Exact coherent structures in stably-stratified plane Couette flow

Under consideration for pulication in J. Fluid Mech. Exact coherent structures in staly-stratified plane Couette flow D. Olvera & R. R. Kerswell School of Mathematics, Bristol University, Bristol, BS8 TW, UK (Received 2 June 27) The existence of exact coherent structures in staly-stratified plane Couette flow (gravity perpendicular to the plates) is investigated over Reynolds-Richardson numer (Re- Ri ) space for a fluid of unit Prandtl numer (P r = ) using a comination of numerical and asymptotic techniques. Two states are repeatedly discovered using edge-tracking - EQ7 and EQ7- in the nomenclature of Gison & Brand (24) - and found to connect with 2-dimensional convective roll solutions when tracked to negative Ri (the Rayleigh- Benard prolem with shear). Both these states and Nagata s (99) original exact solution feel the presence of stale stratification when Ri = O(Re 2 ) or equivalently when the Rayleigh numer Ra := Ri Re 2 P r = O(). This is confirmed via a stratified extension of the Vortex-Wave-Interaction (VWI) theory of Hall & Sherwin (2). If the stratification is increased further, EQ7 is found to progressively spanwise and crossstream localise until a second regime is entered at Ri = O(Re 2/3 ). This corresponds to a stratified version of the Boundary Reduced Equation (BRE) regime of Deguchi, Hall & Walton (23). Increasing the stratification further, appears to lead to a third, ultimate regime where Ri = O() in which the flow fully localises in all three directions at the minimal Kolmogorov scale which then corresponds to the Osmidov scale. Implications for the laminar-turulent oundary in the (Re-Ri ) plane are riefly discussed.. Introduction Our understanding of transition to turulence in linearly-stale shear flows such as plane Couette flow (pcf) and pipe flow has experienced a considerale step forward over the last 25 years ased upon the realisation that such flows have many alternative unstale solutions such as equiliria, travelling waves and periodic states (Nagata 99; Waleffe 998; Faisst & Eckhardt 23; Wedin & Kerswell 24). These states, which start to populate phase space at some critical Reynolds numer and quickly proliferate as the Reynolds numer increases, eventually produce a sufficiently tangled structure of stale and unstale manifolds in phase space to act as a scaffold for the oserved complex dynamics (Kerswell 25; Eckhardt et al. 27; Gison and Cvitanović 2; Kawahara et al. 22). These states (variously referred to as simple invariant solutions or Exact Coherent Structures (ECS) ), can either e emedded in the laminar-turulent oundary (Wang et al. 27; Duguet et al. 28) or sit in the asin of attraction of the turulent state with some uried in the turulent attractor itself (Kerswell & Tutty 27; Gison et al. 28). As a consequence, properties of the ECS have something to say aout how transition is triggered (Viswanath & Cvitanović 29; Duguet et al. 2; Pringle et al. 22), the susequent transitional process (Itano & Toh 2; Skufca et al. 26; Schneider et al. 27; Melliovsky et al. 29) and features of the turulent state itself

2 D. Olvera & R. R. Kerswell (e.g. Kawahara & Kida (2); Hof et al. (24); Viswanath (27); Gison et al. (28); Chandler & Kerswell (23); Willis et al (23); Lucas & Kerswell (25)). The motivation for this study is to extend these investigations to staly-stratified shear flow which is the generic situation in environmental and geophysical flows (e.g. oceanic flows are almost always staly stratified - Thorpe (27)). While there has een previous work on unstaly-stratified shear flow - Rayleigh-Benard convection with imposed shear (Clever et al. 977; Clever & Busse 992), internally-heated shear flow (Generalis & Nagata 23) or natural convection with imposed shear Hall (22) - the only work in computing ECS for stale stratification is that of Clever and Busse (992,2) who estalished that Nagata s (99) now famous first solution in pcf could e continued ack to Rayleigh-Benard convection with shear. This lack of attention might well e ecause stale stratification is perceived as a universally stailizing influence although this is now appreciated as an oversimplification (Howard & Maslowe 973; Huppert 973; Davey & Reid 977) or ecause introducing stratification increases the dimension of parameter space from a very manageale (Re, the Reynolds numer) to a more daunting 3 (Re, Ri and P r where the ulk Richardson and Prandtl numers are defined elow). Mathematically, rather than a complication, stratification actually presents an opportunity to emed the now-well-studied prolem of pcf into a larger framework which can smoothly connect to the even-more-well-studied prolem of Rayleigh-Benard convection. Physically, another interesting aspect of adding further parameters to the prolem is the increased dimension of any laminar-oundary in parameter space. Ignoring the Prandtl numer (set to unity throughout this study) still leaves 2 parameters and then a - dimensional laminar-turulent oundary as opposed to the -dimensional situation in unstratified pcf (e.g. see figure of Deuseio et al. (25)). Understanding exactly how this oundary ehaves for large Re is an important open prolem in stratified turulence which has ovious implications for parametrising turulence in ocean, atmosphere and climate models. In particular, if Ri c (Re) defines the laminar-turulent oundary, does Ri c tend to zero or not as Re? Having an asymptotic regime where the flow is always transitional opens up interesting new opportunities to proe the complex dynamics involved. An ovious place to start addressing this question is to understand where in parameter space ECS exist since they are currently viewed as necessary precursors for turulent dynamics. This is the focus of the current study. The plan of the paper is as follows. Section 2 formulates the prolem of stratified plane Couette flow and discusses the numerical methods to e used. In 3., an initial investigation is descried which identifies where the laminar-turulent oundary is in the (Re, Ri ) plane for two typical (small ox) geometries. 3.2 then descries the results of edge-tracking in a variety of geometries aimed at uncovering new ECS. Two already known, aleit now stratified, ECS are repeatedly found - EQ7 and EQ7- in the nomenclature of Gison & Brand (24). In 3.3 these are continued around in (Re, Ri ) parameter space, including into negative Ri (unstale stratification), along with Nagata s (99) first solution in pcf. In 3.4 the ehaviour of EQ7 (which is most roust to stale stratification) is examined at large Re. A stratified version of the Vortex-Wave-Interaction asymptotics of Hall & Sherwin (2) is then discussed for very weak stratification. As the stratification is increased, a second regime is suggested y the numerical solutions which show EQ7 spanwise-localizing and amounts to a stratified version of the Boundary Reduced Equations of Deguchi, Hall & Walton (23). A third, ultimate regime is also discussed in which the ECS are fully localised. Section 4 provides a final discussion of the results and future directions for further research.

