emergence and secondary instability of ekman …dubos/pub/tdcbpd08.pdfemergence and secondary...

Emergence and Secondary Instability of Ekman Layer Rolls

T. DUBOS

IPSL/Laboratoire de Météorologie Dynamique, École Polytechnique, Palaiseau, France

C. BARTHLOTT

Institut für Meteorologie und Klimaforschung, Universität Karlsruhe/Forschungszentrum Karlsruhe, Karlsruhe, Germany

P. DROBINSKI

IPSL/Service d’Aéronomie, École Polytechnique, Palaiseau, France

(Manuscript received 27 June 2007, in final form 26 November 2007)

ABSTRACT

The authors revisit the idealized scenario by which long-lived rolls are believed to emerge in the neutralplanetary boundary layer, that is, through the saturation of the shear instability of the neutrally stratifiedEkman flow.

First, the nonlinear stages of the primary instability are studied, using a constant turbulent viscosity withReynolds numbers up to 1000. Two-dimensional equilibrated rolls are found to exist, as predicted earlierbased on a weakly nonlinear expansion. However, the flow may not saturate into those equilibrated rolls ifthe turbulent Reynolds number is too high. Second, a linear stability analysis of these equilibrated rolls isperformed, which finds that they are subject to a three-dimensional instability. The growth rate of the mostunstable mode is comparable to the growth rate of the primary instability; the selected horizontal lengthscale is about 4 times shorter. The unstable mode draws its energy by interacting with both across-roll andalong-roll shear, the latter interaction being stronger.

The latitude and the direction of the geostrophic wind affect the dynamics through the horizontalcomponent of the Coriolis vector; their influence is investigated in both studies. At Reynolds numberssufficiently higher than the threshold of the primary instability, the saturated rolls depend negligibly onlatitude and wind direction. However, the growth rate of the secondary instability depends substantially onlatitude and wind direction over the range of Reynolds numbers considered.

1. Introduction

Organized eddies are frequently observed in theplanetary boundary layer (PBL) in convective or near-neutral conditions. In the strongly convective case, notaddressed in this work, convective instability is themain mechanism driving the formation and character-istics of the eddies, which take the form of buoyantplumes whose shape does not distinguish a preferredhorizontal orientation. In near-neutral PBLs, eddies inthe form of rolls spanning the depth of the PBL andpersisting for hours or days are often observed (Etlingand Brown 1993; Drobinski et al. 1998; Young et al.

2002). In this case, the main potential source of insta-bility is the shear. This hypothesis has been confirmedfor the Ekman spiral flow, an exact solution of the in-compressible Navier–Stokes equations in the presenceof rotation and a rigid boundary.

The Ekman flow is useful as a prototype flow foridealized dynamical studies of the PBL. In such studies,the Reynolds number Re is understood as a turbulentReynolds number. Lilly established that the neutrallystratified Ekman flow is unstable for sufficiently highReynolds numbers (Lilly 1966). Because of nonlinearinteractions after the initial stage where the linear ap-proximation is valid, a linearly unstable flow mayevolve according to various scenarios: it may bifurcateto a new equilibrium, to periodic or multiply periodicbehavior, or directly to chaos. The Rayleigh–Benardinstability, for instance, is known to saturate into con-

Corresponding author address: T. Dubos, IPSL/LMD, ÉcolePolytechnique, 91128 Palaiseau, France.E-mail: [email protected]

2326 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 65

DOI: 10.1175/2007JAS2550.1

© 2008 American Meteorological Society

JAS2550

trarotating rolls, with no preferred orientation. Con-cerning the evolution of the instability of the neutralEkman flow in the nonlinear regime near the criticalReynolds number, severely truncated models or weaklynonlinear amplitude expansions have been used to pre-dict the saturation of the instability into a finite ampli-tude roll traveling at a constant speed in a modifiedmean flow (Brown 1970; Foster 1996). Despite somevariability, the observed rolls form at angles compatiblewith such theories (Etling and Brown 1993; Young et al.2002). Shear instability is also invoked to explain therolls observed more recently in hurricane boundary lay-ers (Foster 2005), although inertial instability may alsoplay a key role in the emergence of certain roll struc-tures (Nolan 2005). The atmospheric boundary layer isnever truly neutral, and the observed rolls tend to formin slightly to moderately convective conditions. Theemergence of the observed rolls is therefore usuallyinterpreted as the outcome of a mixed convective–shearinstability, with convection reinforcing rolls whose hori-zontal scale and orientation are selected by the shearinstability.

Values of the turbulent Reynolds number consideredtypical of the PBL can reach Re � 500 (Foster 1997),which is much above the threshold of instability. Im-portant characteristics of the nonlinear development ofthe instability remain unexplored in this regime. In-deed, inconclusive results concerning the Ekman flowhave been obtained by direct numerical simulations(DNS), large-eddy simulations (LES), and semianalyticdevelopments. Equilibrated rolls formed in two-dimensional LES (Mason and Sykes 1980) and DNS. Inthe DNS they formed at Re � 150 but then becameunsteady at Re � 400 (Faller and Kaylor 1966; Cole-man et al. 1990). However, using high-order nonlinearamplitude expansions, Foster found that steady rollsshould exist at least up to Re� 500 (Foster 1996, 2005).These results raise the question of whether (and howquickly) the instability actually saturates as a functionof the Reynolds number. Furthermore, in three-dimensional DNS, small perturbations lead to thebreakdown of time-periodic rolls, and rolls were absentfrom the turbulent fields at all latitudes and wind di-rections (Coleman et al. 1990). In three-dimensionalLES simulations, the same absence of longitudinal rollswas observed, and only short-lived, short-scale, near-ground streaky structures were found (Mason andThomson 1987; Moeng and Sullivan 1994; Lin et al.1996; Drobinski and Foster 2003; Drobinski et al. 2007).This strongly suggests that the equilibrated rolls areunstable with respect to infinitesimal three-dimensionalperturbations, in the same way as, for instance, Kelvin–Helmholtz billows generated in free-shear layers.

Finally, latitude enters the problem through the hori-zontal component of the Coriolis vector, whose magni-tude is inversely proportional to the Reynolds number.The horizontal component of the Coriolis vector in-creases or reduces the growth rate of primary instabilitydepending on its direction with respect to the geo-strophic wind, and this influence diminishes as theReynolds number increases (Leibovich and Lele 1985).However, the turbulent statistics and the coherentstructures still feel this influence at high Reynolds num-bers (Coleman et al. 1990; Esau 2003). Hence, despiteits smallness, the coupling between the different veloc-ity components induced by the horizontal part of theCoriolis vector may have a significant overall effect.

A number of questions remain open or incompletelyanswered, and it is the purpose of the present work toanswer them. They include:

• Do steady rolls exist for all Reynolds numbers, and isthis flow pattern reached in a reasonable time by theperturbed Ekman flow?

