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Under onsideration for publi ation in J. Fluid Me h. 1Strati�ed Kolmogorov ow IIBy N.J. Balmforth1 and Y.-N. Young21Departments of Mathemati s and Earth & O ean S ien es, University of British Columbia2Center for Turbulen e Resear h, Stanford University(Re eived ?? and in revised form ??)For ed, strati�ed ows are shown to su�er two types of linear, long-wave instabili-ties: a \vis ous" instability whi h is related to the lassi al instability of Kolmogorov ow, and a \ ondu tive instability", with the form of a large-s ale, negative thermaldi�usion. The nonlinear dynami s of both instabilities is explored with weakly nonlin-ear theory and numeri al omputations. The introdu tion of strati� ation suppressesthe vis ous instability, but also makes it sub riti al. The se ond instability arises withstronger strati� ation and reates prominent stair ase in the buoyan y �eld; the steps ofthe stair ase evolve over long times ales by oarsening.1. Introdu tionStrati�ed shear ows arise frequently in geophysi al and astrophysi al uid dynami s.A entral issue in su h ontexts is understanding how eddying, unsteady motion arisesfrom a steady ow or for ing, and how that motion an re-arrange and transport the uid properties. In this arti le, we ontinue an exploration of a parti ular model problemin whi h the dynami s is a essible to an unusual degree of analysis. More spe i� ally,we study the fully strati�ed version of the so- alled Kolmogorov ow, whi h was origi-nally advo ated as a onvenient theoreti al edi� e to understand unstrati�ed shear owdynami s and the transition to turbulen e. Instabilities of Kolmogorov ow exhibit theproperty of inverse as ade: although instabilities an be seeded on moderate lengths ales, energy is ontinually transfered via nonlinear me hanisms to longer length s ales.In our previous arti le (Balmforth & Young 2002), we showed how the as ade is arrestedby relatively weak strati� ation. This arrest is also impli it in mu h of the explorationdes ribed in the urrent arti le. However, it does not provide our main fo us, whi h liesin a di�erent dire tion.Laboratory experiments and o eani observations have both revealed that ows instably strati�ed uids an generate \stair ases" of well-mixed layers separated by sharpinterfa es. In the laboratory, stair ases have been reated by dragging grids or barsthrough tanks of salt-strati�ed water (Park et al. 1996, Holford & Linden 2000); in theo ean, mixing by the motions of the ever turbulent environment is assumed to havethe same e�e t (S hmitt 1994). Small-s ale �ngering instability due to double di�usionis also thought to reate large-s ale stair ases without externally driven ows (Radko2003), and turbulent thermohaline onve tion has been seen to generate sta ked layersin the laboratory and solar ponds (e.g. Turner 1985). Although it has never been shownexpli itly, it is ommonly assumed that a turbulent ow �eld is an essential ingredientin the layering problem. That is, that the Reynolds number of the mixing ow must bevery large. Based on this premise, several authors onstru ted rude models of turbulentstrati�ed uids and thereby rationalized the layering pro ess (Phillips 1972, Posmentier
2 N.J. Balmforth and Y.-N. Young1977, Balmforth et al. 1998). These models typi ally rely on simple, sometimes empir-i al, parameterizations of turbulent transport, and formulate a non-monotoni relationbetween the density ux and the density gradient (the \ ux-gradient relation"). The un-derlying notion is that wherever the ux de reases with the gradient, the strati� ation isunstable and small u tuations will seed the growth of sharp steps via negative di�usion.One of our purposes in the present arti le is to show that stair ases an also result formu h lower Reynolds numbers, when the mixing ows are laminar. This opens up theproblem to analyti al explorations based on the governing equations of uid me hani s,rather than rude turbulen e parameterizations. In parti ular, by using the method ofmultiple s ales, we establish that instability an o ur in the form of negative di�usion,and determine the ux-gradient relation expli itly in the vi inity of the onset of insta-bility. Stair ases an then be predi ted to o ur; we examine the robustness of layeringwithin this formulation. Our analysis is similar to that used to ompute eddy di�usiv-ities in homogeneous uid (Gama, Vergolassa & Fris h 1994), and there are analogieswith stability theories of Rossby waves (Lorenz 1972) and internal gravity waves (Drazin1978, Kurgansky 1979 and 1980, Lombard & Riley 1996) whi h have appli ations toatmospheri dynami s and to o eani mixing (Thorpe 1994).Our analysis pro eeds by way of multiple s ales, assuming that instability arises on amu h longer spatial s ale than the intrinsi lengths ale of the steady ow pattern thatis set up by a suitable body for ing of the uid (the Kolmogorov ow). This analysisdete ts linear, long-wave instability (se tion 2) whi h we then ontinue on to explore atthe �nite-amplitude level using weakly nonlinear te hniques (se tion 3) and numeri al omputation (se tion 4).2. Formulation and linear theory2.1. Governing equationsWe begin with the basi equations for a for ed uid in the Boussinesq approximation.After introdu ing a streamfun tion, (x; z; t), and the buoyan y �eld, b(x; z; t) (repre-senting an agent su h as temperature or salinity), whi h des ribe the deviation from themotionless, (linearly) strati�ed state, these equations arer2 t + Jx;z( ;r2 ) = bx + �r4 � �r4' (2.1)and bt + Jx;z( ; b) +N2 x = �r2b; (2.2)where ' represents the for ed sour e of vorti ity,Jr;s(f; g) = frgs � fsgr (2.3)is the Ja obian of the fun tions f and g with respe t to the oordinates r and s, �is the vis osity, � the ondu tivity, and N2 the buoyan y frequen y arising from theba kground, linear strati� ation. For pra ti al purposes, we take' = '0 sin(kx�mz);where '0 is the amplitude of the for ing, and the wavenumbers, (k;m), determined the tiltwith respe t to the verti al. This for ing generates a steady equilibrium ow of the form,(u;w) = (m; k) os(kx � mz), where is a onstant. We impose periodi boundary onditions in the horizontal, and delay dis ussion of the verti al boundary onditionsuntil later.We pla e the equations in a nondimensional form using units given by the for ing;
Strati�ed Kolmogorov ow II 3that is, the lengths ale, K�1, and the times ale, '0K2, where K2 = k2 +m2. We de�nex0 = Kx, z0 = Kz, t0 = '0K2t, 0 = ='0 and b0 = K3'20b. On substitution into(2.1)-(2.2), and dis arding the primes, we arrive atr2 t + Jx;z( ;r2 ) = bx +Re�1(r4 � os�) (2.4)and bt + Jx;z( ; b) + � x = Pe�1r2b; (2.5)where � = x os � + z sin �, with m = K sin � and k = K os �, the dimensionlessgroups, Re = '0=� and Pe = '0=�, denote the Reynolds and Pe let numbers, and� = N2=('20K4) is a strati� ation parameter somewhat like a Ri hardson number.2.2. Multiple-s ale expansionWe introdu e �t ! �2�T ; �z ! �z + ��Z ; (2.6)where T � �2t and Z � �z denote a slow times ale and a long lengths ale (a orrespondinglong s ale for x turns out to be not ne essary be ause the �rst solvability onditions thatone then en ounters demand that there be no variation on su h a s ale, at least for orderone �). With these res alings:�2[�2x + (�z + ��Z)2℄ T + Jx;z( ; xx + (�z + ��Z)2 ) + �Jx;Z( ; xx + (�z + ��Z)2 )= bx +Re�1[�2x + (�z + ��Z)2℄2 �Re�1 os� (2.7)and �2bT + Jx;z( ; b) + �Jx;Z( ; b) + � x = Pe�1[�2x + (�z + ��Z)2℄b: (2.8)Over the shorter spatial s ales, (x; z), we look for solutions that have the same periodi ityas the for ing; these patterns are modulated on the long spatial s ale Z and over theslow time.It is useful to quote the averages over the (x; z)�s ales:� T + x Z = �Re�1 ZZ and �bT + ( xb)Z = �Pe�1bZZ ; (2.9)where the bar denotes the average over a spatial period of the for ing, and we haveintegrated the �rst relation twi e in Z, assuming the integration onstants vanish byvirtue of the verti al boundary onditions.We now introdu e the asymptoti sequen es, = 0 + � 1 + ::: b = b0 + �b1 + :::; (2.10)and olle t together terms of like order. At order one,Re b0x + (�2x + �2z )2 0 �Re Jx;z( 0; 0xx + 0zz) = os� (2.11)and �Pe 0x � (�2x + �2z )b0 +Pe Jx;z( 0; b0) = 0; (2.12)with solution, 0 = 0(Z; T ) + os�1 +G and b0 = b0(Z; T ) + �Pe sin� os �1 +G ; (2.13)where G = �RePe os2 �.At the following order:Re b1x + (�2x + �2z )2 1 = �Re 0Z sin� os �1 +G +N1 (2.14)
4 N.J. Balmforth and Y.-N. Youngand �Pe 1x � (�2x + �2z )b1 = �Pe2 0Z os� os2 �1 +G + Pe b0Z sin� os �1 +G +N2; (2.15)where N1 and N2 are nonlinear terms that vanish for the solution, 1 = 1(Z; T ) + Re os �(1 +G)2 �(�Pe2 os2 � � 1) 0Z sin�+Peb0Z os � os�� (2.16)and b1 = b1(Z; T ) + Pe os �(1 +G)2 �(Re + Pe)� 0Z os � os�� b0Z sin�� : (2.17)We substitute these solutions into the average equations to �nd two di�usion equations,Re 0T = �1� Re2(1� �G) os2 �2(1 +G)3 � 0ZZ (2.18)and Pe b0T = �1� Pe2(G� 1) os2 �2(1 +G)3 � b0ZZ ; (2.19)where � = �=� is the Prandtl number.2.3. Criti al onditionsThe di�usivities in (2.18)-(2.19) are not positive de�nite. Indeed, for ertain hoi es ofthe parameters, these quantities may be ome negative, signifying a long-s ale instability.Ea h equation provides an instability ondition:1 < Re2(1� �G) os2 �2(1 +G)3 and 1 < Pe2(G� 1) os2 �2(1 +G)3 : (2.20)At this stage, we observe that the only e�e t of � is to res ale the Reynolds and Pe letnumbers, and so the in lination of the for ing has a minor e�e t on the linear, long-wavedynami s. For brevity we therefore set � = 0 hereon.Mathemati ally, it is onvenient to sele t Re, � and G as the governing parameters ofthe problem. We may then translate the onditions in (2.20) into the riti al Reynoldsnumbers,Re > Re1 = p2(1 +G)3=2(1� �G)1=2 and Re > Re2 = p2(1 +G)3=2�(G � 1)1=2 : (2.21)If G = 0, Re1 ! p2 whereas Re2 eases to exist. The former is the instability thresholdof the usual Kolmogorov instability (Meshalkin & Sinai, 1960), whi h an be seen from(2.18) to result from a negative e�e tive vis osity. As shown in �gure 1, the instabilitybe omes modi�ed by strati� ation, and even removed, when G is in reased from zero.Thus, Re1 hara terizes a familiar long-wave instability that we refer to as \vis ous".The nonlinear dynami s of the weakly strati�ed, vis ous instability was onsidered byYoung (1999) and Balmforth & Young (2002).The other riti al threshold, Re2, orresponds to a se ond type of instability whi h(2.19) reveals to result from negative ondu tion. This se ond mode of instability ap-pears only at higher strati� ation (G or �), and we refer to it as \ ondu tive". Thetwo instabilities typi ally appear in di�erent parts of parameter spa e, although they an be oin ident when � < 1 (see �gure 1). The instabilities appear simultaneously for
Strati�ed Kolmogorov ow II 50 0.5 1 1.5 2 2.5 3 3.5 4
