, the ‘‘physical reason’’ for the Ekman-induced context of the underlying Hamiltonian structure of the destabilization of inviscidly stable baroclinic quasigeostrophic flow has yet to be given. The principal purpose of this note is to provide the model interpretation for the Corresponding author address: Gordon E. Swaters, Department of Mathematical and Statistical Sciences, University of Alberta, onset of this instability. By exploiting the concept of the Edmonton, AB T6G 2G1, Canada. ‘‘kinematic wave’’ introduced by Lighthill and Whitham E-mail: [email protected] (1955a,b) and further described by Whitham (1974), it is

DOI: 10.1175/2009JPO4311.1

Ó 2010 American Meteorological Society Unauthenticated | Downloaded 09/27/21 07:14 PM UTC APRIL 2010 N O T E S A N D C O R R E S P O N D E N C E 831 shown that the onset of Ekman destabilization of in- as to whether it is generic that the zero dissipation limit viscidly stable baroclinic flow in a zonal channel in the of the marginal stability boundary in a dissipative baro- Phillips model occurs when the dissipative kinematic clinic instability theory does not collapse to the inviscid wave phase velocity lies outside the range of zonal phase result. Finally, in as much as it may be physically desir- velocities spanned by the neutrally stable planetary Rossby able that the zero dissipation limit of the dissipative waves. [In the present context, the kinematic wave is the marginal stability boundary does collapse to the inviscid solution to the long-wave approximation to the governing result, it is remarked that the parameterization in which equations when dissipation is present; see sections 2.2, the dissipation is proportional to the geostrophic po- 3.1, and 10.1 in Whitham (1974) for a general account, tential vorticity (e.g., Klein and Pedlosky 1992; Pedlosky with a more complete description given in section 3.] and Thomson 2003; Flierl and Pedlosky 2007) possesses The onset of dissipative destabilization does not corre- this property. Concluding remarks are given in section 4. spond to a coalescence of the barotropic and baroclinic modes as in inviscid baroclinic instability; indeed, the 2. Governing equations and problem formulation necessary conditions for baroclinic instability need not hold. This is the reason why subcritical shears in the The underlying geometry is a straightforward periodic inviscid theory can be unstable even in the zero dissipa- zonal channel of north–south width L and east–west tion limit when Ekman layers are present. length 2aL (a is nondimensional). The Phillips model The kinematic wave phase velocity stability condition with an upper and lower Ekman layer, in standard nota- described here is conceptually similar to the stability tion (Pedlosky 1987), can be written in the nondimen- condition for the formation of roll waves in gravity- sional forms driven flow down an inclined plane when bottom friction is present (Whitham 1974, section 3.2; Baines 1984, [Du1 Fr(u1 u2)]t1J(u1, Du1 1 Fru2 1by)5 rDu1 1995). Of course in that problem the relevant inviscid and (1) modes are internal gravity waves and not planetary Rossby waves as they are here. Baines (1984) described the [Du2 Fr(u2 u1)]t 1J(u2, Du2 1Fru1 1by)5 rDu2, onset of dissipative destabilization to the development (2) of ‘‘disorder’’ associated with the development, for example, of shock waves and bores (when nonlinearity is taken into account) in shallow water theory, and this where, for convenience, the upper and lower layers have occurs when the phase velocities of the kinematic waves equal scale thickness; the length scale is L so that the [ 2 2 9 9 (when dissipation is present) lie outside the range of the rotational Froude number Fr f L /(g H), where g , H, and f are the reduced gravity, scale layer thickness, and phase velocities allowed by the inviscid modes. Math- 2 ematically, this is equivalent to demanding that the constant Coriolis parameter, respectively; b [ b*L /U, ‘‘characteristics’’ associated with the dissipative kine- where b* is the dimensional beta parameter and U is the matic waves (that arise as solutions to a ‘‘low order’’ velocity scale; and r is the nondimensional (nonnegative) hyperbolic partial differential operator) must lie in the Ekman damping parameter (Pedlosky 1987, section 4.6; span of the characteristics associated with the inviscid again, for convenience, the upper- and lower-layer damp- waves (that arise as solutions to a higher-order hyperbolic ing parameters have been chosen to be the same). partial differential operator, which determines the over- Alphabetical subscripts will denote, unless otherwise all dynamical system properties for wave propagation). specified, partial differentiation, and the 1 and 2 sub- Swaters (2009) presents a similar treatment for the Ekman scripts refer to the upper and lower layers, respectively. destabilization of baroclinic grounded abyssal flow over The geostrophic streamfunctions are given by u1,2, a sloping bottom. J(A, B) [ AxBy 2 AyBx, D[›xx 1 ›yy,andx and y are The plan of this note is as follows: Section 2 gives the oriented eastward and northward, respectively. governing equations and very briefly describes the dis- Because of the underlying Galilean invariance of (1) continuous behavior of the zero dissipation limit of the and (2), it is sufficient to consider the stability of the baro- 5 [2 5 [ marginal stability boundary when Ekman layers are pres- clinic zonal flow u1 u10 Uy and u2 u20 0 ent in the Phillips model for baroclinic instability. In sec- [where U is a constant, which is an exact solution to (1) tion 3, the linear stability problem is recast in a form that and (2) irrespective of whether r 5 0], for which the linear facilitates the introduction of the dissipation-dependent stability problem can be written in the forms kinematic mode and the modal interpretation for the › 1 U› [Df Fr(f f )] 1 (b 1 FrU)› f 5 rDf Ekman destabilization of inviscidly stable baroclinic t x 1 1 2 x 1 1 flow is given. Additional comments are given in relation and (3)