• Are those rolls unstable to three-dimensional pertur-bations, and with what preferred spatial and temporalscales?

• What is the influence of the horizontal component ofthe Coriolis vector on these phenomena?

We expose these investigations in the following man-ner. In section 2 we investigate the saturation of theprimary instability. We present evidence that steadytraveling rolls exist for all the considered Reynoldsnumbers above the threshold of primary instability butsuggest that this state is not necessarily reached in ashort time. Relevant Reynolds numbers range from�200 to �500, and we consider Reynolds numbers upto Re � 1000. Section 3 is devoted to the secondarystability study. The saturated rolls are found unstable tothree-dimensional perturbations above a critical Rey-nolds number Re2� 300. The instability grows near thehyperbolic stagnation point between two rolls. We payspecial attention to the effect of horizontal componentof the Coriolis vector on the secondary instability andfind that its growth rate depends substantially on lati-tude and the direction of the geostrophic wind. Theresults are discussed in section 4, and conclusions andperspectives are given in section 5. The numericalmethods and parameters are described in an appendix.

2. Saturation of the primary instability

a. Position of the problem

We start with dimensional equations, then adimen-sionalize the problem in the classical manner (Lilly

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1966). The representations of time, space, velocity, vor-ticity, and pressure appear with hats (t, x, u, �, P) in thedimensional formulation and lose their hats in the adi-mensionalized problem. The flow dynamics are drivenby the incompressible Navier–Stokes equations in a ro-tating reference frame, with a constant turbulent vis-cosity K, as shown here:

divu � 0, �1�

�u

�t� �� f� � u � ��P

��

u · u2 � � k�2u, �2�

where t is the time; the gradient is relative to the posi-tion x in three-dimensional space; u is the flow velocity,where � � curl u is its vorticity; and P is the deviationof the pressure from its hydrostatic value PH(z) � P0 ��gz, with z being the vertical coordinate, g the accel-eration of gravity, and � the constant fluid density. Theboundary conditions are u(z � 0) � 0 and u(z � ) �ug, where ug is the geostrophic wind vector far aboveground. At latitude , the Coriolis vector is given by

f � fzez � fNeN, where �3�

fz � 2�0 sin� and fN � 2�0 cos�, �4�

ez is the upward vertical unit vector, and eN is the unitvector pointing to the north. We scale velocities by thegeostrophic wind G � �ug � and lengths by the laminarboundary layer depth � ��2k/fz; hence, we scale timeby the advective time scale ad � �/G; that is,

u � u�G, P � P��G2, �5�

x � x��, z � z ��, �6�

t � t��ad, and � � �ad�. �7�

The resulting adimensional equations are divu � 0 and

�� 2

Re�ez � cotg�eN�� u

� ��P �u · u

2 � � 1Re

�2u ��u�t

, �8�

where Re � G�/K is the Reynolds number and thegradient is relative to the adimensional position x. Theadimensional Coriolis parameter is 1/Ro � 2/Re, andthe inertial oscillations have a period of 2�Ro � �Re.

Values typical of a midlatitude neutral atmosphericboundary layer are a Coriolis parameter f � 10�4 s�1

and a geostrophic wind G � 10 m � s�1. The boundarylayer depth is 3–5 times the Ekman length scale �, de-pending on the precise definition, hence typically � �200–500 m. The resulting advective time scale is

roughly ad � � /G � 1 min; thus, the model is mean-ingful for time scales of the order of a few hundreds of ad (a few hours). Furthermore, with these values of Gand � we get for the turbulent viscosity K � f�2/2 � 4� 20 m2� s�1 and Re� 200–1000. This rough estimateof the relevant range of Reynolds numbers can be re-fined. A scaling argument (R. Foster 2007, personalcommunication) indicates that Re� 17 C�1/2

d , where Cd

is the drag coefficient. Over a smooth surface like theocean, small values of Cd (Fairall et al. 1996) lead to amaximum turbulent Reynolds number about 500. Nev-ertheless, we shall consider turbulent Reynolds num-bers up to 1000 to gain a slightly broader view of Reyn-olds-number effects.

Equation (8) admits the well-known Ekman station-ary solution,

u0�z� � U0�z�eE � V0�z�eN, �9�

U0�z� � cos � e�z cos� � z�, and �10�

V0�z� � sin � e�z sin� � z�, �11�

where the angle � (between the vector pointing to theeast eE and the geostrophic wind) is arbitrary (Leibo-vich and Lele 1985). Here, the global parameters of theproblem are the turbulent Reynolds number Re, thelatitude , and the angle �.

Solving the linear stability problem consists of look-ing for eigenmodes of the rotating Navier–Stokes dy-namics linearized around the basic flow u0(z). Becauseof horizontal homogeneity, such eigenmodes vary hori-zontally like exp(ik1 · x) for some horizontal wave vec-tor k1. For given k1, Re, , and �, a few discrete eigen-modes, growing like e� , are usually found. We use thenotation �1(k1; Re, , �) for the largest growth rate.Next, for a given Re, , and �, the growth rate �1

reaches a maximum �1(Re, , �) at a certain k1(Re, ,�). This selects the preferred horizontal length scale l�2�/k1, which emerges when the basic flow is perturbed.

At latitude � 90°, Lilly (1966) shows that unstablemodes appear when the Reynolds number reaches thecritical value Rec � 54. The corresponding instability isviscous and labeled as the type II instability after Faller(1963). A second branch corresponding to an inflec-tion-point instability emerges at Re � 113 and domi-nates at high Reynolds numbers. This is the type I in-stability. In the sequel, we consider the nonlinear satu-ration of the most unstable eigenmode belonging to thetype I branch. Figure 1 reiterates the known fact thatthe primary instability is substantially influenced by thehorizontal Coriolis vector near the onset of instability


(Leibovich and Lele 1985). Specifically, the Ekmanflow due to a westerly geostrophic wind (stars) is lessunstable at � 45° than it would be at the pole(circles), the Ekman flow due to an easterly geostrophicwind (oblique crosses) is more unstable, and the flowsdue to northerly (triangles) and southerly (straightcrosses) wind have almost unchanged stability. How-ever, this effect diminishes as the Reynolds numberincreases.

Calculations are made easier by choosing the x axisalong k1. Rolls then have their axis along the y direc-tion. We follow the convention used in Foster (1997)where the geostrophic wind is �G(sin �ex � cos �ey), �being the angle between the rolls and the geostrophicwind. The eastward- and northward-pointing unit vec-tors are then eE � �sin(� � �)ex � cos(� � �)ey andeN � cos(� � �)ex � sin(� � �)ey, respectively. Theangles and the system of axes are summarized in Fig. 2.The velocity along x would be called streamwise in thecontext of parallel flow instability, but we prefer to callit across-roll in this paper because the geostrophic flowis almost aligned with y. For the same reason, the ve-locity along y is called axial, being along the roll axis.