2. Formulation Exact coherent structures in staly-stratified plane Couette flow 3 2.. Stratified plane Couette flow We consider the usual plane Couette flow set-up of two (horizontal) parallel plates separated y a distance 2h with the top plate moving at Uˆx and the ottom plate moving at U ˆx. Dirichlet oundary conditions are imposed on the density field (Armenio & Sarkar 22; Garcia-Villala & del Alamo 2): the fluid density is ρ ρ at the top plate and ρ + ρ at the ottom plate (gravity g is normal to the plates and directed downwards from the top plate to the ottom plate). Using the Boussinesq approximation ( ρ ρ ), the governing equations can e non-dimensionalised using U, h and ρ to give u t + u u = p Ri ρ ŷ + Re 2 u, (2.) u =, (2.2) ρ t + u ρ = Re P r 2 ρ (2.3) where the ulk Richardson numer Ri, Reynolds numer Re, and the Prandtl numer P r (always set to in this study) are respectively defined as: Ri := ρ g h ρ U 2, U h Re := ν, P r := ν κ. (2.4) Here u = (u, v, w) is the velocity field, κ the thermal diffusivity, the total dimensional density is ρ + ρ ρ, p is the pressure and ν is the kinematic viscosity. The oundary conditions are then which admit the steady D solution u(x, ±, z, t)) = ± & ρ(x, ±, z, t) =. (2.5) u = y ˆx & ρ = y. (2.6) The (possily large) disturance fields away from this asic state, û(x, y, z, t) = u y ˆx, ˆρ(x, y, z, t) = ρ + y, (2.7) conveniently satisfy homogeneous oundary conditions at y = ±. Periodic oundary conditions are used in oth the (x) streamwise and (z) spanwise directions over wavelengths L x h and L z h so that the (non-dimensionalised) computational domain is L x 2 L z. In this geometry, the system has the following symmetries: shift (streamwise)- &-reflect (spanwise) S : (u, v, w, p, ρ)(x, y, z) (u, v, w, p, ρ)(x + 2 L x, y, 2 L z z), (2.8) a rotation of ±π aout the z-axis and spanwise reflection Ω : (u, v, w, p, ρ)(x, y, z) ( u, v, w, p, ρ)( x, y, z), (2.9) Z : (u, v, w, p, ρ)(x, y, z) (u, v, w, p, ρ)(x, y, z). (2.)

4 D. Olvera & R. R. Kerswell The kinetic energy of the disturance (per unit volume) is E k := û 2 dv = (u y ˆx) 2 dv (2.) 2V 2V V which is the sum of the kinetic energy associated with each velocity component, Ek u := û 2 dv, Ek v := ˆv 2 dv and Ek w := ŵ 2 dv. (2.2) 2V 2V 2V V V The potential energy of the disturance (per unit volume) is taken to e E p := Ri (ρ + y) 2 dv (2.3) 2V so that total energy (per unit volume) is V V E := E k + E p = (E u k + E v k + E w k ) + E p. (2.4) An important quantity used for characterising states in what follows is the average wall shear stress deviation away from that of the steady D solution (2.6) normalised y the wall shear stress of the steady D solution (2.6) defined as := L x L z Lx Lz u y dx dz = y= L x L z 2.2. Methods Lx Lz V û y dx dz. (2.5) y= The governing equations are solved numerically using a parallelized DNS code Dialo (Taylor 28 and http://www.damtp.cam.ac.uk/user/jrt5/files.html) which uses a third-order mixed Runge-Kutta-Wray/Crank-Nicolson timestepper. The horizontal directions are periodic and treated pseudospectrally, while a second-order finite-difference discretization is used in the cross-stream direction. The resolution used was typically 64 Fourier modes per 2π in x and z and 28 finite difference points in y. If needed, this resolution was douled to ensure numerical accuracy. Dialo was used for edge-tracking and to isolate ECS (simple solutions of the governing equations) y coupling it to a Newton-Raphson-GMRES algorithm (Viswanath 27). Tracing ECS over parameter space (varying Re, Ri, L x, L z ) was done y two complementary approaches: Dialo coupled to a Newton-Raphson-GMRES scheme and a direct Newton-Raphson solver of the governing equations discretized using spectral methods assuming steadiness in an appropriately chosen Galilean frame. The latter approach is much more efficient if the ECS has many spatial symmetries to optimise the discretization which was via Fourier modes in the (homogeneous) streamwise and spanwise directions and one or two spectral elements in the cross-stream direction. The two-spectral element approach was used to improve accuracy around the mid plane of the domain: details are given in Appendix A. 3. Results 3.. Laminar-Turulent Boundary To set the scene for this investigation, we first choose some specific geometries and identify the laminar-turulent oundary in (Re, Ri, P r) parameter space. In this paper P r = throughout to keep this study focussed although dependency with P r across the range

Exact coherent structures in staly-stratified plane Couette flow 5 Figure : Laminar-turulent oundary for two geometries 2π 2 2π (lue line through the crosses) and 2π 2 π (green line through the stars). There is linear instaility for Ri < 78/(6Re 2 P r) which is the well-known Rayleigh-Benard threshold (unchanged y shear - Kelly (977)). The energy staility limit is also marked as an (almost) vertical lack dashed line..7 (heated air) through 7 (heated water) to 7 (salt in water) is an interesting issue for future examination. A simple protocol was adopted to estimate the position of the laminar-turulent oundary for the two geometries 2π 2 2π and 2π 2 π and Re 5. At a given point in the parameter plane (Re, Ri, ), simulations were performed with random initial conditions with the point assigned to the turulent region if 5% or more of these runs remained turulent after a time h/u and otherwise to the relaminarisation region. Figure shows that the turulent part of parameter space extends to higher Ri at a given Re for the larger domain with the oundary extending as Re for some Ri (Re) (see Deuseio et al. (25) for a similar plot ut at higher Re and larger domain size). Also shown for completeness is the region Ri < (unstale stratification) where the prolem is equivalent to Rayleigh-Benard convection with imposed shear: see appendix B which shows how the usual Rayleigh numer Ra = Ri Re 2 P r and the neutral staility line is given y Ra = 78/6 (the extra factor of 6 ecause the oundary separation has een non-dimensionalised to 2 rather than ). Since the laminar-turulent oundary exists in a 2-dimensional parameter plane, it can e approached in two different ways starting from the turulent region: either decreasing Re at fixed Ri or increasing Ri at fixed Re. Work in unstratified flows indicates that the laminar-turulent oundary (or edge manifold if the turulence is not strictly an attractor - see Skufca et al. (26)) and the turulent attractor collide when turulence ceases to exist (e.g. figure 5, Schneider & Eckhardt 29). This type of ehaviour carries over to weakly stratified (small Ri ) shear flow here: the top plots in Figure 2 show the energy levels of the turulence and edge manifold coming together as Re is decreased towards the oundary. A similar convergence of energy levels also seems to occur for fixed (high) Re and increasing Ri (ottom plots of Figure 2), ut the dominant feature now is the presence of increasingly large and slower fluctuations in the turulent energy level