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Figure 1. Criti al Reynolds numbers, Min(Re1;Re2), against G for � = 0 and several values ofPrandtl number (as labelled).Re1 = Re2, whi h demands that G = 1 + �2�(1 + �) : (2.22)From a physi al perspe tive, G is not a suitable parameter to des ribe the systembe ause G = ��Re2, and � is arguably the dimensionless group that an be pres ribed.However, eliminating G in favour of � ompli ates the riti al onditions, and Re1 andRe2 annot be found in losed form. Instead, we show sample riti al thresholds omputednumeri ally against � for di�erent Prandtl numbers in �gure 2. This �gure illustrateshow strati� ation stabilizes the vis ous instability at large Reynolds number, and om-pletely removes instability for any Reynolds number beyond a ��dependent value. Theinstability window of the ondu tive mode is similar, but typi ally lies at higher Reynoldsnumber and only overlaps the region of vis ous instability for � < 1. The values of � andReynolds number for whi h the two instabilities disappear entirely (that is, the valuesat the nose of the stability boundaries in �gure 2) are shown against Prandtl number in�gure 3. Note that the stabilizing e�e t of strati� ation, and spe i� ally the removal ofthe instability for any Reynolds number beyond a riti al value of �, is reminis ent ofthe elebrated Ri hardson number riterion. However, the basi ow is in the dire tion ofgravity here, and the riti al threshold in � has a novel dependen e on Prandtl number,as illustrated in �gure 3.Note that the theory identi�es only long-wave instabilities. However, instabilities with�nite wavenumber are also possible, and these would lead to more windows of instabil-ity elsewhere in parameter spa e. Some on�den e that long waves are responsible forinstability omes from the numeri al solutions of se tion 4 at isolated parameter values,although these omputations also un over other instabilities. A more systemati approa hwould entail a detailed numeri al exploration of the linear stability problem for arbitrarywavenumber.3. Nonlinear theoryWe next demonstrate that layering is expe ted in the mildly nonlinear stages of theinstability dis ussed above. We pro eed by deriving a Cahn-Hilliard equation througha weakly nonlinear asymptoti expansion, an equation that is well known to possesssolutions in the form of layers. We perform this onstru tion for both the vis ous and ondu tive instabilities (taking � = 0).
6 N.J. Balmforth and Y.-N. Young
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β∼ 1/(8σ2)Figure 3. The Reynolds numbers, Re1 and Re2, and strati� ation parameter, �, plotted against� at the \nose" of the long-wave stability boundaries (i.e. the values of Rej and � for whi hinstability disappears entirely). The limiting values for � ! 0 and 1 are indi ated.For both instabilities, we again introdu e the long s ale, Z = �z, and res ale time, butin a slightly di�erent way: �t ! �4�� (� = �4t). Then, the governing equations be ome�4(�2x + �2�2Z) � + �( x xxZ � Z xxx) + �3( x ZZZ � Z xZZ)= bx +Re�1(�2x + �2�2Z)2 � Re�1 osx (3.1)and �4b� + �( xbZ � Zbx) + � x = Pe�1(�2x + �2�2Z)b: (3.2)3.1. Weak ondu tive instabilityTo derive the Cahn-Hilliard model in the ondu tive ase, we begin with a ow onthe brink of instability, and then ki k the system into the unstable regime by slightly
Strati�ed Kolmogorov ow II 7modifying the parameter values. In parti ular, we fo us on the spe i� parameter hoi es,� = 3Re�1p6(2 + �G1); Pe�1 = 13p6 + �2�2: (3.3)That is, we tune both � and Pe, treating Re as a free parameter. The two hoi es in (3.3)are ne essary for reasons that will be dis ussed later; essentially, we obtain the Cahn-Hilliard equation only at a o-dimension-two point where the oeÆ ient of the leadingquadrati nonlinear term is for ed to vanish.We begin again from the governing equations in the forms (3.1)-(3.2), but now hoosethe leading-order solution, 0 = 13 osx and b0 = B(Z; �) + 23Re sinx: (3.4)The signi� an e of this hoi e is that in (2.18)-(2.19) there are two long-wave modesevolving on the t � ��2 times ale: a vis ous mode and a ondu tive mode. Only these ond of these modes is marginally stable for G1 = �2 = 0, and therefore evolves evenmore slowly on the t � ��4 time. The vis ous mode, on the other hand, at this point ismore heavily damped. As a result, we assume that the mode de ays to low amplitudebefore the ondu tive mode begins to grow (whi h does not, in fa t, always remain trueas shown by omputations reported below), leading us to in lude only the slow mode,B(Z; �).At the next order, we �ndRe b1x + 1xxxx = 0 and 2 1x �Re b1xx = 13(3Rep6BZ +G1) sinx: (3.5)We take 1 = � Rep63 BZ + G19 ! osx; and b1 = p63 BZ + G19Re! sinx: (3.6)At order �2, Re b2x + 2xxxx = 0 (3.7)and 2 2x �Re b2xx = �3ReBZZ os 2x� "�G13 +p6ReBZ�2 + 2�2p6# sinx; (3.8)whi h we solve with 2 = 13 "�G13 +p6ReBZ�2 + 2�2p6# osx� Re12BZZ sin 2x (3.9)and b2 = � 13Re "�G13 +p6ReBZ�2 + 2�2p6# sinx� 23BZZ os 2x: (3.10)The third-order equations and their solution pro eed in mu h the same way. For brevity,key formulae are relegated to the appendix. Finally, we insert the solutions for and binto the horizontal averages of the governing system to arrive at the amplitude equation,B� = 2 �2 + G21p6108 !BZZ � 13p672 + Re108!BZZZZ + G1Re3 (B2Z)Z + p6Re23 (B3Z)Z :(3.11)