Flows considered in this section are two-dimensionaland invariant by translations along the roll axis y. Tostudy their dynamics we shall solve two related prob-lems:

(i) the initial-value problem of finding u(x, z, t) givenan initial condition u(x, z, t� 0)� u0(z)� u�0(x, z),where u�0(x, z) is a small initial perturbation to theEkman flow, and using a smooth perturbation u�0(x,z) with maximum magnitude 10�5 (in adimensionalunits, or 10�5 G in dimensional units); and

(ii) the steady problem of finding a pattern u1(x, z) oftraveling rolls, for instance, such that u(x, z, t) �u1(x � ct, z) is a solution of (8) for a certain non-linear phase speed c(Re, , �).

We now investigate how the saturation of an unstableperturbation takes place at various Reynolds numbers.

b. Transient phase toward saturation

The transient phase, during which a small initial per-turbation grows before possibly settling down in theform of a traveling roll, presents fairly different char-acteristics as a function of the Reynolds number. Wepresent results of numerical simulations performed atlatitude 90° for Re � 150, Re � 500, and Re � 1000.The axes are oriented as in Fig. 2 with the x axis ori-ented along the wave vector k1(Re) corresponding tothe most unstable normal mode. During the simulationwe compute the instantaneous phase velocity c(t) de-fined by

c � ��xu · a�u��xu · �xu�

, �12�

FIG. 2. Top view of the axis system. There is an angle � betweenthe east and the geostrophic wind and an angle � between thegeostrophic wind and the y axis (roll axis).

FIG. 1. (a) Maximum growth rate �1 of the type I primary instability as a function of Re at latitude � 90°(circles) and at latitude � 45°. (b) Selected horizontal wave vector k1.

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where a(u) is the left-hand side of (8) and the brackets� � indicate a spatial integral. This definition is such thatthe instationarity �u/�t � c�u/�x in the reference frametraveling at speed c will be minimal in a least squaressense. In Fig. 3 we plot this phase speed as a function ofthe adimensional time t for the three Reynolds numbersmentioned above.

All three cases start with a linear stage. During thefirst part of this linear stage, the most unstable eigen-mode grows like exp(�1t) and progressively emerges.This takes a few 1 � 1/�1 � 40–100 (not displayed).After this stage where the most unstable mode is se-lected, the phase velocity takes the value c1(Re) asso-ciated with the linear eigenmode (see Table 1). Theplateau c(t)� c1(Re) ends when nonlinear effects comeinto play, after a few more 1, that is, at t � 1500 � 10 1

at Re � 150, and t � 500 � 10 1 at Re � 500.At Re � 150, the linear stage is followed by a short

transition before c(t) settles to a slightly different, time-independent value. Inspection of the flow itself con-firms that traveling rolls have formed and do not evolvesignificantly any more. At Re � 500, the saturationtoward a constant value is delayed by slowly decayingoscillations. Rescaling the time scale by the inertial pe-riod �Re shows that these oscillations are inertial andthat such oscillations are also present at Re � 150 withvery small amplitude. Inspection of the flow revealsthat traveling rolls have in fact formed but that an in-

ertial wave is present at high altitudes, modulating theoverall phase speed. This inertial wave reaches a maxi-mum amplitude at an altitude that increases with timefrom about z � 6 at t � 3000 to z � 40 at t � 105, withthis maximum amplitude decaying with time. Noticethat this essentially means that the inertial wave es-capes from the domain because the Ekman model losesits validity long before z � 40. Apart from the presenceof the inertial wave, the flow does not significantly de-viate from its asymptotic pattern after t � 3000 � 100 1,as does c(t) in Fig. 3. Finally, at Re � 1000, no satura-tion toward a flow pattern traveling at a constant phasespeed takes place. Instead, the phase speed varies er-ratically between �0.3 and 0.1, and the flow alsoevolves in an apparently chaotic manner. This behaviordoes not preclude the existence of a steady pattern atthis Reynolds number. Such a pattern may exist and beunstable to two-dimensional perturbations.

We illustrate the three phases (linear, nonlinear tran-sient, and relaxation) at Re � 500 with Fig. 4, whichpresents isocontours of total streamfunction and totalaxial velocity at times t � 500, t � 700, and t � 2000. Attime t � 500, the most unstable mode has grown suffi-ciently for the total flow to be modified. The formingroll lifts air up, forming a tip in axial velocity near x �0. At t � 700, this tip has evolved into a long and highfilament of low axial vorticity, rolling up into the strong,fully nonlinear roll emphasized by the closed stream-

FIG. 3. Evolution of the phase speed c(t) during the nonlinear saturation of the unstablemode at latitude � 90° and Re � 150, 500, and 1000.


lines. Later, the roll weakens and the axial vorticityfilament equilibrates at a lower altitude. The inertialwave is not visible on these plots. This evolution isaltered at Re � 150 and Re � 1000 (not shown). AtRe � 150, the strong transient (during which a filamentof low axial velocity is ejected at a high altitude) doesnot occur. Conversely, at Re � 1000, the ejected fila-ment is thinner and rolls into a long spiral, but the rollkeeps changing its shape as time passes.

To summarize, the nonlinear evolution of the un-stable normal mode leads to saturation into a travelingroll pattern at Re � 150 and Re � 500 but not at Re �1000. However, while this saturation takes only a few 1 � 1/�1 at Re � 150, it needs a much longer time�100 1 at Re � 500. Furthermore, complete saturationat Re � 500 is reached only after an inertial wave hasdecayed, which is an even slower process.

c. Structure of the equilibrated rolls

If a steady flow pattern exists at Re � 1000 and isunstable to 2D perturbations, it is unreachable by atemporal integration. Hence, we now search exactlyequilibrated flows for 80 values of the Reynolds num-ber in the range 113 Re 1000. As explained in theappendix, to solve this nonlinear problem we use New-ton’s iteration, which requires a first guess not too farfrom the exact solution. At Re � 500, we use the flowreached at t � 105 in the previously described simula-tion as a first guess. Because it has almost reached sta-tionarity, apart from an inertial wave of very small am-plitude, a few Newton iterations are enough to find aflow u1 satisfying �u1/�t � c�u1/�x � 0 to machine pre-cision. Figures 5 and 6 display this flow. The cross-rollflow (u, w) is conveniently described through a stream-function � such that w � �x� and u � ��z�. The rollaxis is perpendicular to the figures.

We first follow the common practice of separatingthe flow u1(x, z) into its horizontal mean u1(z) and thedeviation from this mean u�1 � u1 � u1 (Fig. 5). Thestreamfunction �� (Fig. 5a) then shows two counterro-tating cells, resembling those of the linearly unstablemode (Lilly 1966). The asymmetry between the twocells is, however, the manifestation of nonlinearities.

