The Destabilizatin f Rssby Nrmal Mdes by Meridinal Barclinic Shear by Jseph Pedlsky Wds Hle Oceangraphic Institutin Wds Hle, MA 0543 Abstract The Rssby nrmal mdes f a tw-layer fluid in a meridinal channel f width L * are altered by the presence f a meridinal flw with a small vertical shear. The stability f the mdes in the presence f the weak shear is cnsidered. It is fund that the jint presence f the Rssby mdes and the vertical shear leads t barclinic instability even fr arbitrarily small values f the shear. The results are used t explain previus numerical calculatins f the persistent instability f meridinal flws when the rati βl D / V >1, where V is the magnitude f the shear, β is the planetary vrticity gradient and L D is the defrmatin radius. If the flw were znal it wuld be stable fr such weak shears. The grwth rates are weak when βl D / V >>1 and each unstable mde exists in a narrw range f meridinal wavenumber. The asympttic results qualitatively agree with the earlier numerical results at mderate values f the same parameter. 1. Intrductin In a recent paper Walker and Pedlsky (00), hereafter WP, examined the instability, within the tw-layer mdel, f a meridinal flw n the beta plane in a meridinal channel f width L *. If the channel were unbunded and the meridinal flw were f infinite lateral extent earlier results (Pedlsky, 1987) shw that the flw wuld always be unstable t a wave-like perturbatin independent f the znal directin (x). Such a perturbatin wuld thus be insensitive t the stabilizing presence f β. The finite width f the channel frces an x variatin in the perturbatin stream functin field and thus a meridinal velcity which senses the planetary vrticity. Nevertheless, WP fund that the meridinal shear flw was unstable fr all values f β examined, even fr thse values f βl D / V >1 (symbls have cnventinal meanings and are defined belw) fr which the flw wuld be stable if it were znal instead f meridinal. In additin, the instability extended t wave lengths shrter than the classical shrt- e-mail: jpedlsky@whi.edu
wave cut-ff f the tw-layer mdel. WP speculated that the extended range f instability in shear and wave number was due t the destabilizatin by the presence f weak shear f the Rssby nrmal mdes present in the channel. (Nte that fr a plane wave in an unbunded regin a stability threshld des exist if the wave vectr is nt purely meridinal). Hwever, the difficulty f the numerical analysis when the shear is weak and the grwth rates are small precluded a satisfying verificatin f this hypthesis. In this paper I present an asympttic perturbatin analysis valid fr very weak shear t demnstrate the persistence f at least weak instabilities fr small values f the shear. In additin t its explanatry quality with regard t the earlier results in WP it is suggestive that the ptential energy present in the weak shear can be tapped by the Rssby mde. Althugh the basic flw cnsidered here is cnsiderably simpler than the circulatin in a cmplete subtrpical gyre it is pssible that the instability utlined here can prvide a mechanism t maintain Rssby mdes in such gyres against the inevitable effects f dissipatin whse presence has ften been cited as a reasn fr the unlikelihd fr the existence f Rssby nrmal mdes. In sectin I frmulate the basic prblem and exhibit the perturbatin analysis. Sectin 3 is a presentatin f results and sectin 4 is a brief discussin f the results and their significance.. Frmulatin Cnsider a tw layer, quasi-gestrphic mdel n the beta plane (Pedlsky, 1987). Fr simplicity we will take each f the layers t have the same basic thickness, H, in the absence f mtin. Imagine a channel f width L *, riented nrth suth, as shwn in Figure 1. There is a unifrm nrthward flw, V, in nly the upper layer f the tw layers. In rder fr such a flw t be a cnsistent slutin f the ptential vrticity equatin there must be a vrticity surce n the beta plane t maintain the flw. One can imagine a unifrm wind stress curl being respnsible fr maintaining V. Small perturbatins that are wave-like in y disturb the basic flw and the quasi-gestrphic ptential vrticity equatin is linearized in rder t describe the initial evlutin f the perturbatin field. In dimensinless units the perturbatin equatins are: (v s) [ φ 1xx l φ 1 + (φ φ 1 )]+ 1 il φ 1x + vφ 1 = 0 ( s) [ φ xx l φ + (φ 1 φ )]+ 1 il φ x vφ = 0 (.1 a,b)