Unauthenticated | Downloaded 09/27/21 07:14 PM UTC 832 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 40

›t[Df2 b 1 Fr(f2 f1)] 1 (b FrU)›xf2 5 rDf2, and the ‘‘natural’’ disturbance boundary conditions f1,2 5 0 (4) along y 5 0 and 1, respectively, have been assumed as well as periodicity along x 56a. Plainly, the right-hand side asso- where f1,2 are the disturbance streamfunctions: that is, ciated with the energy balance equation is not definite in u1,2 5 u10,20 1 f1,2. sign. As shown by KM09, depending on the value of r and It is important to appreciate that the Ekman dissipation the mathematical properties of the solutions, it is possible for terms are generally not negative definite in the globally the perturbation field to extract energy out of the mean flow integrated energy balance, and this is what is responsible and amplify even if the background flow is stable when r 5 0. for the potential for dissipative destabilization of invis- This can be more clearly seen by exploiting the under- cidly stable baroclinic flow. Following KM09, if (1) and lying Hamiltonian structure of (1) and (2) when r 5 0 (2) are multiplied through by u1 and u2, respectively, and (see, e.g., Holm et al. 1985). The well-known normal- the result is added together and integrated over the spatial mode stability condition for baroclinic flow in the in- domain, it follows for the mean flow considered here that ðð viscid Phillips model [see Pedlosky (1987) and brief dE description later] is most properly obtained (within the 5 r u Du 1 u Du dx dy dt 1 1 2 2 context of the Hamiltonian formalism) from a variational ððV principle built on constraining the conserved barotropic 5 r u10Df1 $f1 $f1 $f2 $f2 dx dy, zonal momentum with appropriately chosen conserved V Casimir functionals (see Panetta et al. 1987), which in the problem considered here are simply weighted enstrophy where V denotes the spatial domain; the globally in- integrals (see, e.g., Swaters 1999). tegrated energy E is given by ðð When r 5 0, the baroclinic flow u1 5 u10 [2Uy and 1 2 u2 5 u20 [ 0 satisfy the first-order necessary conditions E [ $u1 $u1 1 $u2 $u2 1 Fr(u1 u2) dx dy; 2 V for an extremal to the conserved functional

ðð b FrU 2 b 1 FrU 2 H 5 [Du1 Fr(u1 u2) 1 by] 1 [Du2 Fr(u2 u1) 1 by] V 2 2 2 2 2 y(b Fr U )(Du1 1Du2) dx dy ðð 2 2 (b FrU)q1 (b FrU)q2 2 2 2 5 1 y(b Fr U )(q1 1 q2 2by) dx dy, V 2 2

where q10 [Du10 Fr(u10 u20) 1 by 5 (b 1 FrU)y, q [Du Fr(u u ) 1 by and 1 1 1 2 q20 [Du20 Fr(u20 u10) 1 by 5 (b FrU)y.

q2 [Du2 Fr(u2 u1) 1 by. Furthermore, it follows that The first two terms in H are constant-weighted enstrophy ðð integrals for the upper and lower layers, respectively. 2 2 d Hjq 5q ,q 5q 5 (b FrU)(dq1) The third term in H is a constant-weighted barotropic 1 10 2 20 V zonal momentum integral. It is straightforward to show 2 using (1) and (2) that each of the three terms in H is an 1 (b 1 FrU)(dq2) dx dy. invariant of the full nonlinear motion (so that dH/dt 5 0) when r 5 0. 2 Because it is necessarily the case that d Hjq 5q ,q 5q It follows that 1 10 2 20 is an invariant of the linear stability Eqs. (3) and ðð (4) when r 5 0, where the identifications dq1 5Df1 2 dH 5 (b FrU)[q (b 1 FrU)y]dq 1 1 F(f1 2 f2) and dq2 5Df2 2 F(f2 2 f1) are made, it V 2 follows that, if U is such that d Hjq 5q ,q 5q is definite in 1 (b 1 FrU)[q (b FrU)y]dq dx dy 1 10 2 20 2 2 sign for all perturbations, then linear stability in the sense 0dHj 5 0, of Liapunov can be established for the inviscid problem q15q10,q25q20 (with respect to the enstrophy norm). In fact, for this because particular problem, should such U exist, then nonlinear