The difference between the mean flow u1 (Fig. 5b, solidline) and the Ekman profile (Fig. 5b, dashed line) alsoresults from nonlinear feedback of the velocity fluctua-tions onto the mean flow. This mean-flow modification,however, is small—no more than 10% over the range ofReynolds numbers considered. Indeed, the significanceof the rolls does not reside so much in their impact onthe mean flow as in the momentum fluxes they induce(Brown 1970; Morrison et al. 2005; Foster 2005). Thepattern of axial velocity fluctuations (Fig. 5c), with apositive anomaly in the downdraft around x � �6 anda negative anomaly in the updraft near x � 1, is con-sistent with axial momentum fluxes ��w� � 0 actingagainst the axial geostrophic wind �1(z � ) � 0 (d).

Because the flow trajectories are obtained from thestreamlines of the total flow, computed in the travelingreference frame where it is steady, it is also instructivein the present case to display the total flow instead ofjust the perturbation. This is what we do in Fig. 6.Closed streamlines (Fig. 6a) now indicate the presenceof a single vortex per spatial period. The periodic pat-tern is therefore one of counterclockwise corotatingvortices, consistent with the direction of the geo-strophic wind, which blows here from right to left (Fig.5a). Of course the corotating pattern results from thefact that we plot the total flow (i.e., horizontal meanplus fluctuations), and the difference between the tworepresentations is a matter of convention. The vortex isfairly broad, with a core located around x � �2 in Fig.6. Under this broad vortex, a secondary, intensifiedboundary layer is formed. In between, the intersectionof two streamlines between two consecutive vorticesnear x � 3 and x � �10 indicates a hyperbolic stagna-tion point. In the absence of the Coriolis force, the axialvelocity would be simply advected by the flow in the (x,z) plane and diffused by viscosity. This explains theejection of low axial velocity (in absolute value) be-tween two consecutive rolls, visible near x � 1 (Fig. 6b).

d. Dependence on Reynolds number and latitude

The steady rolls at Re � 500 also offer a good firstguess to the steady problem at slightly smaller or largerReynolds numbers. This way, we solve the steady prob-

TABLE 1. Columns list growth rate �1, phase speed c1, orientation �1 relative to the geostrophic wind, and wave vector k1 of the mostunstable mode of primary instability; nonlinear phase speed c of the saturated rolls; growth rate �2 and wave vector k2 of the mostunstable mode of secondary instability. Negative phase speeds are in the same direction as the geostrophic wind.

�1 � 10�3 c1 � 10�3 �1 � 0.1° k1 � 10�2 c � 10�3 �2 � 10�3 k2 � 0.1

Re � 150 0.008 0.061 9.7 0.54 �0.050 — —Re � 500 0.024 �0.021 16.8 0.50 0.038 0.020 1.9Re � 1000 0.028 �0.039 18.4 0.50 0.048 0.039 2.7

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FIG. 4. Transient evolution at Re� 500. Isocontours of the total streamfunction andaxial velocity are displayed at t � 500 (end of linear phase), t � 700 (rapid nonlineartransient), and t � 2000 (quasi-equilibrated roll).


lem for Reynolds numbers gradually increasing or de-creasing from Re � 500, covering the range 113

Re 1000. At each Reynolds number we search for alocal maximum of �1 (k1; Re); hence follow the type Ibranch of primary instability. The lower boundaryRe � 113 corresponds to the Reynolds number belowwhich this branch is stable. We find that steady rollsexist all over this range. While the structure of the rolls

is quite dependent on the Reynolds number close to thecritical value Re � 113, this dependence is weak atReynolds number larger than about 200 and the flowpattern depicted in Fig. 6 is typical, with gradients be-coming sharper as the Reynolds number increases. Theamplitude of the roll, as measured by the integratedkinetic energy of u1 � u0, decreases smoothly towardzero as the Reynolds number approaches from above

FIG. 6. Spatial structure of the equilibrated rolls for Re� 500 and latitude � 90°. The rollsare perpendicular to the x axis. Two spatial periods 2�/ �k1 � � 12 are presented. (a) Thecontour levels �1 � �0.1 of the streamfunction �1 cross at a hyperbolic stagnation point nearx � 3. (b) Axial velocity �1.

FIG. 5. Horizontal mean u1(z) and deviation u1 � u1 of the equilibrated roll flow for Re � 500 and latitude � 90°. �he roll axisis perpendicular to the x–z plane. (a) Streamfunction �1 � �1 and (b) mean across-roll wind u1(z) (solid line) compared to the Ekmanprofile u0(z) (dashed); (c) axial velocity �1� �1 and (d) mean axial wind �1(z) (solid line) compared to the Ekman profile �0(z) (dashed).

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the threshold of primary instability Re � 113 (notshown). This characterizes the bifurcation as supercriti-cal, as predicted by Foster (1996), based on weakly non-linear expansions. The bifurcation of the other branchat Re� 55 is known to be supercritical (e.g., smooth) aswell (Iooss et al. 1978; Haeusser and Leibovich 2003).

We reproduced the previous steps at latitude � 45°for wind blowing from the four cardinal directions. Asfar as the stationary rolls are concerned, very little de-pendence was observed. The vertical velocity field,which is zero in the Ekman spiral flow, is a good indi-cation of the roll intensity. We found that the differencebetween the vertical velocities at � 45° and � 90°is less than 2% in a root-mean-square sense.

3. Secondary instability of saturated rolls

In the previous section we have found saturated rolls,which are stationary solutions of the rotating Navier–Stokes Eqs. (8) in a Galilean referential traveling witha nonlinear phase speed c. Such columnar vortices aresubject to different families of secondary instability.The Kelvin–Helmholtz rolls that result from the satu-ration of a free-shear instability are known to suffer inneutral or stable stratification from elliptic and hyper-bolic instabilities (Peltier and Caulfield 2003). UnlikeKelvin–Helmholtz rolls, saturated Ekman rolls haveaxial velocity, like swirling jets. This produces shearalong the roll axis and may result in another type ofinstability. The presence of a rigid boundary could af-fect the stability properties as well.

a. Position of the problem

The secondary stability analysis is done in the framemoving at the nonlinear phase speed c(Re, , �). Weconsider the growth or decay of an infinitesimal, three-dimensional perturbation u2(x, y, z, t) evolving accord-ing to rotating Navier–Stokes dynamics (8) linearizedabout the basic flow u1(x, z; Re, , �) � cex. Becausethis linear initial-value problem is homogeneous in they direction, its solution can be written as a superposi-tion of modes of the form

u2 � u�2�x, z, t�ei�y � c.c., �13�

where c.c. indicates the complex conjugate. Further-more, because the basic flow is periodic in the x direc-tion, the mode u�2(x, z, t) obeys a differential equationwhose coefficients are 2�/k periodic in the x direction.In this case, Floquet theory indicates that it is enough toconsider perturbations of the form

u�2 � u2�x, z, t�ei x, �14�

u2 � u2�x, z, t�ei� x��y� � c.c., �15�

where the pattern u2 is now 2�/k1-periodic in the xdirection, and � and � are the cross-roll and axial com-ponents, respectively, of the secondary wavenumberk2 � �ex � �ey (see Peltier and Caulfield 2003 fordetails). For a given � and �, the linear operator L(k2;Re, �, �) driving the evolution of u2, that is,