6 D. Olvera & R. R. Kerswell...8.8.6 E.6 E.4.4 (,.2) (,.5).2.2 5 52 54 56 58 Time 56 58 6 62 64 Time.5.4.5.4.3 E.2.3 E.2 (5,.3). (5,.73). 7 75 8 85 Time 7 75 8 85 Time Figure 2: Edge (red, lower, flatter lines) and turulent (lack, upper, jagged lines) kinetic energy levels either receding from the laminar-turulent oundary at fixed low Ri and Re increasing left-to-right (top) or from fixed Re and Ri decreasing left-to-right ottom (i.e. left figures are approaching laminar-turulent interface compared to right figures). Top left (Re, Ri ) = (4,.2); top right (Re, Ri ) = (,.2) red, (Re, Ri ) = (,.5) green; ottom left Re = 5 and Ri =.73 lack, Ri =.75 green dash-dot, Ri =.77 lue dashed, Ri =.8 lue, Ri =.85 lack dotted line; ottom right (Re, Ri ) = (5,.3). as the oundary is approached. The lines for Ri =.77 (lue dashed) and Ri =.8 (solid lue) in the ottom left plot of figure 2 show how a less-stratified flow can reach the energy levels of the turulent state yet immediately relaminarise while a more-stratified flow can stay turulent aleit with large fluctuations in energy. It seems that sustained turulence is lost in the latter case y the turulent attractor expanding to touch the edge manifold as opposed to moving as a whole towards it in the former case. 3.2. Identifying ECS: Edge Tracking The method of edge tracking was used to identify ECS (Itano & Toh 2; Skufca et al. 26) without any imposed symmetries unless explicitly stated. This method seeks to find attractors on the laminar-turulent oundary y a isection approach. The approach can e hit or miss - the attractor may turn out to e chaotic rather than simple (i.e. an equilirium, travelling wave or periodic state) - ut does have the advantage that if

Exact coherent structures in staly-stratified plane Couette flow 7..9.8 EQ7 (4π 2 2π).7.6 E.5.4.3.2. EQ7 (2π 2 2π) 8 EQ7 (2π 2 π) EQ7 (2π 2 2π) 7 4 2 3 4 5 6 7 8 9 Re Figure 3: Continuation in Re at Ri = starting at Re = for EQ7- in oxes 4π 2 2π (red dashed ) and 2π 2 2π (thin lack for oth, upper and lower disconnected ranches), and continuation for EQ7 in ox 2π 2 π (old green ). Points 4, 7 and 8 are marked in the continuation in Ri in figure 4. 8 x 3 7 6 2..8 E.6.4 5 4 5 E 4 3 2 2 3 8 7 4.2..2.3.4.5.6 Ri 9.5.5.5.5 2 2.5 3 Ri x 3 Figure 4: Continuation in Ri at Re = for solution EQ7- in oxes 4π 2 2π (red dashed ) and 2π 2 2π (old dark ), EQ7 in ox 2π 2 π (old green ) and 2D Rayleigh-Benard solution in ox 4π 2 2π (thin red ). Inset shows how solution EQ7 reaches larger values of stratification close to Ri.6. The curves shown have een traced y continuing from the lower and upper ranch solutions in the unstratified situation and cannot oviously e rought together - see 3.3 for more details. 3 6 5 an ECS is found then it is known to help organise the transition process and therefore e dynamically important. Crucially, the method also offers an uniased approach to finding ECS in contrast to simply taking known ECS at Ri = and continuing them into the region Ri >. Since adding stale stratification to a shear flow introduces the

8 D. Olvera & R. R. Kerswell.4 x 4.2 Ek w.8 E.6 E p.ek u.4 E k v.2 2 4 6 8 2 4 6 8 Ri x 3 Figure 5: Continuation in Ri at Re = for solution EQ7- in ox 4π 2 2π. Red dashed line in figure 4. The kinetic energy is splitted into its three components (E u, E v and E w ) additional phenomenon of internal gravity waves, it is possile that new forms of ECS could exist strictly for Ri >. These would have different underpinning dynamics to that in unstratified flows (known variously as the self-sustaining process (SSP) - Waleffe (995, 997) or Vortex-Wave-Interaction (VWI) - Hall & Smith (99); Hall & Sherwin (2)) and therefore would e of considerale interest. The investigation was started y reproducing Schneider et al. s (28) unstratified edge state calculation for (Re, Ri, L x, L z ) = (4,, 4π, 2π). Adding even a little stratification (Ri = 3 ) moved the system too close to the laminar-turulent oundary to make edge tracking still feasile (the separation etween the edge and the turulent energy levels ecomes too small) so Re was increased to. Here, the unstratified calculation recovers a variant of the edge state at Re = 4 (a memer of the Nagata (99) family of solutions) while taking (Re, Ri, L x, L z ) = (, 3, 4π, 2π) led to a spanwiselocalised equilirium with S and Ω symmetries. Increasing Ri aove 3 led to a chaotic edge state so the search was moved to a shorter domain (Re, Ri, L x, L z ) = (, 3, 2π, 2π) where edge-tracking again revealed a spanwise-localised equilirium. Branch-continuing this state ack to the 4π 2 2π ox recovered the spanwiselocalised state found previously. Continuing this state in the 2π 2 2π ox from Ri = 3 down to and then continuing Re similarly downwards indicated that this spanwise-localised state was the stratified version of EQ7- in the nomenclature of Gison & Brand (24): compare figure 3 with figure 4 of Gison & Brand (24) (note the ordinates are different and we have multiple curves for EQ7-). The results of continuing EQ7- around in Ri at fixed Re = for oth geometries is shown in figure 4. In the 4π 2 2π ox, EQ7- exists at Ri comparale to that at the laminar-turulent oundary (see figure ), while, in the 2π 2 2π ox, EQ7- only reaches to Ri = 2.87 3 efore turning ack. In the igger ox, figure 5 indicates that at the saddle node point (point of maximum Ri ) the potential energy of EQ7- is almost twice the kinetic energy of the cross-stream (vertical) velocity. At this point the