8 N.J. Balmforth and Y.-N. YoungWhen expressed in terms of a new variable ' = BZ , this expression has the form of theCahn-Hilliard equation (the term, B2Z , an be eliminated by setting BZ = '+C, whereC is a suitable onstant, pla ing the system in the usual Cahn-Hilliard form).Note that the nonlinearity is ubi in (3.11). This resulted entirely be ause the oef-� ient of the otherwise leading quadrati term (B2Z)Z is proportional to (G � 2), andwe su essfully pushed that term to the same order as the ubi nonlinearity by mak-ing the o-dimension-two parameter hoi es in (3.3). Quadrati nonlinear terms an-not be dis ounted by symmetry arguments in the urrent expansion be ause the gov-erning system has the re e tion symmetry transformations, (x; ) ! (�x;� ) and(x; z; b)! (�x;�z;�b). From these symmetries alone, we see that the amplitude annotbe the standard Cahn-Hilliard equation, be ause that equation is invariant under theindependent transformations, B ! �B and Z ! �Z.A Lyapunov fun tional exists for the Cahn-Hilliard equation whi h predi ts that theevolution of the system is the inexorable onvergen e to the steady solution with thelargest spatial s ale (e.g. Chapman & Pro tor 1980). The onvergen e, however, an bedelayed for long periods by meta-stable states onsisting of a sequen e of layers separatedby slowly drifting interfa es. (An illustration of the oarsening pro ess is given below.)It is this property of the Cahn-Hilliard model that leads us to predi t that layering anresult in laminar ows. 3.2. Weak vis ous instabilityThe marginal stability ondition for vis ous instability, viewed as a riti al Reynoldsnumber, Re = Re , is Re2 = 2(1 +G)31� �G ; (3.12)whi h also �xes the Pe let number given the Prandtl number. To push the system intoa weakly unstable regime, we set Re = Re + �2Re2, and again perform an asymptoti expansion. We begin with the sequen es, = 0 + � 1 + :::; b = b0 + �b1 + :::; (3.13)and sele t a leading-order solution, 0 = A(Z; �) + 11 +G osx and b0 = GRe(1 +G) sinx; (3.14)whi h, this time, ontains only the slow vis ous mode.At order �2, we �nd the relations, 1xxxx +Re b1x = Re ( 0x 0xxZ � 0Z 0xxx) (3.15)and G 1x �Re b1xx = �Re2 (b0x 0Z � b0Z 0x); (3.16)whi h are solved by 1 = Re(�G � 1)(1 +G)2 AZ sinx; b1 = B1(Z; �) + G(1 + �)(1 +G)2 AZ osx: (3.17)In (3.17), we add a mean buoyan y term; it turns out that this is ne essary be ause theslow vis ous mode for es a mean response in b at order �, as is lear from the verti alaverage of (3.2), whi h provides the relation,�Re ( 1xb1Z � 1Zb1x � 2Zb0x + 0xb2Z � 0Zb2x) = b1ZZ � B1ZZ ; (3.18)
Strati�ed Kolmogorov ow II 9at order �3. The left-hand side does not, in general, vanish, thus for ing B1. We delaythe onstru tion of this quantity until we solve the system at next order. We again pla esome of the details in the Appendix, and arrive at the relation,B1ZZ = �G(1� 2G� + �2)(A2Z)Z(1 +G)(1 + �2 � �G� �2G) � �(A2Z)Z : (3.19)At this stage we must make some statement about boundary onditions in Z; we adoptperiodi ity in this dire tion. The integral of (3.19) then implies thatB1Z = �(A2Z � hA2Zi); (3.20)where the angular bra kets denote the average in Z.Finally, we pro eed to third order and evaluate the verti al average of (3.1). This leadsto the amplitude equation:A� = �Re3 �(5� 2�G+ 3�)6(1 +G)4 �hA2Zi � 2Re2Re �AZZ� 3Re 2(1 +G)5 �1� �G+ �2Re2 G(�G � 8� � 3)24(G+ 1)(G+ 16) �AZZZZ+ Re3 6(1 +G)4 ���(5� 2�G+ 3�) + 2� 6�G+ �2G(G� �G � 3� � 5)1 +G � (A3Z)Z : (3.21)If G = 0, this equation redu es to a Cahn-Hilliard equation for the variable, AZ , and isequivalent to a system derived previously by Sivashinksy (1985). However, with G 6= 0,it is not pre isely of Cahn-Hilliard form be ause of the nonlo al term involving A2Z .The negative di�usion term in (3.21) ampli�es gradients of AZ , and therefore BZ .These gradients ontinue to sharpen as the instability operates, but an saturate whenthe nonlinear di�usion term omes into play. In the standard Cahn-Hilliard system, su ha saturation is guaranteed if the oeÆ ient in front of the ubi nonlinearity, (A3Z)Z , ispositive. For our model in (3.21), the ubi oeÆ ient is indeed positive when the strat-i� ation parameter, G, is small. However, as G in reases, the ubi oeÆ ient de reasesand an eventually hange sign. In this ir umstan e, one anti ipates that the leadingnonlinearity annot saturate the sharpening of the interfa es, but a elerates it untilfurther nonlinear terms be ome important. This situation is analogous to a sub riti albifur ation (a onne tion that an be made �rmer by de omposing A into normal modesin Z and performing a standard amplitude expansion). In other words, by stratifyingthe uid, we an for e the vis ous instability to be ome sub riti al, reating a \harder"transition at onset. Also, with the nonlo al term, hA2Zi, it is not lear what remains of the oarsening dynami s des ribed by the Cahn-Hilliard equation. We solve a more generalversion of the amplitude equation (3.21) below to shed some light on this se ond issue.3.3. Long-wave equations by proje tionAn alternative approa h to the asymptoti expansion above is provided by a Galerkinproje tion of the form, = A(z; t) +P3j=1[a1;j(z; t) os jx+ a2;j(z; t) os jx℄b = B(z; t) +P3j=1[b1;j(z; t) os jx+ b2;j(z; t) os jx℄ ; (3.22)where the oeÆ ients a1;j , a2;j , b1;j and b2;j an be determined in terms of A and Bby introdu ing the proje tion into the governing equations with the time derivativesnegle ted. Substitution of the proje tion into the horizontal averages of the governing