Unauthenticated | Downloaded 09/27/21 07:14 PM UTC APRIL 2010 N O T E S A N D C O R R E S P O N D E N C E 833 stability in the sense of Liapunov (with respect to the Finally, it is remarked that the stability argument pre- disturbance enstrophy norm) is automatically established sented in KM09 exploited a variational principle based for the inviscid problem, because H is quadratic in the on constraining the energy E with appropriately chosen dependent variables u and u so that d2Hj is Casimir functionals. Unfortunately, the sufficient stability 1 2 q15q10,q25q20 an invariant of the full nonlinear Eqs. (1) and (2) when conditions obtained with that approach are, in fact, more r 5 0, where the identifications dq1 5 q1 2 q10 and dq2 5 restrictive (and do not reproduce the known normal- q2 2 q20 are made. mode stability conditions) than those obtained using the There is no value of U for which both b 1 FrU , 0 zonal momentum–based argument described here (the and b 2 FrU , 0. However, both b 1 FrU $ 0 and b 2 KM09 approach yields the stability condition 2b/Fr , FrU $ 0 if and only if jUj # b/Fr (implying that U # 0). The difference is a consequence of the fact that d2Hj . 0 for all dq and dq ), which is the well- the KM09 approach corresponds to Fjørtoft’s stability q15q10,q25q20 1 2 known normal-mode stability condition for the baro- condition, whereas the argument presented here corre- clinic flow u1 5 u10 [2Uy and u2 5 u20 [ 0 in the sponds to Rayleigh’s inflection point stability condition inviscid Phillips problem [see Pedlosky (1987) and brief (see Swaters 1999). description later]. As described earlier, jUj # b/Fr Returning to the case when r . 0, if is sufficient to prove that the flow u1 5 u10 [2Uy ðð 2 and u2 5 u20 [ 0 are nonlinearly stable in the sense of L [ (b FrU)[Df1 Fr(f1 f2)] Liapunov with respect to the disturbance enstrophy norm. V Conversely, of course, inviscid baroclinic instability 2 1 (b 1 FrU)[Df2 Fr(f2 f1)] dx dy only occurs if jUj . b/Fr, in which case the two terms in 2 2 d Hj are of opposite sign. In particular, the is introduced (L is just d Hjq 5q ,q 5q written in terms q15q10,q25q20 1 10 2 20 inviscidly baroclinically unstable normal-mode solutions of f1 and f2; however, when r 6¼ 0, it is not appropriate to the linear stability problem (described later) will satisfy to label it as the second variation of a conserved func- d2Hj 5 0forallt $ 0. tional), it follows from (3) and (4) that q15q10,q25q20

ðð dL 5 2r (b FrU)[Df Fr(f f )] Df 1 (b 1 FrU)[Df Fr(f f )]Df dx dy dt 1 1 2 1 2 2 1 2 ððV 2 2 2 2 2 5 2r f(b FrU)(Df1) 1 (b 1 FrU)(Df2) 1 Fr[bj$(f1 f2)j 1 U(j$f2j j$f1j )]g dx dy. V

Hence, in the baroclinically inviscidly stable case where (f1, f2) 5 (m,1)A sin(npy) exp[ik (x ct)] 1 c.c., jUj # b/F (0L . 0), the dissipation integral on the right-hand side is not deﬁnite in sign and a dissipation- induced instability (i.e., dL/dt . 0) is possible de- where the nondimensional channel walls are located at pending on the properties of the solution to the linear y 5 0 and 1, respectively; c.c. means the complex con- stability problem and the values of r and U (as shown jugate of the preceding term; n 5 1, 2, ... ; the along later). channel wavenumber is k 5 mp/a $ 0 with m 5 0, 1, ...; The linear stability Eqs. (3) and (4) have the normal- A is a free amplitude coefﬁcient; c 5 cR 1 icI is the mode solution [Romea (1977); for the f-plane version of complex-valued zonal phase velocity determined by the this problem, see Pedlosky (1987, section 7.12)] dispersion relation

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ul2(l2 1 2Fr) 2(l2 1 Fr)(b 1 irl2/k) 6 4Fr2(b 1 irl2/k)2 U2l4(4Fr2 l4) c 5 , (5) 2l2(l2 1 2Fr) where l2 [ k2 1 n2p2 . 0; the branch cut is taken along It is interesting to observe that the dispersion relation the negative real axis; and m is given by (5) is identical in form to the inviscid dispersion relation (Pedlosky 1987), with b replaced by b 1 irl2/k. Thus, in 2 2 m 5 [c(l 1 Fr) FrU 1 b 1 irl /k]/(cFr). (6) some sense, Ekman dissipation may be thought of as

Unauthenticated | Downloaded 09/27/21 07:14 PM UTC 834 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 40

equivalent to introducing a complex-valued background presence of the Ekman layer. This was ﬁrst shown by vorticity gradient in the normal-mode equations. Romea (1977) in the context of the linear normal-mode

Stability occurs when cI # 0; that is, stability problem for the Phillips model. Normal-mode qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi solutions associated with baroclinic shears in the interval 2 2 2 2 4 2 4 Im 4Fr (b 1 irl /k) U l (4Fr l Uc0 , jUj # UB have the property that dL/dt . 0, with . 2 2 L 0. # 2rl (l 1 Fr)/k. (7) Examining (8) and (10), it is apparent that, as r increases, eventually U . U so that for large enough After a little algebra [see Eq. (3.5) in Romea (1977) and c B r the viscid stability boundary is ‘‘above’’ the inviscid Eq. (4.6) in KM09; see the appendix for derivation de- stability boundary. It follows from (8) and (10) that tails], this is equivalent to "# U2 Fr2 r2l6(l2 1 2Fr) 4 Fr2b2 r2l2 c 5 1 1 , 2 2 2 2 2 2 2 2 U # Uc [ 1 , (8) U (l 1 Fr) 4k Fr b (2Fr l2) l2(l2 1 Fr)2 k2 B

which implies that Uc . UB whenever if 0 , l2 , 2Fr and the flow is unconditionally stable for 2 l outside this interval. The marginal stability boundary 2bFr2jkj is given by jUj 5 U and in the limit that Ekman dissi- 0 # r , r [ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . c max l3(l2 1 Fr) l2 1 2Fr pation vanishes