Exact coherent structures in staly-stratified plane Couette flow 9.5.5 Y Y.5.5 π 2π Z π 2π Z.5.5 Y Y.5.5 π 2π Z π 2π Z.5.5 Y Y.5.5 π 2π Z π 2π Z Figure 6: Contours of yz cross-sections of streamwise component of perturation velocity û (arrows indicate velocity field of (ˆv, ŵ) plane) of steady solution EQ7- at Re = in ox 2π 2 2π shown as lack circles in figure 4. Top left, pt 7 at Ri = ; top right, pt 8 at Ri = ; middle left, pt 9 at Ri = 3 ; middle right, pt at Ri = 2.88 3 ; ottom left pt at Ri =.22 3 ; ottom right, pt 2 at Ri = 9.9 4. All plots have 8 contours etween [-.3,.3], which are the minimum and maximum of most energetic state at pt (arrows rescaled as well). small streamwise rolls are eing suppressed y the potential energy penalty imposed y the stale stratification: see 3.4 for further discussion of this. There is also a difference in continuing EQ7- etween the two geometries at negative Ri where the stratification is unstale: the 4π 2 2π EQ7- solutions connect to a 2D solution ranch of Rayleigh-Benard rolls whereas the 2π 2 2π EQ7- solutions do not. Figures 6 and 7 indicate how the structure of EQ7- varies around these two solutions ranches. Figure 7 is particularly interesting as EQ7- starts out as spanwise gloal (top left and point on the 2D convective roll ranch in figure 4) and then spanwise localises y point 4 (in figure 4). It achieves this y the flanks of the structure gradually eing away as Ri is increased which is made clear in figure 8. As discussed in 4. of Gison & Brand (24), this decay, at least at small amplitudes distant from the core of the flow structure, can e understood y identifying the least spatially damped steady eigenfunction of the linear operator ased upon the asic state (typically this will e a 2D eigenfunction invariant in the streamwise direction here). One consequence of this is the expectation that the rate of spatial evanescence increases as Ri increases away

D. Olvera & R. R. Kerswell.5.5 Y Y.5.5 π 2π Z π 2π Z.5.5 Y Y.5.5 π 2π Z π 2π Z.5.5 Y Y.5.5 π 2π Z π 2π Z Figure 7: Contours of yz cross-sections of streamwise component of perturation velocity û (arrows indicate velocity field of (ˆv, ŵ) plane) of steady solution EQ7- at Re = in ox 4π 2 2π shown as red triangles in figure 4. Top left pt ; top right, pt 2; middle left pt 3; middle right, pt 4; ottom left pt 5; ottom right, pt 6. All plots have 8 contours etween [-.48,.48], which are the minimum and maximum of state at pt 7 (arrows rescaled as well). from the ifurcation point where the decay rate is zero. Since the spanwise domain is relatively small, we do not attempt any further analysis here ut a similar process is oserved when tracking a stratified version of the snake solution of Schneider et al. (2) and is analysed in Olvera & Kerswell (27). Further edge tracking was done in a domain where the width was halved - so (Re, Ri, L x, L z ) = (, 3, 2π, π) - which led to a gloal steady mode: see Figure 9. This solution, which possesses all three symmetries S, Ω and Z, is the stratified version of the hairpin vortex solution (HVS) of Itano & Generalis (29), EQ7 in the nomenclature of Gison et al. (29) and the mirror-symmetric solution of Deguchi & Hall (24, 25): hereafter we will refer to it as EQ7. The unstratified version has also een found efore y edge tracking at the parameter settings (Re, Ri, L x, L z ) = (,, 4.35π,.5π) (see figure 5 of Rain et al. (22)). Figure 4 shows that EQ7 exists significantly aove the laminar-turulent oundary: figure indicates that Ri.5 at the oundary for the 2π 2 π ox whereas EQ7 extends up to Ri.55. As oserved for EQ7-, the solution ranch for EQ7 turns ack to lower Ri when the potential energy of the state ecomes roughly twice the kinetic energy in the cross-stream (vertical) direction: see

y y y y Exact coherent structures in staly-stratified plane Couette flow.8.8.6.6.4.4.2.2.2.2.4.4.6.6.8.8 2 3 4 5 6 z 2 3 4 5 6 z.8.8.6.6.4.4.2.2.2.2.4.4.6.6.8.8 2 3 4 5 6 z 2 3 4 5 6 z Figure 8: Contours of total velocity u at levels u = {, ±.5} of steady solution EQ7- at Re = in ox 4π 2 2π shown as red triangles in figure 4. Top left, pt ; top right, pt 2; ottom left, pt 3; ottom right, pt 4. All states have een shifted in the spanwise direction y half the domain to highlight the gradual evanescence of the central domain..5.5.5 Y.5.5.5 π Z π Z π Z Figure 9: Contours of yz cross-sections of streamwise component of perturation velocity û (arrows indicate velocity field of (ˆv, ŵ) plane) of steady solution EQ7 at Re = in ox 2π 2 π shown as squares in figure 4. Left, pt 3 at Ri =.; middle, pt 4 at Ri =.53; right, pt 5 at Ri = upper ranch. All plots have 8 contours etween[-.2,.83], which are the minimum and maximum of state at pt 3 (arrows rescaled as well). figure. The EQ7 solution ranch shown also could not e closed which is investigated in 3.3. In the hunt for further ECS, edge-tracking was also performed in small domains with various cominations of symmetries imposed and extended (wider or longer) domains (typically with mirror symmetry Z imposed). However, only chaotic states were found eyond the ECS already identified. For example, edge-tracking in the streamwise-extended domain (Re, Ri, L x, L z ) = (, 3, 6π, 2π), with and without Z-symmetry imposed, lead to streamwise-localised chaotic states as in Schneider et al. (2). Some computations were also done for strong stratification (Ri =.) and Re =, in

2 D. Olvera & R. R. Kerswell x 3 3.5 3 2.5 E 2.5.5 E k w E k v E p.e k u w E k E p E k v.e k u..2.3.4.5.6 Ri Figure : Continuation in Ri at Re = for solution EQ7 in ox 2π 2 π (see inset of figure 4). The kinetic energy is splitted into its three components (E u, E v and E w ). Figure : Parameter space (with P r = ): the (Re, Ri ) plane. The lue line indicates the turulent laminar oundary for ox 2π 2 2π and the green line for ox 2π 2 π. The red line shows the area of existence for EQ7 and its likely continuation is discussed in 3.4.