�

�tu2 � L�k2; Re, �, � · u2, �16�

is obtained by plugging u � u1 � cex � u2(x, z, t)expik2 · x � c.c. into (8) and neglecting the quadraticterms in u2. The linear stability of the rolls with respectto three-dimensional perturbations then depends onthe eigenvalues �2 � i�2 of L. Notice that �2(�) ��2(� � k1), which provides a consistency check of thenumerical method. After this check is passed, it is suf-ficient to study the range 0 � k1. If for some k2 thegrowth rate �2(k2; Re, , �) � 0, then the roll is un-stable with respect to perturbations with wave vectork2. This typically occurs for Reynolds numbers largerthan a critical threshold Re2(, �). As for the primaryinstability, the wave vector k2 (Re, , �) that maximizes�2 is selected together with the corresponding eigen-mode.

As an illustration of the structure of the secondaryperturbation, we display in Fig. 7 a horizontal crosssection of one component of a fabricated field u2. Twoperiods are displayed along the x axis and one along they axis. On the left-hand side, � � 0 and phase planes�x � �y � const are parallel to the (x, z) plane. Con-versely, on the right-hand side � 0 and the phaseplanes are oblique.

b. Secondary stability at latitude � 90°

We first consider the latitude � 90°. The most un-stable mode is found for an axial wavenumber � � 2.For 1.5 � 2.5, �2 depends very weakly on thecross-roll wavenumber � ∈ [0, k1], with relative varia-tions of about 1% (not shown). Hence, in the sequel weset � � 0, � � k2 and consider a purely axial wavenum-ber k2 � k2ey. In Fig. 8 we display the growth rate �2 asa function of k2 for Re � 500. Wave numbers 1.1

k2 2.8 are unstable, with a maximum growth rate�2 � 0.021 reached for k2 � 1.9. Hence, the time scalefor the secondary instability is comparable to that of theprimary instability, but the selected length scale isabout four times shorter because k1 � 0.5 (Table 1).

We display in Fig. 9 the y-averaged amplitude � u2(x,z) �2 of the most unstable mode at Re � 500 and lati-tude � 90°. The mode is very localized along the


streamline emerging from the hyperbolic stagnationpoint near x � 3 and is therefore suggestive of a hyper-bolic instability (Godeferd et al. 2001). Notice, how-ever, that this region also experiences a strong axialshear (i.e., the axial velocity has strong gradients in thisregion). Hence, another possibility would be an axialshear instability. Figure 10 displays the three-dimen-sional structure of the most unstable mode by means oftwo planar cross sections of the velocity field. Bothcross sections contain the line (x � 9.2, z � 1.7) wherethe amplitude of the most unstable mode is largest. Afull period is displayed in the y direction, but only theportion of the flow domain where the unstable modehas significant amplitude is displayed in the x and z

directions. The (a) horizontal cross section reveals anoblique pattern of alternating updrafts and downdrafts(contours), with horizontal motion mostly parallel tothis oblique pattern. Accordingly, the (b) vertical cross-section shows horizontal motion with alternating modu-lations of u and � in phase. The updrafts are in quadra-ture with the horizontal motion, and the maximum ver-tical velocity is about 40% of the maximum horizontalvelocity.

To get some insight into the instability mechanism,we form the integral budget of kinetic energy for thethree-dimensional perturbation u2� u�ex� ��ey� w�ez

to the basic flow u1�Uex� Vey�Wez. It follows from(8) that

�t�12

u2 · u2� � D � �sAX� � �sCR�, �17�

where

sAX � ��u��xV � ��w��zV �18�

and sCR � �u�w��xW � �zU� � u�u��xU

� w�w��zW, �19�

where the brackets � � indicate a spatial integral and Drepresents the viscous dissipation of kinetic energy. Theright-hand side is decomposed into the production ofkinetic energy by axial shear sAX and by cross-roll shearsCR (i.e., shear due to the overturning circulation in theplane perpendicular to the roll). We display in Fig. 11the spatial distributions of sAX and sCR, averaged alongthe roll axis y. The units are arbitrary but identical forboth. Both production terms are positive everywhere;they peak in the region where the unstable mode islocalized, near x � 3 and y � 1.5. The integrated pro-duction by axial shear �sAX� is about 6 times larger thanfor cross-roll shear �sCR�. Hence the unstable modedraws its energy primarily, but not only, through itsinteraction with the axial shear of the roll.

c. Dependence on Reynolds number

We now investigate the characteristics of the second-ary instability at latitude � 90° in the range 113

Re 1000. The growth rate �2 is plotted as a functionof the Reynolds number in Fig. 12a (circles; other sym-bols are for latitude � 45°, discussed later). The sec-ondary instability appears at critical Reynolds numberRe2 � 326, above which the growth rate increases regu-larly to reach �2 � 0.020 � 10�3 at Re � 500 and �2 �0.039 � 10�3 at Re � 1000. These values are close tothose of the primary instability; hence, both instabilitiesevolve on the same time scale. The selected axial wave-number k2 is plotted as a function of the Reynolds num-

FIG. 7. Example pattern of a secondary perturbation (cross sec-tion at z � const). Grayscale: u2, �2, or w2. Dashed line: a phaseplane �x � �y � const. In this example, k1 � 1, � � 2, and (a)� � 0 or (b) � � 1⁄2.

JULY 2008 D U B O S E T A L . 2335

ber in Fig. 12b (circles). Although for the primary in-stability the selected wavenumber slightly decreasesfrom k1 � 0.56 at Re � 113 to k1 � 0.50 at Re � 1000(Fig. 1), the secondary wave vector k2 increases con-tinuously with the Reynolds number from k2 � 1.6 �10�1 at Re � 326 to k2 � 2.7 � 10�1 at Re � 1000.Hence, the horizontal scale selected by the secondaryinstability is about four times smaller than the scaleselected by the primary instability.

d. Dependence on latitude

We next consider the latitude � 45° and wind blow-ing from the four cardinal directions. We first commentthe secondary stability curve �2(k2) at Reynolds num-ber Re � 500 (Fig. 8). All curves have their maximumnear the same axial wavenumber k2 � 1.9; hence, thehorizontal component of the Coriolis vector does not

affect the selected horizontal length scale. Further-more, for wind blowing from (triangles) or to (straightcrosses) the north, the growth rates are nearly identicalto their values at latitude � 90°. However, wind blow-ing from the west (stars) induces growth rates larger byabout one-third, and conversely wind blowing from theeast (oblique crosses) induces growth rates smaller byabout one-third, compared to the other three configu-rations. Hence, westerly winds have a destabilizing ef-fect on the stationary rolls, but easterly winds have astabilizing effect, as is also the case for the primaryinstability.