An estimate for the order of magnitude of rmax in com- 2Fbffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi limUc 5 Uc0 [ p . (9) parison to b and Fr can be obtained by examining its value r!0 l(l2 1 Fr) 2Fr l2 at the point of inviscid marginal stability associatedppffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with l 5 l [ As observed by Romea (1977), the stability boundary the minimum of UB located at min 2Fr for which U j 5 b/Fr (Pedlosky 1987). It follows that (9) is not the same as the stability boundary obtained B l5lmin by setting r 5 0 directly in (5): that is, the classical in- 21/4b pffiffiffiffiffi viscid Phillips instability problem, which is given by (see ffiffiffi ffiffiffiffiffi rmaxjl5l 5 p 3/2p ’ O(b/ Fr), Pedlosky 1987) min (1 1 2) Fr where O(k) ; l has been assumed. For these scalings, 2Frb min jUj 5 U [ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , (10) the dissipation plays an equal role in the dynamics as all B 2 2 4 l 4Fr l the other terms in (3) and (4). Thus, r does not need to be b for 0 , l2 , 2Fr (and the flow is also unconditionally asymptotically small in comparison to and Fr for the stable for l2 outside this interval). Over the range of flow to be inviscidly baroclinically stable but dissipatively destabilized. potentially unstable wavenumbers, it follows that UB . Uc0 (they are never equal unless Fr 5 0) because 3. Modal interpretation 2Frb 2Frb U 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The goal here is to provide the ‘‘physical mechanism’’ B 2 2 4 2 2 4 l 4Fr l l 2Fr l 2Frl 1 l behind the dissipative destabilization of inviscidly stable q2Frffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib flows in terms of the wave modes and its relationship to 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l 2Fr l2 (l2 1 Fr)2 Fr2 the onset of classical baroclinic instability. The linear stability problem (3) and (4) can be combined into the 2Frbffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi single equation, . p 5 Uc (11) l(l2 1 Fr) 2Fr l2 0 2r(l2 1 Fr) (› 1 c › )(› 1 c › )f 5 (› 1 c › )f, [see also Fig. 1 in Romea (1977) and Fig. 5 in KM09]. t x t 1 x (l2 1 2Fr) t 0 x Baroclinic shears in the interval Uc0 , jUj # UB do (12) not satisfy the necessary conditions for inviscid baroclinic instability but are dissipatively destabilized by the where

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ul2(l2 1 2Fr) 2(l2 1 Fr)b 6 4l6(l2 1 2Fr)r2/k2 1 4Fr2b2 U2l4(4Fr2 l4) c [ and (13) 6 2l2(l2 1 2Fr)

Unauthenticated | Downloaded 09/27/21 07:14 PM UTC APRIL 2010 N O T E S A N D C O R R E S P O N D E N C E 835