Exact coherent structures in staly-stratified plane Couette flow 3..9.8.7.6 E.5.4.3.2....2.3.4.5 Ri Figure 2: Energy E verses Ri for Nagata s solution at Re = 3 (thin solid lue line) and Re = 4 (thick solid red line) together with EQ7 at Re = 4 (green dash-dot line) and 2D convective roll solutions (thin and thick dashed lines and the very thick line in the ottom left corner) for Ri <. The plot shows that Nagatas solution ranch is initially (at Re = 3) a loop which starts and finishes on a 2D convective roll ranch whereas, at Re = 4, this loop has roken to connect to the EQ7 solution ranch. The upper Nagata solution ranch arm then extends ack to convectively unstale flows. small domains ut again, only chaotic edge states were found even in really small domains like π 2 4π constrained y the imposition of the mirror symmetry Z. Since oth EQ7- and EQ7 could e continued ack to the unstratified limit, Ri =, (see Figure 4) and looked to have all the features associated with SSP/VWI (dominant streaks with secondary streamwise rolls and wave field), no purely-stratified ECS (ECS with no Ri = limit) were found. Such new ECS proaly only exist at much larger Ri than that found for Re = O(5) since the ratio of internal-gravity-wave timescale to advective timescale (h/u) is / Ri. This presumaly needs to e O() to e important which, as we argue elow, only occurs at large Re when the ECS ecomes fully localised. The fact that EQ7 exists eyond the laminar-turulent oundary for the Re examined (Re - see figure ) at least confirms the general consensus that ECS are a necessary precursor (now as Ri decreases at fixed Re) for turulence to e possile. In fact, the region where EQ7 exists ut turulence does not, possesses a ursting phenomenon - see figure - where certain initial conditions can give rise to large energy growth up to energy levels commensurate with the turulent state at lower Ri (see Olvera & Kerswell (27) for more details). Henceforth we focus on studying EQ7. Before considering the question of how large Ri can ecome and still have ECS, we first examine the stratification of Nagata s (99) first solution and how it relates to EQ7.

4 D. Olvera & R. R. Kerswell 3.3. EQ7 and Nagata s solution Nagata s (99) solution was the first finite amplitude state discovered in plane Couette flow and it has een known since the work of Clever & Busse (992) that this solution actually ifurcates off a 2D (single layer) roll solution in Rayleigh-Benard convection (negative Ri ). It has een found here that EQ7 also ifurcates off a 2D (doule layer) roll solution - see figure 2. Since Nagata s solution has one less symmetry than EQ7, one could expect that Nagata s solution ifurcates off EQ7 in a symmetry-reaking ifurcation. This is indeed the case at least at Re = 4: see figure 2 which also shows that Ri does not need to e increased very far efore Nagata s solution ceases to exist compared to EQ7. Figure further emphasizes this. Before considering EQ7 at large Re, we first return to an issue shown in figure 4. The inset of figure 4 shows two continuations of EQ7 at Re = in a ox 2π 2 π, one starting at the edge-tracked solution at Ri = 3 (the lower ranch) and the other starting from the upper ranch solution at Ri = (otained y continuing the lower ranch solution ack to Ri = and then continuing around the saddle node using the spanwise wavenumer). At lower Re, these two continuations meet (e.g. at Re = 4 as shown in figure 2) when varying Ri to produce a smooth connection etween unstratified upper and lower ranch solutions ut clearly do not at Re =. At Re = 5 - see figure 3 - the connection etween upper and lower unstratified solutions has roken. Tracing the new crossings of the Ri = line reveals what looks to e a previously-unknown ranch of inner solutions coloured red in figure 4 (left) at Re = 3 which ifurcates off a 3D state almost immediately after that ifurcates off a doule roll state in Rayleigh-Benard convection (Ri < ). A further lower ranch reak up occurs etween Re = 7 and Re = 85 as shown in figure 4 (right) which finally explains the complicated continuation curve at Re = in the inset to figure 4. Clearly, stale stratification has a strong influence on the states. We now turn our attention to understanding exactly why. 3.4. EQ7 Asymptotics To gain some understanding of how stratification affects the SSP/VWI states at large Re, we concentrate on understanding how the lower ranch EQ7 solutions ehave as Ri is increased from at large Re. Of particular interest is how the maximum Richardson numer for such states scales with Re. Figure 6 shows how the wall shear stress deviation varies as Ri increases for the lower ranch of EQ7. All curves show an initial drop in stress from the unstratified value, a minimum of stress at some finite Ri min and then two turning points where the curve reaches a local maximum in Ri := Ri max and then a local minimum in Ri. Four states at Re = 4, and various values of Ri (marked y lue dots in figure 6) are shown in figure 7 which indicates that EQ7 transits from a gloal flow state to one showing localisation in the cross-shear direction y and further localisation in the spanwise direction z. The first two states, which appear gloal, can e descried as eing in regime whereas the second two states, which are clearly localised, indicate a second and possily third regime. 3.4.. Regime Regime corresponds to the range of Ri where a stratified version of the VWI structure of Hall & Sherwin (2) is realised. The only prior work to consider stratification is Hall (22) which looked at the natural convection situation where the stratification is parallel to the shear: differences with our case here of stratification perpendicular to the shear will e highlighted elow. A first key oservation is that ecause EQ7 is steady and Ω-symmetric, the critical layer u = is always located on y = and therefore has

Exact coherent structures in staly-stratified plane Couette flow 5 u 5 4.5 4 3.5 3 2.5 2.5.5..2.3.4.5.6 Ri.5.5.5.5 2 2.5 3.5.5.5.5 2 2.5 3 Figure 3: Continuation in Ri of EQ7 at Re = 5 in a ox 2π 2 π. Dots on the Ri = axis indicate known upper and lower unstratified solutions. The upper right plot shows the state (contours of û and arrows for (ˆv, ŵ)) at the upper (red) dot (2 contour levels across [-.2,.2]) and the lower right plot shows the state at the lower (lue) dot (2 contour levels across [-.7,.7]). 5 5 4.5 4.5 4 4 3.5 3.5 3 2 3 u 2.5 2.5 5 3 u 2.5 2.5.5 6 4.5.8.6.4.2.2.4 Ri...2.3.4.5.6 Ri Figure 4: Left: EQ7 continuation in Ri at Re = 3 in a ox 2π 2 π. The inner (red) ranch connects to the dashed green 3D ranch of x-wavelength π just after it itself ifurcates off the lack 2D-roll ranch. The outer (lue) ranch ifurcates directly off the lack 2D-roll ranch. Typical resolution used (M, N, L) = (6, 25, 6) and plots of the various states marked y dots and numers are given in figure 5. Right: Lower ranch continuation for EQ7 in Ri for at Re = 7 (lue dashed line) in a ox 2π 2 π which is comparale with that at Re = 5 and Re = 85 (red solid line). This plot shows that the lower ranch loop at Re=7 has roken up into two disconnected pieces at Re = 85 ( this reak up happens in the interval ( 8, 85) ). Typical resolutions used to confirm this ehaviour were (M, N, L) = (2, 25, 6), (4, 4, 2) and (6, 3, 2).