Figure 1. The meridinal channel cntaining the flw. The upper layer basic state velcity is V and the channel width is L. In (.1) the crss channel crdinate x has been scaled with the defrmatin radius, (g ' H) 1/ / f where the reduced gravity and Crilis parameter are in standard ntatin. Subscripts x dentes derivatives with respect t x. The alng channel wave number l is similarly scaled with the inverse defrmatin radius. Bth the meridinal basic state velcity and the cmplex phase speed f the wave perturbatin, s, are scaled with the characteristic Rssby lng wave speed βl D. Fr details f the derivatin the reader is referred t WP. If the basic state shear is weak s that v = V / βl D << 1 an expansin in an asympttic series in v is suggested. First, hwever, there is a suggestin frm the numerical results f WP that the unstable mdes, if they exist, will have scales that can be shrt cmpared t a defrmatin radius and wavelengths in the y-directin that are als shrt. This suggests that the fllwing transfrmatins fr x,l and c are useful: 3
ξ = x / v 1/, a = lv 1/ (. a,b,c) s = v(s + vs 1 +...) where we have expanded the phase speed in a series in the small parameter v, the perturbatin equatins are nw: [ ] i a φ 1ξ + vφ 1 = 0 (1 s vs 1 +...) φ 1ξξ a φ 1 + v(φ φ 1 ) (.3a,b) ( s vs 1 +...) [ φ ξξ a φ + v(φ 1 φ )] i a φ ξ vφ = 0 Nte that in these stretched crdinates the bundaries f the channel are at ξ = 0 and ξ = L * (L D v 1/ ) L. We als nte that t lwest rder in the shear, v, the tw layers are decupled. This is cnsistent with the results f WP which shwed that unstable mdes exist fr βl D / V >1, with strikingly different crss channel scales in the x-directin. The perturbatin streamfunctin is als expanded in a series in v, as is a, thus: φ n = φ n () + vφ n (1) +..., n =1, a = a (1 + vα +...) (.4) At lwest rder this yields the prblem d (1 s ) dξ φ 1 () a () φ 1 i φ a 1ξ = 0, d ( s ) dξ φ () a φ () i φ a ξ = 0 (.5a,b) 4
whse slutins, subject t the bundary cnditins f vanishing stream functin at the channel walls are: φ 1 () = A 1 e iξ /[a (1 s )] sinmξ, (.6a,b) φ () = A e iξ /[a s ] sin nξ where m Mπ / L, M =1,,3... n = Nπ / L, N = 1,,3... (.7 a,b) Since L is the channel width scaled inversely with v 1/ it fllws that fr cnsistency M 1/ and N, each an integer, must be large, O ( v ) s that m and n are O(1) in accrdance with the assumptins f the asympttics. We nte that the slutins (.6) are each the Rssby nrmal mdes f the upper and lwer layers respectively in which a carrier wave is mdulated by a slutin f the Helmhltz equatin (Pedlsky, 1987). The nrmal mde structure in the upper layer is altered nly by the Dppler shift f the meridinal flw in the upper layer. The slutins (.7) reflect the fact that the hrizntal structure f the mde may be quite different frm ne layer t the ther if M and N are different. The cnditin that bth (.6 a,b) are slutins crrespnding t the same frequency r phase speed is simply that the tw slutins fr s, frm (.6b) and s = s =1 1 a (a + n ) 1/ (.8) 1 a (a + m ) 1/ (.9) frm (.6a), must yield identical results. The cnditin that s be the same in (.8) and (.9), i.e. that we are dealing with a single cmpsite mde, yields the cnditin, 5