U b which is introduced in section 2), propagates zonally c0 [ , (14) 2 l2 1 Fr with phase velocity c0, and diffuses with a dissipation coefficient that is dependent on the wavenumber. Per- where f is either f1 or f2 and a normal-mode form for turbations with ‘‘short’’ zonal wavelength dissipate less the solution has been implicitly assumed so that it is pos- rapidly than waves with a ‘‘long’’ zonal wavelength 2 sible to write D52l . Writing the linear stability equation (recall that l $ p for all k $ 0). in the form of (12) helps to facilitate the comparison with Lighthill and Whitham (1955a,b) introduced the the theory presented in Lighthill and Whitham (1955a,b) concept of the kinematic wave as a traveling wave so- and Whitham (1974). To explicitly focus on the modal lution to a first-order (1 1 1 dimensional) conservation interpretation for the dissipation-induced instability of law in which a functional relationship exists between the inviscidly stable flow, it will henceforth be assumed that ‘‘density’’ and ‘‘flux’’ in the conservation law. Kinematic 2 2 jUj # UB and 0 , l , 2Fr (l outside this interval is not waves are not necessarily ‘‘classical’’ or ‘‘dynamic’’ of interest), which implies that the quantity within the waves that are described by Newton’s second law of square root in (13) is strictly positive and thus c6 are real motion in which the acceleration associated with parti- valued and ordered as c2 # c1. cle displacement occurs against a background restoring The c1 and c2 correspond to Doppler-shifted neutrally force (e.g., gravity waves). The kinematic wave concept stable baroclinic and barotropic planetary Rossby wave has been useful in understanding the dynamics and sta- modes, respectively, which include the strictly neutral bility of, for example, roll wave formation, continuous frequency shift associated with the dissipation [the term models for traffic flow, and spatially variable chemical 2 proportional to r within the square root in (13)]. The reactions (see Whitham 1974, section 2.2). / r 0 limit of c6 is exactly the inviscid Rossby wave In the context of the present problem, the kinematic solution associated with (5). The c0 mode corresponds wave part of the partial differential operator in (12) is to a kinematic wave, as described by Lighthill and the dissipation-dependent first-order wave operator on Whitham (1955a,b) and Whitham (1974). the right-hand side (if r 5 0, the right-hand side is zero). The kinematic wave phase velocity c0 may be inter- Within this paradigm, the second-order partial differ- preted as the average of the individual Rossby wave ential operator on the left-hand side of (12) is the ‘‘dy- velocities associated with the linear stability equations namic wave’’ part that describes the zonally propagating for the upper and lower layers, respectively, in the ‘‘un- vorticity waves that arise as a consequence of meridional coupled’’ inviscid limit. The uncoupled inviscid approxi- displacement of patches of anomalous vorticity against mations to (3) and (4) are, respectively, the background planetary vorticity gradient. In the absence of dissipation, the right-hand side of (12) is zero (› 1 U› )(D Fr)f 1 (b 1 FrU)› f 5 0 and t x 1 x 1 and what remains are only the dynamic waves. ›t(D Fr)f2 1 (b FrU)›xf2 5 0. The low-frequency/wavenumber approximation in (12) would be to neglect the left-hand side and retain only the The upper-layer equation can be explicitly identified as kinematic wave part of the partial differential operator. an equivalent-barotropic approximation, but the pres- Thus, roughly speaking, in the context of (12) and from ence of the FrU coefficient in the lower-layer equation the perspective of the solution to the pure initial-value makes a similar identification for it not exact. Assuming problem, the low-frequency–wavenumber part of the a normal-mode solution, the phase velocity associated solution would be governed by the right-hand side and with the upper layer, denoted as c1, is given by c1 5 U 2 2 everything else would be governed by the left-hand side. (b 1 FrU)/(l 1 Fr); the phase velocity associated with However, as written, (12) corresponds to a hyperbolic the lower layer, denoted as c2, is given by c2 5 (FrU 2 b)/ 2 partial differential equation in which the characteristics (l 1 F ). It follows immediately that c0 5 (c1 1 c2)/2. This are exclusively determined by the second-order dynamic is not just algebraically coincidental. In fact, c0 arises wave terms on the left-hand side. From this perspective, precisely in this manner in the derivation of (12). for the problem to be well posed, the characteristics / In the limit that r ‘ (when dissipation dominates associated with the first-order kinematic wave operator the dynamics), the leading order balance associated with (governing the low-frequency–wavenumber part of the (12) is solution) must be consistent with the characteristics as- l2 sociated with the second-order dynamic wave part of (› 1 c › )f 5 f , t 0 x 2k2(l2 1 Fr) xx the operator [see the discussion of wave hierarchies in Whitham (1974, chapter 10)], which alone determines which is parabolic. Thus, in the limit of ‘‘large’’ dissi- the signal propagation properties for the overall dynam- pation, the disturbance field is stable (because r . rmax, ical system; that is, the phase velocity associated with

Unauthenticated | Downloaded 09/27/21 07:14 PM UTC 836 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 40

dissipative kinematic wave operator must lie in the in- 2 2 2 Uc b[(l 1 Fr) 7 Fr ] terval spanned by the phase velocities associated with cR U5U 5 2 2 2 and (18) c 2 l (l 1 2Fr)(l 1 Fr) the dynamic Rossby wave operator. Thus, the stability condition associated with (12) is that 2 r(1 7 1)(l 1 Fr) cI U5U 5 2 . (19) c k(l 1 2Fr) c # c0 # c1 , (15) At marginal stability, it is the baroclinic mode that is which, if (13) and (14) are substituted in, can be written neutrally stable and the barotropic mode exponentially in the form decays because of the dissipation. Alternatively, it is possible to interpret the stability qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Fr2b condition (15) as the requirement that the ‘‘effective # 4Fr2b2 U2l4(4Fr2 l4)14l6(l2 12Fr)r2, (l2 1Fr) diffusion coefficient’’ associated with (12) be nonnegative, assuming the dynamics is dominated by the kinematic (16) wave (see Whitham 1974, section 3.1). The linear stability Eq. (12) can be written in the form because the left-hand inequality in (15) is satisfied for all parameter values. If both sides of (16) are squared, it 2r(l2 1 Fr) follows that (› 1 c › )f 5 n2f (› 1 c~›x)2f, (l2 1 2Fr) t 0 x xx t 4Fr2b2 4Fr4b2/(l2 1 Fr)2 1 4l6(l2 1 2Fr)r2/k2 U2 # where l4(4Fr2 l4) () Ul2(l2 1 2Fr) 2(l2 1 Fr)b 2b2 l2 1 2 2 r2l2 ~ 4 Fr [( Fr) Fr ] 2 c [ 2 2 , 5 1 5Uc, 2l (l 1 2F) (2Fr l2) l4(l2 1 2Fr)(l2 1 Fr)2 k2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 6 2 2 2 2 4 2 4 (17) 4l (l 1 2Fr)r2/k 1 4Fr b U l (4Fr l ) n [ 2l2(l2 1 2Fr) recovering (8).