6 D. Olvera & R. R. Kerswell.5.5.5.5.5.5 2 2.5 3.5.5 2 2.5 3.5.5.5.5.5.5 2 2.5 3.5.5 2 2.5 3.5.5.5.5.5.5 2 2.5 3.5.5 2 2.5 3 Figure 5: EQ7 flow states (contours of û and arrows for (ˆv, ŵ)) shown over a y-z plane at Re = 3. Upper left state shown in figure 4, upper right state 2, middle left state 3, middle right state 4, lower left state 5 and lower right state 6. There are 2 contours over [-.725,.725] (red-white) and max cross-flow speed is.296 across all plots to show relative strengths of the states. no curvature. To keep the discussion general, we roaden the direction of gravity g to cos θŷ sin θẑ and consider two cases: θ = is cross-stream gravity perpendicular to the oundaries and θ = π/2 is spanwise gravity aligned with the oundaries: in the geophysical literature, these limiting cases are usually referred to as vertical and horizontal stratified shear situations respectively. The governing equations in component

Exact coherent structures in staly-stratified plane Couette flow 7.7.6.5.4.3 2.2..2.4.6.8..2.4 Ri 3 8 6 4 2 Ri Figure 6: Left. The wall shear stress deviation as a function of Ri for the lower ranch of EQ7 at Re = (green solid), 2 (lue dashed), 5 (lack thick dash-dot) and, (cyan thick solid). All curves show an initial drop in stress from the unstratified value, a minimum of stress at some finite Ri min and two turning points where the curve reaches a local maximum in Ri max and then a local minimum in Ri. Right. The same data as on the left ut plotted on a log-log plot with extra lower ranch data (at 5, (red solid), 2, (green dashed), 3, (magenta solid), 4, (lue thick solid, 6, (lack thick solid) and 8, (red thick dashed) ) added. The velocity states corresponding to the 4 lue dots on the lue Re = 4, curve - at Ri 6, 5, the minium stress point Ri min = 4.5 4 and the local max point Ri max =.833 3 - are shown in figure 7. form for the total fields (recall from (2.7) that u = û + yˆx and ρ = ˆρ y) are u t + uu x + vu y + wu z + p x = Re 2 u, (3.) v t + uv x + vv y + wv z + p y = Re 2 v Ri ρ cos θ, (3.2) w t + uw x + vw y + ww z + p z = Re 2 w Ri ρ sin θ, (3.3) ρ t + uρ x + vρ y + wρ z = ReP r 2 ρ, (3.4) u x + v y + w z =. (3.5) Note also that the effect of the transformations S, Ω and Z detailed in (2.8), (2.9) and (2.) is for the (original) θ = case where the transformations experienced y v and ρ match: for θ = π/2 the transformation properties of ρ have to e changed to match those of w and for general θ there are no such symmetries). Following Hall & Sherwin

8 D. Olvera & R. R. Kerswell.5.5.5.5.5.5 2 2.5 3.5.5 2 2.5 3.5.5.5.5.5.5 2 2.5 3.5.5 2 2.5 3 Figure 7: EQ7 flow states (contours of û and arrows for (ˆv, ŵ)) shown over a y-z plane at Re = 4,. The states correspond to the lue dots in figure 6 (right) with the order upper left, upper right, lower left (the stress minimum), lower right corresponding to increasing Ri. The contour levels for the streamwise velocity perturation û = u y (the arrows indicate vŷ+wẑ) are as follows: upper left, contours etween ±6.73 2 (arrow max. 3 ); upper right, contours etween ±3.8 2 (arrow max. 3 ) ; lower left, contours etween ±.8 2 (arrow max.8 3 ); and lower right, contours etween ±2.3 2 (arrow max 2.5 3 ). The dashed lack lines on each plot correspond to the y level of the corresponding x z slices shown in figure 8. (2), we look for steady solutions away from the critical layer of the form u = ū(y, z) +... +δre /3 (U(y, z)e iαx + c.c) +..., v = Re v(y, z) +... +δre /3 (V (y, z)e iαx + c.c) +..., w = Re w(y, z) +... +δre /3 (W (y, z)e iαx + c.c) +..., ρ = ρ(y, z) +... +δre /3 (R(y, z)e iαx + c.c) +..., p = Re 2 p(y, z) +... +δre /3 (P (y, z)e iαx + c.c) +... where c.c. denotes complex conjugate and all components of the wave field (U, V, W, R, P ) are assumed of equal magnitude consistent with spatial scales eing similar in every direction. The small parameter Re /3 is the usual scaling for a critical layer where advection y the mean flow is alanced y viscous terms (e.g. see Hall & Sherwin (2)), and δ is the wave amplitude in the critical layer to e determined later. Sustituting these expansions into equations (3.)-(3.4) gives, to leading order in Re, for the x-independent

Exact coherent structures in staly-stratified plane Couette flow 9 6 6 5 5 4 4 3 3 2 2.5.5 2 2.5 3.5.5 2 2.5 3 6 6 5 5 4 4 3 3 2 2.5.5 2 2.5 3.5.5 2 2.5 3 Figure 8: EQ7 flow states shown over a x-z plane at Re = 4, and the y levels shown in figure 7 (y =.2,.3,.7 and.5 respectively). The contour levels for the streamwise velocity perturation u y are as follows: upper left, contours across [.42,.6]; upper right, contours across [.42,.33] ; lower left, contours across [.669,.66]; and lower right, contours across [.26,.239]. 6 6 5 5 4 4 3 3 2 2.5.5 2 2.5 3.5.5 2 2.5 3 Figure 9: The density perturation ρ + y for the 3rd and 4th EQ7 flow states shown in figures 7 and 8 over a x-z plane at y =.7 (left) and.5 (right). The contour levels are as follows: left, contours across [.67,.692]; and right, contours across [.24,.22].