10 N.J. Balmforth and Y.-N. Youngequations then provides the evolution equations for A and B. A mammoth amount ofalgebra, greatly assisted by MAPLE, eventually yields the system,At = Re�1 �1� Re2(1� �G)2(1 +G)3 �Azz + �Re32(1 +G)4 [2(1� �G)AzzBz + (1 + �)(AzBz)z℄�3Re2(1 +G)4 �1� �G+ �2Re2G(�G� 8� � 3)24(G+ 16)(G+ 1) �Azzzz � 3�2Re52(1 +G)5 (2� �G+ �)B2zAzz� 2�2Re5(1 +G)5 (1+�)AzBzBzz+ Re36(1 +G)5 [2�6�G+�2G2(1��)��2G(3�+5)℄(A3z)z (3.23)andBt = Pe�1 �1� �2Re2(G� 1)2(1 +G)3 �Bzz+ Re2(1 +G)4 [G(2G���2�1)A2z+�2Re2(G�2)B2z ℄z� �Re2(1 +G)4 �5G� 1 + �Re2(9�G� 8� + 6G� �G2)8(G+ 16)(G+ 1) �Bzzzz + �3Re5(3�G)2(1 +G)5 (B3z )z+ �Re32(1 +G)5 [(3G� 1)(�2 + 1) + 4�G(1�G)℄(A2zBz)z : (3.24)This system displays all the symmetries of the original equations and redu es to theasymptoti models above in suitable limits of the parameters. However, it is not ne essaryto restri t the parameter settings to those values. Moreover, the system aptures thedynami s of the situation in whi h both the vis ous and ondu tive instabilities be omeunstable at the same time, taking the same form as equations derivable by asymptoti means. Hen e, (3.23)-(3.24) provide a more ompa t des ription of the dynami s, and wesolve this system numeri ally to gain further insight into the nonlinear behaviour. Notethat the growth of A stimulates B, but B evolves of its own if A(z; 0) = 0.We illustrate the weakly nonlinear dynami s aptured by the model in �gures 4-6.Figure 4 shows a omputation in whi h the ondu tive instability arises, but there is novis ous instability. In this ase, A de ays to zero, and BZ forms a sequen e of alternatinglayers (�ve layers with positive and �ve with negative gradient appear initially, reatingthe pattern of stripes in the �gure). As time pro eeds, the interfa es bordering the layersdrift under mutual intera tion. At various instants, interfa es approa h one another and ollide, thereby removing one of the layers, and widening its neighbours. As a result ofthis pro ess, the hara teristi lengths ale of the pattern gradually in reases with time;this is the oarsening dynami s aptured by Cahn-Hilliard (to whi h the system redu eswhen A ! 0). At the end of the simulation, only four interfa es remain (two pairs oflayers). Another ollision o urs at later times (not shown) to oarsen the pattern to itsultimate �nal state, a pattern with the longest spatial s ale. In other words, the pure ondu tive mode exhibits a ompleted inverse as ade. Asso iated with this as adeare rearrangements of the strati� ation that take the form of a stair ase in the fullbuoyan y distribution. Also, even though the horizontal average of the velo ity �eldde ays, the motions driven by the ondu tive mode do not. The Fourier-series form ofthe solution suggests slowly evolving ellular patterns in the velo ity �eld, as observedin the omputations des ribed in the next se tion.Figure 5 displays the emergen e of the (super riti al) vis ous instability in the absen eof a ondu tive one. When the vis ous instability �rst enters the nonlinear regime, apattern with �ve light and dark stripes appears in Az . These stripes re e t an alter-nating sequen e of horizontally dire ted jets superposed on the underlying verti al ow
Strati�ed Kolmogorov ow II 11z
Bz
0.5 1 1.5 2 2.5 3 3.5 4
x 1010
−40
−20
0
20
40
0 0.5 1 1.5 2 2.5 3 3.5 4
x 1010
10−5
100
Time2, t2
Max
−M
in
B
A
β z + B
z Figure 4. Condu tive instability in a domain of size 100. The top panel shows a greys aleof Bz on the (t2; z)�plane. The middle panel displays the time history of the peak-to-peakamplitudes of A and B. The �nal panel shows the evolving, total buoyan y �eld, �z + b, as asequen e of snapshots su essively o�set to the right. The parameter values: Re = 6, G = 2and � = 2. In this omputation, the sign of the oeÆ ient of the term Azzzz in (3.23) has beenreversed to ensure that the system remains well-posed. The signi� an e of positive sign of this oeÆ ient is that short waves are predi ted to be unstable (an ultra-violet atastrophe), whi his unphysi al and probably an artifa t of the long-wave expansion. Although the swit h of signis a little arbitrary, it has the same e�e t as in luding a regularizing term with higher derivativesprovided there are no instabilities with short wavelengths.(whi h orresponds to a hara teristi meandering motion that is visible in the numeri alsolutions of the next se tion). Sharp negative buoyan y gradients build up in the shearlayers bordering the jets, giving a hara teristi verti al s ale to b whi h is twi e that ofthe velo ity �eld. However, the e�e t on the total buoyan y �eld is less pronoun ed andlittle forms by way of a stair ase. Note that the interfa e ollapses in �gure 5 generate aresponse throughout the entire pattern (the overall shading of the layers and interfa esappears to abruptly hange at the ollisions, espe ially for B(z; t)), in ontrast to the rel-atively lo al e�e ts seen in �gure 4. This re e ts a more nonlo al nature of the dynami swhi h is also expe ted from the nonlo al term in (3.21). Moreover, the pattern does not oarsen further at later times, unlike in �gure 4, and the state ending �gure 5 appearsto be the �nal one. Thus oarsening is arrested, as found in our earlier paper for mu hweaker strati� ation.Figure 6 shows a omputation in whi h both modes are unstable and a steady, non- oarsening pattern emerges. This pi ture illustrates another feature of the dynami s,namely that when the two modes ompete, the vis ous mode dominates and suppressesthe ondu tive mode. In order to emphasize this feature of the dynami s in the ompu-tation, the growth rate of the ondu tive mode was arti� ially in reased and the initial ondition seeded with the unstable ondu tive mode in order to promote that instabilityover the vis ous mode, at least initially. After a period of time, however, the vis ousmode overtakes the ondu tive mode and establishes a steady pattern that shows no oarsening.