a = 1 ( a + m ) 1/ + 1 ( a + n 1/. (.10) ) Fr any pair (m,n) there is a single slutin fr the (scaled) y- wave number a. Figure shws a map in the M, N, plane f a fr the case v =.0, L * =10. Althugh M and N are integers the figure treats the variables as cntinuus fr graphical clarity. It is imprtant t nte that t this rder the phase speed is strictly real s that instability will be apparent nly at the next rder in the expansin in v. At next rder the perturbatin equatins are: (1 s ) (1) φ 1ξξ a (1) [ φ 1 ] i (1) φ a 1ξ = (1 s ) () φ () φ1 [ ] () +s 1 φ 1ξξ a () [ φ 1 ] φ 1 () (.11a) +(1 s ) α a () [ φ 1 ] i αφ a 1 ξ () (1) s φ ξξ a (1) φ i (1) φ a ξ = s () (φ () 1 φ ) () +φ +s 1 φ () a () ξξ φ s a () αφ (.11b) i a αφ ξ (). It is nly at this rder that the cupling between the tw layers enters the perturbatin equatins and the ptential vrticity gradient assciated with the vertical shear is explicitly included in the analysis. If barclinic instability is t ccur these must be essential ingredients. The instability prperties will be cntained in behavir f s 1 whse imaginary part will yield the grwth rate fr the perturbatin (after multiplicatin by the y wave-number). 6
Figure. Cnturs f the critical wave number a in the M, N plane fr v =.0, L =10. T find s 1 it is nly necessary t remve resnant terms frm the right-hand sides f (.11a,b). This is easily dne by multiplying (.11a) by the functin f 1 = e iξ /(a [1 s]) sin mξ which has the frm f the cmplex cnjugate f the nrmal mde f the upper layer, and multiplying (.11b) by f = e i ξ /(a s) sinnξ and integrating ver the width f the channel. After sme algebra this leads t tw equatins relating the amplitudes f the nrmal mdes f the tw layers, viz.: 7
and A = A 1 ( ) s s 1 4a (1 s ) + 1 (1 s )α 4a + m γ 1 (1 s ) L (.1a) A 1 = A s 1 4a s (1 s ) + s α (4a + n ) L γ 1 * s (.1b) where the cupling cnstant γ 1 is given by: γ 1 = i mn(k m + k n ) e i(k m+ kn)l ( 1) M +N [ 1 ] [(m n) (k m + k n ) ][ (m + n) (k m + k n ) ] (.13) where: k m = m + a ( ) 1/ k n = n + a ( ) 1/. Nte that the asterisk in (.1b) dentes the cmplex cnjugate f the cupling cefficient γ 1. Eliminating A 1 and A between (.1 a,b) leads t a quadratic equatin fr s 1. 1 1 s + s B+ C = 0, 1 1 + (1 ) s (1 s ) B = a s s s (1 s) α( a ) ( ) + n α a + m (1 s ) s 3 3 1 16 (1 ) / C = γ s s L 4 4 a s (1 s ) (1 s) sα( a + n ) s (1 s) α( a + m ) (.14 a,b,c) whse slutin determines the stability f the mde whse structure t first-rder is given by (.6 a,b). 8
3. Results In terms f ur riginal nn-dimensinal variables the grwth rate f an unstable mde is given by lc i where c i is the imaginary part f the wave s phase speed. We have already nted that t lwest rder in ur expansin in pwers f v the phase speed is real and an imaginary part, if it exists, will depend n the imaginary part f s 1. It then fllws that the grwth rate σ will be σ = lc i = av 3/ imag (s 1 ) (3.1) where the scaled wave number a is itself given by the expansin (.4) s that the grwth rate is a functin f α. In the fllwing figures the grwth rate is given in terms f its riginal variables, i.e. l and in terms f v r its inverse β =1/v. Figure 3. Grwth rate versus wave number fr v =.0, M = 50, N = 0, L = 5. Nte the narrw windw in l fr which σ I > 0. 9
Figure 3 shws the grwth rate as a functin f l fr v=.0( β = 50 ) fr the mde crrespnding t M=50 and N =0 fr a channel f width 5 L D. There is a narrw range f wavenumbers in which the grwth rate is psitive and the magnitude f the grwth rate is quite small, fr these values f the rder f 10-4 (in units βl D ). There is a symmetry in M,N s that the mde with M and N reversed has the same grwth rate. Any cmbinatin (M,N) will have sme range ver which its grwth rate will be nn zer s that a plt f grwth rate versus l fr small v wuld shw a bewildering superpsitin f intervals f unstable wave numbers. This is precisely what WP fund numerically but the smallness f the grwth rates precluded a clear picture fr v less than 0.5. Fr example, the grwth rate curve fr M = 5, N = 5 is shwn in Figure 4. The maximum grwth rate is slightly larger than in the previus example. Indeed, as shwn belw, the maximum crrespnds t the diagnal slutin M=N but the differences are nt great enugh t truly privilege that mde. Nte t that the interval f wave number is different than the previus example. As WP fund, these mdes are unstable fr values f l which exceed the classical shrt wave cut-ff f 1/4. Figure 4. As in Figure 3 except fr M =5, N = 5. Nte the change in the l interval f instability. 10