2 D. Olvera & R. R. Kerswell part (the roll and streak equations) and for the x-dependent (wave) part vū y + wū z = ū yy + ū zz, (3.6) v v y + w v z + p y = v yy + v zz Re 2 Ri ρ cos θ, (3.7) v w y + w w z + p z = w yy + w zz Re 2 Ri ρ sin θ, (3.8) v ρ y + w ρ z = P r ( ρ yy + ρ zz ), (3.9) v y + w z = (3.) iαūu + V ū y + W ū z + iαp =, (3.) iαūv + P y = Ri R cos θ, (3.2) iαūw + P z = Ri R sin θ, (3.3) iαūr + V ρ y + W ρ z =, (3.4) iαu + V y + W z = (3.5) which is just the inviscid linearised Navier-Stokes aout the streak field with critical layer at ū(y, z) =. In the x-independent equations (3.6)-(3.), the leading nonlinear terms due to the wave field are O(δ 2 Re /3 ) smaller in the ū and ρ equations and O(δ 2 Re 4/3 ) smaller in the v and w equations. Therefore, providing δ = o(re 2/3 ), the rolls and streaks are unforced in the interior and so, if they are not to slowly dissipate, must e forced y the critical layer through matching conditions across it - this is the essence of the Vortex-Wave-Interaction theory of (Hall & Sherwin 2). Also at this point it is clear that stratification first affects the roll equations when Ri /Re 2 as opposed to Ri /Re for stratification (gravity) aligned with the shear direction (Hall 22) where the right hand side of (3.6) would e ū yy + ū zz ReRi ρ). This, of course, is ecause the (streamwise) rolls are O(/Re) smaller than the streaks and therefore more easily affected. In the critical layer around ū(y, z) = (here, conveniently, just the plane y = ), assuming p does not vanish at the critical layer y = and ū(y, z) has a simple zero at y =, equations (3.)-(3.4) indicate that U, V, W /y and R /y 2 as y. Since y = O(Re /3 ) in the critical layer, the U and W fields are O(Re /3 ) larger in the critical layer than outside and R is O(Re 2/3 ) larger. The V field cannot e similarly O(Re /3 ) larger due to incompressiility so remains O(δRe /3 ) and therefore matches to a higher order outer component on the outside. This thinking motivates the expansions inside the critical layer of u = Re /3 Y dū dy (, z) + δ2 Re /3 ū c (Y, z) +... + δ( U(Y, z)e iαx + c.c) +..., v = Re 4/3 Y d v dy (, z) + δ2 v c (Y, z) +... + δre /3 ( V(Y, z)e iαx + c.c) +..., w = Re w(, z) + δ 2 Re /3 w c (Y, z) +... + δ( W(Y, z)e iαx + c.c) +..., ρ = ρ(, z) + δ 2 Re 2/3 ρ c (Y, z) +... + δre /3 ( R(Y, z)e iαx + c.c) +..., p = Re 2 p(, z) + δ 2 Re 2/3 p c (Y, z) +... + δre /3 ( P(Y, z)e iαx + c.c) +... where Y := Re /3 y. For θ =, ū(, z) = v(, z) = ρ(, z) = (y the Ω symmetry) whereas for θ = π/2 ρ(, z) is not zero, so the expansions for u and v reflect this while

Exact coherent structures in staly-stratified plane Couette flow 2 that for ρ is kept general. The critical layer wave equations are to leading order iαy dū dū (, z) U + V dy dy (, z) + WY d2 ū dydz (, z) + iαp = U Y Y, (3.6) iαy dū dy (, z) V + Re2/3 P Y = V Y Y ReRi R cos θ, (3.7) iαy dū dy (, z) W + P z = W Y Y Re 2/3 Ri R sin θ, (3.8) iαy dū dy (, z) R + W ρ(, z) z = /P r R Y Y, (3.9) iα U + V Y + W z = (3.2) assuming δ = o(re /3 ) so that there is no nonlinear feedack. The scalings for the x- independent critical layer variales (indicated with an overar and superscript c ) follow y ensuring that the leading nonlinear interaction of the wave is alanced in the various components of the equations. Specifically, the dominant alances are (iα U U + V U Y + W U Z + c.c.) = ū c Y Y, (3.2) (iα U V + V V Y + W V Z + c.c.) + p c Y = v c Y Y ReRi ρ c cos θ, (3.22) (iα U W + V W Y + W W Z + c.c.) = w c Y Y Re 2/3 Ri ρ c sin θ, (3.23) (iα U R + V R Y + W R z + c.c.) = P r ρc Y Y, (3.24) v c Y + w c Z =. (3.25) Beyond the addition of stratification, the difference here from the analysis of Hall & Sherwin (2) is the lack of critical layer curvature. The presence of such curvature means that a pressure correction p c must exist of O(δ 2 Re /3 ) (so p c y = O(δ 2 ) ) to supply the necessary centripetal force to produce the curvature of the spanwise flow component along the critical layer (see equation (2.22) of Hall & Sherwin (2)). With no curvature, the pressure correction is O(Re /3 ) smaller so there is no longer a pressure jump across the critical layer and the dominant alance is more complicated ( see (3.22) ). This difference, however, has no effect on the key momentum alance along the critical layer which sets the size of δ. Here, regardless of whether p c is O(δ 2 Re /3 ) or O(δ 2 Re 2/3 ) as here, this pressure is sudominant to the other terms shown in (3.23) which is equivalent to equation (2.2) in Hall & Sherwin (2). Now the key realisation is that there must e some jump across the critical layer to energise the rolls otherwise they would decay. This jump occurs first for the tangential rather than normal flow component as δ increases from and can take a numer of forms. The jump could e in the tangential velocity component w itself which would require w c = O(Re ) or in the st normal derivative w c / y which would require w c = O(Re 4/3 ) or even in the 2nd normal derivative 2 w c / y 2 which would require w c = O(Re 5/3 ). The corresponding scalings for δ would e Re 2/3, Re 5/6 and Re respectively with the middle scaling discussed y Hall & Sherwin (2) as VWI (presumaly a jump in the 2nd derivative is too weak to offset the secular damping of the rolls). There is no jump in the density across the critical layer as ρ c is an O(Re ) smaller than the outer field which is also the situation in the natural convection situation (stratification parallel to the plate flow direction) (Hall 22). It is now possile to see that the VWI process first feels the presence of either horizontal or vertical stratification when Ri = O(Re 2 ) in the roll equations (3.7)-(3.8) of the interior away from the critical layer. In the critical layer, stratification needs to e