12 N.J. Balmforth and Y.-N. Youngz
Az
0.5 1 1.5 2 2.5 3 3.5 4
x 1010
−40
−20
0
20
40
z
Bz
0.5 1 1.5 2 2.5 3 3.5 4
x 1010
−40
−20
0
20
40
0 0.5 1 1.5 2 2.5 3 3.5 4
x 1010
10−5
100
Time2, t2
Max
−M
in
A
B
B(z,t) β z + B
z
Figure 5. Vis ous instability in a domain of size 100. The panels show greys ales of Az and Bzon the (t; z)�plane, time series of the peak-to-peak amplitudes of A and B, a greys ale of B onthe (t; z)�plane and a sequen e of snapshots of total buoyan y, �z + b. The parameter values:Re = 3, G = 1=4 and � = 1=2.We lose this se tion by autioning that the long-wave model in (3.23)-(3.24) is notalways well-posed: for ertain parameter hoi es, the hyper-di�usion terms, Azzzz andBzzzz , turn out to have positive oeÆ ients. (In �gure 4, for example, even though theA-mode de ayed away, we arti� ially reversed the sign of the Azzzz term to ensure thesystem remained well-posed.) Moreover, the oeÆ ients of the nonlinear terms, (Bz)3and (Az)3, are not always positive and the system seems able to pass abruptly into aphase where gradients are ontinually sharpened and the omputation breaks down. Of ourse, sin e the original system is unlikely to be prone to the same problems, the faultmust lie in the Galerkin trun ation. We avoid any su h problems below by solving thefull governing equations numeri ally.
Strati�ed Kolmogorov ow II 13z
Az
1 2 3 4 5 6 7 8 9 10
x 105
−10
−5
0
5
z
Bz
1 2 3 4 5 6 7 8 9 10
x 105
−10
−5
0
5
0 1 2 3 4 5 6 7 8 9 10
x 105
10−5
100
Time2, t2
Max
−M
in
A
BFigure 6. Double instability in a domain of size 20. The panels show greys ales of (a) Az and(b) Bz on the (t; z)�plane, and (b) the peak-to-peak amplitudes of A and B. The parametervalues: Re = 75, G = 2:5 and � = 0:11. In order to promote arti� ially the ondu tive mode overthe vis ous mode, the growth rate of the ondu tive instability has been in reased unphysi allyby a fa tor of 20, and the initial ondition seeded mainly with the unstable linear ondu tivemode.4. Dire t Numeri al SimulationsTo ompare the results of the previous se tion with numeri al solutions we sele t threesets of representative parameter values:(i) Condu tive : Re = 6; G = 2; � = 2 (Pe = 12; � = 1=36; � = 0);(ii) V is ous : Re = 3; G = 1=4; � = 1=2 (Pe = 3; � = 1=18; � = 0);(iii) Combination : Re = 17; G = 1:705; � = 1=2 (Pe = 8:5; � = 1=85; � = 0);with a domain of horizontal and verti al lengths, Lx = 2n� and Lz = 32�. We explore ases with n = 1 or 2: If n = 1, there is a single pair of opposed verti al jets in thedomain; for n = 2, there are two pairs.Figure 7 shows the evolution of ase (i), with a single pair of jets. The instability reatesa ellular velo ity pattern that is also hara terized by a \blobby" �eld of buoyan yanomaly. There is weak mean (x�averaged) horizontal ow, but strong re-arrangementof the mean (x�averaged) buoyan y into a stair ase, both of whi h are typi al of the ondu tive instability. The pattern initially forms an array of four ellular stru tures,but these oarsen to two, as expe ted from the underlying Cahn-Hilliard dynami s.Figure 8 shows the evolution of a single jet pair in ase (ii). The instability initiates adistin tive meander of the jet and eddies emerge within parts of the meander; this patternis mu h like that arising in weakly strati�ed Kolmogorov ow onsider in our earlier paper
14 N.J. Balmforth and Y.-N. Young
0
101
6.3
z
x
0 62 3 4 51
Z
100Time (10 )4Figure 7. Condu tive instability in the domain [2�; 32�℄ The panels show snapshots of (a) (x; z; t) and (b) b(x; z; t) as greys ales on the (x; z)�plane. The parameter values: Re = 6,G = 2 and � = 2. The times of the snapshots are 5000, 15000 and 26000. The �nal panel showsthe evolution of the total, horizontally averaged buoyan y �eld, �z +B(z; t).