As shown, the velocity c0 may be interpreted as the (i.e., c6 5 c~6 n), where it is noted that n is real for average of the individual Rossby wave velocities asso- 2 2 jUj # UB in the interval 0 , l , 2Fr (l outside this ciated with the linear stability equations for the upper interval is not of interest). Assuming the dynamics is and lower layers in the uncoupled and inviscid limit, principally determined by the left-hand side, it follows respectively. Thus, we may interpret the stability con- that ›t ’ 2c0›x, which if substituted into the right-hand dition (15) as the requirement that the average of the side yields individual layers’ Rossby wave velocities for the upper and lower layers in the uncoupled and inviscid limit, 2 2r(l 1 Fr) 2 2 respectively, must lie in the interval spanned by the (› 1 c › )f 5 [n (c~ c ) ]f . (l2 1 2Fr) t 0 x 0 xx phase velocities associated with the fully coupled dynamic Rossby waves. For stability, it follows that Hence, the Ekman-induced destabilization of inviscidly stable baroclinic quasigeostrophic ﬂow in the Phil- 2 25 (c~ c0) # n c0 n # c~ # c0 1 n, lips model occurs when the phase velocity associated with the dissipation-created kinematic wave lies outside the which is exactly (15). Of course, instability occurs if interval spanned by the neutral barotropic and baroclinic the effective diffusion coefﬁcient is negative: that is, 2 2 planetary Rossby waves (including the frequency shift (c~ c0) . n . associated with the dissipation). From the point of view Although the current results have been obtained for of normal-mode theory, this is a very different scenario Ekman dissipation, it is natural to speculate whether it is as compared to inviscid baroclinic instability in which generic that the marginal stability boundary associated the onset of destabilization occurs when a coalescence with the zero dissipation limit of a dissipative baroclinic develops between the barotropic and baroclinic Rossby instability theory collapses to the inviscid result. The waves (i.e., the phase velocities become equal). Dissipa- answer would seem to be that the generic situation is tive destabilization does not require any such coalescence that the stability boundary associated with the zero in the barotropic and baroclinic phase velocities; indeed, dissipation limit of a dissipative baroclinic instability it follows from (5) that theory does not collapse to the inviscid result.

Unauthenticated | Downloaded 09/27/21 07:14 PM UTC APRIL 2010 N O T E S A N D C O R R E S P O N D E N C E 837

For example, if Ekman friction in the Phillips model no-slip solutions on Re cannot be expected to be contin- were to be replaced by horizontal turbulent friction, then, uous as Re / ‘; again (for mathematical reasons similar within the context of the quasigeostrophic assumptions to the Orr–Sommerfeld to Rayleigh transition), the sta- implicit in the derivation (Pedlosky 1987), the terms bility boundary associated with the zero dissipation limit

2rDu1,2 on the right-hand side of (1) and (2), re- will not necessarily collapse to the inviscid result. 21 2 spectively, would be replaced by Re D u1,2, where Re In as much as it is desirable that the stability boundary is the Reynolds number. The resulting stability theory associated with the zero dissipation limit of a dissipative for a periodic zonal channel with free slip along the me- baroclinic instability theory does collapse to the inviscid ridional channel walls is exactly as described earlier, with result, the parameterization in which the dissipation is r simply replaced by l2/Re. assumed proportional to the geostrophic potential vortic- However, objections can be raised to this particular ity (e.g., Klein and Pedlosky 1992; Pedlosky and Thomson example. For example, whether no-slip (rather than free- 2003; Flierl and Pedlosky 2007) has this property. This slip) boundary conditions should be applied on the me- parameterization would have (3) and (4) replaced by ridional channel walls y 5 0andL when Re ,‘might be (› 1 U› )[Df Fr(f f )] 1 (b 1 FrU)› f considered an issue. If no-slip boundary conditions are t x 1 1 2 x 1 applied, the resulting dispersion relation for the normal 5 r[Df1 Fr(f1 f2)] and (20) modes will no longer be (5) with r replaced by l2/Re. In the no-slip case, the inﬁnite Re limit of (3) and (4), with ›t[Df2 Fr(f2 f1)] 1 (b FrU)›xf2 horizontal friction present, is singular, because the (spa- 5 r[Df2 Fr(f2 f1)], (21) tial) order of the partial differential equations is reduced in the limit (e.g., from fourth order to second order with for which the dispersion relation associated with the respect to y). Consequently, the dependence of the normal modes is given by

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l2(U 2ir/k)(l2 1 2Fr) 2(l2 1 Fr)b 6 4Fr2b2 U2l4(4Fr2 l4) c 5 . (22) 2l2(l2 1 2Fr)

It follows from (22) that the stability occurs when and unstable otherwise. Unlike the Ekman-layer result qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (8), in the limit that r / 0, UP reduces to the inviscid 2 2 2 4 2 4 2 2 Im 4Fr b U l (4Fr l ) # 2rl (l 1 2Fr) /k. marginal stability boundary UB given by (10). (23) From the perspective of the kinematic stability condition, the analog of (12) for (20) and (21) is Clearly, if l 5 0orl2 $ 2Fr, then (23) is satisfied for all shears. If 0 , l2 , 2Fr, the flow is stable, provided that qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (›t 1 c~›x)(›t 1 c~1 ›x)f 5 2r(›t 1 c~0›x)f, (25) 2 Fr2b2 1 r2l4(l2 1 2Fr)2/k2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jUj # UP [ p , (24) l2 4Fr2 l4 where