22 D. Olvera & R. R. Kerswell 2 8 6 / Re 2/3 4 2 8 6 4 2 2 4 6 Re 2 Ri 2.5.5 2 2.5 3 Re 2/3 Ri Figure 2: Left. Regime : wall shear stress deviation normalised y (its value at Ri = ) verses Re 2 Ri. Right. Regime 2: compensated wall stress Re 2/3 verses Re 2/3 Ri (colours as in figure 6). The dot on each curve marks the position of Ri min while Ri max corresponds to the rightmost turning point on each curve. much stronger to affect matters, specifically Ri = O(Re ) for vertical stratification - see (3.7) and (3.22) - and Ri = O(Re 2/3 ) for horizontal stratification - see (3.8) and (3.23). Physically, stratification inhiits motions directed against gravity imposing a potential energy penalty. For oth horizontal and vertical stratification, the most immediate impact of this is on the weak O(Re ) rolls and the scaling Ri = O(Re 2 ) is precisely when their kinetic energy ecomes comparale to the potential energy which is O(Ri ). Figure 2 (left) confirms this scaling as Ri increases from zero which is just a statement that the Rayleigh numer, Ra := Ri Re 2 P r, is O(). This scaling was proposed y Eaves & Caulfield (25) y heuristic arguments and has also een independently found recently y Deguchi (27). 3.4.2. Regime 2 When Ri O(Re 2 ) regime must give way to another regime. Figures 7 and 8 indicate that û and ˆρ localize in the spanwise and cross-stream directions yet stay gloal in the streamwise direction. This adjustment can e captured y the rescaling ( y, z ) ( y, z )/ɛ and (ū, v, w, ρ) (ɛū, v/ɛ, w/ɛ, ɛ ρ) (3.26) where ɛ := (Re 2 Ri ) /4 for Ri Re 2 indicates the scale of localisation in oth cross-stream and spanwise directions. This preserves the form of the equations for the roll, streak and streamwise-averaged density fields yet accommodates the enlarged uoyancy term in the roll equations (either (3.7) or (3.8)). The rescaling for ū and ρ comes from the fact that the streaks and ρ only now extend over a cross-stream distance of O(ɛ) of the underlying applied shear and density fields. A similar scaling was discussed in Blackurn et al. (23) for unstratified flows when considering the large spanwise wavenumer limit of the SSP/VWI process. Here, the rescaling is driven y increasing Ri ut the overall effect is the same except there is no localisation in the streamwise direction. This rescaling accommodates increasing stratification until the spanwise localisation of the streamwise-independent fields approaches the critical layer scaling of Re /3 i.e. when ɛ = O(Re /3 ) Ri = O(Re 2/3 ). (3.27)

Exact coherent structures in staly-stratified plane Couette flow 23.8.8.6.6.4 u Re /3.4.2 u Re /3.2.2.2.4.6.4.8.6 2 3 4 5 6 x 4 3 2 2 3 4 y Re /3 2.8.6 u Re /3.4.2.2.4 & Ri max Ri min 3.6.8 3 2 2 (z z ) Re /3 c 3 4 5 Re Figure 2: Selected profiles through the streamwise velocity perturation at Ri max for Re/, =, 2, 4, 6 and 8. Upper left. Re /3 u(x, y c (Re/), z c ) verses x where y c () =.675, y c (2) =.575, y c (4) =.5, y c (6) =.4 and y c (8) =.35 are the approximate levels of the streak core shown, for example, in figure 7 (lower right) for Re = 4,. Upper right. Re /3 u(, y, z c ) verses Re /3 y where z c.83 is the spanwise position of the (left) streak core. Lower left. Re /3 u(, y c (Re), z) verses Re /3 (z z c ) (colours as in figure 6). Lower right. Ri min (Re) (lue line with dots) where reaches a minimum as Ri increases from zero and Ri max (Re) (red line with open squares) where the Ri curve turns around verses Re. The dashed lack guideline indicates a scaling of Re 2/3. At this point, the whole perturation velocity and density fields are confined to the critical layer which is actually a region localised in oth cross-stream and spanwise directions. In this, viscosity is important and the rolls and waves are indistinquishale requiring a new asymptotic description. The Boundary Reduced Equations (BREs) of Deguchi, Hall & Walton (23) and Deguchi & Hall (25) then seem the only viale rescaling to reflect this new structure. In these, a new spanwise variale Z := Re /3 (z z c ) (where z c is the centre of the localised structure) is defined along with the cross-shear critical layer variale Y := Re /3 y in regime. Concentrating on the vertical stratification situation

24 D. Olvera & R. R. Kerswell and essentially following Deguchi & Hall (25), we write u = Re /3 [ Y + U(x, Y, Z) ] + Re 2/3 U (x, Y ) + Re d u Y +... (3.28) v = Re 2/3 V (x, Y, Z) + Re V (x, Y ) +... (3.29) w = Re 2/3 W (x, Y, Z) +... (3.3) p = Re 4/3 [ 2 Ri Re 2/3 Y 2 + P (x, Y, Z) ] + Re 5/3 P (x, Y ) +... (3.3) ρ = Re /3 [ Y + R(x, Y, Z) ] + Re 2/3 R (x, Y ) + Re d ρ Y +... (3.32) The rationale ehind this expansion is the realisation that the leading fields (U, V, W, P, R) are concentrated over a reduced scale O(Re /3 ) in z so spanwise-averaging produces fields O(Re /3 ) smaller, that is since y = Re /3 Y and u = y + Re 2/3 [ U (x, Y ) + d u y ] +..., (3.33) v = Re V (x, Y ) +..., (3.34) w = (y symmetry), (3.35) p = 2 Ri y 2 + Re 5/3 P (x, Y ) +..., (3.36) ρ = y + Re 2/3 [ R (x, Y ) + d ρ y ] +..., (3.37) (U, V, W, P, R) = (,,,, ) where ( ) := Lz ( ) dz. (3.38) L z These scalings give the leading alance (Y + U)U x + V ( + U Y ) + W U Z = U Y Y + U ZZ, (3.39) (Y + U)V x + V V Y + W V Z + P Y = V Y Y + V ZZ Ri Re 2/3 R, (3.4) (Y + U)W x + V W Y + W W Z + P Z = W Y Y + W ZZ, (3.4) U x + V Y + W Z =, (3.42) UR x + V ( + R Y ) + W R Z = P r (R Y Y + R ZZ ) (3.43) to e solved with periodic oundary conditions in x and asymptotically decaying oundary conditions that (U, V, W, R) (,,, ) as Y & Z ± since no other possiility with vanishing spanwise average can connect to linear profiles in an unforced exterior. The spanwise-averaged prolem is (U 2 ) x + (UV ) Y = U,Y Y, (3.44) (UV ) x + (V 2 ) Y = V,Y Y P,Y Ri Re 2/3 R, (3.45) = U,x + V,Y, (3.46) (UR) x + (V R) Y = P r R,Y Y. (3.47) The prolem (3.44)-(3.47) is then really just one in U and R which are each computed y two quadratures of (3.44) and (3.47) respectively (V is then found y one quadrature of U,x and P follows from (3.45) ). In fact, all the computed solutions indicate that U = R (recall P r = ) - for example see figure 9 - and hence U = R. The spanwise-averaged prolem then oils down to solving (3.44) which requires (U 2 ) x integrated twice and (UV ) integrated once with respect to Y. Assuming that U and