(although there the underlying jets were horizontally dire ted). The meandering re e tsstronger mean horizontal ow, and the buoyan y �eld shows twi e the verti al s ale ofthe horizontal �eld. The eddying meanders oarsen from six to four, with large transientsprodu ed in the buoyan y �eld as a result of the mergers. At four meanders, oarseningappears to halt. Again, these features re e t the weakly nonlinear dynami s predi tedby the long-wave theory.The single pair of jets for ase (iii) is shown in �gure 9. The forming patterns bearmu h in ommon with those reated by the vis ous instability, although there is littlesign of oarsening in the meanders. As expe ted the ondu tive mode is suppressed.Results for n = 2 are shown in �gures 10-11. The main surprise for ase (i) is thatthe layering pattern of the ondu tive mode be omes repla ed by a pattern with themeandering and eddying hara ter of the vis ous mode. The pathway to this �nal statepro eeds in two stages. First, a pattern emerges with the blobby and ellular featurestypi al of the ondu tive mode, but it is unsteady, irregular and ontains a signi� ant ontribution from a subharmoni , x�wavenumber-one disturban e. This initial phase isinterrupted by the growth of what appears to be a se ondary instability that spawns themeandering and eddying jet pattern. Case (ii) shows no surprises, being a periodi allyrepeated version of the n = 1 single paired jets, and is not therefore displayed. Mean-
Strati�ed Kolmogorov ow II 15
x
z
Figure 8. Vis ous instability in the domain [2�; 32�℄ The panels show snapshots of (a) (x; z; t)and (b) b(x; z; t) as greys ales on the (x; z)�plane. The �nal panels show the horizontal averagesof and b on the (t; z)�plane. The parameter values: Re = 3, G = 1=4 and � = 1=2. The timesof the snapshots are 1:5 � 103, 4:6� 103, and 6� 103, respe tively (from left to right).dering and eddying again predominate in the ase (iii) omputation, but now, somewhatsurprisingly, signi� ant oarsening takes pla e. This redu es the initial number of me-anders from seven down to three. From the trend of the omputation, we suspe t thata further oarsening to two takes pla e around t = 2 � 104 if we were to ontinue the omputation. Amusingly, the oarsening is now mediated by an os illatory intera tioninvolving the horizontal motion of buoyan y anomalies and pulsations of the eddies. Theplots of horizontally averaged streamfun tion and buoyan y anomaly ni ely bring outthis feature, and also illustrate how strong layer migrations o ur in this omputation.Finally, we remark that we have also run some further omputations at either higherReynolds numbers, or with in lined jets. At higher Reynolds number, little visibly remainsof the stru tured patterns seen just above onset, whi h are repla ed by ompli atedtime-dependent motions, at least for small G. Computations for larger G and Reynoldsnumber showed steady states with a strong vis ous mode and weak layering, even under onditions where the vis ous mode was expe ted to be relatively strongly damped. Weattribute this dynami s to the visous mode be oming sub riti al and growing nonlinearly.Simulations with in lined jets reveal a di�erent dynami s still, involving wavy, time-dependent patterns. For fear of opening Pandora's box, we end the exploration here.
16 N.J. Balmforth and Y.-N. Young
x
z
Figure 9. Double instability in the domain [2�; 32�℄ The two panels show the �nal distribution(at t = 1219) of (a) (x; z; t) and (b) b(x; z; t) as greys ales on the (x; z)�plane. The �nal panelsshow the horizontal averages of and b on the (t; z)�plane. The parameter values: Re = 17,G = 1:705 and � = 0:5. The times of the snapshots are 1:04 � 103, 1:12 � 103, and 1:22 � 103,respe tively (from left to right).5. Con luding remarksWe have explored instability of for ed, strati�ed shear ow. The tale is largely one oftwo modes of instability that develop over long, verti al s ales. The �rst is a \vis ous"instability whi h is related to the lassi al instability of Kolmogorov ow. The intro-du tion of strati� ation an suppress this instability, but an also make it sub riti al.The se ond instability is a \ ondu tive one" whi h operates by reating a large-s ale,negative thermal di�usion. This instability arises with stronger strati� ation and reatesprominent stair ases in the buoyan y �eld. The steps of the stair ase have their ownnonlinear dynami s, and often show \ oarsening" { the merger of steps as the dividinginterfa es ollide, with the subsequent lengthening of the s ale of the overall pattern. Thevis ous instability appears able to dominate and suppress the ondu tive mode shouldboth operate, and reates little by way of layering in the density �eld. Moreover, thereis a tenden y for that mode to be ome sub riti al and dominate by growing nonlinearly,even under onditions where it is linearly stable. Thus, although it seems possible thatthe ondu tive instability is the laminar pre ursor of the uid phenomenon that generatesturbulent stair ases, it is essential to remove the vis ous mode. In our previous investi-gation (Balmforth & Young 2002), the on�guration and parameter range are su h that