Finally we consider the Reynolds-number depen-dence of this effect (Fig. 12a). Its magnitude measuredby the difference �WE

2 � �EW2 decreases over the range

300 Re 1000. Because the Coriolis term is inverselyproportional to the Reynolds number, we expect the

FIG. 9. Amplitude � u2(x, z) � 2 (arbitrary units) of the most unstable mode at latitude �90° and Re � 500. Dotted curves: roll streamfunction as in Fig. 6.

FIG. 8. Growth rate �2 of the secondary instability as a function of the axial wavenumber k2,for a Reynolds number Re � 500 at latitude � 90° (circles) and 45°, for a geostrophic windalong the four cardinal directions.


difference �WE2 � �EW

2 to scale like 1/Re at sufficientlyhigh Reynolds numbers. Indeed, �WE

2 � �EW2 � 6.8/Re

provides a good fit in the range 600 Re 1000 (notshown). Hence, although the effect of latitude probablyvanishes asymptotically at higher Reynolds numbers,its magnitude over the range of Reynolds numbers rep-resentative of an atmospheric boundary layer is farfrom negligible compared to the reference growth rate�2(Re, � 90°). The effect on the selected horizontalwavelength, however, is negligible at all Reynolds num-bers except near the onset of secondary instability (Fig.12b).

4. Discussion

a. Saturation of the primary instability

The question of whether or not the flow saturates ina reasonable time is not a purely academic matter. In-deed, if the equilibrated rolls are the basis of param-eterizations of roll effects in the PBL, they should berepresentative of the actual flow (Foster and Brown1994; Foster 2005). Our results complete the scenario ofthe nonlinear saturation of the primary Ekman insta-bility, a work undertaken by several authors before.Faller and Kaylor (1966) produced the first numericalintegrations of the two-dimensional nonlinear initial-value problem. Similar integrations were performed byColeman et al. (1990) at Re � 150 and Re � 400.

The emission of an inertial wave during the nonlineartransient phase was reported by Faller and Kaylor(1966). Inertial oscillations are also excited in the three-dimensional simulations of Coleman et al. (1990). Re-garding the question of whether or not the flow satu-rates in a reasonable time (a few hours, hence a fewhundreds of advective times ad � �/G), our resultsshow that this emission is of little importance. Indeed,at low Reynolds numbers, saturation is reached quicklywith a very small inertial activity, while at high Rey-nolds numbers the erratic behavior of the flow happenson a time scale much smaller than the inertial period.Hence, only at intermediate Reynolds numbers (Re �500) could it possibly be of some importance. However,although the inertial wave decays very slowly in thatcase (Fig. 3), saturation is reached in practice long be-fore the wave vanishes. Indeed, the oscillation is con-fined to altitudes far above the rolls, and the rolls them-selves do not change their structure while the inertialwave fades out.

In high-Reynolds-number, three-dimensional simula-tions, either DNS or LES, near-ground streaks ratherthan rolls are usually observed (Mason and Thomson1987; Moeng and Sullivan 1994; Drobinski and Foster2003). We have found no signs of near-ground streaky

FIG. 10. Cross sections of the most unstable mode of secondaryinstability at latitude � 90° and Re � 500. The amplitude of themode is arbitrary. (a) Horizontal cross section of vertical compo-nent w (contours) and (u, �) components (arrows) at z � 1.7. (b)Vertical cross section of u component (contours) and (�, w) com-ponents (arrows) at x� 2.6. The two panels use the same scale forthe arrows but different grayscales. The secondary flow is periodicalong the y direction with wavelength 2�/k2 � �.

JULY 2008 D U B O S E T A L . 2337

structures in our higher-Re simulations. Notice, how-ever, that the flow geometry is severely constrained inour simulations because along-roll variations are notallowed. This is probably preventing streaks from de-veloping because they usually form with an angle withrespect to the geostrophic wind larger than the angleselected by the maximal primary instability growth rate.This feature is part of the design of our simulations andpermits us to focus on the emergence of the rolls.

For moderate Reynolds numbers, Faller and Kaylor(1966) observed a rapid saturation to a pattern of trav-eling rolls. At Re � 300 the saturation occurred after alarge-amplitude transient, and at Re � 600 it did notseem to occur at all. However, the duration of the in-

tegrations was limited to a few tens of advective timescales, and the results were inconclusive regarding thesaturation toward an equilibrium at Reynolds numbershigher than 600. Using a quasi-stationary approach,with a Reynolds number varying slowly in time, theymanaged to obtain equilibrated patterns up to Re �900 for particular values of the angle e between the rollsand the geostrophic wind. Foster (1996) obtainedequilibrated patterns for arbitrary Reynolds numbers,but the applicability of high-order weakly nonlinear ex-pansions far from the threshold of primary instability isnot guaranteed.

Our results show that equilibria indeed exist over thewhole range 113 Re 1000, thus bringing increased

FIG. 12. (a) Maximum growth rate �2 of the secondary instability as a function of Re at latitude � 90°(circles) and latitude � 45°. (b) Selected axial wavenumber k2.

FIG. 11. Production of energy by (a) axial shear sAX � ��(u��xV � w��zV ) and (b)cross-roll shear sCR � �u�w�(�xW � �zU ) � u�u��xU � w�w��zW for the most unstable modeat latitude � 90° and Re � 500. Both graphs use the same arbitrary unit of energy produc-tion. Dotted curves: isocontours of (a) roll axial velocity and (b) streamfunction, as in Fig. 6.


credibility to the use of such expansions. It would beinteresting to quantitatively assess the accuracy of theirpredictions. On the other hand, our long integrationsshow that these states may not be strong enough attrac-tors at high Reynolds numbers because saturation is notreached, nor even approached in this case.

b. Three-dimensional secondary instability

Two-dimensional vortices are generically subject tothree-dimensional instabilities. For vortices with noaxial flow, universal mechanisms are the elliptic, cen-trifugal, and hyperbolic instabilities (Godeferd et al.2001; Kerswell 2002). With a superimposed axial flow,axial shear provides an additional source of instability.Kelvin–Helmholtz billows belong to the first category,and the second category includes swirling jets. Bothmechanisms are potentially active in the present case,in addition to a rigid boundary and an intensifiedboundary layer below each roll vortex.

Notice that Pier (2004) obtains thresholds of second-ary instability of analogous flows over a rotating disk.However, the relevant criterion for the selection of theprimary wave vector in that problem is based on spa-tiotemporal and absolute instability, which leads to dif-ferent values of the critical Reynolds number and pri-mary wave vector k1(Rec) and to a different basic flow.