An alternative representatin represents the maximum grwth rate fr each M,N in the M,N plane. One can identify the crrespnding y-wavenumber l with the aid f Figure. Figure 5 shws just such a cntur plt. The maximum ccurs very clse t the values chsen fr Figure 4 and the maximum value is abut 1. 10-4 and ccurs n the M,N diagnal. Nte that n the diagnal s = 0.5, a result that culd be anticipated frm symmetry, s that fr such diagnal mdes the structure f the eigenfunctins is, t lwest rder, the same in each layer. There will be a departure at higher rder and the ff-diagnal unstable mdes, whse grwth rates are cmmensurate with the diagnal mdes will have very different structures in tw layers. Figure 5. Cnturs f grwth rate in the M N plane. Nte that the maximum ccurs alng the line M = N. 4. Discussin The examinatin f the instability f meridinal barclinic shear flws prduces sme nvel instability characteristics when cmpared with the classical prblem f the instability f znal flws. Perhaps nne is mre surprising than the absence f a critical threshld fr instabil- 11
ity fr the shear even in thse cases, as studied in this paper, where the gemetry f the flw frces the disturbance t sense the effect f β. In spite f cnsiderable effrt a necessary cnditin fr instability has nt been prven fr the system (.1) and althugh a negative is nt a prf it was thught t suggest that all meridinal flws wuld be unstable. That interesting hypthesis is put n firmer grund by the asympttic result f this paper which shws that very weak meridinal shears can destabilize therwise neutral Rssby nrmal mdes fr the channel. The preexisting mde has its structure slightly altered by the shear allwing the release f the available ptential energy in the shear flw. Since the shears we have cnsidered are weak the crrespnding energy surce is rather feeble and the resulting grwth rates are small. One culd stretch the asympttics t values f v which are nt very small t btain larger grwth rates but that range is already cvered by the detailed numerical analysis f WP which qualitatively agrees with the present results. Frm the pint f view f ceanic applicatins that may be unnecessary. Althugh the gemetry f the channel is simple cmpared t the gemetry f the subtrpical gyre it is still true that instabilities f the flw in the eastern regins f the gyre can be interpreted in terms f the instability f meridinal flw (see fr example Spall, 000) and emphasizes the imprtant rle f znal bundaries (here taken as simple meridians) in affecting the variability f the mid-basin flw. This leads t the interesting pssibility that instabilities f the type described in this paper can extract sufficient energy frm the gyre circulatin t maintain Rssby nrmal mdes against dissipatin as lng as the weak grwth rates pertinent t such mdes exceed the dissipatin rates fr such mdes. Cessi and Primeau (001) and LaCasce (000) have already pinted ut the existence f such special Rssby nrmal mdes with exceedingly lng dissipative times under the influence f scale selective dissipatin. The cupled instability f the Rssby nrmal mdes and the barclinic shear culd prvide a mechanism fr the generatin and maintenance f the nrmal mdes rather that relying n the persistent ringing f the system by repeated external frcing. Acknwledgments. This research was supprted in part by a grant frm the Natinal Science Fundatin, OCE 99-01654 References Cessi, P. and F. Primeau, 001. Dissipative selectin f lw-frequency mdes in a reduced gravity basin. Jurnal f Physical Oceangraphy, 31, 17 137. LaCasce, J. H., 000. Barclinic Rssby waves in a square basin. Jurnal f Physical Oceangraphy, 30, 3161 3178. Pedlsky, J., 1987. Gephysical Fluid Dynamics. Springer-Verlag. New Yrk, pp. 710. Spall, M., 000. Generatin f strng messcale eddies by weak cean gyres. Jurnal f Marine Research, 58, 97 116 Walker, A. and J. Pedlsky, 00. On the instability f meridinal barclinic currents. Jurnal f Physical Oceangraphy (t appear). 1