Strati�ed Kolmogorov ow II 17
x
z
Figure 10. Condu tivity instability with two paired jets. The times of the snapshots are 390,670, and 1:44� 103, respe tively (from left to right).only vis ous instability is present, and thus almost no eviden e of layering is observed inthat ase.Instabilities atalyzed by vis osity or thermal ondu tion are known in a variety ofother ontexts, notably di�erentially rotating annular olumns (Yih 1961) and stellarinteriors (Goldrei h & S hubert 1967), and baro lini vorti es (M Intyre 1970). One istempted to rationalize the urrent instabilities as analogies of those examples. However,there are di�eren es in the urrent theory. For example, a general on lusion rea hedin these other problems is that ondu tive instability is typi al of low Prandtl number(� < 1), and vis ous instability of high Prandtl number (� > 1), whi h is not found here.A knowledgements: This work was supported by the National S ien e Foundation(Collaborations in Mathemati al Geos ien es, grant ATM0222109).Appendix A. Formulae for weakly nonlinear theoryA.1. Weak ondu tive instabilityThe order �3 solution is: 3 = 13Re72 �G13 + Rep6BZ�BZZ sin 2x+ Re5976(10p6 + 3Re)BZZZ os 3x
18 N.J. Balmforth and Y.-N. Young
x
z
Figure 11. Combined instability with two paired jets. The times of the snapshots are 1080,1890, and 1:432 � 104, respe tively (from left to right).�"13 �G13 +p6ReBz�3 + Re72 (Re + 30p6)BZZZ + p69 �2�G13 +p6ReBZ�# osxand b3 = 139 �G13 + Rep6BZ�BZZ os 2x� 12988(135p6�Re)BZZZ sin 3x�" Re36 +r38!BZZZ � 13Re �G13 +p6ReBZ�3 � p69Re�2G1 � 2�2BZ# sinx:A.2. Weak vis ous instabilityAt order �2, we �nd 2xxxx+Re b2x = Re ( 0x 1xxZ+ 1x 0xxZ� 1Z 0xxx� 0Z 1xxx)�2 0xxZZ ; (A 1)G 2x �Re b2xx = �Re2 (b1x 0Z + b0x 1Z � b0Z 1x � b1Z 0x) + Re b0ZZ ; (A 2)whi h gives 2 = �2�G� 1 + �2G1 +G A2Z � �B1Z� Re2 osx(G+ 1)2 � �2GRe2 AZZ os 2x4(1 +G)2(16 +G) (A 3)
Strati�ed Kolmogorov ow II 19andb2 = ��B1Z � GRe (1� �G + � + �2)(G+ 1) A2Z� sinx(1 +G)2 + 2�2Re GAZZ sin 2x(1 +G)2(G+ 16) : (A 4)At order �3: 3 = Q1Re sinx2(1 +G)4 + �10G(G+ 4)(� + 1)(1 +G)(16 +G) AZAZZ �B1ZZ� �2Re3 sin 2x8(G+ 16)(1 +G)2� �2GRe3 (9 + �G+ 8�)48(81 +G)(G + 16)(1 +G)3AZZZ sin 3x; (A 5)b3 = Q2 osx(1 +G)4 ��G[G2 � 3G� 64� 20(4 + �)℄2(G+ 16)(1 +G) AZAZZ +B1ZZ� �2Re2 os 2x(G+ 16)(1 +G)2+ �2GRe2(G� 9�G� 72�)8(81 +G)(G+ 16)(1 +G)3AZZZ os 3x (A 6)withQ1 = �Re2 (1+G)(2��G+�)AZB1ZZ+�G�2(8� � �G+ 3)8(16 +G) � 3(1 +G)(1� �G)�AZZZ+Re2 (�2G2 � �3G� 2�2G� 3�G+ 1)A3Z (A 7)Q2 = �Re2 (1�G2)(1 + �)AZB1Z � �2Re2 G(1 + �)(1� 2�G+ �2)A3Z+�(1 +G)(4� 2�G+G+ �)� �2Re2 (�G� 8� + 3G)8(G+ 16) �GAZZZ (A 8)REFERENCESBalmforth, N. J., Llewellyn Smith, S. G. & Young, W. R. 1998 Dynami s of steps andlayers in a turbulent strati�ed uid. J. Fluid Me h. 400, 500{500.Balmforth, N. J. & Young, Y.-N. 2002 Strati�ed Kolmogorov ow. J. Fluid Me h. 450,131{167.Chapman, C. & Pro tor, M. 1980 Nonlinear Rayleigh-B�enard onve tion with poorly on-du ting boundaries. J. Fluid Me h. 101, 759{782.Drazin, P. G. 1978 On the instability of an internal gravity wave. Pro . Roy. So . London A356, 411{432.Gama, S., Vergassola, M. & Fris h, U. 1994 Negative eddy vis osity in isotropi ally for ed2-dimensional ow - linear and nonlinear dynami s. J. Fluid Me h. 260, 95{126.Goldrei h, P. & S hubert, G. 1967 Di�erential rotation in stars. Astrophys. J. 150, 571{587.Holford, J. & Linden, P. F. 1999 Turbulent mixing in a strati�ed uid. Dynam. Atmos.O eans 30, 173{198.Kurgansky, M. V. 1979 Hydrodynami instability of internal waves in the atmosphere. Izv.A ad. S i., USSR, Atmos. O eani Phys. 15, 707{710.Kurgansky, M. V. 1980 Instability of internal gravity waves propagating at small angles tothe verti al. Izv. A ad. S i., USSR, Atmos. O eani Phys. 16, 758{764.Lombard, P. N. & Riley, J. J. 1996 Instability and breakdown internal gravity waves. Phys.Fluids 8, 3271{3287.Lorenz, E. N. 1972 Barotropi instability of rossby wave motion. J. Atmos. S i. 29, 258{269.M Intyre, M. E. 1970 Di�usive destabilization of the baro lini ir ular vortex. Geophys. FluidDyn. 1, 19{57.Meshalkin, L. & Sinai, Y. 1961 Investigation of the stability of a stationary solution of asystem of equations for the plane movement of an in ompressible vis ous uid. Journal ofAppl. Math. Me h. 25, 1700.
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