We find that the Ekman rolls are subject to a three-dimensional instability. Unlike many secondary stabil-ity studies (Peltier and Caulfield 2003), we do not re-strict ourselves to � � 0 from the start. The across-rollsecondary wavenumber � controls the phase differenceof the perturbation between two successive vortices(Fig. 7). When � � 0, phase planes �x � �y � const areparallel to the (x, z) plane and the perturbation isstrictly x periodic. When � 0, phase planes make anangle with the (x, z) plane; the perturbation is identicalat abscissas x and x � 2�/k1, except for a shift in the ydirection. If there is a strong maximum in the curve�2(�), this preferred value of � is selected, indicating aphase lock and a strong interaction between the per-turbations at x and x � 2�/k1. Here, the converse istrue: the growth rate does not depend on the cross-rollsecondary wavenumber �. This means that the pertur-bation may be shifted along the y axis by a (smooth)x-dependent amount with little effect on the growthrate. Thus, the instability is most probably local, and thefact that the basic flow is periodic is inessential.

Moreover, in the axial direction, a preferred wavevector � is selected. This means that a modulation ofthe wind speed should be observable along rolls. Forexample, Doppler lidar transects through near-horizontal planes (Drobinski et al. 1998) could revealthis modulation. Our results predict that the wave-

length for this modulation should typically be 3 to 4times shorter than the roll scale. Furthermore, thesemodulations are concentrated in the part of the rollwhere the vertical velocity is positive. Accordingly, pro-vided the roll does not break down into turbulence, wecan expect modulations of the intensity of the roll-induced updrafts. If these updrafts produce a cloudstreet, the along-roll modulations may reflect onto thecloud street, perhaps in the form of a pearl string.

Both cross-roll and axial shear transfer kinetic energyfrom the basic flow to the growing three-dimensionalperturbation. Kelvin–Helmholtz billows, for instance,have no axial shear; thus, their three-dimensional insta-bility is entirely due to cross-roll shear. By contrast, theproduction of energy by axial shear is dominant. Thestrong axial shear results from the ejection of low-axial-velocity fluid from the near-ground region into a fila-ment elongated between two consecutive rolls. Thisfeature is therefore associated with both the veeringand the presence of the rigid boundary. However, theenergetics do not provide a straightforward identifica-tion of the instability mechanism. For instance, in thecase of the primary inflection-point instability of theEkman flow, energetics show that the axial shear pro-duction of energy dominates cross-roll shear produc-tion. However, only the latter results from the inflec-tion-point instability because axial shear production isenabled by the buildup of axial velocity by the Coriolisforce (Lilly 1966; Foster 1997). Given the relative com-plexity and lack of symmetry of the equilibrated rollflow compared to, for instance, an axisymmetric swirl-ing jet or Kelvin–Helmholtz billows, it might be a dif-ficult task to label the secondary instability as resultingfrom one of the pure, well-identified instability mecha-nisms.

We do not study the nonlinear phase of the instabil-ity; however, instabilities in columnar vortices andswirling jets are known not to saturate and to lead di-rectly to the turbulent breakdown of the basic flow.This is probably the case here also, which would beconsistent with the absence of persistent rolls in three-dimensional direct numerical simulations (Coleman etal. 1990). Preliminary results of an ongoing study in-cluding the effect of stable stratification indicate thatthe roll follows a recurring cycle of emergence andbreakdown. In this case, the probability of observingthe roll could remain high despite its formal instability.More systematic work is required to provide firm an-swers.

c. Horizontal component of the Coriolis vector

The general mechanism by which the small Coriolisterm influences the integral characteristics of turbu-

JULY 2008 D U B O S E T A L . 2339

lence is indirect: because the Coriolis force does notwork, it does not enter the budget of turbulent kineticenergy. However, it does enter the dynamical equationdriving the buildup of the Reynolds stresses, which gov-ern the transfer of energy from the basic flow to theperturbation (Johnston et al. 1972). This mechanism issometimes referred to as the effect of “small strains”(Coleman et al. 1990). The turbulent statistics and thecoherent structures still feel this influence at high Rey-nolds numbers (Coleman et al. 1990; Esau 2003). In thepresent system, the horizontal Coriolis vector and itsdirection with respect to the geostrophic wind modifythe domain of primary instability (Leibovich and Lele1985), but this influence diminishes as the Reynoldsnumber increases and becomes very small at Reynoldsnumbers relevant for the PBL (200–500). Furthermore,the equilibrated rolls are also almost insensitive to thiseffect in the range of Reynolds numbers considered.

This is not the case, however, for the secondary in-stability. We find that the growth rate of secondaryinstability is substantially sensitive to the horizontal Co-riolis vector. Moreover, the critical Reynolds numberRe2 beyond which the rolls become unstable spans, as afunction of wind direction, the range 270 Re2

400—that is, most of the range relevant to the neutralPBL. Hence for otherwise identical conditions, rollsmay be stable when the geostrophic wind is easterly butunstable when the wind is westerly.

5. Conclusions

From the results discussed above, there may existperfectly equilibrated roll patterns with little resem-blance to the instantaneous flow for sufficiently highturbulent Reynolds numbers. Indeed, even with the im-posed constraint of a purely two-dimensional flow,equilibrated rolls are not always reached even at longtimes; and when three-dimensional perturbations areallowed, they destabilize the two-dimensional rolls. Onthe other hand, observations agree quantitatively withpredictions based on the weakly nonlinear, two-dimensional stationary theory (Young et al. 2002; Fos-ter 2005). The reasons for this success are uncertain. Itmight be the case that the flow pattern, although evolv-ing in a complicated way, remains close enough to theunstable, steady roll that the latter provides a robustestimate of the winds and fluxes; or perhaps the rollvortex may indeed break down but relaminarize after-ward, as does, for instance, a Karman vortex street sub-ject to elliptical instability (Stegner et al. 2005). Thevalidation of such scenarios is, however, probably be-yond the reach of the simple Ekman model used here.

Despite the obvious limitations of the Ekman flow in

view of atmospheric applications, it is a useful toymodel containing a few key ingredients: a rigid bound-ary, the Coriolis force, shear, and veering. These ingre-dients trigger the fairly complex mechanisms discussedabove, and are likely to do so in a realistic PBL as well.A fundamental ingredient not considered in this work isthe (stable or unstable) stratification, which has astrong direct influence on the primary instability, itsnonlinear saturation, and the secondary instability.Work is under way to fill this gap and its results willhopefully be communicated in a forthcoming publica-tion. Furthermore, if quantitative predictions are to bemade, efforts toward a more realistic PBL are needed.A first step is to replace the Ekman spiral with a real-istic profile matching a logarithmic layer near theground, as is done for instance by Foster (2005) or No-lan (2005). However, this will not cure the oversimpli-fied representation of the feedback of small turbulenteddies onto the large-scale flow as a constant eddy dif-fusivity, which is itself the cause of the unrealistic windprofile. Hence, a more consistent way of doing wouldbe to include a less oversimplified representation ofsubgrid stresses, turning the model into a stripped-down LES model. This would allow more direct com-parisons with atmospheric observations. Furthermore,modifications of the flow would induce modifications ofthe turbulent viscosity, an additional and interestingmechanism of interaction between the basic flow andan unstable perturbation.