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4r2l4(l2 1 2Fr)2/k2 1 4Fr2b2 U2l4(4Fr2 l4) c~ [ c~ 6 and (26) 6 0 2l2(l2 1 2Fr)

Ul2(l2 1 2Fr) 2(l2 1 Fr)b mode will always lie in the range of the phase velocities c~ [ . (27) 0 2l2(l2 1 2Fr) spanned by the Rossby waves. From the perspective of the time evolution of the functional L (introduced in section 2), it will follow from It follows that c~ # c~ # c~ when jUj # U if 0 , l2 , 0 1 P (20) and (21) that 2Fr (and unconditionally for l2 outside this interval). There is no dissipative destabilization of inviscidly stable modes with this parameterization because UP $ UB dL 0 5 2rL L 5 Ljt50 exp(2rt), so that, if U # UB, the phase velocity of the kinematic dt

Unauthenticated | Downloaded 09/27/21 07:14 PM UTC 838 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 40 which again underscores the fact that there can be no Acknowledgments. Preparation of this paper was sup- dissipative destabilization of inviscidly stable modes with ported in part by the Natural Sciences and Engineering this parameterization; that is, when the flow is inviscidly Research Council of Canada. stable so that L . 0, there is no (positive) value of r for which dL/dt . 0 and the amplitude of the disturbance APPENDIX field, in the inviscidly stable case, always decays to zero. Derivation of the Dissipative Stability Condition 4. Conclusions For completeness, the dissipative stability condition It is known that the presence of Ekman layers in the (2.8) is derived here. Define the real numbers a and g to Phillips model for baroclinic instability has the property be given by that there are inviscidly (i.e., in the absence of dissipation) 2 2 2 2 4 2 4 stable flows (i.e., subcritical shears) that are dissipatively a exp(ig) 5 4Fr (b 1 irl /k) U l (4Fr l ) destabilized no matter how small the Ekman number is 5 4Fr2(b2 r2l4/k2) U2l4(4Fr2 l4) (Romea 1977). Recent work (KM09) has shown this de- 1 8irbFr2l2/k, (A1) stabilization occurs even if the flow is inviscidly non- linearlystableinthesenseofLiapunovandthatthe where a $ 0 and 0 # g # p (because the imaginary part dissipation induces a nonlinear instability. Dissipative de- is nonnegative). The stability condition (2.7) can be writ- stabilization corresponds to the onset of disorder (Baines ten in the form 1984) and is a consequence of the fact that the propagation pffiffiffi properties of the dissipation-dependent low-frequency– a sin(g/2) # 2rl2(l2 1 Fr) /k. (A2) wavenumber waves are inconsistent with the propagation properties of the overall dynamical system, which are Because both sides of (A2) are necessarily positive, it determined principally by inviscid dynamical processes. follows that In this note, a ‘‘modal interpretation’’ for this dissipation- 2 2 4 2 2 2 induced instability was given. By exploiting the kinematic 2a sin (g/2) 5 a[1 cos(g)] # 8r l (l 1 Fr) /k ; wave concept, introduced by Lighthill and Whitham (1955a,b), it was shown that the dissipative de- substituting in for a and g leads to stabilization of inviscidly stable baroclinic flow corre- qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sponds to the situation where the phase velocity of the G2 1 64b2r2Fr4l4/k2 # G18r2l4(l2 1 Fr)2/k2, (A3) kinematic wave lies outside the range spanned by the least and greatest phase velocities of the neutral plane- where tary Rossby waves. When the phase velocity of the kinematic waves lies inside the range inclusively spanned G[4Fr2(b2 r2l4/k2) U2l4(4Fr2 l4). (A4) by the least and greatest phase velocities of the neutral planetary Rossby waves, the flow is stable. At the point For stability, the right-hand side of (A3) must be of marginal stability, there is no coalescence between positive because the left-hand side is as well [or else the barotropic and baroclinic mode. Thus, dissipative (A3) can never hold and instability necessarily follows]. destabilization of baroclinic quasigeostrophic flow is not The right-hand side of (A3) is positive, provided that a ‘‘variant’’ of classical inviscid baroclinic instability. 2 2 2 4 4 2 2 2 2 4 2 4 It was speculated that the zero dissipation limit of the 4F b 1 4r l (2l 1 4Frl 1 Fr )/k $ U l (4Fr l ). marginal stability boundary in a dissipative baroclinic (A5) instability theory generically does not collapse to the inviscid result. In as much as it may be desirable that the Inequality (A5) is clearly satisfied for all shears (i.e., U) zero dissipation limit of the marginal stability boundary if l 5 0orl2 $ 2Fr. If 0 , l2 , 2Fr, (A5) will be sat- does collapse to the inviscid result, it was shown that the isfied, provided that dissipation parameterization that is proportional to the 2 geostrophic potential vorticity (e.g., Klein and Pedlosky 4Fr2b2 1 4r2l4(2l4 1 4Frl2 1 Fr2)/k U2 # . (A6) 1992; Pedlosky and Thomson 2003; Flierl and Pedlosky l4(4Fr2 l4) 2007) has this property. In this parameterization, the phase velocity of the kinematic mode always lies in the This forms a necessary condition for stability. In a mo- range of the phase velocities spanned by Rossby waves ment, a necessary and sufficient stability condition will for subcritical baroclinic shears. be obtained that will subsume this necessary condition.