Acknowledgments. We thank Ralph Foster for hisuseful comments and suggestions. Christian Barthlottwas supported by a research grant of the École Poly-technique.

APPENDIX

Numerical Methods

a. Spatial discretization

First, we actually solve for the deviation u� � u � u0

from the Ekman profile. The vertical coordinate z goesfrom 0 at the ground to �. The boundary conditionssatisfied by u� are then u� � 0 at z � 0 and z � andperiodicity along x. We work within a finite-dimen-sional space U of nondivergent vector fields, defined inthe two-dimensional case by

u� �!mn

amn�t��n�z�eyeimk1x � c.c.

�!mn�ex�z � ez�x�"bmn�t��n�z�e

imk1x# � c.c.

�A1�


The first sum induces a purely axial circulation and thesecond sum induces a cross-roll circulation in the (x, z)plane only. The basis profiles �n(z) and �n(z) are Jacobipolynomials in the variable $ � 1 � 2e�z/z0 ∈ "�1, 1],multiplied respectively by (1 � $)(1 � $) and (1 �$)2(1 � $) to satisfy the boundary conditions. Theabove sums are truncated to M Fourier modes and Npolynomials, resulting in �4MN degrees of freedomcontained in the complex coefficients amn, bmn. The freeparameter z0 controls the thickness of the well-resolvedregion above z � 0 and is set to z0 � 4.1 in the presentwork. This spectrally accurate spatial discretization isdescribed in more detail in Spalart et al. (1991) and hasbeen used in earlier studies of the Ekman boundarylayer (Coleman et al. 1990; Foster 1997).

b. Temporal discretization

When performing temporal integrations, we solve (8)in the Galerkin sense; that is, we search u� ∈ U such that

�u ∈ U�u · �tu�� u · a�u0 � u��, �A2�

where a is the left-hand side of (8) and u is an arbitrarytest field. Because U is made of nondivergent fieldsonly, computing the pressure is never required. The dotproducts �u · u�� are computed by Legendre–Gaussquadrature. The coefficients amn and bmn are obtainedfrom the values of u� at the quadrature points by hori-zontal Fourier transforms followed by matrix multipli-cation of the vertical profiles and vice versa. For thetemporal discretization, we use a third-order semi-implicit Runge–Kutta scheme (Yoh and Zhong 2004).The implicit part deals with viscous terms with uncon-ditional stability and involves the inversion of operatorsof the form 1 � Re%2, which is done efficiently viabanded-matrix methods (Spalart et al. 1991). The ex-plicit part is subject to a Courant–Friedrich–Levy sta-bility condition. Furthermore, the equilibria of (8) arefixed points of the discrete-time system, an importantproperty for the present stability study.

c. Computation of steady rolls

To solve the steady problem, we search for a Galer-kin solution of

a�u�� 0,

where a�u�� a�u0 � u�� c�u��xu�

and c�u�� xu� · a�u0 � u��

��xu� · �xu��. �A3�

Equation (A3) is a nonlinear equation in terms of theunknowns (amn, bmn), which we solve by a Newton–Krylov method as follows. Given an initial guess u�0, we

compute a sequence (u�0, u�1, . . .) and two auxiliary se-quences (a0, a1, . . .) and (�u�0, �u�1, . . .) as

Ln · �u�n � an, �A4�

u�n�1 � u�n � �u�n, �A5�

where Ln is the linearization of a around u�n, that is,

a�u�n � ��u�� an � �Ln · �u� � O��2�. �A6�

The sequence (u�0, u�1, . . .) converges to the solution of(A3) if the initial guess is close enough. At each itera-tion, an � a(u�n) is computed and the linear problemLn · �un � an is solved for �un. We do this using thegeneralized minimal residual (GMRES) algorithm,which finds an approximation of �un in the subspace ofU spanned by an, Ln · an, . . . , LD

n · an so as to minimizethe norm of residual Ln · �un � an (Saad and Schultz1986). The operator Ln enters the algorithm onlythrough operations of the type � ← Ln · �. This avoidsthe construction and factorization of a large matrix, aswould be the case with a direct method. We set therelative tolerance of the GMRES solver to 10�2 and therelative tolerance of the Newton solver to 10�8.

d. Secondary stability

We search for the three-dimensional perturbation u2

in a finite-dimensional space U2(k2) analogous to U,except that the definition (A1) must be adapted for u2

to be nondivergent (14).Because the dimension of U2 is still quite large

(�104), obtaining the eigenvalues of L by a directmethod operating on the matrix of L is not practical.An acceptable approximation of �2 is obtained by solv-ing (16) with an arbitrary initial condition u0

2 for a suf-ficiently long time . Indeed, the perturbation at time is exp( L) · u0

2. Exponentiation emphasizes the eigen-values of L with largest real part, and eventually themost unstable eigenmode stands out regardless of theinitial condition u0

2. This idea is improved by orthogo-nally projecting L onto the Krylov space spanned by u0

2,exp( L) · u0

2, exp(2 L) · u02, . . . , exp(m L) · u0

2 (Le-houcq et al. 1997). The desired relative accuracy of theeigenvalue can then be obtained by increasing the di-mension m� 1 of the Krylov space, typically a few tens.Here, we set this tolerance to 10�4. As for GMRES, theoperator L enters the algorithm only through opera-tions of the type u�2 ← exp( L) · u2, for example,through temporal integrations of (16) during a time .

e. Numerical convergence

Confidence in the numerical convergence has beenobtained by reproducing the results (primary and sec-

JULY 2008 D U B O S E T A L . 2341

ondary growth rates, selected wavenumbers) at latitude � 90° with several resolutions: M � N � 42 � 42,42 � 84, 84 � 42, 84 � 84. Within the range 0 Re

500, the results coincide with 1% relative tolerance atall these resolutions. For 500 Re 1000, the resultsat resolutions M � N � 42 � 84, 84 � 42, 84 � 84coincide, but resolution M� N� 42� 42 is insufficientand produces quantitatively incorrect, although quali-tatively consistent, results. Inspection of the velocityfields shows that small Gibbs oscillations are present atthis coarser resolution and disappear at higher resolu-tions.

The results presented in the paper have been ob-tained with M � 84 Fourier modes and N � 84 poly-nomials, corresponding to 128 � 128 quadrature pointsbecause of the 2⁄3 dealiasing rule.

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