Unauthenticated | Downloaded 09/27/21 07:14 PM UTC APRIL 2010 N O T E S A N D C O R R E S P O N D E N C E 839

To proceed further, it is assumed that right-hand side l2 $ 2Fr. If 0 , l2 , 2Fr, (A7) will be satisfied, provided of (A3) is positive. Thus, both sides of (A3) can be squared, that "# and it follows that (if r 6¼ 0) 4 Fr2b2 r2l2 U2 # 1 , (A8) "# (2F l2) l2(l2 1 Fr)2 k2 Fr2b2 r2l4 U2l4(4Fr2 l4) # 4l4(2Fr 1 l2) 1 . (l2 1 Fr)2 k2 which is (2.8). All that remains to be shown is that, if (A8) holds, so (A7) does (A6). Thus, (A8) [i.e., (2.8)] is the necessary and sufficient stability condition if 0 , l2 , 2Fr and the flow Again, (A7) is clearly satisfied if l 5 0orl2 $ 2Fr. is unconditionally stable for l2 outside this interval. It Hence, the flow is unconditionally stable for l 5 0or follows from (A8) that

"#"# 4 Fr2b2 r2l2 4(l2 1 2Fr) Fr2b2 r2l4 1 5 1 (2Fr l2) l2(l2 1 Fr)2 k2 l2(4Fr2 l4) (l2 1 Fr)2 k2 "# 4[(l2 1 Fr)2 Fr2] Fr2b2 r2l4 5 1 l4(4Fr2 l4) (l2 1 Fr)2 k2 4 r2l4 # Fr2b2 1 [2(l2 1 Fr)2 Fr2] l4(4Fr2 l4) k2 4Fr2b2 1 4r2l4(2l4 1 4Frl2 1 Fr2)/k2 5 , l4(4Fr2 l4) which is the right-hand side of (A6). Thus, if (A8) holds, ——, and ——, 2009: Dissipation-induced instability phenomena in so does (A6). infinite dimensional systems. Arch. Ration. Mech. Anal., 194, 611–668. REFERENCES Lighthill, M. J., and G. B. Whitham, 1955a: On kinematic waves: I. Flood movement in long rivers. Proc. Roy. Soc. London, Baines, P. G., 1984: A unified description of two-layer flow over 229A, 281–316. topography. J. Fluid Mech., 146, 127–167. ——, and ——, 1955b: On kinematic waves: II. Theory of traffic ——, 1995: Topographic Effects in Stratified Flows.CambridgeUni- flow on long crowded roads. Proc. Roy. Soc. London, 229A, versity Press, 500 pp. 317–345. Drazin, P. G., and W. H. Reid, 1981: Hydrodynamic Stability. Panetta, R. L., I. M. Held, and T. Pierrehumbert, 1987: External Cambridge University Press, 525 pp. Rossby waves in the two-layer model. J. Atmos. Sci., 44, 2924– Flierl, G. R., and J. Pedlosky, 2007: Nonlinear dynamics of time- 2933. dependent subcritical baroclinic currents. J. Phys. Oceanogr., Pedlosky, J., 1987: Geophysical Fluid Dynamics. 2nd ed. Springer- 37, 1001–1021. Verlag, 710 pp. Holm, D. D., J. E. Marsden, T. Ratiu, and A. Weinstein, 1985: ——, and J. Thomson, 2003: Baroclinic instability of time de- Nonlinear stability of fluid and plasma equilibria. Phys. Rep., pendent currents. J. Fluid Mech., 490, 189–215. 123, 1–116. Romea, R. D., 1977: The effects of friction and b on finite-amplitude Holopainen, E. O., 1961: On the effect of friction in baroclinic baroclinic waves. J. Atmos. Sci., 34, 1689–1695. waves. Tellus, 13, 38–52. Swaters, G. E., 1999: Introduction to Hamiltonian Fluid Dynamics Klein, P., and J. Pedlosky, 1992: The role of dissipation mechanisms and Stability Theory. Chapman & Hall/CRC, 274 pp. in the nonlinear dynamics of unstable baroclinic waves. J. Atmos. ——, 2009: Ekman destabilization of inertially stable baroclinic Sci., 49, 29–48. abyssal flow on a sloping bottom. Phys. Fluids, 21, 086601, Krechetnikov, R., and J. E. Marsden, 2007: Dissipation-induced doi:10.1063/1.3211274. instabilities in finite dimensions. Rev. Mod. Phys., 79, Whitham, G. B., 1974: Linear and Nonlinear Waves. John Wiley 519–553. and Sons, 636 pp.

Unauthenticated | Downloaded 09/27/21 07:14 PM UTC